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: [: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 = (1.8e-03, 9.5e-04, 1.5e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (9.540 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.886 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: 13.444 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: 13.839 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: 14.325 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: 14.716 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: 15.164 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: 15.616 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: 16.127 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: 16.470 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: 16.897 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: 17.435 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: 17.721 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: 18.176 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: 18.585 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: 19.008 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: 19.421 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: 19.857 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: 20.417 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: 20.715 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: 21.201 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: 21.576 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: 22.030 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: 22.443 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: 22.884 seconds
[ Info: i: 0480, t: 1.392 hours, Δt: 7.478 seconds, umax = (9.3e-02, 5.2e-02, 4.1e-02) ms⁻¹, wall time: 23.302 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: 23.808 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: 24.227 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: 24.771 seconds
[ Info: i: 0560, t: 1.555 hours, Δt: 7.351 seconds, umax = (9.2e-02, 5.5e-02, 4.1e-02) ms⁻¹, wall time: 25.186 seconds
[ Info: i: 0580, t: 1.596 hours, Δt: 7.264 seconds, umax = (9.8e-02, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 25.661 seconds
[ Info: i: 0600, t: 1.636 hours, Δt: 7.464 seconds, umax = (9.6e-02, 5.5e-02, 4.1e-02) ms⁻¹, wall time: 26.060 seconds
[ Info: i: 0620, t: 1.675 hours, Δt: 6.748 seconds, umax = (1.0e-01, 5.6e-02, 4.2e-02) ms⁻¹, wall time: 26.580 seconds
[ Info: i: 0640, t: 1.712 hours, Δt: 7.014 seconds, umax = (9.8e-02, 5.3e-02, 4.2e-02) ms⁻¹, wall time: 26.945 seconds
[ Info: i: 0660, t: 1.750 hours, Δt: 6.607 seconds, umax = (9.8e-02, 5.4e-02, 4.4e-02) ms⁻¹, wall time: 27.384 seconds
[ Info: i: 0680, t: 1.786 hours, Δt: 6.731 seconds, umax = (1.0e-01, 6.0e-02, 4.3e-02) ms⁻¹, wall time: 27.841 seconds
