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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: 44.8 KiB

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: 44.4 KiB

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, 9.4e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (17.869 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.922 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 19.009 seconds, umax = (3.5e-02, 1.2e-02, 2.0e-02) ms⁻¹, wall time: 21.924 seconds
[ Info: i: 0040, t: 16.835 minutes, Δt: 13.092 seconds, umax = (5.5e-02, 2.0e-02, 2.3e-02) ms⁻¹, wall time: 22.312 seconds
[ Info: i: 0060, t: 21.026 minutes, Δt: 11.274 seconds, umax = (6.2e-02, 3.2e-02, 3.1e-02) ms⁻¹, wall time: 22.816 seconds
[ Info: i: 0080, t: 24.657 minutes, Δt: 11.214 seconds, umax = (6.2e-02, 3.1e-02, 3.4e-02) ms⁻¹, wall time: 23.198 seconds
[ Info: i: 0100, t: 28.521 minutes, Δt: 10.476 seconds, umax = (6.3e-02, 3.4e-02, 3.8e-02) ms⁻¹, wall time: 23.636 seconds
[ Info: i: 0120, t: 32.034 minutes, Δt: 11.097 seconds, umax = (6.5e-02, 3.1e-02, 3.2e-02) ms⁻¹, wall time: 24.076 seconds
[ Info: i: 0140, t: 35.543 minutes, Δt: 10.343 seconds, umax = (6.7e-02, 3.4e-02, 3.0e-02) ms⁻¹, wall time: 24.616 seconds
[ Info: i: 0160, t: 38.987 minutes, Δt: 9.531 seconds, umax = (7.1e-02, 3.5e-02, 3.2e-02) ms⁻¹, wall time: 24.957 seconds
[ Info: i: 0180, t: 42.073 minutes, Δt: 9.749 seconds, umax = (7.7e-02, 3.6e-02, 3.3e-02) ms⁻¹, wall time: 25.402 seconds
[ Info: i: 0200, t: 45.333 minutes, Δt: 9.253 seconds, umax = (7.3e-02, 4.1e-02, 3.5e-02) ms⁻¹, wall time: 26.040 seconds
[ Info: i: 0220, t: 48.354 minutes, Δt: 8.755 seconds, umax = (7.4e-02, 3.9e-02, 3.6e-02) ms⁻¹, wall time: 26.351 seconds
[ Info: i: 0240, t: 51.190 minutes, Δt: 9.198 seconds, umax = (7.3e-02, 4.3e-02, 3.5e-02) ms⁻¹, wall time: 26.824 seconds
[ Info: i: 0260, t: 54.188 minutes, Δt: 8.462 seconds, umax = (7.6e-02, 4.2e-02, 3.4e-02) ms⁻¹, wall time: 27.289 seconds
[ Info: i: 0280, t: 57.071 minutes, Δt: 8.989 seconds, umax = (7.8e-02, 4.0e-02, 3.6e-02) ms⁻¹, wall time: 27.773 seconds
[ Info: i: 0300, t: 1 hour, Δt: 8.724 seconds, umax = (8.3e-02, 4.4e-02, 3.6e-02) ms⁻¹, wall time: 28.201 seconds
[ Info: i: 0320, t: 1.048 hours, Δt: 8.311 seconds, umax = (7.9e-02, 5.2e-02, 4.0e-02) ms⁻¹, wall time: 28.654 seconds
[ Info: i: 0340, t: 1.093 hours, Δt: 8.581 seconds, umax = (8.3e-02, 4.7e-02, 3.7e-02) ms⁻¹, wall time: 29.180 seconds
[ Info: i: 0360, t: 1.139 hours, Δt: 8.495 seconds, umax = (8.4e-02, 4.6e-02, 3.6e-02) ms⁻¹, wall time: 29.532 seconds
