Vorticity and Quasi-Geostrophy

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Quasi-Geostrophy

While geostrophic balance captures the essence of geophysical flows, it is inadequate for capturing mesoscale processes such as eddies. Quasi-geostrophy, or QG, adds one order of ageostrophy and incorporates the time evolution of relative vorticity. It also incorporates vertical velocities that are disregarded under the geostrophic approximation.

While Paul Spence’s lecture refers to the continuously stratified equations, we will continue within the shallow water framework used to derive potential vorticity.

Recall the equations:

$$
\begin{align*}
\frac{Du}{Dt} – fv &= -g\frac{\partial h}{\partial x}\\
\frac{Dv}{Dt} + fu &= -g\frac{\partial h}{\partial y}\\
\frac{Dh}{Dt} + h\nabla \cdot \vec{u} &= 0
\end{align*}
$$

In geostrophic flow, the advection terms are discarded as it is small compared with the Coriolis terms:

$$\begin{align*}
fv &= g\frac{\partial h}{\partial x} \\
fu &=g\frac{\partial h}{\partial y} \\
\end{align*}$$

To talk about “small” in the context of GFD, we must introduce an important geophysical parameter called the Rossby number \(Ro\):

$$ Ro = \frac{U}{f_0 L} $$

This parameter emerges when we nondimensionalize the momentum equations. It is small for geophysical fluids because rates of advection \(\frac{U}{L}\) are small compared the rotation rate \(f_0\). We will make several assumptions when nondimensionalizing our equations:

  • Let \(T=\frac{L}{U}\), so advection sets the characteristic time scale.
  • We will use a \(\beta\)-plane approximation. We will assume that \(\beta \sim \frac{f_0}{L}\). This will make the \(\beta\) term order Rossby.
  • Let \(L_d = \frac{\sqrt{gH}}{f_0} \). This is the Rossby radius of deformation, and refers to the length scale above which rotation influences the dynamics.

The dimensionless horizontal momentum equations then look like:

$$\begin{align*}
Ro \frac{\tilde{D}\tilde{u}}{\tilde{D}\tilde{t}} – (1 + Ro \tilde{y})\tilde{v} &= -\left(\frac{L_d}{L}\right)^2 Ro^{-1}\frac{\partial \tilde{h}}{\partial \tilde{x}}\\
Ro\frac{\tilde{D}\tilde{v}}{\tilde{D}\tilde{t}} + (1 + Ro \tilde{y}) \tilde{u} &= -\left(\frac{L_d}{L}\right)^2 Ro^{-1} \frac{\partial \tilde{h}}{\partial \tilde{y}}
\end{align*}
$$

We will assume that \(L_d\) and \(L\) are on roughly the same order, so their ratio on the RHS cancels.

We can decompose our velocities into an order 1 geostrophic and order Rossby ageostrophic part. For example:

$$ \tilde{u} = \tilde{u}_0 + \tilde{u}_1 Ro $$

For height, it looks more like like \(1 + \tilde{\eta}_0 Ro + \tilde{\eta} Ro^2\), as the height deviations are already assumed to be small.

The first order component of the momentum equation gives us geostrophic balance. The \(\mathcal{O}(Ro)\) component becomes:

$$\begin{align*}
\frac{\tilde{D}_0\tilde{u}_0}{\tilde{D}\tilde{t}} – \tilde{v}_1 – \tilde{y} \tilde{v}_0 &= – \frac{\partial \tilde{\eta}_1}{\partial \tilde{x}}\\
\frac{\tilde{D}_0\tilde{v}_0}{\tilde{D}\tilde{t}} + \tilde{u}_1 +\tilde{y} \tilde{u}_0 &= – \frac{\partial \tilde{\eta}_1}{\partial \tilde{y}}
\end{align*}
$$

where \(\frac{\tilde{D}_0}{\tilde{D}\tilde{t}}\) is material derivative with advection by the horizontal geostrophic velocities.

For mass conservation, our dimensionless equation looks like

$$\begin{align*}
Ro \frac{\tilde{D} \tilde{\eta}}{\tilde{D}\tilde{t}} + \tilde{h}\tilde{\nabla} \cdot\tilde{\underline{u}} &= 0
\end{align*}
$$

When we substitute the expansion for the height and velocity terms, at first order we have the divergence of the horizontal geostrophic velocities equal to zero. At order Rossby:

$$\begin{align*}
\frac{\tilde{D}_0 \tilde{\eta}_0}{\tilde{D}\tilde{t}} + \tilde{\nabla} \cdot \tilde{\underline{u_1}} &= 0
\end{align*}
$$

We can take the curl of the momentum equation to introduce vorticity into the mix:

$$\begin{align*}
\frac{\tilde{D}_0\tilde{\zeta}_0}{\tilde{D}\tilde{t}} +\tilde{\nabla} \cdot \tilde{\underline{u_1}} + \tilde{v}_0 &= 0
\end{align*}
$$

We can substitute the expression for the ageostrophic velocity divergence from the mass conservation equation:

$$\begin{align*}
\frac{\tilde{D}_0\tilde{\zeta}_0}{\tilde{D}\tilde{t}} – \frac{\tilde{D}_0 \tilde{\eta}_0}{\tilde{D}\tilde{t}} + \tilde{v}_0 &= 0
\end{align*}
$$

Now we can introduce the streamfunction \(\psi\), defined such that

$$\begin{align*}
\frac{\partial \psi}{\partial x} &= v\\
\frac{\partial \psi}{\partial y} &= -u\\
\end{align*}
$$

The streamfunction is useful because it can be used to diagnose all the other variables in the system. The vorticity is equal to the Laplacian of the streamfunction, and the meridional velocity is the x-derivative of the streamfunction. As for the height, one can derive this relation from geostrophic balance:

$$\begin{align*}
f\frac{\partial \psi}{\partial x} &= g\frac{\partial \eta}{\partial x}\\
\implies \eta &\sim \frac{f \psi}{g}
\end{align*}
$$

After some additional non-dimensionalization, we arrive at the following equation:

$$\begin{align*}
\left(\frac{\tilde{D}_0}{\tilde{D}\tilde{t}} \left[\tilde{\nabla}^2 – \frac{1}{L_d^2}\right] + \frac{\partial}{\partial \tilde{x}}\right)\tilde{\psi}_0 &= 0
\end{align*}
$$

Re-dimensionalizing the equation:

$$\begin{align*}
\left(\frac{D_0}{Dt} \left[\nabla^2- \frac{1}{L_d^2}\right] + \beta\frac{\partial}{\partial x}\right)\psi_0 &= 0
\end{align*}
$$

Note that from here we can derive the potential vorticity by letting \(v = \frac{Dy}{Dt}\):

$$q = \left(\nabla^2 – \frac{1}{L_d^2}\right)\psi_0 + \beta y = \zeta_0 – \frac{f_0}{H}\eta_0 + \beta y$$

Comparison to shallow water PV

Recall:

$$q = \frac{\zeta + f_0 + \beta y}{H + \eta}$$

It turns out the two potential vorticity forms can be (approximately) linked via a Rossby number expansion of the variables! I will leave this as an exercise.

The important point is: by going one additional order beyond geostrophy, we were able to arrive at a potential vorticity evolution equation close to that of the true shallow water equations. QG is a simple, yet powerful way to add nonlinear contributions to the linear geostrophic equations. We will see in a future blog post that the same can be done for multi-layered or continuously stratified fluids, which will introduce us to the concept of baroclinicity.


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