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))

128×128×64 RectilinearGrid{Float32, 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 , 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-34931/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-34931/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 = (1.5e-03, 8.4e-04, 1.5e-03) m s⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (36.528 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.415 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 19.143 seconds, umax = (3.5e-02, 1.2e-02, 2.2e-02) m s⁻¹, wall time: 39.740 seconds
[ Info: i: 0040, t: 16.838 minutes, Δt: 12.580 seconds, umax = (5.4e-02, 2.1e-02, 2.1e-02) m s⁻¹, wall time: 40.126 seconds
[ Info: i: 0060, t: 20.765 minutes, Δt: 10.802 seconds, umax = (6.2e-02, 2.6e-02, 3.3e-02) m s⁻¹, wall time: 40.390 seconds
[ Info: i: 0080, t: 24.420 minutes, Δt: 10.859 seconds, umax = (6.8e-02, 3.4e-02, 3.0e-02) m s⁻¹, wall time: 40.574 seconds
[ Info: i: 0100, t: 27.901 minutes, Δt: 10.694 seconds, umax = (6.5e-02, 3.1e-02, 3.4e-02) m s⁻¹, wall time: 40.812 seconds
[ Info: i: 0120, t: 31.468 minutes, Δt: 11.305 seconds, umax = (6.4e-02, 3.5e-02, 3.3e-02) m s⁻¹, wall time: 41.056 seconds
[ Info: i: 0140, t: 35 minutes, Δt: 10.955 seconds, umax = (6.9e-02, 3.2e-02, 2.7e-02) m s⁻¹, wall time: 41.268 seconds
[ Info: i: 0160, t: 38.587 minutes, Δt: 8.904 seconds, umax = (7.1e-02, 4.0e-02, 3.0e-02) m s⁻¹, wall time: 41.512 seconds
[ Info: i: 0180, t: 41.632 minutes, Δt: 9.801 seconds, umax = (7.0e-02, 3.7e-02, 3.9e-02) m s⁻¹, wall time: 41.843 seconds
[ Info: i: 0200, t: 44.845 minutes, Δt: 9.630 seconds, umax = (7.2e-02, 4.3e-02, 3.4e-02) m s⁻¹, wall time: 42.052 seconds
[ Info: i: 0220, t: 48.052 minutes, Δt: 9.422 seconds, umax = (7.6e-02, 3.8e-02, 3.6e-02) m s⁻¹, wall time: 42.286 seconds
[ Info: i: 0240, t: 51.120 minutes, Δt: 9.026 seconds, umax = (7.3e-02, 4.3e-02, 3.6e-02) m s⁻¹, wall time: 42.526 seconds
[ Info: i: 0260, t: 54.102 minutes, Δt: 8.382 seconds, umax = (8.4e-02, 4.1e-02, 3.8e-02) m s⁻¹, wall time: 42.719 seconds
[ Info: i: 0280, t: 56.906 minutes, Δt: 8.267 seconds, umax = (7.6e-02, 4.3e-02, 3.7e-02) m s⁻¹, wall time: 42.937 seconds
[ Info: i: 0300, t: 59.779 minutes, Δt: 8.814 seconds, umax = (8.6e-02, 4.6e-02, 4.4e-02) m s⁻¹, wall time: 43.152 seconds
[ Info: i: 0320, t: 1.043 hours, Δt: 8.291 seconds, umax = (8.6e-02, 4.4e-02, 3.7e-02) m s⁻¹, wall time: 43.389 seconds
[ Info: i: 0340, t: 1.085 hours, Δt: 7.786 seconds, umax = (8.5e-02, 4.6e-02, 3.6e-02) m s⁻¹, wall time: 43.703 seconds
[ Info: i: 0360, t: 1.130 hours, Δt: 8.149 seconds, umax = (8.4e-02, 4.2e-02, 3.9e-02) m s⁻¹, wall time: 43.848 seconds
