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
using Zarr

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 in the paper by DocumenterCitations.CitationSiteNode("Wagner2021-cite-1")

,

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 DocumenterCitations.CitationSiteNode("Wagner2021-cite-2")

, 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  , DocumenterCitations.CitationSiteNode("Wagner2021-cite-3")

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)

DocumenterCitations.CitationSiteNode("Wagner2021-cite-4")

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

DocumenterCitations.CitationSiteNode("Wagner2021-cite-5")

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:
│   ├── momentum: WENO{5, Float64, Oceananigans.Utils.BackendOptimizedDivision}(order=9)
│   └── b: 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 DocumenterCitations.CitationSiteNode("Wagner2021-cite-6")

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. We write to a Zarr store — each output becomes a chunked array of shape (Nx, Ny, Nz, Nt) that grows along the time axis. The on-disk layout is friendly to chunked / parallel reads.

output_interval = 5minutes

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

simulation.output_writers[:fields] =
    ZarrWriter(model, fields_to_output,
               schedule = TimeInterval(output_interval),
               filename = "langmuir_turbulence_fields.zarr",
               overwrite_files = true)

ZarrWriter scheduled on TimeInterval(5 minutes):
├── filepath: /var/lib/buildkite-agent/Oceananigans.jl-33941/docs/src/literated/langmuir_turbulence_fields.zarr
├── store: DirectoryStore
├── 4 outputs: (u, v, w, b)
├── array_type: Array{Float32}
├── chunks: auto
├── compressor: none
└── file_splitting: NoFileSplitting

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] =
    ZarrWriter(model, (; U, V, B, wu, wv),
               schedule = AveragedTimeInterval(output_interval, window=2minutes),
               filename = "langmuir_turbulence_averages.zarr",
               overwrite_files = true)

ZarrWriter scheduled on TimeInterval(5 minutes):
├── filepath: /var/lib/buildkite-agent/Oceananigans.jl-33941/docs/src/literated/langmuir_turbulence_averages.zarr
├── store: DirectoryStore
├── 5 outputs: (U, V, B, wu, wv) averaged on AveragedTimeInterval(window=2 minutes, stride=1, interval=5 minutes)
├── array_type: Array{Float32}
├── chunks: auto
├── compressor: none
└── file_splitting: NoFileSplitting

Running the simulation

This part is easy,

run!(simulation)

