Phillips model of Baroclinic Instability

A simulation of the growth of barolinic instability in the Phillips 2-layer model when we impose a vertical mean flow shear as a difference $\Delta U$ in the imposed, domain-averaged, zonal flow at each layer.

Install dependencies

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

using Pkg
pkg"add GeophysicalFlows, CairoMakie, Printf"

Let's begin

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

using GeophysicalFlows, CairoMakie, Printf

using Random: seed!

Choosing a device: CPU or GPU

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

Numerical parameters and time-stepping parameters

n = 128                  # 2D resolution = n²
stepper = "FilteredRK4"  # timestepper
     dt = 2.5e-3         # timestep
 nsteps = 20000          # total number of time-steps
 nsubs  = 50             # number of time-steps for plotting (nsteps must be multiple of nsubs)

Physical parameters

L = 2π                   # domain size
μ = 5e-2                 # bottom drag
β = 5                    # the y-gradient of planetary PV

nlayers = 2              # number of layers
f₀ = 1                   # Coriolis parameter
H = [0.2, 0.8]           # the rest depths of each layer
b = [-1.0, -1.2]         # Boussinesq buoyancy of each layer

U = zeros(nlayers)       # the imposed mean zonal flow in each layer
U[1] = 1.0
U[2] = 0.0

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.

prob = MultiLayerQG.Problem(nlayers, dev; nx=n, Lx=L, f₀, H, b, U, μ, β,
                            dt, stepper, aliased_fraction=0)

and define some shortcuts.

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

Setting initial conditions

Our initial condition is some small-amplitude random noise. We smooth our initial condidtion using the timestepper's high-wavenumber filter.

device_array() function returns the array type appropriate for the device, i.e., Array for dev = CPU() and CuArray for dev = GPU().

seed!(1234) # reset of the random number generator for reproducibility
q₀  = 1e-2 * device_array(dev)(randn((grid.nx, grid.ny, nlayers)))
q₀h = prob.timestepper.filter .* rfft(q₀, (1, 2)) # apply rfft  only in dims=1, 2
q₀  = irfft(q₀h, grid.nx, (1, 2))                 # apply irfft only in dims=1, 2

MultiLayerQG.set_q!(prob, q₀)

Diagnostics

Create Diagnostics – energies function is imported at the top.

E = Diagnostic(MultiLayerQG.energies, prob; nsteps)
diags = [E] # 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_2layer"
plotname = "snapshots"
filename = joinpath(filepath, "2layer.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)
  sol, params, vars, grid = prob.sol, prob.params, prob.vars, prob.grid

  @. vars.qh = sol
  streamfunctionfrompv!(vars.ψh, vars.qh, params, grid)
  @. vars.uh = -im * grid.l * vars.ψh
  invtransform!(vars.u, vars.uh, params)

  return vars.u
end

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

Visualizing the simulation

We create a figure using Makie's Observables that plots the potential vorticity field and the evolution of energy and enstrophy. 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.

Lx, Ly = grid.Lx, grid.Ly

title_KE = Observable(@sprintf("μt = %.2f", μ * clock.t))

q₁ = Observable(Array(vars.q[:, :, 1]))
ψ₁ = Observable(Array(vars.ψ[:, :, 1]))
q₂ = Observable(Array(vars.q[:, :, 2]))
ψ₂ = Observable(Array(vars.ψ[:, :, 2]))

function compute_levels(maxf, nlevels=8)
  # -max(|f|):...:max(|f|)
  levelsf  = @lift collect(range(-$maxf, stop = $maxf, length=nlevels))