[ Info: i: 0380, t: 1.174 hours, Δt: 8.026 seconds, umax = (8.9e-02, 4.7e-02, 4.0e-02) m s⁻¹, wall time: 44.134 seconds
[ Info: i: 0400, t: 1.217 hours, Δt: 7.800 seconds, umax = (8.9e-02, 4.8e-02, 3.9e-02) m s⁻¹, wall time: 44.303 seconds
[ Info: i: 0420, t: 1.259 hours, Δt: 7.916 seconds, umax = (8.5e-02, 4.8e-02, 4.0e-02) m s⁻¹, wall time: 44.584 seconds
[ Info: i: 0440, t: 1.302 hours, Δt: 7.509 seconds, umax = (8.8e-02, 5.0e-02, 4.0e-02) m s⁻¹, wall time: 44.760 seconds
[ Info: i: 0460, t: 1.342 hours, Δt: 7.542 seconds, umax = (9.1e-02, 5.1e-02, 3.9e-02) m s⁻¹, wall time: 45.051 seconds
[ Info: i: 0480, t: 1.384 hours, Δt: 7.707 seconds, umax = (8.8e-02, 5.2e-02, 4.3e-02) m s⁻¹, wall time: 45.228 seconds
[ Info: i: 0500, t: 1.425 hours, Δt: 7.760 seconds, umax = (9.3e-02, 5.3e-02, 3.7e-02) m s⁻¹, wall time: 45.506 seconds
[ Info: i: 0520, t: 1.467 hours, Δt: 7.283 seconds, umax = (9.3e-02, 5.4e-02, 4.0e-02) m s⁻¹, wall time: 45.685 seconds
[ Info: i: 0540, t: 1.506 hours, Δt: 7.435 seconds, umax = (9.4e-02, 5.5e-02, 4.2e-02) m s⁻¹, wall time: 45.982 seconds
[ Info: i: 0560, t: 1.546 hours, Δt: 7.388 seconds, umax = (9.1e-02, 5.4e-02, 4.3e-02) m s⁻¹, wall time: 46.148 seconds
[ Info: i: 0580, t: 1.585 hours, Δt: 6.783 seconds, umax = (9.7e-02, 5.5e-02, 4.5e-02) m s⁻¹, wall time: 46.471 seconds
[ Info: i: 0600, t: 1.625 hours, Δt: 7.551 seconds, umax = (9.8e-02, 5.5e-02, 4.0e-02) m s⁻¹, wall time: 46.617 seconds
[ Info: i: 0620, t: 1.667 hours, Δt: 7.152 seconds, umax = (9.9e-02, 5.1e-02, 4.0e-02) m s⁻¹, wall time: 46.829 seconds
[ Info: i: 0640, t: 1.706 hours, Δt: 6.960 seconds, umax = (9.8e-02, 5.9e-02, 3.9e-02) m s⁻¹, wall time: 47.059 seconds
[ Info: i: 0660, t: 1.745 hours, Δt: 7.444 seconds, umax = (9.7e-02, 5.4e-02, 3.9e-02) m s⁻¹, wall time: 47.273 seconds
[ Info: i: 0680, t: 1.784 hours, Δt: 7.335 seconds, umax = (9.6e-02, 5.6e-02, 3.9e-02) m s⁻¹, wall time: 47.514 seconds
[ Info: i: 0700, t: 1.825 hours, Δt: 6.649 seconds, umax = (9.7e-02, 5.7e-02, 4.4e-02) m s⁻¹, wall time: 47.727 seconds
[ Info: i: 0720, t: 1.861 hours, Δt: 7.043 seconds, umax = (9.7e-02, 5.6e-02, 4.1e-02) m s⁻¹, wall time: 47.978 seconds
[ Info: i: 0740, t: 1.900 hours, Δt: 7.050 seconds, umax = (9.7e-02, 5.9e-02, 4.4e-02) m s⁻¹, wall time: 48.190 seconds
[ Info: i: 0760, t: 1.939 hours, Δt: 7.254 seconds, umax = (1.1e-01, 5.5e-02, 4.0e-02) m s⁻¹, wall time: 48.419 seconds
[ Info: i: 0780, t: 1.978 hours, Δt: 6.607 seconds, umax = (9.9e-02, 5.9e-02, 3.9e-02) m s⁻¹, wall time: 48.627 seconds
[ Info: i: 0800, t: 2.014 hours, Δt: 6.772 seconds, umax = (1.0e-01, 5.9e-02, 4.0e-02) m s⁻¹, wall time: 48.870 seconds
[ Info: i: 0820, t: 2.052 hours, Δt: 6.839 seconds, umax = (9.7e-02, 6.0e-02, 4.0e-02) m s⁻¹, wall time: 49.080 seconds
