Two dimensional turbulence example
In this example, we initialize a random velocity field and observe its turbulent decay in a two-dimensional domain. This example demonstrates:
- How to run a model with no tracers and no buoyancy model.
- How to use computed
Fields to generate output.
Install dependencies
First let's make sure we have all required packages installed.
using Pkg
pkg"add Oceananigans, CairoMakie"Model setup
We instantiate the model with an isotropic diffusivity. We use a grid with 128² points, a fifth-order advection scheme, third-order Runge-Kutta time-stepping, and a small isotropic viscosity. Note that we assign Flat to the z direction.
using Oceananigans
grid = RectilinearGrid(size=(128, 128), extent=(2π, 2π), topology=(Periodic, Periodic, Flat))
model = NonhydrostaticModel(; grid,
advection = UpwindBiased(order=5),
closure = ScalarDiffusivity(ν=1e-5))NonhydrostaticModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 128×128×1 RectilinearGrid{Float64, Periodic, Periodic, Flat} on CPU with 3×3×0 halo
├── timestepper: RungeKutta3TimeStepper
├── advection scheme: UpwindBiased(order=5)
├── tracers: ()
├── closure: ScalarDiffusivity{ExplicitTimeDiscretization}(ν=1.0e-5)
├── buoyancy: Nothing
└── coriolis: NothingRandom initial conditions
Our initial condition randomizes model.velocities.u and model.velocities.v. We ensure that both have zero mean for aesthetic reasons.
using Statistics
u, v, w = model.velocities
uᵢ = rand(size(u)...)
vᵢ = rand(size(v)...)
uᵢ .-= mean(uᵢ)
vᵢ .-= mean(vᵢ)
set!(model, u=uᵢ, v=vᵢ)Setting up a simulation
We set-up a simulation that stops at 50 time units, with an initial time-step of 0.1, and with adaptive time-stepping and progress printing.
simulation = Simulation(model, Δt=0.2, stop_time=50)Simulation of NonhydrostaticModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── Next time step: 200 ms
├── Elapsed wall time: 0 seconds
├── Wall time per iteration: NaN days
├── Stop time: 50 seconds
├── Stop iteration : Inf
├── Wall time limit: Inf
├── Callbacks: OrderedDict with 4 entries:
│ ├── stop_time_exceeded => Callback of stop_time_exceeded on IterationInterval(1)
│ ├── stop_iteration_exceeded => Callback of stop_iteration_exceeded on IterationInterval(1)
│ ├── wall_time_limit_exceeded => Callback of wall_time_limit_exceeded on IterationInterval(1)
│ └── nan_checker => Callback of NaNChecker for u on IterationInterval(100)
├── Output writers: OrderedDict with no entries
└── Diagnostics: OrderedDict with no entriesThe TimeStepWizard helps ensure stable time-stepping with a Courant-Freidrichs-Lewy (CFL) number of 0.7.
wizard = TimeStepWizard(cfl=0.7, max_change=1.1, max_Δt=0.5)
simulation.callbacks[:wizard] = Callback(wizard, IterationInterval(10))Callback of TimeStepWizard(cfl=0.7, max_Δt=0.5, min_Δt=0.0) on IterationInterval(10)Logging simulation progress
We set up a callback that logs the simulation iteration and time every 100 iterations.
using Printf
function progress_message(sim)
max_abs_u = maximum(abs, sim.model.velocities.u)
walltime = prettytime(sim.run_wall_time)
return @info @sprintf("Iteration: %04d, time: %1.3f, Δt: %.2e, max(|u|) = %.1e, wall time: %s\n",
iteration(sim), time(sim), sim.Δt, max_abs_u, walltime)
end
add_callback!(simulation, progress_message, IterationInterval(100))Output
We set up an output writer for the simulation that saves vorticity and speed every 20 iterations.
