# 2024/03/13
## Hydrostatic Equation
### Hydrostatic
This is vertical momentum equation with scale analysis.
There is the scale of the equation:
|Terms|$\frac{Dw}{Dt}$|$\frac{u^2+v^2}{a}$|$-\frac{1}{\rho}\frac{\partial p}{\partial z}$|$2\Omega u cos\phi$|$g$|$\nu\nabla^2 w$|
|:---:|:---:|:---:|:---:|:---:|:---:|:---:|
|Scale|$10^{-7}$|$10^{-5}$|$10$|$10^{-3}$|$10$|$10^{-15}$|

For primary approximation, there is:
$$
\begin{aligned}
0 &\approx -\frac{1}{\rho}\frac{\partial p}{\partial z}-g\\
\\
\Rightarrow \frac{\partial p}{\partial z} &\approx -\rho g
\end{aligned}
$$

This relation shows that hydrostatic condition is almost the equilibrium state of vertical motion in atmosphere.

### Hydrostatic equation
For discussing the motion under equilibrium state, we define pressure and density field as $p_0\;(z)$ and $\rho_0\;(z)$, respectively.
$$
\frac{1}{\rho_0}\frac{dp_0}{dz} = -g
$$

## Hydrostatic Approximation
1. Assumption: Linearization
Consider field variables are composited of mean-state and perturbation. Therefore, pressure and density field can be written as:
$$
\begin{aligned}
&p(x, y, z, t) = p_0(z) + p^\prime (x, y, z ,t)\\
& \rho(x, y, z, t) = \rho_0(z) + \rho^\prime (x, y, z ,t)
\end{aligned}
$$

2. Scale analysis
With hydrostatic equation, there is:
$$
\begin{aligned}
&-\frac{1}{\rho}\frac{\partial p}{\partial z}=g\\
\\
\Rightarrow &-\frac{1}{\rho_0+\rho^\prime}\frac{\partial (p_0+p^\prime)}{\partial z} = g\\
\end{aligned}
$$

With Taylor's series, $(1+x)^{-1} = 1-x$
$$
-\frac{1}{\rho_0}(1-\frac{\rho^\prime}{\rho_0})(\frac{d p_0}{dz}+\frac{\partial p^\prime}{\partial z})=g
$$

Ignoring terms with smaller order of magnitude:
$$
\begin{aligned}
&-\frac{1}{\rho_0}\frac{dp_0}{dz}-\frac{1}{\rho_0}\frac{\partial p^\prime}{\partial z}+\frac{\rho^\prime}{\rho_0^2}\frac{dp_0}{dz}\approx g\\
\\
\Rightarrow &-\frac{1}{\rho_0}\frac{\partial p^\prime}{\partial z}-\rho^\prime g\approx 0
\end{aligned}
$$

Therefore, no matter whether the equilibrium state is reached or not, hydrostatic equation is a good proxy of vertical motion.