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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, 8.5e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (9.343 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.931 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 19.805 seconds, umax = (3.7e-02, 1.2e-02, 2.3e-02) ms⁻¹, wall time: 13.689 seconds
[ Info: i: 0040, t: 16.952 minutes, Δt: 13.541 seconds, umax = (5.5e-02, 2.3e-02, 2.1e-02) ms⁻¹, wall time: 14.255 seconds
[ Info: i: 0060, t: 21.213 minutes, Δt: 11.232 seconds, umax = (6.5e-02, 2.7e-02, 3.0e-02) ms⁻¹, wall time: 14.990 seconds
[ Info: i: 0080, t: 24.941 minutes, Δt: 9.756 seconds, umax = (7.0e-02, 3.2e-02, 3.1e-02) ms⁻¹, wall time: 15.634 seconds
[ Info: i: 0100, t: 28.252 minutes, Δt: 10.712 seconds, umax = (6.1e-02, 3.1e-02, 3.3e-02) ms⁻¹, wall time: 16.301 seconds
[ Info: i: 0120, t: 31.830 minutes, Δt: 10.356 seconds, umax = (6.5e-02, 3.2e-02, 3.0e-02) ms⁻¹, wall time: 16.792 seconds
[ Info: i: 0140, t: 35.362 minutes, Δt: 10.309 seconds, umax = (7.0e-02, 3.6e-02, 3.2e-02) ms⁻¹, wall time: 17.348 seconds
[ Info: i: 0160, t: 38.864 minutes, Δt: 10.093 seconds, umax = (7.2e-02, 3.6e-02, 2.9e-02) ms⁻¹, wall time: 17.682 seconds
[ Info: i: 0180, t: 42.227 minutes, Δt: 10.027 seconds, umax = (6.7e-02, 3.6e-02, 3.2e-02) ms⁻¹, wall time: 18.252 seconds
[ Info: i: 0200, t: 45.483 minutes, Δt: 9.508 seconds, umax = (7.6e-02, 4.5e-02, 3.4e-02) ms⁻¹, wall time: 19.073 seconds
[ Info: i: 0220, t: 48.659 minutes, Δt: 9.242 seconds, umax = (7.8e-02, 3.8e-02, 3.4e-02) ms⁻¹, wall time: 19.638 seconds
[ Info: i: 0240, t: 51.722 minutes, Δt: 8.954 seconds, umax = (7.4e-02, 4.3e-02, 3.5e-02) ms⁻¹, wall time: 20.342 seconds
[ Info: i: 0260, t: 54.622 minutes, Δt: 8.277 seconds, umax = (7.8e-02, 4.6e-02, 3.8e-02) ms⁻¹, wall time: 21.053 seconds
[ Info: i: 0280, t: 57.433 minutes, Δt: 8.150 seconds, umax = (7.6e-02, 4.3e-02, 3.7e-02) ms⁻¹, wall time: 21.724 seconds
[ Info: i: 0300, t: 1.002 hours, Δt: 8.693 seconds, umax = (7.9e-02, 4.6e-02, 3.6e-02) ms⁻¹, wall time: 22.633 seconds
[ Info: i: 0320, t: 1.049 hours, Δt: 8.597 seconds, umax = (7.8e-02, 4.4e-02, 4.1e-02) ms⁻¹, wall time: 23.105 seconds
[ Info: i: 0340, t: 1.094 hours, Δt: 8.549 seconds, umax = (8.1e-02, 4.9e-02, 3.8e-02) ms⁻¹, wall time: 24.001 seconds
[ Info: i: 0360, t: 1.140 hours, Δt: 7.935 seconds, umax = (8.0e-02, 4.9e-02, 3.6e-02) ms⁻¹, wall time: 24.603 seconds
