SQGE.tex

When testing finite element code it is useful to simplify the problem even
though in the real world stationary flows don't exist, the \emph{stationary} QGE
(SQGE) are useful, e.g. in testing code.  Additionally, the time dependence of
the QGE, \eqref{qge_psi_1}, adds additional complexity to the finite element
error analysis. Finite error analysis for time dependent problems usually can be
split into two parts; analysis of the spatial discretization that arises through
the application of finite elements, and the discretization error in time that
arises from the application of the method of lines. Thus, a FE error analysis of
the SQGE is a good push off point for analysis of the time-dependent QGE and is
therefore the main motivation for presenting the SQGE.  It is not only the
theory that motivates the study of the SQGE. In practice, the time
discretization for the QGE is built around that of the SQGE.

The SQGE are obtained by setting $\dfrac{\partial q}{\partial t}$ to $0$ in
\eqref{qge_q_psi_1} and therefore we get the \emph{streamfunction formulation}
of the \emph{one-layer stationary quasi-geostrophic equations}
\begin{eqnarray}
  Re^{-1} \, \Delta^2 \psi + J(\psi , \Delta \psi) - Ro^{-1} \, \frac{\partial
    \psi}{\partial x} = Ro^{-1} \, F .
  \label{eqn:SQGE_Psi}
\end{eqnarray}
Equations \eqref{qge_q_psi_1}-\eqref{qge_q_psi_2}, \eqref{qge_psi_1}, and
\eqref{eqn:SQGE_Psi} are the usual formulations of the one-layer QGE in
streamfunction-vorticity formulation, the one-layer QGE in streamfunction
formulation, and the steady one-layer QGE in streamfunction formulation,
respectively.

To completely specify the equations in \eqref{eqn:SQGE_Psi}, we need to impose
boundary conditions. For consistency and with the QGE \eqref{qge_psi_1}  we
consider
\begin{equation*}
  \psi = \frac{\partial \psi}{\partial \mathbf{n}} = 0 \qquad \text{on } \partial \Omega,
\end{equation*}
which, as stated previously, are also the boundary conditions used in
\cite{Gunzburger89} for the streamfunction formulation of the 2D NSE.