Quasi-Linear forced-dissipative barotropic QG beta-plane turbulence

A simulation of forced-dissipative barotropic quasi-geostrophic turbulence on a beta plane under the quasi-linear approximation. The dynamics include linear drag and stochastic excitation.

Install dependencies

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

using Pkg
pkg"add GeophysicalFlows, CUDA, CairoMakie"

Let's begin

Let's load GeophysicalFlows.jl and some other packages we need.

using GeophysicalFlows, CUDA, Random, Printf, CairoMakie

using Statistics: mean

parsevalsum = FourierFlows.parsevalsum
record = CairoMakie.record                # disambiguate between CairoMakie.record and CUDA.record

Choosing a device: CPU or GPU

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

Numerical parameters and time-stepping parameters

      n = 128            # 2D resolution = n^2
stepper = "FilteredRK4"  # timestepper
     dt = 0.05           # timestep
 nsteps = 8000           # total number of time-steps
 nsubs  = 10             # number of time-steps for intermediate logging/plotting (nsteps must be multiple of nsubs)

Physical parameters

L = 2π        # domain size
β = 10.0      # planetary PV gradient
μ = 0.01      # bottom drag

Forcing

We force the vorticity equation with stochastic excitation that is delta-correlated in time and while spatially homogeneously and isotropically correlated. The forcing has a spectrum with power in a ring in wavenumber space of radius $k_f$ (forcing_wavenumber) and width $δ_f$ (forcing_bandwidth), and it injects energy per unit area and per unit time equal to $\varepsilon$. That is, the forcing covariance spectrum is proportional to $\exp{[-(|\bm{k}| - k_f)^2 / (2 δ_f^2)]}$.

forcing_wavenumber = 14.0 * 2π/L  # the forcing wavenumber, `k_f`, for a spectrum that is a ring in wavenumber space
forcing_bandwidth  = 1.5  * 2π/L  # the width of the forcing spectrum, `δ_f`
ε = 0.001                         # energy input rate by the forcing

grid = TwoDGrid(dev; nx=n, Lx=L)

K = @. sqrt(grid.Krsq)            # a 2D array with the total wavenumber

forcing_spectrum = @. exp(-(K - forcing_wavenumber)^2 / (2 * forcing_bandwidth^2))
@CUDA.allowscalar forcing_spectrum[grid.Krsq .== 0] .= 0 # ensure forcing has zero domain-average

ε0 = parsevalsum(forcing_spectrum .* grid.invKrsq / 2, grid) / (grid.Lx * grid.Ly)
@. forcing_spectrum *= ε/ε0       # normalize forcing to inject energy at rate ε

We reset of the random number generator for reproducibility

if dev==CPU(); Random.seed!(1234); else; CUDA.seed!(1234); end

Next we construct function calcF! that computes a forcing realization every timestep. For that, we call randn! to obtain complex numbers whose real and imaginary part are normally-distributed with zero mean and variance 1/2.

function calcF!(Fh, sol, t, clock, vars, params, grid)
  randn!(Fh)
  @. Fh *= sqrt(forcing_spectrum) / sqrt(clock.dt)
  return nothing
end

Problem setup

We initialize a Problem by providing a set of keyword arguments. We use stepper = "FilteredRK4". Filtered timesteppers apply a wavenumber-filter at every time-step that removes enstrophy at high wavenumbers and, thereby, stabilize the problem, despite that we use the default viscosity coefficient ν=0. Thus, we choose not to do any dealiasing by providing aliased_fraction=0.

prob = BarotropicQGQL.Problem(dev; nx=n, Lx=L, β, μ, dt, stepper,
                              calcF=calcF!, stochastic=true, aliased_fraction=0)

and define some shortcuts.

sol, clock, vars, params, grid = prob.sol, prob.clock, prob.vars, prob.params, prob.grid
x,  y  = grid.x, grid.y
Lx, Ly = grid.Lx, grid.Ly

