Model

The QG model expresses the conservation of potential vorticity on the sphere.

Notations

Symbol	Description
\(\theta\)	co-latitude
\(\phi\)	longitude
\(q(\theta, \phi)\)	potential vorticity on the sphere
\(\psi(\theta, \phi)\)	stream function on the sphere
\(f'\)	bla bla
\(f_0\)	bla bla
\(\sigma\)	bla bla
\(p\)	bla bla

Model equations

The QG model equation reads

\[\frac{\partial q}{\partial t} = - \mathrm{J}(\psi, q) - D + S,\]

where \(\mathrm{J}\) is the Jacobian:

\[\mathrm{J}(\psi, q) \triangleq \frac{1}{R^{2}\mathrm{sin}\theta} \Bigg[ \frac{\partial\psi}{\partial\theta}\frac{\partial q}{\partial\phi} - \frac{\partial\psi}{\partial\phi}\frac{\partial q}{\partial\theta} \Bigg],\]

and where \(D\) and \(S\) encapsulate the dissipative and source processes, respectively. Furthermore, the potential vorticity \(q\) and the stream function \(\psi\) are related by the following Poisson-like equation:

\[q = \Delta \psi + f' + \frac{\partial}{\partial p} \bigg( \frac{f_{0}^{2}}{\sigma}\frac{\partial \psi}{\partial p}\bigg).\]

More details here about this equation + the orography correction

Dissipation processes

Three dissipation processes are included in the model: (i) thermal relaxation, (ii) Ekman friction, and (iii) hyper-diffusion. The diffusion term \(D\) aggregates the contribution of all three processes.

Thermal relaxation

Bla bla.

Ekman friction

Bla bla.

Hyper-diffusion

Bla bla.

Source processes

Bla bla.

Implementation with spherical harmonics

Jacobian term

Thermal relaxation term

Bla bla.

Ekman friction term

Bla bla.

Hyper-diffusion term

Bla bla.

Source term

Bla bla.