# Quasi-Geostrophic Theory
## Background of Q-G Theory
Charney (1948) provided this dynamical framework in his doctorial thesis, which is a great step made in the research field of atmospheric sciences.

## Assumption
1. **Hydrostatic approximation**
Physically, this approximation indicates that the aspectatial is small.
2. **secondary approximation in momentum equation**
Recall note in chapter 1, the result of scale analysis shows that the balanced can be made approximately by ==pressure gradient force==, ==Coriolis force==, and ==Material acceleration==.
This is also one of the reason why this framework is called *Quasi*-Geostrophic, exact geostrophic relation was constructed without term of acceleration, while the secondary approximation improves the framework into prognostic.

## Primitive equation
Based on the two approximation made previously, original governing equations set can be simplified as:

$$
\begin{cases}
\frac{\partial u}{\partial t} + \vec{V_h}\cdot \nabla u + \omega \frac{\partial u}{\partial p} = -\frac{\partial \Phi}{\partial x} + fv \\
\\
\frac{\partial v}{\partial t} + \vec{V_h}\cdot \nabla v + \omega \frac{\partial v}{\partial p} = -\frac{\partial \Phi}{\partial y} - fu\\
\\
\frac{\partial \Phi}{\partial p} = -\alpha = -\frac{RT}{p}\\
\\
\nabla_p\cdot \vec{V_h} + \frac{\partial \omega}{\partial p}=0\\
\\
\frac{\partial T}{\partial t} + \vec{V_h}\cdot \nabla T - S_p\omega = \frac{\dot{Q}}{C_p}
\end{cases}
$$

## QG vort. equ.
To examine an advanced properties of motion, vorticity is one of the target to study on.
Based on the same method of derivation, vorticity equation in QG system can be rewritten as:
$$
\frac{\partial \zeta}{\partial t} = {-\vec{V_h}\cdot \nabla\vec{\zeta_a}} {- \zeta_a\cdot Div} {- (\frac{\partial \omega}{\partial x}\frac{\partial v}{\partial p} - \frac{\partial \omega}{\partial y}\frac{\partial u}{\partial p})}
$$

In this equation, there are four terms, including local vorticity change , vorticity advection , divergence term, and tilting term .

With scale analysis, there are two approximation can be concerned:
1. the deviation of Coriolis parameter $f$ is much smaller than the reference value $f_0$.
2. Tilting term can be ignored.

Thus, the equation can be written as:
$$
\frac{\partial \zeta}{\partial t} = {-\vec{V_h}\cdot \nabla\vec{\zeta_a}} {- f_0\cdot Div}
$$

With continuity equation:
$$
\frac{\partial \zeta}{\partial t} = {-\vec{V_h}\cdot \nabla\vec{\zeta_a}}{+ f_0\cdot \frac{\partial \omega}{\partial p}}
$$

To integrate with discussion later on, we can write vorticity in form of:
$$
\zeta \equiv \frac{\nabla^2\Phi}{f_0}
$$

Also, based on the assumption of geostrophic, and the rewritten form of vorticity, there is a novel version of vorticity equation in QG framework:
$$
\begin{aligned}
&\frac{\partial \zeta_g}{\partial t} = {-\vec{V_g}\cdot \nabla(\vec{\zeta_g}+f)} {+ f_0\cdot \frac{\partial \omega}{\partial p}}\\
\\
\Rightarrow & \frac{\partial }{\partial t} (\frac{\nabla^2 \Phi}{f_0})= {-\vec{V_g}\cdot \nabla(\frac{\nabla^2 \Phi}{f_0}+f)} {+ f_0\cdot \frac{\partial \omega}{\partial p}}
\end{aligned}
$$

## QG Thermal equation
With equation of state, and hydrostatic equation,
$$
T = -\frac{p}{R}\frac{\partial \Phi}{\partial p}
$$

