GitHub

Langmuir turbulence example

This example implements a Langmuir turbulence simulation similar to the one reported in section 4 of

Wagner et al., "Near-inertial waves and turbulence driven by the growth of swell", Journal of Physical Oceanography (2021)

This example demonstrates

  • How to run large eddy simulations with surface wave effects via the Craik-Leibovich approximation.

  • How to specify time- and horizontally-averaged output.

Install dependencies

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

using Pkg
pkg"add Oceananigans, CairoMakie, CUDA"
using Oceananigans
using Oceananigans.Units: minute, minutes, hours
using CUDA

Model set-up

To build the model, we specify the grid, Stokes drift, boundary conditions, and Coriolis parameter.

Domain and numerical grid specification

We use a modest resolution and the same total extent as Wagner et al. (2021),

grid = RectilinearGrid(GPU(), size=(128, 128, 64), extent=(128, 128, 64))
128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CUDAGPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 128.0) regularly spaced with Δx=1.0
├── Periodic y ∈ [0.0, 128.0) regularly spaced with Δy=1.0
└── Bounded  z ∈ [-64.0, 0.0] regularly spaced with Δz=1.0

The Stokes Drift profile

The surface wave Stokes drift profile prescribed in Wagner et al. (2021), corresponds to a 'monochromatic' (that is, single-frequency) wave field.

A monochromatic wave field is characterized by its wavelength and amplitude (half the distance from wave crest to wave trough), which determine the wave frequency and the vertical scale of the Stokes drift profile.

g = Oceananigans.defaults.gravitational_acceleration

amplitude = 0.8 # m
wavelength = 60  # m
wavenumber = 2π / wavelength # m⁻¹
frequency = sqrt(g * wavenumber) # s⁻¹

# The vertical scale over which the Stokes drift of a monochromatic surface wave
# decays away from the surface is `1/2wavenumber`, or
const vertical_scale = wavelength / 4π

# Stokes drift velocity at the surface
const Uˢ = amplitude^2 * wavenumber * frequency # m s⁻¹
0.06791774197745354

The const declarations ensure that Stokes drift functions compile on the GPU. To run this example on the CPU, replace GPU() with CPU() in the RectilinearGrid constructor above.

The Stokes drift profile is

uˢ(z) = Uˢ * exp(z / vertical_scale)
uˢ (generic function with 1 method)

and its z-derivative is

∂z_uˢ(z, t) = 1 / vertical_scale * Uˢ * exp(z / vertical_scale)
∂z_uˢ (generic function with 1 method)
The Craik-Leibovich equations in Oceananigans

Oceananigans implements the Craik-Leibovich approximation for surface wave effects using the Lagrangian-mean velocity field as its prognostic momentum variable. In other words, model.velocities.u is the Lagrangian-mean $x$-velocity beneath surface waves. This differs from models that use the Eulerian-mean velocity field as a prognostic variable, but has the advantage that $u$ accounts for the total advection of tracers and momentum, and that $u = v = w = 0$ is a steady solution even when Coriolis forces are present. See the physics documentation for more information.

Finally, we note that the time-derivative of the Stokes drift must be provided if the Stokes drift and surface wave field undergoes forced changes in time. In this example, the Stokes drift is constant and thus the time-derivative of the Stokes drift is 0.

Boundary conditions

At the surface $z = 0$, Wagner et al. (2021) impose

τx = -3.72e-5 # m² s⁻², surface kinematic momentum flux
u_boundary_conditions = FieldBoundaryConditions(top = FluxBoundaryCondition(τx))
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── top: FluxBoundaryCondition: -3.72e-5
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)

Wagner et al. (2021) impose a linear buoyancy gradient N² at the bottom along with a weak, destabilizing flux of buoyancy at the surface to faciliate spin-up from rest.

Jᵇ = 2.307e-8 # m² s⁻³, surface buoyancy flux
N² = 1.936e-5 # s⁻², initial and bottom buoyancy gradient

b_boundary_conditions = FieldBoundaryConditions(top = FluxBoundaryCondition(Jᵇ),
                                                bottom = GradientBoundaryCondition(N²))
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: GradientBoundaryCondition: 1.936e-5
├── top: FluxBoundaryCondition: 2.307e-8
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
The flux convention in Oceananigans

Note that Oceananigans uses "positive upward" conventions for all fluxes. In consequence, a negative flux at the surface drives positive velocities, and a positive flux of buoyancy drives cooling.

Coriolis parameter

Wagner et al. (2021) use

coriolis = FPlane(f=1e-4) # s⁻¹
FPlane{Float64}(f=0.0001)

which is typical for mid-latitudes on Earth.