[ Info: i: 0700, t: 1.825 hours, Δt: 6.965 seconds, umax = (9.9e-02, 5.6e-02, 4.1e-02) ms⁻¹, wall time: 28.284 seconds
[ Info: i: 0720, t: 1.863 hours, Δt: 6.912 seconds, umax = (1.0e-01, 5.8e-02, 4.2e-02) ms⁻¹, wall time: 28.749 seconds
[ Info: i: 0740, t: 1.900 hours, Δt: 6.811 seconds, umax = (9.9e-02, 6.0e-02, 4.6e-02) ms⁻¹, wall time: 29.194 seconds
[ Info: i: 0760, t: 1.936 hours, Δt: 6.587 seconds, umax = (9.7e-02, 5.9e-02, 4.3e-02) ms⁻¹, wall time: 29.648 seconds
[ Info: i: 0780, t: 1.974 hours, Δt: 6.535 seconds, umax = (1.1e-01, 6.4e-02, 4.7e-02) ms⁻¹, wall time: 30.082 seconds
[ Info: i: 0800, t: 2.009 hours, Δt: 6.913 seconds, umax = (1.0e-01, 6.0e-02, 4.5e-02) ms⁻¹, wall time: 30.587 seconds
[ Info: i: 0820, t: 2.047 hours, Δt: 6.228 seconds, umax = (1.1e-01, 6.4e-02, 4.9e-02) ms⁻¹, wall time: 30.984 seconds
[ Info: i: 0840, t: 2.081 hours, Δt: 6.756 seconds, umax = (1.1e-01, 6.8e-02, 4.6e-02) ms⁻¹, wall time: 31.428 seconds
[ Info: i: 0860, t: 2.116 hours, Δt: 6.947 seconds, umax = (1.0e-01, 6.5e-02, 4.4e-02) ms⁻¹, wall time: 31.890 seconds
[ Info: i: 0880, t: 2.155 hours, Δt: 6.884 seconds, umax = (1.0e-01, 6.5e-02, 4.6e-02) ms⁻¹, wall time: 32.341 seconds
[ Info: i: 0900, t: 2.191 hours, Δt: 6.863 seconds, umax = (1.1e-01, 6.6e-02, 4.5e-02) ms⁻¹, wall time: 32.797 seconds
[ Info: i: 0920, t: 2.228 hours, Δt: 6.282 seconds, umax = (1.0e-01, 6.3e-02, 4.8e-02) ms⁻¹, wall time: 33.248 seconds
[ Info: i: 0940, t: 2.263 hours, Δt: 6.612 seconds, umax = (1.1e-01, 6.4e-02, 4.3e-02) ms⁻¹, wall time: 33.721 seconds
[ Info: i: 0960, t: 2.299 hours, Δt: 6.457 seconds, umax = (1.1e-01, 7.0e-02, 4.3e-02) ms⁻¹, wall time: 34.155 seconds
[ Info: i: 0980, t: 2.333 hours, Δt: 6.491 seconds, umax = (1.1e-01, 6.2e-02, 4.5e-02) ms⁻¹, wall time: 34.610 seconds
[ Info: i: 1000, t: 2.369 hours, Δt: 6.420 seconds, umax = (1.1e-01, 6.5e-02, 4.2e-02) ms⁻¹, wall time: 35.073 seconds
[ Info: i: 1020, t: 2.406 hours, Δt: 6.209 seconds, umax = (1.0e-01, 6.6e-02, 4.2e-02) ms⁻¹, wall time: 35.527 seconds
[ Info: i: 1040, t: 2.439 hours, Δt: 6.694 seconds, umax = (1.1e-01, 6.7e-02, 4.5e-02) ms⁻¹, wall time: 36.005 seconds
[ Info: i: 1060, t: 2.476 hours, Δt: 6.334 seconds, umax = (9.8e-02, 6.2e-02, 4.2e-02) ms⁻¹, wall time: 36.456 seconds
[ Info: i: 1080, t: 2.510 hours, Δt: 6.549 seconds, umax = (1.1e-01, 6.6e-02, 4.3e-02) ms⁻¹, wall time: 36.939 seconds
[ Info: i: 1100, t: 2.547 hours, Δt: 6.186 seconds, umax = (1.2e-01, 6.7e-02, 4.2e-02) ms⁻¹, wall time: 37.353 seconds
[ Info: i: 1120, t: 2.581 hours, Δt: 6.442 seconds, umax = (1.0e-01, 6.5e-02, 4.7e-02) ms⁻¹, wall time: 37.807 seconds
[ Info: i: 1140, t: 2.615 hours, Δt: 6.468 seconds, umax = (1.1e-01, 6.8e-02, 5.2e-02) ms⁻¹, wall time: 38.266 seconds
[ Info: i: 1160, t: 2.650 hours, Δt: 6.413 seconds, umax = (1.1e-01, 7.4e-02, 4.7e-02) ms⁻¹, wall time: 38.715 seconds
