Shear and Vorticity in a Combined Einstein-Cartan-Brans-Dicke Inflationary Lambda-Universe

合集下载

Laboratory experiments for intense vortical structures in turbulence velocity fields

a r X i v :p h y s i c s /0703057v 1 [p h y s i c s .f l u -d y n ] 6 M a r 2007Physics of FluidsLaboratory experiments for intense vortical structures in turbulence velocity ﬁeldsHideaki Mouri,a Akihiro Hori,b and Yoshihide Kawashima bMeteorological Research Institute,Nagamine,Tsukuba 305-0052,Japan(Dated:February 2,2008)Vortical structures of turbulence,i.e.,vortex tubes and sheets,are studied using one-dimensional velocity data obtained in laboratory experiments for duct ﬂows and boundary layers at microscale Reynolds numbers from 332to 1934.We study the mean velocity proﬁle of intense vortical struc-tures.The contribution from vortex tubes is dominant.The radius scales with the Kolmogorov length.The circulation velocity scales with the rms velocity ﬂuctuation.We also study the spatial distribution of intense vortical structures.The distribution is self-similar over small scales and is random over large scales.Since these features are independent of the microscale Reynolds number and of the conﬁguration for turbulence production,they appear to be universal.I.INTRODUCTIONTurbulence contains various classes of structures that are embedded in the background random ﬂuctuation.They are important to intermittency as well as mixing and diﬀusion.Of particular interest are small-scale struc-tures,which could have universal features that are inde-pendent of the Reynolds number and of the large-scale ﬂow.We explore such universality using velocity data obtained in laboratory experiments.We focus on vortical structures,i.e.,vortex tubes and sheets.The former is often regarded as the elementary structure of turbulence.1,2,3At low microscale Reynoldsnumbers,Re λ<∼200,direct numerical simulations de-rived basic parameters of vortex tubes.3,4,5,6,7,8The radii are of the order of the Kolmogorov length η.The total lengths are of the order of the correlation length L .The circulation velocities are of the order of the rms velocity ﬂuctuation u 2 1/2or the Kolmogorov velocity u K .Here · denotes an average.The lifetimes are of the order of the turnover time for energy-containing eddies L/ u 2 1/2.For these vortical structures,however,universality has not been established because the behavior at high Reynolds numbers has not been known.At Re λ>∼200,a direct numerical simulation is not easy for now.The promising approach is velocimetry in laboratory experiments.A probe suspended in the ﬂow is used to obtain a one-dimensional cut of the velocity ﬁeld.The velocity variation is intense at the positions of in-tense structures.Especially at the positions of intense vortical structures,the variation of the velocity compo-nent that is perpendicular to the one-dimensional cut is intense.9,10Thus,the velocity variation oﬀers some infor-mation about intense structures,although it is diﬃcult to specify their geometry.The above approach was taken in several studies.11,12,13,14,15For example,using grid turbulence 14at Re λ=105–329and boundary layers 15at Re λ=295–TABLE I:Experimental conditions and turbulence parameters:duct-exit or incoming-ﬂow velocity U∗,coordinates x andz of the measurement position,mean streamwise velocity U,sampling frequency f s,kinematic viscosityν,mean energydissipation rate ε =15ν (∂x v)2 /2,rms velocityﬂuctuations u2 1/2and v2 1/2,Kolmogorov velocity u K=(ν ε )1/4,rms spanwise-velocity increment over the sampling interval δv2s 1/2= [v(x+U/2f s)−v(x−U/2f s)]2 1/2,correlation lengths L u=R∞0 u(x+r)u(x) / u2 dr and L v=R∞0 v(x+r)v(x) / v2 dr,Taylor microscaleλ=[2 v2 / (∂x v)2 ]1/2,Kolmogorov lengthη=(ν3/ ε )1/4,and microscale Reynolds number Reλ=λ v2 1/2/ν.The velocity derivative was obtained as∂x v=[8v(x+r)−8v(x−r)−v(x+2r)+v(x−2r)]/12r with r=U/f s.Ductﬂow Boundary layerUnits1234567891011FIG.1:Sketch of a vortex tube penetrating the(x,y)plane at a point(x0,y0).The inclination is(θ0,ϕ0).The circulation velocity is uΘ.We consider the spanwise velocity v along the x axis in the mean streamdirection.FIG.2:Mean proﬁles in the streamwise(u)and spanwise (v)velocities for the Burgers vortices with random positions (x0,y0)and inclinations(θ0,ϕ0).The u proﬁle is separately shown for∂x u>0(u+)and∂x u≤0(u−)at x=0.The position x and velocities are normalized by the radius and maximum circulation velocity of the Burgers vortices.The dotted line is the v proﬁle of the Burgers vortex for x0= y0=θ0=0,the peak value of which is scaled to that of the mean v proﬁle.uΘand strainﬁeld(u R,u Z)areuΘ∝ν4ν ,(1a) (u R,u Z)= −a0RRuΘ(R),(2a) v(x−x0)=(x−x0)cosθ0Ru R(R),(4a)v(x−x0)=−(x−x0)sin2θ0sinϕ0cosϕ0+y0(1−sin2θ0sin2ϕ0)FIG.3:Probability density distribution of the absolute spanwise-velocity increment|v(x+U/2f s)−v(x−U/2f s)| at Reλ=719,1304,and1934.The distribution is verti-cally shifted by a factor103.The increment is normalized by δv2s 1/2= [v(x+U/2f s)−v(x−U/2f s)]2 1/2.The arrows indicate the ranges for intense vortical structures,which share 0.1and1%of the total.The dotted line denotes the Gaussian distribution.bulence was almost isotropic becausethe measured ra-tio u2 / v2 is not far from unity(Table I).The vortex tubes induce small-scale variations in the spanwise veloc-ity.If we consider intense velocity variations above a high threshold,their scale and amplitude are close to the ra-dius and circulation velocity of intense vortex tubes with |y0|<∼R0andθ0≃0.To demonstrate this,mean proﬁles are calculated for the circulationﬂows uΘof the Burgers vortices with random positions(x0,y0)and inclinations (θ0,ϕ0).Their radii R0and maximum circulation veloc-ities V0=uΘ(R0)are set to be the same.We consider the Burgers vortices with|∂x v|at x=0being above a threshold,|∂x v|/3at x=0for x0=y0=θ0=0.When ∂x v is negative,the sign of the v signal is inverted before the averaging.The result is shown in Fig.2.Despite the relatively low threshold,the scale and peak amplitude of the mean v proﬁle are still close to those of the v proﬁle for x0=y0=θ0=0(dotted line).The extended tails are due to the Burgers vortices with|y0|≫R0orθ0≫0.IV.MEAN VELOCITY PROFILEMean proﬁles of intense vortical structures in the streamwise(u)and spanwise(v)velocities are extracted,FIG.4:Mean proﬁles of intense vortical structures for the 0.1%threshold in the streamwise(u)and spanwise(v)ve-locities.(a)Reλ=719.(b)Reλ=1098.The u proﬁle is separately shown for∂x u>0(u+)and∂x u≤0(u−) at x=0.The position x is normalized by the Kolmogorov length.The velocities are normalized by the peak value of the v proﬁle.We also show the v proﬁle of the Burgers vortex for x0=y0=θ0=0by a dotted line.by averaging signals centered at the position where the absolute spanwise-velocity increment|v(x+r/2)−v(x−r/2)|is above a threshold.10,13,14,15,19The scale r is the sampling interval U/f s.The threshold is such that0.1% or1%of the increments are used for the averaging(here-after,the0.1%or1%threshold).These increments com-prise the tail of the probability density distribution of all the increments as in Fig.3.20Example of the results are shown in Fig.4.20The v proﬁle in Fig.4is close to the v proﬁle in Fig. 2.Hence,the contribution from vortex tubes is dominant.The contribution from vortex sheets is not dominant.If it were dominant,the v proﬁle should ex-hibit some kind of step.12Direct numerical simulations at Reλ<∼200revealed that intense vorticity tends to be organized into tubes rather than sheets.4,5,6,7,21,22This tendency appears to exist up to Reλ≃2000.Vortex sheets might contribute to the extended tails in Fig. 4. They are more pronounced than those in Fig. 2.Here it should be noted that our discussion is somewhat sim-pliﬁed because there is no strict division between vortex tubes and sheets in real turbulence.Byﬁtting the v proﬁle in Fig.4around its peaks by the v proﬁle of the Burgers vortex for x0=y0=θ0=0 (dotted line),we estimate the radius R0and maximumTABLE II:Parameters for intense vortical structures:radius R 0,maximum circulation velocity V 0,Reynolds number Re 0=R 0V 0/νand small-scale clustering exponent µ0.We also list the threshold level τ0.Duct ﬂowBoundary layer Units1234567891011δx pδx p /2−δx p /2v t (x +r )dr.(5)For allthe data,the R 0and V 0values are summarizedin Table II.They characterize the scale and intensity of vortical structures,even if they are not the Burgers vortices.The radius R 0is several times the Kolmogorov length η.The maximum circulation velocity V 0is several tenths of the rms velocity ﬂuctuation v 2 1/2and several times the Kolmogorov velocity u K .Similar results were obtained from direct numerical simulations 3,4,6,7,8and laboratory experiments 11,12,14,15at the lower Reynolds numbers,Re λ<∼1300.The u proﬁle in Fig.4is separated for ∂x u >0(u +)and ∂x u ≤0(u −)at x =0.Since the contamina-tion with the w component 17induces a symmetric posi-tive excursion,14,23,24we decomposed the u ±proﬁles into symmetric and antisymmetric components and show only the antisymmetric components.15The u ±proﬁles in Fig.4have larger amplitudes than those in Fig. 2.Hence,the u ±proﬁles in Fig.4are dominated by the circu-lation ﬂows u Θof vortex tubes that passed the probe with some incidence angles to the mean ﬂow direction,11tan −1[v/(U +u )].@The radial inﬂow u R of the strain ﬁeld is not discernible,except that the u −proﬁle has a larger amplitude than the u +proﬁle.14,15Unlike the Burgers vortex,a real vortex tube is not always oriented to the stretching direction.4,5,6,7,8,25V.SPATIAL DISTRIBUTIONThe spatial distribution of intense vortical structures is studied using the distribution of interval δx 0between successive intense velocity increments.13,14,15,22The in-tense velocity increment is deﬁned in the same manner as for the mean velocity proﬁles in Sec.IV.Since theyare dominated by vortex tubes,we expect that the distri-bution of intense vortical structures studied here is also essentially the distribution of intense vortex tubes.Ex-amples of the probability density distribution P (δx 0)are shown in Figs.5and 6.20The probability density distribution has an exponen-tial tail 14,15that appears linear on the semi-log plot of Fig.5.This exponential law is characteristic of intervals for a Poisson process of random and independent events.FIG.5:Probability density distribution of interval between intense vortical structures for the 1%threshold at Re λ=719,1304,and 1934.The distribution is normalized by the ampli-tude of the exponential tail (dotted line),and it is vertically shifted by a factor 10.The interval is normalized by the streamwise correlation length L u .The arrow indicates the spanwise correlation length L v .FIG.6:Probability density distribution of interval between intense vortical structures for the1%threshold at Reλ=719, 1304,and1934.The distribution is normalized by the peak value,and it is vertically shifted by a factor10.The dotted line indicates the power-law slope from30ηto300η.The interval is normalized by the Kolmogorov lengthη.The arrow indicates the spanwise correlation length L v.The large-scale distribution of intense vortical structures is random and independent.Below the spanwise correlation length L v,the proba-bility density is enhanced over that for the exponential distribution.15Thus,intense vortical structures cluster together below the energy-containing scale.In fact,di-rect numerical simulations revealed that intense vortex tubes lie on borders of energy-containing eddies.6Over small intervals,the probability density distribu-tion is apower law13,22that appears linear on the log-log plot of Fig.6:P(δx0)∝δx−µ0.(6)Thus,the small-scale clustering of intense vortical struc-tures is self-similar and has no characteristic scale.22Ta-ble II lists the clustering exponentµ0estimated over in-tervals fromδx0=30ηto300η.Its value is close to unity.The exponential law over large intervals and the power law over small intervals were also found in laboratory ex-periments for regions of low pressure.26,27,28,29They are associated with vortex tubes,although their radii tend to be larger than those of intense vortical structures studied here.29VI.SCALING LA WDependence of parameters for intense vortical struc-tures on the microscale Reynolds number Reλand on the conﬁguration for turbulence production,i.e.,ductﬂow or boundary layer,is studied in Fig.7.Each quantity was normalized by its value in the ductﬂow at Reλ=1934 individually for the0.1%and1%thresholds.That is,we avoid the prefactors that depend on the threshold.When the threshold is high,the radius R0is small,the maxi-mum circulation velocity V0is large,and the clustering exponentµ0is small as in Table II.We focus on scaling laws of these quantities.The radius R0scales with the Kolmogorov lengthηas R0∝η[Fig.7(a)].Thus,intense vortical structures remain to be of smallest scales of turbulence.The maximum circulation velocity V0scales with the rms velocityﬂuctuation v2 1/2as V0∝ v2 1/2[Fig. 7(b)].Although the rms velocityﬂuctuation is a charac-FIG.7:Dependence of parameters for intense vortical struc-tures on Reλ.(a)R0/η.(b)V0/ v2 1/2.(c)V0/u K.(d)Re0.(e)Re0/Re1/2λ.(f)µ0.The open andﬁlled circles respec-tively denote the ductﬂows for the0.1%and1%thresholds. The upward and downward triangles respectively denote the boundary layers for the0.1%and1%thresholds.Each quan-tity is normalized by its value in the ductﬂow at Reλ=1934 individually for the0.1%and1%thresholds.teristic of the large-scaleﬂow,vortical structures could be formed via shear instability on borders of energy-containing eddies,6,27,28where a small-scale velocity vari-ation could be comparable to the rms velocityﬂuctua-tion.The maximum circulation velocity does not scale with the Kolmogorov velocity u K,a characteristic of the small-scaleﬂow,as V0∝u K[Fig.7(c)].Direct numerical simulations for intense vortex tubes6,7at Reλ<∼200and laboratory experiments for intense vortical structures11,15at Reλ<∼1300derived the scalings R0∝ηand V0∝ v2 1/2.We have found that these scalings exist up to Reλ≃2000,regardless of the conﬁguration for turbulence production.The scalings of the radius R0and circulation veloc-ity V0lead to a scaling of the Reynolds number Re0= R0V0/νfor the intense vortical structures:6,7Re0∝Re1/2λif R0∝ηand V0∝ v2 1/2,(7a) Re0=constant if R0∝ηand V0∝u K.(7b) Our result favors the former scaling[Fig.7(e)]rather than the latter[Fig.7(d)].With an increase of Reλ, intense vortical structures progressively have higher Re0 and are more unstable.6,7Their lifetimes are shorter.It is known30that theﬂatness factor (∂x v)4 / (∂x v)2 2 scales with Re0.3λ.Since (∂x v)4 is dominated by in-tense vortical structures,it scales with v2 2/η4.Since (∂x v)2 2is dominated by the background randomﬂuc-tuation,it scales with u4K/η4.If the number density of intense vortical structures remains the same,we have (∂x v)4 / (∂x v)2 2∝ v2 2/u4K∝Re2λ.The diﬀerence from the real scaling implies that vortical structures with V0≃ v2 1/2are less numerous at a higher Reynolds num-ber Reλ,albeit energetically more important.The small-scale clustering exponentµ0is constant[Fig. 7(f)].A similar result withµ0≃1was obtained from laboratory experiments of the K´a rm´a nﬂow between two rotating disks22at Reλ≃400–1600.The small-scale clustering of intense vortical structures at high Reynolds numbers Reλis independent of the conﬁguration for tur-bulence production.Lastly,recall that only intense vortical structures are considered here.For all vortical structures with vari-ous intensities,the scalings V0∝ v2 1/2and Re0=R0V0/ν∝Re1/2λare not necessarily expected.For allvortex tubes,in fact,direct numerical simulations3,8at Reλ<∼200derived the scaling V0∝u K.The devel-opment of an experimental method to study all vortical structures is desirable.VII.CONCLUSIONThe spanwise velocity was measured in ductﬂows at Reλ=719–1934and in boundary layers at Reλ=332–1304(Table I).We used these velocity data to study fea-tures of vortical structures,i.e.,vortex tubes and sheets. We studied the mean velocity proﬁles of intense vor-tical structures(Fig.4).The contribution from vortex tubes is dominant.Essentially,our results are those for vortex tubes.The radius R0is several times the Kol-mogorov lengthη.The maximum circulation velocity V0 is several tenths of the rms velocityﬂuctuation v2 1/2 and several times the Kolmogorov velocity u K(Table II).There are the scalings R0∝η,V0∝ v2 1/2,and Re0=R0V0/ν∝Re1/2λ(Fig.7).We also studied the distribution of interval between in-tense vortical structures.Over large intervals,the distri-bution obeys an exponential law(Fig.5),which reﬂects a random and independent distribution of intense vortical structures.Over small intervals,the distribution obeys a power law(Fig.6),which reﬂects self-similar clustering of intense vortical structures.The clustering exponent is constant,µ0≃1(Table II and Fig.7).Direct numerical simulations3,4,6,7,8,9,10and laboratory experiments11,12,13,14,15,22derived some of those features. We have found that they are independent of the Reynolds number and of the conﬁguration for turbulence produc-tion,up to Reλ≃2000that exceeds the Reynolds num-bers of the prior studies.The Reynolds numbers Reλin our study are still lower than those of some turbulence,e.g.,atmospheric turbu-lence at Reλ>∼104.Such turbulence is expected to contain intense vortical structures,because turbulence is more intermittent at a higher Reynolds number Reλand small-scale intermittency is attributable to intense vortical structures.They are expected to have the same features as found in our study.These features appear to have reached asymptotes at Reλ≃2000(Fig.7),regard-less of the conﬁguration for turbulence production,and hence appear to be universal at high Reynolds numbers Reλ.AcknowledgmentsThe authors are grateful to T.Gotoh,S.Kida, F. Moisy,M.Takaoka,and Y.Tsuji for interesting discus-sions.1U.Frisch,Turbulence,The Legacy of A.N.Kolmogorov (Cambridge Univ.Press,Cambridge,1995),Chap.8.2K.R.Sreenivasan and R.A.Antonia,“The phenomenol-ogy of small-scale turbulence,”Annu.Rev.Fluid Mech. 29,435(1997).3T.Makihara,S.Kida,and H.Miura,“Automatic tracking of low-pressure vortex,”J.Phys.Soc.Jpn.71,1622(2002). These authors pushed forward the notion that vortex tubes are the elementary structures of turbulence.4A.Vincent and M.Meneguzzi,“The spatial structure and8statistical properties of homogeneous turbulence,”J.Fluid Mech.225,1(1991).5A.Vincent and M.Meneguzzi,“The dynamics of vorticity tubes in homogeneous turbulence,”J.Fluid Mech.258, 245(1994).6J.Jim´e nez,A.A.Wray,P.G.Saﬀman,and R.S.Rogallo,“The structure of intense vorticity in isotropic turbulence,”J.Fluid Mech.255,65(1993).7J.Jim´e nez and A.A.Wray,“On the characteristics of vor-texﬁlaments in isotropic turbulence,”J.Fluid Mech.373, 255(1998).8M.Tanahashi,S.-J.Kang,T.Miyamoto,S.Shiokawa,and T.Miyauchi,“Scaling law ofﬁne scale eddies in turbulent channelﬂows up to Reτ=800,”Int.J.Heat Fluid Flow 25,331(2004).9A.Pumir,“Small-scale properties of scalar and velocity diﬀerences in three-dimensional turbulence,”Phys.Fluids 6,3974(1994).10H.Mouri,M.Takaoka,and H.Kubotani,“Wavelet iden-tiﬁcation of vortex tubes in a turbulence velocityﬁeld,”Phys.Lett.A261,82(1999).11F.Belin,J.Maurer,P.Tabeling,and H.Willaime,“Obser-vation of intenseﬁlaments in fully developed turbulence,”J.Phys.(Paris)II6,573(1996).They studied turbulence velocityﬁelds at Reλ=151–5040.We do not consider their results at Reλ>∼700,where (∂x u)3 / (∂x u)2 3/2and (∂x u)4 / (∂x u)2 2of their data are known to be inconsis-tent with those from other studies.212A.Noullez,G.Wallace,W.Lempert,es,and U.Frisch,“Transverse velocity increments in turbulentﬂow using the RELIEF technique,”J.Fluid Mech.339,287 (1997).13R.Camussi and G.Guj,“Experimental analysis of inter-mittent coherent structures in the nearﬁeld of a high Re turbulent jetﬂow,”Phys.Fluids11,423(1999).14H.Mouri,A.Hori,and Y.Kawashima,“Vortex tubes in velocityﬁelds of laboratory isotropic turbulence:depen-dence on the Reynolds number,”Phys.Rev.E67,016305 (2003).15H.Mouri,A.Hori,and Y.Kawashima,“Vortex tubes in turbulence velocityﬁelds at Reynolds numbers Reλ≃300–1300,”Phys.Rev.E70,066305(2004).16K.R.Sreenivasan and B.Dhruva,“Is there scaling in high-Reynolds-number turbulence?,”Prog.Theor.Phys.Suppl.130,103(1998).17The two wires individually respond to all the u,v,and w components.Since the measured u component corresponds to the sum of the responses of the two wires,it is contam-inated with the w component.Since the measured v com-ponent corresponds to the diﬀerence of the responses,it is free from the w component.18T.S.Lundgren,“Strained spiral vortex model for turbu-lentﬁne structure,”Phys.Fluids25,2193(1982).19For convenience,when consecutive increments are all above the threshold,each increment is taken to determine the center of a vortex.This is somewhat unreasonable but does not cause serious problems,judging from Fig.2where mean velocity proﬁles were obtained practically in the same manner.20While the experimental curves in Figs.3and4are mere loci of discrete data points,we applied smoothing to the tails of the experimental curves in Figs.5and6.21F.Moisy and J.Jim´e nez,“Geometry and clustering of in-tense structures in isotropic turbulence,”J.Fluid Mech.513,111(2004).22F.Moisy and J.Jim´e nez,“Clustering of intense structures in isotropic turbulence:numerical and experimental ev-idence,”in IUTAM Symposium on Elementary Vortices and Coherent Structures:Signiﬁcance in Turbulence Dy-namics,edited by S.Kida(Springer,Dordrecht,2006),p.3.23K.Sassa and H.Makita,“Reynolds number dependence of elementary vortices in turbulence,”in Engineering Tur-bulence Modelling and Experiments6,edited by W.Rodi and M.Mulas(Elsevier,Oxford,2005),p.431.24The positive excursion might be partially induced byﬂuc-tuation of the instantaneous velocity U+u at which a struc-ture passes the probe.Under Taylor’s frozen-eddy hypoth-esis,the velocity increment over the sampling interval U/f s is more intense for a faster-moving structure,which is more likely to be incorporated in our conditional averaging.14 Other mechanisms might be also at work.25M.Kholmyansky,A.Tsinober,and S.Yorish,“Velocity derivatives in the atmospheric surface layer at Reλ=104,”Phys.Fluids13,311(2001).26P.Abry,S.Fauve,P.Flandrin,and roche,“Analy-sis of pressureﬂuctuations in swirling turbulentﬂows,”J.Phys.(Paris)II4,725(1994).27O.Cadot,S.Douady,and Y.Couder,“Characterization of the low-pressureﬁlaments in a three-dimensional turbulent shearﬂow,”Phys.Fluids7,630(1995).28E.Villermaux,B.Sixou,and Y.Gagne,“Intense vortical structures in grid-generated turbulence,”Phys.Fluids7, 2008(1995). Porta,G.A.Voth,F.Moisy,and E.Bodenschatz,“Using cavitation to measure statistics of low-pressure events in large-Reynolds-number turbulence,”Phys.Fluids 12,1485(2000).30B.R.Pearson and R.A.Antonia,“Reynolds-number de-pendence of turbulent velocity and pressure increments,”J.Fluid Mech.444,343(2001).。

