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
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 ,

grid = RectilinearGrid(GPU(), size=(128, 128, 64), extent=(128, 128, 64), halo=(5, 5, 5))

128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CUDAGPU with 5×5×5 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 , 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

1.013385408689217

The Stokes drift of a monochromatic surface wave decays away from the surface over the vertical scale

const vertical_scale = 1 / 2wavenumber

4.7746482927568605

and has surface velocity

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) = uˢ(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 undergo forced changes in time. In this example, the Stokes drift is constant and thus the time-derivative of the Stokes drift is zero.

Boundary conditions

At the surface  , 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)

impose a linear buoyancy gradient N² at the bottom along with a weak, destabilizing flux of buoyancy at the surface to facilitate 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 (a loss of buoyancy).

Coriolis parameter

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 ninth-order Weighted Essentially Non-Oscillatory (WENO) advection scheme, whose numerical dissipation acts as an implicit large eddy simulation closure. 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),
                            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) = N² * min(z, -initial_mixed_layer_depth)

bᵢ(x, y, z) = stratification(z) + 1e-1 * Ξ(z) * N² * grid.Lz

bᵢ (generic function with 1 method)

The simulation we reproduce from starts from 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–Friedrichs–Lewy (CFL) number of 1,

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

Nice progress messaging

We define a function that prints a helpful message with the maximum absolute value of and the elapsed wall clock time.

using Printf

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

    @info @sprintf("i: %04d, t: %s, Δt: %s, umax = (%.1e, %.1e, %.1e) m s⁻¹, wall time: %s",
                   iteration(simulation),
                   prettytime(simulation),
                   prettytime(simulation.Δt),
                   maximum(abs, u), maximum(abs, v), maximum(abs, w),
                   prettytime(simulation.run_wall_time))

    return nothing
end

add_callback!(simulation, progress, IterationInterval(20))

