# Circulation and Vorticity
## Definition
1. Aiming to describe rotation of fluid.
2. There are 2 ways to reach this target:
* Macroscopic: Circulation ($C$)
* Microscopic: Vorticity ($\vec{\omega}$)
3. Definition
* Circulation
$$
C \equiv \oint_\Gamma \vec{u}\cdot d\vec{l}
$$

* Vorticity
$$
\vec{\omega} =\nabla\times\vec{u}
$$

## Relations between Vort. and Circ.
If area circled by path $\Gamma$ is simple connection, then by Stoke's Thm.:
$$
C \equiv \oint_\Gamma \vec{u}\cdot d\vec{l} = \iint\limits_R (\nabla\times\vec{u})\cdot \hat{n}\;dA
$$

By definition of vorticity:
$$
\begin{aligned}
&C = \iint\limits_R (\nabla\times\vec{u}) \cdot \hat{n}\;dA\\
\\
\Rightarrow & C = \iint\limits_R \vec{\omega} \cdot \hat{n}\;dA
\end{aligned}
$$

This relation tells us that circulation is the collection of vortex tube over simple connected area.

## Kelvin Circ. Thm.
### Abs. Circ., Rel. Circ., and Planetary Circ.
By definition of circulation, absolute circulation can be written as:
$$
C_a \equiv \oint\limits_\Gamma \vec{v_a} \cdot d\vec{l}
$$

By derivation in Chapter 2, absolute velocity can be regarded as the combination of relative velocity and rotation of the system. Therefore, the absolute circulation can be written as:
$$
\begin{aligned}
C_a &\equiv \oint\limits_\Gamma \vec{v_a} \cdot d\vec{l} \\
\\
&=\oint\limits_\Gamma (\vec{v}+\vec{v_e}) \cdot d\vec{l}\\
\\
&=\oint\limits_\Gamma \vec{v} \cdot d\vec{l} + \oint\limits_\Gamma \vec{v_e} \cdot d\vec{l}\\
\\
& = C+C_e
\end{aligned}
$$

### Derivation
From horizontal momentum equation:
$$
\frac{D_a\vec{V_a}}{Dt} = -\frac{1}{\rho}\nabla_h p-\nabla\Phi+\nu\nabla^2 \vec{v}
$$

Intergral equation above on specific path $\Gamma$:
$$
\oint\limits_\Gamma \frac{D_a\vec{V_a}}{Dt} \cdot d\vec{l} = \oint\limits_\Gamma (-\frac{1}{\rho}\nabla_h p-\nabla\Phi+\nu\nabla^2 \vec{v} )\cdot d\vec{l}
$$

Since line integral on potential function is 0, the equation can be simplified as:
$$
\oint\limits_\Gamma \frac{D_a\vec{V_a}}{Dt} \cdot d\vec{l} = \oint\limits_\Gamma (-\frac{1}{\rho}\nabla_h p+\nu\nabla^2 \vec{v} )\cdot d\vec{l}
$$

With chain rule, LHS can be written as:
$$
\begin{aligned}
&\oint\limits_\Gamma \frac{D_a (\vec{V_a}\cdot d\vec{l})}{Dt} - \vec{V_a}\cdot\frac{D_a d\vec{l}}{Dt} = \oint\limits_\Gamma (-\frac{1}{\rho}\nabla_h p+\nu\nabla^2 \vec{v} )\cdot d\vec{l}\\
\\
\Rightarrow & \oint\limits_\Gamma \frac{D_a (\vec{V_a}\cdot d\vec{l})}{Dt} - \vec{V_a}\cdot d\vec{V_a} = \oint\limits_\Gamma (-\frac{1}{\rho}\nabla_h p+\nu\nabla^2 \vec{v} )\cdot d\vec{l}
\end{aligned}
$$