Common Phase Error due to Phase Noise in OFDM - Estimation and Suppression

COMMON PHASE ERROR DUE TO PHASE NOISE IN OFDM-ESTIMATION AND SUPPRESSIONDenis Petrovic,Wolfgang Rave and Gerhard FettweisV odafone Chair for Mobile Communications,Dresden University of Technology,Helmholtzstrasse18,Dresden,Germany{petrovic,rave,fettweis}@ifn.et.tu-dresden.deAbstract-Orthogonal frequency division multiplexing (OFDM)has already become a very attractive modulation scheme for many applications.Unfortunately OFDM is very sensitive to synchronization errors,one of them being phase noise,which is of great importance in modern WLAN systems which target high data rates and tend to use higher frequency bands because of the spectrum availability.In this paper we propose a linear Kalmanﬁlter as a means for tracking phase noise and its suppression.The algorithm is pilot based.The performance of the proposed method is investigated and compared with the performance of other known algorithms.Keywords-OFDM,Synchronization,Phase noise,WLANI.I NTRODUCTIONOFDM has been applied in a variety of digital commu-nications applications.It has been deployed in both wired systems(xDSL)and wireless LANs(IEEE802.11a).This is mainly due to the robustness to frequency selective fading. The basic principle of OFDM is to split a high data rate data stream into a number of lower rate streams which are transmitted simultaneously over a number of orthogonal subcarriers.However this most valuable feature,namely orthogonality between the carriers,is threatened by the presence of phase noise in oscillators.This is especially the case,if bandwidth efﬁcient higher order modulations need to be employed or if the spacing between the carriers is to be reduced.To compensate for phase noise several methods have been proposed.These can be divided into time domain[1][2]and frequency domain approaches[3][4][5].In this paper we propose an algorithm for tracking the average phase noise offset also known as the common phase error(CPE)[6]in the frequency domain using a linear Kalmanﬁlter.Note that CPE estimation should be considered as aﬁrst step within more sophisticated algorithms for phase noise suppression[5] which attempt to suppress also the intercarrier interference (ICI)due to phase noise.CPE compensation only,can however sufﬁce for some system design scenarios to suppress phase noise to a satisfactory level.For these two reasons we consider CPE estimation as an important step for phase noise suppression.II.S YSTEM M ODELAn OFDM transmission system in the presence of phase noise is shown in Fig. 1.Since all phase noise sources can be mapped to the receiver side[7]we assume,without loss of generality that phase noise is present only at the front end of the receiver.Assuming perfect frequency and timing synchronization the received OFDM signal samples, sampled at frequency f s,in the presence of phase noise can be expressed as r(n)=(x(n) h(n))e jφ(n)+ξ(n).Each OFDM symbol is assumed to consist of a cyclic preﬁx of length N CP samples and N samples corresponding to the useful signal.The variables x(n),h(n)andφ(n)denote the samples of the transmitted signal,the channel impulse response and the phase noise process at the output of the mixer,respectively.The symbol stands for convolution. The termξ(n)represents AWGN noise with varianceσ2n. The phase noise processφ(t)is modelled as a Wiener process[8],the details of which are given below,with a certain3dB bandwidth∆f3dB.,0,1,2...m lX l=,0,1,2...m lR l=Fig.1Block diagram of an OFDM transmission chain.At the receiver after removing the N CP samples cor-responding to the cyclic preﬁx and taking the discrete Fourier transform(DFT)on the remaining N samples,the demodulated carrier amplitude R m,lkat subcarrier l k(l k= 0,1,...N−1)of the m th OFDM symbol is given as[4]:R m,lk=X m,lkH m,lkI m(0)+ζm,lk+ηm,lk(1)where X m,lk,H m,lkandηm,lkrepresent the transmitted symbol on subcarrier l k,the channel transfer function andlinearly transformed AWGN with unchanged variance σ2n at subcarrier l k ,respectively.The term ζm,l k represents intercarrier interference (ICI)due to phase noise and was shown to be a gaussian distributed,zero mean,randomvariable with variance σ2ICI =πN ∆f 3dB s[7].The term I m (0)also stems from phase noise.It does not depend on the subcarrier index and modiﬁes all subcarriers of one OFDM symbol in the same manner.As its modulus is in addition very close to one [9],it can be seen as a symbol rotation in the complex plane.Thus it is referred to in the literature as the common phase error (CPE)[6].The constellation rotation due to CPE causes unaccept-able system performance [7].Acceptable performance can be achieved if one estimates I m (0)or its argument and compensates the effect of the CPE by derotating the received subcarrier symbols in the frequency domain (see Eq.(1)),which signiﬁcantly reduces the error rate as compared to the case where no compensation is used.The problem of esti-mating the CPE was addressed by several authors [3][4][10].In [3]the authors concentrated on estimating the argument of I m (0)using a simple averaging over pilots.In [10]the argument of I m (0)was estimated using an extended Kalman ﬁlter,while in [4]the coefﬁcient I m (0)itself was estimated using the LS algorithm.Here we introduce an alternative way for minimum mean square estimation (MMSE)[11]of I m (0)using a linear scalar Kalman ﬁlter.The algorithm is as [4]pilot based.III.P HASE N OISE M ODELFor our purposes we need to consider a discretized phase noise model φ(n )=φ(nT s )where n ∈N 0and T s =1/f s is the sampling period at the front end of the receiver.We adopt a Brownian motion model of the phase noise [8].The samples of the phase noise process are given as φ(n )=2πf c √cB (n )where f c is the carrier frequency,c =∆f 3dB /πf 2c [8]and B (n )represents the discretizied Brownian motion process,Using properties of the Brownian motion [12]the fol-lowing holds:B (0)=0and B (n +1)=B (n )+dB n ,n ∈N 0where each increment dB n is an independent random variable and dB n ∼√T s N (0,1).Noting that φ(n )=2πf c √cB (n )we can write the discrete time phase noise process equation asφ(n +1)=φ(n )+w (n )(2)where w (n )∼N (0,4π2f 2c cT s )is a gaussian randomvariable with zero mean and variance σ2w =4π2f 2c cT s .IV.CPE E STIMATION U SING A K ALMAN F ILTER Since all received subcarriers within one OFDM symbolare affected by the same factor,namely I m (0),the problem at hand can be seen as an example of estimating a constant from several noisy measurements given by Eq.(1)for which purpose a Kalman ﬁlter is well suited [11].For a Kalmanﬁlter to be used we need to deﬁne the state space model of the system.Deﬁne ﬁrst the set L ={l 1,l 2,l 3,...l P }as a subset of the subcarrier set {0,1,...N −1}.Using Eq.(1)one can writeR m,l k =A m,l k I m,l k (0)+εm,l k(3)where A m,l k =X m,l k H m,l k and I m,l k (0)=I m (0)for all k =1,2...,P .Additional indexing of the CPE terms is done here only for convenience of notation.On the other hand one can writeI m,l k +1(0)=I m,l k (0).(4)Equations (3)and (4)are the measurement and processequation of the system state space model,where A m,l k represents the measurement matrix,while the process matrix is equal to 1and I m,l k (0)corresponds to the state of the system.The measuring noise is given by εm,l k which combines the ICI and AWGN terms in Eq.(1),the varianceof which for all l k equals σ2ε=(σ2ICI +σ2n ).The process noise equals zero.Note that the deﬁned state space model is valid only for one OFDM symbol.For the state space model to be fully deﬁned,knowledge of the A m,l k =X m,l k H m,l k is needed.Here we assume to have ideal knowledge of the channel.On the other hand we deﬁne the subset L to correspond to the pilot subcarrier locations within one OFDM symbol so that X m,q ,q ∈L are also known.We assume that at the beginning of each burst perfect timing and frequency synchronization is achieved,so that the phase error at the beginning of the burst equals zero.After the burst reception and demodulation,the demodulated symbols are one by one passed to the Kalman ﬁlter.For a Kalman ﬁlter initialization one needs for eachOFDM symbol an a priori value for ˆI m,l 1(0)and an a priori error variance K −m,1.At the beginning of the burst,when m =1,it is reasonable to adopt ˆI −1,l 1(0)=1.Within each OFDM symbol,say m th,the ﬁlter uses P received pilot subcarriers to recursively update the a priori value ˆI −1m,l 1(0).After all P pilot subcarriers are taken into account ˆI m,l P (0)is obtained,which is adopted as an estimate ofthe CPE within one OFDM symbol,denoted as ˆIm (0).The Kalman ﬁlter also provides an error variance of the estimateof I m,l P (0)as K m,P .ˆI m,l P(0)and K m,P are then used as a priori measures for the next OFDM symbol.The detailed structure of the algorithm is as follows.Step 1:InitializationˆI −m,l 1(0)=E {I −m,l 1(0)}=ˆI m −1(0)K −m,1=E {|I m (0)−ˆIm −1(0)|2}∼=E {|φm −ˆφm −1|2}=σ2CP E +K m −1,Pwhere σ2CP E =4π2N 2+13N +N CP ∆f 3dBf s(see [10]),K 0,P =0and φm =arg {I m (0)}.Repeat Step2and Step3for k=1,2,...,P Step2:a-posteriori estimation(update)G m,k=K−m,kH H m,lkH m,lkK−m,kH Hm,l k+(σ2ICI+σ2n)ˆIm,l k (0)=ˆI−m,l k(0)+G m,k[R m,lk−H m,l kˆI−m,l k(0)]K m,k=(1−G m,k H m,lk )K−m,kStep3:State and error variance propagationK−m,k+1=K m,k(5)ˆI−m,l k+1(0)=ˆI m,lk(0)Note that no matrix inversions are required,since the state space model is purely scalar.V.CPE C ORRECTIONThe easiest approach for CPE correction is to derotate all subcarriers l k of the received m th symbol R m,lkby φm=−arg{ˆI m(0)}.Unambiguity of the arg{·}function plays here no role since any unambiguity which is a multiple of2πrotates the constellation to its equivalent position in terms of its argument.The presented Kalmanﬁlter estimation algorithm is read-ily applicable for the decision feedback(DF)type of algo-rithm presented in[4].The idea there was to use the data symbols demodulated after theﬁrst CPE correction in a DFE manner to improve the quality of the estimate since that is increasing the number of observations of the quantity we want to estimate.In our case that would mean that after theﬁrst CPE correction the set L={l1,l2,l3,...l P}of the subcarriers used for CPE estimation,which previously corresponded to pilot subcarriers,is now extended to a larger set corresponding to all or some of the demodulated symbols. In this paper we have extended the set to all demodulated symbols.The Kalmanﬁlter estimation is then applied in an unchanged form for a larger set L.VI.N UMERICAL R ESULTSThe performance of the proposed algorithm is investigated and compared with the proposal of[4]which is shown to outperform other known approaches.The system model is according to the IEEE802.11a standard,where64-QAM modulation is used.We investigate the performance in AWGN channels and frequency selective channels using as an example the ETSI HiperLAN A-Channel(ETSI A). Transmission of10OFDM symbols per burst is assumed.A.Properties of an EstimatorThe quality of an estimation is investigated in terms of the mean square error(MSE)of the estimator for a range of phase noise bandwidths∆f3dB∈[10÷800]Hz.Table1 can be used to relate the phase noise bandwidth with other quantities.Figures2and3compare the MSE of the LS estimator from[4]and our approach for two channel types and both standard correction and using decision feedback. Note that SNRs are chosen such that the BER of a coded system after the Viterbi algorithm in case of phase noise free transmission is around1·10−4.Kalmanﬁlter shows better performance in all cases and seems to be more effective for small phase noise bandwidths. As expected when DF is used the MSE of an estimator is smaller because we are taking more measurements into account.Fig.2MSE of an estimator for AWGN channel.Fig.3MSE of an estimator for ETSI A channel.Table 1Useful relationsQuantitySymbolRelationTypical values for IEEE802.11aOscillator constant c [1radHz]8.2·10−19÷4.7·10−18Oscillator 3dB bandwidth ∆f 3dB [Hz]∆f 3dB =πf 2cc 70÷400Relative 3dB bandwidth ∆f 3dB ∆f car∆f 3dBfsN 2·10−4÷13·10−4Phase noise energy E PN [rad]E PN =4π∆f 3dB∆fcar0.0028÷0.016Subcarrier spacing∆f car∆f car =f s N312500HzB.Symbol Error Rate DegradationSymbol error rate (SER)degradation due to phase noise is investigated also for a range of phase noise bandwidths ∆f 3dB ∈[10÷800]Hz and compared for different correc-tion algorithms.Ideal CPE correction corresponds to the case when genie CPE values are available.In all cases simpleconstellation derotation with φ=−arg {ˆIm (0)}is used.Fig.4SER degradation for AWGN channel.In Figs.4and 5SER degradation for AWGN and ETSI A channels is plotted,respectively.It is interesting to note that as opposed to the ETSI A channel case in AWGN channel there is a gap between the ideal CPE and both correction approaches.This can be explained if we go back to Eq.(1)where we have seen that phase noise affects the constellation as additive noise.Estimation error of phase noise affects the constellation also in an additive manner.On the other hand the SER curve without phase noise in the AWGN case is much steeper than the corresponding one for the ETSI A channel.A small SNR degradation due to estimation errors will cause therefore large SER variations.This explains why the performance differs much less in the ETSI A channel case.Generally from this discussion a conclusion can be drawn that systems with large order of diversity are more sensitive to CPE estimation errors.Note that this ismeantFig.5SER degradation for ETSI A channel.not in terms of frequency diversity but the SER vs.SNR having closely exponential dependence.It can be seen that our approach shows slightly better performance than [4]especially for small phase noise bandwidths.What is also interesting to note is,that DF is not necessary in the case of ETSI A types of channels (small slope of SER vs.SNR)while in case of AWGN (large slope)it brings performance improvement.VII.C ONCLUSIONSWe investigated the application of a linear Kalman ﬁlter as a means for tracking phase noise and its suppression.The proposed algorithm is of low complexity and its performance was studied in terms of the mean square error (MSE)of an estimator and SER degradation.The performance of an algorithm is compared with other algorithms showing equivalent and in some cases better performance.R EFERENCES[1]R.A.Casas,S.Biracree,and A.Youtz,“Time DomainPhase Noise Correction for OFDM Signals,”IEEE Trans.on Broadcasting ,vol.48,no.3,2002.[2]M.S.El-Tanany,Y.Wu,and L.Hazy,“Analytical Mod-eling and Simulation of Phase Noise Interference in OFDM-based Digital Television Terrestial Broadcast-ing Systems,”IEEE Trans.on Broadcasting,vol.47, no.3,2001.[3]P.Robertson and S.Kaiser,“Analysis of the effects ofphase noise in OFDM systems,”in Proc.ICC,1995.[4]S.Wu and Y.Bar-Ness,“A Phase Noise SuppressionAlgorithm for OFDM-Based WLANs,”IEEE Commu-nications Letters,vol.44,May1998.[5]D.Petrovic,W.Rave,and G.Fettweis,“Phase NoiseSuppression in OFDM including Intercarrier Interfer-ence,”in Proc.Intl.OFDM Workshop(InOWo)03, pp.219–224,2003.[6]A.Armada,“Understanding the Effects of PhaseNoise in Orthogonal Frequency Division Multiplexing (OFDM),”IEEE Trans.on Broadcasting,vol.47,no.2, 2001.[7]E.Costa and S.Pupolin,“M-QAM-OFDM SystemPerformance in the Presence of a Nonlinear Ampliﬁer and Phase Noise,”IEEE mun.,vol.50, no.3,2002.[8]A.Demir,A.Mehrotra,and J.Roychowdhury,“PhaseNoise in Oscillators:A Unifying Theory and Numerical Methods for Characterisation,”IEEE Trans.Circuits Syst.I,vol.47,May2000.[9]S.Wu and Y.Bar-ness,“Performance Analysis of theEffect of Phase Noise in OFDM Systems,”in IEEE 7th ISSSTA,2002.[10]D.Petrovic,W.Rave,and G.Fettweis,“Phase NoiseSuppression in OFDM using a Kalman Filter,”in Proc.WPMC,2003.[11]S.M.Kay,Fundamentals of Statistical Signal Process-ing vol.1.Prentice-Hall,1998.[12]D.J.Higham,“An Algorithmic Introduction to Numer-ical Simulation of Stochastic Differential Equations,”SIAM Review,vol.43,no.3,pp.525–546,2001.。

Scattering of vortex pairs in 2D easy-plane ferromagnets

a r X i v :c o n d -m a t /0112158v 1 10 D e c 2001Scattering of vortex pairs in 2D easy-plane ferromagnetsA.S.KovalevInstitute for Low Temperature Physics and Engineering,47Lenin Ave.,61164Kharkov,UkraineS.Komineas and F.G.MertensPhysikalisches Institut,Universit¨a t Bayreuth,D-95440Bayreuth,Germany(February 1,2008)Vortex-antivortex pairs in 2D easy-plane ferromagnets have characteristics of solitons in two dimensions.We investigate numerically and analytically the dynamics of such vortex pairs.In particular we simulate numerically the head-on collision of two pairs with diﬀerent velocities for a wide range of the total linear momentum of the system.If the momentum diﬀerence of the two pairs is small,the vortices exchange partners,scatter at an angle depending on this diﬀerence,and form two new identical pairs.If it is large,the pairs pass through each other without losing their identity.We also study head-tail collisions.Two identical pairs moving in the same direction are bound into a moving quadrupole in which the two vortices as well as the two antivortices rotate around each other.We study the scattering processes also analytically in the frame of a collective variable theory,where the equations of motion for a system of four vortices constitute an integrable system.The features of the diﬀerent collision scenarios are fully reproduced by the theory.We ﬁnally compare some aspects of the present soliton scattering with the corresponding situation in one dimension.I.INTRODUCTIONThe statics and dynamics of magnetic vortices is al-ready an old subject [1–4].An increasing interest in the problem has arisen again [5–7]which is con-nected with the synthesis and experimental study of new low-dimensional magnetic compounds such as two-dimensional magnetic lipid layers,organic intercalated quasi-2D layered magnets and HTSC-materials in the an-tiferromagnetic state.Magnetic vortices play an important role in easy-plane magnets.They are the main ingredients in the Kosterlitz-Thouless phase transition.At a ﬁnite temperature the density of vortices is large and they should give a consid-erable contribution to the dynamic correlations.We already have a detailed picture of the dynamics in a system with a small number of vortices.An isolated vor-tex in an inﬁnite system can only move together with the background ﬂux [3,8].Two vortices interact and undergo Kelvin motion if they have opposite topological charges while they move one around the other when they have the same charge.We shall call the pair of two vortices with opposite topological charge a vortex-antivortex pair (V-A pair).Such pairs can have the characteristics of a soliton in the sense that they move coherently with some constant velocity.Solitons of this kind have been nu-merically investigated in some magnetic systems [9,10].Analogous V-A pairs have been studied in superﬂuids [11–13],nonlinear optics [14]and hydrodynamics [15].In a system with a lot of vortices the picture becomes accordingly more complicated.If we suppose a dilute vortex gas,the average velocity is ρ∼1/L ,where ρis the density of vortices and L the average distancebetween them.The interaction energy is proportional to the logarithm of the average distance.This picture should be realistic above the Kosterlitz-Thouless temper-ature.At low temperatures vortex-antivortex pairs are expected to form.The energy of a pair is ﬁnite and it is proportional to the logarithm of its size.The interac-tion potential between them is inversely proportional to the second power of their size.One may be tempted to treat the vortex pairs as elementary weakly interacting particles.However,their dynamics is not Newtonian and most importantly they have an internal structure which may change during interaction.In a dense enough gas of vortices,interactions among traveling V-A pairs are unavoidable.In particular,any change in the number of vortices present in the system,or the number of vortices in equilibrium,should be a di-rect or indirect result of the scattering among V-A pairs.Our article is devoted to head-on and head-tail collisions between V-A pairs in easy-axis ferromagnets,i.e.to the case of zero total angular momentum of the system.Our study is both numerical and analytical.A collective co-ordinate theory is found to be particularly successful and provides the basis for a clear picture of the dynamics.Our results should also be relevant for a variety of other systems where V-A pairs have been found.Furthermore comparisons can be made to the well-studied soliton in-teractions in one space dimension.The outline of the rest of the paper is as follows.In Sec.II we give a short description of the system and an account of the dynamics of vortices and vortex-antivortex pairs.In Sec.III we present numerical simulations for collisions between V-A pairs.Sec.IV presents a theory which explains the features of the dynamical behavior of1vortex pairs.Our concluding remarks are contained in the last Sec.V.II.VORTICES AND VORTEX-ANTIVORTEXPAIRSWe consider the classical two-dimensional Heisenberg ferromagnet with a uniaxial anisotropy of the easy-plane type.The corresponding Hamiltonian has the formH =−J(n,m )(S n ·S m )+β2dxdy(∇m )2J/β.m,Φare canonically conjugateﬁelds and the equations of motion have the Hamiltonian form[17]∂Φδm,∂mδΦ,(3)explicitely∂Φ1−m 2−m (∇m )2∂t=(1−m 2)∆Φ−2m ∇m ∇Φ.(4)Our numerical algorithm uses the formulation through the stereographic variableΩ=1+mexp(i Φ).(5)This satisﬁes the equation i ∂ΩΩΩ∂µΩ∂µΩ−1−Ω1+ΩΩdenotes the complex conjugate of Ω.This vari-able was used in the solution of the Landau-Lifshitz equa-tion in one dimension since,in terms of it,the soliton solutions attain their simplest form [17,18].In studying statics and dynamics for the above model,the topological densityγ=∂m∂y−∂m∂x(7)is a most useful quantity.It has been called the ”local vorticity”since it plays here a role analogous to the or-dinary vorticity in ﬂuid dynamics [8,19].The integrated topological densityΓ=γdxdy (8)is an invariant and takes values which are integral mul-tiples of 2πfor the vortex solutions that we shall discusshere.The vorticity density enters the deﬁnitions of the linear and angular momentum of the theory which read [8]P µ=−εµνx νγdxdy,µ,ν=1,2,(9)ℓ=(x 2+y 2)γdxdy.(10)The role of the total vorticity in the dynamics can bealso appreciated through the deﬁnition of the so-called ”gyrocoupling vector”[20]G =−ˆz Γ,(11)where ˆzis the unit vector in the third direction.The gyrocoupling vector enters the equations which describe the dynamics of vortices in a collective coordinate theory.We now turn our attention to the discussion of topo-logical excitations.We take as a boundary condition that the ﬁeld Φis proportional to the polar angle φat spatial inﬁnity.We then obtain vortex solutions which have the form [2,3,5]Φ=κarctany −Yr,r ≫1,(14)where r is the distance from the vortex center,a,b are constants and λ=m (r =0)=±1we call the ”polarity”of the vortex.The radius of the vortex is unity in our2units.We use in our numerical simulations only vortices and antivortices withκ=±1and polarityλ=1.The total vorticity of the vortices(12,13)isΓ=−2πκλ. The structure of the magnetic vortices(12,13)is similar to that of the vortices in a non-ideal Bose gas[11]where the quantity(1−m)is the density of the Bose particles. However,the ferromagnetic vortices diﬀer in that they come with two possible values of the polarityλ.The most impressive characteristic of the dynamics of an isolated vortex is that it is spontaneously pinned in an inﬁnite medium.One can trace the reasons of this dy-namical behavior to their topological complexity which is reﬂected in the nonzero value ofΓ[8].On the other hand,a vortex-antivortex pair undergoes Kelvin motion. This motion was studied in[21,22](see also[23])for a large vortex separation.In general,Kelvin motion sets in when the two vortices have opposite total vorticities:κ1λ1=−κ2λ2.One can argue that the simplest topologically nontriv-ial objects which can be found in free translational mo-tion should have the form of a vortex-antivortex pair with Γ=0.A conclusive numerical and analytical study in an easy-plane ferromagnet was given in[9]where the proﬁles of coherently moving structures were numerically calcu-lated.There is a branch of solitons with velocities varying from zero to unity,which is the velocity of spin waves in the medium in our units.For small velocities the solitons have indeed the form of a V-A pair with a large separation L between the vortex and the antivortex.Their velocity is inversely proportional to the distance between themv=11−f iFIG.1.Head-on collisions between vortex-antivortex pairs.We plot here contours of the third component of the spin using the levels0.1,0.3,0.5,0.7,0.9.The numbers1,4denote the vortices and the3the antivortices.In(a)we have the case of a small diﬀerence between the linear momentum of the two pairs and the vortices exchange partners and scatter at an angle. In(b)the diﬀerence in momentum is larger.The pairs exchange partners,follow a looping orbit andﬁnally rejoin the initial partners and travel along the initial direction of motion.In(c)the momentum diﬀerence is large.The two pairs pass through each other.Note that the boxes presented here have dimensions40×40while the simulations were done in a space100×100.4Fig.1a presents the head-on collision of two pairs with a small diﬀerence between their sizes.We use here the V-A pair solitons which have been numerically calculated in [9].The initial ansatz in our simulation is the product ansatz of two such pairs.We have taken the size of the left pair L 1≃4which corresponds to a velocity v 1=0.27and the size of the pair on the right L 2≃6.3which cor-responds to v 2=0.15.The initial separation measured on the x -axis is δ=25.In the scattering process the vor-tices exchange their partners and two new identical pairs are formed which are scattered at an angle.Varying the sizes L 1,L 2we observe that the angle tends to 90o as the diﬀerence in the velocities (and momentums)of the two pairs is getting smaller.This process generalizes the 90o scattering of two identical solitons [24].According to (9),the larger soliton (pair on the right)has also a larger linear momentum.Conservation of the total momentum implies that each of the resulting iden-tical pairs has a non-zero x -component of the momentum and a velocity to the left.Fig.2a shows the trajectories followed by each vortex and antivortex.Open circles de-note the centers of the antivortices and ﬁlled circles those of the vortices.We defer for later the case of an intermediate diﬀer-ence between the sizes of the two pairs and discuss ﬁrst the case of a large diﬀerence (Fig.1c and 2c).In the latter case one expects that the vortices which belong to diﬀerent pairs will interact loosely with the vortices of the other pair.As a result,the two pairs are expected to travel almost undistracted.We use the ansatz (16)with the parameters L 1=4,L 2=7L 1=28.The initial separa-tion of the pairs is δ=30.The result is close to expecta-tions,that is the small pair passes through the large one.The distortion in the trajectories should become smaller as the size of the large pair becomes larger.This case is thus analogous to soliton interaction in one-dimensional integrable systems,as has been noted by Aref [27].In order to explore further this analogy we have plotted in Fig.5a the x -coordinate of the two pairs as a function of time.(The data of Fig.2c correspond to the data of Fig.5a only until time=345).We observe that the fast pair experiences a delay during the interaction with the slow one,which results in a negative shift (in comparison to the free motion)in its x -position after the collision.On the other hand,the slow pair is accelerated during the in-teraction and thus gains a positive shift in its x -position.The present observation should be contrasted to the sit-uation in one-dimensional soliton collisions where a pos-itive shift for both solitons is observed in all models.An intermediate situation between those in Figs.1a and 1c is presented in Fig.1b.The parameters in the initial ansatz are L 1=4,L 2=6L 1=24,δ=34.The pairs initially exchange partners during the scattering process and form new V-A pairs just as in the simula-tion in Fig.1a.However,after some excursion the new pairs approach each other again,exchange partners once more and the initial pairs re-emerge traveling along their initial direction of motion.The trajectories followed bythe vortices are depicted in Fig.2b.A similar scenario for point vortices in hydrodynamics has been discussed in [28].−2020−20020−20020−20020−20020−20020FIG.2.The orbits of the vortices and antivortices of Fig.1during their head-on collision.The ﬁlled circles denote the position of vortices and the open circles the position of the antivortices at successive and equal times intervals.The num-bers 1,3,4denote the initial position of the vortices and antivortices.5FIG. 3.Head-tail collisions between vortex-antivortex pairs.We plot here contours of the third component of the spin using the levels0.1,0.3,0.5,0.7,0.9.The numbers1,3de-note the vortices and the4the antivortices.In(a)the mo-mentum diﬀerence is large and the pairs pass through each other.In(b)we have a propagating quadrupole state.The boxes have dimensions40×40,the simulations were done in a space100×100.−20020−2020−20020−2020FIG.4.The orbits of the vortices and antivortices of Fig.3 during their head-tail collision.Theﬁlled circles denote the position of vortices and the open circles the position of the antivortices at successive and equal times intervals.The num-bers1,4denote the initial position of the vortices and antivortices.In the next set of simulations we explore the situation of a head-tail collision.That is,both the slow and the fast pair move to the same direction(to the right in Figs.3a, 4a).The parameters here are L1=8,L2=20,δ=25.The two pairs pass through each other,but some diﬀer-ences to the case of Fig.1c should be pointed out.In Fig.5b we give the x-component of the trajectories of the pairs.Weﬁnd that the fast pair is accelerated dur-ing the interaction(positive shift)while the slow pair is decelerated(negative shift).(See,however,the relevant remarks in the next section.)The last simulation,presented in Figs.3b and4b,in-cludes two identical V-A pairs traveling along the same direction.It can be considered as a limiting case to that of Fig.3a when the sizes of the pairs are equal.The pa-rameter values are L1=8,L2=8,δ=6.The system can6be also viewed as a vortex-vortex pair and an antivortex-antivortex pair.Both pairs rotate while at the same timethe magnetic quadrupole which is formed is propagatingalong the x-axis.A similar”leap-frogging”motion wasstudied in hydrodynamics by Love[29]and has also beenobserved with two vortex rings[30].Since the magneticquadrupole is characterized by two parameters(the ve-locity and the internal frequency)it can be considered asan analog of a breather[27,28].We can make an estimateof the mean velocity of propagation of the quadrupole.Suppose that L is the mean distance of the pair of vor-tices from the pair of antivortices.If this is large com-pared to the distance between vortices of the same kindwe may consider the quadrupole as two dipoles on top ofeach other.Then,a straightforward generalization of theresults of[9]gives2v≃2)and(2)and(3FIG.6.Head-on collision between a slow V-A pair withvelocity v2=0.1(right in theﬁrst entry)and a fast solitonwith velocity v1=0.9(left in theﬁrst entry).We present sixsnapshots at times t=0,22,44,66,88,132.The fast solitonis split in a vortex and an antivortex at the time of collision.The vortices exchange partners and two new pairs are formedwhich are scattered at an angle.The value of v1until which scattering at an angle occurs,seems to be close to the value v1=0.9in these cases,too.For increasing v2the minimum scattering angle,that wecan obtain by simulations,increases.The simulations of this section give an overview ofthe possible scattering scenarios.However,the picturewill not be complete until we obtain an analytical under-standing.The theory which we present in the next sec-tion accounts for the simulations presented in Figs.1,3and provides a reasonably satisfactory understanding.Astep towards the understanding of the results of Figs.6,7will also be taken.υ10.20.40.60.81cosθFIG.7.The cosine of the scattering angle for a series ofsimulations where the slow soliton has a velocity v2=0.1and the fast one0.1≤v1<1,is given by the dots which havebeen connected by a solid line.The dashed line results fromconservation of energy and linear momentum.The dottedand the dot-dashed lines give the result of formula(35)fortwo diﬀerent methods of calculating the soliton lengths(seetext for details).IV.COLLISION OF VORTEX-ANTIVORTEXPAIRS:ANALYTICAL DESCRIPTIONA full analytical investigation of interactions betweenvortex pairs appears to be quite complicated.Suﬃce itto say that no analytical formula is known for a singlevortex pair soliton.However,one can employ a rigidshape approximation and suppose that each vortex is acoherently traveling structure and is also well separatedfrom all the others.Then the dynamics of the system ofvortices reduces to that of their centers R i.The latterobey the equations[20,4]d R iE ≃−2πi<jκi κj ln(|R i −R j |),(20)where a constant has been omitted.We put R i =(X i ,Y i )in Eq.(18)and obtaindXi(R i −R j )2,dY i(R i −R j )2.(21)These are the same as the equations of motion of point vortices in hydrodynamics [15,16]when the hydrody-namic ”vortex strengths”,which correspond to κi in the present system,are ±1.One can now see thatP x =iP x i = i2πκi Y i ,P y =iP y i =−i2πκi X i ,(22)as well asℓ=i2πκi (X 2i +Y 2i )(23)are conserved quantities.Eqs.(22)give the two compo-nents of the total momentum and Eq.(23)gives the total angular momentum of the system.They can be derived from the formulas (9,10)when the rigid-shape approxi-mation is used.We further note that Eqs.(18)can be derived as the Hamilton equations associated with the Hamiltonian (20)with the conjugate variables X i and P x i =2πκi Y i .In general,system (21)is integrable only for a system of three point vortices.However,for a system of two V-A pairs,for which condition (19)is satisﬁed,Eqs.(21)can be integrated,when the pairs are initially moving along the same line [32].This is a fortunate situation since we shall be exclusively concerned with such systems in the present paper.The approximations employed so far seem to be quite crude.For instance we expect the rigid-shape approxi-mation to be valid when the distances between the vor-tices are much larger than the size of the out-of-plane structure of each vortex (unity in our units).This could severely restrict the applicability of Eqs.(21).However,the numerical simulations,of the previous section,indi-cate that the results of the present approximate theory could be qualitative correct even for situations beyond the applicability limits of the theory.In the following we give the equations of motion of two V-A pairs.The vortex and antivortex of the ﬁrst pair are placed at positions (x 1,±y 1)and have κ=±1.We denote this pair schematically as (134).(cf.top entries of Fig.1.)We choose the polarity λ=1for all vortices.The system represents two V-A pairs in a head-on collision course.From Eqs.(21)we derive the four equations of motiondx 12y 1+y 1−y 2(x 1−x 2)2+(y 1+y 2)2,(24)dy 1(x 1−x 2)2+(y 1−y 2)2+x 1−x 2dt=−1(x 1−x 2)2+(y 1−y 2)2+y 1+y 2dt=−x 1−x 2(x 1−x 2)2+(y 1+y 2)2.(27)The system (24-27)is completely integrable since thereare four independent conserved quantities.We shall use the energy and the x-component of the linear momentum:y 1y 2(x 1−x 2)2+(y 1−y 2)2L 1,v 2=12)is the ”small”and fast pair and (2≃5.83,the pairs changepartners during the process and scatter at an angle (Fig.1a).(b)In the intermediate case,when α1<α<α2=(√√2− 5−1)≃8.35,the vor-tices change partners during scattering but laterthey rejoin their initial partners and travel along the initial direction of motion (Fig.1b).9(c)When α>α2the fastpair passesthroughtheslowone (Fig.1c).InFig.8we give a schematic representation of the diﬀer-ent regions in the (v 1,v 2)plane where the three scattering processes occur.In case (a)the scattering process can be described by(134)→(124)which is a schematic repre-sentation of the change of partners during collision.In the limit of two identical pairs one can use Love’s [29]equation (28)to obtain the following well-known result1y 21=1L 1L 2.(32)This formula is known for point-like vortices in 2D hy-drodynamics of incompressible ﬂuids [16].The angle of scattering can be found if we use the laws of conservation of energy and linear momentum:E =E 1+E 22,(33)where E is the energy and P the absolute value of the linear momentum on the x -axis of each of the ﬁnal pairs.The angle θis measured from the direction of motion of the slow pair.We obtaincos θ=P 2−P 12√L 2−L 1=−y 212)y (0)1.(36)The above results,should be a good approximation in the limit of large L 1,L 2.There is a simple way to apply Eq.(35)when the two colliding V-A pairs are slow and the vortex and antivor-tex in a pair are well separated.The size of each pair can be taken to be the distance between the two points where the spin variable reaches the north pole and its velocity is the inverse of it.This method gives fairly good results for simulations with pairs of the size used in Fig.1.A comparison of Eq.(35)with numerical results is given in Fig.7.The cosine of the angle θfor 1≤α≤α1,is plotted by a dotted line.Fig.7shows cos θas a func-tion of v 1,for v 2=0.1.(not as a function of α).We have v 1>v 2and we consider α=v 1/v 2.The dotted line is a poor approximation to the simulation results for α>4(v 2>0.4).The deviation from the numericalpoints is even qualitatively wrong already for v 2>∼0.583.We recall at this point that in our simulations of the previous section we obtained results which could be un-derstood as interaction processes between two dipoles even in the cases where the V-A pairs have no apparent dipole character.Exploiting this remark we assume that the V-A pair solitons (even those with large velocities 10and no apparent vortex-antivortex character)are dipoles with a length L and a charge q.To be sure,in the limit that the velocity goes to zero,L should go to the simple deﬁnition of length described in the previous paragraphs and q should go to the vortex numberκ=±1.In gen-eral,however,we use a generalization of Eqs.(22)and (15)and write for the momentum and velocity of such a dipole:P=2πqL,v=qP P v1−v2→∞,that is an in-ﬁnite length of the dipole and also q∼1/√v1arctan(2v1v2t),x2=−v2t−1v2(2v1v2t)α34α4α242)deviates strongly from the rectilinear motion.The shifts in the positions of the vortex pairs as time varies from−∞to∞,are ∆x1=−4π/v1for the fast pair and∆x2=4π/v2α3forthe slow one.(The terms in the dots do not give anyshift in the pair position when times goes from−∞to ∞.)The shifts are measured relative to the direction of motion of each pair.The results of the simulations for the vortex system show that the fast pair is deceleratedduring the collision as if it is repelled by the slow one,in agreement with Eq.(39).The simulations further show that the slow pair is accelerated during the collision as if it is attracted by the fast one.Theﬁnal result is a pos-itive and negative shift in the positions of the fast and slow pair,respectively,a phenomenon which has not been observed in head-on soliton collisions in one dimension. The obtained trajectories are similar to those for the2D Euler equation[34].In case(b),α1<α<α2,the scattering process is rep-resented by the scheme(134)→(124)→(134).This means that the vortices exchange part-ners at aﬁrst stage of the scattering,then the new pairs follow a looping orbit and at theﬁnal stage the initial partners rejoin and travel along the initial direction of motion(cf.Figs.1c,2c).The loop becomes the parabola (36)in the limitα=α1and it is a cusp whenα=α2. We now turn to the head-tail collision.It corresponds to the area denoted(d)in Fig.8and the relevant numer-ical simulations have been given in Figs.3and4.The case has been studied within a hydrodynamical context in[25,29,28].As Love showed,only a slip-through mo-tion(Fig.3a)and a leap-frogging motion(Fig.3b)can occur.Suppose that vortices1and3haveκ=1while4are antivortices and haveκ=−1(cf.Fig.3a).The equa-tions of motion are modiﬁed as follows:the left hand side of(24,25)and theﬁrst term on the right hand side of (26)change their signs.The conserved quantities(28,29) now ready1y2(x1−x2)2+(y1+y2)2(x(0)1−x(0)2)2+(y(0)1−y(0)2)2,(40)y1+y2=y(0)1+y(0)2.(41) The pairs are initially at(x(0)1,±y(0)1)and(x(0)2,±y(0)2). Using the above one canﬁnd that two pairs which start inﬁnitely far apart will pass through each other for any value of y(0)1<y(0)2.For a large diﬀerence in the size of the pairs(α≫1)weﬁnd,by an expansion,the solution of the equations of motion(with an error O 1v1arctan(2v1v2t),x2=v2t−1v2(2v1v2t)+1v2arctan(2v1v2t)+...,y1=y(0)1 1−11+(2v1v2t)2 ,y2=y(0)2 1+11+(2v1v2t)2 .(42) The dots stand for terms of order O 1t,x1= V t+√t.It is interesting to compare this shift with that for solitons in1D systems where the shift is proportional to the logarithm of the diﬀerence of the velocities of the two solitons(∆x1∼ln(v1−v2))and the distance between two identical solitons after collision goes like x1−x2∼ln t.Two V-A pairs which have aﬁnite distance between them,will pass through each other when the following relation holds(y(0)2+y(0)1)2[(y(0)2+y(0)1)2−2(y(0)2−y(0)1)2]δ2−2from the Graduiertenkolleg”Nichtlineare Spektroskopie und Dynamik”.。

