# Extratropical Cyclones
## Properties of ESCs
For general ESC properties, there are 2 features:
1. maximum vorticity $\zeta$
2. minimum pressure

Therefore, this paragraph will start from both vorticity and pressure viewpoints.

## Vorticity frames
### Mechanism provided by vorticity equation in isobaric coordinate
$$
\frac{\partial \zeta_a}{\partial t} = -\vec{v}\cdot \nabla\vec{\zeta_a}-\omega \frac{\partial \zeta_a}{\partial p}-[\frac{\partial \omega}{\partial x}\frac{\partial v}{\partial p}-\frac{\partial \omega}{\partial y}\frac{\partial u}{\partial p}]+\zeta_a\frac{\partial \omega}{\partial p}+(\frac{\partial F_x}{\partial x}-\frac{\partial F_y}{\partial y})
$$

where:
* $-\vec{v}\cdot \nabla\vec{\zeta_a}-\omega \frac{\partial \zeta_a}{\partial p}$: vorticity advection
* $\frac{\partial \omega}{\partial x}\frac{\partial v}{\partial p}-\frac{\partial \omega}{\partial y}\frac{\partial u}{\partial p}$ : twisting terms
* $\zeta_a\frac{\partial \omega}{\partial p}$ : strenching term
* $(\frac{\partial F_x}{\partial x}-\frac{\partial F_y}{\partial y})$: Viscous terms

With scale analysis, the equation can be rewritten as:
$$
\frac{\partial \zeta}{\partial t} = \zeta_a\frac{\partial \omega}{\partial p} = -(\zeta+f)(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y})
$$

where $\zeta_a = (\zeta+f)$.
In this equations, the tendency of vorticity is composed of vorticity and divergence term.

From this equation, the pre-existed vorticity is required for generated positive vorticity tendency.

In atmosphere, great pre-existed vorticity sources are: frontal structure and topography and convective cloud.

Therefore, to monitor rapid spin up of ESCs, the region with pre-existed vorticity system is where to concern.

## Pressure Frames
For ESC, or alternatively, low-pressure system, net mass divergence is needed in the overl;ying air column.

### Math expression
For hydrostatic equation:
$$
\begin{aligned}
&\frac{\partial p}{\partial z} = -\rho g\\
\\
\Rightarrow& p = -g\int \rho dz \\
\\
\Rightarrow& \frac{dp}{dt} = -g\int\frac{\partial \rho}{\partial t} dz
\end{aligned}
$$

By continuity equation:
$$
\frac{dp}{dt} = -g\int\nabla\cdot (\rho\vec{v}) dz
$$

### Physical meaning
Since low-pressure system need net mass divergence, but the surface mass flux is converge to the center of system. However, the divergence aloft must be focused.

Also, it is proven that the aloft high-pressure system is requiredfor ESCs.
Generally, divergence area forward to the troughis likely generate the ESC.