## Vorticity and Quasi-Geostrophy

1
2
3
4

### Shallow Water Potential Vorticity

Potential vorticity, denoted by $$q$$, is a materially conserved quantity in systems without frictional or diabatic processes. The form of $$q$$ depends on the system we are modeling. We will derive the potential vorticity within a shallow water framework. The shallow water equations model the free surface of a fluid as a function of time and horizontal space coordinates. It is a barotropic model, so the velocities do not depend on $$z$$.

Let’s start from the horizontal momentum equations:

\begin{align*} \frac{Du}{Dt} – fv &= -\frac{1}{\rho}\frac{\partial p}{\partial x}\\ \frac{Dv}{Dt} + fu &= -\frac{1}{\rho}\frac{\partial p}{\partial y} \end{align*}

We will assume that pressure is structured hydrostatically: $$p=p_0 + \rho g (h-z)$$, where $$p_0$$ is the pressure at the surface and $$h$$ is the height of the fluid. If we take $$H$$ to be the mean height, we may write $$h=H+\eta$$, where $$\eta$$ is the deviation in free surface height from the mean. Lastly, we will assume that $$f$$ is on a beta plane, so $$f = f_0 + \beta y$$. We get (assuming constant fluid density):

\begin{align*} \frac{Du}{Dt} – fv &= -g\frac{\partial \eta}{\partial x}\\ \frac{Dv}{Dt} + fu &= -g\frac{\partial \eta}{\partial y} \end{align*}

These are the shallow water momentum equations. As for mass conservation, we can derive our shallow water equivalent from the 3D conservation statement:

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$$

We can express $$w$$ as the material derivative of the free surface height:

$$w = \frac{Dh}{Dt}$$

Putting this into our 3D conservation statement:

$$\frac{Dh}{Dt} = -\int\limits_{0}^h \nabla \cdot \vec{u}dz$$

But the horizontal velocity is independent of $$z$$, so we simply get:

$$\frac{Dh}{Dt} + h\nabla \cdot \vec{u} = 0$$

Note that if we take the curl of the horizontal velocity (assigning a value of 0 to the z-component), we get:

$$\require{cancel}\nabla \times \vec{u} = -\cancel{\frac{\partial v}{\partial z}}\hat{i} + \cancel{\frac{\partial u}{\partial z}}\hat{j} +\zeta \hat{k} = \zeta \hat{k}$$

So we are only interested in the vertical component of vorticity.

Going to the momentum equations, we will use the fact that:

$$\vec{u} \cdot \nabla{\vec{u}} = \zeta\hat{k} \times \vec{u} + \frac{1}{2}\nabla \|\vec{u}\|^2$$

where

Taking the curl of the momentum equations and letting $$\zeta = (\nabla \times \vec{u})\cdot\hat{k}$$, we get:

$$\frac{\partial \zeta}{\partial t} + \nabla \times (\zeta\hat{k} \times \vec{u}) \cdot \hat{k} + (f_0 + \beta y)(\nabla \cdot \vec{u}) + \beta v = 0$$

We can now use a calculus identity:

$$\require{cancel}\nabla \times (\zeta\hat{k} \times \vec{u}) = (\nabla \cdot \vec{u})\zeta\hat{k} + (\vec{u} \cdot \nabla) \zeta\hat{k} – \cancel{(\nabla \cdot \zeta\hat{k} )\vec{u}} – \cancel{\zeta\hat{k} \cdot \nabla \vec{u}}$$

The last two components disappear because (1) the divergence of a curl is zero, and (2) the horizontal velocities vary in the xy-plane, which is tangent to the vertical vorticity. We will substitute these back into the equation, with two additional changes:

• The material derivative of height replaces the velocity divergence via the mass conservation equation.
• The $$v$$ from the $$\beta v$$ term is rewritten as $$\frac{Dy}{Dt}$$

We get:

$$\frac{D(\zeta + f_0 + \beta y)}{Dt} – \frac{\zeta + f_0 + \beta y}{h}\frac{Dh}{Dt} = 0$$

with the material derivative of height replacing the velocity divergence via the mass conservation equation. This can be repacked into a conservation term by assigning $$q = \frac{\zeta + f_0 + \beta y}{H + \eta}$$. Then we arrive at our potential vorticity equation:

$$\frac{Dq}{Dt} = 0$$

This is analogous to the person spinning in a chair from solid body rotation. If a parcel of air is vertically compressed, then $$h$$ decreases. This leads to a greater radius by mass conservation. To compensate, the parcel must rotate slower to conserve PV. Just a person slows as they stick their arms out, a parcel slows as its radius increases.

There is the additional factor of latitude not present in the angular momentum example. This is due to the rotation of the earth. As a parcel moves north from the equator, for example, its absolute vorticity $$\zeta + f$$ may be conserved, but its relative vorticity $$\zeta$$ must decrease to compensate for the additional Coriolis force.

Potential vorticity conservation is a mechanism for the propagation of several types of waves, as described on the next page.

1
2
3
4

This site uses Akismet to reduce spam. Learn how your comment data is processed.

### Archive

November 2018 (2)
February 2018 (1)
August 2017 (1)

General (1)
Modeling (1)
Pedagogical (2)

From Wikipedia.