[ Info: Initializing simulation...
[ Info: i: 0000, t: 0 seconds, Δt: 49.500 seconds, umax = (1.7e-03, 8.8e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (37.998 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (1.898 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 18.598 seconds, umax = (3.4e-02, 1.2e-02, 2.2e-02) ms⁻¹, wall time: 40.510 seconds
[ Info: i: 0040, t: 16.850 minutes, Δt: 12.788 seconds, umax = (5.2e-02, 2.0e-02, 2.3e-02) ms⁻¹, wall time: 40.867 seconds
[ Info: i: 0060, t: 20.768 minutes, Δt: 11.141 seconds, umax = (6.2e-02, 2.8e-02, 3.1e-02) ms⁻¹, wall time: 41.160 seconds
[ Info: i: 0080, t: 24.409 minutes, Δt: 10.284 seconds, umax = (7.0e-02, 3.5e-02, 3.1e-02) ms⁻¹, wall time: 41.373 seconds
[ Info: i: 0100, t: 27.798 minutes, Δt: 11.231 seconds, umax = (6.3e-02, 3.5e-02, 3.0e-02) ms⁻¹, wall time: 41.645 seconds
[ Info: i: 0120, t: 31.519 minutes, Δt: 10.535 seconds, umax = (6.2e-02, 3.4e-02, 2.8e-02) ms⁻¹, wall time: 41.905 seconds
[ Info: i: 0140, t: 35 minutes, Δt: 10.518 seconds, umax = (6.6e-02, 3.1e-02, 2.9e-02) ms⁻¹, wall time: 42.161 seconds
[ Info: i: 0160, t: 38.455 minutes, Δt: 10.082 seconds, umax = (6.9e-02, 3.4e-02, 3.5e-02) ms⁻¹, wall time: 42.420 seconds
[ Info: i: 0180, t: 41.559 minutes, Δt: 9.830 seconds, umax = (7.7e-02, 3.5e-02, 3.0e-02) ms⁻¹, wall time: 42.679 seconds
[ Info: i: 0200, t: 44.790 minutes, Δt: 9.106 seconds, umax = (7.2e-02, 3.6e-02, 3.4e-02) ms⁻¹, wall time: 42.928 seconds
[ Info: i: 0220, t: 47.726 minutes, Δt: 9.162 seconds, umax = (7.6e-02, 4.7e-02, 3.4e-02) ms⁻¹, wall time: 43.197 seconds
[ Info: i: 0240, t: 50.591 minutes, Δt: 9.609 seconds, umax = (8.1e-02, 4.2e-02, 3.6e-02) ms⁻¹, wall time: 43.495 seconds
[ Info: i: 0260, t: 53.722 minutes, Δt: 8.498 seconds, umax = (7.8e-02, 4.5e-02, 3.5e-02) ms⁻¹, wall time: 43.704 seconds
[ Info: i: 0280, t: 56.511 minutes, Δt: 8.661 seconds, umax = (8.4e-02, 4.5e-02, 4.2e-02) ms⁻¹, wall time: 43.970 seconds
[ Info: i: 0300, t: 59.347 minutes, Δt: 8.303 seconds, umax = (7.8e-02, 4.9e-02, 4.1e-02) ms⁻¹, wall time: 44.214 seconds
[ Info: i: 0320, t: 1.035 hours, Δt: 8.188 seconds, umax = (8.1e-02, 5.0e-02, 3.9e-02) ms⁻¹, wall time: 44.483 seconds
[ Info: i: 0340, t: 1.080 hours, Δt: 7.920 seconds, umax = (8.9e-02, 4.8e-02, 4.1e-02) ms⁻¹, wall time: 44.734 seconds
[ Info: i: 0360, t: 1.122 hours, Δt: 8.329 seconds, umax = (8.4e-02, 5.2e-02, 4.0e-02) ms⁻¹, wall time: 45.001 seconds