Computing vorticity and speed
To make our equations prettier, we unpack u, v, and w from the NamedTuple model.velocities:
u, v, w = model.velocitiesNamedTuple with 3 Fields on 128×128×1 RectilinearGrid{Float64, Periodic, Periodic, Flat} on CPU with 3×3×0 halo:
├── u: 128×128×1 Field{Face, Center, Center} on RectilinearGrid on CPU
├── v: 128×128×1 Field{Center, Face, Center} on RectilinearGrid on CPU
└── w: 128×128×1 Field{Center, Center, Face} on RectilinearGrid on CPUNext we create two Fields that calculate (i) vorticity that measures the rate at which the fluid rotates and is defined as
\[ω = ∂_x v - ∂_y u \, ,\]
ω = ∂x(v) - ∂y(u)BinaryOperation at (Face, Face, Center)
├── grid: 128×128×1 RectilinearGrid{Float64, Periodic, Periodic, Flat} on CPU with 3×3×0 halo
└── tree:
- at (Face, Face, Center)
├── ∂xᶠᶠᶜ at (Face, Face, Center) via identity
│ └── 128×128×1 Field{Center, Face, Center} on RectilinearGrid on CPU
└── ∂yᶠᶠᶜ at (Face, Face, Center) via identity
└── 128×128×1 Field{Face, Center, Center} on RectilinearGrid on CPUWe also calculate (ii) the speed of the flow,
\[s = \sqrt{u^2 + v^2} \, .\]
s = sqrt(u^2 + v^2)UnaryOperation at (Face, Center, Center)
├── grid: 128×128×1 RectilinearGrid{Float64, Periodic, Periodic, Flat} on CPU with 3×3×0 halo
└── tree:
sqrt at (Face, Center, Center) via identity
└── + at (Face, Center, Center)
├── ^ at (Face, Center, Center)
│ ├── 128×128×1 Field{Face, Center, Center} on RectilinearGrid on CPU
│ └── 2
└── ^ at (Center, Face, Center)
├── 128×128×1 Field{Center, Face, Center} on RectilinearGrid on CPU
└── 2We pass these operations to an output writer below to calculate and output them during the simulation.
filename = "two_dimensional_turbulence"
simulation.output_writers[:fields] = JLD2OutputWriter(model, (; ω, s),
schedule = TimeInterval(0.6),
filename = filename * ".jld2",
overwrite_existing = true)JLD2OutputWriter scheduled on TimeInterval(600 ms):
├── filepath: two_dimensional_turbulence.jld2
├── 2 outputs: (ω, s)
├── array type: Array{Float64}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 28.9 KiBRunning the simulation
Pretty much just
run!(simulation)[ Info: Initializing simulation...
[ Info: Iteration: 0000, time: 0.000, Δt: 1.00e-01, max(|u|) = 7.3e-01, wall time: 0 seconds
[ Info: ... simulation initialization complete (8.104 seconds)
[ Info: Executing initial time step...
[ Info: ... initial time step complete (3.718 seconds).
[ Info: Iteration: 0100, time: 6.060, Δt: 6.45e-02, max(|u|) = 3.7e-01, wall time: 12.727 seconds
[ Info: Iteration: 0200, time: 13.280, Δt: 8.20e-02, max(|u|) = 3.2e-01, wall time: 13.579 seconds
[ Info: Iteration: 0300, time: 20.632, Δt: 7.82e-02, max(|u|) = 2.9e-01, wall time: 14.387 seconds
[ Info: Iteration: 0400, time: 28.160, Δt: 8.08e-02, max(|u|) = 2.7e-01, wall time: 15.142 seconds
[ Info: Iteration: 0500, time: 35.640, Δt: 7.81e-02, max(|u|) = 2.6e-01, wall time: 15.975 seconds
[ Info: Iteration: 0600, time: 43.200, Δt: 8.23e-02, max(|u|) = 2.6e-01, wall time: 16.734 seconds
[ Info: Simulation is stopping after running for 17.443 seconds.
[ Info: Simulation time 50 seconds equals or exceeds stop time 50 seconds.
Visualizing the results
We load the output.
ω_timeseries = FieldTimeSeries(filename * ".jld2", "ω")
s_timeseries = FieldTimeSeries(filename * ".jld2", "s")
times = ω_timeseries.timesand animate the vorticity and fluid speed.
using CairoMakie
set_theme!(Theme(fontsize = 24))
fig = Figure(size = (800, 500))
axis_kwargs = (xlabel = "x",
ylabel = "y",
limits = ((0, 2π), (0, 2π)),
aspect = AxisAspect(1))
ax_ω = Axis(fig[2, 1]; title = "Vorticity", axis_kwargs...)
ax_s = Axis(fig[2, 2]; title = "Speed", axis_kwargs...)We use Makie's Observable to animate the data. To dive into how Observables work we refer to Makie.jl's Documentation.
n = Observable(1)Observable(1)
Now let's plot the vorticity and speed.
ω = @lift ω_timeseries[$n]
s = @lift s_timeseries[$n]
heatmap!(ax_ω, ω; colormap = :balance, colorrange = (-2, 2))
heatmap!(ax_s, s; colormap = :speed, colorrange = (0, 0.2))
title = @lift "t = " * string(round(times[$n], digits=2))
Label(fig[1, 1:2], title, fontsize=24, tellwidth=false)
fig
Finally, we record a movie.
frames = 1:length(times)
@info "Making a neat animation of vorticity and speed..."
record(fig, filename * ".mp4", frames, framerate=24) do i
n[] = i
end[ Info: Making a neat animation of vorticity and speed...
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