Langmuir turbulence example

This example implements a Langmuir turbulence simulation similar to the one reported in section 4 of

• Wagner, G. L.; Chini, G. P.; Ramadhan, A.; Gallet, B. and Ferrari, R. (2021). Near-inertial waves and turbulence driven by the growth of swell. Journal of Physical Oceanography 51, 1337–1351.

This example demonstrates

• How to run large eddy simulations with surface wave effects via the Craik-Leibovich approximation.

• How to specify time- and horizontally-averaged output.

Install dependencies

First let's make sure we have all required packages installed.

using Pkg
pkg"add Oceananigans, CairoMakie, CUDA"

using Oceananigans
using Oceananigans.Units: minute, minutes, hours
using CUDA
using Random

Random.seed!(1337) # for reproducible results

Random.TaskLocalRNG()

Model set-up

To build the model, we specify the grid, Stokes drift, boundary conditions, and Coriolis parameter.

Domain and numerical grid specification

We use a modest resolution and the same total extent as Wagner et al. (2021),

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

128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CUDAGPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 128.0) regularly spaced with Δx=1.0
├── Periodic y ∈ [0.0, 128.0) regularly spaced with Δy=1.0
└── Bounded  z ∈ [-64.0, 0.0] regularly spaced with Δz=1.0

The Stokes Drift profile

The surface wave Stokes drift profile prescribed by Wagner et al. (2021), corresponds to a 'monochromatic' (that is, single-frequency) wave field.

A monochromatic wave field is characterized by its wavelength and amplitude (half the distance from wave crest to wave trough), which determine the wave frequency and the vertical scale of the Stokes drift profile.

g = Oceananigans.defaults.gravitational_acceleration

amplitude = 0.8 # m
wavelength = 60  # m
wavenumber = 2π / wavelength # m⁻¹
frequency = sqrt(g * wavenumber) # s⁻¹

# The vertical scale over which the Stokes drift of a monochromatic surface wave
# decays away from the surface is `1/2wavenumber`, or
const vertical_scale = wavelength / 4π

# Stokes drift velocity at the surface
const Uˢ = amplitude^2 * wavenumber * frequency # m s⁻¹

0.06791774197745354

The const declarations ensure that Stokes drift functions compile on the GPU. To run this example on the CPU, replace GPU() with CPU() in the RectilinearGrid constructor above.

The Stokes drift profile is

uˢ(z) = Uˢ * exp(z / vertical_scale)

uˢ (generic function with 1 method)

and its z-derivative is

∂z_uˢ(z, t) = 1 / vertical_scale * Uˢ * exp(z / vertical_scale)

∂z_uˢ (generic function with 1 method)

The Craik-Leibovich equations in Oceananigans

Oceananigans implements the Craik-Leibovich approximation for surface wave effects using the Lagrangian-mean velocity field as its prognostic momentum variable. In other words, model.velocities.u is the Lagrangian-mean -velocity beneath surface waves. This differs from models that use the Eulerian-mean velocity field as a prognostic variable, but has the advantage that accounts for the total advection of tracers and momentum, and that     is a steady solution even when Coriolis forces are present. See the physics documentation for more information.

Finally, we note that the time-derivative of the Stokes drift must be provided if the Stokes drift and surface wave field undergoes forced changes in time. In this example, the Stokes drift is constant and thus the time-derivative of the Stokes drift is 0.

Boundary conditions

At the surface  , Wagner et al. (2021) impose

τx = -3.72e-5 # m² s⁻², surface kinematic momentum flux
u_boundary_conditions = FieldBoundaryConditions(top = FluxBoundaryCondition(τx))

Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── top: FluxBoundaryCondition: -3.72e-5
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)

Wagner et al. (2021) impose a linear buoyancy gradient N² at the bottom along with a weak, destabilizing flux of buoyancy at the surface to faciliate spin-up from rest.

Jᵇ = 2.307e-8 # m² s⁻³, surface buoyancy flux
N² = 1.936e-5 # s⁻², initial and bottom buoyancy gradient

b_boundary_conditions = FieldBoundaryConditions(top = FluxBoundaryCondition(Jᵇ),
                                                bottom = GradientBoundaryCondition(N²))

Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: GradientBoundaryCondition: 1.936e-5
├── top: FluxBoundaryCondition: 2.307e-8
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)

The flux convention in Oceananigans

Note that Oceananigans uses "positive upward" conventions for all fluxes. In consequence, a negative flux at the surface drives positive velocities, and a positive flux of buoyancy drives cooling.

Coriolis parameter

Wagner et al. (2021) use

coriolis = FPlane(f=1e-4) # s⁻¹

FPlane{Float64}(f=0.0001)

which is typical for mid-latitudes on Earth.

