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Langmuir turbulence example

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

Wagner et al., "Near-inertial waves and turbulence driven by the growth of swell", Journal of Physical Oceanography (2021)

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

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 in 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 $x$-velocity beneath surface waves. This differs from models that use the Eulerian-mean velocity field as a prognostic variable, but has the advantage that $u$ accounts for the total advection of tracers and momentum, and that $u = v = w = 0$ 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 $z = 0$, 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 $x, y$, we use UniformStokesDrift, which expects Stokes drift functions of $z, t$ 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 $u$ and $w$.

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 $u, v, w$ 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 = (1.4e-03, 8.0e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (9.511 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.731 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 19.413 seconds, umax = (3.5e-02, 1.3e-02, 2.3e-02) ms⁻¹, wall time: 13.554 seconds
[ Info: i: 0040, t: 16.797 minutes, Δt: 13.068 seconds, umax = (5.2e-02, 2.1e-02, 2.3e-02) ms⁻¹, wall time: 13.971 seconds
[ Info: i: 0060, t: 21.025 minutes, Δt: 11.424 seconds, umax = (5.9e-02, 2.7e-02, 3.0e-02) ms⁻¹, wall time: 14.472 seconds
[ Info: i: 0080, t: 24.760 minutes, Δt: 10.923 seconds, umax = (6.3e-02, 3.1e-02, 3.4e-02) ms⁻¹, wall time: 14.862 seconds
[ Info: i: 0100, t: 28.212 minutes, Δt: 10.583 seconds, umax = (6.2e-02, 3.0e-02, 2.9e-02) ms⁻¹, wall time: 15.314 seconds
[ Info: i: 0120, t: 31.639 minutes, Δt: 11.187 seconds, umax = (6.6e-02, 3.5e-02, 2.6e-02) ms⁻¹, wall time: 15.781 seconds
[ Info: i: 0140, t: 35.184 minutes, Δt: 10.178 seconds, umax = (7.1e-02, 3.6e-02, 2.9e-02) ms⁻¹, wall time: 16.349 seconds
[ Info: i: 0160, t: 38.591 minutes, Δt: 9.349 seconds, umax = (7.3e-02, 3.4e-02, 3.0e-02) ms⁻¹, wall time: 16.656 seconds
[ Info: i: 0180, t: 41.604 minutes, Δt: 9.244 seconds, umax = (7.0e-02, 3.5e-02, 3.3e-02) ms⁻¹, wall time: 17.191 seconds
[ Info: i: 0200, t: 44.680 minutes, Δt: 9.639 seconds, umax = (7.3e-02, 4.0e-02, 3.4e-02) ms⁻¹, wall time: 17.620 seconds
[ Info: i: 0220, t: 47.876 minutes, Δt: 8.701 seconds, umax = (7.3e-02, 4.0e-02, 3.5e-02) ms⁻¹, wall time: 18.095 seconds
[ Info: i: 0240, t: 50.748 minutes, Δt: 9.171 seconds, umax = (7.9e-02, 3.9e-02, 3.7e-02) ms⁻¹, wall time: 18.652 seconds
[ Info: i: 0260, t: 53.749 minutes, Δt: 8.405 seconds, umax = (8.0e-02, 4.2e-02, 3.5e-02) ms⁻¹, wall time: 19.090 seconds