[ Info: i: 0380, t: 1.182 hours, Δt: 7.980 seconds, umax = (9.1e-02, 4.7e-02, 3.8e-02) ms⁻¹, wall time: 30.014 seconds
[ Info: i: 0400, t: 1.226 hours, Δt: 7.534 seconds, umax = (8.8e-02, 4.7e-02, 4.0e-02) ms⁻¹, wall time: 30.439 seconds
[ Info: i: 0420, t: 1.268 hours, Δt: 7.822 seconds, umax = (8.7e-02, 4.5e-02, 3.9e-02) ms⁻¹, wall time: 31.034 seconds
[ Info: i: 0440, t: 1.310 hours, Δt: 7.510 seconds, umax = (8.9e-02, 5.4e-02, 3.9e-02) ms⁻¹, wall time: 32.231 seconds
[ Info: i: 0460, t: 1.350 hours, Δt: 7.929 seconds, umax = (9.1e-02, 4.7e-02, 4.0e-02) ms⁻¹, wall time: 32.697 seconds
[ Info: i: 0480, t: 1.394 hours, Δt: 7.530 seconds, umax = (9.4e-02, 5.3e-02, 3.8e-02) ms⁻¹, wall time: 33.138 seconds
[ Info: i: 0500, t: 1.435 hours, Δt: 7.329 seconds, umax = (9.1e-02, 5.3e-02, 3.8e-02) ms⁻¹, wall time: 33.660 seconds
[ Info: i: 0520, t: 1.476 hours, Δt: 7.384 seconds, umax = (9.5e-02, 4.9e-02, 3.8e-02) ms⁻¹, wall time: 34.095 seconds
[ Info: i: 0540, t: 1.517 hours, Δt: 7.300 seconds, umax = (9.1e-02, 5.4e-02, 3.7e-02) ms⁻¹, wall time: 34.566 seconds
[ Info: i: 0560, t: 1.557 hours, Δt: 6.530 seconds, umax = (9.4e-02, 5.8e-02, 4.2e-02) ms⁻¹, wall time: 35.005 seconds
[ Info: i: 0580, t: 1.593 hours, Δt: 7.138 seconds, umax = (9.3e-02, 5.5e-02, 3.9e-02) ms⁻¹, wall time: 35.548 seconds
[ Info: i: 0600, t: 1.633 hours, Δt: 7.443 seconds, umax = (9.4e-02, 5.7e-02, 4.1e-02) ms⁻¹, wall time: 35.977 seconds
[ Info: i: 0620, t: 1.672 hours, Δt: 7.509 seconds, umax = (1.1e-01, 6.1e-02, 4.5e-02) ms⁻¹, wall time: 36.616 seconds
[ Info: i: 0640, t: 1.713 hours, Δt: 7.358 seconds, umax = (1.0e-01, 5.5e-02, 4.3e-02) ms⁻¹, wall time: 36.991 seconds
[ Info: i: 0660, t: 1.750 hours, Δt: 7.030 seconds, umax = (1.0e-01, 6.3e-02, 4.5e-02) ms⁻¹, wall time: 37.478 seconds
[ Info: i: 0680, t: 1.787 hours, Δt: 7.095 seconds, umax = (9.8e-02, 5.7e-02, 4.1e-02) ms⁻¹, wall time: 38.003 seconds
[ Info: i: 0700, t: 1.826 hours, Δt: 7.032 seconds, umax = (1.0e-01, 5.9e-02, 4.4e-02) ms⁻¹, wall time: 38.528 seconds
[ Info: i: 0720, t: 1.864 hours, Δt: 7.035 seconds, umax = (1.0e-01, 5.6e-02, 3.9e-02) ms⁻¹, wall time: 39.039 seconds
[ Info: i: 0740, t: 1.903 hours, Δt: 6.975 seconds, umax = (1.1e-01, 5.9e-02, 4.8e-02) ms⁻¹, wall time: 39.554 seconds
[ Info: i: 0760, t: 1.940 hours, Δt: 6.936 seconds, umax = (9.7e-02, 5.8e-02, 4.5e-02) ms⁻¹, wall time: 40.039 seconds
[ Info: i: 0780, t: 1.979 hours, Δt: 6.840 seconds, umax = (9.9e-02, 5.7e-02, 4.9e-02) ms⁻¹, wall time: 40.511 seconds
[ Info: i: 0800, t: 2.015 hours, Δt: 6.774 seconds, umax = (1.1e-01, 6.5e-02, 4.3e-02) ms⁻¹, wall time: 41.017 seconds