Model instantiation

We are ready to build the model. We use a fifth-order Weighted Essentially Non-Oscillatory (WENO) advection scheme and the AnisotropicMinimumDissipation model for large eddy simulation. Because our Stokes drift does not vary in , we use UniformStokesDrift, which expects Stokes drift functions of only.

model = NonhydrostaticModel(grid; coriolis,
                            advection = WENO(order=9),
                            tracers = :b,
                            buoyancy = BuoyancyTracer(),
                            stokes_drift = UniformStokesDrift(∂z_uˢ=∂z_uˢ),
                            boundary_conditions = (u=u_boundary_conditions, b=b_boundary_conditions))

NonhydrostaticModel{CUDAGPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CUDAGPU with 5×5×5 halo
├── timestepper: RungeKutta3TimeStepper
├── advection scheme: WENO{5, Float64, Float32}(order=9)
├── tracers: b
├── closure: Nothing
├── buoyancy: BuoyancyTracer with ĝ = NegativeZDirection()
└── coriolis: FPlane{Float64}(f=0.0001)

Initial conditions

We make use of random noise concentrated in the upper 4 meters for buoyancy and velocity initial conditions,

Ξ(z) = randn() * exp(z / 4)

Our initial condition for buoyancy consists of a surface mixed layer 33 m deep, a deep linear stratification, plus noise,

initial_mixed_layer_depth = 33 # m
stratification(z) = z < - initial_mixed_layer_depth ? N² * z : N² * (-initial_mixed_layer_depth)

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

bᵢ (generic function with 1 method)

The simulation we reproduce from Wagner et al. (2021) is zero Lagrangian-mean velocity. This initial condition is consistent with a wavy, quiescent ocean suddenly impacted by winds. To this quiescent state we add noise scaled by the friction velocity to and .

u★ = sqrt(abs(τx))
uᵢ(x, y, z) = u★ * 1e-1 * Ξ(z)
wᵢ(x, y, z) = u★ * 1e-1 * Ξ(z)

set!(model, u=uᵢ, w=wᵢ, b=bᵢ)

Setting up the simulation

simulation = Simulation(model, Δt=45.0, stop_time=4hours)

Simulation of NonhydrostaticModel{CUDAGPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── Next time step: 45 seconds
├── run_wall_time: 0 seconds
├── run_wall_time / iteration: NaN days
├── stop_time: 4 hours
├── stop_iteration: Inf
├── wall_time_limit: Inf
├── minimum_relative_step: 0.0
├── callbacks: OrderedDict with 4 entries:
│   ├── stop_time_exceeded => Callback of stop_time_exceeded on IterationInterval(1)
│   ├── stop_iteration_exceeded => Callback of stop_iteration_exceeded on IterationInterval(1)
│   ├── wall_time_limit_exceeded => Callback of wall_time_limit_exceeded on IterationInterval(1)
│   └── nan_checker => Callback of NaNChecker for u on IterationInterval(100)
└── output_writers: OrderedDict with no entries

We use the TimeStepWizard for adaptive time-stepping with a Courant-Freidrichs-Lewy (CFL) number of 1.0,

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

Nice progress messaging

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

using Printf

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

    # Print a progress message
    msg = @sprintf("i: %04d, t: %s, Δt: %s, umax = (%.1e, %.1e, %.1e) ms⁻¹, wall time: %s\n",
                   iteration(simulation),
                   prettytime(time(simulation)),
                   prettytime(simulation.Δt),
                   maximum(abs, u), maximum(abs, v), maximum(abs, w),
                   prettytime(simulation.run_wall_time))

    @info msg

    return nothing
end

simulation.callbacks[:progress] = Callback(progress, IterationInterval(20))

Callback of progress on IterationInterval(20)

Output

A field writer

We set up an output writer for the simulation that saves all velocity fields, tracer fields, and the subgrid turbulent diffusivity.

output_interval = 5minutes

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

simulation.output_writers[:fields] =
    JLD2Writer(model, fields_to_output,
               schedule = TimeInterval(output_interval),
               filename = "langmuir_turbulence_fields.jld2",
               overwrite_existing = true)

JLD2Writer scheduled on TimeInterval(5 minutes):
├── filepath: langmuir_turbulence_fields.jld2
├── 4 outputs: (u, v, w, b)
├── array_type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 0 bytes (file not yet created)

An "averages" writer

We also set up output of time- and horizontally-averaged velocity field and momentum fluxes.