Since line integral of scalar is 0:

$$
\begin{aligned}
&\oint\limits_\Gamma \frac{D_a (\vec{V_a}\cdot d\vec{l})}{Dt} = \oint\limits_\Gamma (-\frac{1}{\rho}\nabla_h p+\nu\nabla^2 \vec{v} )\cdot d\vec{l}\\
\\
\Rightarrow & \frac{D_a}{Dt} \oint\limits_\Gamma \vec{V_a}\cdot d\vec{l} = \oint\limits_\Gamma -\frac{1}{\rho}\nabla_h p \cdot d\vec{l}+\oint\limits_\Gamma \nu\nabla^2 \vec{v} \cdot d\vec{l}\\
\\
\Rightarrow & \frac{D_a C_a}{Dt} = \oint\limits_\Gamma -\frac{1}{\rho}\nabla_h p \cdot d\vec{l}+\oint\limits_\Gamma \nu\nabla^2 \vec{v} \cdot d\vec{l}
\end{aligned}
$$

Since area surrouded by $\Gamma$ is simple connected, then by Stoke's thm.:
$$
\begin{aligned}
& \frac{D_a C_a}{Dt} = \oint\limits_\Gamma -\frac{\nabla\rho\times \nabla p}{\rho^2} \cdot \hat{n} dA+C_F
\end{aligned}
$$

The first term on RHS of equation above is solenoidal term, which will only exist in baroclinic atmosphere.
Or alternatively, ==in barotropic atmosphere, absolute circulation is conserved by Kelvin Circ. Thm.==

## Bjerknes Circ. Thm.
### Planetary circ.
By definition of circulation:
$$
\begin{aligned}
&C_e = \oint\limits_\Gamma \vec{V_e}\cdot d\vec{l}\\
\\
\Rightarrow & \oint\limits_\Gamma (\vec{\Omega}\times\vec{r})\cdot d\vec{l}
\end{aligned}
$$

Consider the Earth is simply connected, by Stoke's Thm.:
$$
\begin{aligned}
&\oint\limits_\Gamma (\vec{\Omega}\times\vec{r})\cdot d\vec{l} \\
\\
\Rightarrow & \iint\limits_R \nabla\times(\vec{\Omega}\times\vec{r})\cdot\hat{n} dA
\end{aligned}
$$

Consider the coordinate as cylindral:
$$
\begin{aligned}
&\iint\limits_R \nabla\times(\Omega R\hat{\epsilon_R})\cdot \hat{n}dA \\
\\
\Rightarrow &\iint\limits_R 2\Omega \hat{\epsilon_R}\cdot\hat{n}dA
\end{aligned}
$$

Since $\hat{\epsilon_R}$ points to the axis of Earth rotation, and $\hat{n}$ points to positive vertical driection locally, the angle between the two vector is the same as $90^\circ -\phi$.
Therefore:
$$
\begin{aligned}
&\iint\limits_R 2\Omega \hat{\epsilon_R}\cdot\hat{n}dA \\
\\
\Rightarrow & \iint\limits_R 2\Omega sin\phi\;dA
\end{aligned}
$$

Therefore, there is the relation:
$$
C_e = 2\Omega \iint\limits_R \sin\phi\;dA
$$

For infinitesimal area, the term, $\iint\limits_R\sin\phi\; dA$ can be regarded as projection of an area on eqatorial plane, or alternatively, equivalent area $A_e$ :
$$
\begin{aligned}
& C_e\approx 2\Omega A_e\\
\\
\Rightarrow & C_a \approx C+2\Omega A_e
\end{aligned}
$$

### Derivation
Doing differentiation on equation above:
$$
\begin{aligned}
& \frac{DC_a}{Dt} = \frac{DC}{Dt}+\frac{DC_e}{Dt} \\
\\
\Rightarrow & \frac{DC}{Dt} = \frac{DC_a}{Dt} -\frac{DC_e}{Dt}
\end{aligned}
$$

By Kelvin's Circ. Thm.:
$$
\begin{aligned}
& \frac{DC}{Dt} = \frac{DC_a}{Dt} -\frac{DC_e}{Dt}\\
\\
\Rightarrow & \frac{DC}{Dt} = -\oint\limits_\Gamma \frac{dp}{\rho} -2\Omega \frac{DA_e}{Dt}
\end{aligned}
$$
The equation is called ==Bjerknes Circ. Thm.==.

