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{Float32, 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{Float32}(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{Float32, Periodic, Periodic, Bounded} on CUDAGPU with 5×5×5 halo
├── timestepper: RungeKutta3TimeStepper
├── advection scheme:
│   ├── momentum: WENO{5, Float32, Oceananigans.Utils.BackendOptimizedDivision}(order=9)
│   └── b: WENO{5, Float32, Oceananigans.Utils.BackendOptimizedDivision}(order=9)
├── tracers: b
├── closure: Nothing
├── buoyancy: BuoyancyTracer with ĝ = NegativeZDirection()
└── coriolis: FPlane{Float32}(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-35915/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-35915/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 (26.770 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.484 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: 29.776 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: 29.883 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: 30.022 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: 30.108 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: 30.227 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: 30.358 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: 30.525 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: 30.600 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: 30.738 seconds
[ Info: i: 0200, t: 44.420 minutes, Δt: 9.092 seconds, umax = (7.2e-02, 3.8e-02, 3.3e-02) m s⁻¹, wall time: 30.859 seconds
[ Info: i: 0220, t: 47.351 minutes, Δt: 8.920 seconds, umax = (7.5e-02, 4.0e-02, 4.2e-02) m s⁻¹, wall time: 31.086 seconds
[ Info: i: 0240, t: 50.141 minutes, Δt: 8.763 seconds, umax = (7.8e-02, 4.1e-02, 4.0e-02) m s⁻¹, wall time: 31.243 seconds
[ Info: i: 0260, t: 53.037 minutes, Δt: 8.013 seconds, umax = (8.0e-02, 4.3e-02, 3.8e-02) m s⁻¹, wall time: 31.318 seconds
[ Info: i: 0280, t: 55.727 minutes, Δt: 8.400 seconds, umax = (8.0e-02, 5.0e-02, 3.7e-02) m s⁻¹, wall time: 31.468 seconds
[ Info: i: 0300, t: 58.502 minutes, Δt: 8.275 seconds, umax = (8.0e-02, 4.6e-02, 3.7e-02) m s⁻¹, wall time: 31.561 seconds
[ Info: i: 0320, t: 1.021 hours, Δt: 7.896 seconds, umax = (8.4e-02, 4.6e-02, 3.8e-02) m s⁻¹, wall time: 31.680 seconds
[ Info: i: 0340, t: 1.065 hours, Δt: 8.030 seconds, umax = (8.6e-02, 4.8e-02, 4.1e-02) m s⁻¹, wall time: 31.786 seconds
[ Info: i: 0360, t: 1.109 hours, Δt: 8.051 seconds, umax = (8.9e-02, 5.1e-02, 4.0e-02) m s⁻¹, wall time: 31.904 seconds
[ Info: i: 0380, t: 1.153 hours, Δt: 7.028 seconds, umax = (8.9e-02, 5.0e-02, 4.5e-02) m s⁻¹, wall time: 32.007 seconds
[ Info: i: 0400, t: 1.192 hours, Δt: 7.144 seconds, umax = (1.0e-01, 4.9e-02, 4.3e-02) m s⁻¹, wall time: 32.124 seconds