u, v, w = model.velocities
b = model.tracers.b

 U = Average(u, dims=(1, 2))
 V = Average(v, dims=(1, 2))
 B = Average(b, dims=(1, 2))
wu = Average(w * u, dims=(1, 2))
wv = Average(w * v, dims=(1, 2))

simulation.output_writers[:averages] =
    JLD2Writer(model, (; U, V, B, wu, wv),
               schedule = AveragedTimeInterval(output_interval, window=2minutes),
               filename = "langmuir_turbulence_averages.jld2",
               overwrite_existing = true)

JLD2Writer scheduled on TimeInterval(5 minutes):
├── filepath: langmuir_turbulence_averages.jld2
├── 5 outputs: (U, V, B, wu, wv) averaged on AveragedTimeInterval(window=2 minutes, stride=1, interval=5 minutes)
├── array_type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 0 bytes (file not yet created)

Running the simulation

This part is easy,

run!(simulation)

[ Info: Initializing simulation...
[ Info: i: 0000, t: 0 seconds, Δt: 49.500 seconds, umax = (1.8e-03, 9.5e-04, 1.5e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (9.888 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.908 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 19.470 seconds, umax = (3.6e-02, 1.2e-02, 2.1e-02) ms⁻¹, wall time: 14.027 seconds
[ Info: i: 0040, t: 17.083 minutes, Δt: 12.980 seconds, umax = (5.3e-02, 2.1e-02, 2.5e-02) ms⁻¹, wall time: 14.636 seconds
[ Info: i: 0060, t: 21.181 minutes, Δt: 10.514 seconds, umax = (6.4e-02, 2.9e-02, 3.2e-02) ms⁻¹, wall time: 15.382 seconds
[ Info: i: 0080, t: 24.649 minutes, Δt: 10.835 seconds, umax = (6.4e-02, 3.1e-02, 3.4e-02) ms⁻¹, wall time: 16.056 seconds
[ Info: i: 0100, t: 28.387 minutes, Δt: 11.357 seconds, umax = (6.1e-02, 3.0e-02, 3.0e-02) ms⁻¹, wall time: 16.745 seconds
[ Info: i: 0120, t: 32.015 minutes, Δt: 11.254 seconds, umax = (6.1e-02, 2.9e-02, 2.8e-02) ms⁻¹, wall time: 17.457 seconds
[ Info: i: 0140, t: 35.548 minutes, Δt: 10.870 seconds, umax = (6.6e-02, 3.4e-02, 3.0e-02) ms⁻¹, wall time: 18.291 seconds
[ Info: i: 0160, t: 39.055 minutes, Δt: 10.051 seconds, umax = (6.9e-02, 3.7e-02, 3.0e-02) ms⁻¹, wall time: 18.862 seconds
[ Info: i: 0180, t: 42.196 minutes, Δt: 9.376 seconds, umax = (7.2e-02, 3.6e-02, 3.4e-02) ms⁻¹, wall time: 19.565 seconds
[ Info: i: 0200, t: 45.154 minutes, Δt: 9.332 seconds, umax = (7.0e-02, 3.7e-02, 3.3e-02) ms⁻¹, wall time: 20.456 seconds
[ Info: i: 0220, t: 48.263 minutes, Δt: 8.574 seconds, umax = (7.5e-02, 4.3e-02, 3.5e-02) ms⁻¹, wall time: 20.806 seconds
[ Info: i: 0240, t: 51.238 minutes, Δt: 8.667 seconds, umax = (7.5e-02, 4.1e-02, 3.8e-02) ms⁻¹, wall time: 21.323 seconds
[ Info: i: 0260, t: 54.183 minutes, Δt: 8.782 seconds, umax = (7.8e-02, 3.9e-02, 3.6e-02) ms⁻¹, wall time: 21.830 seconds
[ Info: i: 0280, t: 57.056 minutes, Δt: 8.470 seconds, umax = (7.8e-02, 4.6e-02, 3.7e-02) ms⁻¹, wall time: 22.537 seconds
[ Info: i: 0300, t: 59.841 minutes, Δt: 8.423 seconds, umax = (8.2e-02, 4.1e-02, 4.1e-02) ms⁻¹, wall time: 23.289 seconds
[ Info: i: 0320, t: 1.042 hours, Δt: 7.873 seconds, umax = (8.9e-02, 4.4e-02, 3.6e-02) ms⁻¹, wall time: 24.156 seconds
[ Info: i: 0340, t: 1.086 hours, Δt: 8.292 seconds, umax = (8.2e-02, 4.7e-02, 3.8e-02) ms⁻¹, wall time: 25.115 seconds
[ Info: i: 0360, t: 1.132 hours, Δt: 8.269 seconds, umax = (8.2e-02, 4.3e-02, 4.0e-02) ms⁻¹, wall time: 25.615 seconds
[ Info: i: 0380, t: 1.179 hours, Δt: 8.177 seconds, umax = (8.7e-02, 4.6e-02, 3.6e-02) ms⁻¹, wall time: 26.413 seconds