Output

A field writer

We set up an output writer for the simulation that saves all velocity and tracer fields. 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-35983/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 velocities, buoyancy, and vertical 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-35983/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 = (1.4e-03, 9.0e-04, 1.3e-03) m s⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (22.106 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.604 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 19.409 seconds, umax = (3.7e-02, 1.2e-02, 2.2e-02) m s⁻¹, wall time: 25.406 seconds
[ Info: i: 0040, t: 16.752 minutes, Δt: 13.081 seconds, umax = (5.6e-02, 2.4e-02, 2.7e-02) m s⁻¹, wall time: 25.593 seconds
[ Info: i: 0060, t: 20.777 minutes, Δt: 11.482 seconds, umax = (6.1e-02, 2.8e-02, 3.5e-02) m s⁻¹, wall time: 25.828 seconds
[ Info: i: 0080, t: 24.513 minutes, Δt: 11.112 seconds, umax = (6.3e-02, 3.2e-02, 4.3e-02) m s⁻¹, wall time: 25.998 seconds
[ Info: i: 0100, t: 28.188 minutes, Δt: 11.889 seconds, umax = (6.0e-02, 3.2e-02, 3.3e-02) m s⁻¹, wall time: 26.219 seconds
[ Info: i: 0120, t: 31.932 minutes, Δt: 10.629 seconds, umax = (6.3e-02, 3.0e-02, 2.8e-02) m s⁻¹, wall time: 26.496 seconds
[ Info: i: 0140, t: 35.161 minutes, Δt: 9.969 seconds, umax = (6.8e-02, 3.5e-02, 3.1e-02) m s⁻¹, wall time: 26.754 seconds
[ Info: i: 0160, t: 38.344 minutes, Δt: 9.363 seconds, umax = (7.0e-02, 3.6e-02, 3.6e-02) m s⁻¹, wall time: 26.962 seconds
[ Info: i: 0180, t: 41.341 minutes, Δt: 9.166 seconds, umax = (7.4e-02, 3.7e-02, 3.3e-02) m s⁻¹, wall time: 27.169 seconds
[ Info: i: 0200, t: 44.423 minutes, Δt: 9.088 seconds, umax = (7.2e-02, 3.8e-02, 3.3e-02) m s⁻¹, wall time: 27.366 seconds
[ Info: i: 0220, t: 47.352 minutes, Δt: 8.916 seconds, umax = (7.5e-02, 4.0e-02, 4.2e-02) m s⁻¹, wall time: 27.581 seconds
[ Info: i: 0240, t: 50.141 minutes, Δt: 8.759 seconds, umax = (7.8e-02, 4.1e-02, 4.0e-02) m s⁻¹, wall time: 27.842 seconds
[ Info: i: 0260, t: 53.036 minutes, Δt: 8.021 seconds, umax = (8.0e-02, 4.3e-02, 3.8e-02) m s⁻¹, wall time: 27.976 seconds
[ Info: i: 0280, t: 55.727 minutes, Δt: 8.388 seconds, umax = (8.0e-02, 5.0e-02, 3.7e-02) m s⁻¹, wall time: 28.202 seconds
[ Info: i: 0300, t: 58.496 minutes, Δt: 8.271 seconds, umax = (8.0e-02, 4.6e-02, 3.7e-02) m s⁻¹, wall time: 28.376 seconds
[ Info: i: 0320, t: 1.021 hours, Δt: 7.889 seconds, umax = (8.4e-02, 4.6e-02, 3.8e-02) m s⁻¹, wall time: 28.592 seconds
[ Info: i: 0340, t: 1.065 hours, Δt: 8.035 seconds, umax = (8.7e-02, 4.8e-02, 4.1e-02) m s⁻¹, wall time: 28.781 seconds
[ Info: i: 0360, t: 1.109 hours, Δt: 7.987 seconds, umax = (8.9e-02, 5.0e-02, 3.9e-02) m s⁻¹, wall time: 28.982 seconds
[ Info: i: 0380, t: 1.153 hours, Δt: 7.031 seconds, umax = (8.9e-02, 5.0e-02, 4.5e-02) m s⁻¹, wall time: 29.173 seconds
[ Info: i: 0400, t: 1.192 hours, Δt: 7.227 seconds, umax = (1.0e-01, 4.9e-02, 4.3e-02) m s⁻¹, wall time: 29.376 seconds
[ Info: i: 0420, t: 1.232 hours, Δt: 7.480 seconds, umax = (9.2e-02, 5.3e-02, 4.1e-02) m s⁻¹, wall time: 29.569 seconds
[ Info: i: 0440, t: 1.274 hours, Δt: 7.562 seconds, umax = (8.9e-02, 4.9e-02, 4.3e-02) m s⁻¹, wall time: 29.771 seconds
[ Info: i: 0460, t: 1.314 hours, Δt: 7.336 seconds, umax = (1.0e-01, 5.4e-02, 3.8e-02) m s⁻¹, wall time: 29.976 seconds