[ Info: i: 0420, t: 1.231 hours, Δt: 7.310 seconds, umax = (9.0e-02, 5.7e-02, 4.2e-02) m s⁻¹, wall time: 32.233 seconds
[ Info: i: 0440, t: 1.272 hours, Δt: 7.431 seconds, umax = (8.8e-02, 5.0e-02, 4.3e-02) m s⁻¹, wall time: 32.355 seconds
[ Info: i: 0460, t: 1.312 hours, Δt: 7.286 seconds, umax = (9.2e-02, 6.1e-02, 4.0e-02) m s⁻¹, wall time: 32.458 seconds
[ Info: i: 0480, t: 1.351 hours, Δt: 6.954 seconds, umax = (9.4e-02, 5.6e-02, 4.4e-02) m s⁻¹, wall time: 32.578 seconds
[ Info: i: 0500, t: 1.390 hours, Δt: 6.583 seconds, umax = (9.7e-02, 5.8e-02, 4.4e-02) m s⁻¹, wall time: 32.681 seconds
[ Info: i: 0520, t: 1.426 hours, Δt: 7.311 seconds, umax = (9.4e-02, 5.5e-02, 4.8e-02) m s⁻¹, wall time: 32.874 seconds
[ Info: i: 0540, t: 1.467 hours, Δt: 7.059 seconds, umax = (1.0e-01, 5.4e-02, 4.7e-02) m s⁻¹, wall time: 32.967 seconds
[ Info: i: 0560, t: 1.506 hours, Δt: 6.677 seconds, umax = (9.9e-02, 5.9e-02, 4.7e-02) m s⁻¹, wall time: 33.116 seconds
[ Info: i: 0580, t: 1.544 hours, Δt: 6.984 seconds, umax = (1.0e-01, 6.3e-02, 4.6e-02) m s⁻¹, wall time: 33.196 seconds
[ Info: i: 0600, t: 1.582 hours, Δt: 6.784 seconds, umax = (9.8e-02, 5.6e-02, 4.2e-02) m s⁻¹, wall time: 33.310 seconds
[ Info: i: 0620, t: 1.619 hours, Δt: 6.968 seconds, umax = (1.0e-01, 6.6e-02, 4.1e-02) m s⁻¹, wall time: 33.440 seconds
[ Info: i: 0640, t: 1.658 hours, Δt: 6.752 seconds, umax = (9.7e-02, 5.9e-02, 4.7e-02) m s⁻¹, wall time: 33.565 seconds
[ Info: i: 0660, t: 1.694 hours, Δt: 6.519 seconds, umax = (1.0e-01, 5.9e-02, 4.3e-02) m s⁻¹, wall time: 33.684 seconds
[ Info: i: 0680, t: 1.732 hours, Δt: 6.650 seconds, umax = (1.0e-01, 5.6e-02, 4.4e-02) m s⁻¹, wall time: 33.790 seconds
[ Info: i: 0700, t: 1.767 hours, Δt: 6.454 seconds, umax = (9.9e-02, 5.8e-02, 3.9e-02) m s⁻¹, wall time: 33.918 seconds
[ Info: i: 0720, t: 1.803 hours, Δt: 6.813 seconds, umax = (1.0e-01, 5.9e-02, 4.3e-02) m s⁻¹, wall time: 34.024 seconds
[ Info: i: 0740, t: 1.841 hours, Δt: 6.652 seconds, umax = (9.9e-02, 6.0e-02, 4.3e-02) m s⁻¹, wall time: 34.165 seconds
[ Info: i: 0760, t: 1.877 hours, Δt: 6.757 seconds, umax = (1.0e-01, 6.9e-02, 4.5e-02) m s⁻¹, wall time: 34.247 seconds
[ Info: i: 0780, t: 1.913 hours, Δt: 6.388 seconds, umax = (1.2e-01, 5.9e-02, 4.9e-02) m s⁻¹, wall time: 34.354 seconds
[ Info: i: 0800, t: 1.948 hours, Δt: 6.563 seconds, umax = (1.1e-01, 6.3e-02, 5.2e-02) m s⁻¹, wall time: 34.475 seconds
[ Info: i: 0820, t: 1.985 hours, Δt: 6.917 seconds, umax = (1.1e-01, 6.3e-02, 4.9e-02) m s⁻¹, wall time: 34.580 seconds
[ Info: i: 0840, t: 2.021 hours, Δt: 6.094 seconds, umax = (1.1e-01, 6.1e-02, 4.4e-02) m s⁻¹, wall time: 34.730 seconds
[ Info: i: 0860, t: 2.055 hours, Δt: 6.739 seconds, umax = (1.0e-01, 6.1e-02, 4.5e-02) m s⁻¹, wall time: 34.834 seconds