专业英语

Definition of polymers A simple understanding of polymers can be gained by imaging them to be like a chain or, perhaps, a string of pearls, where the individual pearl represent small molecules that are chemically bonded together. Therefore, a polymer is a molecule made up of smaller molecules that are joined together by chemical bonds. The word polymer means „many parts or units.‟ The parts or units are the small molecules that combine. The result of the combination is, of course, a chainlike molecule (polymer). Usually the polymer chains are long, often consisting of hundreds of units, but polymers consisting of only a few units linked together are also known and can be commercially valuable.
Figure 1.1 Diagram illustrating the definition of plastics.
As Figure 1.1 shows, all materials can be classified as gases, simple liquids, or solids, with the understanding that most materials can be converted from one state to another through heating or cooling. If only materials that are structural solids at normal temperatures are examined, three major types of materials are encountered: metals, polymers, and ceramics. The polymer materials can be further divided into synthetic polymers and natural polymers. Most synthetic polymers are those that do not occur naturally and are represented by materials such as nylon, polyethylene, and polyester. Some synthetic polymers could be manufactured copies of naturally occurring materials (such as

研究生英语泛读翻译第十四单元

失意时期的伦勃朗新的创作荷兰在17世纪是欧洲最兴旺的国家。

但到了世纪中叶，部分由于耗资巨大的战争，泡沫破裂了。

荷兰的艺术市场在最高峰时崩溃了。

有人说：“哦，那只是阶段性的衰落。

”并非如此，荷兰艺术的黄金时代就此结束了。

伦勃朗（Rembrandt)受的打击尤其严重。

十年前他曾是一个明星，想买他绘画的客户能排出一英里长。

当时的阿姆斯特丹人就像是今天的纽约人一样，对艺术渴望的有钱人在家里挂上伦勃朗的绘画是必须的，曾经是必须的。

于是伦勃朗把自己变成了一架艺术制造机，雇佣了大批助手来赶制他的作品，他变的十分富有。

他还变得随心所欲。

他倾其所有去借贷。

除了制造艺术，他也自己销售,他不仅经营自己的绘画，而且卖其他艺术家的作品。

他购买了一幅鲁宾斯的作品，然后转手加价。

他还兜售那些与自己的作品十分相似的，学徒的绘画。

衰落的经济使一切都分崩离析。

客户不见了，债主上门了。

他破产了，不再流行了，成了失败者。

可能其他艺术家认为荷兰的经济恢复只是个时间问题，但我猜想伦勃朗不是这么想的。

他不再像以前那样去画了，他失去的太多了。

他走上了一条自己的路。

这些是我几天前去大都会艺术博物馆参观荷兰绘画时的感想。

我自从在2007年看了“伦勃朗的时代”那次画展后再没有看过这些绘画，说实在的，我那时对这样的艺术有点腻烦了。

但是艺术给人的感受不是从一而终的，而是根据周围的变化而变化。

我现在从一个经济崩溃的角度看这些荷兰绘画。

一个触礁的市场，繁荣假象的幻灭，使这种艺术给人以不同的感受。

这次伦勃朗给我的不同感受尤其震撼，其实我一直是个会被突然的不同发现感动的人。

就像是你多年熟悉的老朋友，你认为他们的所做所为不会出乎你的预料，但是你错了。

因为他们从未像你想的那样一样过。

我对维米尔（Vermeer, 荷兰17世纪画家——译者按）的作品没有这样的感受。

我在博物馆里再次看到的维米尔的绘画与我记忆中的维米尔基本一致。

是不是由于在他的作品中，每一个构图都是那样精确的决定，每个主体都是同样的摆放，每个人物都有清晰的线条，就像是押韵顺口的诗词或是固有的想法那样在脑中挥之不去。

保护鲸鱼公益广告英语作文

Whales are majestic creatures that have roamed the oceans for millions of years, yet they face an existential threat from human activities. The recent public service announcement PSA I saw on protecting whales deeply resonated with me, and it urged me to reflect on our responsibilities towards these gentle giants of the sea.The PSA began with a hauntingly beautiful image of a whale gliding through the ocean, its massive body cutting through the water with grace and ease. The serene scene was quickly juxtaposed with the harsh reality of the threats whales face, such as entanglement in fishing nets, habitat destruction, and the deafening noise pollution from ship traffic. These images were accompanied by a powerful narration that highlighted the plight of whales and the urgent need for action.One of the most impactful parts of the PSA was the segment on whale communication. Whales use a complex system of vocalizations to communicate with each other over vast distances. However, the increasing noise levels in the ocean, caused by human activities, are disrupting this communication, leading to disorientation and separation of families. The PSA included a chilling audio clip of a whales song drowned out by the cacophony of ship engines, a stark reminder of the human impact on these creatures lives.The PSA also touched on the issue of whaling, a practice that has led to the near extinction of some whale species. The images of whales being hunted and killed were heartwrenching, and the narration emphasized the need to protect these animals from such cruelty. It was a stark reminder ofthe devastating impact that human greed can have on the natural world.But the PSA wasnt all doom and gloom. It also highlighted the positive steps that individuals and organizations are taking to protect whales. It showcased the efforts of marine conservation groups, who work tirelessly to rescue entangled whales and advocate for stronger protections. The PSA also encouraged viewers to reduce their plastic consumption, as plastic pollution is a significant threat to marine life, including whales.One of the most memorable parts of the PSA was a call to action, urging viewers to get involved in whale conservation efforts. It provided practical steps that individuals can take, such as supporting organizations that protect whales, reducing plastic waste, and spreading awareness about the issue. The PSA ended with a powerful message: every action counts, and together, we can make a difference.The PSA left a profound impact on me. It opened my eyes to the challenges that whales face and the urgent need for action. It also inspired me to get involved in conservation efforts. Ive started researching ways to reduce my plastic consumption and have joined a local marine conservation group as a volunteer.Moreover, the PSA has made me more aware of the importance of preserving the natural world. Its a reminder that we share this planet with countless other species, and its our responsibility to protect them. The PSA has also made me more mindful of my own actions and their impact on the environment.In conclusion, the PSA on protecting whales was a powerful and moving call to action. It highlighted the challenges that whales face and the urgent need for human intervention. It also provided practical steps that individuals can take to make a difference. The PSA has inspired me to get involved in conservation efforts and has made me more mindful of my own actions and their impact on the environment. Its a reminder that every action counts, and together, we can make a difference in protecting these magnificent creatures.。

美国研制的机器海豚robotuna

Flow,Turbulence and Combustion71:105–118,2003.105©2004Kluwer Academic Publishers.Printed in the Netherlands.Separation and Turbulence Control in Biomimetic FlowsALEXANDRA H.TECHET,FRANZ S.HOVER andMICHAEL S.TRIANTAFYLLOUDepartment of Ocean Engineering,Massachusetts Institute of Technology,Cambridge,MA02139,U.S.A.;E-mail:mistetri@Received2October2002;accepted in revised form3October2003Abstract.The study of theﬂow around live marine animals and robotic mechanisms which emulate ﬁsh motion has revealed a number of mechanisms ofﬂow control,optimised through evolution to minimize the energy required for steady and unsteady motion underwater.We outline some of the mechanisms used to(a)eliminate separation,(b)reduce turbulence,and(c)extract energy from oncoming vorticalﬂows.Key words:biomimeticﬂows,separation control,turbulence control.1.IntroductionWork with swimmingﬁsh and marine mammals has revealed a new paradigm of locomotion,distinctly different from conventional propulsion used in man-made vehicles[33,34].Fish employ relatively large amplitude,rhythmic unsteady body motions.These motions are used by swimming animals to achieve high efﬁciency and outstanding maneuvering agility[16].Biomimetic studies and observations of ﬁsh and cetaceans have provided a wide array of information on the kinematics,i.e. how these animals employ theirﬂapping tails and severalﬁns to produce propulsive and maneuvering forces(see reviews in[13,34]).Recent work with live animals provides important information on the resultingﬂow structures behind swimming ﬁsh[2,8,9,19,21,23,36].Unsteadyﬁsh-like swimming motions have been stud-ied experimentally in order to better understand howﬁsh use their body motions to control separation,turbulence and vorticity;results from these experiments are discussed herein.Unsteady motions have been used in laboratory and theoretical studies in the past to achieveﬂow control.Taneda[26]and Tokomaru and Dimotakis[31]im-posed a rotational oscillation on a two-dimensional cylinder in cross-ﬂow,reducing the width of the wake and hence the drag coefﬁcient signiﬁcantly,for properly se-lected parametric values.Injection of unsteady vorticity is the mechanism through which theﬂow control was implemented.Ffowcs-Williams and Zhao[12]also have shown that is possible to obtain efﬁcientﬂow control through the unsteady106 A.H.TECHET ET AL. motion of a body inﬂuid.Maxworthy[20],Ellington[10,11],Freymuth[14]and Dickinson[6,7]studied the aerodynamics related to theﬂight of hovering insects and concluded that unsteadyﬂow mechanisms play a very important role.The possibility of extracting energy from oncoming vortices through an os-cillating foil was investigated experimentally by Gopalkrishnan et al.[15];and theoretically by Wu and Chwang[37],Sparenberg and Wiersma[22],and Streitlien and Triantafyllou[24,25].It was shown that energy contained in large-scale ed-dies can be retrieved hence enhancing propulsive efﬁciency,or amplifying thrust production.We outline evidence ofﬂow control achieved inﬁsh-like swimming,including separation elimination and turbulence reduction,up to Reynolds numbers of1.5×106;as well as energy extraction from oncoming vorticalﬂows.2.Separation and Turbulence ControlAﬂat plate undergoing transverse oscillations in the form of a traveling wave, placed in an oncoming steadyﬂow,was found to exhibit reduced turbulence in-tensity and separation as the phase speed of the traveling wave is increased to reach values comparable to the free stream velocity[26].The same mechanism of unsteady vorticity injection appears to control separation and turbulence produc-tion.The Reynolds number based onﬂat plate length was below250,000,so it is important for applications to investigate much higher Reynolds numbers.2.1.METHODOLOGYSeveral mechanisms have been designed to studyﬁsh-like swimming motions. These mechanisms emulate the motion observed in live swimmingﬁsh,but do not attempt to model the feedbackﬂow control that a liveﬁsh employs through the use of its lateral line sensing mechanisms.Theﬁrst mechanism is the MIT RoboTuna, an eight link,tendon and pulley driven robot,whose external shape has the form of a Blueﬁn Tuna[32],capable of emulating the swimming motion of a live tuna.The second two mechanisms,a waving plate mechanism and aﬂapping foil device,aim to investigate respectively:traveling wave motion effects on near boundaryﬂow and the energy extraction from oncoming vorticalﬂows byﬂapping foil devices.The RoboTuna is an eight link,pulley-driven robot modeled after the swimming motion and geometry of a Blueﬁn Tuna[32].It is mounted on a carriage at the MIT Testing Tank,a towing tank100feet long,eight feet wide andﬁve feet deep on average capable of towing speeds up to1.5m/s.Figure1shows the RoboTuna attached to the tank carriage(at left)with its lycra skin and the image at right shows the inside linkages that comprise part of the robot skeletal system.Experimental work with a1-meterﬂapping plate in a recirculating water tunnel at Reynolds numbers up to2,000,000shows reduction or even elimination of sep-aration,and signiﬁcant reduction of turbulence intensity as ration of the travelingSEPARATION AND TURBULENCE CONTROL IN BIOMIMETIC FLOWS107Figure1.MIT RoboTuna.At left the robot swimming in the MIT Testing Tank attached toa towing carriage.At right,the inside structure of the RoboTuna allows the complex bodymotion of a swimmingﬁsh to be emulated.(Original in colour)wave speed,c,to the freestreamﬂow speed,U,approaches c/U=1.2[30].Fig-ure2shows a sketch of the experimental apparatus,which allows for a traveling wave motion using a series of scotch yokes driven by a single motor,activating eight pistons.The rods are connected to a reinforced rubber mat that forms the ﬂapping plate.The waving plate mechanism is designed using aﬂexible neoprene mat with eight pistons driving a traveling wave down the length of the mat.The apparatus was mounted in a recirculating water tunnel capable of speeds up to9m/s.The tunnel has a working section of0.5m by0.5m by1.2m long.The drive mechanism shown in Figure2atop the tunnel section is comprised of eight piston rods,which pass through a sealed window in the top of the tunnel section,controlled by a drive shaft coupled to the piston cranks arms by chain linkage.The mat is a structurally reinforced neoprene mat that is1.4m long and0.5m wide,spanning the tunnel.At the leading edge of the mat is an aluminum plate rigidly attached to the tunnel such that the amplitude at this point is zero.The motion has an amplitude envelope with slope1/16,increasing to the stern.The mat length is1.25times the traveling wave wavelength.The phase speed,c,of the traveling wave is dictated by the speed of the drive shaft.Data on the waving plate and RoboTuna mechanism were taken using a Digital Particle Image Velocimetry technique,where a laser sheet illuminates neutrally buoyant particles seeded throughout theﬂuid and a CCD camera records the mo-tion of the particles in order to calculate the velocity vectors.An overview of this technique can be found in[1,35].In the case of the RoboTuna a Pulnix TM-1040108 A.H.TECHET ET AL.Figure2.Sketch of the waving plate apparatus showing the drive mechanism and the piston conﬁguration.The waving mat can be seen through the tunnel section side window.Flow is from left to right and the traveling wave propagates in the direction of theﬂow,with increasing amplitude,A(A=x/16,where x=0m is the leading edge of the mat).The drive mechanism is atop the tunnel and connected to the mat via eight piston rods.camera,with1024×1024pixels and30Hz,was submerged in a watertight hous-ing looking down on the robot with a light sheet,generated by a Spectra Physics PIV400mJ/pulse ND:Yag laser,entering the tank on a horizontal plane along the ﬁsh’s lateral line(the mid-line between the dorsal(top)and anal(bottom)sides of theﬁsh).For the waving plate the laser sheet,from a NewWave Gemini PIV 120mJ/pulse ND:Yag laser,was oriented vertical coming in through the bottom window in the tunnel section and the Pulnix camera viewed the particle motions through the side window.In addition to PIV,a Dantec Laser Dopplar Velocimetry (LDV)was used to measure the boundary layer proﬁle on the waving mat.2.2.RESULTSFlow visualization with the1.20m long robotic mechanism,the RoboTuna,ex-hibited lack of separation effects along its body,even when the motion amplitude reached values10%of the body length.Also,measurement of theﬂow charac-teristics of the boundary layer,using particle image velocimetry(PIV)techniques, showed apparent re-laminarization of theﬂow when the traveling wave phase speed was close in value to the stream velocity[3,27–29].Under conditions of towing the mechanism without transverse motion,the boundary layer was characterized by a turbulent velocity proﬁle.Figure3demon-strates the difference between the two measured average velocity proﬁles:The form of the velocity curve has the characteristic law of the wall shape when there is no transverse motion,while a laminarized shape appears when there is active swimming motion with c/U=1.14,where c is the speed of the traveling wave along the body length and U the speed of the external stream.The solid lineﬁtsSEPARATION AND TURBULENCE CONTROL IN BIOMIMETIC FLOWS109parison of the near boundary average streamwise velocity proﬁles obtained with digital particle image velocimetry(DPIV)about the MIT RoboTuna at Reynolds number 800,000,based on theﬁsh body length.Red(solid)symbols represent the non-swimming pro-ﬁle and are indicative of a turbulent boundaryﬂow.The blue symbols represent the swimming case at c/U=1.14;this proﬁle indicates a laminarization of the near bodyﬂow.The solid lineﬁts the‘law-of-the-wall’proﬁle to the boundary layer data for the non-swimmingﬁsh fora baseline comparison.(Original in colour)the‘law-of-the-wall’proﬁle to the boundary layer data for the non-swimmingﬁsh for a baseline comparison.Computational studies on aﬂat plate undergoing traveling wave oscillations within an oncoming stream,show reduction of separation as the phase speed c increases,with complete elimination of separation at c/U close to1[38].Similarly, turbulence intensity is found to decrease non-uniformly across the length of the plate,with increasing c/U,up to a value of about1.5.The study was conducted at Reynolds number,based on plate length,of6,000and then18,000.Although theﬂow features around the plate change with Reynolds number for a non-vibrating plate,theﬂow remains qualitatively similar at the Reynolds numbers studied when c/U is near1,with a preferred value of c/U=1.2.The energy to propel theﬂapping plate,deﬁned as the energy to tow the plate(which can be negative if the plate produces thrust)plus the energy to oscillate the plate,is minimal at a value of c/U=1.2[38,39].Figure4shows the turbulence intensity along the length of aﬂapping plate,for two values of c/U,0.4and1.2.There is non-uniform,but substantial reduction of turbulence intensity in the case of c/U= 1.2.110 A.H.TECHET ET AL.Figure4.Direct Numerical Simulation(DNS)results show the turbulence intensity over a waving plate at Reynolds number6,000.Plot at left is for phase speed,c=0.4U and at right is c=1.2U.The red color indicates high levels of turbulence and the blue,low levels.As phase speed increases turbulence energy is progressively reduced,as indicated by the reduction of red color in the right image.(Original in colour)parison of PIV data from the waving plate experiment(left)at Reynolds num-bers106,based on mat length,and direct numerical simulations(DNS)(right)at Reynolds number6,000.Theﬁgures show persistence of laminarization through traveling wave motion at higher Reynolds number than previously investigated.Data sets represent boundary layer proﬁles at a mat crest.Three wave phase speeds are represented by each of the data sets.For the experimental data the phase speed to freestream velocity rations are c/U=0.3,0.6and1.2are shown in red,blue,and green respectively;for the DNS data the ratios shown arec/U=0.0,0.4,and1.2,in circle,squares and triangles respectively.The solid line(left)and dashed line(right)is aﬁt to the‘law of the wall’turbulent boundary layer proﬁle.(Original in colour)SEPARATION AND TURBULENCE CONTROL IN BIOMIMETIC FLOWS111 Figure5shows a comparison between digital particle image velocimetry(DPIV)data at Reynolds numbers106and direct numerical simulations at Reynolds number6,000,demonstrating the qualitative similarity between the twocases,despite the large difference in Reynolds number.Figure6shows the phaseaveraged turbulence intensity of the horizontal(streamwise)and vertical compo-nents of the velocity normalized with respect to U2,the square of the averagefreestreamﬂow speed.The turbulence intensity is calculated from ensemble aver-aged laser Doppler velocimetry(LDV)records.The total velocity is U t=u+U, where u is the turbulenceﬂuctuation and U is the averageﬂow velocity obtainedfrom phase averaged velocity samples(sample size N=1000).The velocity mea-sured4mm under the mat boundary at piston#5(ﬁfth piston aft of leading edge).The inﬂow velocity is U=1.0m/s and the phase speed varies from c/U=0.3to2.0.As shown,the local turbulence intensity is reduced for c/U up to1.5,butincreases for phase speeds beyond this value.Recent studies with a three-dimensional body,in the form of a water snake,undergoing traveling wave oscillations,identical to those studied with theﬂapping plate mechanism,shows non-uniform turbulence reduction along the body,but the clear trends observed with a two-dimensional plate could not be established[17].3.Energy Extraction from Oncoming Vortical FlowsAnecdotal evidence of energy recovery by liveﬁsh in turbulentﬂow containing large scale eddies is substantiated by experimental work with liveﬁsh[5];and with simpler experiments using the controlled motion of aﬂapping foil within the wake of a bluff body,such as a D-section cylinder.The foil is used to extract energy from theﬂow by properly timing its motion relative to the arrival of large eddies [15,4,5].Gopalkrishnan et al.[15]placed a two-dimensional foil in the wake of a cylinder within an oncoming stream to measure the forces and torque needed to oscillate the foil so as to produce thrust;and to observe theﬂow patterns around the oscillating foil.The foil would undergo aﬂapping motion,i.e.harmonic,controlled heave (transverse)and pitch(angular)motions with adjustable amplitude and relative phase.They found that depending on the timing of the motion of the foil relative to the arrival of oncoming vortices,three distinct patterns could be observed:•A destructive interference pattern,where vortices shed by the foil wouldﬁrst pair and then coalesce with opposite-sign vortices shed by the upstream cylin-der to form a wake consisting of weak vortices aligned near the centerline of the wake.The efﬁciency of thrust production was found to be increased compared to other conditions.•A constructive interference pattern,where vortices shed by the foil would coa-lesce with same-signed cylinder vortices forming strong vortices arranged in a112 A.H.TECHET ET AL.Figure6.Normalized horizontal, uu /U2,and vertical, uu /U2,turbulent energy,respec-tively top and bottom,calculated from ensemble averaged LDV data recorded4mm underthe mat boundary at piston#5.u is theﬂuctuation velocity and U is the meanﬂow speed.Inﬂow speed is U=1.0m/s and phase speed varies from c/U=0.3–2.0.Local turbulenceenergetics are reduced for phase speed up to1.5times inﬂow speed,but increase for phasespeeds beyond this value up to c/U=2.0.Red data is taken in the mat trough and blue at themat crest.(Original in colour)von Kármán street or reverse von Kármán street,depending on the parametricconditions.Efﬁciency was found to be minimal under such conditions.•An intermediate condition,where foil vortices would pair with opposite signed cylinder vortices forming‘mushroom’-like structures,expanding thewake width.Theoretical studies by Streitlien&Triantafyllou[24,25]showed that energyextraction is possible when a foil operates in the wake of a bluff body,in an‘intercepting’mode,where the foil would intercept with its leading edge oncomingvortices.Efﬁciency,deﬁned as the ratio of the useful energy(thrust times freestream velocity)over expended energy,could exceed100%under conditions ofenergy extraction.Theﬂow patterns associated with the intercepting mode consist of opposite-signed vortices,one from the cylinder and the other from the foil,which are pushed close together,resulting in an effective mutual elimination(the inviscidSEPARATION AND TURBULENCE CONTROL IN BIOMIMETIC FLOWS113Figure7.The foil-cylinder apparatus is shown above.The foil and cylinder are attached to an overhead carriage and move together down the tank.The D-cylinder moves sinusoidally in heave only,creating a von K´a rm´a n vortex street.The foil moves in heave and pitch and the phase of the motion follows the cylinder determining the interaction of the foil with the von K´a rm´n wake.(Original in colour)code could not predict destructive vortex interference,since two inviscid vortices of the opposite sign do not coalesce).A different mode,named‘slaloming mode’,was characterized by the foil avoiding to intercept oncoming vortices(slaloming around them):Energy was not recovered and the resulting wake consisted of vigorous vortices resulting from coalescence of same signed vortices,one from the cylinder and the other from the foil.3.1.METHODOLOGYWe employ a specially constructed apparatus(see Figure7)at the MIT Testing Tank Facility,consisting of two inverted-U frames:The front frame supporting a bluff cylinder of diameter7.5cm,span60cm,and with a D-shaped cross-section (theﬂat portion facing downstream),capable of oscillating in an up and down mo-tion;and the rear frame supporting a two dimensional NACA0012foil,with chord 10cm and span60cm,capable of executing a heave(up and down linear motion) and a pitch(angular motion about an axis located at about1/3of the chord from the leading edge).The cylinder and foil are towed at constant speed one behind the other,while executing independently-driven harmonic motions of given amplitude and phase.Circular end plates wereﬁtted at both ends of the cylinder and foil to reduce end effects.The dimensions of the Tank are30m by2.6m by1.14m.Pairs of lead-screws and linear tables were employed to achieve the linear mo-tions of both the cylinder and the foil,powered by Parker ML3475B direct-drive114 A.H.TECHET ET AL.Figure8.Foil thrust coefﬁcient,C t,and propulsive efﬁciency,??when placed behind a cylin-der generating large scale eddies.The horizontal axis is the phase angle between foil motion and arrival of a cylinder vortex,which causes large changes in C t andηdue to vortex-foil interactions.The cylinder motion has a heave amplitude,H cyl=0.4D,where D is the cylinder diameter(D=7.5cm).The foil amplitude of motion is H foil=0.5D,amplitudeof pitch angle isα0=0◦,and heave-pitch phase angle isψ=90◦.The Strouhal number is St=0.3and the Reynolds number R c=40,000,based on the foil chord length(c=10cm).(Original in colour)motors and ampliﬁers;while the foil pitch motion was actuated through a chain and sprocket arrangement powered by a Pittman GM14900geared servomotor.Forces were measured at the ends of the cylinder and foil through Kistler9601force cells and a9069torque cell.A potentiometer returned the pitch angle,and the linear motion was measured through an LVDT.The foil was placedﬁve cylinder diameters behind the cylinder,while the phase between the heave and pitch motion wasﬁxed at90◦,while the heave amplitude of both the cylinder and foil wasﬁxed at7.5cm.The frequency of motion was chosen to close to the Strouhal frequency of the cylinder,while the phase angle between the cylinder and foil motion,and the angle of attack of the foil were the major parameters of the testing program[5].The phase angle between cylinder and foil motion controls the timing between the arrival of a von Kármán vortex generated by the cylinder and the position of the leading edge of the foil,which can,for example,intercept the vortex,or avoid it.3.2.RESULTSFigure8shows the thrust coefﬁcient and efﬁciency of aﬂapping foil within thewake of a vortex-shedding cylinder.The thrust coefﬁcient,C t,is calculated fromthe measured thrust,T,as C t=2T/(U2cs),where U is the towing speed,c is the foil chord and s is the foil span.The foil efﬁciency is calculated asη=T U/P foil,where P foil is the power generated by theﬂapping foil.The foil power is calculatedfrom the measured lift,L,and torque,τ,and heave,h,and pitch,θ,accelerations:P foil=L(d h/d t)+τ(dθ/d t).As seen,both thrust coefﬁcient and efﬁciency are affected signiﬁcantly bythe timing of vortex arrival.The relative timing between the arrival of cylinder-generated eddies and eddies shed by the trailing edge of the foil is crucial indetermining the type of vortex interaction that will prevail.Visualization of theﬂow shows that the three principal patterns identiﬁed in[15]can be associatedwith the efﬁciency peaks(destructive interference mode),efﬁciency troughs(con-structive interference mode),and efﬁciency nodes(vortex pairing).Efﬁciency can even exceed100%due to energy recovery by the foil[4].In the case of liveﬁsh,work with swimming trout in a waterﬂume shows thatwhen a bluff body is placed upstream of the animals,their regular swimming pat-terns change,and the frequency of body motion is synchronized with the frequencyof vortex shedding from the cylinder.The body motion also changes,with theapparent trend thatﬁsh arrange for their body to intercept oncoming vortices,inagreement with the experimentalﬁndings from the airfoils[5].Also,the wave-length of the body motion is changed to accommodate the frequency requirementsof the upstream bluff body.4.ConclusionsExperimental and numerical investigations ofﬁsh-like swimming motion explorethe beneﬁts of unsteady motion for the control of separation,turbulence,and vor-ticity.A traveling wave motion can reduce local turbulence levels signiﬁcantlyas shown through experiments and numerical simulations,especially in regionsof high separation such as the wave trough;and experiments with an unsteadyﬂapping foil in the wake of a heaving cylinder revealed energy extraction from anoncoming vortical wake.To study separation elimination and turbulence reduction,ﬁrst a roboticﬁshwas constructed and tested,modeled after the blueﬁn tuna.The robot emulatesthe measuredﬁsh kinematics in water,which have the form of a structural wavetraveling from the head to the tail.Flow visualization using DPIV near theﬁsh’sbody,at an optimal wave speed of c/U=1.14,reveals that the boundary layer ofthe actively swimming robot has an average velocity proﬁle closely resembling alaminar boundary layer;whereas when towed rigid-straight the velocity proﬁle has a turbulent boundary layer shape.Further investigation of the effect of traveling wave motion on the boundary layer of aﬂat plate was performed experimentallyand numerically.The two-dimensional plate undergoing traveling wave motion is a simpler structure than the three-dimensional surface of aﬁsh-like body,which nonetheless captures the essence of the phenomena involved in turbulence reduc-tion and separation elimination.It was shown that,over a wide range of Reynolds numbers,qualitatively similar mechanisms are at work,providingﬂow without separation,as well as spatially non-uniform reduction in turbulence intensity;these effects were found to be optimal when the phase speed is close to1.2times the free stream velocity.To study energy extraction from oncoming turbulentﬂow,a special apparatus was used,consisting of an upstream bluff cylinder,which produces a regular von Kármán street when towed at constant speed;and a downstream foil,capable of executing a harmonic heaving and pitching motion.Under properly selected con-ditions,the foil can extract energy from the oncoming cylinder-generated eddies, hence augmenting its thrust and efﬁciency.Subsequent experiments with liveﬁsh andﬂexibly mounted foils,show that energy extraction is feasible both forﬂapping two-dimensional foils as well as swimmingﬁsh,andﬁsh-like three-dimensional bodies.Theﬂapping foil motion can be tuned to maximize propulsive efﬁciency, while liveﬁsh adapt to oncomingﬂow structures to minimize the required power to maintain position within an oncomingﬂow.These experimental and numerical studies show thatﬁsh-like locomotion can employ mechanisms ofﬂow control to achieve(a)separation elimination,(b)turbu-lence reduction and(c)energy extraction from oncomingﬂow,in order to minimize the energy needed for locomotion.AcknowledgementsSupport by ONR under contract N00014-00-1-0198and by the MIT Sea Grant Program under Grant NA46RG0434is gratefully acknowledged.References1.Adrian,R.J.,Particle imaging techniques for experimentalﬂuid mechanics.Annual Rev.FluidMech.23(1991)261–304.2.Anderson,J.M.,V ortex control for efﬁcient propulsion.Ph.D.Thesis,Joint Program,Massa-chusetts Institute of Technology&Woods Hole Oceanographic Institution(1996).3.Anderson,E.J.,McGillis,W.R.,Grosenbaugh,M.A.,Techet,A.H.and Triantafyllou,M.S.,Visualization and analysis of boundary layerﬂow in swimmingﬁsh.In:Proceedings of the First International Symposium on Turbulence&Shear Flow Phenomena,Santa Barbara,CA (1999).4.Beal,D.N.,Hover,F.S.and Triantafyllou,M.S.,The effect of a vortex wake on the thrustand efﬁciency of an oscillating foil.In:Proceedings of the12th International Symposium on Unmanned Untethered Submersible Technology(UUST01),Durham,NH(2001).5.Beal,D.N.,Propulsion through wake synchronization using aﬂapping foil.Ph.D.Thesis,MIT,Cambridge,MA(2003).6.Dickinson,M.H.,The effect of wing rotation on unsteady aerodynamic performance at lowReynolds numbers.J.Exp.Biol.192(1994)179–206.7.Dickinson,M.H.,Lehmann,F.O.and Sane,S.P.,Wing rotation and the aerodynamic basis ofinsectﬂight.Science284(1999)1954–1960.8.Drucker,E.G.and Lauder,G.V.,Locomotor forces on a swimmingﬁsh:Three dimensionalvortex wake dynamics quantiﬁed using DPIV.J.Exp.Biol.202(1999)2393–2412.9.Drucker,E.G.and Lauder,G.V.,A hydrodynamic analysis ofﬁsh swimming speed:Wakestructure and locomotor force in slow and fast labriform swimmers.J.Exp.Biol.203(2000) 2379–2393.10.Ellington,C.P.,The aerodynamics of hovering insectﬂight.Philos.Trans.Roy.Soc.London,Ser.B305(1984)17–181.11.Ellington,C.P.,The novel aerodynamics of insectﬂight:Applications to micro-air vehicles.J.Exp.Biol.202(1999)3439–3448.12.Ffowcs-Williams,J.and Zhao,B.,The active control of vortex shedding.J.Fluids Struct.3(1989)115–122.13.Fish,F.E.and Hui,C.A.,Dolphin swimming–A review.Mammal Rev.21,181–195.14.Freymuth P.,Thrust generation by an airfoil in hover modes.Exps.Fluids9(1990)17–24.15.Gopalkrishnan,R.,Triantafyllou,M.S.,Triantafyllou,G.S.and Barett,D.S.,Active vorticitycontrol in a shearﬂow using aﬂapping foil.J.Fluid Mech.274(1994)1–21.16.Harper,D.G.and Blake,R.W.,Prey capture and the fast-start performance of northern pike(Essox Lucius).J.Exp.Biol.155(1991)175–192.17.Michel,A.P.M.,Techet,A.H.,Hover,F.S.and Triantafyllou,M.S.,Experiments with anundulating snake robot.In:Proceedings Oceans2001,Honolulu,HI(2001).18.Kumph,J.M.,Techet,A.H.,Yue,D.K.P.and Triantafyllou,M.S.,Flow control ofﬂexiblehull vehicles.In:Proceedings of the11th International Symposium on Unmanned Untethered Submersible Technology(UUST99),Durham,NH,August22–25(1999).uder,G.V.,2000,Function of the caudalﬁn during locomotion inﬁshes:Kinematics,ﬂowvisualization and evolutionary patterns.Amer.Zool.40(2000)101–122.20.Maxworthy,T.,Experiments on the Weis-Fogh mechanism of lift generation by insects inhoveringﬂight.Part I.Dynamics of theﬂing.J.Fluid Mech.93(1979)47–63.21.Mueller,U.,van den Heuvel,B.,Stamhuis,E.and Videler,J.,Fish foot prints:Morphologyand energetics of the wake behind a continuously swimming mullet(Chelon Labrosus Risso).J.Exp.Biol.200(1997)2893–2806.22.Sparenberg,J.A.and Wiersma,A.K.,On the efﬁciency increasing interaction of thrust produc-ing lifting surfaces.In:Wu,T.,Brokaw,C.W.and Brennen,C.(eds.),Swimming and Flying in Nature,V ol.2(1975)pp.891–917.23.Stamhuis,E.and Videler,J.,Quantitativeﬂow analysis around aquatic animals using laser sheetparticle image velocimetry.J.Exp.Biol.198(1995)283–294.24.Streitlien,K.and Triantafyllou,M.S.,Force and moment on a Joukowski proﬁle in the presenceof point vortices.AIAA J.33(1995)603–610.25.Streitlien,K.,Triantafyllou,G.S.and Triantafyllou,M.S.,Efﬁcient foil propulsion throughvortex control.AIAA J.34(1996)2315–2319.26.Taneda,S.,Visual study of unsteady separatedﬂows around bodies.Progr.Aerosp.Sci.17(1977)287–348.27.Techet,A.H.and Triantafyllou,M.S.,Experimental study of a waving plate.In:52nd AnnualMeeting of the Division of Fluid Dynamics,American Physical Society,New Orleans,LA, 21–24November(1999).28.Techet,A.H.and Triantafyllou,M.S.,Near boundary visualization of theﬂow aboutﬁsh-likeswimming bodies.In:53rd Annual Meeting of the Division of Fluid Dynamics,American Physical Society,Washington,DC,November19–21(2000).。