[ Info: i: 0480, t: 1.354 hours, Δt: 6.836 seconds, umax = (9.2e-02, 5.4e-02, 4.4e-02) m s⁻¹, wall time: 30.179 seconds
[ Info: i: 0500, t: 1.392 hours, Δt: 6.909 seconds, umax = (9.7e-02, 5.5e-02, 4.8e-02) m s⁻¹, wall time: 30.370 seconds
[ Info: i: 0520, t: 1.430 hours, Δt: 7.035 seconds, umax = (9.2e-02, 5.4e-02, 4.9e-02) m s⁻¹, wall time: 30.574 seconds
[ Info: i: 0540, t: 1.470 hours, Δt: 7.107 seconds, umax = (9.1e-02, 5.4e-02, 4.7e-02) m s⁻¹, wall time: 30.760 seconds
[ Info: i: 0560, t: 1.508 hours, Δt: 6.854 seconds, umax = (9.7e-02, 5.8e-02, 4.9e-02) m s⁻¹, wall time: 31.007 seconds
[ Info: i: 0580, t: 1.545 hours, Δt: 6.793 seconds, umax = (9.9e-02, 5.7e-02, 5.1e-02) m s⁻¹, wall time: 31.160 seconds
[ Info: i: 0600, t: 1.581 hours, Δt: 6.803 seconds, umax = (9.8e-02, 5.9e-02, 4.5e-02) m s⁻¹, wall time: 31.355 seconds
[ Info: i: 0620, t: 1.617 hours, Δt: 7.131 seconds, umax = (9.6e-02, 5.8e-02, 5.0e-02) m s⁻¹, wall time: 31.555 seconds
[ Info: i: 0640, t: 1.657 hours, Δt: 6.830 seconds, umax = (9.8e-02, 5.9e-02, 4.1e-02) m s⁻¹, wall time: 31.746 seconds
[ Info: i: 0660, t: 1.692 hours, Δt: 6.968 seconds, umax = (9.7e-02, 6.3e-02, 4.2e-02) m s⁻¹, wall time: 31.949 seconds
[ Info: i: 0680, t: 1.730 hours, Δt: 7.294 seconds, umax = (9.9e-02, 6.0e-02, 4.3e-02) m s⁻¹, wall time: 32.141 seconds
[ Info: i: 0700, t: 1.769 hours, Δt: 6.594 seconds, umax = (1.1e-01, 6.1e-02, 4.8e-02) m s⁻¹, wall time: 32.363 seconds
[ Info: i: 0720, t: 1.807 hours, Δt: 6.512 seconds, umax = (1.0e-01, 5.7e-02, 4.6e-02) m s⁻¹, wall time: 32.613 seconds
[ Info: i: 0740, t: 1.843 hours, Δt: 6.571 seconds, umax = (1.0e-01, 5.9e-02, 4.3e-02) m s⁻¹, wall time: 32.993 seconds
[ Info: i: 0760, t: 1.880 hours, Δt: 6.490 seconds, umax = (1.0e-01, 6.5e-02, 4.4e-02) m s⁻¹, wall time: 33.325 seconds
[ Info: i: 0780, t: 1.917 hours, Δt: 6.369 seconds, umax = (1.0e-01, 6.0e-02, 4.6e-02) m s⁻¹, wall time: 33.705 seconds
[ Info: i: 0800, t: 1.952 hours, Δt: 6.380 seconds, umax = (1.0e-01, 6.0e-02, 4.8e-02) m s⁻¹, wall time: 34.057 seconds
[ Info: i: 0820, t: 1.987 hours, Δt: 6.341 seconds, umax = (1.1e-01, 6.0e-02, 4.8e-02) m s⁻¹, wall time: 34.434 seconds
[ Info: i: 0840, t: 2.021 hours, Δt: 5.871 seconds, umax = (1.0e-01, 6.6e-02, 4.3e-02) m s⁻¹, wall time: 34.814 seconds
[ Info: i: 0860, t: 2.056 hours, Δt: 6.570 seconds, umax = (1.1e-01, 6.2e-02, 4.3e-02) m s⁻¹, wall time: 35.184 seconds
[ Info: i: 0880, t: 2.090 hours, Δt: 6.463 seconds, umax = (1.0e-01, 6.6e-02, 5.1e-02) m s⁻¹, wall time: 35.605 seconds
[ Info: i: 0900, t: 2.125 hours, Δt: 6.399 seconds, umax = (1.1e-01, 6.2e-02, 4.5e-02) m s⁻¹, wall time: 35.910 seconds
[ Info: i: 0920, t: 2.161 hours, Δt: 6.833 seconds, umax = (1.0e-01, 6.3e-02, 4.4e-02) m s⁻¹, wall time: 36.284 seconds
[ Info: i: 0940, t: 2.198 hours, Δt: 6.034 seconds, umax = (1.1e-01, 6.8e-02, 4.6e-02) m s⁻¹, wall time: 36.631 seconds
[ Info: i: 0960, t: 2.232 hours, Δt: 6.150 seconds, umax = (1.1e-01, 7.1e-02, 5.0e-02) m s⁻¹, wall time: 37.003 seconds
[ Info: i: 0980, t: 2.264 hours, Δt: 5.757 seconds, umax = (1.1e-01, 7.1e-02, 4.7e-02) m s⁻¹, wall time: 37.397 seconds
[ Info: i: 1000, t: 2.297 hours, Δt: 6.440 seconds, umax = (1.1e-01, 6.5e-02, 4.7e-02) m s⁻¹, wall time: 37.610 seconds
[ Info: i: 1020, t: 2.332 hours, Δt: 6.172 seconds, umax = (1.0e-01, 7.1e-02, 4.8e-02) m s⁻¹, wall time: 37.805 seconds
