Vorticity and Quasi-Geostrophy

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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.


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