  # only positive
  levelsf⁺ = @lift collect(range($maxf/(nlevels-1), stop = $maxf, length=Int(nlevels/2)))

  # only negative
  levelsf⁻ = @lift collect(range(-$maxf, stop = -$maxf/(nlevels-1), length=Int(nlevels/2)))

  return levelsf, levelsf⁺, levelsf⁻
end

maxψ₁ = Observable(maximum(abs, vars.ψ[:, :, 1]))
maxψ₂ = Observable(maximum(abs, vars.ψ[:, :, 2]))

levelsψ₁, levelsψ₁⁺, levelsψ₁⁻ = compute_levels(maxψ₁)
levelsψ₂, levelsψ₂⁺, levelsψ₂⁻ = compute_levels(maxψ₂)

KE₁ = Observable(Point2f[(μ * E.t[1], E.data[1][1][1])])
KE₂ = Observable(Point2f[(μ * E.t[1], E.data[1][1][2])])
PE  = Observable(Point2f[(μ * E.t[1], E.data[1][2])])

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

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

axq₁ = Axis(fig[1, 1]; title = "q₁", axis_kwargs...)

axψ₁ = Axis(fig[2, 1]; title = "ψ₁", axis_kwargs...)

axq₂ = Axis(fig[1, 2]; title = "q₂", axis_kwargs...)

axψ₂ = Axis(fig[2, 2]; title = "ψ₂", axis_kwargs...)

axKE = Axis(fig[1, 3],
            xlabel = "μ t",
            ylabel = "KE",
            title = title_KE,
            yscale = log10,
            limits = ((-0.1, 2.6), (1e-9, 5)))

axPE = Axis(fig[2, 3],
            xlabel = "μ t",
            ylabel = "PE",
            yscale = log10,
            limits = ((-0.1, 2.6), (1e-9, 5)))

heatmap!(axq₁, x, y, q₁; colormap = :balance)

heatmap!(axq₂, x, y, q₂; colormap = :balance)

contourf!(axψ₁, x, y, ψ₁;
          levels = levelsψ₁, colormap = :viridis, extendlow = :auto, extendhigh = :auto)
 contour!(axψ₁, x, y, ψ₁;
          levels = levelsψ₁⁺, color=:black)
 contour!(axψ₁, x, y, ψ₁;
          levels = levelsψ₁⁻, color=:black, linestyle = :dash)

contourf!(axψ₂, x, y, ψ₂;
          levels = levelsψ₂, colormap = :viridis, extendlow = :auto, extendhigh = :auto)
 contour!(axψ₂, x, y, ψ₂;
          levels = levelsψ₂⁺, color=:black)
 contour!(axψ₂, x, y, ψ₂;
          levels = levelsψ₂⁻, color=:black, linestyle = :dash)

ke₁ = lines!(axKE, KE₁; linewidth = 3)
ke₂ = lines!(axKE, KE₂; linewidth = 3)
Legend(fig[1, 4], [ke₁, ke₂,], ["KE₁", "KE₂"])

lines!(axPE, PE; linewidth = 3)

fig

Time-stepping the Problem forward

Finally, we time-step the Problem forward in time.

startwalltime = time()

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

record(fig, "multilayerqg_2layer.mp4", frames, framerate = 18) do j
  if j % (1000 / nsubs) == 0
    cfl = clock.dt * maximum([maximum(vars.u) / grid.dx, maximum(vars.v) / grid.dy])

    log = @sprintf("step: %04d, t: %.1f, cfl: %.2f, KE₁: %.3e, KE₂: %.3e, PE: %.3e, walltime: %.2f min",
                   clock.step, clock.t, cfl, E.data[E.i][1][1], E.data[E.i][1][2], E.data[E.i][2][1], (time()-startwalltime)/60)

    println(log)
  end

  q₁[] = vars.q[:, :, 1]
  ψ₁[] = vars.ψ[:, :, 1]
  q₂[] = vars.q[:, :, 2]
  ψ₂[] = vars.ψ[:, :, 2]

  maxψ₁[] = maximum(abs, vars.ψ[:, :, 1])
  maxψ₂[] = maximum(abs, vars.ψ[:, :, 2])