[ Info: i: 1040, t: 2.367 hours, Δt: 6.345 seconds, umax = (1.1e-01, 6.7e-02, 4.9e-02) m s⁻¹, wall time: 38.008 seconds
[ Info: i: 1060, t: 2.402 hours, Δt: 6.270 seconds, umax = (1.0e-01, 7.9e-02, 4.5e-02) m s⁻¹, wall time: 38.200 seconds
[ Info: i: 1080, t: 2.435 hours, Δt: 6.050 seconds, umax = (1.1e-01, 7.4e-02, 4.9e-02) m s⁻¹, wall time: 38.403 seconds
[ Info: i: 1100, t: 2.469 hours, Δt: 6.156 seconds, umax = (1.1e-01, 7.0e-02, 5.5e-02) m s⁻¹, wall time: 38.593 seconds
[ Info: i: 1120, t: 2.502 hours, Δt: 5.917 seconds, umax = (1.1e-01, 6.8e-02, 5.0e-02) m s⁻¹, wall time: 38.869 seconds
[ Info: i: 1140, t: 2.533 hours, Δt: 5.639 seconds, umax = (1.1e-01, 7.2e-02, 4.5e-02) m s⁻¹, wall time: 39.004 seconds
[ Info: i: 1160, t: 2.566 hours, Δt: 5.962 seconds, umax = (1.1e-01, 7.2e-02, 4.7e-02) m s⁻¹, wall time: 39.201 seconds
[ Info: i: 1180, t: 2.598 hours, Δt: 5.564 seconds, umax = (1.1e-01, 7.5e-02, 4.7e-02) m s⁻¹, wall time: 39.406 seconds
[ Info: i: 1200, t: 2.630 hours, Δt: 5.942 seconds, umax = (1.1e-01, 7.1e-02, 4.6e-02) m s⁻¹, wall time: 39.600 seconds
[ Info: i: 1220, t: 2.663 hours, Δt: 5.873 seconds, umax = (1.1e-01, 7.6e-02, 4.8e-02) m s⁻¹, wall time: 39.795 seconds
[ Info: i: 1240, t: 2.695 hours, Δt: 5.481 seconds, umax = (1.1e-01, 7.9e-02, 4.7e-02) m s⁻¹, wall time: 40.000 seconds
[ Info: i: 1260, t: 2.727 hours, Δt: 5.867 seconds, umax = (1.0e-01, 7.1e-02, 5.5e-02) m s⁻¹, wall time: 40.218 seconds
[ Info: i: 1280, t: 2.758 hours, Δt: 5.883 seconds, umax = (1.0e-01, 7.5e-02, 5.0e-02) m s⁻¹, wall time: 40.499 seconds
[ Info: i: 1300, t: 2.792 hours, Δt: 5.839 seconds, umax = (1.1e-01, 7.3e-02, 4.4e-02) m s⁻¹, wall time: 40.692 seconds
[ Info: i: 1320, t: 2.825 hours, Δt: 5.939 seconds, umax = (1.1e-01, 6.7e-02, 4.8e-02) m s⁻¹, wall time: 40.927 seconds
[ Info: i: 1340, t: 2.857 hours, Δt: 6.018 seconds, umax = (1.2e-01, 7.7e-02, 5.0e-02) m s⁻¹, wall time: 41.168 seconds
[ Info: i: 1360, t: 2.889 hours, Δt: 5.581 seconds, umax = (1.1e-01, 7.3e-02, 5.0e-02) m s⁻¹, wall time: 41.359 seconds
[ Info: i: 1380, t: 2.922 hours, Δt: 5.966 seconds, umax = (1.1e-01, 7.5e-02, 5.1e-02) m s⁻¹, wall time: 41.601 seconds
[ Info: i: 1400, t: 2.955 hours, Δt: 5.925 seconds, umax = (1.1e-01, 7.4e-02, 4.7e-02) m s⁻¹, wall time: 41.751 seconds
[ Info: i: 1420, t: 2.989 hours, Δt: 6.393 seconds, umax = (1.1e-01, 7.8e-02, 4.8e-02) m s⁻¹, wall time: 41.945 seconds
[ Info: i: 1440, t: 3.022 hours, Δt: 5.508 seconds, umax = (1.1e-01, 7.8e-02, 4.7e-02) m s⁻¹, wall time: 42.168 seconds
[ Info: i: 1460, t: 3.053 hours, Δt: 5.660 seconds, umax = (1.1e-01, 7.0e-02, 5.0e-02) m s⁻¹, wall time: 42.354 seconds
[ Info: i: 1480, t: 3.083 hours, Δt: 5.586 seconds, umax = (1.1e-01, 7.6e-02, 5.0e-02) m s⁻¹, wall time: 42.561 seconds
[ Info: i: 1500, t: 3.115 hours, Δt: 5.755 seconds, umax = (1.2e-01, 7.6e-02, 4.8e-02) m s⁻¹, wall time: 42.765 seconds
[ Info: i: 1520, t: 3.146 hours, Δt: 5.197 seconds, umax = (1.1e-01, 7.9e-02, 4.3e-02) m s⁻¹, wall time: 43.003 seconds
[ Info: i: 1540, t: 3.176 hours, Δt: 6.009 seconds, umax = (1.1e-01, 7.1e-02, 4.7e-02) m s⁻¹, wall time: 43.408 seconds
[ Info: i: 1560, t: 3.209 hours, Δt: 5.966 seconds, umax = (1.1e-01, 7.1e-02, 4.6e-02) m s⁻¹, wall time: 43.733 seconds
[ Info: i: 1580, t: 3.241 hours, Δt: 5.682 seconds, umax = (1.2e-01, 6.9e-02, 4.7e-02) m s⁻¹, wall time: 44.084 seconds
[ Info: i: 1600, t: 3.272 hours, Δt: 5.784 seconds, umax = (1.1e-01, 7.9e-02, 4.8e-02) m s⁻¹, wall time: 44.477 seconds