[ Info: i: 0820, t: 2.054 hours, Δt: 6.820 seconds, umax = (9.8e-02, 6.1e-02, 4.1e-02) ms⁻¹, wall time: 41.502 seconds
[ Info: i: 0840, t: 2.091 hours, Δt: 6.785 seconds, umax = (1.0e-01, 6.1e-02, 4.3e-02) ms⁻¹, wall time: 42.150 seconds
[ Info: i: 0860, t: 2.127 hours, Δt: 6.776 seconds, umax = (1.1e-01, 5.9e-02, 4.5e-02) ms⁻¹, wall time: 42.590 seconds
[ Info: i: 0880, t: 2.165 hours, Δt: 5.980 seconds, umax = (9.8e-02, 6.0e-02, 4.1e-02) ms⁻¹, wall time: 43.133 seconds
[ Info: i: 0900, t: 2.197 hours, Δt: 6.085 seconds, umax = (1.0e-01, 6.8e-02, 4.3e-02) ms⁻¹, wall time: 43.695 seconds
[ Info: i: 0920, t: 2.232 hours, Δt: 6.361 seconds, umax = (1.0e-01, 6.9e-02, 4.3e-02) ms⁻¹, wall time: 44.238 seconds
[ Info: i: 0940, t: 2.267 hours, Δt: 6.536 seconds, umax = (1.1e-01, 6.2e-02, 4.3e-02) ms⁻¹, wall time: 44.837 seconds
[ Info: i: 0960, t: 2.304 hours, Δt: 6.961 seconds, umax = (1.0e-01, 6.0e-02, 4.4e-02) ms⁻¹, wall time: 45.364 seconds
[ Info: i: 0980, t: 2.340 hours, Δt: 6.420 seconds, umax = (1.0e-01, 6.5e-02, 4.5e-02) ms⁻¹, wall time: 46.031 seconds
[ Info: i: 1000, t: 2.377 hours, Δt: 6.191 seconds, umax = (1.0e-01, 6.6e-02, 4.6e-02) ms⁻¹, wall time: 46.462 seconds
[ Info: i: 1020, t: 2.410 hours, Δt: 6.202 seconds, umax = (9.9e-02, 6.5e-02, 4.4e-02) ms⁻¹, wall time: 47.008 seconds
[ Info: i: 1040, t: 2.445 hours, Δt: 6.360 seconds, umax = (1.1e-01, 6.8e-02, 4.4e-02) ms⁻¹, wall time: 47.569 seconds
[ Info: i: 1060, t: 2.479 hours, Δt: 6.475 seconds, umax = (1.0e-01, 6.6e-02, 4.7e-02) ms⁻¹, wall time: 48.114 seconds
[ Info: i: 1080, t: 2.515 hours, Δt: 6.381 seconds, umax = (1.1e-01, 7.1e-02, 4.6e-02) ms⁻¹, wall time: 48.664 seconds
[ Info: i: 1100, t: 2.551 hours, Δt: 6.214 seconds, umax = (1.1e-01, 8.5e-02, 4.4e-02) ms⁻¹, wall time: 49.188 seconds
[ Info: i: 1120, t: 2.583 hours, Δt: 6.282 seconds, umax = (1.0e-01, 7.3e-02, 4.7e-02) ms⁻¹, wall time: 49.745 seconds
[ Info: i: 1140, t: 2.618 hours, Δt: 6.016 seconds, umax = (1.0e-01, 8.0e-02, 4.4e-02) ms⁻¹, wall time: 50.322 seconds
[ Info: i: 1160, t: 2.652 hours, Δt: 6.134 seconds, umax = (1.1e-01, 7.1e-02, 4.7e-02) ms⁻¹, wall time: 50.877 seconds
[ Info: i: 1180, t: 2.685 hours, Δt: 6.489 seconds, umax = (1.1e-01, 6.3e-02, 4.3e-02) ms⁻¹, wall time: 51.455 seconds
[ Info: i: 1200, t: 2.720 hours, Δt: 6.265 seconds, umax = (1.1e-01, 6.8e-02, 4.7e-02) ms⁻¹, wall time: 51.992 seconds
[ Info: i: 1220, t: 2.754 hours, Δt: 6.421 seconds, umax = (1.1e-01, 6.6e-02, 4.7e-02) ms⁻¹, wall time: 52.708 seconds
[ Info: i: 1240, t: 2.789 hours, Δt: 6.274 seconds, umax = (1.1e-01, 6.9e-02, 4.9e-02) ms⁻¹, wall time: 53.105 seconds
