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Kelvin-Helmholtz instability

This example simulates a simple 2D Kelvin-Helmholtz instability and is based on the similar Oceananigans example.

Before starting, make sure you have the required packages installed for this example, which can be done with

using Pkg
pkg"add Oceananigans, Oceanostics, CairoMakie, Rasters"

Model and simulation setup

We begin by creating a model with an isotropic diffusivity and fifth-order advection on a xz 128² grid using a buoyancy b as the active scalar. We'll work here with nondimensional quantities.

using Oceananigans

N = 128
L = 10
grid = RectilinearGrid(size=(N, N), x=(-L/2, +L/2), z=(-L/2, +L/2), topology=(Periodic, Flat, Bounded))

model = NonhydrostaticModel(grid; timestepper = :RungeKutta3,
                            advection = UpwindBiased(order=5),
                            closure = ScalarDiffusivity(ν=2e-5, κ=2e-5),
                            buoyancy = BuoyancyTracer(), tracers = :b)
NonhydrostaticModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 128×1×128 RectilinearGrid{Float64, Periodic, Flat, Bounded} on CPU with 3×0×3 halo
├── timestepper: RungeKutta3TimeStepper
├── advection scheme: UpwindBiased(order=5)
├── tracers: b
├── closure: ScalarDiffusivity{ExplicitTimeDiscretization}(ν=2.0e-5, κ=(b=2.0e-5,))
├── buoyancy: BuoyancyTracer with ĝ = NegativeZDirection()
└── coriolis: Nothing

We use hyperbolic tangent functions for the initial conditions and set the maximum Richardson number below the threshold of 1/4. We also add some grid-scale small-amplitude noise to u to kick the instability off:

noise(x, z) = 2e-2 * randn()
shear_flow(x, z) = tanh(z) + noise(x, z)

Ri₀ = 0.1; h = 1/4
stratification(x, z) = h * Ri₀ * tanh(z / h)

set!(model, u=shear_flow, b=stratification)

Next create an adaptive-time-step simulation using the model above:

simulation = Simulation(model, Δt=0.1, stop_time=100)

wizard = TimeStepWizard(cfl=0.8, max_Δt=1)
simulation.callbacks[:wizard] = Callback(wizard, IterationInterval(2))
Callback of TimeStepWizard(cfl=0.8, max_Δt=1.0, min_Δt=0.0) on IterationInterval(2)

Model diagnostics

We set-up a progress messenger using the TimedMessenger, which displays, among other information, the time step duration

using Oceanostics

progress = ProgressMessengers.TimedMessenger()
simulation.callbacks[:progress] = Callback(progress, IterationInterval(200))
Callback of Oceanostics.ProgressMessengers.TimedMessenger{Oceanostics.ProgressMessengers.AbstractProgressMessenger} on IterationInterval(200)

We can also define some useful diagnostics for of the flow, starting with the RichardsonNumber

Ri = RichardsonNumber(model)
KernelFunctionOperation at (Center, Center, Face)
├── grid: 128×1×128 RectilinearGrid{Float64, Periodic, Flat, Bounded} on CPU with 3×0×3 halo
├── kernel_function: richardson_number_ccf (generic function with 1 method)
└── arguments: ("Field", "Field", "Field", "Field", "Tuple")

We also set-up the QVelocityGradientTensorInvariant, which is usually used for visualizing vortices in the flow:

Q = QVelocityGradientTensorInvariant(model)
KernelFunctionOperation at (Center, Center, Center)
├── grid: 128×1×128 RectilinearGrid{Float64, Periodic, Flat, Bounded} on CPU with 3×0×3 halo
├── kernel_function: Q_velocity_gradient_tensor_invariant_ccc (generic function with 1 method)
└── arguments: ("Field", "Field", "Field")

Q is one of the velocity gradient tensor invariants and it measures the amount of vorticity versus the strain in the flow and, when it's positive, indicates a vortex. This method of vortex visualization is called the Q-criterion.

Let's also keep track of the amount of buoyancy mixing by measuring the buoyancy variance dissipation rate and diffusive term. When volume-integrated, these two quantities should be equal.

