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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 = (1.5e-03, 8.3e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (11.846 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.804 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 17.921 seconds, umax = (3.5e-02, 1.3e-02, 2.2e-02) ms⁻¹, wall time: 15.840 seconds
[ Info: i: 0040, t: 16.828 minutes, Δt: 12.617 seconds, umax = (5.3e-02, 2.1e-02, 2.3e-02) ms⁻¹, wall time: 16.253 seconds
[ Info: i: 0060, t: 20.822 minutes, Δt: 11.032 seconds, umax = (6.4e-02, 2.9e-02, 3.2e-02) ms⁻¹, wall time: 16.795 seconds
[ Info: i: 0080, t: 24.481 minutes, Δt: 10.633 seconds, umax = (6.4e-02, 3.2e-02, 3.1e-02) ms⁻¹, wall time: 17.157 seconds
[ Info: i: 0100, t: 28.006 minutes, Δt: 11.194 seconds, umax = (6.4e-02, 2.9e-02, 3.1e-02) ms⁻¹, wall time: 17.622 seconds
[ Info: i: 0120, t: 31.613 minutes, Δt: 10.844 seconds, umax = (6.7e-02, 3.2e-02, 2.8e-02) ms⁻¹, wall time: 18.109 seconds
[ Info: i: 0140, t: 35.180 minutes, Δt: 10.662 seconds, umax = (6.6e-02, 3.2e-02, 3.1e-02) ms⁻¹, wall time: 18.700 seconds
[ Info: i: 0160, t: 38.652 minutes, Δt: 10.116 seconds, umax = (6.9e-02, 3.4e-02, 3.4e-02) ms⁻¹, wall time: 19.002 seconds
[ Info: i: 0180, t: 41.856 minutes, Δt: 9.369 seconds, umax = (7.4e-02, 3.9e-02, 3.5e-02) ms⁻¹, wall time: 19.478 seconds
[ Info: i: 0200, t: 44.996 minutes, Δt: 9.276 seconds, umax = (7.4e-02, 4.2e-02, 3.4e-02) ms⁻¹, wall time: 19.896 seconds
[ Info: i: 0220, t: 47.863 minutes, Δt: 8.837 seconds, umax = (7.6e-02, 4.1e-02, 3.7e-02) ms⁻¹, wall time: 20.365 seconds
[ Info: i: 0240, t: 50.712 minutes, Δt: 8.746 seconds, umax = (7.7e-02, 4.0e-02, 3.7e-02) ms⁻¹, wall time: 20.894 seconds
[ Info: i: 0260, t: 53.599 minutes, Δt: 8.837 seconds, umax = (8.1e-02, 4.6e-02, 3.9e-02) ms⁻¹, wall time: 21.268 seconds
[ Info: i: 0280, t: 56.476 minutes, Δt: 9.001 seconds, umax = (7.8e-02, 4.4e-02, 3.5e-02) ms⁻¹, wall time: 21.739 seconds
[ Info: i: 0300, t: 59.348 minutes, Δt: 8.678 seconds, umax = (8.0e-02, 4.4e-02, 3.5e-02) ms⁻¹, wall time: 22.156 seconds
[ Info: i: 0320, t: 1.036 hours, Δt: 8.080 seconds, umax = (8.2e-02, 4.5e-02, 3.7e-02) ms⁻¹, wall time: 22.631 seconds
[ Info: i: 0340, t: 1.083 hours, Δt: 8.235 seconds, umax = (8.8e-02, 4.1e-02, 3.6e-02) ms⁻¹, wall time: 23.053 seconds
[ Info: i: 0360, t: 1.126 hours, Δt: 7.591 seconds, umax = (8.7e-02, 4.5e-02, 3.6e-02) ms⁻¹, wall time: 23.562 seconds
