# Barotropic quasigeostrophic model

### Model equations
The QGModel class solves the two-dimensional equation

\begin{equation}\tag{1}
q_t + \beta \psi_x + J(\psi,q) =  \nu \triangle q -\nu_4 \triangle^2 q - \mu q  \,,
\end{equation}

where the vorticity, $q$, is related to the streamfunction via the inversion relation

\begin{equation}\tag{2}
q = \triangle \psi\, .
\end{equation}


### Power integrals

The kinetic energy, $K_e \equiv \langle |\nabla \psi|^2 \rangle/2$, satisfies

\begin{equation}\tag{3}
\dot K_e = \underbrace{-\nu \langle|\nabla \psi|^2\rangle + \nu_4 \langle q \triangle q\rangle - \mu \langle |\nabla \psi|^2 \rangle}_{\equiv\,\,\varepsilon_\psi}\,.
\end{equation}

The enstrophy, $S_e \equiv \langle q^2\rangle/2$, satisfies

\begin{equation}\tag{4}
\dot S_e = \underbrace{-\beta\langle q \psi_x\rangle}_{\equiv\,\,\Gamma_q}\,\,\, \underbrace{-\nu \langle|\nabla q|^2\rangle - \nu_4 \langle (\triangle q)^2\rangle - \mu \langle q^2 \rangle}_{\equiv\,\, \chi_q}\,.
\end{equation}

f97e8b6435cd2d004810df8f945aa10fb2a65030
ipynb
Jupyter Notebook
docs/QGModel.ipynb
crocha700/NIWQG
17971b8b7e1e16261a023515f9142d0e89fb1b16
MIT
2019-03-25T23:03:30.000Z
2020-07-01T08:25:13.000Z
docs/QGModel.ipynb
crocha700/NIWQG
17971b8b7e1e16261a023515f9142d0e89fb1b16
MIT
docs/QGModel.ipynb
crocha700/NIWQG
17971b8b7e1e16261a023515f9142d0e89fb1b16
MIT
2020-07-01T08:24:37.000Z
2020-07-01T08:24:37.000Z
Qwen/Qwen-72B
1. YES
2. YES