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33This module solves the quasi-geostrophic barotropic vorticity equation on a
44beta-plane of variable fluid depth $H-h(x,y)$. The flow is obtained through a
5- streamfunction $\psi$ as $(u,v ) = (-\partial_y\psi, \partial_x\psi)$. All flow
5+ streamfunction $\psi$ as $(u,\upsilon ) = (-\partial_y\psi, \partial_x\psi)$. All flow
66fields can be obtained from the quasi-geostrophic potential vorticity (QGPV).
77Here the QGPV is
88
9- $$ \underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{(\partial_x v
9+ $$ \underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{(\partial_x \upsilon
1010 - \partial_y u)}_{\text{relative vorticity}} +
1111 \underbrace{\frac{f_0 h}{H}}_{\text{topographic PV}}. $$
1212
3737
3838$$ \mathcal{L} = \beta\frac{\mathrm{i}k_x}{k^2} - \mu - \nu k^{2n_\nu}\ , $$
3939$$ \mathcal{N}(\widehat{\zeta}) = - \mathrm{i}k_x \mathrm{FFT}(u q)-
40- \mathrm{i}k_y \mathrm{FFT}(v q)\ . $$
40+ \mathrm{i}k_y \mathrm{FFT}(\upsilon q)\ . $$
Original file line number Diff line number Diff line change 11# TwoDTurb Module
22
33This module solves two-dimensional incompressible turbulence. The flow is given
4- through a streamfunction $\psi$ as $(u,v ) = (-\partial_y\psi, \partial_x\psi)$.
4+ through a streamfunction $\psi$ as $(u,\upsilon ) = (-\partial_y\psi, \partial_x\psi)$.
55The dynamical variable used here is the component of the vorticity of the flow
6- normal to the plane of motion, $q=\partial_x v - \partial_y u = \nabla^2\psi$.
6+ normal to the plane of motion, $q=\partial_x \upsilon - \partial_y u = \nabla^2\psi$.
77The equation solved by the module is:
88
99$$ \partial_t q + J(\psi, q) = \underbrace{-\left[\mu(-1)^{n_\mu} \nabla^{2n_\mu}
2626
2727$$ \mathcal{L} = -\mu k^{-2n_\mu} - \nu k^{2n_\nu}\ , $$
2828$$ \mathcal{N}(\widehat{q}) = - \mathrm{i}k_x \mathrm{FFT}(u q)-
29- \mathrm{i}k_y \mathrm{FFT}(v q)\ . $$
29+ \mathrm{i}k_y \mathrm{FFT}(\upsilon q)\ . $$
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