[ Info: i: 0380, t: 1.169 hours, Δt: 7.876 seconds, umax = (8.7e-02, 4.8e-02, 4.1e-02) ms⁻¹, wall time: 24.167 seconds
[ Info: i: 0400, t: 1.212 hours, Δt: 7.231 seconds, umax = (8.5e-02, 4.7e-02, 4.2e-02) ms⁻¹, wall time: 24.466 seconds
[ Info: i: 0420, t: 1.254 hours, Δt: 7.843 seconds, umax = (8.6e-02, 5.1e-02, 4.1e-02) ms⁻¹, wall time: 25.049 seconds
[ Info: i: 0440, t: 1.297 hours, Δt: 7.775 seconds, umax = (9.0e-02, 4.9e-02, 4.2e-02) ms⁻¹, wall time: 25.365 seconds
[ Info: i: 0460, t: 1.337 hours, Δt: 7.719 seconds, umax = (9.2e-02, 4.9e-02, 3.6e-02) ms⁻¹, wall time: 25.953 seconds
[ Info: i: 0480, t: 1.380 hours, Δt: 7.463 seconds, umax = (9.7e-02, 5.1e-02, 3.8e-02) ms⁻¹, wall time: 26.271 seconds
[ Info: i: 0500, t: 1.421 hours, Δt: 7.388 seconds, umax = (9.3e-02, 5.2e-02, 4.2e-02) ms⁻¹, wall time: 26.869 seconds
[ Info: i: 0520, t: 1.462 hours, Δt: 7.546 seconds, umax = (9.1e-02, 5.2e-02, 3.9e-02) ms⁻¹, wall time: 27.186 seconds
[ Info: i: 0540, t: 1.502 hours, Δt: 7.229 seconds, umax = (9.7e-02, 5.2e-02, 4.2e-02) ms⁻¹, wall time: 27.794 seconds
[ Info: i: 0560, t: 1.543 hours, Δt: 6.969 seconds, umax = (9.2e-02, 5.1e-02, 4.4e-02) ms⁻¹, wall time: 28.100 seconds
[ Info: i: 0580, t: 1.581 hours, Δt: 7.277 seconds, umax = (9.8e-02, 5.0e-02, 4.1e-02) ms⁻¹, wall time: 28.534 seconds
[ Info: i: 0600, t: 1.619 hours, Δt: 6.914 seconds, umax = (9.5e-02, 5.8e-02, 4.2e-02) ms⁻¹, wall time: 29.023 seconds
[ Info: i: 0620, t: 1.656 hours, Δt: 7.210 seconds, umax = (9.1e-02, 5.9e-02, 4.3e-02) ms⁻¹, wall time: 29.463 seconds
[ Info: i: 0640, t: 1.695 hours, Δt: 6.850 seconds, umax = (9.9e-02, 6.0e-02, 4.2e-02) ms⁻¹, wall time: 29.960 seconds
[ Info: i: 0660, t: 1.734 hours, Δt: 7.134 seconds, umax = (9.2e-02, 5.3e-02, 4.3e-02) ms⁻¹, wall time: 30.400 seconds
[ Info: i: 0680, t: 1.774 hours, Δt: 7.357 seconds, umax = (9.9e-02, 5.8e-02, 4.1e-02) ms⁻¹, wall time: 30.895 seconds
[ Info: i: 0700, t: 1.814 hours, Δt: 6.571 seconds, umax = (1.0e-01, 6.3e-02, 4.2e-02) ms⁻¹, wall time: 31.334 seconds
[ Info: i: 0720, t: 1.851 hours, Δt: 7.074 seconds, umax = (9.6e-02, 6.2e-02, 4.3e-02) ms⁻¹, wall time: 31.836 seconds
[ Info: i: 0740, t: 1.889 hours, Δt: 6.915 seconds, umax = (9.5e-02, 5.9e-02, 4.5e-02) ms⁻¹, wall time: 32.273 seconds
[ Info: i: 0760, t: 1.926 hours, Δt: 6.930 seconds, umax = (9.6e-02, 5.7e-02, 4.2e-02) ms⁻¹, wall time: 32.829 seconds
[ Info: i: 0780, t: 1.964 hours, Δt: 6.409 seconds, umax = (1.0e-01, 5.6e-02, 4.5e-02) ms⁻¹, wall time: 33.221 seconds
[ Info: i: 0800, t: 2 hours, Δt: 6.976 seconds, umax = (1.0e-01, 5.6e-02, 4.4e-02) ms⁻¹, wall time: 33.674 seconds