Bring temperature into thermal equation:
$$
\begin{aligned}
&\frac{\partial}{\partial t}(-\frac{p}{R}\frac{\partial \Phi}{\partial p}) + u\frac{\partial}{\partial x}(-\frac{p}{R}\frac{\partial \Phi}{\partial p}) + v\frac{\partial}{\partial y}(-\frac{p}{R}\frac{\partial \Phi}{\partial p}) - S_p \omega = \frac{\dot{Q}}{C_p}\\
\\
\Rightarrow& \frac{\partial}{\partial t}(-\frac{\partial \Phi}{\partial p}) + u\frac{\partial}{\partial x}(-\frac{\partial \Phi}{\partial p}) + v\frac{\partial}{\partial y}(-\frac{\partial \Phi}{\partial p}) - \frac{R}{p}S_p \omega = \frac{\dot{RQ}}{pC_p}\\
\\
\Rightarrow & \frac{\partial}{\partial t}(-\frac{\partial \Phi}{\partial p}) + u\frac{\partial}{\partial x}(-\frac{\partial \Phi}{\partial p}) + v\frac{\partial}{\partial y}(-\frac{\partial \Phi}{\partial p}) - \sigma \omega = \frac{\dot{RQ}}{pC_p}\\
\\
\Rightarrow & \frac{\partial}{\partial t}(-\frac{\partial \Phi}{\partial p}) = - \vec{V}\cdot \nabla (-\frac{\partial \Phi}{\partial p}) - {\sigma \omega} + {\frac{\dot{RQ}}{pC_p}}
\end{aligned}
$$

The last second term on the rhs of equation shows the adiabatic heating/cooling, and the last term shows diabatic heating/cooling.

Under geostrophic approximation, the QG thermal equation can be written as:
$$
\frac{\partial}{\partial t}(-\frac{\partial \Phi}{\partial p}) = - \vec{V_g}\cdot \nabla (-\frac{\partial \Phi}{\partial p}) - \sigma \omega + \frac{\dot{RQ}}{pC_p}
$$

## QG equation
With equations discussed above, there is an equation set obtaining vorticity and thermal equations:
$$
\begin{cases}
& \frac{\partial }{\partial t} (\frac{\nabla^2 \Phi}{f_0})= -\vec{V_g}\cdot \nabla(\frac{\nabla^2 \Phi}{f_0}+f)+ f_0\cdot \frac{\partial \omega}{\partial p}\\
\\
& \frac{\partial}{\partial t}(-\frac{\partial \Phi}{\partial p}) = - \vec{V_g}\cdot \nabla (-\frac{\partial \Phi}{\partial p}) - \sigma \omega + \frac{R\dot{Q}}{pC_p}
\end{cases}
$$

In following discussion, the term $\frac{\partial \Phi}{\partial t}$ will be written as $\chi$, which represents the tendency of geopotential height change.
With this notation, the equations can be rewritten as:

$$
\begin{cases}
& \frac{1}{f_0} (\nabla^2 \chi)= -\vec{V_g}\cdot \nabla(\frac{\nabla^2 \Phi}{f_0}+f)+ f_0\cdot \frac{\partial \omega}{\partial p}\\
\\
& -\frac{\partial \chi}{\partial p} = - \vec{V_g}\cdot \nabla (-\frac{\partial \Phi}{\partial p}) - \sigma \omega + \frac{R\dot{Q}}{pC_p}
\end{cases}
$$

In this format, there is only two independent variables, which is $\chi$ and $\omega$, in equations, constructed a complete set of equations.

In QG system, we usually use geopotential height tendency and vertical velocity as out prognostic variables. And thus constructed two noval equations: ==tendency equation== and ==$\omega$ equation==.

