# SingleLayerQG decaying 2D turbulence with and without finite Rossby radius of deformation

This example can be viewed as a Jupyter notebook via .

We use here the SingleLayerQG module to simulate decaying two-dimensional turbulence and investigate how does a finite Rossby radius of deformation affects its evolution.

## Install dependencies

First let's make sure we have all required packages installed.

using Pkg
pkg"add GeophysicalFlows, Printf, Random, Plots"

## Let's begin

Let's load GeophysicalFlows.jl and some other needed packages.

using GeophysicalFlows, Printf, Random, Plots

using GeophysicalFlows: peakedisotropicspectrum
using Random: seed!

## Choosing a device: CPU or GPU

dev = CPU()     # Device (CPU/GPU)

## Numerical, domain, and simulation parameters

First, we pick some numerical and physical parameters for our model.

n, L  = 128, 2π             # grid resolution and domain length

# Then we pick the time-stepper parameters
dt = 1e-2  # timestep
nsteps = 4000  # total number of steps
nsubs = 20    # number of steps between each plot

## Problem setup

We initialize two problems by providing a set of keyword arguments to the Problem constructor. The two problems are otherwise the same, except one has an infinite deformation radius, prob_bqg, and the other has finite deformation radius, prob_eqbqg.

prob_bqg = SingleLayerQG.Problem(dev; nx=n, Lx=L,
dt=dt, stepper="FilteredRK4", aliased_fraction=0)
dt=dt, stepper="FilteredRK4", aliased_fraction=0)

## Setting initial conditions

For initial condition we construct a relative vorticity with energy most energy around total wavenumber $k_0$.

seed!(1234)
k₀, E₀ = 6, 0.5
∇²ψ₀ = peakedisotropicspectrum(prob_bqg.grid, k₀, E₀, mask=prob_bqg.timestepper.filter)

SingleLayerQG allows us to set up the initial $q$ for each problem via set_q!() function. To initialize both prob_bqg and prob_eqbqg with the same flow, we first use function SingleLayerQG.streamfunctionfrompv! to get the streamfunction that corresponds to the relative vorticity we computed above. This works in the purely barotropic problem, prob_bqg since in that case the QGPV is simply the relative vorticity.

∇²ψ₀h = rfft(∇²ψ₀)
ψ₀h = @. 0 * ∇²ψ₀h
SingleLayerQG.streamfunctionfrompv!(ψ₀h, ∇²ψ₀h, prob_bqg.params, prob_bqg.grid)

and then use the streamfunction to compute the corresponding $q_0$ for each problem,

q₀_bqg   = irfft(-prob_bqg.grid.Krsq .* ψ₀h, prob_bqg.grid.nx)
q₀_eqbqg = irfft(-(prob_eqbqg.grid.Krsq .+ 1/prob_eqbqg.params.deformation_radius^2) .* ψ₀h, prob_bqg.grid.nx)

Now we can initialize our problems with the same flow.

SingleLayerQG.set_q!(prob_bqg, q₀_bqg)
SingleLayerQG.set_q!(prob_eqbqg, q₀_eqbqg)

Let's plot the initial vorticity field for each problem. A function that returns relative vorticity from each problem's state variable will prove useful. Note that when plotting, we decorate the variable to be plotted with Array() to make sure it is brought back on the CPU when the variable lives on the GPU.

relativevorticity(prob) = irfft(-prob.grid.Krsq .* prob.vars.ψh, prob.grid.nx)

x, y = prob_bqg.grid.x, prob_bqg.grid.y

p_bqg = heatmap(x, y, Array(relativevorticity(prob_bqg)'),
aspectratio = 1,
c = :balance,
clim = (-40, 40),
xlims = (-L/2, L/2),
ylims = (-L/2, L/2),
xticks = -3:3,
yticks = -3:3,
xlabel = "x",
ylabel = "y",
title = "barotropic\n ∇²ψ, t=" * @sprintf("%.2f", prob_bqg.clock.t),
framestyle = :box)

p_eqbqg = heatmap(x, y, Array(relativevorticity(prob_eqbqg)'),
aspectratio = 1,
c = :balance,
clim = (-40, 40),
xlims = (-L/2, L/2),
ylims = (-L/2, L/2),
xticks = -3:3,
yticks = -3:3,
xlabel = "x",
ylabel = "y",
title = "equivalent barotropic; deformation radius: " * @sprintf("%.2f", prob_eqbqg.params.deformation_radius) * "\n ∇²ψ, t=" * @sprintf("%.2f", prob_eqbqg.clock.t),
framestyle = :box)

l = @layout Plots.grid(1, 2)
p = plot(p_bqg, p_eqbqg, layout = l, size = (800, 380))

## Time-stepping the Problem forward

Now we time-step both problems forward and animate the relative vorticity in each case.

startwalltime = time()

cfl(prob) = prob.clock.dt * maximum([maximum(prob.vars.u) / prob.grid.dx, maximum(prob.vars.v) / prob.grid.dy])

anim = @animate for j = 0:Int(nsteps/nsubs)
if j % (1000 / nsubs) == 0
log_bqg = @sprintf("barotropic; step: %04d, t: %d, cfl: %.2f, walltime: %.2f min",
prob_bqg.clock.step, prob_bqg.clock.t, cfl(prob_bqg), (time()-startwalltime)/60)
println(log_bqg)

log_eqbqg = @sprintf("equivalent barotropic; step: %04d, t: %d, cfl: %.2f, walltime: %.2f min",
prob_eqbqg.clock.step, prob_eqbqg.clock.t, cfl(prob_eqbqg), (time()-startwalltime)/60)
println(log_eqbqg)
end

p[:z] = Array(relativevorticity(prob_bqg))
p[:title] = "barotropic\n ∇²ψ, t=" * @sprintf("%.2f", prob_bqg.clock.t)
p[:z] = Array(relativevorticity(prob_eqbqg))
p[:title] = "equivalent barotropic; deformation radius: " * @sprintf("%.2f", prob_eqbqg.params.deformation_radius) * "\n ∇²ψ, t=" * @sprintf("%.2f", prob_eqbqg.clock.t)

stepforward!(prob_bqg, nsubs)
SingleLayerQG.updatevars!(prob_bqg)

stepforward!(prob_eqbqg, nsubs)
SingleLayerQG.updatevars!(prob_eqbqg)
end

mp4(anim, "singlelayerqg_barotropic_equivalentbarotropic.mp4", fps=18)