Tannakian Categories attached to abelian Varieties

a r Xiv:074.1238v2[mat h.A G]13J u n27Tannakian Categories attached to abelian Varieties Rainer Weissauer February 1,2008Let k be an algebraically closed ﬁeld k ,where k is either the algebraic closure of a ﬁnite ﬁeld or a ﬁeld of characteristic zero.Let l be a prime different from the characteristic of k .Notations.For a variety X over k let D b c (X,Q l -sheaves on X in the sense of [5].For a complex K ∈D b c (X,Q l -sheaves with respect to the standard t -structure.The abelian subcategory P erv (X )of middle perverse sheaves is the full subcategory of all K ∈D b c (X,Q l )is semi-perverse if and only if dim (S ν)≤νholds for all integers ν∈Z ,where S νdenotes the support of the cohomology sheaf H −ν(L )of L .If k is the algebraic closure of a ﬁnite ﬁeld κ,then a complex K of etaleQ l -sheaves with upper weights w (H ν(K ))−ν≤w for all integers ν.It is called pure of weight w ,if K and its Verdier dual D (K )are mixed of weight ≤w .Concerning base ﬁelds of characteristic zero,we assume mixed sheaves to be sheaves of geometric origin in the sense of the last chapter of [1],so we still dispose over the notion of the weight ﬁltration and purity and Gabber’s decompo-sition theorem in this case.In this sense let P erv m (X )denote the abelian category of mixed perverse sheaves on X .The full subcategory P (X )of P erv m (X )of pure perverse sheaves is a semisimple abelian category.1Abelian varieties.Let X be an abelian variety X of dimension g over an alge-braically closedﬁeld k.The addition law of the abelian variety a:X×X→X deﬁnes the convolution product K∗L∈D b c(X,Q l)by the direct imageK∗L=Ra∗(K⊠L).For the skyscraper sheafδ0concentrated at the zero element0notice K∗δ0=K. Translation-invariant sheaf complexes.More generally K∗δx=T∗−x(K),where x is a closed k-valued point in X,δx the skyscraper sheaf with support in{x}and where T x(y)=y+x denotes the translation T x:X→X by x.In fact T∗y(K∗L)∼= T∗y(K)∗L∼=K∗T∗y(L)holds for all y∈X(k).For K∈D b c(X,Q l) with a translation-invariant complex on X is a translation-invariant complex.A translation-invariant perverse sheaf K on X is of the form K=E[g],for an or-dinary etale translation-invariantQ l)the irreducible constituents of the perverse cohomology sheaves p Hν(K)are translation-invariant.Multipliers.The subcategory T(X)of P erv(X)of all perverse sheaves,whose ir-reducible perverse constituents are translation-invariant,is a Serre subcategory of the abelian category P erv(X).Let denoteP(X)the image of P(X),which is a full subcategory of semisimple objects. The full subcategory of D b c(X,Q l).LetQ l)be the corresponding triangulated quotient category,which containsD b c(X,D b c(X,D b c(X,Deﬁnition.A perverse sheaf K on X is called a multiplier,if the convolution induced by K∗K:Q l)→Q l)preserves the abelian subcategory:1)Skyscraper sheaves are multipliers2)If i:C֒→X is a projective curve,which generates the abelian variety X,and E is an etaleQ l[g] andλY=i∗P erv(X).The morphismπ=a◦(j×id X)is afﬁne.Indeed W=π−1(V)is afﬁne for afﬁne subsets V of X,W being isomorphic under the isomorphism(u,v)→(u,u+v)of X2to the afﬁne product U×V.By the afﬁne vanishing theorem of Artin:For perverse sheaves L∈P erv(X)we getλU⊠L∈P erv(X2)and p Hν(Rπ!(λU⊠L))=0for allν<0.The distinguished triangle Ra∗(λY⊠L),Rπ!(λU⊠L),Ra∗(δX⊠L) forδX=P erv(X).For smooth Y the intersection cohomology sheaf isλY=i∗Q l,Y[g−1]∗L has image in Q l,Y[g−1]is a multiplier.Let M(X)⊆P(X)denote the full category of semisimple multipliers.LetP(X)of P(X).Then,by the3deﬁnition of multipliers,the convolution product preservesM(X)×M(X).Theorem.With respect to this convolution product the categoryQ l-linear tensor category,hence as a tensor cate-gory(K,L)∼=Γ{0}(X,H0(K∗L∨)∗),M(X)where H0denotes the degree zero cohomology sheaf andΓ{0}(X,−)sections with support in the neutral element.Let L=K be simple and nonzero.Then the left side becomes End Q l.On the other hand K∗L∨is a direct sum of a perverse sheaf P and translates of translation-invariant perverse sheaves.Hence H0(K∗L∨)∨)is the direct sum of a skyscraper sheaf S and translation-invariant etale sheaves.ThereforeΓ{0}(X,H0(K∗L∨)∨)=Γ{0}(X,S).By a comparison of both sides therefore S=δ0.Noticeδ0is the unit element1of the convolution ing the formula above we not only get(K∗L∨,1),HomM(X)but alsoﬁnd a nontrivial morphismev K:K∗K∨→1.By semisimplicityδ0is a direct summand of the complex K∗K∨.In particular the K¨u nneth formula implies,that the etale cohomology groups do not all vanish identicallyH•(X,K)=0.Therefore the arguments of[7]2.6show,that the simple perverse sheaf K is du-alizable.Hence Q l-linear tensor category.Let T be aﬁnitely4⊗-generated tensor subcategory with generator say A .To show T is super-Tannakian,by [4]it is enough to show for all nlenght T (A ∗n )≤N n ,where N is a suitable constant.For any B ∈Q l (H ν(X,B ))t ν.Thenlenght T (B )≤h (B,1),since every summand of B is a multiplier and there-fore has nonvanishing cohomology.For B =A ∗n the K¨u nneth formula gives h (B,1)=h (A,1)n .Therefore the estimate above holds for N =h (A,1).This completes the outline for the proof of the theorem.Principally polarized abelian varieties .Suppose Y is a divisor in X deﬁning a principal polarization.Suppose the intersection cohomology sheaf δY of Y is a multiplier.Then a suitable translate of Y is symmetric,and again a multiplier.So we may assume Y =−Y is symmetric.LetM (X )generated by δY .The corresponding super-group G (X,Y )attached toM (X ).By assumption δY is selfdual in the sense,that there exists an isomorphism ϕ:δ∨Y ∼=δY.Obviously ϕ∨=±ϕ.This deﬁnesa nondegenerate pairing on W ,and the action of G (X,Y )on W respects this pairing.Curves .If X is the Jacobian of smooth projective curve C of genus g over k ,X car-ries a natural principal polarization Y =W g −1.If we replace this divisor by a sym-metric translate,then Y is a multiplier.The corresponding group G (X,Y )is the semisimple algebraic group G =Sp (2g −2,Q l )/µg −1depending on whether the curve C is hyperelliptic or not.The representation W of G (X,Y )deﬁned by δY as above is the unique irreducibleQ l )/µg −1[2]or Sl (2g −2,References[1]Beilinson A.,Bernstein J.,Deligne P.,Faisceaux pervers,Asterisque100(1982)[2]Deligne P.,Milne J.S.,Tannakian categories,in Lecture Notes in Math900,p.101–228[3]Deligne P.,Categories tannakiennes,The Grothendieck Festschrift,vol II,Progr.Math,vol.87,Birkh¨a user(1990),111–195[4]Deligne P.,Categories tensorielles,Moscow Math.Journal2(2002)no.2,227–248[5]Kiehl R.,Weissauer R.,Weil conjectures,perverse sheaves and l-adic Fouriertransform,Ergebnisse der Mathematik und ihrer Grenzgebiete42,Springer (2001)[6]Weissauer R.,Torelli’s theorem from the topological point of view,arXivmath.AG/0610460[7]Weissauer R.,Brill-Noether Sheaves,arXiv math.AG/06109236。

剑桥雅思阅读11真题及答案解析（test3）

剑桥雅思阅读11真题及答案解析(test3)雅思阅读部分一直都是中国考生比较重视的题目，并且也是很有难度的题目，针对于雅思阅读真题资料也是大家需要重点分析的。

今天小编就给大家带来了关于剑桥雅思阅读11原文参考译文以及答案(test3)的内容，一起来分析一下吧。

剑桥雅思阅读11原文(test3)READING PASSAGE 1You should spend about 20 minutes on Questions 1-13, which are based on Reading Passage 1 below.THE STORY OF SILKThe history of the world’s most luxurious fabric, from ancient China to the present daySilk is a fine, smooth material produced from the cocoons —soft protective shells —that are made by mulberry silkworms (insect larvae). Legend has it that it was Lei Tzu, wife of the Yellow Emperor, ruler of China in about 3000 BC, who discovered silkworms. One account of the story goes that as she was taking a walk in her husband’s gardens, she discovered that silkworms were responsible for the destruction of several mulberry trees. She collected a number of cocoons and sat down to have a rest. It just so happened that while she was sipping some tea, one of the cocoons that she had collected landed in the hot tea and started to unravel into a fine thread. Lei Tzu found that she could wind this thread around her fingers. Subsequently, she persuaded her husband to allow her to rear silkworms on a grove of mulberry trees. She also devised a special reel to draw the fibres from the cocoon into a single thread so that they would be strong enough to be woven into fabric. While it is unknown just how much of this is true, it is certainly known that silk cultivationhas existed in China for several millennia.Originally, silkworm farming was solely restricted to women, and it was they who were responsible for the growing, harvesting and weaving. Silk quickly grew into a symbol of status, and originally, only royalty were entitled to have clothes made of silk. The rules were gradually relaxed over the years until finally during the Qing Dynasty (1644-1911 AD), even peasants, the lowest caste, were also entitled to wear silk. Sometime during the Han Dynasty (206 BC-220 AD), silk was so prized that it was also used as a unit of currency. Government officials were paid their salary in silk, and farmers paid their taxes in grain and silk. Silk was also used as diplomatic gifts by the emperor. Fishing lines, bowstrings, musical instruments and paper were all made using silk. The earliest indication of silk paper being used was discovered in the tomb of a noble who is estimated to have died around 168 AD.Demand for this exotic fabric eventually created the lucrative trade route now known as the Silk Road, taking silk westward and bringing gold, silver and wool to the East. It was named the Silk Road after its most precious commodity, which was considered to be worth more than gold. The Silk Road stretched over 6,000 kilometres from Eastern China to the Mediterranean Sea, following the Great Wall of China, climbing the Pamir mountain range, crossing modern-day Afghanistan and going on to the Middle East, with a major trading market in Damascus. From there, the merchandise was shipped across the Mediterranean Sea. Few merchants travelled the entire route; goods were handled mostly by a series of middlemen.With the mulberry silkworm being native to China, the country was the world’s sole producer of silk for many hundreds of years. The secret of silk-making eventually reached the rest ofthe world via the Byzantine Empire, which ruled over the Mediterranean region of southern Europe, North Africa and the Middle East during the period 330-1453 AD. According to another legend, monks working for the Byzantine emperor Justinian smuggled silkworm eggs to Constantinople (Istanbul in modern-day Turkey) in 550 AD, concealed inside hollow bamboo walking canes. The Byzantines were as secretive as the Chinese, however, and for many centuries the weaving and trading of silk fabric was a strict imperial monopoly. Then in the seventh century, the Arabs conquered Persia, capturing their magnificent silks in the process. Silk production thus spread through Africa, Sicily and Spain as the Arabs swept through these lands. Andalusia in southern Spain was Europe’s main silk-producing centre in the tenth century. By the thirteenth century, however, Italy had become Europe’s leader in silk production and export. Venetian merchants traded extensively in silk and encouraged silk growers to settle in Italy. Even now, silk processed in the province of Como in northern Italy enjoys an esteemed reputation.The nineteenth century and industrialisation saw the downfall of the European silk industry. Cheaper Japanese silk, trade in which was greatly facilitated by the opening of the Suez Canal, was one of the many factors driving the trend. Then in the twentieth century, new manmade fibres, such as nylon, started to be used in what had traditionally been silk products, such as stockings and parachutes. The two world wars, which interrupted the supply of raw material from Japan, also stifled the European silk industry. After the Second World War, Japan’s silk production was restored, with improved production and quality of raw silk. Japan was to remain the world’s biggest producer of raw silk, and practically the only major exporter of raw silk, untilthe 1970s. However, in more recent decades, China has gradually recaptured its position as the world’s biggest producer and exporter of raw silk and silk yarn. Today, around 125,000 metric tons of silk are produced in the world, and almost two thirds of that production takes place in China.Questions 1-9Complete the notes below.Choose ONE WORD ONLY from the passage for each answer.Write your answers in boxes 1-9 on your answer sheet.THE STORY OF SILKEarly silk production in ChinaAround 3000 BC, according to legend:- silkworm cocoon fell into emperor’s wife’s 1 __________- emperor’s wife invented a 2 __________ to pull out silk fibresOnly 3 __________ were allowed to produce silkOnly 4 __________ were allowed to wear silkSilk used as a form of 5 __________- e.g. farmers’ taxes consisted partly of silkSilk used for many purposes- e.g. evidence found of 6 __________ made from silk around 168 ADSilk reaches rest of worldMerchants use Silk Road to take silk westward and bring back 7 __________ and precious metals550 AD: 8 __________ hide silkworm eggs in canes and take them to ConstantinopleSilk production spreads across Middle East and Europe20th century: 9 __________ and other manmade fibres cause decline in silk productionQuestions 10-13Do the following statements agree with the information in Reading Passage 1?In boxes 10-13 on your answer sheet, writeTRUE if the statement agrees with the informationFALSE if the statement contradicts the informationNOT GIVEN if there is no information on this10 Gold was the most valuable material transported along the Silk Road.11 Most tradesmen only went along certain sections of the Silk Road.12 The Byzantines spread the practice of silk production across the West.13 Silk yarn makes up the majority of silk currently exported from China.READING PASSAGE 2You should spend about 20 minutes on Questions 14-26, which are based on Reading Passage 2 below.Great MigrationsAnimal migration, however it is defined, is far more than just the movement of animals. It can loosely be described as travel that takes place at regular intervals ?— often in an annual cycle — that may involve many members of a species, and is rewarded only after a long journey. It suggests inherited instinct. The biologist Hugh Dingle has identified five characteristics that apply, in varying degrees and combinations, to all migrations. They are prolonged movements that carry animals outside familiar habitats; they tend to be linear, not zigzaggy; they involve special behaviours concerning preparation (such as overfeeding) and arrival; they demand special allocations of energy. And onemore: migrating animals maintain an intense attentiveness to the greater mission, which keeps them undistracted by temptations and undeterred by challenges that would turn other animals aside.An arctic tern, on its 20,000 km flight from the extreme south of South America to the Arctic circle, will take no notice of a nice smelly herring offered from a bird-watcher’s boat along the way. While local gulls will dive voraciously for such handouts, the tern flies on. Why? The arctic tern resists distraction because it is driven at that moment by an instinctive sense of something we humans find admirable: larger purpose. In other words, it is determined to reach its destination. The bird senses that it can eat, rest and mate later. Right now it is totally focused on the journey; its undivided intent is arrival.Reaching some gravelly coastline in the Arctic, upon which other arctic terns have converged, will serve its larger purpose as shaped by evolution: finding a place, a time, and a set of circumstances in which it can successfully hatch and rear offspring.But migration is a complex issue, and biologists define it differently, depending in part on what sorts of animals they study. Joe! Berger, of the University of Montana, who works on the American pronghorn and other large terrestrial mammals, prefers what he calls a simple, practical definition suited to his beasts: ‘movements from a seasonal home area away to another home area and back again’. Generally the reason for such seasonal back-and-forth movement is to seek resources that aren’t available within a single area year-round.But daily vertical movements by zooplankton in the ocean —upward by night to seek food, downward by day to escapepredators —can also be considered migration. So can the movement of aphids when, having depleted the young leaves on one food plant, their offspring then fly onward to a different host plant, with no one aphid ever returning to where it started.Dingle is an evolutionary biologist who studies insects. His definition is more intricate than Berger’s, citing those five features that distinguish migration from other forms of movement. They allow for the fact that, for example, aphids will become sensitive to blue light (from the sky) when it’s time for takeoff on their big journey, and sensitive to yellow light (reflected from tender young leaves) when it’s appropriate to land. Birds will fatten themselves with heavy feeding in advance of a long migrational flight. The value of his definition, Dingle argues, is that it focuses attention on what the phenomenon of wildebeest migration shares with the phenomenon of the aphids, and therefore helps guide researchers towards understanding how evolution has produced them all.Human behaviour, however, is having a detrimental impact on animal migration. The pronghorn, which resembles an antelope, though they are unrelated, is the fastest land mammal of the New World. One population, which spends the summer in the mountainous Grand Teton National Park of the western USA, follows a narrow route from its summer range in the mountains, across a river, and down onto the plains. Here they wait out the frozen months, feeding mainly on sagebrush blown clear of snow. These pronghorn are notable for the invariance of their migration route and the severity of its constriction at three bottlenecks. If they can’t pass through each of the three during their spring migration, they can’t reach their bounty of summer grazing; if they can’t pass through again in autumn, escaping south ontothose windblown plains, they are likely to die trying to overwinter in the deep snow. Pronghorn, dependent on distance vision and speed to keep safe from predators, traverse high, open shoulders of land, where they can see and run. At one of the bottlenecks, forested hills rise to form a V, leaving a corridor of open ground only about 150 metres wide, filled with private homes. Increasing development is leading toward a crisis for the pronghorn, threatening to choke off their passageway.Conservation scientists, along with some biologists and land managers within the USA’s National Park Service and other agencies, are now working to preserve migrational behaviours, not just species and habitats. A National Forest has recognised the path of the pronghorn, much of which passes across its land, as a protected migration corridor. But neither the Forest Service nor the Park Service can control what happens on private land at a bottleneck. And with certain other migrating species, the challenge is complicated further —by vastly greater distances traversed, more jurisdictions, more borders, more dangers along the way. We will require wisdom and resoluteness to ensure that migrating species can continue their journeying a while longer.Questions 14-18Do the following statements agree with the information given in Reading Passage 2?In boxes 14-18 on your answer sheet, writeTRUE if the statement agrees with the informationFALSE if the statement contradicts the informationNOT GIVEN if there is no information on this14 Local gulls and migrating arctic terns behave in the same way when offered food.15 Experts’ definitions of migration tend to vary accordingto their area of study.16 Very few experts agree that the movement of aphids can be considered migration.17 Aphids’ journeys are affected b y changes in the light that they perceive.18 Dingle’s aim is to distinguish between the migratory behaviours of different species.Questions 19-22Complete each sentence with the correct ending, A-G, below.Write the correct letter, A-G, in boxes 19-22 on your answer sheet.19 According to Dingle, migratory routes are likely to20 To prepare for migration, animals are likely to21 During migration, animals are unlikely to22 Arctic terns illustrate migrating animals’ ability toA be discouraged by difficulties.B travel on open land where they can look out for predators.C eat more than they need for immediate purposes.D be repeated daily.E ignore distractions.F be governed by the availability of water.G follow a straight line.Questions 23-26Complete the summary below.Choose ONE WORD ONLY from the passage for each answer.Write your answers in boxes 23-26 on your answer sheet.The migration of pronghornsPronghorns rely on their eyesight and 23 __________ to avoid predators. One particular pop ulation’s summer habitat is a national park, and their winter home is on the 24 __________,where they go to avoid the danger presented by the snow at that time of year. However, their route between these two areas contains three 25 __________. One problem is the construction of new homes in a narrow 26 __________ of land on the pronghorns’ route.READING PASSAGE 3You should spend about 20 minutes on Questions 27-40, which are based on Reading Passage 3 below.Preface to ‘How the other half thinks: Adventu res in mathematical reasoning’A Occasionally, in some difficult musical compositions, there are beautiful, but easy parts — parts so simple a beginner could play them. So it is with mathematics as well. There are some discoveries in advanced mathematics that do not depend on specialized knowledge, not even on algebra, geometry, or trigonometry. Instead they may involve, at most, a little arithmetic, such as ‘the sum of two odd numbers is even’, and common sense. Each of the eight chapters in this book illustrates this phenomenon. Anyone can understand every step in the reasoning.The thinking in each chapter uses at most only elementary arithmetic, and sometimes not even that. Thus all readers will have the chance to participate in a mathematical experience, to appreciate the beauty of mathematics, and to become familiar with its logical, yet intuitive, style of thinking.B One of my purposes in writing this book is to give readers who haven’t had the opportunity to see and enjoy real mathematics the chance to appreciate the mathematical way of thinking. I want to reveal not only some of the fascinating discoveries, but, more importantly, the reasoning behind them.In that respect, this book differs from most books on mathematics written for the general public. Some present the lives of colorful mathematicians. Others describe important applications of mathematics. Yet others go into mathematical procedures, but assume that the reader is adept in using algebra.C I hope this book will help bridge that notorious gap that separates the two cultures: the humanities and the sciences, or should I say the right brain (intuitive) and the left brain (analytical, numerical). As the chapters will illustrate, mathematics is not restricted to the analytical and numerical; intuition plays a significant role. The alleged gap can be narrowed or completely overcome by anyone, in part because each of us is far from using the full capacity of either side of the brain. To illustrate our human potential, I cite a structural engineer who is an artist, an electrical engineer who is an opera singer, an opera singer who published mathematical research, and a mathematician who publishes short stories.D Other scientists have written books to explain their fields to non-scientists, but have necessarily had to omit the mathematics, although it provides the foundation of their theories. The reader must remain a tantalized spectator rather than an involved participant, since the appropriate language for describing the details in much of science is mathematics, whether the subject is expanding universe, subatomic particles, or chromosomes. Though the broad outline of a scientific theory can be sketched intuitively, when a part of the physical universe is finally understood, its description often looks like a page in a mathematics text.E Still, the non-mathematical reader can go far in understanding mathematical reasoning. This book presents thedetails that illustrate the mathematical style of thinking, which involves sustained, step-by-step analysis, experiments, and insights. You will turn these pages much more slowly than when reading a novel or a newspaper. It may help to have a pencil and paper ready to check claims and carry out experiments.F As I wrote, I kept in mind two types of readers: those who enjoyed mathematics until they were turned off by an unpleasant episode, usually around fifth grade, and mathematics aficionados, who will find much that is new throughout the book.This book also serves readers who simply want to sharpen their analytical skills. Many careers, such as law and medicine, require extended, precise analysis. Each chapter offers practice in following a sustained and closely argued line of thought. That mathematics can develop this skill is shown by these two testimonials:G A physician wrote, ‘The discipline of analytical thought processes [in mathematics] prepared me extremely well for medical school. In medicine one is faced with a problem which must be thoroughly analyzed before a solution can be found. The proces s is similar to doing mathematics.’A lawyer made the same point, ‘Although I had no background in law — not even one political science course — I did well at one of the best law schools. I attribute much of my success there to having learned, through the study of mathematics, and, in particular, theorems, how to analyze complicated principles. Lawyers who have studied mathematics can master the legal principles in a way that most others cannot.’I hope you will share my delight in watching as simple, even na?ve, questions lead to remarkable solutions and purely theoretical discoveries find unanticipated applications.Questions 27-34Reading Passage 3 has seven sections, A-G.Which section contains the following information?Write the correct letter, A-G, in boxes 27-34 on your answer sheet.NB You may use any letter more than once.27 a reference to books that assume a lack of mathematical knowledge28 the way in which this is not a typical book about mathematics29 personal examples of being helped by mathematics30 examples of people who each had abilities that seemed incompatible31 mention of different focuses of books about mathematics32 a contrast between reading this book and reading other kinds of publication33 a claim that the whole of the book is accessible to everybody34 a reference to different categories of intended readers of this bookQuestions 35-40Complete the sentences below.Choose ONE WORD ONLY from the passage for each answer.Write your answers in boxes 35-40 on your answer sheet.35 Some areas of both music and mathematics are suitable for someone who is a __________.36 It is sometimes possible to understand advanced mathematics using no more than a limited knowledge of __________.37 The writer intends to show that mathematics requires__________ thinking, as well as analytical skills.38 Some books written by __________ have had to leave out the mathematics that is central to their theories.39 The writer advises non-mathematical readers to perform __________ while reading the book.40 A lawyer found that studying __________ helped even more than other areas of mathematics in the study of law.剑桥雅思阅读11原文参考译文(test3)PASSAGE 1 参考译文：丝绸的故事世上最昂贵奢华织物的历史，从古代中国直到今天丝绸是种细软、光滑的布料，产自桑蚕(该昆虫的幼体形态)制作出的蚕茧——即其柔软的保护性外壳。

