托卡马克位形优化2
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Nonlinearity of LH wave absorption
The plasma temperature in HL-2A is much lower than that in future reactor. To establish RS configuration, the LH driven current should be located off-axis where the plasma temperature is even lower, and the plasma absorption of high phase velocity LH waves is too weak to ensure the waves are damped during their first pass.
R0 / R n// n// 0 ˆ ) 1 ( pe / ) /(q
ˆ q a/ R . with , where x
10
q cyl
Propagation Domain At the boundary of the propagation domain, (7) As the tokamak equilibrium is toroidal axisymetric, the toroidal mode number n is conserved. From the definition of n// ,
6.5 n// Te [kev] k//c
n// - Upshift Boundary In the simulated quasi-stationary RS discharges, it turns out 19 3 Te01.4kev (Ti02.8kev) with n e 2.32 10 m . In such conditions there is a spectral gap between the parallel LHW phase velocity and the electron thermal velocity.
Here // d / // (d / ) follows from the cold plasma dispersion relation. The damping rate e for electron Landau damping can be written as follows:
Dr k ห้องสมุดไป่ตู้ k k 0
4 2 2 // //
For cold plasmas ( 0) ,
k k ( / ) /
2 2 // 2 pe 2
The consistency condition (3) is satisfied for n /
Fig. 10 Radial profiles of P and |E//|2 for multipass absorption of a single field harmonic (m = 100, n = 450) in the cylindrical approximation. 3
In the tokamak toroidal geometry intrinsic poloidal asymmetry breaks the invariance of m, causing formation of a thick stochastic layer in the ray phase space. In the LHCD discharges on Tore Supra, a regime with stationary oscillation behavior has been observed because of the nonlinearly coupling effect of both wave-plasma interaction and turbulence suppression by the RS q profile. It is interpreted as that the current density and electron temperature profiles behave as a predator-prey system [Giruzzi, G., et al., Phys. Rev. Lett. 91 Fig, 11 Surface-s of section in the (m, ) plane for two parameter sets on Tore Supra. (2003) 135001].
THE CYLINDRICAL APPROXIMATION Appropriate canonical coordinates in tokamak geometry are (x, k)(r, , , kr, m, n). We consider the source S = Pin(2)-2 (r - ro) (kr – kr0), the solution of ( 1) is
9
B B 2 2 n// n n nr , B B
where n is the wave vector component perpendicular to the magnetic field. We are interested in the maximum upshift factor of n// , taking nr 0 , which applies at a radial turning point. In tokamak plasmas, n// n n B / B . By using the cold electrostatic approximation of the dispersion relation,
2 pi , j 2
,
3 4
2 pe 4 ce
v 3
2 Te j
2 2 pi , j Ti , j
v
4
// 1 / , K zz ,i
2 pe 2
2 pe
f e dv// v// v// ( k // v// )
6
If n kc / 1 , a simplified dispersion relation can be found from the matrix equation by asymptotically expanding
0 0 0 E n E
I I nn , so that
7
In first order, there arises the solubility condition:
0 0 n K n 0
This gives the simplified dispersion relation (electrostatic limit)
After determining U( x, k), we can calculate physical quantities such as the (timeaveraged) absorbed power density,
1
The (time-averaged) energy density in the rf parallel electric field,
0 1 1 1 n ~ n 2 n o 4 n n ,
With the assumption
0 1 1 1 E ~ E 2 E o 4 n n
0 1 E KE
0 E n
2
,
(3) where
0 to the lowest order, I E 0 ,
(2)
5
For LH waves,
ci ce
2 pe
K xx K yy k , K yx K xy i xy i , ce
K zz // iK zz ,i
where
1 j
2 pe 2 ce
4
LH wave absorption in a quasi-stationary RS plasma I. Simplified dispersion relation When the WKB approximation is valid, the wave matrix equation is (1)
2 // 2 pe
2 ce
(~0.5-0.12)
8
LH wave absorption regime
Strong Landau Damping Limit If the LH wave phase velocity is higher than 3.5 times the electron thermal velocity, there are too few velocity-resonant electrons to carry driven current density comparable with the ohmic current density.
2 2 [kk I k k0 K (r , k ,)] E 0
wher k0 / c , K is the dielectric tensor. For a non-
trivial solution, 2 2 D( , k , r ) [k k I k k0 K (r , k , )] 0
In the weak electron Landau damping condition LH wave rays make many passes through the wave propagation domain in plasma and undergo numerous reflections at the propagation boundaries. Considering the propagation of lower-hybrid waves in a tokamak. The phase space energy density of the rf field is denoted by U( x, k, t), where x is the position vector, k is the wave vector. U obeys the WKE,
where
r 2 dr2 ( r ) dr ( r) y( r ) , with r1 r1 v ( r) vr ( r ) r
1
r
The absorbed power density,
2
The above solution of the WKE can be classified into two distinct para-meter regimes: (i) the multipass regime (for 1), when U tends to be uniform along the entire ray orbit in the (r, k) plane, and (ii) the singlepass regime (for > 5) when nearly all the power is absorbed before the ray reaches the caustic. In the multi-pass regime, the absorption due to electron Landau damping is strongly peaked at the caustic.
