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

$$ \psi = 0, \qquad \text{at} \qquad z \rightarrow -\infty\,, $$
$$
\hat \psi = \frac{f_0}{N} \frac{1}{\kappa}\hat b\,, \qquad \text{at} \qquad z = 0\,,
$$

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