爱因斯坦全英文

1905，爱因斯坦开始发表一系列论文震动了整个科学和
当希特勒接管了德国在
1933，他必须离开国家和最后定居在美国。他继续他的研究结构的宇宙，直到他死在1955。
知识的世界，以及他的理论
建立在他的论文获得了诺贝尔物理学奖1921。
Thanks to his theory of relativity, Albert Einstein became the most famous scientist of the 20th century. In 1905, while working in a Swiss patent office, Einstein published a paper proposing a "special theory of relativity," a groundbreaking notion which laid the foundation for much of modern physics theory. (The theory included his famous equation e=mc².)
3. 一个人从未犯错是因为他不曾尝试新鲜事物。 Anyone who has never made a mistake has never tried anything new.
4. 态度上的弱点会变成性格上的弱点。 Weakness of attitude becomes weakness of Character. 5. 并不是我很聪明，而只是我和问题相处得比较久一点。 It's not that I'm so smart, it's just that I stay with problems
longer.

Evaluation of 2D coupled galloping oscillations of slender structures

EVALUATION OF 2D COUPLED GALLOPING OSCILLATIONS OF SLENDER STRUCTURES
Q. S. Li{, J. Q. Fang and A. P. Jeary
Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong (Received 24 February 1997; accepted 8 October 1997) AbstractÐThe necessary conditions for the occurrence of 2D coupled galloping oscillations and a description of the set of general dierential equations for 2D coupled galloping vibrations are established in this paper. A procedure to predict the 2D coupled galloping oscillations is proposed. Wind loading was treated as a stochastic process including mean and ¯uctuating components caused by atmospheric turbulence. The formulae and the procedure presented permit a better assessment of galloping oscillations of a prismatic structure which has the same natural frequencies in the directions of the two orthogonal horizontal axes. The eects of turbulence on 2D coupled galloping are investigated. It is shown that 2D coupled galloping oscillations mainly occur in the cross-wind direction. The eect of the turbulence integral scale on the onset wind velocity of 2D coupled galloping oscillations is negligible. However, the turbulence integral scale has certain in¯uences on the rms response both in the along-wind and the cross-wind directions when the wind velocity is equal or close to the onset wind velocity for galloping. An increase in the turbulence integral scale in the range of 10±40 m considered in this paper increases the galloping response. # 1998 Elsevier Science Ltd. All rights reserved

Vortices in Schwinger-Boson Mean-Field Theory of Two-Dimensional Quantum Antiferromagnets

a rXiv:c ond-ma t/95471v119Apr1995Vortices in Schwinger-Boson Mean-Field Theory of Two-Dimensional Quantum Antiferromagnets Tai-Kai Ng department of Physics,Hong Kong University of Science and Technology,Clear Water Bay Road,Kowloon,Hong Kong (February 6,2008)Abstract In this paper we study the properties of vortices in two dimensional quantum antiferromagnets with spin magmitude S on a square lattice within the frame-work of Schwinger-boson mean-ﬁeld theory.Based on a continuum descrip-tion,we show that vortices are stable topological excitations in the disordered state of quantum antiferromagnets.Furthermore,we argue that vortices can be divided into two kinds:the ﬁrst kind always carries zero angular momen-tum and are bosons,whereas the second kind carries angular momentum S under favourable conditions and are fermions if S is half-integer.A plausible consequence of our results relating to the RVB theories of High-T c supercon-ductors is pointed out.PACS Nos,75.10.Jm,75.30.Ds,79.60.-iTypeset using REVT E XI.INTRODUCTIONIn the past few years there has been a lot of interests in the study of quantum anti-ferromagnets in two dimension on a square lattice,stimulated by the discovery of High-T c superconductors[1,2].Among others,one approximated approach to the quantum antiferro-magnet problems is the Schwinger-Boson Mean-ﬁeld Theory(SBMFT).[2].In this approach, the quantum spins are represented as Schwinger bosons and an approximated ground state is constructed by bose condensation of spin-singlet pairs.The mean-ﬁeld theory can also be formulated in a large-N expansion as the saddle point solution to a generalized SU(N) quantum spin model[2,3].In the limit N→∞,the mean-ﬁeld theory becomes exact. SBMFT predicts that in one dimension,Heisenberg antiferromagnet is always disordered, with a non-zero spin gap in the excitation spectrum.The theory is found to give an adequate description for integer spin chains,while it fails to describe the massless state of half-integer spin chains.This is because the topological Berry phase term which plays crucial role in the latter case is not taken into account correctly in SBMFT.[4]However,in two dimension topological term does not exist and SBMFT is more reliable.In particular,the theory pre-dicts that the disordered(spin gap)phase exists only when the spin magnitude S is small enough,when S<S c∼0.19.It turns out that SBMFT oﬀers a fairly accurate description for the magnetic properties of the High-T c compounds in the low doping regime[5].However, at large doping levels,the generalization of SBMFT which includes holes as fermions[6], does not seem to describe the High-T c compounds correctly.For example,photo-emission experiments[7]indicate that the compounds have”large”Fermi-surfaces which satisfy Lut-tinger theorem,whereas the generalization of SBMFT to include holes predicts a”small”Fermi surface with area proportional to concentration of holes.In the large doping regime, it turns out that the slave-boson mean-ﬁeld theory[8]which treats the spins as fermions and holes as bosons produces a lot of properties of High-T c compounds correctly[9]and the SBMFT is only superier in describing the low-doping,antiferromagnetic state.Thus a natural question is:what is the relation between the two theories?Can one understand thetwo theories within a single framework?How does nature crossover from one description tothe other as concentration of holes increases?In SBMFT,the spin operator S i on site i is expressed in terms of Schwinger bosons S=¯Z i σZ i,where σis the Pauli matrix,Z i(¯Z i)are two component spinors¯Z i=(¯Z i↑,¯Z i↓), ietc.Notice that in order to represent a spin with magnitude S,there should be2S bosonsper site[2].The Hamiltonian can then be represented in terms of Schwinger bosons,and amean-ﬁeld theory can be formulated by introducing order parameters∆i,i+η=<Z i↑Z i+η↓−Z i↓Z i+η↑>(η=±ˆx,±ˆy)[2].Alternatively,SBMFT can also be formulated in a large-N expansion as the saddle point solution to a generalized SU(N)quantum spin model.To derive the large-N theory,we divide the square lattice into A and B sublattices and consider the following transformation for the Schwinger bosons on B-sublattice,Z B j↑(¯Z B j↑)→−Z B j↓(¯Z B j↓),Z B j↓(¯Z B j↓)→Z B j↑(¯Z B j↑),for all sites j on the B sublattice.The Schwinger bosons on the A sublattice remainsunchanged in the above transformation.The Lagrangian of the Heisenberg model can thenbe represented in the transformed boson coordinates as[3]L= iσ[¯Z A iσ(d dτ+iλj)Z B jσ−i2Sλj]+J i,η=±ˆx,±ˆy|∆i,i+η|2−J iσ,η=±ˆx,±ˆy[∆∗i,i+ηZ A iσZ B i+ησ+H.C.](1) whereλi’s are Lagrange multiplierﬁelds enforcing the constraint that there are2S bosons per site,∆i,i+η’s are Hubbard-Stratonovichﬁelds introduced in decoupling of H.σis the spin index.A SU(N)spin theory can be formulated with the above Lagrangian if the usual SU(2)spins↑and↓are extented to the SU(N)case where N-spin species are introduced. SBMFT can be considered as the saddle point solution of the path integral over L which becomes exact in the limit N→∞[3].For inﬁnite system with no defects,the saddle point solution has position independent∆i,i+η’s andλ’s and the solution is formally verysimilar to the BCS solution for superconductors,except that in present case the pairing objects are Schwinger bosons but not electrons.Another important diﬀerence is that in the present case,the pairing bosons are always located on diﬀerent sublattices,whereas no such distinction is found in the case of superconductors.The similarity between SBMFT and BCS theory leads us to ask the natural question of whether vortex-like solutions can be found in SBMFT as in the case of superconductors. More precisely,one may ask the question of whether one can construct a stable solution in SBMFT where∆i,i+ηhas a phase structure∆i,i+η∼|∆i,i+η|e iθ(i+η/2),whereθ( x)is a smooth function of x and has a singular point at x′∼0such that for distance| x|>>0,θ( x)→θ( x)+2πif one moves vector x around a close loop C which enclosed the point x′=0.As in the case of superconductors,the simplest way to address this question is to construct a continuum description for SBMFT(analogous to Ginsburg-Landau theory for superconductors)and study the possibility of having vortex solution in the continuum approximation.In the following we shall perform such a study in the disordered state in SBMFT where the Schwinger bosons are not bose-condensed and spin is a good quantum number.In realistic High-T c cuprates long range antiferromagnetic order is destroyed by introduction of charge carriers(holes).We shall not consider complications introduced by holes here and shall assume simply a disordered magnetic state with realistic spin magnitudes which can be described by the spin gap phase of SBMFT.We shall examine the properties of vortices within the continuum description.In section II we shall derive the continuum theory.Based on the continuum equations,we shallﬁrst study the properties of a single,non-magnetic impurity in the disordered state of SBMFT,where we shall point out the important diﬀerence between perturbations which are symmetric to the A and B sublattices,and perturbations which distinguish the two sublattices.In section III,we shall study static vortex excitations in our model.Based on the continuum theory,we shall argue that vortices are stable topological excitations in the disordered state of2D quantumantiferromagnets.We shall then show that there exists two kind of vortices in the continuum theory,corresponding to vortices centered on the mid-point of a plaquette,and vortices centered on a lattice point.We shall show that the properties of these two kinds of vortices are very diﬀerent,because of the diﬀerent symmetry with respect to the two sublattices.The ﬁrst kind of vortex which is symmetric to the two sublattices can carry only zero angular momentum and is a boson,whereas the second kind of vortex distinguishes between the two sublattices can carry angular momentum S under favourble conditions and is a fermion if S is half-integer.In section IV we shall concentrate ourselves at the case S=1/2which is the physical case of interests and shall examine properties of a”liquid”of fermionic vortices. Based on simple symmetry arguments,we shall argue that the eﬀective theory of a liquid of fermionic vortices has precisely the same form as the slave-boson mean-ﬁeld theory for spins in the undoped limit.The case withﬁnite concentration of holes will also be discussed.The ﬁndings in this paper will be summarized in section V,where we shall discuss a plausible scenerio of how the system crossover from a state described by SBMFT to a state described by slave-boson meanﬁeld theory upon doping.II.CONTINUUM DESCRIPTION FOR SBMFTTo derive the continuum theory we following Read and Sachdev[3]and consider the representation where our system has two sites per unit cell and introduce the’uniform’and ’staggered’components in the∆andλﬁelds,where∆i,i±η=12)+q±η(i±η2)+A±η(i±η2[λA( k)+λB( k)],(2b) Aτ( k)=1Theφ( x)ﬁeld describes uniform antiferromagnetic correlations whereas q( x)ﬁeld de-scribes spin dimerization(spin-Peierls)eﬀects[3].θ( x)and Aη( x)areﬁelds describing the corresponding uniform and staggered phaseﬂuctuations,respectively.Allﬁelds are slowly varying on the scale of a unit cell.Notice that the reason why four independent(real)ﬁelds are needed to describeﬂuctuations in one unit cell is precisely because we have divided the system into A and B sublattices.In the case of usual superconductors such a distinction is not present and only two realﬁelds(or one complex scalarﬁeld)enters into the Ginsburg-Landau equation.Notice that Vµ( x)=∂µθ( x)and Aµ( x)can be considered as U(1)gauge ﬁelds coupling to the Schwinger boson Z’s.The’uniform’gaugeﬁeld Vµ( x)couples to the bosons on the two sublattices with same gauge charge,whereas the gauge charges for the ’staggered’gaugeﬁeld Aµ( x)are opposite on the two sublattices.A single vortex solution centered at x′=0corresponds to a solution of SBMFT with boundary conditionVµ( x)dxµ=2nπ,or in terms of the gaugeﬁeld,there is a’uniform’gaugeﬂux of n/2ﬂux quantum passing through the origin,similar to the case of superconductors.Notice that in SBMFT where<∆i,i+η>=0,the gauge symmetry associated with the’uniform’gaugeﬁeld Vµ( x) (Z A→Z A e iΓ,Z B→Z B e iΓ,Vµ→Vµ−2∂µΓ)is broken,whereas the’staggered’gauge symmetry(Z A→Z A e iΓ,Z B→Z B e−iΓ,Aµ→Aµ−∂µΓ)remains intact in SBMFT.The existence of vortex solution is tied with the broken symmetry gaugeﬁeld Vµ( x)as in usual superconductors.At distance much larger than lattice spacing,ﬂuctuations associated with the’uniform’variablesφandθτare unimportant.Thus we shall neglect the∇φ(x)and∇θτ(x)terms in the following and derive a continuum theory for the rest of the variables.Notice that the Vµvariable is kept in our continuum theory since the term is essential for studying of vortices. In the continuum limit,the Lagrangian L becomesL→dτd 2x σ{¯Z A σ(x )∂∂τZ B σ(x )−µ=ˆx ,ˆy φ(x )(1−12φ(x )(D µ¯Z A σ(x ))(D ∗µ¯Z B σ(x ))+12 µ[φ(x )2+q µ(x )2]−4Sθτ(x ) ,(3)where D µ=∂µ+iA µand D ∗µ=∂µ−iA µ.The continuum Lagrangian Eq.(3)can befurther simpliﬁed by introducing new boson ﬁeldsψσ=12(Z A σ−¯Z B σ).The πσﬁeld can be integrated out safely at large distance and low energy [3],leaving eﬀective Lagrangian for ψσﬁeld,L =σ dτd 2x φ(x ) |D ∗µψσ|2+m (x )2|ψσ|2 +(φ1(x ))−1(D τ−q µD µ)ψ+σ(D ∗τ+q µD ∗µ)ψσ+18(V µ(x )V µ(x )))+m (x )2 (4)where φ( x )m ( x )2=2θτ( x )−4[1−(1/8)(V µ( x )V µ( x )]φ( x )and and φ1( x )=(1/2){θτ( x )+2[1−(1/8)(V µ( x )V µ( x )]φ( x )}.In the limit q µ=0and A µ=0,and neglecting gradient terms ∇φ,∇θτ,the above Lagrangian can be easily diagonalized resulting in an eﬀective Lagrangian L eﬀin terms of the φ( x )and V µ( x )ﬁelds.The dynamical eﬀect of the remaining terms can be obtained by looking at Gaussian ﬂuctuations of the ﬁelds around the saddle point solution A µ,q µ=0.[3]In particular,in the small m ( x )∼m →0limit,we obtain (see Appendix)L eﬀ=dτ d2x (a−4(2S+1))φ+φ2+(1φ)qµqµ+(2S+1)φF2µν+icFµτqµ+O(m2) ,(5) 2e2where e2∼m,a(<4(2S+1)),b and c are constants of order O(1).The precise values of a,b,c depend on the underlying lattice structure and cannot be obtained in a continuum theory.We have consider the realistic case N=2in deriving the above expression.Notice that similar eﬀective Lagrangian has been obtained by Read and Sachdev previously in studying eﬀect of instantons[3].The only diﬀerence here is that the VµVµterm which was not considered by Read and Sachdev is now retained.Recall that Vµis the’uniform’U(1) gaugeﬁeld coupling to the bosons and the VµVµterm in Eq.(5)just represents the Meissner eﬀect associated with nonzero order parameter∆in SBMFT.Notice that as in case of usual superconductors,we have choosen the London gauge∇. V=0in our derivation.The term can also be written in a gauge invariant way by replacing Vµby the gauge invariant object Vµ−2∂µΓin Eq.(5).This term does not contribute to the instanton eﬀect discussed by Read and Sachdev[3]but is crucial to the study of vortices.To understand more bout the continuum theory weﬁrst consider the properties of a singlenon-magnetic impurity in the disordered state of SBMFT.[10]We shall assume that a non-magnetic impurity simply replaces a spin at site i by a non-magnetic object.The simplestway to model this in our formalism is to is replace the constraint equation¯ZZ=2S onsite i by¯ZZ=2S(1−n i)=0,where n i is the number of non-magnetic impurity on site i.It is than easy to see that the eﬀect of nonzero n i in our eﬀective Lagrangian Eq.(5)is tointroduce in the Coulomb gauge an extra term−(2Sn i e i)Aτ,where e i=±1,depending onwhether the impurity is located on the A-or B-sublattices,[10]i.e.,non-magnetic impuritiesappear as eﬀective(staggered)gauge charges of magnitude2Se localized at the impuritysites.A corresponding electrostatic potential V(r)∼(2Se i)ln(r/ξ)(whereξ∼m−1)will beinduced around the impurity.To lower the electrostatic energy,2S bosons will be nucleatedout from the vacuum to screen the electricﬁeld,resulting in formation of local magneticmoment of magnitude S around a non-magnetic impurity[10].Notice that this eﬀect hasbeen observed experimentally in the High-T c compounds in the underdoped region upon substitution of Cu by Zn in the conducting planes[11].This result can also be understood in an alternative way by observing that the same physical eﬀect should have occurred if instead of replacing a spin on site i by an non-magnetic impurity,we let the Heisenberg coupling J i to go to zero for those bonds joining to site i. In this case the’impurity’site i behaves as an non-magnetic impurity as far as the rest of the system is concerned.The only diﬀerence is that a free spin of magnitude S now remains on site i.However,the eﬀect of this approach on our eﬀective Lagrangian Eq.(5)looks rather diﬀerent compared with our previous approach.Instead of introducing a’staggered’electric charge of magnitude2Se on site i,a boundary condition∆i,i+η=0(η=±ˆx,±ˆy)is now imposed on our ing Eq.(2a),the corresponding boundary condition in the continuum theory becomesqˆn(x i)=±ˆnφ(x i)where i→x i in the continuum limit.ˆn is a unit vector.±depends on whether site i is on A-or B-sublattices.The properties of the system around the non-magnetic impurity is obtained by minimizing the free energy Eq.(5)under this boundary condition.Performing the calculation,weﬁndﬁrst of all thatφ(x i)=0and correspondingly q(x i)= 0.The reason for this behaviour can be understood easily from Eq.(5)by noticing that the coeﬃcient in front of the qµqµterm becomes negative asφbecomes smaller than2b, implying that spontaneously dimerization will occur whenφbecomes small enough.Because of stability requirement,it can be shown that dimmerization| q|cannot have value largerthan q o=(√2)φ(see Appendix).Anyway bothφ(xi)and q(x i)∼ q o should be nonzeroin the continuum solution.The reason why q should be nonzero can also be understood from symmetry consideration.Notice that theφ(x)ﬁeld is symmetric under exchange of A-and B-sublattices whereas qµ(x)ﬁeld is antisymmetric.Thus a solution with qµ(x)ﬁeld identically zero would not distinguish between the two sublattices.However,the single non-magnetic impurity problem certainly distinguishes the two sublattices since the impuritycan only be placed on either one of the two sublattices.Thus we expect qµ(x)to be nonzero around a single impurity.Minimizing the free energy with respect to the qµﬁeld weﬁnd also that E(x)∼ q(x),where Eµ=iFµτis the’staggered’electricﬁeld in the system.With the boundary condition q∼(±ˆn)φaround the impurity weﬁnd that electricﬁeld radiating out (or in)from the impurity is obtained in our solution,i.e.,depending on whether the impurity is located on the A-or B-sublattices,it behaves as source or sink for the electricﬁeld,in exact agreement with our previous approach which predicts that non-magnetic impurity behaves as a single staggered gauge charge added to the system.One may also extend our approach to discuss the case when the Heisenberg coupling J′joining to the site i is small but nonzero,so that the spin at site i still couples weakly with the surrounding environment.The previous discussions still apply,except that the spin at site i appears now as a localized spinon state occupied by2S spinons in the system (notice that at two dimension,an arbitrary small attractive interaction is enough to generate a bound state[12]).The spinon state again behaves as a(staggered)gauge charge(with magnitude2S),and will bind2S other spinons with opposite gauge charge(or localized on opposite sublattice)next to itself as in the case with J′=0.However for nonzero J′the localized spinons on the two sublattices will interact antiferromagnetically because of the underlying antiferromagnetic correlation in the system,[13]forming at the end a localized spin singlet around site i.It is only in the limit J′→0that the two spinons are decoupled and free local magnetic moments appear.Our discussion can also be extended to study other forms of local defects.For example, one may study the situation when the Heisenberg coupling J i,i+ˆx between sites i and i+ˆx is set to zero,i.e.,a single bond is being removed from the system.Notice that the perturbation is symmetric with respect to interchange of A-and B-sublattices.It is straightforward to generalize our previous discussion to this case where weﬁnd that the perturbation now appears as an eﬀective electric dipole moment in L eﬀ.No local moment is expected to form around the impurity bond in this case because of the follow reason:because of the symmetry between A-and B-sublattices bosons(spinons)can be nucleated out from vaccuum only inpairs,with one localized on the A sublattice and one on the B sublattice.The interaction between the nucleated spinon pairs will again be antiferromagnetic and has a magnitude ∼Je−x/ξ,where x is the distance between the two spinons andξ∼m−1is the’size’of the spinon wavefunction.For the single bond defect,the distance x between the two spinons is of order of lattice spacing<<ξ.Thus the eﬀective interaction between the two spinons is of order J,i.e.,they will form a strong local spin singlet and no isolated magnetic moment will appear.Experimentally,the Cu ion in the conducting plane of high-T c cupartes can be substituted by Zn(non-magnetic impurity)[11]and magnitude of Heisenberg exchange J can be modiﬁed locally by introducing impurities out of conduction planes or substitution of O ions.It is observed that while substitution of Cu by Zn in the conducting plane introduces local magnetic moments in the underdoped(spin gap)phases of High-T c cuprates[11],no such eﬀect is observed in other ways of introducing defects.Our theoretical investigation on disordered state of quantum antiferromagnets based on continuum description of SBMFT is in satisfying agreement with experimental results.[10]III.STATIC VORTICES IN DISORDERED STATE OF SBMFTIn the continuum theory,a unit vortex centered as x=0corresponds to a solution ofδL eﬀ=0,withθ(r,Ω)=Ω,where r andΩare the distance and angle in the(2D) polar co-ordinate.The nonzero vorticity introduces a diverging kinetic energy through the VµVµ∼(∇θ)2term in L eﬀ,which supresses the magnitude ofφaround the vortex core r→0,as in the case of usual superconductors.Minimizing the free energy with respect√2)φ,weﬁnd thatφ=| q|=0for toφ,and keeping in mind that for smallφ,| q|∼(r<r o∼[(8(2S+1)+4b−2a)/(2S+1)]1/2,but not going to zero smoothly as r→0, as in the case of usual superconductors.This result is an artifact of our continuum theory where the(∇φ)2term is not included in L eﬀ.Nevertheless,the qualitative eﬀect whereφis being suppressed around the vortex core is clear.For large r,φ(r)→φo=(4(2S+1)−a)/2and q→0,indicating that vortices are stable topological excitations in SBMFT.The behaviour of vortices in SBMFT is very similar to vortices in usual superconductors at large r.However,the small r behaviours are very diﬀerent.The suppression ofφand qﬁelds around vortex core reﬂects the fact that around the center of vortex,the bond amplitude ∆i,i+µ’s are being suppressed.As has been discussed in the previous section,suppression of local bond amplitudes may lead to generation of localized magnetic moments,depending on the detailed bond conﬁguration.To understand the properties of a vortex,it is thus important to understand the underlying bond structure at the vortex core.Before studying the bond structure at vortex core,let us explainﬁrst how local magnetic moment binding to vortex core modiﬁes the properties of a vortex.For a unit vortex the magneticﬂux seen by a Schwinger boson is halfﬂux quantum.(recall that Vµis an’uniform’gaugeﬁeld which has same gauge charge for bosons on both sublattices)Therefore the orbital angular momentum of a Schwinger boson around a unit vortex is1the VµVµterm.Following similar analysis as in previous section weﬁnd that the sup-pressed bonds introduce in the continuum theory an eﬀective electric quadrapole structure surrounding the vortex center.Notice that as in the case of single broken bond the structure is symmetric with respect to the A-and B-sublattices.Thus we expect that localized spinon pairs may form around the center of vortex but isolated magnetic moments cannot occur. Thus this kind of vortex carries zero angular momentum and is a boson.The second kind of vortex is centered at a lattice point and as a result,it is expected that the four bonds joining to the vortex center(see Fig.1b)will be largely suppressed. This situation is very similar to the case of single non-magnetic impurity discussed in the last section.First of all,2S spinons will be found localized at the vortex center because of the suppressed bonds.The spinons behaves as a’staggered’gauge charge with magnitide 2S and2S other spinons with opposite gauge charge will be nucleated from the vacuum to screen the’staggered’electricﬁeld generated by the spinons localized at the vortex cen-ter.The spinons on opposite sublattices interact antiferromagnetically through an eﬀective Heisenberg exchange of order J′∼J|∆c|2,where∆c is the bond amplitude of the four bonds joining to the vortex center.As a result,a spin singlet will be formed.The vortex carries zero angular momentum and is again a boson.Notice that this is true even in the limit J′→0since integer number of bosons=4S are found binding to the vortex.The situation is however quite diﬀerent if we consider aﬁnite density of the second kind of vortices.We shall show that when density of vortices is large enough,it may become energetically unfavourable to nucleate spinons from vacuum to screen the staggered gauge ﬁeld and as a result,only2S spinons will be found binding to vortex center resulting in fermionic vortices when S is half-integer(odd number of bound bosons).For simplicity we shallﬁrst consider two vortices seperated by distance l,with one vortex on each sublattice. First let us consider the energy when spinons are nucleated from the vacuum to screen the staggered gauge charge on each vortex.The total energy will be sum of three terms:(i) the electrostatic energy which is of order2(2Se)2ln(l o/ξ),where l o∼ξis the’size’of the nucleated spinon wavefunction,(ii)The exchange energy between the spinons localized asvortex center and nucleated spinons,which is of order−2S(S+1)J′e−l o/ξ∼−2S(S+1)J′, and(iii)The energy needed to nucleate the spinons from the vacuum,which is of order 4Sm.The sum of the three terms is of order2S(2m−(S+1)J′).The energy for the second case when no spinons are nucleated from vacuum consists of two terms:(i)the electrostatics energy which is of order(2Se)2ln(l/ξ)and(ii)the exchange energy between spinons located at the center of the two vortices which is or order−S(S+1)J′e−l/ξ.In2D,e2∼m and the electrostatic energy is of order4S2m.ln(l/ξ).For l>>ξ,it is certainly energtically more favourable to nucleate bosons from vacuum to screen the vortex gauge charge.However, when l≤ξ,the energy cost of the second case is of order−S(S+1)J′,and is energetically more favourable if S(S+1)J′<4Sm.Notice that J′∼J|∆c|2is expected to be very small because of suppresion eﬀect around vortex core.Thus the condition can be satisﬁed easily even with a relatively small m.Forﬁnite density of vorticesδ,l∼δ1which are rather easy to acheive.We cannot,however,be absolutely sure whether’fermionic’vortices exist because of the natural limitation of a continuum theory which gives only or-der of magnitude estimates.Notice also one important distinction between the vortices we study in this paper and usual vortices in superﬂuids.In the present case,vortices cannot be distingished by sign of vorticity since vortices carrying1/2(orπ)and−1/2ﬂux quanta should be considered as identical.On a lattice where the order parameterﬁeld∆i,i+µis a link-variable,πand−πvortices can be related by a pure gauge transformation.The usual singularity encountered in the gauge transformation in continous space does not arise here because of the discretized lattice structure.It is important to clarify the diﬀerent roles played by the’uniform’and’staggered’gaugeﬁelds in deciding the properties of vortices.The gaugeﬁelds arise fromﬂuctuation in phases of the order parameter∆i,i+µ’s.The uniform gaugeﬁeld couples to bosons on the two sublattices with same gauge charge,and the corresponding gauge symmetry is broken in SBMFT.Vortices are stable because of this broken gauge symmetry,and corresponds to solutions with”uniform”magneticﬂux penetrating center of vortices.The’staggered’gaugeﬁeld couples to bosons on opposite sublattices with opposite gauge charge.It plays no role in generating a stable vortex solution,but has strong eﬀects in determining the precise properties of the vortex core.Without the’staggered’gaugeﬁeld,spinons need not be nucleated from vacuum to screen the’staggered’charges generated at,for example,the center of the second kind of vortex.In this case,for half-integer spin systems,the second kind of vortices will always be fermions independent of density,since only odd number of bosons are found binding to center of each vortex(corresponding to the spin S at center of vortex).Similarly,local magnetic moments will not be generated by non-magnetic impurities if’staggered’gaugeﬁeld does not exist.It is also interesting to point out that similar vortex excitations have been considered by Read and Chakraborty[15]in a short-ranged RVB wavefunction for S=1/2quantum antiferromagnets.They considered also the two kinds of vortices we discussed here.The statistics of the vortices were examined by direct Berry phase computations where similar。