[ Info: i: 0380, t: 1.182 hours, Δt: 7.360 seconds, umax = (8.5e-02, 5.0e-02, 4.1e-02) ms⁻¹, wall time: 25.341 seconds
[ Info: i: 0400, t: 1.224 hours, Δt: 7.028 seconds, umax = (8.6e-02, 5.2e-02, 3.8e-02) ms⁻¹, wall time: 26.016 seconds
[ Info: i: 0420, t: 1.265 hours, Δt: 7.731 seconds, umax = (8.7e-02, 5.2e-02, 3.8e-02) ms⁻¹, wall time: 26.763 seconds
[ Info: i: 0440, t: 1.309 hours, Δt: 7.806 seconds, umax = (9.2e-02, 5.2e-02, 3.8e-02) ms⁻¹, wall time: 27.444 seconds
[ Info: i: 0460, t: 1.351 hours, Δt: 7.235 seconds, umax = (8.9e-02, 5.2e-02, 3.9e-02) ms⁻¹, wall time: 28.193 seconds
[ Info: i: 0480, t: 1.392 hours, Δt: 7.050 seconds, umax = (8.9e-02, 5.2e-02, 4.0e-02) ms⁻¹, wall time: 28.855 seconds
[ Info: i: 0500, t: 1.432 hours, Δt: 7.789 seconds, umax = (9.4e-02, 5.3e-02, 3.9e-02) ms⁻¹, wall time: 29.627 seconds
[ Info: i: 0520, t: 1.475 hours, Δt: 7.101 seconds, umax = (9.1e-02, 5.4e-02, 4.3e-02) ms⁻¹, wall time: 30.312 seconds
[ Info: i: 0540, t: 1.514 hours, Δt: 7.524 seconds, umax = (9.1e-02, 5.2e-02, 4.4e-02) ms⁻¹, wall time: 31.126 seconds
[ Info: i: 0560, t: 1.556 hours, Δt: 7.174 seconds, umax = (1.0e-01, 4.9e-02, 4.5e-02) ms⁻¹, wall time: 31.647 seconds
[ Info: i: 0580, t: 1.596 hours, Δt: 7.150 seconds, umax = (9.6e-02, 5.2e-02, 4.2e-02) ms⁻¹, wall time: 32.300 seconds
[ Info: i: 0600, t: 1.636 hours, Δt: 7.426 seconds, umax = (9.6e-02, 5.7e-02, 4.4e-02) ms⁻¹, wall time: 32.912 seconds
[ Info: i: 0620, t: 1.674 hours, Δt: 6.685 seconds, umax = (9.4e-02, 6.0e-02, 4.3e-02) ms⁻¹, wall time: 33.832 seconds
[ Info: i: 0640, t: 1.713 hours, Δt: 7.144 seconds, umax = (9.3e-02, 5.6e-02, 4.1e-02) ms⁻¹, wall time: 34.431 seconds
[ Info: i: 0660, t: 1.750 hours, Δt: 6.915 seconds, umax = (9.8e-02, 5.4e-02, 4.3e-02) ms⁻¹, wall time: 35.130 seconds
[ Info: i: 0680, t: 1.790 hours, Δt: 7.124 seconds, umax = (9.8e-02, 5.1e-02, 4.2e-02) ms⁻¹, wall time: 35.849 seconds
[ Info: i: 0700, t: 1.828 hours, Δt: 7.057 seconds, umax = (1.0e-01, 6.4e-02, 4.5e-02) ms⁻¹, wall time: 36.585 seconds
[ Info: i: 0720, t: 1.867 hours, Δt: 6.798 seconds, umax = (1.0e-01, 5.9e-02, 4.0e-02) ms⁻¹, wall time: 37.322 seconds
[ Info: i: 0740, t: 1.905 hours, Δt: 6.714 seconds, umax = (1.0e-01, 5.3e-02, 4.1e-02) ms⁻¹, wall time: 38.066 seconds
[ Info: i: 0760, t: 1.941 hours, Δt: 6.837 seconds, umax = (1.0e-01, 6.9e-02, 4.2e-02) ms⁻¹, wall time: 38.815 seconds
[ Info: i: 0780, t: 1.978 hours, Δt: 6.327 seconds, umax = (1.1e-01, 6.0e-02, 4.3e-02) ms⁻¹, wall time: 39.545 seconds