### Physical Scinerio
1. **Barotropic atmosphere**
By Kelvin's Circ. Thm., abs. circ. is conserved in barotropic atmosphere, and the Bjerknes circ. thm. can be simplified as:
$$
\begin{aligned}
& \frac{DC}{Dt} = -2\Omega \frac{DA_e}{Dt}
\end{aligned}
$$

There are 2 possible changes in the relation above:
* equivalent area increase
If $A_e$ increse, and the rel. circ. decrease, inducing anticyclone.

* Meridional displacement
If the area is constant, then the equivalent area can change with merdional displacement.
When the circulation system displaces northward, the term $A_e$ increases, then the anticyclonic flow is induced.
This relation can be one of the mechanism of Rossby wave generation.

2. **Zonal/Small-scale motion**
When the planetary circ. can be ignored, the change of rel. circ. is governed by solenoidal term.
There are some examples for better comprehension:
* Land-Sea Breeze

As the schematic above, there is a simple condition for land-sea breeze to construct.
With Kelvin's Circ. Thm.:
$$
\begin{aligned}
\frac{DC}{Dt} = -[\int_B^C \frac{dp}{\rho}+\int_D^A \frac{dp}{\rho}]
\end{aligned}
$$

The other 2 terms is droped due to the path is isobaric, and by hypsometric relation:
$$
\begin{aligned}
\frac{DC}{Dt} &= -[\int_B^C R\overline{T} d(\ln p)+\int_D^A R\overline{T} d(\ln p)]\\
\\
& = R(T_H-T_L)\ln(\frac{p_1}{p_2})
\end{aligned}
$$

In this case, the term, $\frac{DC}{Dt}$, is positive, so the direction of wind is sea breeze.
In real cases, the wind speed will not grow w/o a limit. This result is due to the temperature advection the breeze brings to balance the temperature difference between land and sea.

* East Asia Main Trough
At East Asia, in boreal winter, the direction of density gradient and pressure gradient is not parelle, so the atmosphere is baroclinic. And it's easy to induce cyclonic system, so it;s also called ==Storm Track==.

## Vorticity
### Definition
As circ., vorticity can also be distinguished into 3 types:
1. Rel. vort.
$$
\begin{aligned}
&\vec{\omega} = \nabla \times \vec{v}\\
\\
\Rightarrow & \vec{\omega} =
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k}\\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
u & v & w
\end{vmatrix}\\
\\
\Rightarrow & \vec{\omega} = <(\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}), (\frac{\partial w}{\partial x}-\frac{\partial u}{\partial z}), (\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y})>
\end{aligned}
$$

2. Abs. vort.
$$
\vec{\omega_a} = \nabla \times \vec{v_a}
$$

3. planetary vort.
$$
\begin{aligned}
\vec{\omega_e} &= \nabla \times \vec{v_e}\\
\\
&= \nabla \times (\vec{\Omega} \times \vec{r})\\
\\
&=2\Omega \hat{\epsilon_z}
\end{aligned}
$$

### Vertical component of vorticity
After scale analysis, vort. can only obtaining vertical component.
And therefore, for discussion later on, the name "vorticity" refers to its vertical component.
1. rel. vort.
$$
\begin{aligned}
\zeta & = \hat{k} \cdot \vec{\omega}\\
\\
& = (\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y})
\end{aligned}
$$

2. planetary vort.
$$
\begin{aligned}
\zeta_e & = \hat{k} \cdot \vec{\omega_e}\\
\\
& = \hat{k} \cdot 2\Omega \hat{\epsilon_z}\\
\\
& = 2\Omega \sin\phi\\
\\
& = f
\end{aligned}
$$