[ Info: i: 0880, t: 2.091 hours, Δt: 6.109 seconds, umax = (1.0e-01, 6.8e-02, 5.2e-02) m s⁻¹, wall time: 34.976 seconds
[ Info: i: 0900, t: 2.126 hours, Δt: 6.672 seconds, umax = (1.1e-01, 6.3e-02, 5.4e-02) m s⁻¹, wall time: 35.061 seconds
[ Info: i: 0920, t: 2.161 hours, Δt: 5.900 seconds, umax = (1.1e-01, 6.8e-02, 4.8e-02) m s⁻¹, wall time: 35.167 seconds
[ Info: i: 0940, t: 2.194 hours, Δt: 6.187 seconds, umax = (1.1e-01, 6.5e-02, 5.4e-02) m s⁻¹, wall time: 35.287 seconds
[ Info: i: 0960, t: 2.229 hours, Δt: 6.525 seconds, umax = (1.1e-01, 6.6e-02, 4.3e-02) m s⁻¹, wall time: 35.390 seconds
[ Info: i: 0980, t: 2.264 hours, Δt: 6.068 seconds, umax = (1.1e-01, 7.0e-02, 4.7e-02) m s⁻¹, wall time: 35.514 seconds
[ Info: i: 1000, t: 2.297 hours, Δt: 6.081 seconds, umax = (1.1e-01, 7.0e-02, 4.5e-02) m s⁻¹, wall time: 35.617 seconds
[ Info: i: 1020, t: 2.331 hours, Δt: 6.180 seconds, umax = (1.1e-01, 6.9e-02, 4.5e-02) m s⁻¹, wall time: 35.723 seconds
[ Info: i: 1040, t: 2.364 hours, Δt: 6.007 seconds, umax = (1.1e-01, 6.8e-02, 4.2e-02) m s⁻¹, wall time: 35.839 seconds
[ Info: i: 1060, t: 2.399 hours, Δt: 6.159 seconds, umax = (1.1e-01, 6.8e-02, 4.4e-02) m s⁻¹, wall time: 35.943 seconds
[ Info: i: 1080, t: 2.433 hours, Δt: 6.118 seconds, umax = (1.1e-01, 6.9e-02, 4.6e-02) m s⁻¹, wall time: 36.083 seconds
[ Info: i: 1100, t: 2.467 hours, Δt: 6.354 seconds, umax = (1.1e-01, 7.6e-02, 5.2e-02) m s⁻¹, wall time: 36.193 seconds
[ Info: i: 1120, t: 2.500 hours, Δt: 5.991 seconds, umax = (1.1e-01, 8.1e-02, 4.8e-02) m s⁻¹, wall time: 36.322 seconds
[ Info: i: 1140, t: 2.534 hours, Δt: 6.355 seconds, umax = (1.1e-01, 7.1e-02, 4.7e-02) m s⁻¹, wall time: 36.458 seconds
[ Info: i: 1160, t: 2.569 hours, Δt: 6.287 seconds, umax = (1.1e-01, 7.4e-02, 4.5e-02) m s⁻¹, wall time: 36.577 seconds
[ Info: i: 1180, t: 2.602 hours, Δt: 6.368 seconds, umax = (1.1e-01, 6.9e-02, 5.2e-02) m s⁻¹, wall time: 36.695 seconds
[ Info: i: 1200, t: 2.636 hours, Δt: 5.965 seconds, umax = (1.1e-01, 7.3e-02, 4.8e-02) m s⁻¹, wall time: 36.799 seconds
[ Info: i: 1220, t: 2.668 hours, Δt: 6.159 seconds, umax = (1.2e-01, 7.6e-02, 4.8e-02) m s⁻¹, wall time: 36.952 seconds
[ Info: i: 1240, t: 2.702 hours, Δt: 6.052 seconds, umax = (1.1e-01, 7.9e-02, 5.1e-02) m s⁻¹, wall time: 37.026 seconds
[ Info: i: 1260, t: 2.736 hours, Δt: 6.108 seconds, umax = (1.1e-01, 7.7e-02, 5.1e-02) m s⁻¹, wall time: 37.137 seconds
[ Info: i: 1280, t: 2.768 hours, Δt: 5.543 seconds, umax = (1.1e-01, 7.9e-02, 4.6e-02) m s⁻¹, wall time: 37.255 seconds
[ Info: i: 1300, t: 2.799 hours, Δt: 5.895 seconds, umax = (1.1e-01, 7.4e-02, 4.8e-02) m s⁻¹, wall time: 37.357 seconds
[ Info: i: 1320, t: 2.833 hours, Δt: 6.278 seconds, umax = (1.1e-01, 7.4e-02, 4.7e-02) m s⁻¹, wall time: 37.466 seconds