[ Info: i: 0380, t: 1.167 hours, Δt: 7.911 seconds, umax = (8.7e-02, 5.2e-02, 4.1e-02) ms⁻¹, wall time: 45.244 seconds
[ Info: i: 0400, t: 1.210 hours, Δt: 8.001 seconds, umax = (8.7e-02, 4.9e-02, 3.7e-02) ms⁻¹, wall time: 45.503 seconds
[ Info: i: 0420, t: 1.254 hours, Δt: 8.090 seconds, umax = (9.2e-02, 4.9e-02, 3.8e-02) ms⁻¹, wall time: 45.822 seconds
[ Info: i: 0440, t: 1.298 hours, Δt: 7.725 seconds, umax = (9.0e-02, 5.1e-02, 4.2e-02) ms⁻¹, wall time: 46.006 seconds
[ Info: i: 0460, t: 1.340 hours, Δt: 7.677 seconds, umax = (8.8e-02, 5.1e-02, 4.1e-02) ms⁻¹, wall time: 46.322 seconds
[ Info: i: 0480, t: 1.383 hours, Δt: 8.000 seconds, umax = (9.2e-02, 4.8e-02, 4.4e-02) ms⁻¹, wall time: 46.517 seconds
[ Info: i: 0500, t: 1.425 hours, Δt: 7.632 seconds, umax = (9.5e-02, 5.2e-02, 4.4e-02) ms⁻¹, wall time: 46.812 seconds
[ Info: i: 0520, t: 1.467 hours, Δt: 7.414 seconds, umax = (9.1e-02, 5.0e-02, 3.9e-02) ms⁻¹, wall time: 47.020 seconds
[ Info: i: 0540, t: 1.506 hours, Δt: 6.738 seconds, umax = (9.7e-02, 5.7e-02, 4.1e-02) ms⁻¹, wall time: 47.376 seconds
[ Info: i: 0560, t: 1.545 hours, Δt: 7.314 seconds, umax = (8.9e-02, 5.5e-02, 4.3e-02) ms⁻¹, wall time: 47.603 seconds
[ Info: i: 0580, t: 1.583 hours, Δt: 7.318 seconds, umax = (9.4e-02, 5.3e-02, 4.0e-02) ms⁻¹, wall time: 47.879 seconds
[ Info: i: 0600, t: 1.624 hours, Δt: 7.037 seconds, umax = (9.8e-02, 5.5e-02, 4.1e-02) ms⁻¹, wall time: 48.145 seconds
[ Info: i: 0620, t: 1.663 hours, Δt: 7.113 seconds, umax = (9.7e-02, 5.7e-02, 4.2e-02) ms⁻¹, wall time: 48.390 seconds
[ Info: i: 0640, t: 1.700 hours, Δt: 7.165 seconds, umax = (9.8e-02, 5.6e-02, 4.1e-02) ms⁻¹, wall time: 48.652 seconds
[ Info: i: 0660, t: 1.740 hours, Δt: 7.612 seconds, umax = (9.8e-02, 5.4e-02, 4.2e-02) ms⁻¹, wall time: 48.898 seconds
[ Info: i: 0680, t: 1.781 hours, Δt: 6.963 seconds, umax = (9.9e-02, 5.4e-02, 4.9e-02) ms⁻¹, wall time: 49.170 seconds
[ Info: i: 0700, t: 1.820 hours, Δt: 7.004 seconds, umax = (9.6e-02, 5.6e-02, 4.8e-02) ms⁻¹, wall time: 49.413 seconds
[ Info: i: 0720, t: 1.859 hours, Δt: 6.745 seconds, umax = (9.9e-02, 6.6e-02, 4.8e-02) ms⁻¹, wall time: 49.673 seconds
[ Info: i: 0740, t: 1.896 hours, Δt: 6.843 seconds, umax = (9.6e-02, 6.2e-02, 4.8e-02) ms⁻¹, wall time: 49.918 seconds
[ Info: i: 0760, t: 1.932 hours, Δt: 6.800 seconds, umax = (9.7e-02, 5.6e-02, 4.4e-02) ms⁻¹, wall time: 50.178 seconds
[ Info: i: 0780, t: 1.970 hours, Δt: 6.738 seconds, umax = (9.6e-02, 5.9e-02, 4.4e-02) ms⁻¹, wall time: 50.419 seconds