[ Info: i: 0400, t: 1.224 hours, Δt: 7.848 seconds, umax = (8.3e-02, 4.8e-02, 3.6e-02) ms⁻¹, wall time: 27.052 seconds
[ Info: i: 0420, t: 1.267 hours, Δt: 7.613 seconds, umax = (8.8e-02, 5.3e-02, 4.3e-02) ms⁻¹, wall time: 27.789 seconds
[ Info: i: 0440, t: 1.310 hours, Δt: 7.647 seconds, umax = (9.1e-02, 5.1e-02, 3.9e-02) ms⁻¹, wall time: 28.499 seconds
[ Info: i: 0460, t: 1.350 hours, Δt: 7.444 seconds, umax = (9.2e-02, 5.5e-02, 3.8e-02) ms⁻¹, wall time: 29.226 seconds
[ Info: i: 0480, t: 1.392 hours, Δt: 7.478 seconds, umax = (9.3e-02, 5.2e-02, 4.1e-02) ms⁻¹, wall time: 29.950 seconds
[ Info: i: 0500, t: 1.433 hours, Δt: 7.334 seconds, umax = (9.3e-02, 5.3e-02, 4.0e-02) ms⁻¹, wall time: 30.700 seconds
[ Info: i: 0520, t: 1.474 hours, Δt: 7.630 seconds, umax = (9.0e-02, 5.6e-02, 3.8e-02) ms⁻¹, wall time: 31.411 seconds
[ Info: i: 0540, t: 1.515 hours, Δt: 7.617 seconds, umax = (9.1e-02, 5.6e-02, 4.6e-02) ms⁻¹, wall time: 31.965 seconds
[ Info: i: 0560, t: 1.555 hours, Δt: 7.351 seconds, umax = (9.2e-02, 5.5e-02, 4.1e-02) ms⁻¹, wall time: 32.473 seconds
[ Info: i: 0580, t: 1.596 hours, Δt: 7.263 seconds, umax = (9.8e-02, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 33.041 seconds
[ Info: i: 0600, t: 1.636 hours, Δt: 7.465 seconds, umax = (9.6e-02, 5.5e-02, 4.1e-02) ms⁻¹, wall time: 33.532 seconds
[ Info: i: 0620, t: 1.675 hours, Δt: 6.749 seconds, umax = (1.0e-01, 5.6e-02, 4.2e-02) ms⁻¹, wall time: 34.155 seconds
[ Info: i: 0640, t: 1.712 hours, Δt: 7.015 seconds, umax = (9.8e-02, 5.3e-02, 4.2e-02) ms⁻¹, wall time: 34.604 seconds
[ Info: i: 0660, t: 1.750 hours, Δt: 6.605 seconds, umax = (9.8e-02, 5.4e-02, 4.4e-02) ms⁻¹, wall time: 35.144 seconds
[ Info: i: 0680, t: 1.786 hours, Δt: 6.732 seconds, umax = (1.0e-01, 6.0e-02, 4.3e-02) ms⁻¹, wall time: 35.694 seconds
[ Info: i: 0700, t: 1.825 hours, Δt: 6.976 seconds, umax = (9.9e-02, 5.6e-02, 4.1e-02) ms⁻¹, wall time: 36.233 seconds
[ Info: i: 0720, t: 1.863 hours, Δt: 6.923 seconds, umax = (1.0e-01, 5.8e-02, 4.2e-02) ms⁻¹, wall time: 36.791 seconds
[ Info: i: 0740, t: 1.900 hours, Δt: 6.811 seconds, umax = (9.9e-02, 6.0e-02, 4.6e-02) ms⁻¹, wall time: 37.410 seconds
[ Info: i: 0760, t: 1.936 hours, Δt: 6.555 seconds, umax = (9.7e-02, 5.9e-02, 4.3e-02) ms⁻¹, wall time: 38.020 seconds
[ Info: i: 0780, t: 1.973 hours, Δt: 6.500 seconds, umax = (1.1e-01, 6.4e-02, 4.7e-02) ms⁻¹, wall time: 38.854 seconds
[ Info: i: 0800, t: 2.009 hours, Δt: 6.813 seconds, umax = (1.0e-01, 6.0e-02, 4.5e-02) ms⁻¹, wall time: 39.823 seconds
[ Info: i: 0820, t: 2.046 hours, Δt: 6.296 seconds, umax = (1.1e-01, 6.7e-02, 4.9e-02) ms⁻¹, wall time: 40.524 seconds
[ Info: i: 0840, t: 2.081 hours, Δt: 6.792 seconds, umax = (1.1e-01, 6.8e-02, 4.6e-02) ms⁻¹, wall time: 41.129 seconds
[ Info: i: 0860, t: 2.119 hours, Δt: 6.410 seconds, umax = (1.0e-01, 6.3e-02, 4.2e-02) ms⁻¹, wall time: 41.623 seconds
[ Info: i: 0880, t: 2.156 hours, Δt: 6.841 seconds, umax = (1.0e-01, 6.0e-02, 4.5e-02) ms⁻¹, wall time: 42.108 seconds
[ Info: i: 0900, t: 2.193 hours, Δt: 6.527 seconds, umax = (1.1e-01, 6.7e-02, 4.4e-02) ms⁻¹, wall time: 42.599 seconds
[ Info: i: 0920, t: 2.229 hours, Δt: 6.373 seconds, umax = (1.0e-01, 6.4e-02, 4.6e-02) ms⁻¹, wall time: 43.069 seconds
[ Info: i: 0940, t: 2.263 hours, Δt: 6.554 seconds, umax = (1.1e-01, 6.6e-02, 4.3e-02) ms⁻¹, wall time: 43.627 seconds