[ Info: i: 1340, t: 2.866 hours, Δt: 6.226 seconds, umax = (1.1e-01, 7.1e-02, 5.3e-02) m s⁻¹, wall time: 37.599 seconds
[ Info: i: 1360, t: 2.901 hours, Δt: 6.056 seconds, umax = (1.1e-01, 7.7e-02, 5.2e-02) m s⁻¹, wall time: 37.704 seconds
[ Info: i: 1380, t: 2.933 hours, Δt: 6.155 seconds, umax = (1.1e-01, 7.6e-02, 4.4e-02) m s⁻¹, wall time: 37.835 seconds
[ Info: i: 1400, t: 2.966 hours, Δt: 5.978 seconds, umax = (1.1e-01, 7.1e-02, 4.8e-02) m s⁻¹, wall time: 37.941 seconds
[ Info: i: 1420, t: 2.999 hours, Δt: 5.515 seconds, umax = (1.1e-01, 7.9e-02, 4.8e-02) m s⁻¹, wall time: 38.049 seconds
[ Info: i: 1440, t: 3.029 hours, Δt: 5.742 seconds, umax = (1.1e-01, 7.7e-02, 4.4e-02) m s⁻¹, wall time: 38.166 seconds
[ Info: i: 1460, t: 3.061 hours, Δt: 5.788 seconds, umax = (1.2e-01, 9.8e-02, 4.9e-02) m s⁻¹, wall time: 38.269 seconds
[ Info: i: 1480, t: 3.093 hours, Δt: 5.848 seconds, umax = (1.1e-01, 8.2e-02, 4.7e-02) m s⁻¹, wall time: 38.399 seconds
[ Info: i: 1500, t: 3.127 hours, Δt: 6.093 seconds, umax = (1.1e-01, 7.4e-02, 4.9e-02) m s⁻¹, wall time: 38.491 seconds
[ Info: i: 1520, t: 3.160 hours, Δt: 5.888 seconds, umax = (1.1e-01, 7.1e-02, 4.8e-02) m s⁻¹, wall time: 38.596 seconds
[ Info: i: 1540, t: 3.190 hours, Δt: 5.463 seconds, umax = (1.2e-01, 7.7e-02, 4.8e-02) m s⁻¹, wall time: 38.719 seconds
[ Info: i: 1560, t: 3.221 hours, Δt: 5.744 seconds, umax = (1.1e-01, 7.2e-02, 4.9e-02) m s⁻¹, wall time: 38.821 seconds
[ Info: i: 1580, t: 3.253 hours, Δt: 5.923 seconds, umax = (1.2e-01, 7.5e-02, 4.7e-02) m s⁻¹, wall time: 38.968 seconds
[ Info: i: 1600, t: 3.286 hours, Δt: 5.881 seconds, umax = (1.2e-01, 7.2e-02, 4.6e-02) m s⁻¹, wall time: 39.045 seconds
[ Info: i: 1620, t: 3.319 hours, Δt: 5.969 seconds, umax = (1.2e-01, 7.8e-02, 5.2e-02) m s⁻¹, wall time: 39.149 seconds
[ Info: i: 1640, t: 3.351 hours, Δt: 5.781 seconds, umax = (1.1e-01, 7.6e-02, 5.0e-02) m s⁻¹, wall time: 39.276 seconds
[ Info: i: 1660, t: 3.382 hours, Δt: 5.467 seconds, umax = (1.1e-01, 8.4e-02, 4.9e-02) m s⁻¹, wall time: 39.401 seconds
[ Info: i: 1680, t: 3.413 hours, Δt: 5.528 seconds, umax = (1.1e-01, 7.6e-02, 4.4e-02) m s⁻¹, wall time: 39.534 seconds
[ Info: i: 1700, t: 3.443 hours, Δt: 5.399 seconds, umax = (1.2e-01, 8.3e-02, 4.7e-02) m s⁻¹, wall time: 39.676 seconds
[ Info: i: 1720, t: 3.473 hours, Δt: 5.589 seconds, umax = (1.1e-01, 7.8e-02, 4.9e-02) m s⁻¹, wall time: 39.804 seconds
[ Info: i: 1740, t: 3.503 hours, Δt: 4.947 seconds, umax = (1.2e-01, 8.0e-02, 5.1e-02) m s⁻¹, wall time: 39.974 seconds
[ Info: i: 1760, t: 3.530 hours, Δt: 5.224 seconds, umax = (1.2e-01, 7.6e-02, 4.8e-02) m s⁻¹, wall time: 40.064 seconds