Model instantiation

We are ready to build the model. We use a fifth-order Weighted Essentially Non-Oscillatory (WENO) advection scheme and the AnisotropicMinimumDissipation model for large eddy simulation. Because our Stokes drift does not vary in $x, y$, we use UniformStokesDrift, which expects Stokes drift functions of $z, t$ only.

model = NonhydrostaticModel(grid; coriolis,
                            advection = WENO(order=9),
                            tracers = :b,
                            buoyancy = BuoyancyTracer(),
                            stokes_drift = UniformStokesDrift(∂z_uˢ=∂z_uˢ),
                            boundary_conditions = (u=u_boundary_conditions, b=b_boundary_conditions))
NonhydrostaticModel{CUDAGPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CUDAGPU with 5×5×5 halo
├── timestepper: RungeKutta3TimeStepper
├── advection scheme: WENO{5, Float64, Float32}(order=9)
├── tracers: b
├── closure: Nothing
├── buoyancy: BuoyancyTracer with ĝ = NegativeZDirection()
└── coriolis: FPlane{Float64}(f=0.0001)

Initial conditions

We make use of random noise concentrated in the upper 4 meters for buoyancy and velocity initial conditions,

Ξ(z) = randn() * exp(z / 4)

Our initial condition for buoyancy consists of a surface mixed layer 33 m deep, a deep linear stratification, plus noise,

initial_mixed_layer_depth = 33 # m
stratification(z) = z < - initial_mixed_layer_depth ? N² * z : N² * (-initial_mixed_layer_depth)

bᵢ(x, y, z) = stratification(z) + 1e-1 * Ξ(z) * N² * model.grid.Lz
bᵢ (generic function with 1 method)

The simulation we reproduce from Wagner et al. (2021) is zero Lagrangian-mean velocity. This initial condition is consistent with a wavy, quiescent ocean suddenly impacted by winds. To this quiescent state we add noise scaled by the friction velocity to $u$ and $w$.

u★ = sqrt(abs(τx))
uᵢ(x, y, z) = u★ * 1e-1 * Ξ(z)
wᵢ(x, y, z) = u★ * 1e-1 * Ξ(z)

set!(model, u=uᵢ, w=wᵢ, b=bᵢ)

Setting up the simulation

simulation = Simulation(model, Δt=45.0, stop_time=4hours)
Simulation of NonhydrostaticModel{CUDAGPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── Next time step: 45 seconds
├── run_wall_time: 0 seconds
├── run_wall_time / iteration: NaN days
├── stop_time: 4 hours
├── stop_iteration: Inf
├── wall_time_limit: Inf
├── minimum_relative_step: 0.0
├── 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

We use the TimeStepWizard for adaptive time-stepping with a Courant-Freidrichs-Lewy (CFL) number of 1.0,

conjure_time_step_wizard!(simulation, cfl=1.0, max_Δt=1minute)

Nice progress messaging

We define a function that prints a helpful message with maximum absolute value of $u, v, w$ and the current wall clock time.

using Printf

function progress(simulation)
    u, v, w = simulation.model.velocities

    # Print a progress message
    msg = @sprintf("i: %04d, t: %s, Δt: %s, umax = (%.1e, %.1e, %.1e) ms⁻¹, wall time: %s\n",
                   iteration(simulation),
                   prettytime(time(simulation)),
                   prettytime(simulation.Δt),
                   maximum(abs, u), maximum(abs, v), maximum(abs, w),
                   prettytime(simulation.run_wall_time))

    @info msg

    return nothing
end

simulation.callbacks[:progress] = Callback(progress, IterationInterval(20))
Callback of progress on IterationInterval(20)

Output

A field writer

We set up an output writer for the simulation that saves all velocity fields, tracer fields, and the subgrid turbulent diffusivity.

output_interval = 5minutes

fields_to_output = merge(model.velocities, model.tracers)

simulation.output_writers[:fields] =
    JLD2Writer(model, fields_to_output,
               schedule = TimeInterval(output_interval),
               filename = "langmuir_turbulence_fields.jld2",
               overwrite_existing = true)
JLD2Writer scheduled on TimeInterval(5 minutes):
├── filepath: langmuir_turbulence_fields.jld2
├── 4 outputs: (u, v, w, b)
├── array_type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 0 bytes (file not yet created)

An "averages" writer

We also set up output of time- and horizontally-averaged velocity field and momentum fluxes.

