Langmuir turbulence example

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

• Wagner, G. L.; Chini, G. P.; Ramadhan, A.; Gallet, B. and Ferrari, R. (2021). Near-inertial waves and turbulence driven by the growth of swell. Journal of Physical Oceanography 51, 1337–1351.

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
using Random

Random.seed!(1337) # for reproducible results

Random.TaskLocalRNG()

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 by 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 -velocity beneath surface waves. This differs from models that use the Eulerian-mean velocity field as a prognostic variable, but has the advantage that accounts for the total advection of tracers and momentum, and that     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  , 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 , we use UniformStokesDrift, which expects Stokes drift functions of 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, Oceananigans.Utils.BackendOptimizedDivision}(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 and .

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 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: [: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: [: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 = (1.8e-03, 9.5e-04, 1.5e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (9.879 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.720 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 19.470 seconds, umax = (3.6e-02, 1.2e-02, 2.1e-02) ms⁻¹, wall time: 13.646 seconds
[ Info: i: 0040, t: 17.083 minutes, Δt: 12.980 seconds, umax = (5.3e-02, 2.1e-02, 2.5e-02) ms⁻¹, wall time: 13.931 seconds
[ Info: i: 0060, t: 21.181 minutes, Δt: 10.514 seconds, umax = (6.4e-02, 2.9e-02, 3.2e-02) ms⁻¹, wall time: 14.253 seconds
[ Info: i: 0080, t: 24.649 minutes, Δt: 10.835 seconds, umax = (6.4e-02, 3.1e-02, 3.4e-02) ms⁻¹, wall time: 14.498 seconds
[ Info: i: 0100, t: 28.387 minutes, Δt: 11.357 seconds, umax = (6.1e-02, 3.0e-02, 3.0e-02) ms⁻¹, wall time: 14.799 seconds
[ Info: i: 0120, t: 32.015 minutes, Δt: 11.254 seconds, umax = (6.1e-02, 2.9e-02, 2.8e-02) ms⁻¹, wall time: 15.108 seconds
[ Info: i: 0140, t: 35.548 minutes, Δt: 10.870 seconds, umax = (6.6e-02, 3.4e-02, 3.0e-02) ms⁻¹, wall time: 15.466 seconds
[ Info: i: 0160, t: 39.055 minutes, Δt: 10.051 seconds, umax = (6.9e-02, 3.7e-02, 3.0e-02) ms⁻¹, wall time: 15.683 seconds
[ Info: i: 0180, t: 42.196 minutes, Δt: 9.376 seconds, umax = (7.2e-02, 3.6e-02, 3.4e-02) ms⁻¹, wall time: 15.988 seconds
[ Info: i: 0200, t: 45.154 minutes, Δt: 9.332 seconds, umax = (7.0e-02, 3.7e-02, 3.3e-02) ms⁻¹, wall time: 16.363 seconds
[ Info: i: 0220, t: 48.263 minutes, Δt: 8.574 seconds, umax = (7.5e-02, 4.3e-02, 3.5e-02) ms⁻¹, wall time: 16.550 seconds
[ Info: i: 0240, t: 51.238 minutes, Δt: 8.667 seconds, umax = (7.5e-02, 4.1e-02, 3.8e-02) ms⁻¹, wall time: 16.838 seconds