3. Abs. vort.
$$
\begin{aligned}
\eta & = \zeta + f
\end{aligned}
$$

### Vorticity in natural coordinate
By definition of relative vorticity:
$$
\begin{aligned}
\zeta &= \hat{k} \cdot (\nabla \times \vec{v}) \\
\\
&= \hat{k} \cdot (\nabla \times (V \hat{t})) \\
\\
&= \hat{k} \cdot (\color{red}{V(\nabla\times \hat{t})}+\color{blue}{\nabla V\times \hat{t}})
\end{aligned}
$$

1. Red term
$$
\begin{aligned}
\nabla \times \hat{t} &=
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k}\\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
\cos\phi & \sin\phi & 0
\end{vmatrix}\\
\\
&= <-\frac{\partial}{\partial z}(\sin\phi), \frac{\partial}{\partial z}(\cos\phi), \frac{\partial}{\partial x}(\sin\phi)-\frac{\partial}{\partial y}(\cos\phi)>
\end{aligned}
$$

for its vertical component:
$$
\begin{aligned}
\hat{k} \cdot (\nabla\times\hat{t}) &= \frac{\partial}{\partial x}(\sin\phi)-\frac{\partial}{\partial y}(\cos\phi) \\
\\
&=\cos\phi\frac{\partial \phi}{\partial x}+sin\phi\frac{\partial \phi}{\partial y}\\
\\
&= \hat{t} \cdot \nabla\phi = \hat{t} \cdot <\frac{\partial \phi}{\partial s}, \frac{\partial \phi}{\partial n}> = \frac{\partial \phi}{\partial s} = K_s
\end{aligned}
$$

2. Blue term
$$
\begin{aligned}
\hat{k} \cdot (\nabla V\times \hat{t}) &=
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k}\\
\cos\phi & \sin\phi & 0 \\
\frac{\partial V}{\partial x} & \frac{\partial V}{\partial y} & \frac{\partial V}{\partial z} \\
\end{vmatrix}\\
\\
&= \hat{k} \cdot <\frac{\partial V}{\partial z}\sin\phi, -\frac{\partial V}{\partial z} \cos\phi, \frac{\partial V}{\partial y} \cos\phi-\frac{\partial V}{\partial x} \sin\phi>\\
\\
&= \frac{\partial V}{\partial y} \cos\phi-\frac{\partial V}{\partial x} \sin\phi = \hat{n}\cdot\nabla V = \frac{\partial V}{\partial n}
\end{aligned}
$$

The vorticity can be rewriten as:
$$
\zeta = \color{red}{VK_s}-\color{blue}{\frac{\partial V}{\partial n}}
$$

For physical meaning, there are 2 sources that can generate vorticity:
1. $VK_s$
This term states that the vorticity can be generated by curve-line motion.
By definition of curvature, if the flow is counterclockwise, then this term is positive, which flows that the vorticity is positive when counterclockwise flow.

2. $-\frac{\partial V}{\partial n}$
This term states that the vorticity can be generated by horizontal wind shear.

## Potential Vorticity (PV)
### Conservation of abs. circ. on isentropic surface

By Kelvin's circ. thm.:
$$
\frac{DC_a}{Dt} = -\oint\limits_\Gamma \frac{dp}{\rho}
$$

Consider the path $\Gamma$ locates at isentropic surface, then the density can be regarded as function of pressure.

Therefore, the abs. circ. is conserved when the motion is along isentropic surface.

Therefore:
$$
\color{red}{C_a} = \iint\limits_R \vec{v_a}\cdot d\vec{l} = \color{red}{(\zeta_\theta + f)\delta A} = const.
$$

### Ertel's PV
By hydrostatic equation:
$$
\begin{aligned}
&\delta M = \rho \delta A \delta z = -\frac{\delta p}{g} \delta A\\
\\
\Rightarrow & \delta A = -g\frac{\delta M}{\delta p} = -g\frac{\delta M}{\delta \theta}\frac{\delta \theta}{\delta p}
\end{aligned}
$$