[ Info: i: 1180, t: 2.684 hours, Δt: 6.400 seconds, umax = (1.3e-01, 7.0e-02, 4.4e-02) ms⁻¹, wall time: 39.190 seconds
[ Info: i: 1200, t: 2.719 hours, Δt: 5.806 seconds, umax = (1.1e-01, 6.9e-02, 4.4e-02) ms⁻¹, wall time: 39.619 seconds
[ Info: i: 1220, t: 2.750 hours, Δt: 5.988 seconds, umax = (1.0e-01, 7.3e-02, 4.7e-02) ms⁻¹, wall time: 40.077 seconds
[ Info: i: 1240, t: 2.783 hours, Δt: 5.843 seconds, umax = (1.1e-01, 7.6e-02, 4.4e-02) ms⁻¹, wall time: 40.561 seconds
[ Info: i: 1260, t: 2.816 hours, Δt: 5.964 seconds, umax = (1.1e-01, 7.2e-02, 4.5e-02) ms⁻¹, wall time: 41.015 seconds
[ Info: i: 1280, t: 2.848 hours, Δt: 6.192 seconds, umax = (1.0e-01, 8.3e-02, 4.6e-02) ms⁻¹, wall time: 41.487 seconds
[ Info: i: 1300, t: 2.882 hours, Δt: 6.175 seconds, umax = (1.1e-01, 7.0e-02, 4.5e-02) ms⁻¹, wall time: 41.923 seconds
[ Info: i: 1320, t: 2.914 hours, Δt: 6.068 seconds, umax = (1.1e-01, 7.7e-02, 5.1e-02) ms⁻¹, wall time: 42.378 seconds
[ Info: i: 1340, t: 2.946 hours, Δt: 5.647 seconds, umax = (1.1e-01, 7.4e-02, 5.0e-02) ms⁻¹, wall time: 42.844 seconds
[ Info: i: 1360, t: 2.979 hours, Δt: 5.976 seconds, umax = (1.1e-01, 7.0e-02, 4.9e-02) ms⁻¹, wall time: 43.294 seconds
[ Info: i: 1380, t: 3.011 hours, Δt: 5.861 seconds, umax = (1.1e-01, 7.0e-02, 4.9e-02) ms⁻¹, wall time: 43.769 seconds
[ Info: i: 1400, t: 3.044 hours, Δt: 6.251 seconds, umax = (1.1e-01, 7.7e-02, 5.2e-02) ms⁻¹, wall time: 44.202 seconds
[ Info: i: 1420, t: 3.077 hours, Δt: 5.908 seconds, umax = (1.2e-01, 8.1e-02, 4.9e-02) ms⁻¹, wall time: 44.658 seconds
[ Info: i: 1440, t: 3.110 hours, Δt: 5.867 seconds, umax = (1.1e-01, 7.5e-02, 4.7e-02) ms⁻¹, wall time: 45.118 seconds
[ Info: i: 1460, t: 3.143 hours, Δt: 6.262 seconds, umax = (1.1e-01, 6.7e-02, 4.3e-02) ms⁻¹, wall time: 45.566 seconds
[ Info: i: 1480, t: 3.176 hours, Δt: 6.122 seconds, umax = (1.1e-01, 7.3e-02, 5.3e-02) ms⁻¹, wall time: 46.055 seconds
[ Info: i: 1500, t: 3.210 hours, Δt: 6.044 seconds, umax = (1.1e-01, 7.5e-02, 4.8e-02) ms⁻¹, wall time: 46.470 seconds
[ Info: i: 1520, t: 3.243 hours, Δt: 6.127 seconds, umax = (1.1e-01, 7.7e-02, 4.5e-02) ms⁻¹, wall time: 46.925 seconds
[ Info: i: 1540, t: 3.276 hours, Δt: 5.872 seconds, umax = (1.0e-01, 7.7e-02, 4.9e-02) ms⁻¹, wall time: 47.391 seconds
[ Info: i: 1560, t: 3.310 hours, Δt: 5.991 seconds, umax = (1.1e-01, 7.7e-02, 4.6e-02) ms⁻¹, wall time: 47.837 seconds
[ Info: i: 1580, t: 3.341 hours, Δt: 6.071 seconds, umax = (1.1e-01, 7.4e-02, 4.7e-02) ms⁻¹, wall time: 48.357 seconds
[ Info: i: 1600, t: 3.375 hours, Δt: 6.414 seconds, umax = (1.1e-01, 7.4e-02, 5.1e-02) ms⁻¹, wall time: 48.751 seconds
[ Info: i: 1620, t: 3.411 hours, Δt: 5.699 seconds, umax = (1.1e-01, 6.8e-02, 4.4e-02) ms⁻¹, wall time: 49.198 seconds