[ Info: i: 1620, t: 3.303 hours, Δt: 5.652 seconds, umax = (1.1e-01, 7.8e-02, 5.1e-02) m s⁻¹, wall time: 44.838 seconds
[ Info: i: 1640, t: 3.333 hours, Δt: 6.004 seconds, umax = (1.1e-01, 7.8e-02, 5.1e-02) m s⁻¹, wall time: 45.215 seconds
[ Info: i: 1660, t: 3.367 hours, Δt: 6.160 seconds, umax = (1.1e-01, 7.6e-02, 4.7e-02) m s⁻¹, wall time: 45.583 seconds
[ Info: i: 1680, t: 3.400 hours, Δt: 5.754 seconds, umax = (1.1e-01, 8.2e-02, 4.6e-02) m s⁻¹, wall time: 45.951 seconds
[ Info: i: 1700, t: 3.431 hours, Δt: 6.030 seconds, umax = (1.2e-01, 7.6e-02, 4.8e-02) m s⁻¹, wall time: 46.342 seconds
[ Info: i: 1720, t: 3.464 hours, Δt: 6.098 seconds, umax = (1.2e-01, 7.5e-02, 5.1e-02) m s⁻¹, wall time: 46.691 seconds
[ Info: i: 1740, t: 3.497 hours, Δt: 5.837 seconds, umax = (1.2e-01, 7.6e-02, 4.7e-02) m s⁻¹, wall time: 47.080 seconds
[ Info: i: 1760, t: 3.528 hours, Δt: 5.399 seconds, umax = (1.1e-01, 8.0e-02, 4.4e-02) m s⁻¹, wall time: 47.456 seconds
[ Info: i: 1780, t: 3.559 hours, Δt: 5.874 seconds, umax = (1.2e-01, 8.3e-02, 5.0e-02) m s⁻¹, wall time: 47.833 seconds
[ Info: i: 1800, t: 3.590 hours, Δt: 6.003 seconds, umax = (1.2e-01, 8.0e-02, 5.0e-02) m s⁻¹, wall time: 48.187 seconds
[ Info: i: 1820, t: 3.623 hours, Δt: 5.559 seconds, umax = (1.1e-01, 7.8e-02, 5.1e-02) m s⁻¹, wall time: 48.349 seconds
[ Info: i: 1840, t: 3.654 hours, Δt: 5.697 seconds, umax = (1.1e-01, 8.5e-02, 5.2e-02) m s⁻¹, wall time: 48.543 seconds
[ Info: i: 1860, t: 3.684 hours, Δt: 5.611 seconds, umax = (1.2e-01, 8.6e-02, 4.9e-02) m s⁻¹, wall time: 48.825 seconds
[ Info: i: 1880, t: 3.715 hours, Δt: 5.274 seconds, umax = (1.2e-01, 9.0e-02, 4.6e-02) m s⁻¹, wall time: 49.010 seconds
[ Info: i: 1900, t: 3.745 hours, Δt: 5.706 seconds, umax = (1.1e-01, 8.0e-02, 4.9e-02) m s⁻¹, wall time: 49.206 seconds
[ Info: i: 1920, t: 3.775 hours, Δt: 5.420 seconds, umax = (1.2e-01, 8.7e-02, 5.0e-02) m s⁻¹, wall time: 49.424 seconds
[ Info: i: 1940, t: 3.806 hours, Δt: 5.416 seconds, umax = (1.2e-01, 8.5e-02, 5.0e-02) m s⁻¹, wall time: 49.614 seconds
[ Info: i: 1960, t: 3.837 hours, Δt: 5.647 seconds, umax = (1.2e-01, 8.8e-02, 5.4e-02) m s⁻¹, wall time: 49.864 seconds
[ Info: i: 1980, t: 3.867 hours, Δt: 5.832 seconds, umax = (1.2e-01, 9.1e-02, 5.9e-02) m s⁻¹, wall time: 50.008 seconds
[ Info: i: 2000, t: 3.899 hours, Δt: 5.635 seconds, umax = (1.2e-01, 8.1e-02, 5.5e-02) m s⁻¹, wall time: 50.211 seconds
[ Info: i: 2020, t: 3.929 hours, Δt: 5.430 seconds, umax = (1.1e-01, 8.7e-02, 5.6e-02) m s⁻¹, wall time: 50.458 seconds
[ Info: i: 2040, t: 3.960 hours, Δt: 5.731 seconds, umax = (1.2e-01, 8.0e-02, 5.6e-02) m s⁻¹, wall time: 50.814 seconds
[ Info: i: 2060, t: 3.992 hours, Δt: 5.869 seconds, umax = (1.1e-01, 7.7e-02, 5.0e-02) m s⁻¹, wall time: 51.187 seconds
[ Info: Simulation is stopping after running for 51.290 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.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-35983/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-35983/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-35983/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-35983/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-35983/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-35983/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-35983/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)