[ Info: i: 0840, t: 2.089 hours, Δt: 6.533 seconds, umax = (9.9e-02, 6.8e-02, 4.2e-02) m s⁻¹, wall time: 49.367 seconds
[ Info: i: 0860, t: 2.125 hours, Δt: 6.923 seconds, umax = (1.0e-01, 6.5e-02, 4.8e-02) m s⁻¹, wall time: 49.530 seconds
[ Info: i: 0880, t: 2.164 hours, Δt: 6.557 seconds, umax = (9.9e-02, 6.2e-02, 4.7e-02) m s⁻¹, wall time: 49.745 seconds
[ Info: i: 0900, t: 2.200 hours, Δt: 6.574 seconds, umax = (9.8e-02, 6.7e-02, 4.6e-02) m s⁻¹, wall time: 49.982 seconds
[ Info: i: 0920, t: 2.237 hours, Δt: 6.765 seconds, umax = (1.0e-01, 6.0e-02, 4.1e-02) m s⁻¹, wall time: 50.196 seconds
[ Info: i: 0940, t: 2.272 hours, Δt: 6.408 seconds, umax = (1.0e-01, 6.1e-02, 4.4e-02) m s⁻¹, wall time: 50.418 seconds
[ Info: i: 0960, t: 2.308 hours, Δt: 6.818 seconds, umax = (1.0e-01, 6.5e-02, 4.4e-02) m s⁻¹, wall time: 50.626 seconds
[ Info: i: 0980, t: 2.344 hours, Δt: 6.477 seconds, umax = (1.0e-01, 6.3e-02, 4.9e-02) m s⁻¹, wall time: 51.050 seconds
[ Info: i: 1000, t: 2.380 hours, Δt: 6.487 seconds, umax = (1.0e-01, 7.0e-02, 5.0e-02) m s⁻¹, wall time: 51.241 seconds
[ Info: i: 1020, t: 2.416 hours, Δt: 5.985 seconds, umax = (1.1e-01, 6.6e-02, 4.3e-02) m s⁻¹, wall time: 51.456 seconds
[ Info: i: 1040, t: 2.448 hours, Δt: 6.074 seconds, umax = (1.1e-01, 6.5e-02, 4.5e-02) m s⁻¹, wall time: 51.683 seconds
[ Info: i: 1060, t: 2.481 hours, Δt: 6.158 seconds, umax = (1.1e-01, 7.0e-02, 4.3e-02) m s⁻¹, wall time: 51.890 seconds
[ Info: i: 1080, t: 2.516 hours, Δt: 6.590 seconds, umax = (1.1e-01, 6.3e-02, 4.1e-02) m s⁻¹, wall time: 52.113 seconds
[ Info: i: 1100, t: 2.551 hours, Δt: 6.350 seconds, umax = (1.1e-01, 6.8e-02, 4.4e-02) m s⁻¹, wall time: 52.316 seconds
[ Info: i: 1120, t: 2.585 hours, Δt: 6.361 seconds, umax = (1.0e-01, 6.7e-02, 4.6e-02) m s⁻¹, wall time: 53.830 seconds
[ Info: i: 1140, t: 2.620 hours, Δt: 6.608 seconds, umax = (1.1e-01, 6.9e-02, 4.7e-02) m s⁻¹, wall time: 53.964 seconds
[ Info: i: 1160, t: 2.656 hours, Δt: 6.168 seconds, umax = (1.1e-01, 8.1e-02, 4.6e-02) m s⁻¹, wall time: 54.168 seconds
[ Info: i: 1180, t: 2.688 hours, Δt: 5.977 seconds, umax = (1.1e-01, 7.6e-02, 4.7e-02) m s⁻¹, wall time: 54.399 seconds
[ Info: i: 1200, t: 2.722 hours, Δt: 5.406 seconds, umax = (1.1e-01, 7.3e-02, 5.3e-02) m s⁻¹, wall time: 54.614 seconds
[ Info: i: 1220, t: 2.752 hours, Δt: 5.702 seconds, umax = (1.1e-01, 7.0e-02, 4.6e-02) m s⁻¹, wall time: 54.905 seconds
[ Info: i: 1240, t: 2.783 hours, Δt: 5.919 seconds, umax = (1.0e-01, 7.1e-02, 4.3e-02) m s⁻¹, wall time: 55.044 seconds
[ Info: i: 1260, t: 2.816 hours, Δt: 6.184 seconds, umax = (1.1e-01, 7.8e-02, 4.4e-02) m s⁻¹, wall time: 55.243 seconds