[ Info: i: 0800, t: 2.006 hours, Δt: 6.108 seconds, umax = (1.1e-01, 6.7e-02, 4.6e-02) ms⁻¹, wall time: 50.724 seconds
[ Info: i: 0820, t: 2.040 hours, Δt: 6.603 seconds, umax = (1.1e-01, 6.8e-02, 4.2e-02) ms⁻¹, wall time: 50.919 seconds
[ Info: i: 0840, t: 2.076 hours, Δt: 6.670 seconds, umax = (1.0e-01, 6.2e-02, 4.1e-02) ms⁻¹, wall time: 51.167 seconds
[ Info: i: 0860, t: 2.113 hours, Δt: 6.656 seconds, umax = (1.0e-01, 6.4e-02, 4.1e-02) ms⁻¹, wall time: 51.418 seconds
[ Info: i: 0880, t: 2.149 hours, Δt: 5.828 seconds, umax = (1.0e-01, 6.7e-02, 4.2e-02) ms⁻¹, wall time: 51.661 seconds
[ Info: i: 0900, t: 2.183 hours, Δt: 6.554 seconds, umax = (1.0e-01, 6.0e-02, 4.2e-02) ms⁻¹, wall time: 51.917 seconds
[ Info: i: 0920, t: 2.218 hours, Δt: 6.615 seconds, umax = (1.0e-01, 6.7e-02, 4.5e-02) ms⁻¹, wall time: 52.153 seconds
[ Info: i: 0940, t: 2.253 hours, Δt: 6.269 seconds, umax = (1.0e-01, 6.7e-02, 4.4e-02) ms⁻¹, wall time: 52.467 seconds
[ Info: i: 0960, t: 2.290 hours, Δt: 6.547 seconds, umax = (1.1e-01, 6.3e-02, 4.9e-02) ms⁻¹, wall time: 52.649 seconds
[ Info: i: 0980, t: 2.325 hours, Δt: 6.538 seconds, umax = (1.2e-01, 6.5e-02, 4.6e-02) ms⁻¹, wall time: 52.895 seconds
[ Info: i: 1000, t: 2.361 hours, Δt: 6.385 seconds, umax = (1.0e-01, 6.7e-02, 4.3e-02) ms⁻¹, wall time: 53.159 seconds
[ Info: i: 1020, t: 2.396 hours, Δt: 5.843 seconds, umax = (1.0e-01, 7.2e-02, 4.2e-02) ms⁻¹, wall time: 53.402 seconds
[ Info: i: 1040, t: 2.429 hours, Δt: 6.330 seconds, umax = (1.0e-01, 6.5e-02, 4.7e-02) ms⁻¹, wall time: 53.666 seconds
[ Info: i: 1060, t: 2.464 hours, Δt: 6.290 seconds, umax = (1.0e-01, 6.5e-02, 4.5e-02) ms⁻¹, wall time: 53.905 seconds
[ Info: i: 1080, t: 2.499 hours, Δt: 6.313 seconds, umax = (1.0e-01, 6.8e-02, 4.4e-02) ms⁻¹, wall time: 54.155 seconds
[ Info: i: 1100, t: 2.534 hours, Δt: 6.233 seconds, umax = (1.0e-01, 7.2e-02, 5.1e-02) ms⁻¹, wall time: 54.420 seconds
[ Info: i: 1120, t: 2.569 hours, Δt: 6.413 seconds, umax = (1.1e-01, 6.6e-02, 4.8e-02) ms⁻¹, wall time: 54.664 seconds
[ Info: i: 1140, t: 2.603 hours, Δt: 6.186 seconds, umax = (1.1e-01, 6.9e-02, 4.7e-02) ms⁻¹, wall time: 54.932 seconds
[ Info: i: 1160, t: 2.636 hours, Δt: 5.778 seconds, umax = (1.1e-01, 7.4e-02, 4.7e-02) ms⁻¹, wall time: 55.172 seconds
[ Info: i: 1180, t: 2.668 hours, Δt: 6.009 seconds, umax = (1.2e-01, 7.3e-02, 4.9e-02) ms⁻¹, wall time: 55.503 seconds
[ Info: i: 1200, t: 2.703 hours, Δt: 5.994 seconds, umax = (1.1e-01, 7.3e-02, 5.0e-02) ms⁻¹, wall time: 55.675 seconds