Bₙ = @lift time_series.B[$n]
Uₙ = @lift time_series.U[$n]
Vₙ = @lift time_series.V[$n]
wuₙ = @lift time_series.wu[$n]
wvₙ = @lift time_series.wv[$n]

k = searchsortedfirst(znodes(grid, Face()), -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")

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

Colorbar(fig[3, 3], hm_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 1.13.2
Commit 45b6d15e34b (2026-10-14 20:33 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-1.13.2+0.x64.linux.gnu/local/share/julia:/var/lib/buildkite-agent/.julia/juliaup/julia-1.13.2+0.x64.linux.gnu/share/julia
  JULIA_VERSION_ENZYME = 1.13.1
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
  LD_LIBRARY_PATH = 
  JULIA_MAX_NUM_PRECOMPILE_FILES = 24
  JULIA_VERSION = 1.13.2
  JULIA_CUDA_USE_COMPAT = false
  JULIA_PROJECT = /var/lib/buildkite-agent/Oceananigans.jl-35983/docs/
  JULIA_DEBUG = Literate

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

import Pkg
Pkg.status()

Status `~/Oceananigans.jl-35983/docs/Project.toml`
  [79e6a3ab] Adapt v4.8.0
⌃ [052768ef] CUDA v6.4.1
  [13f3f980] CairoMakie v0.15.17
  [8e50ac2c] ConservativeRegridding v0.2.20
  [e30172f5] Documenter v1.20.0
  [daee34ce] DocumenterCitations v1.5.0
  [4710194d] DocumenterVitepress v0.3.9
  [7da242da] Enzyme v0.13.218
  [033835bb] JLD2 v0.6.7
⌅ [63c18a36] KernelAbstractions v0.9.45
  [98b081ad] Literate v2.21.0
  [da04e1cc] MPI v0.20.29
  [85f8d34a] NCDatasets v0.14.15
  [9e8cae18] Oceananigans v0.117.1 `..`
  [429524aa] Optim v2.3.2
  [f27b6e38] Polynomials v4.1.6
  [3c362404] Reactant v0.2.307
  [6038ab10] Rotations v1.7.1
  [d496a93d] SeawaterPolynomials v0.3.11
  [09ab397b] StructArrays v0.7.4
  [bdfc003b] TimesDates v0.3.3
  [0a941bbe] Zarr v0.11.2
  [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.