The plasma temperature in HL-2A is much lower than that in future reactor. To establish RS configuration, the LH driven current should be located off-axis where the plasma temperature is even lower, and the plasma absorption of high phase velocity LH waves is too weak to ensure the waves are damped during their first pass.
R0 / R n// n// 0 ˆ ) 1 ( pe / ) /(q
ˆ q a/ R . with , where x
10
q cyl
Propagation Domain At the boundary of the propagation domain, (7) As the tokamak equilibrium is toroidal axisymetric, the toroidal mode number n is conserved. From the definition of n// ,
6.5 n// Te [kev] k//c
n// - Upshift Boundary In the simulated quasi-stationary RS discharges, it turns out 19 3 Te01.4kev (Ti02.8kev) with n e 2.32 10 m . In such conditions there is a spectral gap between the parallel LHW phase velocity and the electron thermal velocity.
Here // d / // (d / ) follows from the cold plasma dispersion relation. The damping rate e for electron Landau damping can be written as follows:
Dr k ห้องสมุดไป่ตู้ k k 0
4 2 2 // //
For cold plasmas ( 0) ,
k k ( / ) /
2 2 // 2 pe 2
The consistency condition (3) is satisfied for n /
Fig. 10 Radial profiles of P and |E//|2 for multipass absorption of a single field harmonic (m = 100, n = 450) in the cylindrical approximation. 3
In the tokamak toroidal geometry intrinsic poloidal asymmetry breaks the invariance of m, causing formation of a thick stochastic layer in the ray phase space. In the LHCD discharges on Tore Supra, a regime with stationary oscillation behavior has been observed because of the nonlinearly coupling effect of both wave-plasma interaction and turbulence suppression by the RS q profile. It is interpreted as that the current density and electron temperature profiles behave as a predator-prey system [Giruzzi, G., et al., Phys. Rev. Lett. 91 Fig, 11 Surface-s of section in the (m, ) plane for two parameter sets on Tore Supra. (2003) 135001].
THE CYLINDRICAL APPROXIMATION Appropriate canonical coordinates in tokamak geometry are (x, k)(r, , , kr, m, n). We consider the source S = Pin(2)-2 (r - ro) (kr – kr0), the solution of ( 1) is
9
B B 2 2 n// n n nr , B B
where n is the wave vector component perpendicular to the magnetic field. We are interested in the maximum upshift factor of n// , taking nr 0 , which applies at a radial turning point. In tokamak plasmas, n// n n B / B . By using the cold electrostatic approximation of the dispersion relation,
2 pi , j 2
,
3 4
2 pe 4 ce
v 3
2 Te j
2 2 pi , j Ti , j
v
4
// 1 / , K zz ,i
2 pe 2
2 pe
f e dv// v// v// ( k // v// )
6
If n kc / 1 , a simplified dispersion relation can be found from the matrix equation by asymptotically expanding
0 0 0 E n E
I I nn , so that
7
In first order, there arises the solubility condition:
0 0 n K n 0
This gives the simplified dispersion relation (electrostatic limit)
After determining U( x, k), we can calculate physical quantities such as the (timeaveraged) absorbed power density,
1
The (time-averaged) energy density in the rf parallel electric field,
0 1 1 1 n ~ n 2 n o 4 n n ,
With the assumption
0 1 1 1 E ~ E 2 E o 4 n n
0 1 E KE
0 E n
2
,
(3) where
0 to the lowest order, I E 0 ,
(2)
5
For LH waves,
ci ce
2 pe
K xx K yy k , K yx K xy i xy i , ce
K zz // iK zz ,i
where
1 j
2 pe 2 ce
4
LH wave absorption in a quasi-stationary RS plasma I. Simplified dispersion relation When the WKB approximation is valid, the wave matrix equation is (1)
2 // 2 pe
2 ce
(~0.5-0.12)
8
LH wave absorption regime
Strong Landau Damping Limit If the LH wave phase velocity is higher than 3.5 times the electron thermal velocity, there are too few velocity-resonant electrons to carry driven current density comparable with the ohmic current density.
2 2 [kk I k k0 K (r , k ,)] E 0
wher k0 / c , K is the dielectric tensor. For a non-
trivial solution, 2 2 D( , k , r ) [k k I k k0 K (r , k , )] 0
In the weak electron Landau damping condition LH wave rays make many passes through the wave propagation domain in plasma and undergo numerous reflections at the propagation boundaries. Considering the propagation of lower-hybrid waves in a tokamak. The phase space energy density of the rf field is denoted by U( x, k, t), where x is the position vector, k is the wave vector. U obeys the WKE,
where
r 2 dr2 ( r ) dr ( r) y( r ) , with r1 r1 v ( r) vr ( r ) r
1
r
The absorbed power density,
2
The above solution of the WKE can be classified into two distinct para-meter regimes: (i) the multipass regime (for 1), when U tends to be uniform along the entire ray orbit in the (r, k) plane, and (ii) the singlepass regime (for > 5) when nearly all the power is absorbed before the ray reaches the caustic. In the multi-pass regime, the absorption due to electron Landau damping is strongly peaked at the caustic.