[ Info: i: 1220, t: 2.736 hours, Δt: 6.216 seconds, umax = (1.2e-01, 7.2e-02, 4.7e-02) ms⁻¹, wall time: 55.926 seconds
[ Info: i: 1240, t: 2.768 hours, Δt: 6.526 seconds, umax = (1.1e-01, 7.1e-02, 4.6e-02) ms⁻¹, wall time: 56.191 seconds
[ Info: i: 1260, t: 2.804 hours, Δt: 6.498 seconds, umax = (1.1e-01, 6.8e-02, 4.5e-02) ms⁻¹, wall time: 56.432 seconds
[ Info: i: 1280, t: 2.838 hours, Δt: 6.262 seconds, umax = (1.1e-01, 7.5e-02, 4.9e-02) ms⁻¹, wall time: 56.735 seconds
[ Info: i: 1300, t: 2.872 hours, Δt: 6.064 seconds, umax = (1.0e-01, 7.3e-02, 4.8e-02) ms⁻¹, wall time: 56.931 seconds
[ Info: i: 1320, t: 2.906 hours, Δt: 5.983 seconds, umax = (1.1e-01, 7.5e-02, 4.8e-02) ms⁻¹, wall time: 57.178 seconds
[ Info: i: 1340, t: 2.938 hours, Δt: 5.893 seconds, umax = (1.1e-01, 7.7e-02, 4.8e-02) ms⁻¹, wall time: 57.430 seconds
[ Info: i: 1360, t: 2.971 hours, Δt: 5.956 seconds, umax = (1.1e-01, 7.8e-02, 4.6e-02) ms⁻¹, wall time: 57.672 seconds
[ Info: i: 1380, t: 3.002 hours, Δt: 6.063 seconds, umax = (1.1e-01, 7.3e-02, 4.6e-02) ms⁻¹, wall time: 58.006 seconds
[ Info: i: 1400, t: 3.035 hours, Δt: 6.036 seconds, umax = (1.1e-01, 7.2e-02, 4.8e-02) ms⁻¹, wall time: 58.184 seconds
[ Info: i: 1420, t: 3.068 hours, Δt: 6.078 seconds, umax = (1.1e-01, 7.1e-02, 4.5e-02) ms⁻¹, wall time: 58.436 seconds
[ Info: i: 1440, t: 3.100 hours, Δt: 5.861 seconds, umax = (1.1e-01, 7.3e-02, 5.1e-02) ms⁻¹, wall time: 58.694 seconds
[ Info: i: 1460, t: 3.133 hours, Δt: 6.101 seconds, umax = (1.1e-01, 7.0e-02, 5.8e-02) ms⁻¹, wall time: 58.930 seconds
[ Info: i: 1480, t: 3.167 hours, Δt: 6.293 seconds, umax = (1.1e-01, 6.9e-02, 4.9e-02) ms⁻¹, wall time: 59.186 seconds
[ Info: i: 1500, t: 3.201 hours, Δt: 5.680 seconds, umax = (1.0e-01, 7.4e-02, 4.4e-02) ms⁻¹, wall time: 59.456 seconds
[ Info: i: 1520, t: 3.232 hours, Δt: 5.464 seconds, umax = (1.1e-01, 7.5e-02, 4.7e-02) ms⁻¹, wall time: 59.708 seconds
[ Info: i: 1540, t: 3.262 hours, Δt: 5.957 seconds, umax = (1.1e-01, 7.7e-02, 4.7e-02) ms⁻¹, wall time: 59.969 seconds
[ Info: i: 1560, t: 3.296 hours, Δt: 6.205 seconds, umax = (1.1e-01, 7.0e-02, 4.4e-02) ms⁻¹, wall time: 1.003 minutes
[ Info: i: 1580, t: 3.331 hours, Δt: 6.068 seconds, umax = (1.2e-01, 7.1e-02, 5.3e-02) ms⁻¹, wall time: 1.008 minutes
[ Info: i: 1600, t: 3.364 hours, Δt: 6.197 seconds, umax = (1.0e-01, 7.3e-02, 5.3e-02) ms⁻¹, wall time: 1.012 minutes