u, v, w = model.velocities
b = model.tracers.b

 U = Average(u, dims=(1, 2))
 V = Average(v, dims=(1, 2))
 B = Average(b, dims=(1, 2))
wu = Average(w * u, dims=(1, 2))
wv = Average(w * v, dims=(1, 2))

simulation.output_writers[:averages] =
    JLD2Writer(model, (; U, V, B, wu, wv),
               schedule = AveragedTimeInterval(output_interval, window=2minutes),
               filename = "langmuir_turbulence_averages.jld2",
               overwrite_existing = true)
JLD2Writer scheduled on TimeInterval(5 minutes):
├── filepath: langmuir_turbulence_averages.jld2
├── 5 outputs: (U, V, B, wu, wv) averaged on AveragedTimeInterval(window=2 minutes, stride=1, interval=5 minutes)
├── array_type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 0 bytes (file not yet created)

Running the simulation

This part is easy,

run!(simulation)
[ Info: Initializing simulation...
[ Info: i: 0000, t: 0 seconds, Δt: 49.500 seconds, umax = (1.6e-03, 9.9e-04, 1.3e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (9.527 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.977 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 19.099 seconds, umax = (3.6e-02, 1.3e-02, 2.3e-02) ms⁻¹, wall time: 13.756 seconds
[ Info: i: 0040, t: 16.830 minutes, Δt: 13.057 seconds, umax = (5.3e-02, 2.2e-02, 2.2e-02) ms⁻¹, wall time: 14.256 seconds
[ Info: i: 0060, t: 20.792 minutes, Δt: 10.695 seconds, umax = (6.0e-02, 2.9e-02, 3.1e-02) ms⁻¹, wall time: 14.740 seconds
[ Info: i: 0080, t: 24.355 minutes, Δt: 11.053 seconds, umax = (6.2e-02, 3.2e-02, 3.3e-02) ms⁻¹, wall time: 15.107 seconds
[ Info: i: 0100, t: 27.979 minutes, Δt: 11.068 seconds, umax = (6.0e-02, 3.9e-02, 3.0e-02) ms⁻¹, wall time: 16.110 seconds
[ Info: i: 0120, t: 31.676 minutes, Δt: 10.328 seconds, umax = (6.1e-02, 2.9e-02, 3.3e-02) ms⁻¹, wall time: 23.291 seconds
[ Info: i: 0140, t: 35.187 minutes, Δt: 10.907 seconds, umax = (6.4e-02, 3.5e-02, 2.9e-02) ms⁻¹, wall time: 23.944 seconds
[ Info: i: 0160, t: 38.785 minutes, Δt: 10.246 seconds, umax = (6.9e-02, 3.4e-02, 3.3e-02) ms⁻¹, wall time: 24.253 seconds
[ Info: i: 0180, t: 41.883 minutes, Δt: 9.364 seconds, umax = (7.1e-02, 3.8e-02, 3.3e-02) ms⁻¹, wall time: 24.783 seconds
[ Info: i: 0200, t: 45 minutes, Δt: 8.939 seconds, umax = (6.9e-02, 3.8e-02, 3.5e-02) ms⁻¹, wall time: 25.206 seconds
[ Info: i: 0220, t: 47.984 minutes, Δt: 9.241 seconds, umax = (7.3e-02, 3.9e-02, 3.6e-02) ms⁻¹, wall time: 25.730 seconds
[ Info: i: 0240, t: 50.898 minutes, Δt: 8.548 seconds, umax = (7.3e-02, 3.9e-02, 3.5e-02) ms⁻¹, wall time: 26.277 seconds
[ Info: i: 0260, t: 53.837 minutes, Δt: 9.063 seconds, umax = (8.0e-02, 3.8e-02, 3.8e-02) ms⁻¹, wall time: 26.670 seconds
[ Info: i: 0280, t: 56.787 minutes, Δt: 7.772 seconds, umax = (8.2e-02, 4.7e-02, 3.6e-02) ms⁻¹, wall time: 27.193 seconds
[ Info: i: 0300, t: 59.480 minutes, Δt: 8.535 seconds, umax = (8.2e-02, 4.5e-02, 3.7e-02) ms⁻¹, wall time: 27.613 seconds
[ Info: i: 0320, t: 1.038 hours, Δt: 8.419 seconds, umax = (8.3e-02, 4.7e-02, 4.0e-02) ms⁻¹, wall time: 28.132 seconds
[ Info: i: 0340, t: 1.083 hours, Δt: 8.180 seconds, umax = (8.4e-02, 4.6e-02, 3.7e-02) ms⁻¹, wall time: 28.556 seconds
[ Info: i: 0360, t: 1.129 hours, Δt: 8.344 seconds, umax = (8.5e-02, 5.2e-02, 3.6e-02) ms⁻¹, wall time: 29.067 seconds