[ Info: i: 0260, t: 54.183 minutes, Δt: 8.782 seconds, umax = (7.8e-02, 3.9e-02, 3.6e-02) ms⁻¹, wall time: 17.101 seconds
[ Info: i: 0280, t: 57.056 minutes, Δt: 8.470 seconds, umax = (7.8e-02, 4.6e-02, 3.7e-02) ms⁻¹, wall time: 17.399 seconds
[ Info: i: 0300, t: 59.841 minutes, Δt: 8.423 seconds, umax = (8.2e-02, 4.1e-02, 4.1e-02) ms⁻¹, wall time: 17.669 seconds
[ Info: i: 0320, t: 1.042 hours, Δt: 7.873 seconds, umax = (8.9e-02, 4.4e-02, 3.6e-02) ms⁻¹, wall time: 17.962 seconds
[ Info: i: 0340, t: 1.086 hours, Δt: 8.292 seconds, umax = (8.2e-02, 4.7e-02, 3.8e-02) ms⁻¹, wall time: 18.338 seconds
[ Info: i: 0360, t: 1.132 hours, Δt: 8.269 seconds, umax = (8.2e-02, 4.3e-02, 4.0e-02) ms⁻¹, wall time: 18.526 seconds
[ Info: i: 0380, t: 1.179 hours, Δt: 8.177 seconds, umax = (8.7e-02, 4.6e-02, 3.6e-02) ms⁻¹, wall time: 18.850 seconds
[ Info: i: 0400, t: 1.224 hours, Δt: 7.848 seconds, umax = (8.3e-02, 4.8e-02, 3.6e-02) ms⁻¹, wall time: 19.081 seconds
[ Info: i: 0420, t: 1.267 hours, Δt: 7.613 seconds, umax = (8.8e-02, 5.3e-02, 4.3e-02) ms⁻¹, wall time: 19.384 seconds
[ Info: i: 0440, t: 1.310 hours, Δt: 7.647 seconds, umax = (9.1e-02, 5.1e-02, 3.9e-02) ms⁻¹, wall time: 19.646 seconds
[ Info: i: 0460, t: 1.350 hours, Δt: 7.444 seconds, umax = (9.2e-02, 5.5e-02, 3.8e-02) ms⁻¹, wall time: 19.941 seconds
[ Info: i: 0480, t: 1.392 hours, Δt: 7.477 seconds, umax = (9.3e-02, 5.2e-02, 4.1e-02) ms⁻¹, wall time: 20.204 seconds
[ Info: i: 0500, t: 1.433 hours, Δt: 7.334 seconds, umax = (9.3e-02, 5.3e-02, 4.0e-02) ms⁻¹, wall time: 20.526 seconds
[ Info: i: 0520, t: 1.474 hours, Δt: 7.630 seconds, umax = (9.0e-02, 5.6e-02, 3.8e-02) ms⁻¹, wall time: 20.783 seconds
[ Info: i: 0540, t: 1.515 hours, Δt: 7.618 seconds, umax = (9.1e-02, 5.6e-02, 4.6e-02) ms⁻¹, wall time: 21.095 seconds
[ Info: i: 0560, t: 1.555 hours, Δt: 7.352 seconds, umax = (9.2e-02, 5.5e-02, 4.1e-02) ms⁻¹, wall time: 21.343 seconds
[ Info: i: 0580, t: 1.596 hours, Δt: 7.262 seconds, umax = (9.8e-02, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 21.758 seconds
[ Info: i: 0600, t: 1.636 hours, Δt: 7.464 seconds, umax = (9.6e-02, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 21.998 seconds
[ Info: i: 0620, t: 1.675 hours, Δt: 6.746 seconds, umax = (1.0e-01, 5.6e-02, 4.2e-02) ms⁻¹, wall time: 22.391 seconds
[ Info: i: 0640, t: 1.712 hours, Δt: 7.012 seconds, umax = (9.8e-02, 5.3e-02, 4.2e-02) ms⁻¹, wall time: 22.606 seconds
[ Info: i: 0660, t: 1.750 hours, Δt: 6.606 seconds, umax = (9.8e-02, 5.4e-02, 4.4e-02) ms⁻¹, wall time: 22.871 seconds
[ Info: i: 0680, t: 1.786 hours, Δt: 6.730 seconds, umax = (1.0e-01, 6.0e-02, 4.3e-02) ms⁻¹, wall time: 23.178 seconds
[ Info: i: 0700, t: 1.825 hours, Δt: 6.990 seconds, umax = (9.9e-02, 5.6e-02, 4.1e-02) ms⁻¹, wall time: 23.448 seconds
[ Info: i: 0720, t: 1.863 hours, Δt: 6.936 seconds, umax = (1.0e-01, 5.8e-02, 4.2e-02) ms⁻¹, wall time: 23.746 seconds