[ Info: i: 1620, t: 3.397 hours, Δt: 5.501 seconds, umax = (1.1e-01, 7.7e-02, 5.5e-02) ms⁻¹, wall time: 1.016 minutes
[ Info: i: 1640, t: 3.427 hours, Δt: 5.856 seconds, umax = (1.1e-01, 7.4e-02, 5.2e-02) ms⁻¹, wall time: 1.020 minutes
[ Info: i: 1660, t: 3.459 hours, Δt: 5.605 seconds, umax = (1.2e-01, 8.2e-02, 4.9e-02) ms⁻¹, wall time: 1.024 minutes
[ Info: i: 1680, t: 3.491 hours, Δt: 5.969 seconds, umax = (1.1e-01, 7.6e-02, 5.0e-02) ms⁻¹, wall time: 1.029 minutes
[ Info: i: 1700, t: 3.523 hours, Δt: 5.890 seconds, umax = (1.1e-01, 7.7e-02, 5.2e-02) ms⁻¹, wall time: 1.033 minutes
[ Info: i: 1720, t: 3.556 hours, Δt: 5.805 seconds, umax = (1.1e-01, 7.6e-02, 5.0e-02) ms⁻¹, wall time: 1.037 minutes
[ Info: i: 1740, t: 3.588 hours, Δt: 5.961 seconds, umax = (1.1e-01, 7.7e-02, 5.6e-02) ms⁻¹, wall time: 1.042 minutes
[ Info: i: 1760, t: 3.621 hours, Δt: 5.959 seconds, umax = (1.1e-01, 7.7e-02, 5.3e-02) ms⁻¹, wall time: 1.046 minutes
[ Info: i: 1780, t: 3.654 hours, Δt: 6.054 seconds, umax = (1.1e-01, 8.1e-02, 4.9e-02) ms⁻¹, wall time: 1.050 minutes
[ Info: i: 1800, t: 3.687 hours, Δt: 6.184 seconds, umax = (1.1e-01, 8.1e-02, 4.6e-02) ms⁻¹, wall time: 1.054 minutes
[ Info: i: 1820, t: 3.721 hours, Δt: 6.019 seconds, umax = (1.1e-01, 7.8e-02, 4.6e-02) ms⁻¹, wall time: 1.058 minutes
[ Info: i: 1840, t: 3.753 hours, Δt: 5.551 seconds, umax = (1.1e-01, 7.7e-02, 4.6e-02) ms⁻¹, wall time: 1.064 minutes
[ Info: i: 1860, t: 3.784 hours, Δt: 5.776 seconds, umax = (1.1e-01, 7.6e-02, 4.6e-02) ms⁻¹, wall time: 1.067 minutes
[ Info: i: 1880, t: 3.816 hours, Δt: 5.459 seconds, umax = (1.2e-01, 9.4e-02, 4.8e-02) ms⁻¹, wall time: 1.071 minutes
[ Info: i: 1900, t: 3.844 hours, Δt: 5.201 seconds, umax = (1.1e-01, 9.3e-02, 5.3e-02) ms⁻¹, wall time: 1.075 minutes
[ Info: i: 1920, t: 3.874 hours, Δt: 5.883 seconds, umax = (1.1e-01, 8.9e-02, 4.9e-02) ms⁻¹, wall time: 1.079 minutes
[ Info: i: 1940, t: 3.906 hours, Δt: 5.927 seconds, umax = (1.1e-01, 7.6e-02, 5.1e-02) ms⁻¹, wall time: 1.083 minutes
[ Info: i: 1960, t: 3.938 hours, Δt: 5.937 seconds, umax = (1.1e-01, 7.9e-02, 4.6e-02) ms⁻¹, wall time: 1.087 minutes
[ Info: i: 1980, t: 3.970 hours, Δt: 5.332 seconds, umax = (1.0e-01, 7.9e-02, 4.7e-02) ms⁻¹, wall time: 1.091 minutes
[ Info: Simulation is stopping after running for 1.096 minutes.
[ Info: Simulation time 4 hours equals or exceeds stop time 4 hours.
[ Info: i: 2000, t: 4 hours, Δt: 5.884 seconds, umax = (1.2e-01, 7.4e-02, 5.1e-02) ms⁻¹, wall time: 1.096 minutes

