Model equations

Here we provide that governing equations of the two-layer QG channel model.

Governing equations

The governing equations for the present two-layer QG model are

\[\begin{equation} \partial_{t} q_{1} + \mathrm{J} (\psi_{1}, \, q_{1} ) = \kappa_\mathrm{T} \left ( \frac{\psi_{1} - \psi_{2}}{2} - \tau_\mathrm{eq.} \right ) - \nu \nabla^4 q_{1} \end{equation}\]

\[\begin{equation} \partial_{t} q_{2} + \mathrm{J} (\psi_{2}, \, q_{2} ) = - \kappa_\mathrm{T} \left ( \frac{\psi_{1} - \psi_{2}}{2} - \tau_\mathrm{eq.} \right ) - \mu \nabla^2 \psi_{2} - \nu \nabla^4 q_{2} \end{equation}\]

Linear friction in the lower layer has the coefficient \(\mu\), with dimensions of inverse time. Hyperviscosity is applied to the streamfunction in both layers and has the coefficient \(\nu\) with dimensions of \(\mathrm{L}^{4} \mathrm{T}\). \(\kappa_\mathrm{T}\) is an inverse time scale for damping the baroclinic flow to the background streamfunction, \(\tau_\mathrm{eq.}\).

We now describe a specific example.