[ Info: i: 0740, t: 1.900 hours, Δt: 6.806 seconds, umax = (9.9e-02, 6.0e-02, 4.6e-02) ms⁻¹, wall time: 24.009 seconds
[ Info: i: 0760, t: 1.936 hours, Δt: 6.523 seconds, umax = (9.7e-02, 5.8e-02, 4.3e-02) ms⁻¹, wall time: 24.305 seconds
[ Info: i: 0780, t: 1.973 hours, Δt: 6.451 seconds, umax = (1.1e-01, 6.3e-02, 4.7e-02) ms⁻¹, wall time: 24.564 seconds
[ Info: i: 0800, t: 2.009 hours, Δt: 6.920 seconds, umax = (1.0e-01, 6.0e-02, 4.4e-02) ms⁻¹, wall time: 24.896 seconds
[ Info: i: 0820, t: 2.047 hours, Δt: 6.326 seconds, umax = (1.1e-01, 6.7e-02, 4.8e-02) ms⁻¹, wall time: 25.120 seconds
[ Info: i: 0840, t: 2.083 hours, Δt: 6.776 seconds, umax = (1.1e-01, 6.5e-02, 4.6e-02) ms⁻¹, wall time: 25.384 seconds
[ Info: i: 0860, t: 2.118 hours, Δt: 6.519 seconds, umax = (1.0e-01, 6.4e-02, 4.4e-02) ms⁻¹, wall time: 25.675 seconds
[ Info: i: 0880, t: 2.155 hours, Δt: 6.976 seconds, umax = (1.0e-01, 6.0e-02, 4.5e-02) ms⁻¹, wall time: 25.952 seconds
[ Info: i: 0900, t: 2.194 hours, Δt: 6.707 seconds, umax = (1.1e-01, 6.2e-02, 4.6e-02) ms⁻¹, wall time: 26.286 seconds
[ Info: i: 0920, t: 2.230 hours, Δt: 6.503 seconds, umax = (1.0e-01, 6.8e-02, 4.8e-02) ms⁻¹, wall time: 26.546 seconds
[ Info: i: 0940, t: 2.264 hours, Δt: 6.505 seconds, umax = (1.0e-01, 6.1e-02, 4.2e-02) ms⁻¹, wall time: 26.922 seconds
[ Info: i: 0960, t: 2.300 hours, Δt: 6.427 seconds, umax = (1.0e-01, 6.6e-02, 4.3e-02) ms⁻¹, wall time: 27.177 seconds
[ Info: i: 0980, t: 2.335 hours, Δt: 6.331 seconds, umax = (1.2e-01, 6.1e-02, 4.3e-02) ms⁻¹, wall time: 27.628 seconds
[ Info: i: 1000, t: 2.370 hours, Δt: 6.251 seconds, umax = (1.1e-01, 6.5e-02, 4.3e-02) ms⁻¹, wall time: 27.823 seconds
[ Info: i: 1020, t: 2.405 hours, Δt: 6.129 seconds, umax = (1.1e-01, 7.0e-02, 4.5e-02) ms⁻¹, wall time: 28.086 seconds
[ Info: i: 1040, t: 2.440 hours, Δt: 6.361 seconds, umax = (1.0e-01, 7.4e-02, 4.3e-02) ms⁻¹, wall time: 28.398 seconds
[ Info: i: 1060, t: 2.476 hours, Δt: 6.646 seconds, umax = (1.0e-01, 6.4e-02, 4.7e-02) ms⁻¹, wall time: 28.659 seconds
[ Info: i: 1080, t: 2.511 hours, Δt: 6.524 seconds, umax = (1.1e-01, 7.1e-02, 4.7e-02) ms⁻¹, wall time: 28.972 seconds
[ Info: i: 1100, t: 2.545 hours, Δt: 5.673 seconds, umax = (1.1e-01, 7.2e-02, 4.7e-02) ms⁻¹, wall time: 29.207 seconds
[ Info: i: 1120, t: 2.576 hours, Δt: 5.317 seconds, umax = (1.2e-01, 6.9e-02, 4.7e-02) ms⁻¹, wall time: 29.472 seconds
[ Info: i: 1140, t: 2.605 hours, Δt: 6.248 seconds, umax = (1.2e-01, 6.4e-02, 4.6e-02) ms⁻¹, wall time: 29.781 seconds
[ Info: i: 1160, t: 2.641 hours, Δt: 6.212 seconds, umax = (1.2e-01, 6.5e-02, 4.7e-02) ms⁻¹, wall time: 30.045 seconds
[ Info: i: 1180, t: 2.675 hours, Δt: 6.337 seconds, umax = (1.1e-01, 6.7e-02, 4.8e-02) ms⁻¹, wall time: 30.366 seconds
[ Info: i: 1200, t: 2.710 hours, Δt: 6.355 seconds, umax = (1.1e-01, 7.5e-02, 5.0e-02) ms⁻¹, wall time: 30.593 seconds