[ Info: i: 0380, t: 1.173 hours, Δt: 7.793 seconds, umax = (9.1e-02, 5.0e-02, 3.7e-02) ms⁻¹, wall time: 29.666 seconds
[ Info: i: 0400, t: 1.218 hours, Δt: 8.066 seconds, umax = (8.6e-02, 4.3e-02, 3.7e-02) ms⁻¹, wall time: 30.001 seconds
[ Info: i: 0420, t: 1.261 hours, Δt: 7.916 seconds, umax = (9.1e-02, 4.8e-02, 3.6e-02) ms⁻¹, wall time: 30.560 seconds
[ Info: i: 0440, t: 1.305 hours, Δt: 8.118 seconds, umax = (9.5e-02, 4.9e-02, 3.8e-02) ms⁻¹, wall time: 30.941 seconds
[ Info: i: 0460, t: 1.349 hours, Δt: 7.712 seconds, umax = (8.9e-02, 5.4e-02, 4.4e-02) ms⁻¹, wall time: 31.465 seconds
[ Info: i: 0480, t: 1.392 hours, Δt: 7.454 seconds, umax = (9.3e-02, 5.1e-02, 4.2e-02) ms⁻¹, wall time: 31.883 seconds
[ Info: i: 0500, t: 1.431 hours, Δt: 7.517 seconds, umax = (9.2e-02, 5.0e-02, 3.9e-02) ms⁻¹, wall time: 32.409 seconds
[ Info: i: 0520, t: 1.472 hours, Δt: 7.224 seconds, umax = (9.4e-02, 5.8e-02, 4.1e-02) ms⁻¹, wall time: 32.832 seconds
[ Info: i: 0540, t: 1.512 hours, Δt: 6.872 seconds, umax = (9.4e-02, 5.1e-02, 4.2e-02) ms⁻¹, wall time: 33.386 seconds
[ Info: i: 0560, t: 1.552 hours, Δt: 7.088 seconds, umax = (9.5e-02, 5.4e-02, 4.1e-02) ms⁻¹, wall time: 33.785 seconds
[ Info: i: 0580, t: 1.589 hours, Δt: 7.020 seconds, umax = (9.3e-02, 6.2e-02, 5.1e-02) ms⁻¹, wall time: 34.382 seconds
[ Info: i: 0600, t: 1.630 hours, Δt: 7.081 seconds, umax = (9.4e-02, 5.5e-02, 4.6e-02) ms⁻¹, wall time: 34.727 seconds
[ Info: i: 0620, t: 1.669 hours, Δt: 6.928 seconds, umax = (9.6e-02, 5.5e-02, 4.1e-02) ms⁻¹, wall time: 35.366 seconds
[ Info: i: 0640, t: 1.707 hours, Δt: 6.745 seconds, umax = (1.1e-01, 5.7e-02, 3.9e-02) ms⁻¹, wall time: 35.680 seconds
[ Info: i: 0660, t: 1.744 hours, Δt: 6.981 seconds, umax = (1.1e-01, 5.6e-02, 4.3e-02) ms⁻¹, wall time: 36.132 seconds
[ Info: i: 0680, t: 1.783 hours, Δt: 6.962 seconds, umax = (9.8e-02, 6.0e-02, 4.2e-02) ms⁻¹, wall time: 36.633 seconds
[ Info: i: 0700, t: 1.822 hours, Δt: 6.586 seconds, umax = (1.0e-01, 6.0e-02, 4.4e-02) ms⁻¹, wall time: 37.088 seconds
[ Info: i: 0720, t: 1.858 hours, Δt: 7.002 seconds, umax = (9.8e-02, 6.3e-02, 4.2e-02) ms⁻¹, wall time: 37.604 seconds
[ Info: i: 0740, t: 1.897 hours, Δt: 6.935 seconds, umax = (9.9e-02, 5.9e-02, 4.4e-02) ms⁻¹, wall time: 38.058 seconds
[ Info: i: 0760, t: 1.933 hours, Δt: 7.016 seconds, umax = (1.0e-01, 6.0e-02, 5.0e-02) ms⁻¹, wall time: 38.563 seconds
[ Info: i: 0780, t: 1.972 hours, Δt: 6.555 seconds, umax = (9.7e-02, 6.2e-02, 4.2e-02) ms⁻¹, wall time: 39.008 seconds
[ Info: i: 0800, t: 2.008 hours, Δt: 7.020 seconds, umax = (1.0e-01, 5.9e-02, 4.4e-02) ms⁻¹, wall time: 39.586 seconds
[ Info: i: 0820, t: 2.046 hours, Δt: 6.403 seconds, umax = (1.0e-01, 6.3e-02, 4.1e-02) ms⁻¹, wall time: 39.965 seconds
[ Info: i: 0840, t: 2.082 hours, Δt: 6.818 seconds, umax = (1.0e-01, 6.0e-02, 4.7e-02) ms⁻¹, wall time: 40.424 seconds
[ Info: i: 0860, t: 2.119 hours, Δt: 6.174 seconds, umax = (9.9e-02, 6.3e-02, 4.2e-02) ms⁻¹, wall time: 40.941 seconds
[ Info: i: 0880, t: 2.154 hours, Δt: 6.848 seconds, umax = (1.0e-01, 6.5e-02, 4.1e-02) ms⁻¹, wall time: 41.401 seconds
[ Info: i: 0900, t: 2.191 hours, Δt: 6.729 seconds, umax = (1.0e-01, 6.3e-02, 4.5e-02) ms⁻¹, wall time: 41.911 seconds