[ Info: i: 0800, t: 2.013 hours, Δt: 6.361 seconds, umax = (1.0e-01, 6.5e-02, 4.7e-02) ms⁻¹, wall time: 40.294 seconds
[ Info: i: 0820, t: 2.048 hours, Δt: 6.456 seconds, umax = (1.0e-01, 5.9e-02, 4.3e-02) ms⁻¹, wall time: 40.981 seconds
[ Info: i: 0840, t: 2.083 hours, Δt: 6.566 seconds, umax = (9.9e-02, 6.2e-02, 5.1e-02) ms⁻¹, wall time: 41.722 seconds
[ Info: i: 0860, t: 2.121 hours, Δt: 6.789 seconds, umax = (9.5e-02, 5.6e-02, 4.3e-02) ms⁻¹, wall time: 42.558 seconds
[ Info: i: 0880, t: 2.159 hours, Δt: 6.600 seconds, umax = (9.8e-02, 6.1e-02, 4.1e-02) ms⁻¹, wall time: 43.312 seconds
[ Info: i: 0900, t: 2.194 hours, Δt: 6.616 seconds, umax = (1.0e-01, 6.3e-02, 4.6e-02) ms⁻¹, wall time: 44.053 seconds
[ Info: i: 0920, t: 2.231 hours, Δt: 6.281 seconds, umax = (1.0e-01, 6.4e-02, 4.2e-02) ms⁻¹, wall time: 44.844 seconds
[ Info: i: 0940, t: 2.266 hours, Δt: 6.868 seconds, umax = (1.0e-01, 6.3e-02, 4.3e-02) ms⁻¹, wall time: 45.578 seconds
[ Info: i: 0960, t: 2.304 hours, Δt: 6.358 seconds, umax = (1.0e-01, 6.3e-02, 4.5e-02) ms⁻¹, wall time: 46.033 seconds
[ Info: i: 0980, t: 2.339 hours, Δt: 6.634 seconds, umax = (1.0e-01, 6.9e-02, 4.5e-02) ms⁻¹, wall time: 46.712 seconds
[ Info: i: 1000, t: 2.374 hours, Δt: 6.452 seconds, umax = (1.0e-01, 6.2e-02, 4.5e-02) ms⁻¹, wall time: 47.078 seconds
[ Info: i: 1020, t: 2.410 hours, Δt: 6.741 seconds, umax = (1.0e-01, 6.5e-02, 4.7e-02) ms⁻¹, wall time: 47.559 seconds
[ Info: i: 1040, t: 2.445 hours, Δt: 6.344 seconds, umax = (1.1e-01, 6.6e-02, 4.2e-02) ms⁻¹, wall time: 48.145 seconds
[ Info: i: 1060, t: 2.478 hours, Δt: 6.181 seconds, umax = (1.1e-01, 7.2e-02, 4.9e-02) ms⁻¹, wall time: 48.674 seconds
[ Info: i: 1080, t: 2.512 hours, Δt: 6.640 seconds, umax = (1.0e-01, 6.4e-02, 5.0e-02) ms⁻¹, wall time: 49.209 seconds
[ Info: i: 1100, t: 2.549 hours, Δt: 6.223 seconds, umax = (1.1e-01, 6.3e-02, 4.5e-02) ms⁻¹, wall time: 49.704 seconds
[ Info: i: 1120, t: 2.583 hours, Δt: 6.098 seconds, umax = (1.0e-01, 6.7e-02, 4.4e-02) ms⁻¹, wall time: 50.176 seconds
[ Info: i: 1140, t: 2.617 hours, Δt: 6.192 seconds, umax = (1.1e-01, 6.8e-02, 4.6e-02) ms⁻¹, wall time: 50.659 seconds
[ Info: i: 1160, t: 2.650 hours, Δt: 6.103 seconds, umax = (1.1e-01, 6.4e-02, 4.3e-02) ms⁻¹, wall time: 51.128 seconds
[ Info: i: 1180, t: 2.683 hours, Δt: 5.466 seconds, umax = (1.1e-01, 6.5e-02, 4.6e-02) ms⁻¹, wall time: 51.687 seconds
[ Info: i: 1200, t: 2.715 hours, Δt: 6.612 seconds, umax = (1.1e-01, 7.0e-02, 4.8e-02) ms⁻¹, wall time: 52.141 seconds