[ Info: i: 0960, t: 2.299 hours, Δt: 6.348 seconds, umax = (1.1e-01, 6.9e-02, 4.2e-02) ms⁻¹, wall time: 44.359 seconds
[ Info: i: 0980, t: 2.333 hours, Δt: 6.414 seconds, umax = (1.1e-01, 5.8e-02, 4.2e-02) ms⁻¹, wall time: 45.116 seconds
[ Info: i: 1000, t: 2.369 hours, Δt: 6.484 seconds, umax = (1.1e-01, 6.7e-02, 4.2e-02) ms⁻¹, wall time: 45.884 seconds
[ Info: i: 1020, t: 2.406 hours, Δt: 6.169 seconds, umax = (1.1e-01, 6.5e-02, 4.4e-02) ms⁻¹, wall time: 46.542 seconds
[ Info: i: 1040, t: 2.439 hours, Δt: 6.086 seconds, umax = (1.0e-01, 6.8e-02, 4.1e-02) ms⁻¹, wall time: 47.035 seconds
[ Info: i: 1060, t: 2.472 hours, Δt: 6.222 seconds, umax = (1.0e-01, 6.3e-02, 4.5e-02) ms⁻¹, wall time: 47.497 seconds
[ Info: i: 1080, t: 2.505 hours, Δt: 5.960 seconds, umax = (1.0e-01, 6.8e-02, 4.6e-02) ms⁻¹, wall time: 48.082 seconds
[ Info: i: 1100, t: 2.539 hours, Δt: 5.881 seconds, umax = (1.1e-01, 6.4e-02, 4.4e-02) ms⁻¹, wall time: 48.470 seconds
[ Info: i: 1120, t: 2.573 hours, Δt: 6.108 seconds, umax = (1.1e-01, 7.0e-02, 4.4e-02) ms⁻¹, wall time: 49.231 seconds
[ Info: i: 1140, t: 2.608 hours, Δt: 5.906 seconds, umax = (1.1e-01, 6.4e-02, 4.7e-02) ms⁻¹, wall time: 50.005 seconds
[ Info: i: 1160, t: 2.641 hours, Δt: 6.188 seconds, umax = (1.1e-01, 6.7e-02, 4.4e-02) ms⁻¹, wall time: 50.769 seconds
[ Info: i: 1180, t: 2.673 hours, Δt: 6.001 seconds, umax = (1.1e-01, 6.9e-02, 4.8e-02) ms⁻¹, wall time: 51.531 seconds
[ Info: i: 1200, t: 2.707 hours, Δt: 5.776 seconds, umax = (1.1e-01, 6.6e-02, 4.7e-02) ms⁻¹, wall time: 51.925 seconds
[ Info: i: 1220, t: 2.740 hours, Δt: 5.836 seconds, umax = (1.0e-01, 7.1e-02, 4.1e-02) ms⁻¹, wall time: 52.400 seconds
[ Info: i: 1240, t: 2.772 hours, Δt: 5.910 seconds, umax = (1.1e-01, 7.5e-02, 4.6e-02) ms⁻¹, wall time: 52.953 seconds
[ Info: i: 1260, t: 2.805 hours, Δt: 6.184 seconds, umax = (1.2e-01, 7.5e-02, 4.1e-02) ms⁻¹, wall time: 53.695 seconds
[ Info: i: 1280, t: 2.839 hours, Δt: 5.916 seconds, umax = (1.1e-01, 7.6e-02, 4.6e-02) ms⁻¹, wall time: 54.615 seconds
[ Info: i: 1300, t: 2.873 hours, Δt: 6.259 seconds, umax = (1.1e-01, 7.1e-02, 4.9e-02) ms⁻¹, wall time: 55.440 seconds
[ Info: i: 1320, t: 2.907 hours, Δt: 5.953 seconds, umax = (1.1e-01, 7.5e-02, 4.5e-02) ms⁻¹, wall time: 56.271 seconds
[ Info: i: 1340, t: 2.940 hours, Δt: 5.856 seconds, umax = (1.1e-01, 8.0e-02, 4.7e-02) ms⁻¹, wall time: 57.075 seconds
[ Info: i: 1360, t: 2.972 hours, Δt: 5.595 seconds, umax = (1.1e-01, 7.5e-02, 5.2e-02) ms⁻¹, wall time: 57.843 seconds
[ Info: i: 1380, t: 3.002 hours, Δt: 5.740 seconds, umax = (1.1e-01, 8.0e-02, 5.0e-02) ms⁻¹, wall time: 58.832 seconds
[ Info: i: 1400, t: 3.033 hours, Δt: 5.708 seconds, umax = (1.1e-01, 7.3e-02, 4.5e-02) ms⁻¹, wall time: 59.389 seconds
[ Info: i: 1420, t: 3.066 hours, Δt: 5.881 seconds, umax = (1.1e-01, 7.7e-02, 4.7e-02) ms⁻¹, wall time: 1.003 minutes
[ Info: i: 1440, t: 3.097 hours, Δt: 5.948 seconds, umax = (1.1e-01, 7.0e-02, 5.3e-02) ms⁻¹, wall time: 1.011 minutes
[ Info: i: 1460, t: 3.131 hours, Δt: 6.003 seconds, umax = (1.1e-01, 7.1e-02, 4.9e-02) ms⁻¹, wall time: 1.019 minutes
[ Info: i: 1480, t: 3.164 hours, Δt: 5.787 seconds, umax = (1.1e-01, 7.5e-02, 4.9e-02) ms⁻¹, wall time: 1.027 minutes
[ Info: i: 1500, t: 3.195 hours, Δt: 5.765 seconds, umax = (1.1e-01, 7.3e-02, 5.1e-02) ms⁻¹, wall time: 1.039 minutes
[ Info: i: 1520, t: 3.228 hours, Δt: 5.894 seconds, umax = (1.2e-01, 7.3e-02, 5.3e-02) ms⁻¹, wall time: 1.052 minutes