[ Info: i: 1260, t: 2.822 hours, Δt: 5.863 seconds, umax = (1.0e-01, 7.4e-02, 4.7e-02) ms⁻¹, wall time: 53.661 seconds
[ Info: i: 1280, t: 2.855 hours, Δt: 6.052 seconds, umax = (1.0e-01, 7.2e-02, 4.4e-02) ms⁻¹, wall time: 54.233 seconds
[ Info: i: 1300, t: 2.889 hours, Δt: 5.611 seconds, umax = (1.1e-01, 6.7e-02, 4.9e-02) ms⁻¹, wall time: 54.767 seconds
[ Info: i: 1320, t: 2.920 hours, Δt: 6.223 seconds, umax = (1.1e-01, 7.3e-02, 4.6e-02) ms⁻¹, wall time: 56.358 seconds
[ Info: i: 1340, t: 2.954 hours, Δt: 6.024 seconds, umax = (1.1e-01, 7.1e-02, 4.3e-02) ms⁻¹, wall time: 56.757 seconds
[ Info: i: 1360, t: 2.989 hours, Δt: 6.115 seconds, umax = (1.1e-01, 7.1e-02, 4.4e-02) ms⁻¹, wall time: 57.320 seconds
[ Info: i: 1380, t: 3.022 hours, Δt: 5.853 seconds, umax = (1.1e-01, 6.8e-02, 4.6e-02) ms⁻¹, wall time: 57.887 seconds
[ Info: i: 1400, t: 3.056 hours, Δt: 5.990 seconds, umax = (1.1e-01, 7.2e-02, 4.8e-02) ms⁻¹, wall time: 58.434 seconds
[ Info: i: 1420, t: 3.088 hours, Δt: 5.651 seconds, umax = (1.1e-01, 7.0e-02, 4.6e-02) ms⁻¹, wall time: 59.124 seconds
[ Info: i: 1440, t: 3.121 hours, Δt: 5.933 seconds, umax = (1.1e-01, 7.0e-02, 4.3e-02) ms⁻¹, wall time: 59.561 seconds
[ Info: i: 1460, t: 3.155 hours, Δt: 5.989 seconds, umax = (1.0e-01, 7.1e-02, 4.5e-02) ms⁻¹, wall time: 1.002 minutes
[ Info: i: 1480, t: 3.187 hours, Δt: 5.785 seconds, umax = (1.1e-01, 7.9e-02, 4.5e-02) ms⁻¹, wall time: 1.011 minutes
[ Info: i: 1500, t: 3.219 hours, Δt: 6.073 seconds, umax = (1.1e-01, 8.5e-02, 4.5e-02) ms⁻¹, wall time: 1.020 minutes
[ Info: i: 1520, t: 3.252 hours, Δt: 5.526 seconds, umax = (1.1e-01, 8.9e-02, 4.3e-02) ms⁻¹, wall time: 1.033 minutes
[ Info: i: 1540, t: 3.282 hours, Δt: 5.614 seconds, umax = (1.1e-01, 8.9e-02, 4.5e-02) ms⁻¹, wall time: 1.039 minutes
[ Info: i: 1560, t: 3.315 hours, Δt: 6.190 seconds, umax = (1.1e-01, 8.4e-02, 4.9e-02) ms⁻¹, wall time: 1.048 minutes
[ Info: i: 1580, t: 3.348 hours, Δt: 5.994 seconds, umax = (1.1e-01, 8.1e-02, 4.7e-02) ms⁻¹, wall time: 1.058 minutes
[ Info: i: 1600, t: 3.381 hours, Δt: 5.984 seconds, umax = (1.1e-01, 7.9e-02, 4.9e-02) ms⁻¹, wall time: 1.066 minutes
[ Info: i: 1620, t: 3.415 hours, Δt: 5.738 seconds, umax = (1.2e-01, 7.7e-02, 4.6e-02) ms⁻¹, wall time: 1.076 minutes
[ Info: i: 1640, t: 3.445 hours, Δt: 5.645 seconds, umax = (1.1e-01, 7.5e-02, 5.1e-02) ms⁻¹, wall time: 1.085 minutes
[ Info: i: 1660, t: 3.477 hours, Δt: 5.699 seconds, umax = (1.1e-01, 7.8e-02, 5.1e-02) ms⁻¹, wall time: 1.094 minutes
[ Info: i: 1680, t: 3.508 hours, Δt: 5.607 seconds, umax = (1.1e-01, 7.4e-02, 4.9e-02) ms⁻¹, wall time: 1.105 minutes
[ Info: i: 1700, t: 3.540 hours, Δt: 6.163 seconds, umax = (1.1e-01, 7.7e-02, 5.3e-02) ms⁻¹, wall time: 1.112 minutes