[ Info: i: 1640, t: 3.444 hours, Δt: 6.070 seconds, umax = (1.0e-01, 7.2e-02, 4.9e-02) ms⁻¹, wall time: 49.667 seconds
[ Info: i: 1660, t: 3.478 hours, Δt: 6.082 seconds, umax = (1.1e-01, 7.6e-02, 5.4e-02) ms⁻¹, wall time: 50.118 seconds
[ Info: i: 1680, t: 3.510 hours, Δt: 5.974 seconds, umax = (1.1e-01, 6.9e-02, 4.8e-02) ms⁻¹, wall time: 50.631 seconds
[ Info: i: 1700, t: 3.543 hours, Δt: 6.257 seconds, umax = (1.2e-01, 7.2e-02, 4.5e-02) ms⁻¹, wall time: 51.031 seconds
[ Info: i: 1720, t: 3.577 hours, Δt: 5.690 seconds, umax = (1.1e-01, 7.4e-02, 4.9e-02) ms⁻¹, wall time: 51.482 seconds
[ Info: i: 1740, t: 3.608 hours, Δt: 5.329 seconds, umax = (1.1e-01, 7.8e-02, 5.6e-02) ms⁻¹, wall time: 51.954 seconds
[ Info: i: 1760, t: 3.639 hours, Δt: 6.231 seconds, umax = (1.0e-01, 7.9e-02, 5.0e-02) ms⁻¹, wall time: 52.404 seconds
[ Info: i: 1780, t: 3.674 hours, Δt: 5.979 seconds, umax = (1.1e-01, 8.0e-02, 4.7e-02) ms⁻¹, wall time: 52.949 seconds
[ Info: i: 1800, t: 3.707 hours, Δt: 5.675 seconds, umax = (1.1e-01, 8.1e-02, 4.8e-02) ms⁻¹, wall time: 53.311 seconds
[ Info: i: 1820, t: 3.738 hours, Δt: 5.824 seconds, umax = (1.3e-01, 8.2e-02, 4.8e-02) ms⁻¹, wall time: 53.760 seconds
[ Info: i: 1840, t: 3.770 hours, Δt: 5.863 seconds, umax = (1.1e-01, 8.8e-02, 5.1e-02) ms⁻¹, wall time: 54.229 seconds
[ Info: i: 1860, t: 3.802 hours, Δt: 5.815 seconds, umax = (1.1e-01, 9.1e-02, 5.4e-02) ms⁻¹, wall time: 54.690 seconds
[ Info: i: 1880, t: 3.833 hours, Δt: 5.491 seconds, umax = (1.1e-01, 8.3e-02, 5.8e-02) ms⁻¹, wall time: 55.131 seconds
[ Info: i: 1900, t: 3.865 hours, Δt: 5.329 seconds, umax = (1.1e-01, 9.9e-02, 5.5e-02) ms⁻¹, wall time: 55.600 seconds
[ Info: i: 1920, t: 3.894 hours, Δt: 4.855 seconds, umax = (1.0e-01, 1.0e-01, 5.0e-02) ms⁻¹, wall time: 56.049 seconds
[ Info: i: 1940, t: 3.921 hours, Δt: 5.620 seconds, umax = (1.1e-01, 8.6e-02, 5.1e-02) ms⁻¹, wall time: 56.586 seconds
[ Info: i: 1960, t: 3.954 hours, Δt: 5.982 seconds, umax = (1.1e-01, 8.9e-02, 4.9e-02) ms⁻¹, wall time: 56.939 seconds
[ Info: i: 1980, t: 3.987 hours, Δt: 5.893 seconds, umax = (1.1e-01, 7.8e-02, 5.5e-02) ms⁻¹, wall time: 57.397 seconds
[ Info: Simulation is stopping after running for 57.603 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
  JULIA_PROJECT = /var/lib/buildkite-agent/Oceananigans.jl-30867/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-30867/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-30867/docs/Project.toml`
  [79e6a3ab] Adapt v4.5.2
⌃ [052768ef] CUDA v5.11.0
  [13f3f980] CairoMakie v0.15.9
  [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 v0.20.25
  [85f8d34a] NCDatasets v0.14.15
  [9e8cae18] Oceananigans v0.106.6 `..`
  [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
Info Packages marked with ⌃ have new versions available and may be upgradable.

---

This page was generated using Literate.jl.