[ Info: i: 1540, t: 3.260 hours, Δt: 5.710 seconds, umax = (1.1e-01, 7.8e-02, 4.8e-02) ms⁻¹, wall time: 1.065 minutes
[ Info: i: 1560, t: 3.291 hours, Δt: 6.022 seconds, umax = (1.1e-01, 7.3e-02, 4.8e-02) ms⁻¹, wall time: 1.079 minutes
[ Info: i: 1580, t: 3.324 hours, Δt: 5.785 seconds, umax = (1.1e-01, 8.2e-02, 5.0e-02) ms⁻¹, wall time: 1.092 minutes
[ Info: i: 1600, t: 3.354 hours, Δt: 5.557 seconds, umax = (1.1e-01, 7.8e-02, 4.8e-02) ms⁻¹, wall time: 1.105 minutes
[ Info: i: 1620, t: 3.385 hours, Δt: 5.785 seconds, umax = (1.1e-01, 6.9e-02, 5.1e-02) ms⁻¹, wall time: 1.117 minutes
[ Info: i: 1640, t: 3.417 hours, Δt: 5.560 seconds, umax = (1.1e-01, 8.1e-02, 5.0e-02) ms⁻¹, wall time: 1.130 minutes
[ Info: i: 1660, t: 3.448 hours, Δt: 5.971 seconds, umax = (1.1e-01, 8.2e-02, 5.1e-02) ms⁻¹, wall time: 1.143 minutes
[ Info: i: 1680, t: 3.481 hours, Δt: 6.045 seconds, umax = (1.1e-01, 8.4e-02, 5.1e-02) ms⁻¹, wall time: 1.158 minutes
[ Info: i: 1700, t: 3.514 hours, Δt: 6.295 seconds, umax = (1.1e-01, 8.1e-02, 5.0e-02) ms⁻¹, wall time: 1.168 minutes
[ Info: i: 1720, t: 3.547 hours, Δt: 5.801 seconds, umax = (1.2e-01, 8.2e-02, 5.0e-02) ms⁻¹, wall time: 1.178 minutes
[ Info: i: 1740, t: 3.579 hours, Δt: 5.658 seconds, umax = (1.1e-01, 7.9e-02, 5.1e-02) ms⁻¹, wall time: 1.191 minutes
[ Info: i: 1760, t: 3.608 hours, Δt: 5.464 seconds, umax = (1.1e-01, 8.8e-02, 5.7e-02) ms⁻¹, wall time: 1.206 minutes
[ Info: i: 1780, t: 3.639 hours, Δt: 5.626 seconds, umax = (1.1e-01, 7.5e-02, 4.6e-02) ms⁻¹, wall time: 1.220 minutes
[ Info: i: 1800, t: 3.668 hours, Δt: 5.321 seconds, umax = (1.2e-01, 7.3e-02, 5.4e-02) ms⁻¹, wall time: 1.237 minutes
[ Info: i: 1820, t: 3.697 hours, Δt: 5.535 seconds, umax = (1.2e-01, 7.3e-02, 4.9e-02) ms⁻¹, wall time: 1.247 minutes
[ Info: i: 1840, t: 3.729 hours, Δt: 6.015 seconds, umax = (1.2e-01, 8.3e-02, 4.9e-02) ms⁻¹, wall time: 1.260 minutes
[ Info: i: 1860, t: 3.761 hours, Δt: 6.072 seconds, umax = (1.1e-01, 7.4e-02, 4.7e-02) ms⁻¹, wall time: 1.276 minutes
[ Info: i: 1880, t: 3.793 hours, Δt: 5.748 seconds, umax = (1.1e-01, 7.6e-02, 4.6e-02) ms⁻¹, wall time: 1.289 minutes
[ Info: i: 1900, t: 3.826 hours, Δt: 5.781 seconds, umax = (1.1e-01, 8.0e-02, 4.5e-02) ms⁻¹, wall time: 1.303 minutes
[ Info: i: 1920, t: 3.858 hours, Δt: 5.864 seconds, umax = (1.2e-01, 8.0e-02, 5.0e-02) ms⁻¹, wall time: 1.314 minutes
[ Info: i: 1940, t: 3.889 hours, Δt: 5.897 seconds, umax = (1.1e-01, 8.1e-02, 5.0e-02) ms⁻¹, wall time: 1.324 minutes
[ Info: i: 1960, t: 3.922 hours, Δt: 5.976 seconds, umax = (1.1e-01, 8.1e-02, 4.8e-02) ms⁻¹, wall time: 1.337 minutes
[ Info: i: 1980, t: 3.954 hours, Δt: 5.815 seconds, umax = (1.1e-01, 7.7e-02, 5.6e-02) ms⁻¹, wall time: 1.349 minutes
[ Info: i: 2000, t: 3.986 hours, Δt: 5.763 seconds, umax = (1.1e-01, 8.5e-02, 5.7e-02) ms⁻¹, wall time: 1.363 minutes
[ Info: Simulation is stopping after running for 1.370 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.jld2", "w"),
     u = FieldTimeSeries("langmuir_turbulence_fields.jld2", "u"),
     B = FieldTimeSeries("langmuir_turbulence_averages.jld2", "B"),
     U = FieldTimeSeries("langmuir_turbulence_averages.jld2", "U"),
     V = FieldTimeSeries("langmuir_turbulence_averages.jld2", "V"),
    wu = FieldTimeSeries("langmuir_turbulence_averages.jld2", "wu"),
    wv = FieldTimeSeries("langmuir_turbulence_averages.jld2", "wv"))

