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
Field
s 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,
timestepper = :RungeKutta3,
advection = UpwindBiasedFifthOrder(),
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
├── tracers: ()
├── closure: ScalarDiffusivity{ExplicitTimeDiscretization}(ν=1.0e-5)
├── buoyancy: Nothing
└── coriolis: Nothing
Random 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)
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 entries
Logging simulation progress
We set up a callback that logs the simulation iteration and time every 100 iterations.
progress(sim) = @info string("Iteration: ", iteration(sim), ", time: ", time(sim))
add_callback!(simulation, progress, 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.velocities
NamedTuple 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 CPU
Next we create two Field
s 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 CPU
We 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
└── 2
We 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]
└── max filesize: Inf YiB
Running the simulation
Pretty much just
run!(simulation)
[ Info: Initializing simulation...
[ Info: Iteration: 0, time: 0.0
[ Info: ... simulation initialization complete (5.130 seconds)
[ Info: Executing initial time step...
[ Info: ... initial time step complete (5.082 seconds).
[ Info: time = 18.0, iteration = 100: NaN found in field u. Stopping simulation.
[ Info: Iteration: 100, time: 18.0
Visualizing the results
We load the output.
ω_timeseries = FieldTimeSeries(filename * ".jld2", "ω")
s_timeseries = FieldTimeSeries(filename * ".jld2", "s")
times = ω_timeseries.times
0.0:0.6:18.0
Construct the $x, y, z$ grid for plotting purposes,
xω, yω, zω = nodes(ω_timeseries)
xs, ys, zs = nodes(s_timeseries)
and animate the vorticity and fluid speed.
using CairoMakie
set_theme!(Theme(fontsize = 24))
@info "Making a neat movie of vorticity and speed..."
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...)
[ Info: Making a neat movie of vorticity and speed...
We use Makie's Observable
to animate the data. To dive into how Observable
s work we refer to Makie.jl's Documentation.
n = Observable(1)
Observable(1)
Now let's plot the vorticity and speed.
ω = @lift interior(ω_timeseries[$n], :, :, 1)
s = @lift interior(s_timeseries[$n], :, :, 1)
heatmap!(ax_ω, xω, yω, ω; colormap = :balance, colorrange = (-2, 2))
heatmap!(ax_s, xs, ys, 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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