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本文意在介绍发生在海洋中的动力过程的方程组，阅读本文需要基本的牛顿力学知识即可 4 \0 x9 c4 l$ u5 n4 l
动量方程E1-E3 8 h: `: d- ~5 `) w) i
E1:∂u/∂t+u∂u/∂x+v∂u/∂y+w∂u/∂z=−1/ρ⋅∂p/∂x+fv+υΔu+∂(AH∂u/∂x)/∂x+∂(AH∂u/∂y)/∂y+∂(Az∂u/∂z)/∂z+FxE1:\partial u/\partial t+u\partial u/\partial x+v\partial u/\partial y+w\partial u/\partial z=-1/\rho\cdot\partial p/\partial x+fv+\upsilon\Delta u+\partial (A_H \partial u/\partial x)/\partial x+\partial (A_H \partial u/\partial y)/\partial y+\partial (A_z \partial u/\partial z)/\partial z+F_x
, q U; k0 K9 y# m' W E2:∂v/∂t+u∂v/∂x+v∂v/∂y+w∂v/∂z=−1/ρ⋅∂p/∂y−fu+υΔv+∂(AH∂v/∂x)/∂x+∂(AH∂v/∂y)/∂y+∂(Az∂v/∂z)/∂z+FyE2:\partial v/\partial t+u\partial v/\partial x+v\partial v/\partial y+w\partial v/\partial z=-1/\rho\cdot\partial p/\partial y-fu+\upsilon\Delta v+\partial (A_H \partial v/\partial x)/\partial x+\partial (A_H \partial v/\partial y)/\partial y+\partial (A_z \partial v/\partial z)/\partial z+F_y 4 a1 A) m$ p/ U; ~
E3:∂w/∂t+u∂w/∂x+v∂w/∂y+w∂w/∂z=g−1/ρ⋅∂p/∂z+υΔw+∂(AH∂w/∂x)/∂x+∂(AH∂w/∂y)/∂y+∂(Az∂w/∂z)/∂z+FzE3:\partial w/\partial t+u\partial w/\partial x+v\partial w/\partial y+w\partial w/\partial z=g-1/\rho\cdot\partial p/\partial z+\upsilon\Delta w+\partial (A_H \partial w/\partial x)/\partial x+\partial (A_H \partial w/\partial y)/\partial y+\partial (A_z \partial w/\partial z)/\partial z+F_z
3 P* s5 H- m6 a! Q \/ P 上述三个方程分别是动量方程的x、y、z分量形式 * o- ]& l8 c, y H( r2 j
也可以写成矢量形式： ; D0 P! x, N( R
dV¯/dt=g−1/ρ⋅(hamilton)P+Ω×V¯+υΔ(hamilton)barV+Ft+Frd\bar{V}/dt=g-1/\rho\cdot(hamilton)P+\Omega \times \bar{V}+\upsilon\Delta(hamilton)bar{V}+F_t+F_r 3 k. M- V! L/ U
以下我将逐个解释各项含义
, m; b+ x0 y6 o9 a 等式左边为速度对时间的全导数，以E1为例，u为速度的x方向分量，u是（x,y,z,t)的函数
8 s" r" b" k& s% m" X- f, A 等式右边包括重力、压强梯度力、科氏力、黏性力、湍应力、天体引潮力
4 \8 I" i8 y( j* e7 ~3 N 重力不用过多分析，仅存在于z方向 8 z2 |/ L0 @ X& I: @
压强梯度力：x方向为例，
$ O+ |* _: N0 E. u( ~& G a=F/m=(p−(p+δp))⋅δyδz/ρ⋅δxδyδz=−1/ρ⋅∂p/∂xa=F/m=(p-(p+\delta p))\cdot\delta y\delta z/\rho\cdot \delta x\delta y\delta z=-1/\rho\cdot \partial p/\partial x
7 s! [6 h7 X+ l( M. z8 E+ F 科氏力： F=−2Ω×VF=-2\Omega\times V ' P6 y- [ F G
Ω=2π/day=7.27÷105m/s\Omega=2\pi/day =7.27\div10^5 m/s
- G. ^, _% Q: \% H" b Ω(0,Ωcosφ,Ωsinφ)\Omega (0,\Omega cos\varphi,\Omega sin\varphi)
! W: U. \ A8 r" X8 y5 M" S: f7 f φ=latitude\varphi=latitude ~; V. l* ]0 \; f+ s/ }1 Z' L
近似计算 5 ]( k0 n$ n% N7 B& ~5 P! s
Fx=fvF_x=fv 5 O# b% }" J9 u! M3 ?% x
Fy=−fuF_y=-fu
7 L% l' Q$ h! O% J0 P ff 为科氏系数 f=2Ωsinφf=2\Omega sin\varphi & z" |- ?7 R Q" T0 n: |2 u# Y3 c
黏性力为黏合系数与梯度的乘积，湍应力由湍流的脉冲造成的，天体引潮力过于复杂（与日月等天体有关，暂不介绍） 3 e- r' }5 F$ Y" J% ^, ~
E4 连续性方程
. e9 S$ G0 M1 H m; x ∂u/∂x+∂v/∂y+∂w/∂z=0\partial u/\partial x+\partial v/\partial y+\partial w/\partial z=0
- T: A9 i2 z$ v* T4 |( c- Z Eularian观点：定点处观察经过的流体质量变化 3 x1 ~9 Y5 u' f* `/ |7 C
∂ρ/∂t+(∂(ρu)∂x+∂(ρv)/∂y+∂(ρw)/∂z=0\partial \rho/\partial t+(\partial(\rho u)\partial x+\partial(\rho v)/\partial y+\partial (\rho w)/\partial z=0 ! p! z* F& T. E9 z- M6 }
转化为Lagrange观点：跟踪流体微团 4 s8 J4 z3 Q; R- ~, `
1/ρDρ/Dt+(∂u/∂x+∂v/∂y+∂w/∂z)=01/\rho D\rho /Dt +(\partial u/\partial x+\partial v/\partial y+\partial w/\partial z)=0
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E7 状态方程
" V8 p; |4 d' m- s u ∂s/∂t+u∂s/∂x+v∂s/∂y+w∂s/∂z=kDΔs+∂(kH∂s/∂x)/∂x+∂(kH∂s/∂y)/∂y+∂(kH∂s/∂z)/∂z\partial s/\partial t+u\partial s/\partial x+v\partial s/\partial y+w\partial s/\partial z=k_D\Delta s+\partial(k_H \partial s/ \partial x)/\partial x+\partial(k_H \partial s/ \partial y)/\partial y+\partial(k_H \partial s/ \partial z)/\partial z
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