First let's see how a forcing realization looks like. Note that when plotting, we decorate the variable to be plotted with Array() to make sure it is brought back on the CPU when vars live on the GPU.

calcF!(vars.Fh, sol, 0.0, clock, vars, params, grid)

fig = Figure()

ax = Axis(fig[1, 1],
          xlabel = "x",
          ylabel = "y",
          aspect = 1,
          title = "a forcing realization",
          limits = ((-Lx/2, Lx/2), (-Ly/2, Ly/2)))

heatmap!(ax, x, y, Array(irfft(vars.Fh, grid.nx));
         colormap = :balance, colorrange = (-8, 8))

fig

Setting initial conditions

Our initial condition is simply fluid at rest.

BarotropicQGQL.set_zeta!(prob, device_array(dev)(zeros(grid.nx, grid.ny)))

Diagnostics

Create Diagnostics – energy and enstrophy are functions imported at the top.

E = Diagnostic(BarotropicQGQL.energy, prob; nsteps)
Z = Diagnostic(BarotropicQGQL.enstrophy, prob; nsteps)

We can also define our custom diagnostics via functions.

zetaMean(prob) = prob.sol[1, :]

zMean = Diagnostic(zetaMean, prob; nsteps, freq=10)  # the zonal-mean vorticity

We combile all diags in a list.

diags = [E, Z, zMean] # A list of Diagnostics types passed to "stepforward!" will  be updated every timestep.

Output

We choose folder for outputing .jld2 files and snapshots (.png files).

filepath = "."
plotpath = "./plots_forcedbetaQLturb"
plotname = "snapshots"
filename = joinpath(filepath, "forcedbetaQLturb.jld2")

Do some basic file management,

if isfile(filename); rm(filename); end
if !isdir(plotpath); mkdir(plotpath); end

and then create Output.

get_sol(prob) = prob.sol # extracts the Fourier-transformed solution

function get_u(prob)
  grid, vars = prob.grid, prob.vars

  @. vars.uh = im * grid.l  * grid.invKrsq * sol
  ldiv!(vars.u, grid.rfftplan, deepcopy(vars.uh))

  return  vars.u
end

out = Output(prob, filename, (:sol, get_sol), (:u, get_u))

Output
  ├──── prob: FourierFlows.Problem{DataType, Matrix{ComplexF64}, Float64, Matrix{ComplexF64}}
  ├──── path: ./forcedbetaQLturb.jld2
  └── fields: Dict{Symbol, Function}(:sol => Main.var"##251".get_sol, :u => Main.var"##251".get_u)

Visualizing the simulation

We define a function that plots the vorticity and streamfunction fields, the corresponding zonal-mean vorticity and zonal-mean zonal velocity and timeseries of energy and enstrophy.

title_ζ = Observable(@sprintf("vorticity, μt = %.2f", μ * clock.t))
title_ψ = "streamfunction ψ"

fig = Figure(resolution=(1000, 600))

axis_kwargs = (xlabel = "x",
               ylabel = "y",
               aspect = 1,
               limits = ((-Lx/2, Lx/2), (-Ly/2, Ly/2)))

axζ = Axis(fig[1, 1]; title = title_ζ, axis_kwargs...)

axψ = Axis(fig[2, 1]; title = title_ψ, axis_kwargs...)

axζ̄ = Axis(fig[1, 2],
           xlabel = "zonal mean ζ",
           ylabel = "y",
           aspect = 1,
           limits = ((-3, 3), (-Ly/2, Ly/2)))

axū = Axis(fig[2, 2],
           xlabel = "zonal mean u",
           ylabel = "y",
           aspect = 1,
           limits = ((-0.5, 0.5), (-Ly/2, Ly/2)))

axE = Axis(fig[1, 3],
           xlabel = "μ t",
           ylabel = "energy",
           aspect = 1,
           limits = ((-0.1, 4.1), (0, 0.05)))

axZ = Axis(fig[2, 3],
           xlabel = "μ t",
           ylabel = "enstrophy",
           aspect = 1,
           limits = ((-0.1, 4.1), (0, 5)))