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

times = time_series.w.times

┌ Warning: Reading boundary conditions from Zarr stores is not supported. Using default FieldBoundaryConditions for `grid` and `location`.
└ @ OceananigansZarrExt ~/Oceananigans.jl-33941/ext/OceananigansZarrExt/output_readers.jl:84
┌ Warning: Reading boundary conditions from Zarr stores is not supported. Using default FieldBoundaryConditions for `grid` and `location`.
└ @ OceananigansZarrExt ~/Oceananigans.jl-33941/ext/OceananigansZarrExt/output_readers.jl:84
┌ Warning: Reading boundary conditions from Zarr stores is not supported. Using default FieldBoundaryConditions for `grid` and `location`.
└ @ OceananigansZarrExt ~/Oceananigans.jl-33941/ext/OceananigansZarrExt/output_readers.jl:84
┌ Warning: Reading boundary conditions from Zarr stores is not supported. Using default FieldBoundaryConditions for `grid` and `location`.
└ @ OceananigansZarrExt ~/Oceananigans.jl-33941/ext/OceananigansZarrExt/output_readers.jl:84
┌ Warning: Reading boundary conditions from Zarr stores is not supported. Using default FieldBoundaryConditions for `grid` and `location`.
└ @ OceananigansZarrExt ~/Oceananigans.jl-33941/ext/OceananigansZarrExt/output_readers.jl:84
┌ Warning: Reading boundary conditions from Zarr stores is not supported. Using default FieldBoundaryConditions for `grid` and `location`.
└ @ OceananigansZarrExt ~/Oceananigans.jl-33941/ext/OceananigansZarrExt/output_readers.jl:84
┌ Warning: Reading boundary conditions from Zarr stores is not supported. Using default FieldBoundaryConditions for `grid` and `location`.
└ @ OceananigansZarrExt ~/Oceananigans.jl-33941/ext/OceananigansZarrExt/output_readers.jl:84

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 1.13.0
Commit d1c37793dd2 (2026-09-09 19:00 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-20.1.8 (ORCJIT, znver4)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 128 virtual cores)
Environment:
  JULIA_LOAD_PATH = @:@v#.#:@stdlib
  JULIA_DEPOT_PATH = /var/lib/buildkite-agent/.julia:/var/lib/buildkite-agent/.julia/juliaup/julia-1.13.0+0.x64.linux.gnu/local/share/julia:/var/lib/buildkite-agent/.julia/juliaup/julia-1.13.0+0.x64.linux.gnu/share/julia
  JULIA_VERSION_ENZYME = 1.11.9
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
  LD_LIBRARY_PATH = 
  JULIA_MAX_NUM_PRECOMPILE_FILES = 24
  JULIA_VERSION = 1.13.0
  JULIA_CUDA_USE_COMPAT = false
  JULIA_PROJECT = /var/lib/buildkite-agent/Oceananigans.jl-33941/docs/
  JULIA_DEBUG = Literate

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

import Pkg
Pkg.status()

Status `~/Oceananigans.jl-33941/docs/Project.toml`
  [79e6a3ab] Adapt v4.7.1
⌃ [052768ef] CUDA v5.11.3
  [13f3f980] CairoMakie v0.15.15
⌅ [e30172f5] Documenter v1.17.0
  [daee34ce] DocumenterCitations v1.5.0
  [4710194d] DocumenterVitepress v0.3.6
  [7da242da] Enzyme v0.13.205
  [033835bb] JLD2 v0.6.7
  [63c18a36] KernelAbstractions v0.9.42
  [98b081ad] Literate v2.21.0
  [da04e1cc] MPI v0.20.27
  [85f8d34a] NCDatasets v0.14.15
  [9e8cae18] Oceananigans v0.113.2 `..`
  [429524aa] Optim v2.3.2
  [f27b6e38] Polynomials v4.1.3
  [3c362404] Reactant v0.2.288
  [6038ab10] Rotations v1.7.1
  [d496a93d] SeawaterPolynomials v0.3.10
  [09ab397b] StructArrays v0.7.3
  [bdfc003b] TimesDates v0.3.3
  [0a941bbe] Zarr v0.10.2
  [b77e0a4c] InteractiveUtils v1.11.0
  [37e2e46d] LinearAlgebra v1.13.0
  [44cfe95a] Pkg v1.13.0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`

---

This page was generated using Literate.jl.