This module solves the quasi-linear quasi-geostrophic barotropic vorticity equation on a beta-plane of variable fluid depth
where overline above denotes a zonal mean,
- Farrell, B. F. and Ioannou, P. J. (2003). Structural stability of turbulent jets. J. Atmos. Sci., 60, 2101-2118.
- Srinivasan, K. and Young, W. R. (2012). Zonostrophic instability. Phys. Rev. Lett., 69 (5), 1633-1656.
- Constantinou, N. C., Farrell, B. F., and Ioannou, P. J. (2014). Emergence and equilibration of jets in beta-plane turbulence: applications of Stochastic Structural Stability Theory. J. Atmos. Sci., 71 (5), 1818-1842.
As in the BarotropicQG module, the flow is obtained through a streamfunction
$$\underbrace{f_0 + \beta y}{\text{planetary PV}} + \underbrace{(\partial_x \upsilon - \partial_y u)}{\text{relative vorticity}} + \underbrace{\frac{f_0 h}{H}}_{\text{topographic PV}}.$$
The dynamical variable is the component of the vorticity of the flow normal to the plane of motion,
$$\partial_t \overline{\zeta} + \mathsf{J}(\overline{\psi}, \underbrace{\overline{\zeta} + \overline{\eta}}{\equiv \overline{q}}) + \overline{\mathsf{J}(\psi', \underbrace{\zeta' + \eta'}{\equiv q'})} = \underbrace{-\left[\mu + \nu(-1)^{n_\nu} \nabla^{2n_\nu} \right] \overline{\zeta} }_{\textrm{dissipation}} \ .$$
$$\partial_t \zeta' + \mathsf{J}(\psi', \overline{q}) + \mathsf{J}(\overline{\psi}, q') + \underbrace{\mathsf{J}(\psi', q') - \overline{\mathsf{J}(\psi', q')}}{\textrm{EENL}} + \beta\partial_x\psi' = \underbrace{-\left[\mu + \nu(-1)^{n\nu} \nabla^{2n_\nu} \right] \zeta'}_{\textrm{dissipation}} + f\ .$$
where
Quasi-linear dynamics neglect the term eddy-eddy nonlinearity (EENL) term above.
The equation is time-stepped forward in Fourier space:
In doing so the Jacobian is computed in the conservative form:
Thus:
examples/barotropicqgql_betaforced.jl
: A script that simulates forced-dissipative quasi-linear quasi-geostrophic flow on a beta-plane demonstrating zonation. The forcing is temporally delta-correlated and its spatial structure is isotropic with power in a narrow annulus of total radiuskf
in wavenumber space. This example demonstrates that the anisotropic inverse energy cascade is not required for zonation.