### Tendency equation
#### equation
For constructing equation for geopotential height tendency term, we will eliminate $\omega$ in the equation, and therefore get (derivation is left for the readers):
$$
(\nabla^2 + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2})\chi = -f_0 \vec{V_g}\cdot \nabla (\frac{1}{f_0}\nabla^2\Phi + f) + \frac{f_0^2}{\sigma}\frac{\partial }{\partial p}(-\vec{V_g}\cdot \nabla (\frac{\partial \Phi}{\partial p}))- \frac{f_0^2}{\sigma}\frac{\partial}{\partial p}(\frac{R}{C_p}\frac{\dot{Q}}{p})
$$

#### Physical meaning
Supposing $\chi$ and $\Phi$ can be written as Fourier series:
$$
\begin{cases}
&\chi = X(t)\exp{i(kx+ly+mp)}\\
\\
& \Phi = P(t)\exp{i(kx+ly+mp)}\\
\end{cases}
$$

Bring the two variables into tendency equation, LHS of the equation, it can be written as:
$$
(\nabla^2 +\frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2})\chi = -(k^2 + l^2 +\frac{f_0^2}{\sigma} m^2)\chi
$$

This relation shows that:
1. Sign of this term will change sign.
2. This term will enhance short wave signal.

Now, we will dicuss effect of each term in the equation:
1. **Vorticity advection**
This term will be significant when the short wave signal domanant, which means that the vorticity is more important than Earth rotation.
In math:
$$
(\nabla^2 + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2})\chi = -f_0 \vec{V_g}\cdot \nabla \zeta_g
$$

* positive vorticity advection (PVA, $-\vec{V_g}\cdot \nabla\zeta_g > 0$), the LHS will be positive.
But since the LHS is proportional to $-\chi$, then $\chi$ will be negative.
* negative vorticity advection (NVA), the LHS will be negative.
The LHS will proportional to $-\chi$, and $\chi$ will be positive.


At the left of trough part, the flow is from negative vorticity (clockwise flow) to which is positive (anticlockwise flow), meaning NVA.
And at the right of trough part, the PVA occurs.

Combining with concept above, we can know that geopotential height will decrease at the right, and increase at the left. The trough part will move to eastward.

2. **$\beta$-effect**
This condition will happen where the longwave is dominating.
In math:
$$
(\nabla^2 + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2})\chi = -f_0 \vec{V_g}\cdot \nabla f = -f_0\vec{V_g}\cdot \beta
$$

The RHS of the equation shows that when then flow move southward, the planetary vorticity advection will be positive, and flow moves northward will be negative.

Combining will concept above, $\chi$ will decrease at the left of trough, and increase at te right of trough, driving the system moves westward.

3. **Temperature Advection**
In math:
$$
(\nabla^2 + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2})\chi = \frac{f_0^2}{\sigma}\frac{\partial }{\partial p}(-\vec{V_g}\cdot \nabla (\frac{\partial \Phi}{\partial p}))
$$

In brief words, the RHS can be regarded as change of temperature advection vertically.

There are two conditions:
* if there is cold advection occurs at lower level, the term on RHS will be positive, and $\chi$ will be negative at higher level, triggering a trough structure (low-pressure center).
Please attention that the cold advection will happen at the left of lower-level low-pressure center, so the higher-level low-pressure center will be at left of which at lower-level, which is also called ==up-shear tilt==, or ==lean into shear==.

* In opposite side, warm advection will cause high pressure center at higher-level.

### $\omega$ equation
#### Equation
In such an equation, in order to obtain $\omega$ as an independent variable, terms of $\chi$ will be eliminated:

$$
(\nabla^2 + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}) \omega = \frac{f_0}{\sigma}\frac{\partial}{\partial p} [\vec{V_g}\cdot\nabla(\frac{\nabla^2\Phi}{f_0}+f)] + \nabla^2 [-\vec{V_g}\cdot \nabla(\frac{\partial \Phi}{\partial p})] + \nabla^2(\frac{R}{C_p}\frac{\dot{Q}}{p})
$$

#### Physical interpretation
1. vorticity advection
With concept built up previously, there is the relation:
$$
-\omega\propto \frac{\partial}{\partial p}(\vec{V_g}\cdot \nabla\zeta_a) = \frac{(\vec{V_g}\cdot \nabla\zeta_a)_{500} - (\vec{V_g}\cdot \nabla\zeta_a)_{1000}}{500-1000}
$$

By thermal wind relation, vertical wind shear is perpendicular to gradient of Corilois parameter, and also consider the vorticity advection near surface is 0, so:
$$
-\omega \propto \frac{(\vec{V_g}\cdot \nabla\zeta_a)_{500}}{500-1000}
$$

Therefore, when PVA ($-\vec{V_g}\cdot \nabla\zeta > 0$) occurs at 500 hPa layer, $\omega <0$, which representing upward motion. Or if NVA occurs, the downward motion will be triggered.