times = time_series.w.times

We are now ready to animate using Makie. We use Makie's Observable to animate the data. To dive into how Observables work we refer to Makie.jl's Documentation.

n = Observable(1)

wxy_title = @lift string("w(x, y, t) at z=-8 m and t = ", prettytime(times[$n]))
wxz_title = @lift string("w(x, z, t) at y=0 m and t = ", prettytime(times[$n]))
uxz_title = @lift string("u(x, z, t) at y=0 m and t = ", prettytime(times[$n]))

fig = Figure(size = (850, 850))

ax_B = Axis(fig[1, 4];
            xlabel = "Buoyancy (m s⁻²)",
            ylabel = "z (m)")

ax_U = Axis(fig[2, 4];
            xlabel = "Velocities (m s⁻¹)",
            ylabel = "z (m)",
            limits = ((-0.07, 0.07), nothing))

ax_fluxes = Axis(fig[3, 4];
                 xlabel = "Momentum fluxes (m² s⁻²)",
                 ylabel = "z (m)",
                 limits = ((-3.5e-5, 3.5e-5), nothing))

ax_wxy = Axis(fig[1, 1:2];
              xlabel = "x (m)",
              ylabel = "y (m)",
              aspect = DataAspect(),
              limits = ((0, grid.Lx), (0, grid.Ly)),
              title = wxy_title)