[ Info: i: 1720, t: 3.573 hours, Δt: 6.226 seconds, umax = (1.1e-01, 7.6e-02, 5.3e-02) ms⁻¹, wall time: 1.122 minutes
[ Info: i: 1740, t: 3.606 hours, Δt: 5.505 seconds, umax = (1.1e-01, 9.2e-02, 4.8e-02) ms⁻¹, wall time: 1.131 minutes
[ Info: i: 1760, t: 3.637 hours, Δt: 5.798 seconds, umax = (1.3e-01, 8.7e-02, 4.2e-02) ms⁻¹, wall time: 1.140 minutes
[ Info: i: 1780, t: 3.668 hours, Δt: 5.555 seconds, umax = (1.1e-01, 8.5e-02, 5.3e-02) ms⁻¹, wall time: 1.152 minutes
[ Info: i: 1800, t: 3.700 hours, Δt: 5.909 seconds, umax = (1.1e-01, 8.1e-02, 4.7e-02) ms⁻¹, wall time: 1.159 minutes
[ Info: i: 1820, t: 3.731 hours, Δt: 5.712 seconds, umax = (1.1e-01, 8.7e-02, 4.9e-02) ms⁻¹, wall time: 1.168 minutes
[ Info: i: 1840, t: 3.763 hours, Δt: 6.100 seconds, umax = (1.1e-01, 8.1e-02, 4.9e-02) ms⁻¹, wall time: 1.177 minutes
[ Info: i: 1860, t: 3.796 hours, Δt: 5.981 seconds, umax = (1.2e-01, 8.4e-02, 4.9e-02) ms⁻¹, wall time: 1.186 minutes
[ Info: i: 1880, t: 3.829 hours, Δt: 5.680 seconds, umax = (1.1e-01, 8.6e-02, 5.0e-02) ms⁻¹, wall time: 1.195 minutes
[ Info: i: 1900, t: 3.861 hours, Δt: 5.591 seconds, umax = (1.1e-01, 8.7e-02, 4.8e-02) ms⁻¹, wall time: 1.205 minutes
[ Info: i: 1920, t: 3.893 hours, Δt: 5.794 seconds, umax = (1.1e-01, 8.6e-02, 5.0e-02) ms⁻¹, wall time: 1.214 minutes
[ Info: i: 1940, t: 3.923 hours, Δt: 6.009 seconds, umax = (1.1e-01, 8.5e-02, 5.3e-02) ms⁻¹, wall time: 1.225 minutes
[ Info: i: 1960, t: 3.956 hours, Δt: 5.862 seconds, umax = (1.1e-01, 8.1e-02, 5.0e-02) ms⁻¹, wall time: 1.233 minutes
[ Info: i: 1980, t: 3.988 hours, Δt: 5.776 seconds, umax = (1.1e-01, 8.4e-02, 5.2e-02) ms⁻¹, wall time: 1.242 minutes
[ Info: Simulation is stopping after running for 1.245 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 1.12.2
Commit ca9b6662be4 (2025-11-20 16:25 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-27802/docs/
  JULIA_VERSION = 1.12.2
  JULIA_LOAD_PATH = @:@v#.#:@stdlib
  JULIA_VERSION_ENZYME = 1.10.10
  JULIA_PYTHONCALL_EXE = /var/lib/buildkite-agent/Oceananigans.jl-27802/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-27802/docs/Project.toml`
  [79e6a3ab] Adapt v4.4.0
  [052768ef] CUDA v5.9.5
  [13f3f980] CairoMakie v0.15.8
  [e30172f5] Documenter v1.16.1
  [daee34ce] DocumenterCitations v1.4.1
  [033835bb] JLD2 v0.6.3
  [98b081ad] Literate v2.21.0
  [da04e1cc] MPI v0.20.23
  [85f8d34a] NCDatasets v0.14.10
  [9e8cae18] Oceananigans v0.103.1 `~/Oceananigans.jl-27802`
  [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 v1.12.0

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