# 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 $n_\nu = 1$.

In the case that the imposed background zonal flow is just a constant, 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 advection of $q$ can be 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!`

function.

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,

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

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!`

— Function`updatevars!(sol, vars, params, grid)`

Update the variables in `vars`

with the solution in `sol`

.

`GeophysicalFlows.SingleLayerQG.set_q!`

— Function`set_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`

— 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

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

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,

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

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.

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.