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

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

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

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

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