Surface Quasi-geostrophic Model

Surface quasi-geostrophy (SQG) is a relatively simple model that describes surface intensified flows due to buoyancy. One of it’s advantages is that it only has two spatial dimensions but describes a three-dimensional solution.

The evolution equation is

\[\partial_t b + \mathsf{J}(\psi, b) = 0\,, \qquad \text{at} \qquad z = 0\,,\]

where \(b = \psi_z\) is the buoyancy.

The interior potential vorticity is zero. Hence

\[\frac{\partial }{\partial z}\left(\frac{f_0^2}{N^2}\frac{\partial \psi}{\partial z}\right) + \nabla^2\psi = 0\,,\]

where \(N\) is the buoyancy frequency and \(f_0\) is the Coriolis parameter. In the SQG model both \(N\) and \(f_0\) are constants. The boundary conditions for this elliptic problem in a semi-infinite vertical domain are

\[b = \psi_z\,, \qquad \text{and} \qquad z = 0\,,\]

and

\[\psi = 0, \qquad \text{at} \qquad z \rightarrow -\infty\,.\]

The solutions to the elliptic problem above*, in horizontal Fourier space, gives the inversion relationship between surface buoyancy and surface streamfunction

\[\widehat{\psi} = \frac{f_0}{N} \frac{1}{\kappa} \widehat{b}\,, \qquad \text{at} \qquad z = 0\,.\]

The SQG evolution equation is marched forward similarly to the two-layer model.

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* Since understanding this step is key to making your own modifications to the model, in more detail:

\[\frac{\partial }{\partial z}\left(\frac{f_0^2}{N^2}\frac{\partial \psi(x,y,z)}{\partial z}\right) + \nabla^2\psi(x,y,z) = 0\,\]

Taking the Fourier transform in the x and y directions with \(\kappa^2 = k^2 + l^2\) we get

\[\frac{f_0^2}{N^2}\frac{\partial }{\partial z}\left(\frac{\partial \hat \psi}{\partial z}\right) = \kappa^2 \hat\psi\,,\]

which has solution

\[\hat \psi = Ae^{\frac{\kappa N}{f_0}z} + Be^{-\frac{\kappa N}{f_0}z},.\]

Our decay at negative infinity immediately tells us that \(B = 0\). Differentiating with respect to \(z\) and evaluating at the surface tells us \(A = f_0 \hat b / \kappa N\) so that we have:

\[\hat \psi(k,l,z) = \frac{f_0}{N}\frac{1}{\kappa} \hat b(k,l,z) e^{\frac{\kappa N}{f_0}z},.\]

Evaluating at \(z = 0\) gives the inversion relation given above.