[ Info: i: 0280, t: 56.681 minutes, Δt: 8.919 seconds, umax = (7.7e-02, 4.0e-02, 3.5e-02) ms⁻¹, wall time: 19.562 seconds
[ Info: i: 0300, t: 59.624 minutes, Δt: 8.528 seconds, umax = (8.2e-02, 4.1e-02, 3.8e-02) ms⁻¹, wall time: 19.994 seconds
[ Info: i: 0320, t: 1.039 hours, Δt: 8.911 seconds, umax = (8.3e-02, 4.7e-02, 3.9e-02) ms⁻¹, wall time: 20.461 seconds
[ Info: i: 0340, t: 1.086 hours, Δt: 7.790 seconds, umax = (8.9e-02, 4.7e-02, 3.9e-02) ms⁻¹, wall time: 21.023 seconds
[ Info: i: 0360, t: 1.130 hours, Δt: 7.842 seconds, umax = (8.3e-02, 5.0e-02, 3.9e-02) ms⁻¹, wall time: 21.325 seconds
[ Info: i: 0380, t: 1.173 hours, Δt: 7.964 seconds, umax = (8.1e-02, 4.6e-02, 4.1e-02) ms⁻¹, wall time: 21.879 seconds
[ Info: i: 0400, t: 1.219 hours, Δt: 7.963 seconds, umax = (8.3e-02, 4.9e-02, 4.0e-02) ms⁻¹, wall time: 22.242 seconds
[ Info: i: 0420, t: 1.261 hours, Δt: 7.443 seconds, umax = (8.8e-02, 5.2e-02, 3.7e-02) ms⁻¹, wall time: 22.768 seconds
[ Info: i: 0440, t: 1.303 hours, Δt: 6.817 seconds, umax = (1.0e-01, 5.8e-02, 3.8e-02) ms⁻¹, wall time: 23.158 seconds
[ Info: i: 0460, t: 1.339 hours, Δt: 6.956 seconds, umax = (9.2e-02, 5.3e-02, 4.1e-02) ms⁻¹, wall time: 23.715 seconds
[ Info: i: 0480, t: 1.379 hours, Δt: 7.305 seconds, umax = (9.3e-02, 5.5e-02, 4.3e-02) ms⁻¹, wall time: 24.062 seconds
[ Info: i: 0500, t: 1.417 hours, Δt: 7.328 seconds, umax = (8.8e-02, 5.0e-02, 4.7e-02) ms⁻¹, wall time: 24.507 seconds
[ Info: i: 0520, t: 1.458 hours, Δt: 7.543 seconds, umax = (9.1e-02, 5.2e-02, 4.2e-02) ms⁻¹, wall time: 24.961 seconds
[ Info: i: 0540, t: 1.500 hours, Δt: 7.665 seconds, umax = (9.1e-02, 5.2e-02, 4.6e-02) ms⁻¹, wall time: 25.408 seconds
[ Info: i: 0560, t: 1.542 hours, Δt: 7.453 seconds, umax = (9.3e-02, 5.3e-02, 4.3e-02) ms⁻¹, wall time: 25.886 seconds
[ Info: i: 0580, t: 1.583 hours, Δt: 7.597 seconds, umax = (9.5e-02, 5.6e-02, 4.2e-02) ms⁻¹, wall time: 26.400 seconds
[ Info: i: 0600, t: 1.624 hours, Δt: 7.192 seconds, umax = (9.8e-02, 5.3e-02, 4.3e-02) ms⁻¹, wall time: 26.968 seconds
[ Info: i: 0620, t: 1.664 hours, Δt: 7.342 seconds, umax = (1.0e-01, 5.2e-02, 4.2e-02) ms⁻¹, wall time: 27.505 seconds
[ Info: i: 0640, t: 1.702 hours, Δt: 7.103 seconds, umax = (1.0e-01, 5.2e-02, 4.0e-02) ms⁻¹, wall time: 28.073 seconds
[ Info: i: 0660, t: 1.741 hours, Δt: 6.875 seconds, umax = (9.7e-02, 5.7e-02, 4.5e-02) ms⁻¹, wall time: 28.613 seconds
[ Info: i: 0680, t: 1.779 hours, Δt: 6.855 seconds, umax = (9.7e-02, 5.6e-02, 4.3e-02) ms⁻¹, wall time: 29.183 seconds
[ Info: i: 0700, t: 1.818 hours, Δt: 6.397 seconds, umax = (9.3e-02, 5.8e-02, 4.1e-02) ms⁻¹, wall time: 29.730 seconds
[ Info: i: 0720, t: 1.854 hours, Δt: 7.099 seconds, umax = (9.4e-02, 6.4e-02, 4.3e-02) ms⁻¹, wall time: 30.296 seconds
[ Info: i: 0740, t: 1.893 hours, Δt: 7.171 seconds, umax = (9.9e-02, 5.5e-02, 4.8e-02) ms⁻¹, wall time: 30.768 seconds