[ Info: i: 1280, t: 2.849 hours, Δt: 6.354 seconds, umax = (1.2e-01, 6.9e-02, 4.8e-02) m s⁻¹, wall time: 55.471 seconds
[ Info: i: 1300, t: 2.884 hours, Δt: 6.313 seconds, umax = (1.1e-01, 7.4e-02, 4.6e-02) m s⁻¹, wall time: 55.678 seconds
[ Info: i: 1320, t: 2.918 hours, Δt: 6.517 seconds, umax = (1.1e-01, 6.7e-02, 4.7e-02) m s⁻¹, wall time: 56.000 seconds
[ Info: i: 1340, t: 2.954 hours, Δt: 6.018 seconds, umax = (1.1e-01, 7.3e-02, 4.8e-02) m s⁻¹, wall time: 56.137 seconds
[ Info: i: 1360, t: 2.987 hours, Δt: 5.998 seconds, umax = (1.1e-01, 6.7e-02, 4.7e-02) m s⁻¹, wall time: 56.365 seconds
[ Info: i: 1380, t: 3.020 hours, Δt: 6.310 seconds, umax = (1.1e-01, 6.9e-02, 4.7e-02) m s⁻¹, wall time: 56.600 seconds
[ Info: i: 1400, t: 3.055 hours, Δt: 5.954 seconds, umax = (1.1e-01, 7.8e-02, 4.7e-02) m s⁻¹, wall time: 56.807 seconds
[ Info: i: 1420, t: 3.087 hours, Δt: 5.809 seconds, umax = (1.1e-01, 8.2e-02, 5.1e-02) m s⁻¹, wall time: 57.107 seconds
[ Info: i: 1440, t: 3.119 hours, Δt: 6.094 seconds, umax = (1.1e-01, 7.2e-02, 4.9e-02) m s⁻¹, wall time: 57.239 seconds
[ Info: i: 1460, t: 3.153 hours, Δt: 5.591 seconds, umax = (1.0e-01, 7.0e-02, 4.6e-02) m s⁻¹, wall time: 57.456 seconds
[ Info: i: 1480, t: 3.184 hours, Δt: 5.715 seconds, umax = (1.0e-01, 7.0e-02, 5.4e-02) m s⁻¹, wall time: 57.726 seconds
[ Info: i: 1500, t: 3.216 hours, Δt: 6.253 seconds, umax = (1.1e-01, 7.4e-02, 5.5e-02) m s⁻¹, wall time: 57.944 seconds
[ Info: i: 1520, t: 3.250 hours, Δt: 5.962 seconds, umax = (1.2e-01, 7.6e-02, 4.7e-02) m s⁻¹, wall time: 58.179 seconds
[ Info: i: 1540, t: 3.283 hours, Δt: 6.120 seconds, umax = (1.1e-01, 7.4e-02, 4.6e-02) m s⁻¹, wall time: 58.483 seconds
[ Info: i: 1560, t: 3.317 hours, Δt: 6.092 seconds, umax = (1.1e-01, 7.1e-02, 4.5e-02) m s⁻¹, wall time: 58.694 seconds
[ Info: i: 1580, t: 3.350 hours, Δt: 6.142 seconds, umax = (1.1e-01, 7.5e-02, 4.7e-02) m s⁻¹, wall time: 58.940 seconds
[ Info: i: 1600, t: 3.385 hours, Δt: 6.050 seconds, umax = (1.1e-01, 7.1e-02, 4.8e-02) m s⁻¹, wall time: 59.144 seconds
[ Info: i: 1620, t: 3.418 hours, Δt: 5.961 seconds, umax = (1.1e-01, 7.4e-02, 5.0e-02) m s⁻¹, wall time: 59.444 seconds
[ Info: i: 1640, t: 3.452 hours, Δt: 6.212 seconds, umax = (1.1e-01, 7.0e-02, 4.3e-02) m s⁻¹, wall time: 59.589 seconds
[ Info: i: 1660, t: 3.486 hours, Δt: 6.103 seconds, umax = (1.2e-01, 7.1e-02, 4.6e-02) m s⁻¹, wall time: 59.801 seconds
[ Info: i: 1680, t: 3.518 hours, Δt: 6.060 seconds, umax = (1.0e-01, 7.9e-02, 4.4e-02) m s⁻¹, wall time: 1.002 minutes
[ Info: i: 1700, t: 3.552 hours, Δt: 6.115 seconds, umax = (1.1e-01, 7.4e-02, 5.0e-02) m s⁻¹, wall time: 1.005 minutes
[ Info: i: 1720, t: 3.585 hours, Δt: 6.122 seconds, umax = (1.1e-01, 7.8e-02, 4.9e-02) m s⁻¹, wall time: 1.010 minutes