[ Info: i: 1780, t: 3.561 hours, Δt: 5.625 seconds, umax = (1.3e-01, 8.0e-02, 4.7e-02) m s⁻¹, wall time: 40.182 seconds
[ Info: i: 1800, t: 3.592 hours, Δt: 5.772 seconds, umax = (1.1e-01, 7.8e-02, 5.3e-02) m s⁻¹, wall time: 40.315 seconds
[ Info: i: 1820, t: 3.624 hours, Δt: 5.702 seconds, umax = (1.2e-01, 7.7e-02, 4.8e-02) m s⁻¹, wall time: 40.404 seconds
[ Info: i: 1840, t: 3.657 hours, Δt: 5.968 seconds, umax = (1.1e-01, 8.1e-02, 4.7e-02) m s⁻¹, wall time: 40.511 seconds
[ Info: i: 1860, t: 3.689 hours, Δt: 5.879 seconds, umax = (1.1e-01, 8.2e-02, 4.7e-02) m s⁻¹, wall time: 40.634 seconds
[ Info: i: 1880, t: 3.719 hours, Δt: 5.054 seconds, umax = (1.2e-01, 8.7e-02, 4.9e-02) m s⁻¹, wall time: 40.734 seconds
[ Info: i: 1900, t: 3.748 hours, Δt: 5.861 seconds, umax = (1.1e-01, 8.1e-02, 5.2e-02) m s⁻¹, wall time: 40.843 seconds
[ Info: i: 1920, t: 3.779 hours, Δt: 5.046 seconds, umax = (1.1e-01, 8.5e-02, 5.0e-02) m s⁻¹, wall time: 41.000 seconds
[ Info: i: 1940, t: 3.807 hours, Δt: 5.203 seconds, umax = (1.1e-01, 9.0e-02, 5.5e-02) m s⁻¹, wall time: 41.104 seconds
[ Info: i: 1960, t: 3.836 hours, Δt: 5.359 seconds, umax = (1.1e-01, 9.1e-02, 5.3e-02) m s⁻¹, wall time: 41.249 seconds
[ Info: i: 1980, t: 3.867 hours, Δt: 5.792 seconds, umax = (1.1e-01, 8.2e-02, 5.0e-02) m s⁻¹, wall time: 41.327 seconds
[ Info: i: 2000, t: 3.899 hours, Δt: 5.564 seconds, umax = (1.2e-01, 7.4e-02, 4.9e-02) m s⁻¹, wall time: 41.435 seconds
[ Info: i: 2020, t: 3.930 hours, Δt: 5.466 seconds, umax = (1.2e-01, 8.2e-02, 5.4e-02) m s⁻¹, wall time: 41.563 seconds
[ Info: i: 2040, t: 3.961 hours, Δt: 5.690 seconds, umax = (1.2e-01, 7.9e-02, 4.8e-02) m s⁻¹, wall time: 41.669 seconds
[ Info: i: 2060, t: 3.993 hours, Δt: 5.891 seconds, umax = (1.1e-01, 8.1e-02, 4.8e-02) m s⁻¹, wall time: 41.776 seconds
[ Info: Simulation is stopping after running for 41.814 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-35915/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-35915/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-35915/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-35915/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-35915/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-35915/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-35915/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-35915/docs/
  JULIA_DEBUG = Literate

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

import Pkg
Pkg.status()

Status `~/Oceananigans.jl-35915/docs/Project.toml`
  [79e6a3ab] Adapt v4.7.4
⌃ [052768ef] CUDA v6.2.2
  [13f3f980] CairoMakie v0.15.17
  [8e50ac2c] ConservativeRegridding v0.2.20
  [e30172f5] Documenter v1.19.0
  [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 v0.2.292
  [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.