ax_wxz = Axis(fig[2, 1:2];
              xlabel = "x (m)",
              ylabel = "z (m)",
              aspect = AxisAspect(2),
              limits = ((0, grid.Lx), (-grid.Lz, 0)),
              title = wxz_title)

ax_uxz = Axis(fig[3, 1:2];
              xlabel = "x (m)",
              ylabel = "z (m)",
              aspect = AxisAspect(2),
              limits = ((0, grid.Lx), (-grid.Lz, 0)),
              title = uxz_title)


wₙ = @lift time_series.w[$n]
uₙ = @lift time_series.u[$n]
Bₙ = @lift view(time_series.B[$n], 1, 1, :)
Uₙ = @lift view(time_series.U[$n], 1, 1, :)
Vₙ = @lift view(time_series.V[$n], 1, 1, :)
wuₙ = @lift view(time_series.wu[$n], 1, 1, :)
wvₙ = @lift view(time_series.wv[$n], 1, 1, :)

k = searchsortedfirst(znodes(grid, Face(); with_halos=true), -8)
wxyₙ = @lift view(time_series.w[$n], :, :, k)
wxzₙ = @lift view(time_series.w[$n], :, 1, :)
uxzₙ = @lift view(time_series.u[$n], :, 1, :)

wlims = (-0.03, 0.03)
ulims = (-0.05, 0.05)

lines!(ax_B, Bₙ)

lines!(ax_U, Uₙ; label = L"\bar{u}")
lines!(ax_U, Vₙ; label = L"\bar{v}")
axislegend(ax_U; position = :rb)

lines!(ax_fluxes, wuₙ; label = L"mean $wu$")
lines!(ax_fluxes, wvₙ; label = L"mean $wv$")
axislegend(ax_fluxes; position = :rb)

hm_wxy = heatmap!(ax_wxy, wxyₙ;
                  colorrange = wlims,
                  colormap = :balance)

Colorbar(fig[1, 3], hm_wxy; label = "m s⁻¹")

hm_wxz = heatmap!(ax_wxz, wxzₙ;
                  colorrange = wlims,
                  colormap = :balance)

Colorbar(fig[2, 3], hm_wxz; label = "m s⁻¹")

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

Colorbar(fig[3, 3], ax_uxz; label = "m s⁻¹")

fig

And, finally, we record a movie.

frames = 1:length(times)

CairoMakie.record(fig, "langmuir_turbulence.mp4", frames, framerate=8) do i
    n[] = i
end

---

Julia version and environment information

This example was executed with the following version of Julia:

using InteractiveUtils: versioninfo
versioninfo()

Julia Version 1.12.4
Commit 01a2eadb047 (2026-01-06 16:56 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 9374F 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver4)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 128 virtual cores)
Environment:
  LD_LIBRARY_PATH = 
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
  JULIA_DEPOT_PATH = /var/lib/buildkite-agent/.julia
  JULIA_PROJECT = /var/lib/buildkite-agent/Oceananigans.jl-30795/docs/
  JULIA_VERSION = 1.12.4
  JULIA_LOAD_PATH = @:@v#.#:@stdlib
  JULIA_VERSION_ENZYME = 1.10.10
  JULIA_PYTHONCALL_EXE = /var/lib/buildkite-agent/Oceananigans.jl-30795/docs/.CondaPkg/.pixi/envs/default/bin/python
  JULIA_DEBUG = Literate

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

import Pkg
Pkg.status()

Status `~/Oceananigans.jl-30795/docs/Project.toml`
  [79e6a3ab] Adapt v4.5.0
  [052768ef] CUDA v5.11.0
  [13f3f980] CairoMakie v0.15.9
  [e30172f5] Documenter v1.17.0
  [daee34ce] DocumenterCitations v1.4.1
  [4710194d] DocumenterVitepress v0.3.2
  [033835bb] JLD2 v0.6.4
  [63c18a36] KernelAbstractions v0.9.41
  [98b081ad] Literate v2.21.0
  [da04e1cc] MPI v0.20.24
  [85f8d34a] NCDatasets v0.14.15
  [9e8cae18] Oceananigans v0.106.5 `..`
  [f27b6e38] Polynomials v4.1.1
  [6038ab10] Rotations v1.7.1
  [d496a93d] SeawaterPolynomials v0.3.10
  [09ab397b] StructArrays v0.7.3
  [bdfc003b] TimesDates v0.3.3
  [2e0b0046] XESMF v0.1.6
  [b77e0a4c] InteractiveUtils v1.11.0
  [37e2e46d] LinearAlgebra v1.12.0
  [44cfe95a] Pkg v1.12.1

---

This page was generated using Literate.jl.