[ Info: i: 1740, t: 3.619 hours, Δt: 5.891 seconds, umax = (1.1e-01, 7.3e-02, 4.8e-02) m s⁻¹, wall time: 1.014 minutes
[ Info: i: 1760, t: 3.652 hours, Δt: 6.035 seconds, umax = (1.1e-01, 7.3e-02, 5.1e-02) m s⁻¹, wall time: 1.018 minutes
[ Info: i: 1780, t: 3.685 hours, Δt: 6.010 seconds, umax = (1.1e-01, 7.5e-02, 5.4e-02) m s⁻¹, wall time: 1.022 minutes
[ Info: i: 1800, t: 3.720 hours, Δt: 5.866 seconds, umax = (1.1e-01, 7.6e-02, 5.4e-02) m s⁻¹, wall time: 1.025 minutes
[ Info: i: 1820, t: 3.752 hours, Δt: 6.166 seconds, umax = (1.2e-01, 7.5e-02, 5.2e-02) m s⁻¹, wall time: 1.030 minutes
[ Info: i: 1840, t: 3.784 hours, Δt: 5.621 seconds, umax = (1.0e-01, 7.0e-02, 5.2e-02) m s⁻¹, wall time: 1.032 minutes
[ Info: i: 1860, t: 3.815 hours, Δt: 5.515 seconds, umax = (1.1e-01, 7.5e-02, 5.0e-02) m s⁻¹, wall time: 1.036 minutes
[ Info: i: 1880, t: 3.846 hours, Δt: 5.581 seconds, umax = (1.1e-01, 7.8e-02, 5.2e-02) m s⁻¹, wall time: 1.040 minutes
[ Info: i: 1900, t: 3.876 hours, Δt: 5.729 seconds, umax = (1.1e-01, 7.7e-02, 5.5e-02) m s⁻¹, wall time: 1.043 minutes
[ Info: i: 1920, t: 3.909 hours, Δt: 5.959 seconds, umax = (1.1e-01, 6.6e-02, 5.0e-02) m s⁻¹, wall time: 1.047 minutes
[ Info: i: 1940, t: 3.942 hours, Δt: 6.009 seconds, umax = (1.1e-01, 7.1e-02, 4.7e-02) m s⁻¹, wall time: 1.052 minutes
[ Info: i: 1960, t: 3.975 hours, Δt: 5.816 seconds, umax = (1.1e-01, 7.3e-02, 4.4e-02) m s⁻¹, wall time: 1.055 minutes
[ Info: Simulation is stopping after running for 1.058 minutes.
[ 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-34931/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-34931/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-34931/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-34931/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-34931/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-34931/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-34931/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 1.13.0
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 = 1.13.0
  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-34931/docs/
  JULIA_DEBUG = Literate

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

import Pkg
Pkg.status()

Status `~/Oceananigans.jl-34931/docs/Project.toml`
  [79e6a3ab] Adapt v4.7.4
⌃ [052768ef] CUDA v6.2.2
  [13f3f980] CairoMakie v0.15.16
  [8e50ac2c] ConservativeRegridding v0.2.16
⌅ [e30172f5] Documenter v1.17.0
  [daee34ce] DocumenterCitations v1.5.0
  [4710194d] DocumenterVitepress v0.3.8
  [7da242da] Enzyme v0.13.216
  [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 v0.114.0 `..`
  [429524aa] Optim v2.3.2
  [f27b6e38] Polynomials v4.1.3
⌅ [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.