[ Info: i: 0920, t: 2.227 hours, Δt: 5.965 seconds, umax = (1.0e-01, 6.2e-02, 4.4e-02) ms⁻¹, wall time: 42.379 seconds
[ Info: i: 0940, t: 2.260 hours, Δt: 6.661 seconds, umax = (1.1e-01, 7.0e-02, 4.7e-02) ms⁻¹, wall time: 42.913 seconds
[ Info: i: 0960, t: 2.297 hours, Δt: 6.427 seconds, umax = (1.0e-01, 6.6e-02, 4.4e-02) ms⁻¹, wall time: 43.335 seconds
[ Info: i: 0980, t: 2.333 hours, Δt: 6.035 seconds, umax = (1.1e-01, 6.9e-02, 4.6e-02) ms⁻¹, wall time: 43.797 seconds
[ Info: i: 1000, t: 2.365 hours, Δt: 6.391 seconds, umax = (1.1e-01, 6.6e-02, 4.5e-02) ms⁻¹, wall time: 45.142 seconds
[ Info: i: 1020, t: 2.401 hours, Δt: 6.363 seconds, umax = (1.1e-01, 6.7e-02, 4.9e-02) ms⁻¹, wall time: 45.603 seconds
[ Info: i: 1040, t: 2.435 hours, Δt: 6.383 seconds, umax = (1.1e-01, 6.7e-02, 4.5e-02) ms⁻¹, wall time: 51.152 seconds
[ Info: i: 1060, t: 2.470 hours, Δt: 6.252 seconds, umax = (1.0e-01, 6.6e-02, 4.5e-02) ms⁻¹, wall time: 51.597 seconds
[ Info: i: 1080, t: 2.503 hours, Δt: 6.156 seconds, umax = (1.0e-01, 6.6e-02, 4.4e-02) ms⁻¹, wall time: 52.342 seconds
[ Info: i: 1100, t: 2.536 hours, Δt: 6.286 seconds, umax = (1.0e-01, 7.0e-02, 4.4e-02) ms⁻¹, wall time: 52.687 seconds
[ Info: i: 1120, t: 2.571 hours, Δt: 6.281 seconds, umax = (1.1e-01, 7.3e-02, 4.0e-02) ms⁻¹, wall time: 53.154 seconds
[ Info: i: 1140, t: 2.605 hours, Δt: 6.164 seconds, umax = (1.1e-01, 7.5e-02, 4.3e-02) ms⁻¹, wall time: 53.716 seconds
[ Info: i: 1160, t: 2.638 hours, Δt: 6.113 seconds, umax = (1.0e-01, 7.2e-02, 4.3e-02) ms⁻¹, wall time: 54.171 seconds
[ Info: i: 1180, t: 2.672 hours, Δt: 6.521 seconds, umax = (1.1e-01, 6.9e-02, 4.4e-02) ms⁻¹, wall time: 54.828 seconds
[ Info: i: 1200, t: 2.707 hours, Δt: 6.387 seconds, umax = (1.1e-01, 7.1e-02, 4.4e-02) ms⁻¹, wall time: 55.191 seconds
[ Info: i: 1220, t: 2.742 hours, Δt: 6.223 seconds, umax = (1.1e-01, 6.7e-02, 4.9e-02) ms⁻¹, wall time: 55.655 seconds
[ Info: i: 1240, t: 2.776 hours, Δt: 5.597 seconds, umax = (1.1e-01, 7.0e-02, 4.6e-02) ms⁻¹, wall time: 56.179 seconds
[ Info: i: 1260, t: 2.807 hours, Δt: 5.906 seconds, umax = (1.1e-01, 6.9e-02, 4.3e-02) ms⁻¹, wall time: 56.639 seconds
[ Info: i: 1280, t: 2.838 hours, Δt: 6.405 seconds, umax = (1.1e-01, 7.7e-02, 4.8e-02) ms⁻¹, wall time: 57.255 seconds
[ Info: i: 1300, t: 2.874 hours, Δt: 6.432 seconds, umax = (1.1e-01, 7.4e-02, 4.4e-02) ms⁻¹, wall time: 57.619 seconds
[ Info: i: 1320, t: 2.908 hours, Δt: 5.957 seconds, umax = (1.1e-01, 7.9e-02, 4.8e-02) ms⁻¹, wall time: 58.085 seconds
[ Info: i: 1340, t: 2.941 hours, Δt: 6.455 seconds, umax = (1.1e-01, 6.9e-02, 4.8e-02) ms⁻¹, wall time: 58.612 seconds
[ Info: i: 1360, t: 2.976 hours, Δt: 5.925 seconds, umax = (1.1e-01, 8.3e-02, 4.7e-02) ms⁻¹, wall time: 59.077 seconds
[ Info: i: 1380, t: 3.008 hours, Δt: 5.655 seconds, umax = (1.1e-01, 8.5e-02, 5.1e-02) ms⁻¹, wall time: 59.650 seconds
[ Info: i: 1400, t: 3.041 hours, Δt: 6.287 seconds, umax = (1.1e-01, 7.5e-02, 4.5e-02) ms⁻¹, wall time: 1.001 minutes
[ Info: i: 1420, t: 3.076 hours, Δt: 5.826 seconds, umax = (1.1e-01, 7.8e-02, 4.8e-02) ms⁻¹, wall time: 1.009 minutes
[ Info: i: 1440, t: 3.108 hours, Δt: 6.051 seconds, umax = (1.1e-01, 7.2e-02, 5.3e-02) ms⁻¹, wall time: 1.018 minutes
[ Info: i: 1460, t: 3.140 hours, Δt: 4.991 seconds, umax = (1.2e-01, 7.9e-02, 4.8e-02) ms⁻¹, wall time: 1.025 minutes