2. temperature advection
Simply, the relation can be written as:
$$
-\omega \propto \vec{V_g}\cdot \nabla (\frac{\partial \Phi}{\partial p})
$$

If warm advection occurs, the term $-\omega$ will be positive, then the vertical velocity $\omega < 0$, which representing upward motion.
On the other hand, if cold advection occurs, downward motion will be triggered.

## Primary and Secondary Circulation <This part will also be mentioned in AD (II)>

### Primary circulation
By definition, primary circulation will be the circulation that follows the isobaric lines, which means that the primary circulation will be geostrophic flow.

### Secondary circulation
By tendency equation and $\omega$ equation, there are some motions, such as vertical motion and divergence, to maintaining hydrostatic and geostrophic approximation (thermal relation).

Acer Kuo:一見次環流，已達熱力風平衡

## Environment favors disturbance
* upshear tilt
With discussion above, upshear tilting structure is needed for coupling of upper and lower level of atmosphere.

Based on the concept of secondary circulation, there are two main branch of the circulation:

* meridonal warm advection
If meridional temperature advection is positive, available potential energy will increase.

* vertical warm advection
if vertical temperature advection is positive, available potential energy will release into kinetic energy, growing the disturbence.

## QG PV
By tendency equation:
$$
(\nabla^2 + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2})\chi = -f_0 \vec{V_g}\cdot \nabla (\frac{1}{f_0}\nabla^2\Phi + f) + \frac{f_0^2}{\sigma}\frac{\partial }{\partial p}(-\vec{V_g}\cdot \nabla (\frac{\partial \Phi}{\partial p}))- \frac{f_0^2}{\sigma}\frac{\partial}{\partial p}(\frac{R}{C_p}\frac{\dot{Q}}{p})
$$

Expanding vertical temperature advection, and the Coriolis parameter will be consistent, the equation can be rewritten as:

$$
\begin{aligned}
&\frac{1}{f_0}(\nabla^2 + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2})\frac{\partial \Phi}{\partial t} + \frac{\partial f}{\partial t} = - \vec{V_g}\cdot \nabla (\frac{1}{f_0}\nabla^2\Phi + f) + \frac{f_0}{\sigma}\frac{\partial }{\partial p}(-\vec{V_g}\cdot \nabla (\frac{\partial \Phi}{\partial p}))- \frac{f_0}{\sigma}\frac{\partial}{\partial p}(\frac{R}{C_p}\frac{\dot{Q}}{p})\\
\\
\Rightarrow & \frac{\partial}{\partial t} (\frac{1}{f_0}\nabla^2 \Phi + f + \frac{f_0}{\sigma}\frac{\partial^2 \Phi}{\partial p^2}) = -\vec{V_g}\cdot \nabla(\frac{1}{f_0}\nabla^2 \Phi + f + \frac{f_0}{\sigma}\frac{\partial^2 \Phi}{\partial p^2}) - \frac{f_0}{\sigma}\frac{\partial}{\partial p}(\frac{R}{C_p}\frac{\dot{Q}}{p}) \\
\\
\Rightarrow & \frac{D}{Dt} (\frac{1}{f_0}\nabla^2 \Phi + f + \frac{f_0}{\sigma}\frac{\partial^2 \Phi}{\partial p^2}) = \frac{Dq}{Dt} = - \frac{f_0}{\sigma}\frac{\partial}{\partial p}(\frac{R}{C_p}\frac{\dot{Q}}{p})
\end{aligned}
$$

In the equation, $q$ is so-called potential vorticity in QG framework (QGPV).