# The Advection-diffusion Equations

## 7.1 Shallow-fluid and Conservation of Mass

Shallow fluids 的 vorticity 的定义是这样的： $\omega = \omega_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$

## 7.2 The Advection-Diffusion Equation

\begin{eqnarray} \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} &=& 0
\frac{\partial u}{\partial t} + 2u \frac{\partial u}{\partial x} + \frac{\partial}{y}(u v ) &=& f v
\frac{\partial v}{\partial t} + 2v \frac{\partial v}{\partial x} + \frac{\partial}{y}(u v ) &=& -f v \end{eqnarray}

v &=& \frac{\partial \psi }{\partial x} \end{eqnarray}

\nabla^2 \psi &=& \omega \end{eqnarray} 其中 $[ \psi, \omega ] = \psi_x \omega_y - \psi_y \omega_x$.

## 7.3 Solution Techniques and Characteristics of Advection-Diffusion

JUST A COMMENT OF THREE KINDS OF SYSTEMS：

PARABOLIC （扩散方程）: $\frac{\partial \omega}{\partial t} = v \nabla^2\omega$ HYPERBOLIC （波动方程）: $\frac{\partial \omega}{\partial t} + [\psi, \omega] = 0$ ELLIPTIC : $\nabla^2 \omega =0$

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