[ Info: i: 1480, t: 3.168 hours, Δt: 5.520 seconds, umax = (1.2e-01, 7.8e-02, 4.3e-02) ms⁻¹, wall time: 1.036 minutes
[ Info: i: 1500, t: 3.200 hours, Δt: 5.569 seconds, umax = (1.2e-01, 7.7e-02, 4.6e-02) ms⁻¹, wall time: 1.042 minutes
[ Info: i: 1520, t: 3.231 hours, Δt: 5.682 seconds, umax = (1.1e-01, 8.3e-02, 4.9e-02) ms⁻¹, wall time: 1.049 minutes
[ Info: i: 1540, t: 3.263 hours, Δt: 5.824 seconds, umax = (1.1e-01, 8.1e-02, 5.1e-02) ms⁻¹, wall time: 1.059 minutes
[ Info: i: 1560, t: 3.296 hours, Δt: 6.183 seconds, umax = (1.1e-01, 7.5e-02, 4.8e-02) ms⁻¹, wall time: 1.066 minutes
[ Info: i: 1580, t: 3.330 hours, Δt: 5.661 seconds, umax = (1.1e-01, 8.6e-02, 5.0e-02) ms⁻¹, wall time: 1.074 minutes
[ Info: i: 1600, t: 3.361 hours, Δt: 5.657 seconds, umax = (1.1e-01, 8.5e-02, 5.3e-02) ms⁻¹, wall time: 1.084 minutes
[ Info: i: 1620, t: 3.394 hours, Δt: 5.939 seconds, umax = (1.1e-01, 7.6e-02, 5.3e-02) ms⁻¹, wall time: 1.092 minutes
[ Info: i: 1640, t: 3.426 hours, Δt: 6.131 seconds, umax = (1.1e-01, 7.4e-02, 5.0e-02) ms⁻¹, wall time: 1.103 minutes
[ Info: i: 1660, t: 3.460 hours, Δt: 6.350 seconds, umax = (1.1e-01, 7.7e-02, 5.4e-02) ms⁻¹, wall time: 1.110 minutes
[ Info: i: 1680, t: 3.493 hours, Δt: 5.617 seconds, umax = (1.1e-01, 7.7e-02, 5.5e-02) ms⁻¹, wall time: 1.118 minutes
[ Info: i: 1700, t: 3.524 hours, Δt: 5.587 seconds, umax = (1.1e-01, 7.2e-02, 5.2e-02) ms⁻¹, wall time: 1.126 minutes
[ Info: i: 1720, t: 3.555 hours, Δt: 5.737 seconds, umax = (1.1e-01, 7.2e-02, 5.1e-02) ms⁻¹, wall time: 1.134 minutes
[ Info: i: 1740, t: 3.587 hours, Δt: 6.095 seconds, umax = (1.0e-01, 7.8e-02, 5.0e-02) ms⁻¹, wall time: 1.145 minutes
[ Info: i: 1760, t: 3.621 hours, Δt: 5.937 seconds, umax = (1.1e-01, 7.2e-02, 4.5e-02) ms⁻¹, wall time: 1.151 minutes
[ Info: i: 1780, t: 3.654 hours, Δt: 5.729 seconds, umax = (1.2e-01, 8.0e-02, 4.8e-02) ms⁻¹, wall time: 1.158 minutes
[ Info: i: 1800, t: 3.686 hours, Δt: 5.689 seconds, umax = (1.0e-01, 8.0e-02, 4.5e-02) ms⁻¹, wall time: 1.168 minutes
[ Info: i: 1820, t: 3.717 hours, Δt: 5.812 seconds, umax = (1.1e-01, 7.7e-02, 4.8e-02) ms⁻¹, wall time: 1.176 minutes
[ Info: i: 1840, t: 3.750 hours, Δt: 5.616 seconds, umax = (1.0e-01, 7.8e-02, 5.0e-02) ms⁻¹, wall time: 1.183 minutes
[ Info: i: 1860, t: 3.780 hours, Δt: 5.850 seconds, umax = (1.1e-01, 7.9e-02, 4.9e-02) ms⁻¹, wall time: 1.194 minutes
[ Info: i: 1880, t: 3.813 hours, Δt: 5.684 seconds, umax = (1.1e-01, 8.1e-02, 5.0e-02) ms⁻¹, wall time: 1.202 minutes
[ Info: i: 1900, t: 3.842 hours, Δt: 5.672 seconds, umax = (1.1e-01, 7.8e-02, 4.6e-02) ms⁻¹, wall time: 1.212 minutes
[ Info: i: 1920, t: 3.875 hours, Δt: 6.145 seconds, umax = (1.1e-01, 7.7e-02, 5.5e-02) ms⁻¹, wall time: 1.219 minutes
[ Info: i: 1940, t: 3.909 hours, Δt: 5.542 seconds, umax = (1.1e-01, 8.1e-02, 4.8e-02) ms⁻¹, wall time: 1.227 minutes
[ Info: i: 1960, t: 3.940 hours, Δt: 5.486 seconds, umax = (1.2e-01, 9.1e-02, 4.9e-02) ms⁻¹, wall time: 1.236 minutes
[ Info: i: 1980, t: 3.971 hours, Δt: 5.796 seconds, umax = (1.1e-01, 8.3e-02, 4.8e-02) ms⁻¹, wall time: 1.243 minutes
[ Info: Simulation is stopping after running for 1.250 minutes.
[ Info: Simulation time 4 hours equals or exceeds stop time 4 hours.