[ Info: i: 1220, t: 2.750 hours, Δt: 6.337 seconds, umax = (1.0e-01, 7.4e-02, 4.5e-02) ms⁻¹, wall time: 52.609 seconds
[ Info: i: 1240, t: 2.785 hours, Δt: 5.796 seconds, umax = (1.0e-01, 7.0e-02, 5.4e-02) ms⁻¹, wall time: 53.105 seconds
[ Info: i: 1260, t: 2.818 hours, Δt: 6.134 seconds, umax = (1.0e-01, 7.3e-02, 5.3e-02) ms⁻¹, wall time: 53.555 seconds
[ Info: i: 1280, t: 2.850 hours, Δt: 6.151 seconds, umax = (1.1e-01, 7.0e-02, 5.0e-02) ms⁻¹, wall time: 54.149 seconds
[ Info: i: 1300, t: 2.884 hours, Δt: 6.168 seconds, umax = (1.2e-01, 6.6e-02, 4.6e-02) ms⁻¹, wall time: 54.605 seconds
[ Info: i: 1320, t: 2.918 hours, Δt: 6.146 seconds, umax = (1.1e-01, 7.2e-02, 4.5e-02) ms⁻¹, wall time: 55.265 seconds
[ Info: i: 1340, t: 2.952 hours, Δt: 6.283 seconds, umax = (1.1e-01, 7.0e-02, 4.4e-02) ms⁻¹, wall time: 55.603 seconds
[ Info: i: 1360, t: 2.987 hours, Δt: 6.186 seconds, umax = (1.1e-01, 6.9e-02, 4.6e-02) ms⁻¹, wall time: 56.084 seconds
[ Info: i: 1380, t: 3.021 hours, Δt: 5.905 seconds, umax = (1.1e-01, 7.5e-02, 4.6e-02) ms⁻¹, wall time: 56.586 seconds
[ Info: i: 1400, t: 3.054 hours, Δt: 5.669 seconds, umax = (1.1e-01, 7.5e-02, 4.9e-02) ms⁻¹, wall time: 57.044 seconds
[ Info: i: 1420, t: 3.085 hours, Δt: 5.677 seconds, umax = (1.0e-01, 7.5e-02, 4.7e-02) ms⁻¹, wall time: 57.674 seconds
[ Info: i: 1440, t: 3.117 hours, Δt: 5.743 seconds, umax = (1.0e-01, 7.3e-02, 4.8e-02) ms⁻¹, wall time: 58.017 seconds
[ Info: i: 1460, t: 3.149 hours, Δt: 5.711 seconds, umax = (1.1e-01, 7.5e-02, 5.4e-02) ms⁻¹, wall time: 58.485 seconds
[ Info: i: 1480, t: 3.180 hours, Δt: 6.323 seconds, umax = (1.0e-01, 8.3e-02, 4.2e-02) ms⁻¹, wall time: 58.973 seconds
[ Info: i: 1500, t: 3.214 hours, Δt: 6.113 seconds, umax = (1.1e-01, 7.8e-02, 5.0e-02) ms⁻¹, wall time: 59.439 seconds
[ Info: i: 1520, t: 3.247 hours, Δt: 6.108 seconds, umax = (1.1e-01, 7.6e-02, 5.5e-02) ms⁻¹, wall time: 59.905 seconds
[ Info: i: 1540, t: 3.280 hours, Δt: 6.516 seconds, umax = (1.0e-01, 7.4e-02, 5.1e-02) ms⁻¹, wall time: 1.007 minutes
[ Info: i: 1560, t: 3.315 hours, Δt: 6.207 seconds, umax = (1.1e-01, 8.1e-02, 4.5e-02) ms⁻¹, wall time: 1.014 minutes
[ Info: i: 1580, t: 3.349 hours, Δt: 6.154 seconds, umax = (1.1e-01, 7.7e-02, 4.5e-02) ms⁻¹, wall time: 1.022 minutes
[ Info: i: 1600, t: 3.383 hours, Δt: 5.572 seconds, umax = (1.2e-01, 8.6e-02, 4.3e-02) ms⁻¹, wall time: 1.030 minutes
[ Info: i: 1620, t: 3.415 hours, Δt: 5.959 seconds, umax = (1.1e-01, 6.9e-02, 4.6e-02) ms⁻¹, wall time: 1.039 minutes