[ Info: i: 1220, t: 2.745 hours, Δt: 6.149 seconds, umax = (1.1e-01, 7.0e-02, 4.3e-02) ms⁻¹, wall time: 30.861 seconds
[ Info: i: 1240, t: 2.779 hours, Δt: 5.933 seconds, umax = (1.1e-01, 7.5e-02, 4.5e-02) ms⁻¹, wall time: 31.168 seconds
[ Info: i: 1260, t: 2.813 hours, Δt: 5.836 seconds, umax = (1.1e-01, 6.8e-02, 4.6e-02) ms⁻¹, wall time: 31.432 seconds
[ Info: i: 1280, t: 2.845 hours, Δt: 6.275 seconds, umax = (1.1e-01, 6.9e-02, 4.8e-02) ms⁻¹, wall time: 31.727 seconds
[ Info: i: 1300, t: 2.881 hours, Δt: 6.020 seconds, umax = (1.0e-01, 7.0e-02, 5.0e-02) ms⁻¹, wall time: 31.975 seconds
[ Info: i: 1320, t: 2.915 hours, Δt: 5.797 seconds, umax = (1.2e-01, 7.0e-02, 5.8e-02) ms⁻¹, wall time: 32.242 seconds
[ Info: i: 1340, t: 2.947 hours, Δt: 6.053 seconds, umax = (1.1e-01, 7.2e-02, 4.8e-02) ms⁻¹, wall time: 32.553 seconds
[ Info: i: 1360, t: 2.980 hours, Δt: 5.834 seconds, umax = (1.1e-01, 7.5e-02, 4.8e-02) ms⁻¹, wall time: 32.815 seconds
[ Info: i: 1380, t: 3.012 hours, Δt: 6.315 seconds, umax = (1.1e-01, 7.0e-02, 4.6e-02) ms⁻¹, wall time: 33.102 seconds
[ Info: i: 1400, t: 3.046 hours, Δt: 6.344 seconds, umax = (1.0e-01, 7.5e-02, 4.7e-02) ms⁻¹, wall time: 33.347 seconds
[ Info: i: 1420, t: 3.082 hours, Δt: 6.107 seconds, umax = (1.1e-01, 7.1e-02, 4.4e-02) ms⁻¹, wall time: 33.610 seconds
[ Info: i: 1440, t: 3.115 hours, Δt: 5.896 seconds, umax = (1.1e-01, 8.1e-02, 4.5e-02) ms⁻¹, wall time: 33.904 seconds
[ Info: i: 1460, t: 3.147 hours, Δt: 5.930 seconds, umax = (1.1e-01, 6.9e-02, 4.7e-02) ms⁻¹, wall time: 34.162 seconds
[ Info: i: 1480, t: 3.181 hours, Δt: 6.383 seconds, umax = (1.1e-01, 6.8e-02, 4.7e-02) ms⁻¹, wall time: 34.459 seconds
[ Info: i: 1500, t: 3.216 hours, Δt: 6.221 seconds, umax = (1.1e-01, 7.6e-02, 5.0e-02) ms⁻¹, wall time: 34.708 seconds
[ Info: i: 1520, t: 3.250 hours, Δt: 6.377 seconds, umax = (1.1e-01, 7.1e-02, 4.9e-02) ms⁻¹, wall time: 34.972 seconds
[ Info: i: 1540, t: 3.283 hours, Δt: 6.197 seconds, umax = (1.2e-01, 7.5e-02, 5.0e-02) ms⁻¹, wall time: 35.269 seconds
[ Info: i: 1560, t: 3.318 hours, Δt: 6.162 seconds, umax = (1.1e-01, 7.0e-02, 4.6e-02) ms⁻¹, wall time: 35.529 seconds
[ Info: i: 1580, t: 3.350 hours, Δt: 5.624 seconds, umax = (1.3e-01, 7.6e-02, 5.0e-02) ms⁻¹, wall time: 35.828 seconds
[ Info: i: 1600, t: 3.383 hours, Δt: 6.203 seconds, umax = (1.1e-01, 7.5e-02, 4.7e-02) ms⁻¹, wall time: 36.081 seconds
[ Info: i: 1620, t: 3.416 hours, Δt: 5.613 seconds, umax = (1.1e-01, 7.3e-02, 4.6e-02) ms⁻¹, wall time: 36.344 seconds
[ Info: i: 1640, t: 3.448 hours, Δt: 6.160 seconds, umax = (1.0e-01, 8.3e-02, 4.6e-02) ms⁻¹, wall time: 36.651 seconds
[ Info: i: 1660, t: 3.481 hours, Δt: 5.797 seconds, umax = (1.1e-01, 8.9e-02, 4.7e-02) ms⁻¹, wall time: 36.911 seconds
[ Info: i: 1680, t: 3.513 hours, Δt: 5.718 seconds, umax = (1.1e-01, 9.3e-02, 4.6e-02) ms⁻¹, wall time: 37.223 seconds