[ Info: i: 0760, t: 1.932 hours, Δt: 6.678 seconds, umax = (9.5e-02, 6.2e-02, 4.3e-02) ms⁻¹, wall time: 31.289 seconds
[ Info: i: 0780, t: 1.969 hours, Δt: 6.760 seconds, umax = (9.7e-02, 6.3e-02, 4.0e-02) ms⁻¹, wall time: 31.737 seconds
[ Info: i: 0800, t: 2.006 hours, Δt: 6.829 seconds, umax = (1.0e-01, 6.0e-02, 4.1e-02) ms⁻¹, wall time: 32.312 seconds
[ Info: i: 0820, t: 2.044 hours, Δt: 6.869 seconds, umax = (1.0e-01, 6.4e-02, 4.0e-02) ms⁻¹, wall time: 32.677 seconds
[ Info: i: 0840, t: 2.081 hours, Δt: 6.715 seconds, umax = (1.1e-01, 7.1e-02, 4.3e-02) ms⁻¹, wall time: 33.142 seconds
[ Info: i: 0860, t: 2.118 hours, Δt: 6.809 seconds, umax = (1.0e-01, 6.3e-02, 4.3e-02) ms⁻¹, wall time: 33.631 seconds
[ Info: i: 0880, t: 2.153 hours, Δt: 6.540 seconds, umax = (1.0e-01, 6.4e-02, 4.3e-02) ms⁻¹, wall time: 34.091 seconds
[ Info: i: 0900, t: 2.188 hours, Δt: 6.565 seconds, umax = (1.0e-01, 6.9e-02, 4.0e-02) ms⁻¹, wall time: 34.594 seconds
[ Info: i: 0920, t: 2.225 hours, Δt: 6.038 seconds, umax = (1.0e-01, 7.0e-02, 4.1e-02) ms⁻¹, wall time: 35.058 seconds
[ Info: i: 0940, t: 2.256 hours, Δt: 6.214 seconds, umax = (1.0e-01, 6.6e-02, 4.2e-02) ms⁻¹, wall time: 35.614 seconds
[ Info: i: 0960, t: 2.291 hours, Δt: 6.595 seconds, umax = (1.0e-01, 6.1e-02, 4.4e-02) ms⁻¹, wall time: 36.009 seconds
[ Info: i: 0980, t: 2.327 hours, Δt: 6.503 seconds, umax = (1.0e-01, 6.4e-02, 4.9e-02) ms⁻¹, wall time: 36.477 seconds
[ Info: i: 1000, t: 2.362 hours, Δt: 6.686 seconds, umax = (1.1e-01, 6.1e-02, 4.6e-02) ms⁻¹, wall time: 36.985 seconds
[ Info: i: 1020, t: 2.399 hours, Δt: 6.792 seconds, umax = (1.0e-01, 6.2e-02, 4.6e-02) ms⁻¹, wall time: 37.451 seconds
[ Info: i: 1040, t: 2.435 hours, Δt: 6.631 seconds, umax = (1.0e-01, 6.5e-02, 4.4e-02) ms⁻¹, wall time: 37.936 seconds
[ Info: i: 1060, t: 2.471 hours, Δt: 6.318 seconds, umax = (1.0e-01, 6.6e-02, 4.3e-02) ms⁻¹, wall time: 38.396 seconds
[ Info: i: 1080, t: 2.505 hours, Δt: 6.568 seconds, umax = (1.1e-01, 6.4e-02, 4.5e-02) ms⁻¹, wall time: 38.991 seconds
[ Info: i: 1100, t: 2.541 hours, Δt: 6.305 seconds, umax = (1.1e-01, 6.8e-02, 5.3e-02) ms⁻¹, wall time: 39.366 seconds
[ Info: i: 1120, t: 2.575 hours, Δt: 6.121 seconds, umax = (1.1e-01, 6.7e-02, 4.7e-02) ms⁻¹, wall time: 39.837 seconds
[ Info: i: 1140, t: 2.609 hours, Δt: 6.301 seconds, umax = (1.1e-01, 6.3e-02, 4.5e-02) ms⁻¹, wall time: 40.326 seconds
[ Info: i: 1160, t: 2.644 hours, Δt: 6.494 seconds, umax = (1.0e-01, 6.7e-02, 4.9e-02) ms⁻¹, wall time: 40.794 seconds
[ Info: i: 1180, t: 2.680 hours, Δt: 6.207 seconds, umax = (1.0e-01, 6.6e-02, 4.7e-02) ms⁻¹, wall time: 41.357 seconds
[ Info: i: 1200, t: 2.715 hours, Δt: 6.473 seconds, umax = (1.0e-01, 7.2e-02, 4.7e-02) ms⁻¹, wall time: 41.806 seconds