Making a neat movie

We look at the results by loading data from file with FieldTimeSeries, and plotting vertical slices of $u$ and $w$, and a horizontal slice of $w$ to look for Langmuir cells.

using CairoMakie

time_series = (;
     w = FieldTimeSeries("langmuir_turbulence_fields.jld2", "w"),
     u = FieldTimeSeries("langmuir_turbulence_fields.jld2", "u"),
     B = FieldTimeSeries("langmuir_turbulence_averages.jld2", "B"),
     U = FieldTimeSeries("langmuir_turbulence_averages.jld2", "U"),
     V = FieldTimeSeries("langmuir_turbulence_averages.jld2", "V"),
    wu = FieldTimeSeries("langmuir_turbulence_averages.jld2", "wu"),
    wv = FieldTimeSeries("langmuir_turbulence_averages.jld2", "wv"))

times = time_series.w.times

We are now ready to animate using Makie. 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)

wxy_title = @lift string("w(x, y, t) at z=-8 m and t = ", prettytime(times[$n]))
wxz_title = @lift string("w(x, z, t) at y=0 m and t = ", prettytime(times[$n]))
uxz_title = @lift string("u(x, z, t) at y=0 m and t = ", prettytime(times[$n]))

fig = Figure(size = (850, 850))

ax_B = Axis(fig[1, 4];
            xlabel = "Buoyancy (m s⁻²)",
            ylabel = "z (m)")

ax_U = Axis(fig[2, 4];
            xlabel = "Velocities (m s⁻¹)",
            ylabel = "z (m)",
            limits = ((-0.07, 0.07), nothing))

ax_fluxes = Axis(fig[3, 4];
                 xlabel = "Momentum fluxes (m² s⁻²)",
                 ylabel = "z (m)",
                 limits = ((-3.5e-5, 3.5e-5), nothing))

ax_wxy = Axis(fig[1, 1:2];
              xlabel = "x (m)",
              ylabel = "y (m)",
              aspect = DataAspect(),
              limits = ((0, grid.Lx), (0, grid.Ly)),
              title = wxy_title)

