SingleLayerQG
Basic Equations
This module solves the barotropic or equivalent barotropic quasi-geostrophic vorticity equation on a beta plane of variable fluid depth $H - h(x, y)$. 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
\[ \underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{\partial_x v - \partial_y u}_{\text{relative vorticity}} \underbrace{ - \frac{1}{\ell^2} \psi}_{\text{vortex stretching}} + \underbrace{\frac{f_0 h}{H}}_{\text{topographic PV}} ,\]
where $\ell$ is the Rossby radius of deformation. Purely barotropic dynamics corresponds to infinite Rossby radius of deformation ($\ell = \infty$), while a flow with a finite Rossby radius follows is said to obey equivalent-barotropic dynamics. We denote the sum of the relative vorticity and the vortex stretching contributions to the QGPV with $q \equiv \nabla^2 \psi - \psi / \ell^2$. Also, we denote the topographic PV with $\eta \equiv f_0 h / H$.
The dynamical variable is $q$. Thus, the equation solved by the module is:
\[\partial_t q + \mathsf{J}(\psi, q + \eta) + \beta \partial_x \psi = \underbrace{-\left[\mu + \nu(-1)^{n_\nu} \nabla^{2n_\nu} \right] q}_{\textrm{dissipation}} + F .\]
where $\mathsf{J}(a, b) = (\partial_x a)(\partial_y b)-(\partial_y a)(\partial_x b)$ is the two-dimensional Jacobian. On the right hand side, $F(x, y, t)$ is forcing, $\mu$ is linear drag, and $\nu$ is hyperviscosity of order $n_\nu$. Plain old viscosity corresponds to $n_\nu = 1$.
Implementation
The equation is time-stepped forward in Fourier space:
\[\partial_t \widehat{q} = - \widehat{\mathsf{J}(\psi, q + \eta)} + \beta \frac{i k_x}{|𝐤|^2 + 1/\ell^2} \widehat{q} - \left(\mu + \nu |𝐤|^{2n_\nu} \right) \widehat{q} + \widehat{F} .\]
The state variable sol is the Fourier transform of the sum of relative vorticity and vortex stretching (when the latter is applicable), qh.
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 linear operator is constructed in Equation
GeophysicalFlows.SingleLayerQG.Equation — FunctionEquation(params::BarotropicQGParams, grid)Return the equation for a barotropic QG problem with params and grid. Linear operator $L$ includes bottom drag $μ$, (hyper)-viscosity of order $n_ν$ with coefficient $ν$ and the $β$ term:
\[L = - μ - ν |𝐤|^{2 n_ν} + i β k_x / |𝐤|² .\]
The nonlinear term is computed via calcN! function.
Equation(params::EquivalentBarotropicQGParams, grid)Return the equation for an equivalent-barotropic QG problem with params and grid. Linear operator $L$ includes bottom drag $μ$, (hyper)-viscosity of order $n_ν$ with coefficient $ν$ and the $β$ term:
\[L = -μ - ν |𝐤|^{2 n_ν} + i β k_x / (|𝐤|² + 1/ℓ²) .\]
The nonlinear term is computed via calcN! function.
The nonlinear terms are computed via
GeophysicalFlows.SingleLayerQG.calcN! — FunctioncalcN!(N, sol, t, clock, vars, params, grid)Calculate the nonlinear term, that is the advection term and the forcing,
\[N = - \widehat{𝖩(ψ, q + η)} + F̂ .\]
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
Some helper functions included in the module are:
GeophysicalFlows.SingleLayerQG.updatevars! — Functionupdatevars!(sol, vars, params, grid)Update the variables in vars with the solution in sol.
GeophysicalFlows.SingleLayerQG.set_q! — Functionset_q!(prob, q)Set the solution of problem, prob.sol as the transform of $q$ and update variables prob.vars.
Diagnostics
The kinetic energy of the fluid is computed via:
GeophysicalFlows.SingleLayerQG.kinetic_energy — Functionkinetic_energy(prob)Return the problem's (prob) domain-averaged kinetic energy of the fluid. Since $u² + v² = |{\bf ∇} ψ|²$, the domain-averaged kinetic energy is
\[\int \frac1{2} |{\bf ∇} ψ|² \frac{𝖽x 𝖽y}{L_x L_y} = \sum_{𝐤} \frac1{2} |𝐤|² |ψ̂|² .\]
while the potential energy, for an equivalent barotropic fluid, is computed via:
GeophysicalFlows.SingleLayerQG.potential_energy — Functionpotential_energy(prob)Return the problem's (prob) domain-averaged potential energy of the fluid,
\[\int \frac1{2} \frac{ψ²}{ℓ²} \frac{𝖽x 𝖽y}{L_x L_y} = \sum_{𝐤} \frac1{2} \frac{|ψ̂|²}{ℓ²} .\]
The total energy is:
GeophysicalFlows.SingleLayerQG.energy — Functionenergy(prob)Return the problem's (prob) domain-averaged total energy of the fluid, that is, the kinetic energy for a pure barotropic flow or the sum of kinetic and potential energies for an equivalent barotropic flow.
Other diagnostic include: energy_dissipation, energy_drag, energy_work, enstrophy_dissipation, and enstrophy_drag, enstrophy_work.
Examples
examples/singlelayerqg_betadecay.jl: Simulate decaying quasi-geostrophic flow on a beta plane demonstrating zonation.examples/singlelayerqg_betaforced.jl: Simulate forced-dissipative quasi-geostrophic flow on a beta plane demonstrating zonation. The forcing is temporally delta-correlated with isotropic spatial structure with power in a narrow annulus in wavenumber space with total wavenumber $k_f$.examples/singlelayerqg_decay_topography.jl: Simulate two dimensional turbulence (barotropic quasi-geostrophic flow with $\beta=0$) above topography.examples/singlelayerqg_decaying_barotropic_equivalentbarotropic.jl: Simulate two dimensional turbulence ($\beta=0$) with both infinite and finite Rossby radius of deformation and compares the evolution of the two.