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: NothingWe 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 haloNow 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: 31.2 KiBRun the simulation and process results
To run the simulation:
run!(simulation)[ Info: Initializing simulation...
┌ Info: iter = 0, [000.00%] time = 0 seconds, Δt = 59.158 ms, walltime = 28.705 seconds, walltime / timestep = 0 seconds
└ |u⃗|ₘₐₓ = [1.04e+00, 0.00e+00, 3.11e-02] m/s, advective CFL = 0.8, diffusive CFL = 0.00019, νₘₐₓ = 2e-05 m²/s
[ Info: ... simulation initialization complete (12.353 seconds)
[ Info: Executing initial time step...
[ Info: ... initial time step complete (5.631 seconds).
┌ Info: iter = 200, [011.79%] time = 11.792 seconds, Δt = 60.931 ms, walltime = 45.450 seconds, walltime / timestep = 83.724 ms
└ |u⃗|ₘₐₓ = [1.02e+00, 0.00e+00, 4.04e-02] m/s, advective CFL = 0.8, diffusive CFL = 0.0002, νₘₐₓ = 2e-05 m²/s
┌ Info: iter = 400, [023.43%] time = 23.429 seconds, Δt = 61.209 ms, walltime = 47.757 seconds, walltime / timestep = 11.536 ms
└ |u⃗|ₘₐₓ = [1.02e+00, 0.00e+00, 4.44e-02] m/s, advective CFL = 0.8, diffusive CFL = 0.0002, νₘₐₓ = 2e-05 m²/s
┌ Info: iter = 600, [035.18%] time = 35.181 seconds, Δt = 60.253 ms, walltime = 49.982 seconds, walltime / timestep = 11.125 ms
└ |u⃗|ₘₐₓ = [1.03e+00, 0.00e+00, 7.74e-02] m/s, advective CFL = 0.8, diffusive CFL = 0.0002, νₘₐₓ = 2e-05 m²/s
┌ Info: iter = 800, [046.38%] time = 46.384 seconds, Δt = 54.701 ms, walltime = 52.098 seconds, walltime / timestep = 10.578 ms
└ |u⃗|ₘₐₓ = [1.09e+00, 0.00e+00, 2.33e-01] m/s, advective CFL = 0.8, diffusive CFL = 0.00018, νₘₐₓ = 2e-05 m²/s
┌ Info: iter = 1000, [056.34%] time = 56.341 seconds, Δt = 48.707 ms, walltime = 54.116 seconds, walltime / timestep = 10.093 ms
└ |u⃗|ₘₐₓ = [1.17e+00, 0.00e+00, 3.88e-01] m/s, advective CFL = 0.8, diffusive CFL = 0.00016, νₘₐₓ = 2e-05 m²/s
┌ Info: iter = 1200, [065.84%] time = 1.097 minutes, Δt = 49.850 ms, walltime = 56.045 seconds, walltime / timestep = 9.644 ms
└ |u⃗|ₘₐₓ = [1.16e+00, 0.00e+00, 3.57e-01] m/s, advective CFL = 0.8, diffusive CFL = 0.00016, νₘₐₓ = 2e-05 m²/s
┌ Info: iter = 1400, [075.90%] time = 1.265 minutes, Δt = 53.297 ms, walltime = 58.079 seconds, walltime / timestep = 10.169 ms
└ |u⃗|ₘₐₓ = [1.10e+00, 0.00e+00, 2.80e-01] m/s, advective CFL = 0.8, diffusive CFL = 0.00017, νₘₐₓ = 2e-05 m²/s
┌ Info: iter = 1600, [086.10%] time = 1.435 minutes, Δt = 50.941 ms, walltime = 1.003 minutes, walltime / timestep = 10.544 ms
└ |u⃗|ₘₐₓ = [1.14e+00, 0.00e+00, 3.35e-01] m/s, advective CFL = 0.8, diffusive CFL = 0.00017, νₘₐₓ = 2e-05 m²/s
┌ Info: iter = 1800, [095.69%] time = 1.595 minutes, Δt = 49.723 ms, walltime = 1.035 minutes, walltime / timestep = 9.630 ms
└ |u⃗|ₘₐₓ = [1.16e+00, 0.00e+00, 3.65e-01] m/s, advective CFL = 0.8, diffusive CFL = 0.00016, νₘₐₓ = 2e-05 m²/s
[ Info: Simulation is stopping after running for 38.359 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{Float32} [-4.9609375f0, …, 4.9609375f0] ForwardOrdered Regular Points,
→ z_aac Sampled{Float32} [-4.9609375f0, …, 4.9609375f0] ForwardOrdered Regular Points,
↗ z_aaf Sampled{Float32} [-5.0f0, …, 5.0f0] ForwardOrdered Regular Points,
⬔ Ti Sampled{Float64} [0.0, …, 100.0] ForwardOrdered Regular Points,
◩ x_faa Sampled{Float32} [-5.0f0, …, 4.921875f0] 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 2025-10-12T01:10:23.989…
"schedule" => "TimeInterval"
├────────────────────────────────────────────────────────────────────── raster ┤
missingval: missing
extent: Extent(x_caa = (-4.9609375f0, 4.9609375f0), z_aac = (-4.9609375f0, 4.9609375f0), z_aaf = (-5.0f0, 5.0f0), Ti = (0.0, 100.0), x_faa = (-5.0f0, 4.921875f0))
└──────────────────────────────────────────────────────────────────────────────┘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...);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=$n, y_aca=Near(0)], :x_caa => X, :z_aac => Z)
hm3 = heatmap!(ax3, bₙ; colormap = :balance, colorrange = (-2.5e-2, +2.5e-2))
Colorbar(fig[3, 3], hm3, vertical=false, height=8);┌ Warning: (DimensionalData.Dimensions.Dim{:y_aca},) dims were not found in object.
└ @ DimensionalData.Dimensions ~/.julia/packages/DimensionalData/hv9KC/src/Dimensions/primitives.jl:852
┌ Warning: (DimensionalData.Dimensions.Dim{:y_aca},) dims were not found in object.
└ @ DimensionalData.Dimensions ~/.julia/packages/DimensionalData/hv9KC/src/Dimensions/primitives.jl:852
┌ Warning: (DimensionalData.Dimensions.Dim{:y_aca},) dims were not found in object.
└ @ DimensionalData.Dimensions ~/.julia/packages/DimensionalData/hv9KC/src/Dimensions/primitives.jl:852We now plot the time evolution of our integrated quantities
axb = Axis(fig[4, 1:3]; xlabel="Time", height=100)
times = dims(ds, :Ti)
lines!(axb, Array(times), Array(ds.∫χ), label = "∫χdV")
lines!(axb, Array(times), Array(ds.∫χᴰ), label = "∫χᴰdV", linestyle=:dash)
axislegend(position=:lb, labelsize=14)Makie.Legend()Now we mark the time by placing a vertical line in the bottom panel and adding a helpful title
tₙ = @lift times[$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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