Basic Equations

This module solves the quasi-linear quasi-geostrophic barotropic vorticity equation on a beta plane of variable fluid depth $H - h(x, y)$. Quasi-linear refers to the dynamics that neglects the eddy–eddy interactions in the eddy evolution equation after an eddy–mean flow decomposition, e.g.,

\[\phi(x, y, t) = \overline{\phi}(y, t) + \phi'(x, y, t) ,\]

where overline above denotes a zonal mean, $\overline{\phi}(y, t) = \int \phi(x, y, t) \, 𝖽x / L_x$, and prime denotes deviations from the zonal mean. This approximation is used in many process-model studies of zonation, e.g., Farrell and Ioannou (2003), Srinivasan and Young (2012), and Constantinou et al. (2014).

As in the SingleLayerQG module, the flow is obtained through a streamfunction $\psi$ as $(u, v) = (-\partial_y \psi, \partial_x \psi)$. All flow fields can be obtained from the quasi-geostrophic potential vorticity (QGPV). Here, the QGPV is

\[\underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{\partial_x v - \partial_y u}_{\text{relative vorticity}} + \underbrace{\frac{f_0 h}{H}}_{\text{topographic PV}} .\]

The dynamical variable is the component of the vorticity of the flow normal to the plane of motion, $\zeta \equiv \partial_x v - \partial_y u = \nabla^2 \psi$. Also, we denote the topographic PV with $\eta \equiv f_0 h / H$. After we apply the eddy-mean flow decomposition above, the QGPV dynamics are:

\[\begin{aligned} \partial_t \overline{\zeta} + \mathsf{J}(\overline{\psi}, \overline{\zeta} + \overline{\eta}) + \overline{\mathsf{J}(\psi', \zeta' + \eta')} & = \underbrace{- \left[\mu + \nu(-1)^{n_\nu} \nabla^{2n_\nu} \right] \overline{\zeta} }_{\textrm{dissipation}} , \\ \partial_t \zeta' + \mathsf{J}(\psi', \overline{\zeta} + \overline{\eta}) + \mathsf{J}(\overline{\psi}, \zeta' + \eta') + & \underbrace{\mathsf{J}(\psi', \zeta' + \eta') - \overline{\mathsf{J}(\psi', \zeta' + \eta')}}_{\textrm{EENL}} + \beta \partial_x \psi' = \\ & = \underbrace{-\left[\mu + \nu(-1)^{n_\nu} \nabla^{2n_\nu} \right] \zeta'}_{\textrm{dissipation}} + F . \end{aligned}\]

where $\mathsf{J}(a, b) = (\partial_x a)(\partial_y b) - (\partial_y a)(\partial_x b)$. On the right hand side, $F(x, y, t)$ is forcing (which is assumed to have zero zonal mean, $\overline{F} = 0$), $\mu$ is linear drag, and $\nu$ is hyperviscosity. Plain old viscosity corresponds to $n_{\nu} = 1$.

Quasi-linear dynamics neglects the term eddy-eddy nonlinearity (EENL) term above.


The equation is time-stepped forward in Fourier space:

\[\partial_t \widehat{\zeta} = - \widehat{\mathsf{J}(\psi, \zeta + \eta)}^{\textrm{QL}} + \beta \frac{i k_x}{|𝐤|^2} \widehat{\zeta} - \left ( \mu + \nu |𝐤|^{2n_\nu} \right ) \widehat{\zeta} + \widehat{F} .\]

The state variable sol is the Fourier transform of vorticity, ζh.

The Jacobian is computed in the conservative form: $\mathsf{J}(f, g) = \partial_y [ (\partial_x f) g] - \partial_x [ (\partial_y f) g]$. The superscript QL on the Jacobian term above denotes that triad interactions that correspond to the EENL term are removed.

The linear operator is constructed in Equation

Equation(params, grid)

Return the equation for two-dimensional barotropic QG QL problem with parameters params and on grid. Linear operator $L$ includes bottom drag $μ$, (hyper)-viscosity of order $n_ν$ with coefficient $ν$ and the $β$ term:

\[L = - μ - ν |𝐤|^{2 n_ν} + i β k_x / |𝐤|² .\]

Nonlinear term is computed via calcN! function.


and the nonlinear terms are computed via

calcN!(N, sol, t, clock, vars, params, grid)

Calculate the nonlinear term, that is the advection term and the forcing,

\[N = - \widehat{𝖩(ψ, ζ + η)}^{\mathrm{QL}} + F̂ .\]


which in turn calls calcN_advection! and addforcing!.

Parameters and Variables

All required parameters are included inside Params and all module variables are included inside Vars.

For the decaying case (no forcing, $F = 0$), variables are constructed with Vars. For the forced case ($F \ne 0$) variables are constructed with either ForcedVars or StochasticForcedVars.

Helper functions


The kinetic energy of the fluid is obtained via:

while the enstrophy via:

Other diagnostic include: dissipation, drag, and work.


  • examples/barotropicqgql_betaforced.jl: Simulate 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 radius $k_f$ in wavenumber space. This example demonstrates that the anisotropic inverse energy cascade is not required for zonation.