∫χᴰ = Integral(TracerVarianceEquation.DissipationRate(model, :b))
∫χ = Integral(TracerVarianceEquation.Diffusion(model, :b))
Integral of BinaryOperation at (Center, Center, Center) over dims (1, 2, 3)
└── operand: BinaryOperation at (Center, Center, Center)
    └── grid: 128×1×128 RectilinearGrid{Float64, Periodic, Flat, Bounded} on CPU with 3×0×3 halo

Now we write these quantities, along with b, to a NetCDF:

output_fields = (; Ri, Q, model.tracers.b, ∫χ, ∫χᴰ)

using NCDatasets
filename = "kelvin_helmholtz"
simulation.output_writers[:nc] = NetCDFWriter(model, output_fields,
                                              filename = joinpath(@__DIR__, filename),
                                              schedule = TimeInterval(1),
                                              overwrite_existing = true)
NetCDFWriter scheduled on TimeInterval(1 second):
├── filepath: build/generated/kelvin_helmholtz.nc
├── dimensions: time(0), x_faa(128), x_caa(128), z_aaf(129), z_aac(128)
├── 5 outputs: (Q, ∫χᴰ, Ri, b, ∫χ)
├── array_type: Array{Float32}
├── file_splitting: NoFileSplitting
└── file size: 29.4 KiB

Run the simulation and process results

To run the simulation:

run!(simulation)
[ Info: Initializing simulation...
┌ Info: iter =      0,  [000.00%] time = 0 seconds,  Δt = 58.625 ms,  walltime = 50.093 seconds,  walltime / timestep = 0 seconds
└           |u⃗|ₘₐₓ = [1.04e+00,  0.00e+00,  2.88e-02] m/s,  advective CFL = 0.8,  diffusive CFL = 0.00019,  νₘₐₓ = 2e-05 m²/s
[ Info:     ... simulation initialization complete (19.661 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (3.906 seconds).
┌ Info: iter =    200,  [011.80%] time = 11.795 seconds,  Δt = 61.183 ms,  walltime = 1.213 minutes,  walltime / timestep = 113.373 ms
└           |u⃗|ₘₐₓ = [1.02e+00,  0.00e+00,  2.78e-02] m/s,  advective CFL = 0.8,  diffusive CFL = 0.0002,  νₘₐₓ = 2e-05 m²/s
┌ Info: iter =    400,  [023.55%] time = 23.550 seconds,  Δt = 61.120 ms,  walltime = 1.249 minutes,  walltime / timestep = 10.802 ms
└           |u⃗|ₘₐₓ = [1.02e+00,  0.00e+00,  4.77e-02] m/s,  advective CFL = 0.8,  diffusive CFL = 0.0002,  νₘₐₓ = 2e-05 m²/s
┌ Info: iter =    600,  [035.30%] time = 35.296 seconds,  Δt = 59.151 ms,  walltime = 1.284 minutes,  walltime / timestep = 10.542 ms
└           |u⃗|ₘₐₓ = [1.04e+00,  0.00e+00,  9.61e-02] m/s,  advective CFL = 0.8,  diffusive CFL = 0.00019,  νₘₐₓ = 2e-05 m²/s
┌ Info: iter =    800,  [046.38%] time = 46.377 seconds,  Δt = 53.574 ms,  walltime = 1.316 minutes,  walltime / timestep = 9.753 ms
└           |u⃗|ₘₐₓ = [1.11e+00,  0.00e+00,  2.49e-01] m/s,  advective CFL = 0.8,  diffusive CFL = 0.00018,  νₘₐₓ = 2e-05 m²/s
┌ Info: iter =   1000,  [056.20%] time = 56.196 seconds,  Δt = 48.925 ms,  walltime = 1.348 minutes,  walltime / timestep = 9.602 ms
└           |u⃗|ₘₐₓ = [1.17e+00,  0.00e+00,  3.80e-01] m/s,  advective CFL = 0.8,  diffusive CFL = 0.00016,  νₘₐₓ = 2e-05 m²/s
┌ Info: iter =   1200,  [065.76%] time = 1.096 minutes,  Δt = 50.551 ms,  walltime = 1.379 minutes,  walltime / timestep = 9.095 ms
└           |u⃗|ₘₐₓ = [1.15e+00,  0.00e+00,  3.53e-01] m/s,  advective CFL = 0.8,  diffusive CFL = 0.00017,  νₘₐₓ = 2e-05 m²/s
┌ Info: iter =   1400,  [075.85%] time = 1.264 minutes,  Δt = 53.218 ms,  walltime = 1.410 minutes,  walltime / timestep = 9.460 ms
└           |u⃗|ₘₐₓ = [1.11e+00,  0.00e+00,  2.89e-01] m/s,  advective CFL = 0.8,  diffusive CFL = 0.00017,  νₘₐₓ = 2e-05 m²/s
┌ Info: iter =   1600,  [086.05%] time = 1.434 minutes,  Δt = 50.714 ms,  walltime = 1.444 minutes,  walltime / timestep = 9.963 ms
└           |u⃗|ₘₐₓ = [1.14e+00,  0.00e+00,  3.47e-01] m/s,  advective CFL = 0.8,  diffusive CFL = 0.00017,  νₘₐₓ = 2e-05 m²/s
┌ Info: iter =   1800,  [095.70%] time = 1.595 minutes,  Δt = 50.096 ms,  walltime = 1.474 minutes,  walltime / timestep = 9.059 ms
└           |u⃗|ₘₐₓ = [1.15e+00,  0.00e+00,  3.48e-01] m/s,  advective CFL = 0.8,  diffusive CFL = 0.00016,  νₘₐₓ = 2e-05 m²/s
[ Info: Simulation is stopping after running for 42.606 seconds.
[ Info: Simulation time 1.667 minutes equals or exceeds stop time 1.667 minutes.

