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