Equations of motions, grids

Continuous form

Central state variables are the geostrophic streamfunction \(\psi\) and potential vorticity \(q\) which are related according to:

\[q(x,y,z) = f-f_0 + \Delta \psi + \partial_z \Big ( \frac{f_0^2}{N^2} \partial_z \psi \Big )\]

where \(f_0\) is the averaged Coriolis parameter and \(N(z)\) is the buoyancy frequency.

Density anomalies and geostrophic currents are related to the streamfunction according to:

\[\begin{split}\partial_z \psi &= - \frac{g\rho}{\rho_0 f_0} \\ (u,v) &= (-\partial_y \psi, \partial_x \psi)\end{split}\]

The evolution of the system is governed by the advection of potential vorticity and top and bottom densities by geostrophic currents:

\[\begin{split}\partial_t q + J(\psi,q) + J(\Psi,q) + J(\psi,Q) &= 0 \\ \partial_t \partial_z \psi + J(\psi,\partial_z \psi) + J(\Psi,\partial_z \psi) + J(\psi,\partial_z \Psi) &= 0 \mathrm{\;at\;} z=0,-h\end{split}\]

where capitals represent the large scale - slowly evolving background.

Following Arakawa and Moorthi 1988, we solve for a generalized potential vorticity \(\tilde{q}\):

\[\begin{split}\tilde{q}(x,y,z) &= f-f_0 + \Delta \psi + \partial_z \Big ( \frac{f_0^2}{N^2} \partial_z \psi \Big ) - \frac{f_0^2}{N^2} \partial_z \psi \delta(z=0) + \frac{f_0^2}{N^2} \partial_z \psi \delta(z=-h) \\ &= f-f_0 + \Delta \psi + \partial_z \Big ( \frac{f_0^2}{N^2} \partial_z \psi \Big ) + \frac{f_0}{N^2} \frac{g\rho}{\rho_0} \delta(z=0) - \frac{f_0}{N^2} \frac{g\rho}{\rho_0} \delta(z=-h)\end{split}\]

where \(\delta(z=0)=1/dz\) at \(z=0\) (corresponds to \(\rho_{kup}\), see description of the vertical grid) and \(\delta(z=-h)=1/dz\) at \(z=-h\) (corresponds to \(\rho_{kdown}\))

The quasi-geostrophic evolution is then solely described by the advection of \(\tilde{q}\):

\[\partial_t \tilde{q} + J(\psi,\tilde{q}) + J(\Psi,\tilde{q}) + J(\psi,\tilde{Q}) = 0\]

Vertical grid

The vertical grid is Charney-Phillips type, meaning streamfunction and potential vorticity are on identical vertical levels while density is at intermediate levels.

Horizontal grid