学术英语_社科Unit5五单元原文及翻译

UNIT 5 Sociology Matters1.Culture is the totality of learned,socially transmitted customs,knowledge,material objects,and behavior.It includes the ideas,values,customs,and artifacts of groups of people.Though culture differ in their customs,artifacts,and languages,they all share certain basic characteristics.Furthermore,cultural characteristics change as cultures develop ,and cultures infuence one another through their technological ,commercial, and artistic achievements.文化是指社会传播学，海关，知识，材料的对象，和行为。

它包括思想，价值观，习俗，和人群的文物。

尽管文化在他们的习俗，文物，和语言不同，但是他们都有一些共同的基本特性。

此外，当文化发展时文化特征也在变化，并且文化通过他们的技术，商业，艺术成就相互影响。

Cultural universals文化共性2.All societies,despite their differences,have developed certain general practices known as cultural universals.Many cultural universals are ,in fact,adaptations to meet essential human needs ,such as people’s need for food ,shelter,and clothing. Anthropologist George murdock compiled a list of cultural that included athletic sports, cooking ,funeral ceremonies,medicine,and sexual restrictions.所有的社会，尽管他们的差别，已经形成了一定的一般做法被称为文化的共性。

63nozzle

SOUND ANDVIBRATION/locate/jsvi*Corresponding author.Tel.:+32-2-359-96-15;fax:+32-2-359-96-00.E-mail address:anthoine@vki.ac.be(J.Anthoine).0022-460X/03/$-see front matter r2002Elsevier Science Ltd.All rights reserved.PII:S0022-460X(02)01034-9as in musical instruments.In most cases,aeroacoustic instabilities perturb the operation and have to be suppressed.This article describes experiments aiming at the active reduction of aeroacoustic instabilities by adaptive techniques.We will speciﬁcally analyze the effect of control on the structure of the ﬂow using detailed measurements and in particular particle image velocimetry.The present study focuses on a speciﬁc class of instabilities involving vortex shedding and acoustic resonances.These instabilities have received considerable attention in the past 20years in relation with developments of large solid propellant motors for space vehicles and launchers.The phenomenon develops in the conﬁned ﬂow established in the motor and involves a coupling between vortex shedding and longitudinal acoustic resonant motion.The motor usually features a set of solid propellant grains separated by inhibitor rings ensuring thermal protection (see Fig.1).There are three propellant segments in the Ariane 5solid propellant accelerators (P-230).Propellant is consumed during the ﬂight at a rate which exceeds that of the separation rings (also designated as inhibitor rings).Regression of the burning surface is faster than the consumption of the inhibitor.The inhibitor rings protrude in the ﬂow after a certain time producing regions of high shear where vortex shedding may be generated.This mechanism drives oscillation if the shedding is coupled to one of the acoustic resonant modes of the motor chamber.A considerable amount of experimental data has been gathered on full-scale rockets (Space Shuttle and Titan 4)static tests or in ﬂight (see Refs.[1–4]).Pressure oscillations at frequencies close to those of the ﬁrst two longitudinal eigenmodes were also found during the development full-scale static hot ﬁre tests of the Ariane-5solid propellant motors.Vortex-driven acoustically coupled oscillations were later observed on sub-scale model rockets (see Refs.[5–7]).These various studies point out that the frequency of oscillation decreases as the grain surface regresses.This observation typiﬁes processes which depend on the velocity of the ﬂow.Observations also indicate that the pressure oscillations reach a maximum when the vortex shedding frequency is close to one of the natural frequencies of the motor chamber.An early interpretation of the phenomenon is due to Flandro [8],who linked the oscillations to the hydrodynamic instability of the sheared regions of the ﬂow and to the coupled response of the motor.This viewpoint is supported by experiments at various scales and for a variety of geometries and operating conditions.It is now well established that unstable oscillations areVortexAcoustic Fig.1.Internal geometry of the Ariane 5solid rocket motor and ﬂow-acoustic coupling.J.Anthoine et al./Journal of Sound and Vibration 262(2003)1009–10461010produced by the combination of the following mechanisms (see Fig.1)[9,10,4]:a hydrodynamic instability of shear regions of the ﬂow;the roll-up,growth and advection of vortices;impingement of the vortices on a surface located downstream such as the nozzle head and generation of an acoustic perturbation;acoustic propagation from the downstream source;transfer of energy from the acoustic mode to the shear ﬂow instability.The hydrodynamic instability created by the inhibitor leads to vortex formation.The coherence degree of these structures depends on the presence of an external acoustic ﬁeld.When the vortices hit a surface like the nozzle head a dipole source is generated and energy passes from the vortex to the acoustic ﬁeld.This energy is fed back to the shear ﬂow instability by one of the acoustic modes of the system.The receptivity of the shear region to external modulations closes the loop and a new vortex is shed and advected downstream.Three frequencies must be considered in the process:the characteristic frequencies of the shear layer,f s ;the feedback frequencies of the impinging shear layer,f b ;the acoustic resonance frequencies of the full system,f a :Among the available shear layer frequencies,f s is selected by a feedback mechanism involving an interaction of the shear layer with the downstream end of the system.The characteristic length scale is the distance l between the shear layer origin and the impingement point.The feedback frequency f b is obtained if certain phase relations are satisﬁed.It is generally agreed that this frequency is determined by the expression [11–13,4]mT ¼l U c þl c ÀU þt ;ð1Þwhere T ¼1=f b is the period of the process,m is the number of vortices at a given instant between the shedding region and the impingement surface separated by a distance l :U and U c are the ﬂow velocity and the vortex transport velocity,respectively.c is the speed of sound and t designates the delay between the vortex impingement and the generation of a pressure pulse at the downstream surface.One generally assumes that t may be expressed as a fraction of the period,t ¼a T ;and that the transport velocity is proportional to the ﬂow velocity,U c ¼kU :Upon using these expressions and assuming that the Mach number of the ﬂow is small,the feedback frequency is given byf b C U l m Àa M þ1=k:ð2ÞThis expression was ﬁrst obtained by Rossiter [11]in an analysis of acoustic radiation by an airﬂow over a cavity.Its application to segmented solid rockets provides a suitable estimate of the frequencies observed (see for example Ref.[4]for a clear validation).Data also indicate that the pressure oscillation reaches large amplitudes when the feedback frequency f b ;given by Eq.(2),matches one of the eigenfrequencies f a of an acoustic mode of the system ðf b C f a Þ:Other experiments indicate that the process is also controlled by the relative positions of the vortex shedding region and the acoustic resonant mode structure [3,14].In general the acoustic mode triggers vortex shedding more effectively at locations where the pressure features a node and the velocity an anti-node.J.Anthoine et al./Journal of Sound and Vibration 262(2003)1009–10461011Numerical simulations [15–18]carried out in the last 10years have brought further information on the aeroacoustic processes which lead to vortex-driven oscillations in solid rocket motors.Simulations show that vortices may also develop from edges of the solid propellant (for example,from the upstream edge of a cavity separating two segments or from the downstream edge of the last segment).The bending ﬂow resulting from the propellant combustion also features a natural instability which is potentially capable of generating vortices [19].These ﬁndings are supported by hot and cold ﬂow experiments [20].Because combustion is believed to take a minor part in the instability mechanism,most laboratory experiments were carried out with models operating with cold gases.A conﬁguration involving a duct with one or two diaphragms is used for example by Culick [21],Dunlap [14],Hourigan [22],Huang [23],Planquart [24],Anthoine [25],Mettenleiter [26],Stubos [27]and Anthoine et al.[28].The vortex shedding is produced at the upstream diaphragm and the duct acts as a resonator through its natural eigenmodes.Culick [21]showed that the resonant oscillation could not be sustained with a single diaphragm.An obstacle located downstream of the shedding point is required and serves to transfer energy from the impinging vortices into the resonant acoustic mode [29,30].Another set-up designed by Couton [31]features a rectangular two-dimensional channel with wall injection,obstacles simulating inhibitor bafﬂes and a submerged nozzle.The channel resonance was linked to the excitation of the ﬁrst longitudinal modes and the ampliﬁcation of the acoustic level was found to depend on the injected mass ﬂow rate.In the present study an axisymmetric duct representing the geometry of the real motor is used.Such an axial cold ﬂow model includes only one inhibitor where the vortices are formed,and a submerged nozzle where the vortices collide and generate sound.This conﬁguration correctly simulates the interaction of the advected vortices with the nozzle head.An active control device implemented with a sensor,an actuator and an adaptive controller is ﬁtted to the system to suppress the pressure oscillations when needed.It is thus possible to study a naturally resonant ﬂow under non-resonant conditions.The operating parameters remaining ﬁxed,the control system is switched on and the acoustic pressure oscillation is suppressed.One may then examine the changes in the ﬂow pattern induced by the presence or absence of acoustic resonance.This is done with particle image velocimetry downstream of the inhibitor.The combined use of a resonant model,active control and PIV is aimed at applying adaptive techniques to control aeroacoustic instability phenomena,providing new information on the processes leading to instability in a geometry which typiﬁes the back end of a solid rocket motor,and getting an understanding of how the controller modiﬁes the ﬂow to reduce the pressure oscillations.Because active control of the ﬂow is an essential aspect of this study it is worth giving a brief overview of research in this area.Since the subject is quite broad,we will consider only the topic of instability control.Control of combustion instability using feedback concepts is suggested in the early work of Tsien [32],Crocco [33]and Marble [34].Practical demonstrations have been completed more recently by Dines [35],Heckl [36],Bloxsidge et al.[37]and Lang et al.[38].A review of this subject is given by McManus et al.[39].Active control of ﬂow instabilities was demonstrated experimentally by Roussopoulos [40].Closed loop control experimental demonstrations have concerned ﬂow over cavities,aeroacoustic resonances in ducts[23,26],vortex shedding in low Reynolds number ﬂows [40],impacting ﬂows [41],compressor surge [42],etc.J.Anthoine et al./Journal of Sound and Vibration 262(2003)1009–10461012J.Anthoine et al./Journal of Sound and Vibration262(2003)1009–10461013 Most of the earlier work was based onﬁxed parameter controllers and had a limited range of applicability.Because combustion orﬂow instabilities change with the operating parameters,it is important to use adaptive techniques to achieve control over a broad range ofﬂow conditions. This point was demonstrated by Billoud et al.[43]and later by Koshigoe et al.[44].Adaptive techniques have led to many applications in the control of sound and vibration,a subject of considerable technical importance which has been investigated extensively[45,46].It is believed that technical applications inﬂow control will require self-adjusting systems of the kind tested in this article.The paper begins with a brief presentation of the methods used to identify and characterize vortices in aﬂowﬁeld(Section2).These methods will be used later in this article to examine the ﬂow structure.The experimental conﬁguration is then described(Section3).Some aspects are only brieﬂy covered because they are discussed in more detail in other publications[25,26].The results of three experiments are reported in Section4.A low speed case which is naturally free of oscillation is considered.After that,a higher speed case which is naturally resonant is envisioned. When control is applied to thisﬂow,resonance vanishes.Theﬂow patterns in the three cases are examined with particle image velocimetry(PIV).The data are used in Section4to describe the changes which take place under controlled operation and to improve the current view of the process leading to instability.2.Characterization and identiﬁcation of vorticesThe study of coherent structures in turbulentﬂows remains a central subject inﬂuid dynamics. A‘‘universal deﬁnition’’of coherent structures is still missing.Coherent patterns of a particular ﬂow have common features but they are far from being identical.Moreover,they do not appear with precise regularity in time and space.This lack of regularity makes them so difﬁcult to be deﬁned and described.According to Robinson[47]‘‘a vortex exists when instantaneous streamlines mapped onto a plane normal to the vortex core exhibit a roughly circular or spiral pattern,when viewed from a reference frame moving with the vortex core’’.Following Hussain [48]‘‘a coherent structure is a connected turbulentﬂuid mass with instantaneously phase-correlated vorticity over its spatial extent’’.In other words,a vortex is a region where the instantaneousﬂow is rotating around its center,and which can be advected with a certain velocity.Coherent structures are three-dimensional and the random nature of turbulence makes their‘‘signal-to-noise’’ratio low.As a consequence,their eduction from noisy data is a difﬁcult task.2.1.Characteristics of a vortexThe vortices considered in this research are created by an annular obstacle.The vortex rings shed by the obstacle may not be perfectly axisymmetric but their behavior can still be investigated in a plane O xy perpendicular to the vortex core(or nearly so).In this plane,the characteristic parameters of a vortex are mainly:the position of its centerðx0;y0Þ;its velocity distribution ðV¼uþvÞ;the transport velocityðu v;v vÞ;diameterðD vÞ;vorticity peakðOÞ;vorticity distributionðo Þ;circulation ðG Þand enstrophy ðE Þwhereo ¼@v @x À@u @y;ð3ÞG ¼I v Ád l ¼Z Z o d S ;ð4ÞE ¼Z Z o 2d S :ð5ÞAmong standard vortex models,the Oseen vortex is often used to typify vortex motion.The corresponding velocity distribution is,in polar co-ordinates r ;y :v y ¼G 02p rð1Àe Àr 2=s 2Þ;v r ¼0;ð6Þwhere s is a characteristic dimension of the vortex.Fig.2(a)shows the 2-D-pattern of such a vortex ﬂow.2.2.Identiﬁcation of a vortexExpression (6)should not be taken as an exact representation.It merely serves as a guide in the vortex identiﬁcation process and it will be used to synthesize the data gathered.Jeong and Hussain [49]list the different criteria used until now to recognize vortex motion.First of all,the center of a vortex corresponds to a local pressure minimum.Unfortunately,a pressure map is difﬁcult to obtain experimentally.A straightforward criterion is to detect the presence of a vorticity peak at the center of the pattern.Indeed,vorticity is a Galilean invariant of the velocity,which means that it is not affected by an additional transport velocity.However,while a center of a vortex leads to a peak of vorticity,a vorticity peak does not necessarily correspond to avortex(a) 2D _ pattern (b) Velocity ( ), vorticity ( )and λ ( ) distributions along a diameterFig.2.The Oseen vortex ðr 2¼x 2þy 2Þ:J.Anthoine et al./Journal of Sound and Vibration 262(2003)1009–10461014center,like,for example,in a shear layer.Thus,an analysis based on the vorticityﬁeld can be limited when studying aﬂow with obstacle.In the more elegant method presented by Jeong and Hussain[49]a vortex in an incompressibleﬂow is deﬁned in terms of the eigenvalues of the symmetric tensor S2þO2;where S and O are,respectively,the symmetric and antisymmetric parts of the velocity gradient tensor r V(S and O are deformation and rotation tensors, respectively).In planarﬂows,the vortex core corresponds to a connected region within which l o0;where l is the non-zero eigenvalue of S2þO2:lðx;yÞ¼@u@x2þ@u@y@v@xo0:ð7ÞThis criterion means that within a vortex the derivatives of the velocity components must be of opposite signs.Note that the formulation in cylindrical co-ordinates,more complex to be implemented,has been tested on typical images and led to very close results compared to the2-D approximation given by Eq.(7).The diameter D v of the vortex is deﬁned by the zero crossing contour of l:This contour corresponds to the maximum velocity in Oseen’s vortex.Finally,the eigenvalue l is Galilean invariant as well as the vorticity because it only involves velocity derivatives.Fig.2(b)shows the velocity,vorticity and l distributions along a diameter for an Oseen vortex.While the lﬁeld is used for the identiﬁcation of the vortex and for the determination of its diameter,the other characteristics,such as G and E are computed from the vorticityﬁeld.ction of a vortexParticle image velocimetry provides instantaneous velocity vectorﬁelds,as explained in Section3.2.From theseﬁelds,the instantaneous vorticity and l mappings can be computed and vortices can be detected from the l contours.To make statistical averages on the vortices,their eduction from theﬂowﬁeld should be automatic and it is done here by using a wavelet analysis applied to the lﬁeld.The vortex detection algorithm,based on the property of selectivity in space and scale of the continuous wavelet transform,allows one to determine the vortex position and size.The other characteristics can be calculated from these two informations.The processing of the wavelet transformation[50]is carried out to localize a speciﬁc scale(size of the vortex)within the2-D l-ﬁeld signal(position of the vortex).The signature of a vortex in a l map is a Gaussian curve detected by a2-D wavelet designated as the‘‘Mexican Hat’’.The mother wavelet is given bycðx;yÞ¼Àð2Àx2Ày2ÞeÀðx2þy2Þ=2:ð8ÞMore details about the wavelet analysis can be found in Ref.[51].Fig.3shows the comparison between the lﬁeld and the Mexican Hat.The point where the Mexican Hat crosses the zero axes deﬁnes the diameter of the vortex.The Mexican Hat and the l distribution do not have exactly the same shapes.The relation between the diameter of the Mexican Hat and that of Oseen’s vortex is obtained by calibration.From a PIV velocity vectorﬁeld,the lﬁeld is computed by a standard four points centered scheme.The Mexican Hat is dilated for different scales.Each scale scans the whole lﬁeld.The convolution gives a4-D information:2-D for the locationðx;yÞ;1-D for the scaleðD vÞ;and1-D for the wavelets coefﬁcientðj W jÞ[52].This last information allows detection of J.Anthoine et al./Journal of Sound and Vibration262(2003)1009–10461015a vortex if its value is higher than a chosen threshold,linked to the energy carried by the candidate coherent structure.In other words,the vortex detection algorithm yields the wavelet coefﬁcient j W j as a function of x ;y and D v ;a large value of the coefﬁcient implies a good ﬁt between the wavelet of that scale and the data at that point.When a region of high values of the wavelet coefﬁcient is identiﬁed by the vortex detection algorithm,the center of the vortex core corresponds to the local maximum of that coefﬁcient.A vortex is identiﬁed if the following conditions are fulﬁlled:its enstrophy must be higher than a given percentage of the total enstrophy of the vorticity ﬁeld and the vortex must not overlap a previously found vortex,i.e.,the investigation is restricted to distinct vortices.Therefore,the wavelet coefﬁcient values higher than the threshold but located within the scale D v of the previously found vortex are not considered.3.Experimental conﬁguration3.1.Experimental set-upThe experimental facility is composed of a cylindrical test section,a fan,and an oriﬁce plate.The test section is designed to reproduce the geometry of the solid propellant accelerator of Ariane 5(see Fig.4).As indicated previously,the Ariane 5accelerator is made of three propellant grains (Fig.1).The ﬁrst segment (S1)is active during the ﬁrst 30s of the combustion,while segments S2and S3are active during the total combustion time of 129s :Therefore,by restricting the investigation to times larger than T c =4;where T c is the total combustion time,the head (S1)can be assumed inactive and replaced by an aerodynamically shaped inlet and a honeycomb to get a uniform one-dimensional ﬂow.Behind the inlet,the set-up consists of two cylindrical segments,with an inhibitor,and a submerged nozzle.The internal diameter of the segments,equal to 0:15m ;is 1/15th model scale of the real motor when 50%of the propellant is burnt,and the dynamic similarity is achieved by matching the model and full-scale Mach numbers.Each of the two segments has a length L =2of 0:63m :The nozzle has a throat diameter of 0:059m :Its geometry is sketched in Fig.4.The main characteristic of this nozzle is the appearance of a cavity aroundtheparison between the l ﬁeld (—-)and the Mexican Hat (.......)ðr 2¼x 2þy 2Þ:J.Anthoine et al./Journal of Sound and Vibration 262(2003)1009–10461016convergent section.During combustion,the cavity volume varies as explained in Ref.[53].At 50%of the combustion process,the geometry is close to that drawn in Fig.4.The total length of the test section including the inlet and the nozzle cavity is equal to 1:75m :As the vortices are expected to generate sound when interacting with the nozzle,the inhibitor–nozzle distance l has to be small,in the case of an axial ﬂow.This will guarantee that the vortices reach the nozzle.The value chosen for the inhibitor–nozzle distance of 0:14m corresponds to an optimized ﬂow-acoustic coupling [54].The inhibitor has a thickness of 0:5mm at the leading edge (2:5mm at its base)and an internal diameter of 113:2mm yielding an inhibitor height h of 17:4mm :It is ﬁxed by means of a thin silicon seal.The ﬂow is injected axially in the test section from the head-end (inlet).The model operates with a ﬂow of cold air sucked by a fan placed downstream of the test section.The centrifugal fan produces a mean ﬂow velocity U 0in the test section varying from 8to 28m =s :This velocity is deduced from the mass ﬂow rate measured with the oriﬁce plate.The corresponding Reynolds number based on the inhibitor height h is changing from 9000to 32000.3.2.Instrumentation and post-processingA piezoelectric transducer model 106B50from PCB Piezotronics Inc.is mounted in the test section for pressure ﬂuctuation measurements.This unit is placed at the wall just upstream of the nozzle.A calibrated hot wire of 9m m in diameter,built in-house,is also mounted between the inhibitor and the nozzle.The pressure sensor and the hot wire are connected to aspectrumMeasurement (b) Sketch of the test section(a) Sketch of the experimental facilityFig.4.Sketch of the experimental facility.J.Anthoine et al./Journal of Sound and Vibration 262(2003)1009–10461017analyzer (Hewlett Packard model 35660)or to a DAS1601acquisition card controlled by TESTPOINT TM :A Particle Image Velocimetry (PIV)technique is used to measure the velocity ﬁeld between the inhibitor and the nozzle.For such experiments,the test section consists of a Plexiglas TM tube of 25mm thickness.Repellin [55]has shown that such a thickness is not suitable for PIV measurements because of optical distortion.To avoid this difﬁculty,the Plexiglas TM tube is replaced by a Makrolon TM tube of 1mm thickness on a length of 0:59m from the nozzle.The laser sheet and the camera are placed in front of the thin tube.Tests prove that such a thickness is sufﬁciently small to reduce optical distortion to an acceptable level.Fig.5shows a schematic of the PIV instrumentation.The light sheet is generated by a Nd:YAG double pulsed laser,operating at 10Hz and a set of cylindrical lenses.The laser provides 4ns pulses with energy of 400mJ :A TSI cross-correlation CCD camera model 630044(640Â480pixels)records the successive images which are acquired and saved by the TSI acquisition software INSIGHT TM :The optical calibration in both directions is achieved by measuring the ﬁeld of view of the CCD camera at the location of the laser sheet.The PIV ﬁeld of view of 98Â76mm is situated just downstream of the inhibitor.The laser working at 10Hz and the camera operating at 30Hz are synchronized by the TSI unit 610030.The synchronizer is also used to set a time delay D T between the two laser pulses of around 50m s ;depending on the ﬂow velocity.Because of the stagnation region at the end of the recirculation bubble,the use of standard oil in the form of seeding droplets leads to a ﬁlm deposit on the wall behind the inhibitor and through which the camera is looking.To reduce this effect,oil has been replaced by the global mix smoke ﬂuid manufactured by Le Maitre Ltd.(England),which has the property to condense less.Moreover,an additional supply of compressed air is also injected after the smoke generator and the mixture is introduced in the ﬂow by means of a vertical tube comprising several small oriﬁces,placed in front of the test section.This improves mixing with air and the uniformity of the seeding.Two successive images are processed with the cross-correlation program WiDIM using Fast Fourier Transform algorithms [56,57].The two images are divided into interrogation windows FanFig.5.Instrumentation for PIV measurements.J.Anthoine et al./Journal of Sound and Vibration 262(2003)1009–10461018from which a prediction velocity vector is deduced by using a cross-correlation function [58].Then,the instantaneous velocity ﬁelds of N images obtained from WiDIM are processed to determine the reduced mean velocity components U mean =U 0and V mean =U 0;the turbulentintensities ðÞ1=2=U 0and ð1=2=U 0;and the reduced Reynolds stresses u 0v 0=U 2;with U mean ¼1N XN i ¼1U i ðand a similar expression for V mean Þ;ð9Þﬃﬃﬃﬃﬃﬃu 02p ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1NX N i ¼1ðu i ÀU mean Þ2r ðand a similar expression for ﬃﬃﬃﬃﬃv 02p Þ;ð10Þu 0v 0¼1N X N i ¼1½ðu i ÀU mean Þðv i ÀV mean Þ :ð11ÞThe number of images N varies from 240to 384.The instantaneous velocity vector ﬁelds are then used to calculate the instantaneous vorticity and l maps.Vortices are then detected by applying the wavelet analysis described in Section 2.3.For each vortex identiﬁed by the wavelet analysis,the previous treatment provides the centerlocation ðx 0;y 0Þand the diameter D v :Cartesian indices ði v ;j v Þof the points belonging to a given vortex are deduced from the condition r ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðx Àx 0Þ2þðy Ày 0Þ2q p D v =2and the vortexcharacteristics are then computed:vorticity peak,O ¼max i v ;j v ðo Þ;ð12Þcirculation,G ¼IV Ád l ¼Z Zo d S -X i vXj voD x D y ;ð13Þenstrophy,E ¼Z Zo 2d S -X i vXj vo 2D x D y ;ð14Þtransport velocity,u c ¼u ðx 0;y 0Þand v c ¼v ðx 0;y 0Þ:ð15ÞThe integration in expressions (13)and (14)is carried out over the vortex core.A smallpercentage of vorticity or enstrophy is omitted.This has little inﬂuence on the results.On the other hand it is important to deal with compact vortices.A statistical analysis of these characteristics is developed in Section 4.4.3.3.Control systemTo achieve control,it is ﬁrst important to select a suitable class of algorithms.The controller structure naturally depends on the instability phenomenon which is considered or on the model used to describe the phenomenon.。