[ Info: i: 0820, t: 2.039 hours, Δt: 6.686 seconds, umax = (9.8e-02, 6.0e-02, 4.2e-02) ms⁻¹, wall time: 34.178 seconds
[ Info: i: 0840, t: 2.076 hours, Δt: 6.666 seconds, umax = (1.0e-01, 6.1e-02, 4.5e-02) ms⁻¹, wall time: 34.632 seconds
[ Info: i: 0860, t: 2.113 hours, Δt: 6.821 seconds, umax = (1.0e-01, 6.3e-02, 4.4e-02) ms⁻¹, wall time: 35.138 seconds
[ Info: i: 0880, t: 2.151 hours, Δt: 6.836 seconds, umax = (1.1e-01, 6.1e-02, 4.4e-02) ms⁻¹, wall time: 35.589 seconds
[ Info: i: 0900, t: 2.187 hours, Δt: 6.186 seconds, umax = (1.0e-01, 5.9e-02, 4.5e-02) ms⁻¹, wall time: 36.095 seconds
[ Info: i: 0920, t: 2.220 hours, Δt: 6.296 seconds, umax = (1.1e-01, 6.0e-02, 4.6e-02) ms⁻¹, wall time: 36.540 seconds
[ Info: i: 0940, t: 2.255 hours, Δt: 6.606 seconds, umax = (9.9e-02, 5.7e-02, 4.5e-02) ms⁻¹, wall time: 37.137 seconds
[ Info: i: 0960, t: 2.292 hours, Δt: 6.594 seconds, umax = (1.1e-01, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 37.498 seconds
[ Info: i: 0980, t: 2.329 hours, Δt: 6.677 seconds, umax = (1.0e-01, 6.6e-02, 4.3e-02) ms⁻¹, wall time: 37.957 seconds
[ Info: i: 1000, t: 2.364 hours, Δt: 6.359 seconds, umax = (1.1e-01, 6.1e-02, 4.5e-02) ms⁻¹, wall time: 38.476 seconds
[ Info: i: 1020, t: 2.399 hours, Δt: 6.682 seconds, umax = (1.1e-01, 6.5e-02, 4.2e-02) ms⁻¹, wall time: 38.935 seconds
[ Info: i: 1040, t: 2.434 hours, Δt: 6.385 seconds, umax = (1.0e-01, 6.2e-02, 4.3e-02) ms⁻¹, wall time: 39.454 seconds
[ Info: i: 1060, t: 2.470 hours, Δt: 5.969 seconds, umax = (1.1e-01, 6.3e-02, 4.5e-02) ms⁻¹, wall time: 39.896 seconds
[ Info: i: 1080, t: 2.503 hours, Δt: 6.248 seconds, umax = (1.0e-01, 6.5e-02, 4.4e-02) ms⁻¹, wall time: 40.530 seconds
[ Info: i: 1100, t: 2.538 hours, Δt: 6.318 seconds, umax = (1.1e-01, 6.2e-02, 4.7e-02) ms⁻¹, wall time: 40.873 seconds
[ Info: i: 1120, t: 2.573 hours, Δt: 5.663 seconds, umax = (1.1e-01, 7.0e-02, 4.3e-02) ms⁻¹, wall time: 41.335 seconds
[ Info: i: 1140, t: 2.605 hours, Δt: 6.409 seconds, umax = (1.1e-01, 6.7e-02, 4.2e-02) ms⁻¹, wall time: 41.845 seconds
[ Info: i: 1160, t: 2.640 hours, Δt: 6.418 seconds, umax = (1.1e-01, 7.1e-02, 4.4e-02) ms⁻¹, wall time: 42.302 seconds
[ Info: i: 1180, t: 2.675 hours, Δt: 5.877 seconds, umax = (1.1e-01, 6.6e-02, 4.6e-02) ms⁻¹, wall time: 42.871 seconds
[ Info: i: 1200, t: 2.709 hours, Δt: 6.100 seconds, umax = (1.1e-01, 6.9e-02, 4.5e-02) ms⁻¹, wall time: 43.269 seconds
[ Info: i: 1220, t: 2.743 hours, Δt: 5.946 seconds, umax = (1.0e-01, 7.7e-02, 4.8e-02) ms⁻¹, wall time: 43.731 seconds