[ Info: i: 1220, t: 2.750 hours, Δt: 5.886 seconds, umax = (1.0e-01, 6.8e-02, 4.9e-02) ms⁻¹, wall time: 42.291 seconds
[ Info: i: 1240, t: 2.783 hours, Δt: 5.896 seconds, umax = (1.1e-01, 6.7e-02, 4.6e-02) ms⁻¹, wall time: 45.175 seconds
[ Info: i: 1260, t: 2.817 hours, Δt: 5.760 seconds, umax = (1.1e-01, 6.7e-02, 4.6e-02) ms⁻¹, wall time: 45.721 seconds
[ Info: i: 1280, t: 2.848 hours, Δt: 5.903 seconds, umax = (1.1e-01, 7.6e-02, 4.7e-02) ms⁻¹, wall time: 46.457 seconds
[ Info: i: 1300, t: 2.881 hours, Δt: 6.134 seconds, umax = (1.0e-01, 7.4e-02, 4.9e-02) ms⁻¹, wall time: 46.988 seconds
[ Info: i: 1320, t: 2.915 hours, Δt: 5.712 seconds, umax = (1.0e-01, 8.0e-02, 5.0e-02) ms⁻¹, wall time: 47.544 seconds
[ Info: i: 1340, t: 2.947 hours, Δt: 5.726 seconds, umax = (1.1e-01, 7.3e-02, 5.3e-02) ms⁻¹, wall time: 48.241 seconds
[ Info: i: 1360, t: 2.979 hours, Δt: 5.743 seconds, umax = (1.1e-01, 6.9e-02, 5.0e-02) ms⁻¹, wall time: 48.781 seconds
[ Info: i: 1380, t: 3.011 hours, Δt: 5.610 seconds, umax = (1.1e-01, 6.9e-02, 5.0e-02) ms⁻¹, wall time: 49.432 seconds
[ Info: i: 1400, t: 3.043 hours, Δt: 6.051 seconds, umax = (1.1e-01, 8.0e-02, 4.6e-02) ms⁻¹, wall time: 49.940 seconds
[ Info: i: 1420, t: 3.076 hours, Δt: 5.994 seconds, umax = (1.1e-01, 7.1e-02, 4.3e-02) ms⁻¹, wall time: 50.497 seconds
[ Info: i: 1440, t: 3.108 hours, Δt: 6.252 seconds, umax = (1.1e-01, 6.8e-02, 4.6e-02) ms⁻¹, wall time: 51.529 seconds
[ Info: i: 1460, t: 3.143 hours, Δt: 6.039 seconds, umax = (1.1e-01, 7.8e-02, 4.8e-02) ms⁻¹, wall time: 52.075 seconds
[ Info: i: 1480, t: 3.175 hours, Δt: 6.123 seconds, umax = (1.1e-01, 7.5e-02, 4.6e-02) ms⁻¹, wall time: 53.111 seconds
[ Info: i: 1500, t: 3.210 hours, Δt: 6.394 seconds, umax = (1.0e-01, 7.5e-02, 5.1e-02) ms⁻¹, wall time: 53.594 seconds
[ Info: i: 1520, t: 3.245 hours, Δt: 5.812 seconds, umax = (1.0e-01, 9.0e-02, 4.7e-02) ms⁻¹, wall time: 54.150 seconds
[ Info: i: 1540, t: 3.276 hours, Δt: 5.919 seconds, umax = (1.0e-01, 8.1e-02, 4.1e-02) ms⁻¹, wall time: 54.819 seconds
[ Info: i: 1560, t: 3.309 hours, Δt: 5.984 seconds, umax = (1.1e-01, 7.5e-02, 4.7e-02) ms⁻¹, wall time: 55.370 seconds
[ Info: i: 1580, t: 3.339 hours, Δt: 5.645 seconds, umax = (1.1e-01, 7.3e-02, 5.2e-02) ms⁻¹, wall time: 56.105 seconds
[ Info: i: 1600, t: 3.372 hours, Δt: 5.993 seconds, umax = (1.1e-01, 7.4e-02, 5.3e-02) ms⁻¹, wall time: 56.568 seconds
[ Info: i: 1620, t: 3.405 hours, Δt: 6.115 seconds, umax = (1.1e-01, 7.7e-02, 5.0e-02) ms⁻¹, wall time: 57.125 seconds
[ Info: i: 1640, t: 3.438 hours, Δt: 5.626 seconds, umax = (1.1e-01, 8.8e-02, 4.7e-02) ms⁻¹, wall time: 57.754 seconds
[ Info: i: 1660, t: 3.469 hours, Δt: 5.863 seconds, umax = (1.1e-01, 8.4e-02, 4.9e-02) ms⁻¹, wall time: 58.239 seconds
[ Info: i: 1680, t: 3.502 hours, Δt: 5.667 seconds, umax = (1.1e-01, 8.4e-02, 5.2e-02) ms⁻¹, wall time: 58.977 seconds