[ Info: i: 1640, t: 3.448 hours, Δt: 6.260 seconds, umax = (9.9e-02, 7.2e-02, 4.4e-02) ms⁻¹, wall time: 1.047 minutes
[ Info: i: 1660, t: 3.482 hours, Δt: 6.111 seconds, umax = (1.0e-01, 7.5e-02, 4.9e-02) ms⁻¹, wall time: 1.055 minutes
[ Info: i: 1680, t: 3.515 hours, Δt: 6.148 seconds, umax = (1.1e-01, 7.5e-02, 4.4e-02) ms⁻¹, wall time: 1.063 minutes
[ Info: i: 1700, t: 3.549 hours, Δt: 5.889 seconds, umax = (1.1e-01, 7.9e-02, 4.9e-02) ms⁻¹, wall time: 1.071 minutes
[ Info: i: 1720, t: 3.580 hours, Δt: 5.796 seconds, umax = (1.0e-01, 7.1e-02, 4.5e-02) ms⁻¹, wall time: 1.080 minutes
[ Info: i: 1740, t: 3.612 hours, Δt: 5.817 seconds, umax = (1.1e-01, 7.2e-02, 4.9e-02) ms⁻¹, wall time: 1.088 minutes
[ Info: i: 1760, t: 3.645 hours, Δt: 6.075 seconds, umax = (1.1e-01, 7.4e-02, 4.4e-02) ms⁻¹, wall time: 1.096 minutes
[ Info: i: 1780, t: 3.677 hours, Δt: 5.584 seconds, umax = (1.3e-01, 7.8e-02, 4.9e-02) ms⁻¹, wall time: 1.105 minutes
[ Info: i: 1800, t: 3.708 hours, Δt: 6.011 seconds, umax = (1.2e-01, 7.7e-02, 4.6e-02) ms⁻¹, wall time: 1.112 minutes
[ Info: i: 1820, t: 3.741 hours, Δt: 5.866 seconds, umax = (1.2e-01, 7.6e-02, 4.5e-02) ms⁻¹, wall time: 1.120 minutes
[ Info: i: 1840, t: 3.772 hours, Δt: 5.562 seconds, umax = (1.1e-01, 7.7e-02, 4.9e-02) ms⁻¹, wall time: 1.128 minutes
[ Info: i: 1860, t: 3.803 hours, Δt: 5.655 seconds, umax = (1.1e-01, 7.8e-02, 5.6e-02) ms⁻¹, wall time: 1.136 minutes
[ Info: i: 1880, t: 3.833 hours, Δt: 5.777 seconds, umax = (1.1e-01, 7.1e-02, 4.8e-02) ms⁻¹, wall time: 1.144 minutes
[ Info: i: 1900, t: 3.866 hours, Δt: 6.033 seconds, umax = (1.1e-01, 7.6e-02, 5.2e-02) ms⁻¹, wall time: 1.153 minutes
[ Info: i: 1920, t: 3.899 hours, Δt: 5.809 seconds, umax = (1.2e-01, 8.0e-02, 4.7e-02) ms⁻¹, wall time: 1.160 minutes
[ Info: i: 1940, t: 3.932 hours, Δt: 5.921 seconds, umax = (1.1e-01, 7.4e-02, 4.9e-02) ms⁻¹, wall time: 1.170 minutes
[ Info: i: 1960, t: 3.965 hours, Δt: 5.900 seconds, umax = (1.0e-01, 7.0e-02, 4.9e-02) ms⁻¹, wall time: 1.177 minutes
[ Info: i: 1980, t: 3.998 hours, Δt: 6.033 seconds, umax = (1.1e-01, 7.6e-02, 4.8e-02) ms⁻¹, wall time: 1.185 minutes
[ Info: Simulation is stopping after running for 1.188 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-29180/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-29180/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-29180/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 v0.104.3 `..`
  [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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