ζ̄, ζ′= prob.vars.Zeta, prob.vars.zeta
ζ = Observable(Array(@. ζ̄ + ζ′))
ψ̄, ψ′= prob.vars.Psi,  prob.vars.psi
ψ = Observable(Array(@. ψ̄ + ψ′))
ζ̄ₘ = Observable(Array(vec(mean(ζ̄, dims=1))))
ūₘ = Observable(Array(vec(mean(prob.vars.U, dims=1))))

μt = Observable(μ * E.t[1:1])
energy = Observable(E.data[1:1])
enstrophy = Observable(Z.data[1:1])

heatmap!(axζ, x, y, ζ;
         colormap = :balance, colorrange = (-8, 8))

heatmap!(axψ, x, y, ψ;
         colormap = :viridis, colorrange = (-0.22, 0.22))

lines!(axζ̄, ζ̄ₘ, y; linewidth = 3)
lines!(axζ̄, 0y, y; linewidth = 1, linestyle=:dash)

lines!(axū, ūₘ, y; linewidth = 3)
lines!(axū, 0y, y; linewidth = 1, linestyle=:dash)

lines!(axE, μt, energy; linewidth = 3)
lines!(axZ, μt, enstrophy; linewidth = 3, color = :red)

Time-stepping the Problem forward

We step the Problem forward in time.

startwalltime = time()

frames = 0:round(Int, nsteps / nsubs)

record(fig, "barotropicqgql_betaforced.mp4", frames, framerate = 18) do j
  if j % (1000 / nsubs) == 0
    cfl = clock.dt * maximum([maximum(vars.u .+ vars.U) / grid.dx, maximum(vars.v) / grid.dy])

    log = @sprintf("step: %04d, t: %d, cfl: %.2f, E: %.4f, Q: %.4f, walltime: %.2f min",
      clock.step, clock.t, cfl, E.data[E.i], Z.data[Z.i],
      (time()-startwalltime)/60)

    println(log)
  end

  ζ[] = @. ζ̄ + ζ′
  ψ[] = @. ψ̄ + ψ′
  ζ̄ₘ[] = vec(mean(ζ̄, dims=1))
  ūₘ[] = vec(mean(prob.vars.U, dims=1))

  μt.val = μ * E.t[1:E.i]
  energy[] = E.data[1:E.i]
  enstrophy[] = Z.data[1:E.i]

  title_ζ[] = @sprintf("vorticity, μt = %.2f", μ * clock.t)

  stepforward!(prob, diags, nsubs)
  BarotropicQGQL.updatevars!(prob)
end

step: 0000, t: 0, cfl: 0.00, E: 0.0000, Q: 0.0000, walltime: 0.00 min
step: 1000, t: 50, cfl: 0.65, E: 0.0303, Q: 4.0371, walltime: 0.13 min
step: 2000, t: 100, cfl: 0.77, E: 0.0411, Q: 4.0760, walltime: 0.24 min
step: 3000, t: 150, cfl: 0.88, E: 0.0442, Q: 3.9000, walltime: 0.35 min
step: 4000, t: 200, cfl: 0.88, E: 0.0454, Q: 3.8017, walltime: 0.46 min
step: 5000, t: 250, cfl: 0.90, E: 0.0498, Q: 4.6021, walltime: 0.58 min
step: 6000, t: 300, cfl: 0.93, E: 0.0485, Q: 3.9843, walltime: 0.71 min
step: 7000, t: 350, cfl: 0.90, E: 0.0475, Q: 3.9335, walltime: 0.84 min
step: 8000, t: 400, cfl: 0.80, E: 0.0473, Q: 3.9313, walltime: 0.98 min

Save

Finally, we can save, e.g., the last snapshot via

savename = @sprintf("%s_%09d.png", joinpath(plotpath, plotname), clock.step)
savefig(savename)

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