[ Info: i: 1700, t: 3.534 hours, Δt: 5.644 seconds, umax = (1.1e-01, 8.7e-02, 4.8e-02) ms⁻¹, wall time: 59.311 seconds
[ Info: i: 1720, t: 3.565 hours, Δt: 4.937 seconds, umax = (1.1e-01, 8.8e-02, 5.0e-02) ms⁻¹, wall time: 59.790 seconds
[ Info: i: 1740, t: 3.592 hours, Δt: 5.474 seconds, umax = (1.2e-01, 7.9e-02, 5.1e-02) ms⁻¹, wall time: 1.006 minutes
[ Info: i: 1760, t: 3.622 hours, Δt: 6.036 seconds, umax = (1.1e-01, 7.9e-02, 5.2e-02) ms⁻¹, wall time: 1.014 minutes
[ Info: i: 1780, t: 3.655 hours, Δt: 5.390 seconds, umax = (1.1e-01, 8.1e-02, 5.0e-02) ms⁻¹, wall time: 1.022 minutes
[ Info: i: 1800, t: 3.686 hours, Δt: 5.975 seconds, umax = (1.2e-01, 9.1e-02, 4.9e-02) ms⁻¹, wall time: 1.031 minutes
[ Info: i: 1820, t: 3.719 hours, Δt: 5.828 seconds, umax = (1.1e-01, 8.6e-02, 5.1e-02) ms⁻¹, wall time: 1.039 minutes
[ Info: i: 1840, t: 3.750 hours, Δt: 6.023 seconds, umax = (1.1e-01, 8.8e-02, 5.6e-02) ms⁻¹, wall time: 1.047 minutes
[ Info: i: 1860, t: 3.783 hours, Δt: 6.014 seconds, umax = (1.1e-01, 8.0e-02, 4.8e-02) ms⁻¹, wall time: 1.057 minutes
[ Info: i: 1880, t: 3.817 hours, Δt: 6.185 seconds, umax = (1.1e-01, 7.8e-02, 5.6e-02) ms⁻¹, wall time: 1.065 minutes
[ Info: i: 1900, t: 3.850 hours, Δt: 5.356 seconds, umax = (1.3e-01, 7.7e-02, 5.2e-02) ms⁻¹, wall time: 1.074 minutes
[ Info: i: 1920, t: 3.880 hours, Δt: 5.734 seconds, umax = (1.2e-01, 7.6e-02, 5.8e-02) ms⁻¹, wall time: 1.082 minutes
[ Info: i: 1940, t: 3.911 hours, Δt: 5.107 seconds, umax = (1.3e-01, 8.3e-02, 5.2e-02) ms⁻¹, wall time: 1.090 minutes
[ Info: i: 1960, t: 3.939 hours, Δt: 5.588 seconds, umax = (1.1e-01, 7.9e-02, 6.0e-02) ms⁻¹, wall time: 1.099 minutes
[ Info: i: 1980, t: 3.971 hours, Δt: 5.460 seconds, umax = (1.1e-01, 7.6e-02, 5.0e-02) ms⁻¹, wall time: 1.107 minutes
[ Info: Simulation is stopping after running for 1.116 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 $u$ and $w$, and a horizontal slice of $w$ 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-oceananigans
  JULIA_PROJECT = /var/lib/buildkite-agent/Oceananigans.jl-29255/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-29255/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-29255/docs/Project.toml`
  [79e6a3ab] Adapt v4.4.0
  [052768ef] CUDA v5.9.6
  [13f3f980] CairoMakie v0.15.8
  [e30172f5] Documenter v1.16.1
  [daee34ce] DocumenterCitations v1.4.1
  [033835bb] JLD2 v0.6.3
  [63c18a36] KernelAbstractions v0.9.39
  [98b081ad] Literate v2.21.0
  [da04e1cc] MPI v0.20.23
  [85f8d34a] NCDatasets v0.14.11
  [9e8cae18] Oceananigans v0.104.4 `..`
  [f27b6e38] Polynomials v4.1.0
  [6038ab10] Rotations v1.7.1
  [d496a93d] SeawaterPolynomials v0.3.10
  [09ab397b] StructArrays v0.7.2
  [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

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