译林版高中英语学案选择性必修第一册精品课件 Unit 4 Section A

［尝试翻译］__________________________________________________________________________________________
即使你也许把握不住这首诗的真正含义,但是当你大声朗读时，你总是可以说说这首诗听上去如何。
were
slowly
a
to interpret
it
Poetry's combination of “sound” and “sense” 9._______ （make） you see the world in a new way and allows you to go beyond normal 10._______ （real） for the everlasting beauty.
rhyme
clue
cage
aside
logical
inner
frozen
freeze
freezing
imply
implication
17. ______ vt.发现，查明，侦察出→_________ n.侦查，探测；察觉→_________ adj.侦探的 n.侦探→________ n.检测器；侦察器18. ________ vt.解释，阐释；把……理解为，领会；演绎 vi.口译→____________ n.解释；翻译→___________ adj.解释的；作为说明的→__________ n.解释者；口译者19. ________ n.耐心,忍耐力→_______ adj.忍耐的；耐心的 n.病人→_________ adv.耐心地20. _______ n.现实,实际情况；事实，实际经历→________ adj.现实的；实际的→_____ adj.真的→______ adv.真正地；实际上21. _____ n.薄雾，水汽→______ adj.模糊的；有雾的

鹦鹉英语作文

Parrots are known for their vibrant colors and remarkable ability to mimic human speech.Here are some points to consider when writing an essay about parrots:1.Introduction to Parrots:Begin your essay by introducing parrots as a group of birds known for their intelligence and social nature.Mention their wide variety of species, ranging from the small lovebirds to the large macaws.2.Physical Characteristics:Describe the physical features of parrots,such as their beaks, which are adapted for cracking seeds and nuts,and their strong feet for perching and climbing.3.Coloration and Species:Discuss the diversity in coloration among parrots,from the green of the budgerigar to the vibrant hues of the scarlet macaw.Mention the different species and their unique characteristics.4.Habitat and Distribution:Explain where parrots are typically found,such as tropical and subtropical regions,and how their distribution is affected by factors like climate and habitat loss.5.Diet and Feeding Habits:Describe the diet of parrots,which includes seeds,nuts,fruits, and sometimes insects.Highlight their feeding habits,such as their use of their beaks to manipulate food.6.Social Behavior:Parrots are known for their social behavior.Discuss how they live in flocks and communicate with each other using a variety of calls and sounds.7.Mimicry and Communication:One of the most fascinating aspects of parrots is their ability to mimic sounds,including human speech.Explain how this ability is used for communication within their social groups and how it has made them popular pets.8.Conservation Status:Discuss the conservation status of parrots,highlighting the threats they face such as habitat destruction,poaching for the pet trade,and climate change.9.Captive Care and Ethical Considerations:If youre writing about parrots as pets,its important to address the ethical considerations of keeping such intelligent and social creatures in captivity.Discuss the importance of providing them with a suitable environment,social interaction,and mental stimulation.10.Conclusion:Conclude your essay by summarizing the key points and emphasizing the importance of protecting parrots in the wild and providing proper care for those kept aspets.Remember to use descriptive language and,if possible,include anecdotes or personal experiences to make your essay more engaging.Additionally,ensure that your essay is wellstructured,with a clear introduction,body,and conclusion.。

Vortex Dynamics in a Coarsening Two Dimensional XY Model

a r X i v :c o n d -m a t /0304346v 3 [c o n d -m a t .s o f t ] 3 J u n 2003Vortex Dynamics in a Coarsening Two Dimensional XY ModelHai Qian and Gene F.MazenkoJames Franck Institute and Department of Physics,University of Chicago,Chicago,Illinois 60637(04/14/2003)The vortex velocity distribution function for a 2-dimensional coarsening non-conserved O (2)time-dependent Ginzburg-Landau model is determined numerically and compared to theoretical predictions.In agreement with these predictions the distribution function scales with the average vortex speed which is inversely proportional to t x ,where t is the time after the quench and x is near to 1/2.We ﬁnd the entire curve,including a large speed algebraic tail,in good agreement with the theory.I.INTRODUCTIONIt is important to understand the role of defects in phase ordering [1]problems.We investigate here the growth kinetics of the non-conserved O (2)symmetric time-dependent Ginzburg-Landau (TDGL)model in two dimensions after a quench from a disordered high tem-perature state to zero temperature.The dominant struc-tures in the ordering kinetics of this system are vortices with charges ±1.Vortices with higher order charges are unstable.Various aspects of the defect structure have been explored in some detail before [2,3].We focus here on a numerical determination of the velocity distribution of the vortices as a function of time t after the quench.Theory predicts [4]that the distribution function scales with the average vortex speed which is inversely propor-tional to a length scale L (t ),which grows with time t af-ter the quenches.One also ﬁnds a large speed algebraic tail in good agreement with predictions of an exponent of −3.In terms of the velocity distribution this corre-sponds to an exponent of −4.The number of vortices is also counted and its evolution in time is found to be consistent with previous work [5].The disordering agents in the phase ordering of the n =d =2non-conserved TDGL model (2d XY model)are well known,where d is the spatial dimension and n is the number of the components of the order parame-ter.One has unit one charged vortices and,at nonzero temperature,spin waves.If we focus on quenches to zero temperature we have only the vortices to consider.Thus a typical vortex conﬁguration is shown in Fig.1.As the time evolves one has vortex anti-vortex annihilation un-til ﬁnally there are no surviving vortices and the system is fully ordered (we only consider the zero temperature case with no thermal ﬂuctuations,where the system does eventually order).For general n =d Bray and Ruten-berg [6]have shown that the growth law for such systems is given by L (t )≈t 1/2.The exception is for n =d =2where their method is mute.The growth law for this case was treated by Pargellis et al.[7],and checked numeri-cally by Yurke et al.[5].There is a logarithmic correction to the scaling law:L (t )≈(t/log(t ))1/2.FIG.1.A typical vortex conﬁguration in a 256×256sys-tem with lattice spacing ∆r =π/4.The arrow on each each site represents the order parameter at that point.Not all the lattice sites are shown.The squares and triangles are in the core regions of +1and −1vortices respectively,where the magnitude of the order parameter is near zero.The vortex core regions are picked out by using the method described in the text.There has also been theoretical work on the dynamics of these vortices.Mazenko [4]showed that if the order parameter can be assumed to be a gaussian ﬁeld [8]when constrained to be near a vortex core then the vortex ve-locity probability distribution,for n =d [4,9],has the simple form:P ( v )d n v =Γ(1+n/2)(1+v 2/s 2)(n +2)/2d n v ,(1)where the scaling speed s varies as L −1for long times.If we only care about the magnitude of the velocity and integrate out the directions,then in the case of n =d =2we have the speed probability distributionP (v )dv =2(1+v 2/s 2)2dv ,1or equivalentlyP ( v )d v =2a v∂t=ǫψi +∇2ψi − ψ2ψi ,(4)where i =1,2are the indices for the two components ofthe order parameter ψ.The noise term is zero because the system is quenched to zero temperature.By choos-ing proper units for the time and space and rescaling the order parameter,ǫcan take on any positive value.The equation is put on a square lattice with periodic bound-ary conditions and driven by the ﬁnite diﬀerence scheme,i.e.replacing ∂t ψi (r ,t )by ψm +1i (kl )−ψm i (kl ) /∆t with m being the time step number,and ∇2ψi (r ,t )by∇2ψi (kl )=13 NN+13ψi (kl )(5)where NN and NNN mean the nearest neighbors andnext-nearest neighbors respectively and the lattice point is r /∆r =(k,l ).Here ∆t is the time step and ∆r is the lattice spacing.Both are dimensionless.We have studied two systems in some detail.In both we choose ǫ=0.1and use 1024×1024lattice sites.Insystem one,or the bigger system,we use ∆t =0.02and∆r =π/4.In system two,or the smaller system,we use ∆t =0.01and ∆r =π/8.In both we measured the vor-tices number and the system energy.We measured the vortex speed distribution only in the bigger system.We prepare the system initially in a completely dis-ordered state.The average magnitude ¯Mof the vector order parameter ψat time t is calculated and the vortices’core regions are identiﬁed with those sites on which theorder parameter magnitudes | ψ|<¯M/ually each core has about 10sites in it.Here the coeﬃcient 1/4is appropriately chosen so that no vortices are missed and no irrelevant points are picked.A circular integration around each vortex produces 2πor −2π,which corre-sponds to two types of vortices of topological charges +1and −1respectively.In fact there are situations where we obtain charge 0.This is due to the non-zero area of the integration circle.When a pair of +1and −1vor-tices annihilate and the distance between them becomes smaller than the size of the integration circle,the circular integration will reﬂect the sum of the two charges,which is 0.However,the lifetime of these 0charges are much smaller than the lifetime of the ±1vortices.So they do not aﬀect our statistics.We ﬁnd in our simulations that the numbers of positive and negative vortices are equal.The position of a vortex is given by the center of its core region.Suppose the order parameter’s magnitude at the core region is described by M (x i ,y i )with (x i ,y i )belonging to the core region.Then by ﬁtting M (x i ,y i )to the function M (x,y )=A +B [(x −x 0)2+(y −y 0)2]we can ﬁnd the center (x 0,y 0).The positions of each vortex at diﬀerent times are recorded,and the speed is calculated using v =∆d/∆τ.Here ∆d is the distance that the vortex travels in time ∆τ.We have,simultaneously,recorded the number of vortices as a function of time and checked the scaling result given by Eq.(3).III.NUMERICAL RESULTSEvery δt =10we compute the speed of each vortex with ∆τ=5,i.e.v =|r (t +∆τ)−r (t )|/∆τwith t in-creasing by step length δt and r being the position of the vortex.We found the average speed of the vortices ¯v (t )is proportional to t −x with x =0.51±0.01for the bigger system (∆r =π/4)as shown in Fig. 2.Unlike the case for L v (t )extracted from n v and (E −E 0),we do not observe logarithmic correction for ¯v (t ).We can not rule out such corrections appearing on a longer time scale but they do not enter on the same time scale as for L v (t ).2FIG.2.The average speed¯v(t)of the vortices for the bigger system(∆r=π/4).The speed of each vortex is calculated with∆τ=5.t−x is used toﬁt the data.x=0.51±0.01for the system with∆r=π/4.The data are averaged over60 diﬀerent initial conditions.We compared the speed distribution at diﬀerent times. They have approximately the same shape after being rescaled by the average speed.At early times this is clear.At late times there are fewer data points and the similarity between the two distributions at diﬀerent times is not so apparent.Even at early times the number of data points are not enough to give a goodﬁt to the dis-tribution’s long tail.So we rescale the speed data with the bestﬁt to the average speed¯v(t)∝t−0.51and put the data for all times into one histogram as shown in Fig.3. By this means we obtain better statistics.Weﬁnd that the distribution can be wellﬁt to the function2a vP( v)=FIG.4.The total number of the vortices per unit area n v(t)/S times t,where S=(1024×∆r)2.In both systems n v·t/S can beﬁt to a log(t/t0).The data from the large and the small systems are averaged over68and58diﬀerent initial conditions respectively.FIG.5.The product of the systems energy per unit area (E−E0)/S with t.The ground state energy is E0=−ǫ2S/4. The data can also beﬁt to a ln(t/t0).The data from the large and the small systems are averaged over68and58diﬀerent initial conditions respectively.In both the larger and smaller systems the vortex num-ber densities have the same time dependence.We obtain n v∼[t/log(t/t0)]−1,where n v is the number density of vortices(positive or negative)and t0is a constant.In Fig.4,we show the data for n v·t/S.S is the area of the system.In Fig.5,we show the data for the energy per unit area(E−E0)/S times t.We conclude that the energy density is proportional to the number density of the vortices.IV.CONCLUSIONSWe have studied the growth kinetics of the non-conserved TDGL model in the case of n=d=2.We measured the speed probability distribution for the±1 vortices.At any given time with relatively few vortices, the statistics are poor.However the accumulated data for all times when scaled gives a scaling function with good statistics.Although the scaling exponent z is0.5with signiﬁcant uncertainty,the large speed tail does takes the form of(v/¯v)−y with the exponent y=3.This is consistent with the theoretical prediction.The form of the distribution P( v)is quantitatively consistent with the theoretical prediction.Why does the theory do so well?It was shown by Mazenko and Wickham[11],that one can construct a nontrivial self-consistent gaussian theory for the order parameter if it is constrained to be evaluated near a vor-tex core.Such constraints occur naturally,for example, in averages over the vortex density.This suggests that the theory developed in Ref.[4]may be on aﬁrmer foot-ing thanﬁrst thought.According to the theory,the average speed should be proportional to the inverse of the correlation length L(t). We did not observe any logarithmic correction in the av-erage speed,although it appears in the correlation length when n=d=2.However a logarithmic behavior may still exist.The time we used in our simulations may not be long enough to see the eﬀect.We also measured the number density of the vortices and the energy density of the two systems.Their time dependences both have a logarithmic correction,which is consistent with previous work.Acknowledgments:This work was supported by the National Science Foundation under Contract No.DMR-0099324.。

专八英语阅读

英语专业八级考试TEM-8阅读理解练习册（1）(英语专业2012级)UNIT 1Text AEvery minute of every day, what ecologist生态学家James Carlton calls a global ―conveyor belt‖, redistributes ocean organisms生物.It’s planetwide biological disruption生物的破坏that scientists have barely begun to understand.Dr. Carlton —an oceanographer at Williams College in Williamstown,Mass.—explains that, at any given moment, ―There are several thousand marine species traveling… in the ballast water of ships.‖ These creatures move from coastal waters where they fit into the local web of life to places where some of them could tear that web apart. This is the larger dimension of the infamous无耻的，邪恶的invasion of fish-destroying, pipe-clogging zebra mussels有斑马纹的贻贝.Such voracious贪婪的invaders at least make their presence known. What concerns Carlton and his fellow marine ecologists is the lack of knowledge about the hundreds of alien invaders that quietly enter coastal waters around the world every day. Many of them probably just die out. Some benignly亲切地，仁慈地—or even beneficially — join the local scene. But some will make trouble.In one sense, this is an old story. Organisms have ridden ships for centuries. They have clung to hulls and come along with cargo. What’s new is the scale and speed of the migrations made possible by the massive volume of ship-ballast water压载水— taken in to provide ship stability—continuously moving around the world…Ships load up with ballast water and its inhabitants in coastal waters of one port and dump the ballast in another port that may be thousands of kilometers away. A single load can run to hundreds of gallons. Some larger ships take on as much as 40 million gallons. The creatures that come along tend to be in their larva free-floating stage. When discharged排出in alien waters they can mature into crabs, jellyfish水母, slugs鼻涕虫，蛞蝓, and many other forms.Since the problem involves coastal species, simply banning ballast dumps in coastal waters would, in theory, solve it. Coastal organisms in ballast water that is flushed into midocean would not survive. Such a ban has worked for North American Inland Waterway. But it would be hard to enforce it worldwide. Heating ballast water or straining it should also halt the species spread. But before any such worldwide regulations were imposed, scientists would need a clearer view of what is going on.The continuous shuffling洗牌of marine organisms has changed the biology of the sea on a global scale. It can have devastating effects as in the case of the American comb jellyfish that recently invaded the Black Sea. It has destroyed that sea’s anchovy鳀鱼fishery by eating anchovy eggs. It may soon spread to western and northern European waters.The maritime nations that created the biological ―conveyor belt‖ should support a coordinated international effort to find out what is going on and what should be done about it. (456 words)1.According to Dr. Carlton, ocean organism‟s are_______.A.being moved to new environmentsB.destroying the planetC.succumbing to the zebra musselD.developing alien characteristics2.Oceanographers海洋学家are concerned because_________.A.their knowledge of this phenomenon is limitedB.they believe the oceans are dyingC.they fear an invasion from outer-spaceD.they have identified thousands of alien webs3.According to marine ecologists, transplanted marinespecies____________.A.may upset the ecosystems of coastal watersB.are all compatible with one anotherC.can only survive in their home watersD.sometimes disrupt shipping lanes4.The identified cause of the problem is_______.A.the rapidity with which larvae matureB. a common practice of the shipping industryC. a centuries old speciesD.the world wide movement of ocean currents5.The article suggests that a solution to the problem__________.A.is unlikely to be identifiedB.must precede further researchC.is hypothetically假设地，假想地easyD.will limit global shippingText BNew …Endangered‟ List Targets Many US RiversIt is hard to think of a major natural resource or pollution issue in North America today that does not affect rivers.Farm chemical runoff残渣, industrial waste, urban storm sewers, sewage treatment, mining, logging, grazing放牧,military bases, residential and business development, hydropower水力发电,loss of wetlands. The list goes on.Legislation like the Clean Water Act and Wild and Scenic Rivers Act have provided some protection, but threats continue.The Environmental Protection Agency (EPA) reported yesterday that an assessment of 642,000 miles of rivers and streams showed 34 percent in less than good condition. In a major study of the Clean Water Act, the Natural Resources Defense Council last fall reported that poison runoff impairs损害more than 125,000 miles of rivers.More recently, the NRDC and Izaak Walton League warned that pollution and loss of wetlands—made worse by last year’s flooding—is degrading恶化the Mississippi River ecosystem.On Tuesday, the conservation group保护组织American Rivers issued its annual list of 10 ―endangered‖ and 20 ―threatened‖ rivers in 32 states, the District of Colombia, and Canada.At the top of the list is the Clarks Fork of the Yellowstone River, whereCanadian mining firms plan to build a 74-acre英亩reservoir水库，蓄水池as part of a gold mine less than three miles from Yellowstone National Park. The reservoir would hold the runoff from the sulfuric acid 硫酸used to extract gold from crushed rock.―In the event this tailings pond failed, the impact to th e greater Yellowstone ecosystem would be cataclysmic大变动的，灾难性的and the damage irreversible不可逆转的.‖ Sen. Max Baucus of Montana, chairman of the Environment and Public Works Committee, wrote to Noranda Minerals Inc., an owner of the ― New World Mine‖.Last fall, an EPA official expressed concern about the mine and its potential impact, especially the plastic-lined storage reservoir. ― I am unaware of any studies evaluating how a tailings pond尾矿池，残渣池could be maintained to ensure its structural integrity forev er,‖ said Stephen Hoffman, chief of the EPA’s Mining Waste Section. ―It is my opinion that underwater disposal of tailings at New World may present a potentially significant threat to human health and the environment.‖The results of an environmental-impact statement, now being drafted by the Forest Service and Montana Department of State Lands, could determine the mine’s future…In its recent proposal to reauthorize the Clean Water Act, the Clinton administration noted ―dramatically improved water quality since 1972,‖ when the act was passed. But it also reported that 30 percent of riverscontinue to be degraded, mainly by silt泥沙and nutrients from farm and urban runoff, combined sewer overflows, and municipal sewage城市污水. Bottom sediments沉积物are contaminated污染in more than 1,000 waterways, the administration reported in releasing its proposal in January. Between 60 and 80 percent of riparian corridors (riverbank lands) have been degraded.As with endangered species and their habitats in forests and deserts, the complexity of ecosystems is seen in rivers and the effects of development----beyond the obvious threats of industrial pollution, municipal waste, and in-stream diversions改道to slake消除the thirst of new communities in dry regions like the Southwes t…While there are many political hurdles障碍ahead, reauthorization of the Clean Water Act this year holds promise for US rivers. Rep. Norm Mineta of California, who chairs the House Committee overseeing the bill, calls it ―probably the most important env ironmental legislation this Congress will enact.‖ (553 words)6.According to the passage, the Clean Water Act______.A.has been ineffectiveB.will definitely be renewedC.has never been evaluatedD.was enacted some 30 years ago7.“Endangered” rivers are _________.A.catalogued annuallyB.less polluted than ―threatened rivers‖C.caused by floodingD.adjacent to large cities8.The “cataclysmic” event referred to in paragraph eight would be__________.A. fortuitous偶然的，意外的B. adventitious外加的，偶然的C. catastrophicD. precarious不稳定的，危险的9. The owners of the New World Mine appear to be______.A. ecologically aware of the impact of miningB. determined to construct a safe tailings pondC. indifferent to the concerns voiced by the EPAD. willing to relocate operations10. The passage conveys the impression that_______.A. Canadians are disinterested in natural resourcesB. private and public environmental groups aboundC. river banks are erodingD. the majority of US rivers are in poor conditionText CA classic series of experiments to determine the effects ofoverpopulation on communities of rats was reported in February of 1962 in an article in Scientific American. The experiments were conducted by a psychologist, John B. Calhoun and his associates. In each of these experiments, an equal number of male and female adult rats were placed in an enclosure and given an adequate supply of food, water, and other necessities. The rat populations were allowed to increase. Calhoun knew from experience approximately how many rats could live in the enclosures without experiencing stress due to overcrowding. He allowed the population to increase to approximately twice this number. Then he stabilized the population by removing offspring that were not dependent on their mothers. He and his associates then carefully observed and recorded behavior in these overpopulated communities. At the end of their experiments, Calhoun and his associates were able to conclude that overcrowding causes a breakdown in the normal social relationships among rats, a kind of social disease. The rats in the experiments did not follow the same patterns of behavior as rats would in a community without overcrowding.The females in the rat population were the most seriously affected by the high population density: They showed deviant异常的maternal behavior; they did not behave as mother rats normally do. In fact, many of the pups幼兽，幼崽, as rat babies are called, died as a result of poor maternal care. For example, mothers sometimes abandoned their pups,and, without their mothers' care, the pups died. Under normal conditions, a mother rat would not leave her pups alone to die. However, the experiments verified that in overpopulated communities, mother rats do not behave normally. Their behavior may be considered pathologically 病理上，病理学地diseased.The dominant males in the rat population were the least affected by overpopulation. Each of these strong males claimed an area of the enclosure as his own. Therefore, these individuals did not experience the overcrowding in the same way as the other rats did. The fact that the dominant males had adequate space in which to live may explain why they were not as seriously affected by overpopulation as the other rats. However, dominant males did behave pathologically at times. Their antisocial behavior consisted of attacks on weaker male,female, and immature rats. This deviant behavior showed that even though the dominant males had enough living space, they too were affected by the general overcrowding in the enclosure.Non-dominant males in the experimental rat communities also exhibited deviant social behavior. Some withdrew completely; they moved very little and ate and drank at times when the other rats were sleeping in order to avoid contact with them. Other non-dominant males were hyperactive; they were much more active than is normal, chasing other rats and fighting each other. This segment of the rat population, likeall the other parts, was affected by the overpopulation.The behavior of the non-dominant males and of the other components of the rat population has parallels in human behavior. People in densely populated areas exhibit deviant behavior similar to that of the rats in Calhoun's experiments. In large urban areas such as New York City, London, Mexican City, and Cairo, there are abandoned children. There are cruel, powerful individuals, both men and women. There are also people who withdraw and people who become hyperactive. The quantity of other forms of social pathology such as murder, rape, and robbery also frequently occur in densely populated human communities. Is the principal cause of these disorders overpopulation? Calhoun’s experiments suggest that it might be. In any case, social scientists and city planners have been influenced by the results of this series of experiments.11. Paragraph l is organized according to__________.A. reasonsB. descriptionC. examplesD. definition12．Calhoun stabilized the rat population_________.A. when it was double the number that could live in the enclosure without stressB. by removing young ratsC. at a constant number of adult rats in the enclosureD. all of the above are correct13.W hich of the following inferences CANNOT be made from theinformation inPara. 1?A. Calhoun's experiment is still considered important today.B. Overpopulation causes pathological behavior in rat populations.C. Stress does not occur in rat communities unless there is overcrowding.D. Calhoun had experimented with rats before.14. Which of the following behavior didn‟t happen in this experiment?A. All the male rats exhibited pathological behavior.B. Mother rats abandoned their pups.C. Female rats showed deviant maternal behavior.D. Mother rats left their rat babies alone.15. The main idea of the paragraph three is that __________.A. dominant males had adequate living spaceB. dominant males were not as seriously affected by overcrowding as the otherratsC. dominant males attacked weaker ratsD. the strongest males are always able to adapt to bad conditionsText DThe first mention of slavery in the statutes法令，法规of the English colonies of North America does not occur until after 1660—some forty years after the importation of the first Black people. Lest we think that existed in fact before it did in law, Oscar and Mary Handlin assure us, that the status of B lack people down to the 1660’s was that of servants. A critique批判of the Handlins’ interpretation of why legal slavery did not appear until the 1660’s suggests that assumptions about the relation between slavery and racial prejudice should be reexamined, and that explanation for the different treatment of Black slaves in North and South America should be expanded.The Handlins explain the appearance of legal slavery by arguing that, during the 1660’s, the position of white servants was improving relative to that of black servants. Thus, the Handlins contend, Black and White servants, heretofore treated alike, each attained a different status. There are, however, important objections to this argument. First, the Handlins cannot adequately demonstrate that t he White servant’s position was improving, during and after the 1660’s; several acts of the Maryland and Virginia legislatures indicate otherwise. Another flaw in the Handlins’ interpretation is their assumption that prior to the establishment of legal slavery there was no discrimination against Black people. It is true that before the 1660’s Black people were rarely called slaves. But this shouldnot overshadow evidence from the 1630’s on that points to racial discrimination without using the term slavery. Such discrimination sometimes stopped short of lifetime servitude or inherited status—the two attributes of true slavery—yet in other cases it included both. The Handlins’ argument excludes the real possibility that Black people in the English colonies were never treated as the equals of White people.The possibility has important ramifications后果，影响.If from the outset Black people were discriminated against, then legal slavery should be viewed as a reflection and an extension of racial prejudice rather than, as many historians including the Handlins have argued, the cause of prejudice. In addition, the existence of discrimination before the advent of legal slavery offers a further explanation for the harsher treatment of Black slaves in North than in South America. Freyre and Tannenbaum have rightly argued that the lack of certain traditions in North America—such as a Roman conception of slavery and a Roman Catholic emphasis on equality— explains why the treatment of Black slaves was more severe there than in the Spanish and Portuguese colonies of South America. But this cannot be the whole explanation since it is merely negative, based only on a lack of something. A more compelling令人信服的explanation is that the early and sometimes extreme racial discrimination in the English colonies helped determine the particular nature of the slavery that followed. (462 words)16. Which of the following is the most logical inference to be drawn from the passage about the effects of “several acts of the Maryland and Virginia legislatures” (Para.2) passed during and after the 1660‟s?A. The acts negatively affected the pre-1660’s position of Black as wellas of White servants.B. The acts had the effect of impairing rather than improving theposition of White servants relative to what it had been before the 1660’s.C. The acts had a different effect on the position of white servants thandid many of the acts passed during this time by the legislatures of other colonies.D. The acts, at the very least, caused the position of White servants toremain no better than it had been before the 1660’s.17. With which of the following statements regarding the status ofBlack people in the English colonies of North America before the 1660‟s would the author be LEAST likely to agree?A. Although black people were not legally considered to be slaves,they were often called slaves.B. Although subject to some discrimination, black people had a higherlegal status than they did after the 1660’s.C. Although sometimes subject to lifetime servitude, black peoplewere not legally considered to be slaves.D. Although often not treated the same as White people, black people,like many white people, possessed the legal status of servants.18. According to the passage, the Handlins have argued which of thefollowing about the relationship between racial prejudice and the institution of legal slavery in the English colonies of North America?A. Racial prejudice and the institution of slavery arose simultaneously.B. Racial prejudice most often the form of the imposition of inheritedstatus, one of the attributes of slavery.C. The source of racial prejudice was the institution of slavery.D. Because of the influence of the Roman Catholic Church, racialprejudice sometimes did not result in slavery.19. The passage suggests that the existence of a Roman conception ofslavery in Spanish and Portuguese colonies had the effect of _________.A. extending rather than causing racial prejudice in these coloniesB. hastening the legalization of slavery in these colonies.C. mitigating some of the conditions of slavery for black people in these coloniesD. delaying the introduction of slavery into the English colonies20. The author considers the explanation put forward by Freyre andTannenbaum for the treatment accorded B lack slaves in the English colonies of North America to be _____________.A. ambitious but misguidedB. valid有根据的but limitedC. popular but suspectD. anachronistic过时的，时代错误的and controversialUNIT 2Text AThe sea lay like an unbroken mirror all around the pine-girt, lonely shores of Orr’s Island. Tall, kingly spruce s wore their regal王室的crowns of cones high in air, sparkling with diamonds of clear exuded gum流出的树胶; vast old hemlocks铁杉of primeval原始的growth stood darkling in their forest shadows, their branches hung with long hoary moss久远的青苔;while feathery larches羽毛般的落叶松,turned to brilliant gold by autumn frosts, lighted up the darker shadows of the evergreens. It was one of those hazy朦胧的, calm, dissolving days of Indian summer, when everything is so quiet that the fainest kiss of the wave on the beach can be heard, and white clouds seem to faint into the blue of the sky, and soft swathing一长条bands of violet vapor make all earth look dreamy, and give to the sharp, clear-cut outlines of the northern landscape all those mysteries of light and shade which impart such tenderness to Italian scenery.The funeral was over,--- the tread鞋底的花纹/ 踏of many feet, bearing the heavy burden of two broken lives, had been to the lonely graveyard, and had come back again,--- each footstep lighter and more unconstrained不受拘束的as each one went his way from the great old tragedy of Death to the common cheerful of Life.The solemn black clock stood swaying with its eternal ―tick-tock, tick-tock,‖ in the kitchen of the brown house on Orr’s Island. There was there that sense of a stillness that can be felt,---such as settles down on a dwelling住处when any of its inmates have passed through its doors for the last time, to go whence they shall not return. The best room was shut up and darkened, with only so much light as could fall through a little heart-shaped hole in the window-shutter,---for except on solemn visits, or prayer-meetings or weddings, or funerals, that room formed no part of the daily family scenery.The kitchen was clean and ample, hearth灶台, and oven on one side, and rows of old-fashioned splint-bottomed chairs against the wall. A table scoured to snowy whiteness, and a little work-stand whereon lay the Bible, the Missionary Herald, and the Weekly Christian Mirror, before named, formed the principal furniture. One feature, however, must not be forgotten, ---a great sea-chest水手用的储物箱,which had been the companion of Zephaniah through all the countries of the earth. Old, and battered破旧的，磨损的, and unsightly难看的it looked, yet report said that there was good store within which men for the most part respect more than anything else; and, indeed it proved often when a deed of grace was to be done--- when a woman was suddenly made a widow in a coast gale大风，狂风, or a fishing-smack小渔船was run down in the fogs off the banks, leaving in some neighboring cottage a family of orphans,---in all such cases, the opening of this sea-chest was an event of good omen 预兆to the bereaved丧亲者;for Zephaniah had a large heart and a large hand, and was apt有…的倾向to take it out full of silver dollars when once it went in. So the ark of the covenant约柜could not have been looked on with more reverence崇敬than the neighbours usually showed to Captain Pennel’s sea-chest.1. The author describes Orr‟s Island in a(n)______way.A.emotionally appealing, imaginativeB.rational, logically preciseC.factually detailed, objectiveD.vague, uncertain2.According to the passage, the “best room”_____.A.has its many windows boarded upB.has had the furniture removedC.is used only on formal and ceremonious occasionsD.is the busiest room in the house3.From the description of the kitchen we can infer that thehouse belongs to people who_____.A.never have guestsB.like modern appliancesC.are probably religiousD.dislike housework4.The passage implies that_______.A.few people attended the funeralB.fishing is a secure vocationC.the island is densely populatedD.the house belonged to the deceased5.From the description of Zephaniah we can see thathe_________.A.was physically a very big manB.preferred the lonely life of a sailorC.always stayed at homeD.was frugal and saved a lotText BBasic to any understanding of Canada in the 20 years after the Second World War is the country' s impressive population growth. For every three Canadians in 1945, there were over five in 1966. In September 1966 Canada's population passed the 20 million mark. Most of this surging growth came from natural increase. The depression of the 1930s and the war had held back marriages, and the catching-up process began after 1945. The baby boom continued through the decade of the 1950s, producing a population increase of nearly fifteen percent in the five years from 1951 to 1956. This rate of increase had been exceeded only once before in Canada's history, in the decade before 1911 when the prairies were being settled. Undoubtedly, the good economic conditions of the 1950s supported a growth in the population, but the expansion also derived from a trend toward earlier marriages and an increase in the average size of families; In 1957 the Canadian birth rate stood at 28 per thousand, one of the highest in the world. After the peak year of 1957, thebirth rate in Canada began to decline. It continued falling until in 1966 it stood at the lowest level in 25 years. Partly this decline reflected the low level of births during the depression and the war, but it was also caused by changes in Canadian society. Young people were staying at school longer, more women were working; young married couples were buying automobiles or houses before starting families; rising living standards were cutting down the size of families. It appeared that Canada was once more falling in step with the trend toward smaller families that had occurred all through theWestern world since the time of the Industrial Revolution. Although the growth in Canada’s population had slowed down by 1966 (the cent), another increase in the first half of the 1960s was only nine percent), another large population wave was coming over the horizon. It would be composed of the children of the children who were born during the period of the high birth rate prior to 1957.6. What does the passage mainly discuss?A. Educational changes in Canadian society.B. Canada during the Second World War.C. Population trends in postwar Canada.D. Standards of living in Canada.7. According to the passage, when did Canada's baby boom begin?A. In the decade after 1911.B. After 1945.C. During the depression of the 1930s.D. In 1966.8. The author suggests that in Canada during the 1950s____________.A. the urban population decreased rapidlyB. fewer people marriedC. economic conditions were poorD. the birth rate was very high9. When was the birth rate in Canada at its lowest postwar level?A. 1966.B. 1957.C. 1956.D. 1951.10. The author mentions all of the following as causes of declines inpopulation growth after 1957 EXCEPT_________________.A. people being better educatedB. people getting married earlierC. better standards of livingD. couples buying houses11.I t can be inferred from the passage that before the IndustrialRevolution_______________.A. families were largerB. population statistics were unreliableC. the population grew steadilyD. economic conditions were badText CI was just a boy when my father brought me to Harlem for the first time, almost 50 years ago. We stayed at the hotel Theresa, a grand brick structure at 125th Street and Seventh avenue. Once, in the hotel restaurant, my father pointed out Joe Louis. He even got Mr. Brown, the hotel manager, to introduce me to him, a bit punchy强力的but still champ焦急as fast as I was concerned.Much has changed since then. Business and real estate are booming. Some say a new renaissance is under way. Others decry责难what they see as outside forces running roughshod肆意践踏over the old Harlem. New York meant Harlem to me, and as a young man I visited it whenever I could. But many of my old haunts are gone. The Theresa shut down in 1966. National chains that once ignored Harlem now anticipate yuppie money and want pieces of this prime Manhattan real estate. So here I am on a hot August afternoon, sitting in a Starbucks that two years ago opened a block away from the Theresa, snatching抓取，攫取at memories between sips of high-priced coffee. I am about to open up a piece of the old Harlem---the New York Amsterdam News---when a tourist。