[ Info: i: 1240, t: 2.774 hours, Δt: 6.125 seconds, umax = (1.1e-01, 7.4e-02, 4.9e-02) ms⁻¹, wall time: 44.257 seconds
[ Info: i: 1260, t: 2.808 hours, Δt: 6.021 seconds, umax = (1.1e-01, 7.0e-02, 5.3e-02) ms⁻¹, wall time: 44.713 seconds
[ Info: i: 1280, t: 2.840 hours, Δt: 6.036 seconds, umax = (1.1e-01, 7.1e-02, 5.0e-02) ms⁻¹, wall time: 45.299 seconds
[ Info: i: 1300, t: 2.874 hours, Δt: 6.106 seconds, umax = (1.1e-01, 7.2e-02, 4.6e-02) ms⁻¹, wall time: 45.680 seconds
[ Info: i: 1320, t: 2.908 hours, Δt: 5.707 seconds, umax = (1.1e-01, 7.7e-02, 4.3e-02) ms⁻¹, wall time: 46.141 seconds
[ Info: i: 1340, t: 2.940 hours, Δt: 6.050 seconds, umax = (1.1e-01, 6.8e-02, 5.0e-02) ms⁻¹, wall time: 46.660 seconds
[ Info: i: 1360, t: 2.974 hours, Δt: 6.388 seconds, umax = (1.1e-01, 6.7e-02, 5.1e-02) ms⁻¹, wall time: 47.118 seconds
[ Info: i: 1380, t: 3.009 hours, Δt: 6.305 seconds, umax = (1.0e-01, 6.5e-02, 4.9e-02) ms⁻¹, wall time: 47.689 seconds
[ Info: i: 1400, t: 3.044 hours, Δt: 6.123 seconds, umax = (1.0e-01, 6.7e-02, 5.0e-02) ms⁻¹, wall time: 48.090 seconds
[ Info: i: 1420, t: 3.077 hours, Δt: 5.829 seconds, umax = (1.0e-01, 7.6e-02, 4.6e-02) ms⁻¹, wall time: 48.555 seconds
[ Info: i: 1440, t: 3.109 hours, Δt: 6.156 seconds, umax = (1.1e-01, 6.8e-02, 4.8e-02) ms⁻¹, wall time: 49.081 seconds
[ Info: i: 1460, t: 3.144 hours, Δt: 6.173 seconds, umax = (1.1e-01, 7.1e-02, 4.6e-02) ms⁻¹, wall time: 49.542 seconds
[ Info: i: 1480, t: 3.177 hours, Δt: 6.139 seconds, umax = (1.1e-01, 7.5e-02, 4.7e-02) ms⁻¹, wall time: 50.083 seconds
[ Info: i: 1500, t: 3.210 hours, Δt: 6.086 seconds, umax = (1.1e-01, 7.8e-02, 4.6e-02) ms⁻¹, wall time: 50.506 seconds
[ Info: i: 1520, t: 3.243 hours, Δt: 5.849 seconds, umax = (1.1e-01, 7.1e-02, 4.9e-02) ms⁻¹, wall time: 50.968 seconds
[ Info: i: 1540, t: 3.275 hours, Δt: 5.839 seconds, umax = (1.1e-01, 8.0e-02, 5.2e-02) ms⁻¹, wall time: 51.482 seconds
[ Info: i: 1560, t: 3.308 hours, Δt: 5.896 seconds, umax = (1.1e-01, 8.6e-02, 5.2e-02) ms⁻¹, wall time: 51.940 seconds
[ Info: i: 1580, t: 3.340 hours, Δt: 6.143 seconds, umax = (1.1e-01, 7.1e-02, 4.7e-02) ms⁻¹, wall time: 52.518 seconds
[ Info: i: 1600, t: 3.374 hours, Δt: 5.824 seconds, umax = (1.1e-01, 8.6e-02, 4.5e-02) ms⁻¹, wall time: 52.899 seconds
[ Info: i: 1620, t: 3.407 hours, Δt: 5.684 seconds, umax = (1.1e-01, 7.9e-02, 4.5e-02) ms⁻¹, wall time: 53.364 seconds
[ Info: i: 1640, t: 3.438 hours, Δt: 6.074 seconds, umax = (1.1e-01, 8.1e-02, 4.5e-02) ms⁻¹, wall time: 53.887 seconds