ax_wxz = Axis(fig[2, 1:2];
              xlabel = "x (m)",
              ylabel = "z (m)",
              aspect = AxisAspect(2),
              limits = ((0, grid.Lx), (-grid.Lz, 0)),
              title = wxz_title)

ax_uxz = Axis(fig[3, 1:2];
              xlabel = "x (m)",
              ylabel = "z (m)",
              aspect = AxisAspect(2),
              limits = ((0, grid.Lx), (-grid.Lz, 0)),
              title = uxz_title)


wₙ = @lift time_series.w[$n]
uₙ = @lift time_series.u[$n]
Bₙ = @lift view(time_series.B[$n], 1, 1, :)
Uₙ = @lift view(time_series.U[$n], 1, 1, :)
Vₙ = @lift view(time_series.V[$n], 1, 1, :)
wuₙ = @lift view(time_series.wu[$n], 1, 1, :)
wvₙ = @lift view(time_series.wv[$n], 1, 1, :)

k = searchsortedfirst(znodes(grid, Face(); with_halos=true), -8)
wxyₙ = @lift view(time_series.w[$n], :, :, k)
wxzₙ = @lift view(time_series.w[$n], :, 1, :)
uxzₙ = @lift view(time_series.u[$n], :, 1, :)

wlims = (-0.03, 0.03)
ulims = (-0.05, 0.05)

lines!(ax_B, Bₙ)

lines!(ax_U, Uₙ; label = L"\bar{u}")
lines!(ax_U, Vₙ; label = L"\bar{v}")
axislegend(ax_U; position = :rb)

lines!(ax_fluxes, wuₙ; label = L"mean $wu$")
lines!(ax_fluxes, wvₙ; label = L"mean $wv$")
axislegend(ax_fluxes; position = :rb)

hm_wxy = heatmap!(ax_wxy, wxyₙ;
                  colorrange = wlims,
                  colormap = :balance)

Colorbar(fig[1, 3], hm_wxy; label = "m s⁻¹")

hm_wxz = heatmap!(ax_wxz, wxzₙ;
                  colorrange = wlims,
                  colormap = :balance)

Colorbar(fig[2, 3], hm_wxz; label = "m s⁻¹")

ax_uxz = heatmap!(ax_uxz, uxzₙ;
                  colorrange = ulims,
                  colormap = :balance)

Colorbar(fig[3, 3], ax_uxz; label = "m s⁻¹")

fig

And, finally, we record a movie.

frames = 1:length(times)

CairoMakie.record(fig, "langmuir_turbulence.mp4", frames, framerate=8) do i
    n[] = i
end

Julia version and environment information

This example was executed with the following version of Julia:

using InteractiveUtils: versioninfo
versioninfo()
Julia Version 1.12.4
Commit 01a2eadb047 (2026-01-06 16:56 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 9374F 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver4)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 128 virtual cores)
Environment:
  LD_LIBRARY_PATH = 
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
  JULIA_DEPOT_PATH = /var/lib/buildkite-agent/.julia-oceananigans
  JULIA_PROJECT = /var/lib/buildkite-agent/Oceananigans.jl-29455/docs/
  JULIA_VERSION = 1.12.4
  JULIA_LOAD_PATH = @:@v#.#:@stdlib
  JULIA_VERSION_ENZYME = 1.10.10
  JULIA_PYTHONCALL_EXE = /var/lib/buildkite-agent/Oceananigans.jl-29455/docs/.CondaPkg/.pixi/envs/default/bin/python
  JULIA_DEBUG = Literate

These were the top-level packages installed in the environment:

import Pkg
Pkg.status()
Status `~/Oceananigans.jl-29455/docs/Project.toml`
  [79e6a3ab] Adapt v4.4.0
  [052768ef] CUDA v5.9.6
  [13f3f980] CairoMakie v0.15.8
  [e30172f5] Documenter v1.16.1
  [daee34ce] DocumenterCitations v1.4.1
  [033835bb] JLD2 v0.6.3
  [63c18a36] KernelAbstractions v0.9.40
  [98b081ad] Literate v2.21.0
  [da04e1cc] MPI v0.20.23
  [85f8d34a] NCDatasets v0.14.11
  [9e8cae18] Oceananigans v0.105.0 `..`
  [f27b6e38] Polynomials v4.1.0
  [6038ab10] Rotations v1.7.1
  [d496a93d] SeawaterPolynomials v0.3.10
  [09ab397b] StructArrays v0.7.2
  [bdfc003b] TimesDates v0.3.3
  [2e0b0046] XESMF v0.1.6
  [b77e0a4c] InteractiveUtils v1.11.0
  [37e2e46d] LinearAlgebra v1.12.0
  [44cfe95a] Pkg v1.12.1

This page was generated using Literate.jl.