Baroclinic Vorticity Production in Protoplanetary Disks Part II Vortex Growth and Longevity

a r X i v :a s t r o -p h /0611526v 2 5 D e c 2006Baroclinic Vorticity Production in Protoplanetary DisksPart II:Vortex Growth and LongevityMark R.PetersenLos Alamos National LaboratoryComputer and Computational Science Div.and Center for Nonlinear Studiesmpetersen@Glen R.StewartLaboratory for Atmospheric and Space Physics,University of Colorado at BoulderKeith JulienDept.of Applied Mathematics,University of Colorado at BoulderABSTRACTThe factors aﬀecting vortex growth in convectively stable protoplanetarydisks are explored using numerical simulations of a two-dimensional anelastic-gas model which includes baroclinic vorticity production and radiative cooling.The baroclinic feedback,where anomalous temperature gradients produce vor-ticity through the baroclinic term and vortices then reinforce these temperature gradients,is found to be an important process in the rate of growth of vortices in the disk.Factors which strengthen the baroclinic feedback include fast radiative cooling,high thermal diﬀusion,and large radial temperature gradients in the background temperature.When the baroclinic feedback is suﬃciently strong,anticyclonic vortices form from initial random perturbations and maintain their strength for the duration of the simulation,for over 600orbital periods.Based on both simulations and a simple vortex model,we ﬁnd that the localangular momentum transport due to a single vortex may be inward or outward,depending its orientation.The global angular momentum transport is highly variable in time,and is sometimes negative and sometimes positive.This result is for an anelastic gas model,and does not include shocks that could aﬀect angular momentum transport in a compressible-gas disk.Subject headings:accretion,accretion disks,circumstellar matter,hydrodynam-ics,instabilities,methods:numerical,turbulence,solar system:formation1.IntroductionThe baroclinic term is a source of vorticity in the vorticity equation,and is derived by taking the curl of the pressure gradient in the Navier-Stokes equation,∇× −1ρ2∇ρ×∇p,(1)where p is the pressure andρis the density.The baroclinic term is nonzero when pressure and density gradients are not aligned.An intuitive example of baroclinicity is the land-sea breeze,which is initiated when air temperatures above the land rise more than over the nearby ocean.The warm air over the land expands,isobars rise relative to those over the ocean,and consequently the isobars tilt towards the ocean.At the same time,the colder air over the ocean has a higher density than over the land,so the isopycnals tilt towards the land.The tilting of isobars and isopycnals in opposite directions is a baroclinic source of vorticity,which causes a circulation in the vertical plane that blows from the ocean to the land near the surface.Thus the potential energy of the tilted isopycnals is converted into kinetic energy of the land-sea breeze,which dissipates through surface friction and reduces the land-sea temperature contrast through temperature advection(see,e.g.Holton2004).A related concept is the baroclinic instability,which is of central importance to the production of vortices and Rossby waves at midlatitudes.Here the decrease in solar insolation from equator to pole causes colder temperatures,and consequently higher density,at the surface at higher latitudes;thus the isopycnal surfaces are tilted towards the equator.A system with tilted isopycnals has more potential energy than one with level isopycnals,just like an inclined free surface has more potential energy than a level one.This potential energy is available to processes which canﬂatten out the isopycnals.For example,vortices in the atmosphere and ocean convert the potential energy of the inclined isopycnals to the kinetic energy of their meso-scale motion.Vorticesﬂatten the isopycnals by transferring heat poleward through their mixing action.The baroclinic instability is so-named because of the tilted isopycnals,but the physics is fundamentally diﬀerent from the land-sea breeze:in the land-sea breeze the circulation is in the vertical plane and caused directly by the baroclinic term,i.e.by non-aligned density and pressure gradients in the vertical;in the baroclinic instability the isopycnals are titled in the vertical,but the vortices are in the horizontal plane,so they could not be produced by the baroclinic term directly.Rather,the tilted isopycnals present an unstable conﬁguration which is ripe for processes which can convert the potential energy to kinetic energy,much like how an avalanche levels out a steep incline of snow.The baroclinic processes discussed in this paper for a protoplanetary disk are similar to the land-sea breeze,but in radial geometry.Due to the gravity and radiation of the central star,the density,temperature,and pressure of the disk’s gas all decrease radially. Any azimuthal variations in temperature(and thus density or pressure by the ideal gas law) would lead to an increase in vertical vorticity due to the baroclinic term(1).The focus of this work is the baroclinic feedback,where a vortex enhances azimuthal temperature gradients to reinforce the vortex itself.Under the right conditions,the baroclinic feedback strengthens vortices so that they can exist for long periods.These vortices could play a crucial role in planetary formation,as they are eﬃcient at collecting particles from the disk(Tanga et al. 1996;Johansen et al.2004;Barge&Sommeria1995;Klahr&Bodenheimer2006).The high density of solids in the vortex would speed the formation by core accretion,which is so slow in the rest of the disk that it may not be a feasible theory of planetary formation there (Wetherill1990).A vortex which collects solids is also a potential site of gravitational instability,where the matter is dense enough that it simply collapses into a planet through gravitation self-attraction(Boss1997).Our study was motivated by Klahr&Bodenheimer(2003),who investigated the eﬀects of baroclinicity in a radially stratiﬁed disk using aﬁnite diﬀerence model of the compressible Navier-Stokes equation combined with a radiative transfer model.They found the baroclinic instability to be a source of vigorous turbulence which leads to the formation of long-lasting vortices and positive angular momentum transport.Barotropic simulations,where the en-tropy(temperature)is constant in the radial direction did not develop turbulence,even with large initial perturbations.To explain these results,Klahr(2004)performed a local linear analysis for a disk with constant surface density,and found that modes do not grow if the growth time of the instability is longer than the shear time.The issue of whether the baroclinic instability is a mechanism for nonlinear growth and the formation of vortices has been a recent source of debate.Johnson and Gammie are critical of theﬁndings of Klahr&Bodenheimer(2003)and Klahr(2004).Their linear analysis found no exponentially growing instabilities,except for convective instabilities in the absence of shear(Johnson&Gammie2005a).Furthermore,they use a shearing sheet numerical model to show that disks with a nearly-Keplerian rotation proﬁle and radial gradients on the order of the disk radius are stable to local nonaxisymmetric disturbances(Johnson&Gammie 2006).The goal of this study is to understand the eﬀects of baroclinic instabilities and ra-diative cooling on the generation of turbulence,vortex formation,and vortex longevity in protoplanetary disks.One of our motivations is to shed light on the conﬂicting observations of baroclinic instabilities by Klahr&Bodenheimer(2003)and Johnson&Gammie(2006).This work is presented in two parts.Part I,which precedes this article,presents the equation set,details of the numerical model,and results of the small-domain simulations,which are used to study the process of vortex formation.This paper,Part II,explores the parameters which aﬀect the baroclinic feedback during the growth phase of the vortices;these simulations use the larger quarter annulus domain and are run for hundreds of orbital periods to observe the long-term behavior of the vortices.We begin with a quick review of the equation set in §2.The results in §3discuss the evolution of a typical simulation,the process of the baroclinic feedback,the Richardson number as a diagnostic,and the alpha viscosity.In §4we discuss the angular momentum transport in our simulations,which is highly variable and depends on the orientation of individual vortices.In §5we conclude that the baroclinic instability is an important mechanism for vortex generation and persistence,and review the conditions which aﬀect the instability.For conciseness there is little repetition between Parts I and II,so the reader is advised to read both together.2.Description of the Equation SetThe model equations are described fully in Part I of this work,and are only brieﬂy reviewed here.They model an anelastic gas,which ﬁlter out pressure waves that restrict the timestep of the numerical model but do not impact the physics of interest here.Our equations set is similar to those in Bannon (1996)and Scinocca &Shepherd (1992),which are anelastic models of the atmosphere derived from conservation of momentum,conservation of mass,the second law of thermodynamics,and the ideal gas law.Our equations use two-dimensional polar coordinates (r,φ)where temperature and density are stratiﬁed in the radial direction.Variables such as the vertical component of vorticity ζ,streamfunction Ψ,potential temperature θ,thermal temperature T ,surface density Σ,and Exner pressure πare written as the sum of a background and perturbation term,e.g.θ=θ0(r )+θ′(r,φ,t ),where the background functions only vary radially.The model equations areζ′=1∂r r ∂r +1∂φ2(2)∂ζ′Σ0ζ =c p ∂r∂θ′∂t +1τ+κe ∇2θ′.(4)The ﬁrst is the relationship between the perturbation streamfunction Φ′and the perturbation vorticity ζ′;the second and third are prognostic equations for perturbation vorticity ζ′andperturbation potential temperatureθ′.Radial and azimuthal velocities u=(u,v)are related to the streamfunction byΣ0u=−∇×Ψˆz.Other variables include the radiative cooling time τ,speciﬁc heat at constant pressure c p,time t,viscosityνe,thermal dissipationκe,vertical unit vectorˆz,and the Jacobian∂(a,b)=(∂r a∂φb−∂φa∂r b)/r.The baroclinic term,c p∂r∂θ′r ∂dr =c p∂φdπ0∂r∼∂Σ0T0∂r+T0∂Σ0νe t sc,P e=L2sc(Table1).These simulations capture the salient features of the physics of the anelastic equation set.The topic of Part I was vortex formation,and thus used a smaller domain for onlyﬁve orbital periods.Here we are interested in vortex growth and longevity due to the baroclinic feedback,and have chosen a larger domain and durations of300to600 orbital periods.(This is6,200to12,400years for a solar-mass star.)The simulations were performed on the quarter annulus with a radial extent from5AU to10AU and a resolution of256×256and512×512gridpoints.The background surface density and background temperature are constant in time and are power functions in the radial,Σ0(r)=a r r in d,(8) where r in=5AU is the inner radius of the annulus.The coeﬃcients are a=1000g cm−2, c=600K for the quarter annulus domain;b and d are varied and shown in Table1.For example,for simulation A1,the background surface density varies from1000g cm−2to350g cm−2and the background temperature decreases radially from600K to150K.This range of temperatures can only be achieved in a realistic disk when the radius ranges from1AU to 10AU(Boss1998).We have artiﬁcially enhanced the radial temperature gradient in order to compensate for the lower resolution of our global simulations.We have demonstrated in paper I using a higher-resolution local simulation that more realistic temperature gradients can still produce vortices.Most simulations were run to300orbital periods,measured as a full(2π)orbit at r mid=7.5AU.This is6,200years for a solar-mass star.Thermal temperature T and potential temperatureθare related byθ=T p0(r in)π,(9) where R is the gas constant andπis the Exner pressure.All results in this paper are expressed in terms of the thermal temperature T in order to compare to observations.The potential temperature is a measure of entropy.If entropy increases radially(dθ0/dr>0) then the disk is convectively stable—this is the Schwarzschild criterion(Schwarzschild1958). If the entropy gradient as accompanied by diﬀerential rotation,the Solberg-Høiland criterion (Tassoul2000;R¨u diger et al.2002)is used to test convective stability(see Results section of Part I).For the simulations presented in this paper,the Solberg-Høiland value is positive (0.035–0.299years−2),indicating that they are convectively stable.The initial condition for the perturbation temperature is shown in Fig.1.It is created with a speciﬁed wavenumber distribution in Fourier space,transformed to Cartesian coordi-nates,and interpolated to the Fourier-Chebyshev annular grid(see Part I,section3.2).Theinitial vorticity perturbation is created in a similar fashion.The magnitude of the initial conditions is25%of the maximum of the background function.The small domain simulations in Part I(r∈[9.5,10],φ∈[0,π/32])required a much smaller initial perturbation to initiate vortices—a temperature perturbation of only5%, and an initial vorticity perturbation of zero.This is possible because the small domain is a higher resolution relative to the background shear.The sensitivity analysis in Part I showed that smaller initial perturbations are required to initiate vortices with progressively higher resolution and Reynolds number.The same is true of the background temperature; the quarter annulus domain uses higher temperatures(c=600K)and steeper gradients (d=−2)than the small domain simulations.Again,the sensitivity analysis in Part I showed that at higher resolutions vortices can be formed with progressively cooler disks and shallower background gradients.The evolution of a typical simulation can be described as follows:The initial distribution of vorticity shears due to the diﬀerential rotation of the nearly-Keplerian rotational proﬁle (Fig.2).Even at these early times,the perturbation vorticity and perturbation kinetic energy grow due to the baroclinic term(Fig5).After aboutﬁve orbital periods,anticyclonic vortices begin to form,and by ten orbital periods the domain is populated by numerous small anti-cyclones.Cyclonic(anti-cyclonic)ﬂuid rotates in the same(opposite)direction as the backgroundﬂuid,and is denoted by positive(negative)vorticity perturbation in theﬁgures. It is well-known that anti-cyclones can be long-lived in a Keplerian disk,while cyclones shear out into thinﬁlaments that eventually dissipate away(Godon&Livio1999;Marcus1990; Marcus et al.2000).An anticyclonic vortex has a positive azimuthal velocity at small inner radii and a negative azimuthal velocity at large outer radii.This means that anticyclonic vortices can smoothly match the background shearﬂow,and therefore extract energy from the Keplerian shear.Cyclonic vortices cannot smoothly match the background shearﬂow and are therefore sheared apart.After the initial period of vortex formation,the vortices merge and grow in strength (Figs.3,4).This merging behavior is similar to the merging of like-signed vortices in two-dimensional isotropic turbulence,which transfers energy from smaller to larger scales(the inverse cascade).However,in shearingﬂows vortices do not merge as readily and must be suﬃciently close in the radial direction.It is not at all clear that this merging of vortices can occur in a fully three-dimensional disk if the initial radial vortex scale is small compared to the disk scale height.On the other hand,if vortices primarily form on the upper and lower surfaces of a vertical stratiﬁed disk as found by Barranco&Marcus(2005),then it may be possible for small-scale vortices to merge in these surface layers.Further discussion and images of vortex merger,longevity,and distribution can be found in Godon&Livio(1999)and Umurhan&Regev(2004).There is a clear“sandwich”pattern of temperature perturbations around each vortex (Fig.4):the vortex advects warmerﬂuid towards the outside of the disk and coolerﬂuid towards the inside of the disk.In the sandwich analogy,the temperature perturbations are the bread and the vortex is the meat between the bread.These perturbations have azimuthal temperature gradients that play a role in the baroclinic feedback.3.1.Baroclinic vorticity productionThe model equations for vorticity and temperature perturbations are coupled by the baroclinic term in the vorticity equation(3)and the advection term in the temperature equation(4).This coupling is required to support long-lived vortices;without it,vorticity and temperature perturbations simply decay to zero.The baroclinic feedback operates as follows:1.Azimuthal gradients in the perturbation temperatureﬁeld,∂θ′/∂φ,make the baroclinicterm in the vorticity equation non-zero.2.The baroclinic term is a source of vorticity which strengthens anticyclonic vortices.3.Vortices stir theﬂuid,moving warmﬂuid from the inner annulus outward and coolﬂuid from the outer annulus inward.4.This local advective heat transport enhances azimuthal temperature gradients,∂θ′/∂φ,completing the feedback cycle.In order to show that the vortex growth is indeed due to this baroclinic feedback,the baroclinic term was turned oﬀat various times in simulation set B(Fig.5).In all of these trials,perturbation vorticity and kinetic energy drop oﬀimmediately when the baroclinic term is turned oﬀ.This is particularly striking at t=10and t=100,when vortex strength is growing quickly in the reference simulation.The kinetic energy in these plots is computed from the perturbation velocityﬁelds.The rate of thermal dissipation,τ,plays a crucial role in the formation and growth of vortices.Fig.6shows that there are two distinct stages in these simulations:vortex formation,from t=0to about t=5orbital periods,and vortex growth,which occurs after t=5.During vortex formation,small thermal dissipation(largeτ)allows the strongest vortices to form.That is because the initial temperature perturbation dies oﬀquickly whenthermal dissipation is large,so that azimuthal temperature gradients gradients are smaller and the baroclinic term produces less vorticity.This is not yet the baroclinic feedback because steps3and4are missing—it is just baroclinic vorticity production from the initial temperature gradients,which are steps1and2.Once vortices form they advectﬂuid about them(step3),creating the distinctive“sand-wich”pattern of cool(warm)temperature perturbations to the inside(outside)of the vortex, as shown in Fig. 4.These temperature perturbations create local azimuthal temperature gradients(step4),completing the cycle of the baroclinic feedback.Sometime afterﬁve orbital periods,the vortices have formed and the simulation transitions from the vortex formation stage to the vortex growth stage.Now that the baroclinic feedback is operating, thermal dissipation has the opposite eﬀect than at early times(Fig.6).If the disk cools quickly(τsmall),then the warm and cool temperature perturbations can remain tight about each vortex,so that∂θ′/∂φin the baroclinic term is large,and the baroclinic feedback is strong.If the disk cools slowly(τlarge),the perturbation temperature responds sluggishly to mixing by vortices,∂θ′/∂φis small,the baroclinic feedback is weak,and vortices simply dissipate away(Fig.7).Quantitative measures of disk activity like kinetic energy,and max-imum temperature and vorticity clearly show that the strength of the feedback and rate of growth of vortices is strongly dependent onτ(Fig.6).In simulations where the radiative cooling rate was suﬃciently fast,the baroclinic feedback counters dissipation and vortices remain strong and coherent for hundreds of orbital periods(Fig.4).The longest running simulation,T3where d=−0.75,ended at600orbital periods,at which point all of the vortices had merged into a single anticyclonic vortex.There are two dissipative terms in the temperature equation(4):the Laplacian term κe∇2θ′,which dissipates most quickly at small scales;and the radiative cooling term−θ′/τwhich dissipates equally at all scales.Can the Laplacian term play the same role as the radia-tive cooling term in the baroclinic feedback?Simulations Pe1-Pe3,where P e ranges from104 to2×107,show that the Laplacian term can indeed play that role(Fig.8);higher thermal diﬀusion(smaller Peclet number)produces a stronger baroclinic feedback.Higher diﬀusion produces warm and cool areas around each vortex which are more localized azimuthally,and therefore have larger azimuthal temperature gradients(Fig.9).The azimuthal temperature gradients then produce more vorticity through the baroclinic term(step1of the baroclinic feedback).Other simulations explore the role of background temperature(T1-T5)and background surface density(D1-D3).Larger background temperature gradients in simulations T1-T5 result in larger and stronger vortices(Fig.10).Quantitative measures such as the kinetic energy,maximum vorticity,and maximum temperature all grow faster with larger tempera-ture gradients(Fig.11).The evolution of these quantities does not change as the background density gradient is varied(simulations D1-D3,Fig12).It is somewhat surprising that the baroclinic feedback responds strongly to the background temperature gradient but not the background density gradient,when these gradients seem to be on equal footing in the baro-clinic term(see eqn.7).The background temperature gradient is a source of available potential energy that can be transformed into the kinetic energy of vortex motion as the vortices transport heat from the hot inner disk to the cold outer disk.This nonlinear heat advection cannot be captured in a linear stability analysis.Since the surface density is time-independent in our anelastic model,the background surface density gradient cannot providea source of potential energy for vortex formation.3.2.Richardson numberSeveral previous studies have used the Richardson number to characterize instabilities in protoplaneary disks(Johnson&Gammie2005a,2006),and we compute the Richardson number here for comparison.We believe that the Solberg-Høiland criterion(Tassoul(2000); R¨u diger et al.(2002),also see§4of Part I),which was speciﬁcally created for diﬀerentially rotating astrophysicalﬂuids,is the best way to judge whether a disk is convectively unstable. For the simulations presented in this paper,the Solberg-Høiland values are positive(0.035–0.299years−2),indicating that they are all convectively stable.However,the Richardson number also provides useful information about the instability.We found that the baro-clinic feedback is stronger(i.e.vortex growth is faster)in simulations with more negative Richardson numbers.The Richardson number is often evoked in geophysical turbulence to quantify the re-lationship between stratiﬁcation and shear.For the atmosphere this dimensionless ratio is typicallyRi(z)=N2(z)∂z 2=−g dz∂z 2(10)where N(z)is the local Brunt-V¨a is¨a¨a buoyancy frequency,u is the horizontal velocity,z is the vertical coordinate,ρis the density,and g is the gravitational force(Turner1973).The numerator N2gives the strength of stratiﬁcation,where N2is negative for a convectively unstableﬂuid,is positive and small for weakly stable stratiﬁcation,and is positive and large for strongly stable stratiﬁcation.The denominator gives the strength of the shear.In our equation set the Richardson number isRi=N2∂r 2=−c p dθ0dr∂r 2.(11)By comparing the Richardson number(Fig.13)with kinetic energy or maximum vorticity (Fig.11)for simulations T1–T5,it is clear that the Richardson number is an excellent way to predict the strength of the baroclinic feedback.When Ri≥0(T4and T5,where d=−0.5 and−0.25),kinetic energy and vorticity simply decay away.When Ri<0(T1–T3,where d=−2–−0.75),the baroclinic feedback operates and kinetic energy and vorticity grow.In fact,the simulation with the most negative Richardson number(T1,where d=−2)also has the fastest vortex growth.Johnson&Gammie(2006)found that disks with a nearly-Keplerian rotation proﬁle and radial gradients on the order of the disk radius have Ri≥−0.01,and are stable to local nonaxisymmetric disturbances.Our simulations are not restricted to this Ri≥−0.01crite-rion,as simulations T1-T3have quickly-growing instabilities but have Richardson numbers in the range of−5×10−5to−5×10−4.The most likely diﬀerence between the two models that accounts for this disagreement is that our simulation allows small initial temperature perturbations to evolve into strong local vorticity perturbations that can produce stable vor-tices.This initial evolution can only occur if the viscous dissipation is suﬃciently low(high Reynolds number).3.3.Alpha viscosityProtoplanetary disks are often described by the dimensionless numberα,which is used to parameterize an eﬀective viscosityν=αc s H p where H p is the vertical pressure scale height of the disk and c s is the local sound speed.This simple description was used to calculate the density structure,temperature structure,and mean components of laminar and turbulent gasﬂow in a disk(Shakura&Sunyaev1973,Lynden-Bell&Pringle1974,Lin&Papaloizou 1980).Alpha viscosity,rather than Reynolds number,is commonly reported in the astrophysi-cal literature to characterize the dissipation of energy in the disk.If the pressure scale height is scaled as H p=c s/Ω0,whereΩ0is the background angular velocity,the alpha viscosity can be calculated asα(r)=νeΩ0(r)/c2s.In our anelastic model,this measure cannot be used directly because c s>>|u′|and pressure waves are temporally constrained to adjust instan-taneously.In order to compare the alpha viscosity with other protoplanetary disk modelswe report the ratio of the alpha viscosity to the Mach number squared,α|u′|2=˜ΩJohnson&Gammie(2005b)also found positive angular momentumﬂux in their compress-ible shearing sheet model when they used strong initial vorticity perturbations to trigger vortex formation.As a simple example,consider locally Cartesian coordinates in the radial and azimuthal direction.A slanted vortex of this local coordinate system could have the stream function Ψ′=αexp − φr0 2 .(13)Each streamline is a rotated ellipse centered at the origin with radial extent r0and azimuthal extentφ0.The angle of the ellipse is only aﬀected byβ.This vortex is superimposed on some backgroundﬂow,so the perturbation velocities in locally Cartesian coordinates are u′=−∂φΨ′and v′=∂rΨ′.The angular momentum transport of this vortex isΣ0 ∞−∞u′v′dφ=−βα2φ0Σ0√4exp 1r0 2 −4+β2φ20r20 .(14)The sign of this quantity depends only onβ,the angle of the vortex.Positive and negative vortices with the same orientation have the same angular momentum transport,as the sign of the vortex only aﬀectsα,a squared quantity in(14).This indicates that it is only the orientation of the vortex within theﬂow that aﬀects whether momentum travels towards the inside or outside of the disk;the direction of rotation of the vortex is inconsequential.Our simple analytic example is shown in Fig.15forβ=−0.5(top left)andβ=0.5 (top right),where the other constants areφ0=1,r0=2,andα=1.The bottom panels show vortices with similar orientations in the full numerical model.Clearly,the direction of the angular momentum transport only depends on the angle of the vortex,as in the analytic example.These vortices are not from the simulations in Table1,but are from short simulations that were speciﬁcally designed to produce these orientations.What is the eﬀect of vortices on angular momentum transport when they are imbedded in a turbulentﬂow populated withﬁlaments and other interacting vortices?To investigate this,the angular momentum transport,Σ0u′v′,was recorded using azimuthal and global averages in the numerical model.In typical simulations,like A1,the angular momentum transport is highly variable in space and time(Fig.16).Speciﬁcally,the global angular momentum transport cycles chaotically between positive and negative periods as the vor-ticityﬁeld evolves.The angular momentum transport in these simulations is inﬂuenced by the interaction of numerous vortices and vorticityﬁlaments,which is much more compli-cated than the single vortex case.We would expect that individual vortices within thisﬂow would contribute angular momentum based on their orientation,and that these individual。