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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 $q$. Including an imposed zonal flow $U(y)$, the equation of motion is:

\[\partial_t q + \mathsf{J}(\psi, q) + (U - \partial_y \psi) \partial_x Q + U \partial_x q + (\partial_y Q)(\partial_x \psi) = \underbrace{-\left[\mu + \nu(-1)^{n_\nu} \nabla^{2n_\nu} \right] q}_{\textrm{dissipation}} + F ,\]

with

\[\begin{aligned} \partial_y Q & \equiv \beta - \partial_y^2 U + \partial_y \eta , \\ \partial_x Q & \equiv \partial_x \eta , \end{aligned}\]

the background PV gradient components, and with $\mathsf{J}(a, b) = (\partial_x a)(\partial_y b) - (\partial_y a)(\partial_x b)$ 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 $n_\nu = 1$.

If the imposed background zonal flow is constant over the whole domain (doesn't depend on $y$), the above simplifies to:

\[\partial_t q + \mathsf{J}(\psi, q + \eta) + U \partial_x (q + \eta) + β \partial_x \psi = \underbrace{-\left[\mu + \nu(-1)^{n_\nu} \nabla^{2n_\nu} \right] q}_{\textrm{dissipation}} + F ,\]

and thus the mean advection of $q$ by $U$ is incorporated in the linear term $L$.

Implementation

The equation is time-stepped forward in Fourier space:

\[\partial_t \widehat{q} = - \widehat{\mathsf{J}(\psi, q)} - \widehat{U \partial_x Q} - \widehat{U \partial_x q} + \widehat{(\partial_y \psi) (\partial_x Q)} - \widehat{(\partial_x \psi)(\partial_y Q)} - \left(\mu + \nu |𝐤|^{2n_\nu} \right) \widehat{q} + \widehat{F} .\]

In doing so 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 state variable sol is the Fourier transform of the sum of relative vorticity and vortex stretching (when the latter is applicable), qh.

The linear operator is constructed in Equation

GeophysicalFlows.SingleLayerQG.Equation — Function
Equation(params::BarotropicQGParams, grid)
Equation(params::EquivalentBarotropicQGParams, grid)

Return the equation for a SingleLayerQG 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. If there is a constant background flow $U$ that does not vary in $y$ then the linear term $L$ includes also the mean advection term by $U$, namely $-i k_x U$`. That is:

\[L = -μ - ν |𝐤|^{2 n_ν} + i β k_x / (|𝐤|² + 1/ℓ²) - i k_x U .\]

The nonlinear term is computed via calcN!.

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The nonlinear terms are computed via

GeophysicalFlows.SingleLayerQG.calcN! — Function
calcN!(N, sol, t, clock, vars, params, grid)

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

\[N = - \widehat{𝖩(ψ, q + η)} - \widehat{U ∂_x (q + η)} + \widehat{(∂_x ψ)(∂_y² U)} + F̂ .\]

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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:

Diagnostics

The kinetic energy of the fluid is computed via:

GeophysicalFlows.SingleLayerQG.kinetic_energy — Function
kinetic_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

\[\int \frac1{2} |{\bf ∇} ψ|² \frac{𝖽x 𝖽y}{L_x L_y} = \sum_{𝐤} \frac1{2} |𝐤|² |ψ̂|² .\]

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while the potential energy, for an equivalent barotropic fluid, is computed via:

GeophysicalFlows.SingleLayerQG.potential_energy — Function
potential_energy(prob)

Return the problem's (prob) domain-averaged potential energy of the fluid,

\[\int \frac1{2} \frac{ψ²}{ℓ²} \frac{𝖽x 𝖽y}{L_x L_y} = \sum_{𝐤} \frac1{2} \frac{|ψ̂|²}{ℓ²} .\]

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The total energy is:

GeophysicalFlows.SingleLayerQG.energy — Function
energy(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.

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Other diagnostic include: energy_dissipation, energy_drag, energy_work, enstrophy_dissipation, and enstrophy_drag, enstrophy_work.

Examples