Now we'll read the results using Rasters.jl, which works somewhat similarly to Python's Xarray and can speed-up the work the workflow

using Rasters

ds = RasterStack(simulation.output_writers[:nc].filepath)
┌ 128×128×129×101×128 RasterStack ┐
├─────────────────────────────────┴────────────────────────────────────── dims ┐
  ↓ x_caa Sampled{Float64} [-4.9609375, …, 4.9609375] ForwardOrdered Regular Points,
  → z_aac Sampled{Float64} [-4.9609375, …, 4.9609375] ForwardOrdered Regular Points,
  ↗ z_aaf Sampled{Float64} [-5.0, …, 5.0] ForwardOrdered Regular Points,
  ⬔ Ti Sampled{Float64} [0.0, …, 100.0] ForwardOrdered Regular Points,
  ◩ x_faa Sampled{Float64} [-5.0, …, 4.921875] ForwardOrdered Regular Points
├────────────────────────────────────────────────────────────────────── layers ┤
  :Δx_caa eltype: Union{Missing, Float32} dims: x_caa size: 128
  :Δx_faa eltype: Union{Missing, Float32} dims: x_faa size: 128
  :Δz_aac eltype: Union{Missing, Float32} dims: z_aac size: 128
  :Δz_aaf eltype: Union{Missing, Float32} dims: z_aaf size: 129
  :Q      eltype: Union{Missing, Float32} dims: x_caa, z_aac, Ti size: 128×128×101
  :Ri     eltype: Union{Missing, Float32} dims: x_caa, z_aaf, Ti size: 128×129×101
  :b      eltype: Union{Missing, Float32} dims: x_caa, z_aac, Ti size: 128×128×101
  :∫χ     eltype: Union{Missing, Float32} dims: Ti size: 101
  :∫χᴰ    eltype: Union{Missing, Float32} dims: Ti size: 101
├──────────────────────────────────────────────────────────────────── metadata ┤
  Metadata{Rasters.NCDsource} of Dict{String, Any} with 6 entries:
  "interval"             => 1.0
  "Oceananigans"         => "This file was generated using "
  "Julia"                => "This file was generated using "
  "output time interval" => "Output was saved every 1 second."
  "date"                 => "This file was generated on 2026-02-20T13:11:40.191…
  "schedule"             => "TimeInterval"
├────────────────────────────────────────────────────────────────────── raster ┤
  missingval: missing
  extent: Extent(x_caa = (-4.9609375, 4.9609375), z_aac = (-4.9609375, 4.9609375), z_aaf = (-5.0, 5.0), Ti = (0.0, 100.0), x_faa = (-5.0, 4.921875))
└──────────────────────────────────────────────────────────────────────────────┘