[ Info: i: 1700, t: 3.543 hours, Δt: 5.523 seconds, umax = (1.1e-01, 9.3e-02, 4.6e-02) ms⁻¹, wall time: 37.473 seconds
[ Info: i: 1720, t: 3.574 hours, Δt: 6.224 seconds, umax = (1.1e-01, 8.4e-02, 4.7e-02) ms⁻¹, wall time: 37.739 seconds
[ Info: i: 1740, t: 3.606 hours, Δt: 5.945 seconds, umax = (1.2e-01, 7.3e-02, 5.2e-02) ms⁻¹, wall time: 38.055 seconds
[ Info: i: 1760, t: 3.640 hours, Δt: 6.031 seconds, umax = (1.1e-01, 7.9e-02, 4.5e-02) ms⁻¹, wall time: 38.309 seconds
[ Info: i: 1780, t: 3.671 hours, Δt: 5.698 seconds, umax = (1.2e-01, 8.4e-02, 5.0e-02) ms⁻¹, wall time: 38.652 seconds
[ Info: i: 1800, t: 3.703 hours, Δt: 5.860 seconds, umax = (1.1e-01, 9.0e-02, 4.7e-02) ms⁻¹, wall time: 38.856 seconds
[ Info: i: 1820, t: 3.736 hours, Δt: 6.153 seconds, umax = (1.1e-01, 8.4e-02, 5.0e-02) ms⁻¹, wall time: 39.120 seconds
[ Info: i: 1840, t: 3.768 hours, Δt: 5.361 seconds, umax = (1.1e-01, 8.4e-02, 5.1e-02) ms⁻¹, wall time: 39.427 seconds
[ Info: i: 1860, t: 3.798 hours, Δt: 5.955 seconds, umax = (1.1e-01, 9.3e-02, 4.9e-02) ms⁻¹, wall time: 39.679 seconds
[ Info: i: 1880, t: 3.830 hours, Δt: 5.617 seconds, umax = (1.1e-01, 9.0e-02, 4.7e-02) ms⁻¹, wall time: 39.948 seconds
[ Info: i: 1900, t: 3.860 hours, Δt: 5.547 seconds, umax = (1.1e-01, 8.2e-02, 4.8e-02) ms⁻¹, wall time: 40.230 seconds
[ Info: i: 1920, t: 3.892 hours, Δt: 6.215 seconds, umax = (1.1e-01, 8.6e-02, 5.4e-02) ms⁻¹, wall time: 40.492 seconds
[ Info: i: 1940, t: 3.925 hours, Δt: 5.946 seconds, umax = (1.3e-01, 8.2e-02, 4.7e-02) ms⁻¹, wall time: 40.820 seconds
[ Info: i: 1960, t: 3.957 hours, Δt: 6.025 seconds, umax = (1.1e-01, 8.7e-02, 4.9e-02) ms⁻¹, wall time: 41.045 seconds
[ Info: i: 1980, t: 3.990 hours, Δt: 5.745 seconds, umax = (1.2e-01, 7.6e-02, 4.8e-02) ms⁻¹, wall time: 41.307 seconds
[ Info: Simulation is stopping after running for 41.397 seconds.
[ 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 and , and a horizontal slice of 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:/var/lib/buildkite-agent/.julia/juliaup/julia-1.12.4+0.x64.linux.gnu/local/share/julia:/var/lib/buildkite-agent/.julia/juliaup/julia-1.12.4+0.x64.linux.gnu/share/julia
  JULIA_PROJECT = /var/lib/buildkite-agent/Oceananigans.jl-31219/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-31219/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-31219/docs/Project.toml`
  [79e6a3ab] Adapt v4.5.2
  [052768ef] CUDA v6.1.0
  [13f3f980] CairoMakie v0.15.10
  [e30172f5] Documenter v1.17.0
  [daee34ce] DocumenterCitations v1.4.1
  [4710194d] DocumenterVitepress v0.3.3
  [033835bb] JLD2 v0.6.4
  [63c18a36] KernelAbstractions v0.9.41
  [98b081ad] Literate v2.21.0
  [da04e1cc] MPI v0.20.26
  [85f8d34a] NCDatasets v0.14.15
  [9e8cae18] Oceananigans v0.107.6 `..`
  [f27b6e38] Polynomials v4.1.1
  [6038ab10] Rotations v1.7.1
  [d496a93d] SeawaterPolynomials v0.3.10
  [09ab397b] StructArrays v0.7.3
  [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.