物理海洋学 运动方程

• We can transform equations of motion from velocity-based into vorticity-based.
f u v u v w v ( f )( ) t x y z y x y x y
Friction (Molecular and Eddy Viscosity)
• See appendix notes for detailed derivations.
du 1 2u 2u 2 2 dt z z
However, this equation may not be often used in the ocean, because the ocean is fully turbulent. Molecular vs Eddy
The equations of Motion
Chapter 7
Dominant Forces
• • • • • Gravity Coriolis Friction Pressure Gradient Force Other forces
– Atmospheric Pressure – Seismic
Inertial Motion
The result is a circular motion. See notes for detailed derivation.
du vf dt
dv uf dt
Coriolis Force
• So, the Coriolis force brings a circular pattern of motion in the ocean. • We will know a term “vorticity” later. • Now we want to know how we get its mathematical expression.
Coriolis Force
• On northern Hemisphere
– Right to the velocity direction
• On southern Hemisphere
– Left to the velocity direction
• Proportional to the velocity
Pressure Gradient
i Δz
z
ρ
z
Δx A B
Pressure Gradient
i Δz
z
ρ
z
Δx A B
PA gZ
PB g (Z Z )
1 P 1 PA PB Z g gi x x x
Pressure Gradient
i
Δz ρ1 z ρ2 Δx A B j z
Total Derivative
particle acceleration = local acceleration + field acceleration
Forces
F=ma a = f, here f=F/m is force per mass
Acceleration=Gravity+Pressure gradient+Coriolis+friction+others Gravity Pressure gradient Coriolis Friction Others
Forces
F=ma a = f, here f=F/m is force per mass
Acceleration=Gravity+Pressure gradient+Coriolis+friction Gravity Pressure gradient Coriolis Friction
Pressure Gradient
du u u u u 1 p u v w other forces dt t x y z x dv v v v v 1 p u v w other forces dt t x y z y dw w w w w 1 p u v w other forces dt t x y z z
Case Study
P E
Si=36.2 0.79 Sv
Vi Vo
So=38.3
Mediterranean Sill
Case Study
P E Δx Si=36.2 0.79 Sv Vi Vo
So=38.3
Mediterranean Sill
Case Study
P E dx
Mediterranean Sill
A case study of Conservation
Goal: Estimate R+P-E
Case Study
P E
Si=36.2 0.79 Sv
Vi Vo
R
So=38.3
Atlantic Ocean Mediterranean Sill
Case Study
Assume the difference between density can be ignored. Vi+R+P = Vo+E ViSi = VoSo Gives R+P-E=-4.6X104m3/s Vo-Vi = R+P-E
Coriolis Force
• See notes for detailed derivations.
d d () () () dt space dt
dR dR R dt space dt
d 2R dR dR d ( R ) ( R) 2 dt space dt dt d 2R d 2R 2 2 R ( R) 2 dt space dt
Continuity Equation
u v w 0 x y z
Vertically Integrated Equation of Continuity
η
h
h0
u v ( )h 0 t x y
Forces
F=ma a = f, here f=F/m is force per mass
Acceleration=Gravity+Pressure gradient+Coriolis+friction+others Gravity Pressure gradient Coriolis Friction Others
• Note that the Coriolis force brings circular motions. • We define vorticity as
v u x y
Vorticity
• Try to understand
v u x y
y
x
Equations of Motion
Equator rotation speed
465m/s
Coriolis Force
• Not a real force! • Observation system influenced by the rotation of earth. • Pseudo force/Apparent force
Coriolis Force
• See notes for detailed derivations.
i j k sin w
2 R 2 0 cos u v
f 2 sin
d 2R du dv dw i ( fv 2 w cos ) j ( fu ) k ( 2u cos ) 2 dt space dt dt dt
Coordinate
N
ቤተ መጻሕፍቲ ባይዱ
z
y
x
y
x
Newton’s second law
du 1 F dt m
if we consider force per volume
du 1 F dt
Momentum Equation
du 1 Fx dt
dv 1 Fy dt
dw 1 Fz dt
Coordinate System
• Cartesian Coordinate System
– f plane – beta plane
• Spherical Coordinates
Conservation
• • • • Conservation Conservation Conservation Conservation of of of of Mass Energy Momentum Angular Momentum
Coriolis Force
y
du vf dt
x
dv uf dt
Check the sign to understand the equation.
Coriolis Force
y
du vf dt
x
dv uf dt
Can you solve these equations? Even if you were not allowed to use mathematics, can you image how the result looks like?
Pressure Gradient
Δz p1 Δy Δx p2
1 p y
See notes for detailed derivation.
Coriolis Force
N
150m/s 356m/s 400m/s From earth From space
Move southward
Eddy Viscosity
du 2u 2u 2u Az 2 Ah ( 2 2 ) dt z x y
where Az and Ah are vertical and horizontal eddy viscosity. Review Reynolds Stress Terms in the textbook.
Coriolis Force
• See notes for detailed derivations.
d d () () () dt space dt
For example, a particle on earth moves around a circle observed from space, but on earth it is static.
dw/dt Fz
Fy Fx
dv/dt du/dt
du/dt
acceleration
B
A
u 0 t u 0 t
Velocity
Time
du/dt
du u u u u u v w dt t x y z dv v v v v u v w dt t x y z dw w w w w u v w dt t x y z
Coriolis Force
• f is the Coriolis parameter.
du u u u u 1 p u v w vf other forces dt t x y z x dv v v v v 1 p u v w uf other forces dt t x y z y
Equations of Motion
u u u u 1 p u v w fv t x y z x
v v v v 1 p u v w fu t x y z y
1 p g z
Vorticity