  KE₁[] = push!(KE₁[], Point2f(μ * E.t[E.i], E.data[E.i][1][1]))
  KE₂[] = push!(KE₂[], Point2f(μ * E.t[E.i], E.data[E.i][1][2]))
  PE[]  = push!(PE[] , Point2f(μ * E.t[E.i], E.data[E.i][2]))

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

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

step: 0000, t: 0.0, cfl: 0.00, KE₁: 1.057e-08, KE₂: 4.999e-08, PE: 3.110e-09, walltime: 0.00 min
step: 1000, t: 2.5, cfl: 0.00, KE₁: 4.790e-08, KE₂: 4.955e-08, PE: 6.252e-08, walltime: 0.29 min
step: 2000, t: 5.0, cfl: 0.00, KE₁: 1.611e-07, KE₂: 8.767e-08, PE: 1.824e-07, walltime: 0.41 min
step: 3000, t: 7.5, cfl: 0.00, KE₁: 6.691e-07, KE₂: 2.782e-07, PE: 7.464e-07, walltime: 0.54 min
step: 4000, t: 10.0, cfl: 0.00, KE₁: 3.107e-06, KE₂: 1.241e-06, PE: 3.361e-06, walltime: 0.66 min
step: 5000, t: 12.5, cfl: 0.00, KE₁: 1.570e-05, KE₂: 6.251e-06, PE: 1.685e-05, walltime: 0.79 min
step: 6000, t: 15.0, cfl: 0.00, KE₁: 8.230e-05, KE₂: 3.282e-05, PE: 8.810e-05, walltime: 0.92 min
step: 7000, t: 17.5, cfl: 0.01, KE₁: 4.412e-04, KE₂: 1.760e-04, PE: 4.723e-04, walltime: 1.06 min
step: 8000, t: 20.0, cfl: 0.02, KE₁: 2.394e-03, KE₂: 9.554e-04, PE: 2.564e-03, walltime: 1.20 min
step: 9000, t: 22.5, cfl: 0.05, KE₁: 1.274e-02, KE₂: 5.083e-03, PE: 1.365e-02, walltime: 1.34 min
step: 10000, t: 25.0, cfl: 0.09, KE₁: 5.764e-02, KE₂: 2.314e-02, PE: 6.095e-02, walltime: 1.48 min
step: 11000, t: 27.5, cfl: 0.12, KE₁: 1.720e-01, KE₂: 7.441e-02, PE: 1.710e-01, walltime: 1.62 min
step: 12000, t: 30.0, cfl: 0.19, KE₁: 3.204e-01, KE₂: 1.522e-01, PE: 3.567e-01, walltime: 1.75 min
step: 13000, t: 32.5, cfl: 0.23, KE₁: 3.819e-01, KE₂: 2.204e-01, PE: 3.942e-01, walltime: 1.87 min
step: 14000, t: 35.0, cfl: 0.30, KE₁: 4.390e-01, KE₂: 3.152e-01, PE: 4.753e-01, walltime: 1.99 min
step: 15000, t: 37.5, cfl: 0.27, KE₁: 5.285e-01, KE₂: 3.980e-01, PE: 6.448e-01, walltime: 2.11 min
step: 16000, t: 40.0, cfl: 0.31, KE₁: 5.657e-01, KE₂: 4.813e-01, PE: 6.384e-01, walltime: 2.22 min
step: 17000, t: 42.5, cfl: 0.30, KE₁: 5.449e-01, KE₂: 5.176e-01, PE: 5.810e-01, walltime: 2.34 min
step: 18000, t: 45.0, cfl: 0.31, KE₁: 5.547e-01, KE₂: 5.822e-01, PE: 5.594e-01, walltime: 2.45 min
step: 19000, t: 47.5, cfl: 0.28, KE₁: 5.410e-01, KE₂: 6.639e-01, PE: 5.192e-01, walltime: 2.56 min
step: 20000, t: 50.0, cfl: 0.26, KE₁: 5.316e-01, KE₂: 6.543e-01, PE: 5.425e-01, walltime: 2.67 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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