[ Info: i: 1660, t: 3.472 hours, Δt: 5.548 seconds, umax = (1.1e-01, 7.8e-02, 4.9e-02) ms⁻¹, wall time: 54.342 seconds
[ Info: i: 1680, t: 3.502 hours, Δt: 5.321 seconds, umax = (1.2e-01, 8.1e-02, 4.7e-02) ms⁻¹, wall time: 55.002 seconds
[ Info: i: 1700, t: 3.533 hours, Δt: 6.132 seconds, umax = (1.2e-01, 7.2e-02, 5.2e-02) ms⁻¹, wall time: 55.329 seconds
[ Info: i: 1720, t: 3.567 hours, Δt: 6.093 seconds, umax = (1.1e-01, 8.3e-02, 4.9e-02) ms⁻¹, wall time: 55.792 seconds
[ Info: i: 1740, t: 3.601 hours, Δt: 6.237 seconds, umax = (1.1e-01, 7.1e-02, 5.1e-02) ms⁻¹, wall time: 56.311 seconds
[ Info: i: 1760, t: 3.635 hours, Δt: 6.080 seconds, umax = (1.1e-01, 7.1e-02, 5.2e-02) ms⁻¹, wall time: 56.756 seconds
[ Info: i: 1780, t: 3.668 hours, Δt: 6.083 seconds, umax = (1.0e-01, 7.2e-02, 5.1e-02) ms⁻¹, wall time: 57.405 seconds
[ Info: i: 1800, t: 3.702 hours, Δt: 5.315 seconds, umax = (1.1e-01, 8.8e-02, 5.0e-02) ms⁻¹, wall time: 57.733 seconds
[ Info: i: 1820, t: 3.733 hours, Δt: 5.863 seconds, umax = (1.1e-01, 7.6e-02, 5.5e-02) ms⁻¹, wall time: 58.194 seconds
[ Info: i: 1840, t: 3.765 hours, Δt: 5.793 seconds, umax = (1.1e-01, 8.2e-02, 5.2e-02) ms⁻¹, wall time: 58.711 seconds
[ Info: i: 1860, t: 3.798 hours, Δt: 5.968 seconds, umax = (1.1e-01, 7.9e-02, 4.9e-02) ms⁻¹, wall time: 59.158 seconds
[ Info: i: 1880, t: 3.831 hours, Δt: 5.877 seconds, umax = (1.2e-01, 7.6e-02, 4.9e-02) ms⁻¹, wall time: 59.622 seconds
[ Info: i: 1900, t: 3.862 hours, Δt: 5.690 seconds, umax = (1.1e-01, 8.0e-02, 5.4e-02) ms⁻¹, wall time: 1.002 minutes
[ Info: i: 1920, t: 3.894 hours, Δt: 5.971 seconds, umax = (1.1e-01, 8.0e-02, 5.6e-02) ms⁻¹, wall time: 1.010 minutes
[ Info: i: 1940, t: 3.926 hours, Δt: 5.070 seconds, umax = (1.1e-01, 8.2e-02, 4.9e-02) ms⁻¹, wall time: 1.019 minutes
[ Info: i: 1960, t: 3.955 hours, Δt: 5.975 seconds, umax = (1.2e-01, 8.6e-02, 4.4e-02) ms⁻¹, wall time: 1.026 minutes
[ Info: i: 1980, t: 3.988 hours, Δt: 5.896 seconds, umax = (1.1e-01, 7.4e-02, 4.7e-02) ms⁻¹, wall time: 1.034 minutes
[ Info: Simulation is stopping after running for 1.037 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-28839/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-28839/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-28839/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 v0.14.10
  [9e8cae18] Oceananigans v0.104.2 `..`
  [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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