We now use Makie to create the figure and its axes

using CairoMakie

set_theme!(Theme(fontsize = 24))
fig = Figure()

kwargs = (xlabel="x", ylabel="z", height=150, width=250)
ax1 = Axis(fig[2, 1]; title = "Ri", kwargs...)
ax2 = Axis(fig[2, 2]; title = "Q", kwargs...)
ax3 = Axis(fig[2, 3]; title = "b", kwargs...);
Precompiling packages...
    539.8 ms  ✓ FilePathsGlobExt (serial)
  1 dependency successfully precompiled in 1 seconds
Precompiling packages...
    440.3 ms  ✓ DistancesChainRulesCoreExt (serial)
  1 dependency successfully precompiled in 0 seconds
Precompiling packages...
   1009.1 ms  ✓ TaylorSeriesIAExt (serial)
  1 dependency successfully precompiled in 1 seconds
Precompiling packages...
   5920.8 ms  ✓ OceananigansMakieExt (serial)
  1 dependency successfully precompiled in 6 seconds

Next we use Observables to lift the values and plot heatmaps and their colorbars

n = Observable(1)

Riₙ = @lift set(ds.Ri[Ti=$n, y_aca=Near(0)], :x_caa => X, :z_aaf => Z)
hm1 = heatmap!(ax1, Riₙ; colormap = :bwr, colorrange = (-1, +1))
Colorbar(fig[3, 1], hm1, vertical=false, height=8)

Qₙ = @lift set(ds.Q[Ti=$n, y_aca=Near(0)], :x_caa => X, :z_aac => Z)
hm2 = heatmap!(ax2, Qₙ; colormap = :inferno, colorrange = (0, 0.2))
Colorbar(fig[3, 2], hm2, vertical=false, height=8)

bₙ = @lift set(ds.b[Ti=$n, y_aca=Near(0)], :x_caa => X, :z_aac => Z)
hm3 = heatmap!(ax3, bₙ; colormap = :balance, colorrange = (-2.5e-2, +2.5e-2))
Colorbar(fig[3, 3], hm3, vertical=false, height=8);
┌ Warning: (DimensionalData.Dimensions.Dim{:y_aca},) dims were not found in object.
└ @ DimensionalData.Dimensions ~/.julia/packages/DimensionalData/vUpVz/src/Dimensions/primitives.jl:852
┌ Warning: (DimensionalData.Dimensions.Dim{:y_aca},) dims were not found in object.
└ @ DimensionalData.Dimensions ~/.julia/packages/DimensionalData/vUpVz/src/Dimensions/primitives.jl:852
┌ Warning: (DimensionalData.Dimensions.Dim{:y_aca},) dims were not found in object.
└ @ DimensionalData.Dimensions ~/.julia/packages/DimensionalData/vUpVz/src/Dimensions/primitives.jl:852

We now plot the time evolution of our integrated quantities

axb = Axis(fig[4, 1:3]; xlabel="Time", height=100)
times = dims(ds, :Ti)
lines!(axb, Array(times), Array(ds.∫χ),  label = "∫χdV")
lines!(axb, Array(times), Array(ds.∫χᴰ), label = "∫χᴰdV", linestyle=:dash)
axislegend(position=:lb, labelsize=14)
Makie.Legend()

Now we mark the time by placing a vertical line in the bottom panel and adding a helpful title

tₙ = @lift times[$n]
vlines!(axb, tₙ, color=:black, linestyle=:dash)

title = @lift "Time = " * string(round(times[$n], digits=2))
fig[1, 1:3] = Label(fig, title, fontsize=24, tellwidth=false);

Finally, we adjust the figure dimensions to fit all the panels and record a movie

resize_to_layout!(fig)

@info "Animating..."
record(fig, filename * ".mp4", 1:length(times), framerate=10) do i
       n[] = i
end
"kelvin_helmholtz.mp4"

Similarly to the kinetic energy dissipation rate (see the Two-dimensional turbulence example), TracerVarianceDissipationRate and TracerVarianceDiffusion are implemented with a energy-conserving formulation, which means that (for NoFlux boundary conditions) their volume-integral should be exactly (up to machine precision) the same.

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