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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, 8.9e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (17.386 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.775 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 18.135 seconds, umax = (3.6e-02, 1.4e-02, 2.2e-02) ms⁻¹, wall time: 21.314 seconds
[ Info: i: 0040, t: 16.781 minutes, Δt: 13.419 seconds, umax = (5.3e-02, 2.0e-02, 2.4e-02) ms⁻¹, wall time: 21.722 seconds
[ Info: i: 0060, t: 21.025 minutes, Δt: 10.911 seconds, umax = (6.2e-02, 2.9e-02, 3.3e-02) ms⁻¹, wall time: 22.317 seconds
[ Info: i: 0080, t: 24.679 minutes, Δt: 11.437 seconds, umax = (6.3e-02, 3.2e-02, 3.5e-02) ms⁻¹, wall time: 22.760 seconds
[ Info: i: 0100, t: 28.428 minutes, Δt: 11.109 seconds, umax = (5.8e-02, 3.2e-02, 3.0e-02) ms⁻¹, wall time: 23.262 seconds
[ Info: i: 0120, t: 32.115 minutes, Δt: 11.064 seconds, umax = (6.7e-02, 3.2e-02, 2.9e-02) ms⁻¹, wall time: 23.792 seconds
[ Info: i: 0140, t: 35.728 minutes, Δt: 10.280 seconds, umax = (6.4e-02, 3.3e-02, 3.0e-02) ms⁻¹, wall time: 24.341 seconds
[ Info: i: 0160, t: 39.164 minutes, Δt: 10.030 seconds, umax = (7.1e-02, 3.4e-02, 3.4e-02) ms⁻¹, wall time: 24.707 seconds
[ Info: i: 0180, t: 42.324 minutes, Δt: 9.878 seconds, umax = (7.0e-02, 3.8e-02, 3.5e-02) ms⁻¹, wall time: 25.159 seconds
[ Info: i: 0200, t: 45.475 minutes, Δt: 9.244 seconds, umax = (7.2e-02, 3.9e-02, 3.7e-02) ms⁻¹, wall time: 25.731 seconds
[ Info: i: 0220, t: 48.571 minutes, Δt: 8.789 seconds, umax = (7.7e-02, 3.9e-02, 3.1e-02) ms⁻¹, wall time: 26.077 seconds
[ Info: i: 0240, t: 51.422 minutes, Δt: 8.879 seconds, umax = (7.6e-02, 4.3e-02, 3.3e-02) ms⁻¹, wall time: 26.570 seconds
[ Info: i: 0260, t: 54.374 minutes, Δt: 9.053 seconds, umax = (7.7e-02, 4.2e-02, 3.4e-02) ms⁻¹, wall time: 27.021 seconds
[ Info: i: 0280, t: 57.270 minutes, Δt: 8.479 seconds, umax = (8.0e-02, 4.5e-02, 3.8e-02) ms⁻¹, wall time: 27.504 seconds
[ Info: i: 0300, t: 1 hour, Δt: 8.262 seconds, umax = (8.3e-02, 4.8e-02, 3.5e-02) ms⁻¹, wall time: 27.945 seconds
[ Info: i: 0320, t: 1.046 hours, Δt: 8.132 seconds, umax = (8.3e-02, 4.3e-02, 3.9e-02) ms⁻¹, wall time: 28.403 seconds
[ Info: i: 0340, t: 1.090 hours, Δt: 8.015 seconds, umax = (8.3e-02, 4.4e-02, 3.7e-02) ms⁻¹, wall time: 28.988 seconds
[ Info: i: 0360, t: 1.136 hours, Δt: 8.542 seconds, umax = (8.6e-02, 4.3e-02, 3.9e-02) ms⁻¹, wall time: 29.342 seconds
[ Info: i: 0380, t: 1.180 hours, Δt: 8.229 seconds, umax = (8.3e-02, 4.5e-02, 3.7e-02) ms⁻¹, wall time: 29.874 seconds
[ Info: i: 0400, t: 1.226 hours, Δt: 7.909 seconds, umax = (8.7e-02, 5.0e-02, 3.8e-02) ms⁻¹, wall time: 30.271 seconds
[ Info: i: 0420, t: 1.267 hours, Δt: 7.665 seconds, umax = (8.9e-02, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 30.747 seconds
[ Info: i: 0440, t: 1.308 hours, Δt: 7.335 seconds, umax = (8.8e-02, 6.3e-02, 4.4e-02) ms⁻¹, wall time: 31.193 seconds
[ Info: i: 0460, t: 1.347 hours, Δt: 7.708 seconds, umax = (9.2e-02, 5.4e-02, 3.9e-02) ms⁻¹, wall time: 31.966 seconds
[ Info: i: 0480, t: 1.390 hours, Δt: 7.732 seconds, umax = (9.4e-02, 5.3e-02, 4.0e-02) ms⁻¹, wall time: 33.240 seconds
[ Info: i: 0500, t: 1.431 hours, Δt: 7.722 seconds, umax = (9.3e-02, 5.1e-02, 4.0e-02) ms⁻¹, wall time: 33.742 seconds
[ Info: i: 0520, t: 1.474 hours, Δt: 7.257 seconds, umax = (8.9e-02, 4.8e-02, 4.4e-02) ms⁻¹, wall time: 34.170 seconds
[ Info: i: 0540, t: 1.514 hours, Δt: 6.855 seconds, umax = (1.0e-01, 5.1e-02, 4.8e-02) ms⁻¹, wall time: 34.710 seconds
[ Info: i: 0560, t: 1.552 hours, Δt: 7.010 seconds, umax = (9.2e-02, 6.3e-02, 4.3e-02) ms⁻¹, wall time: 35.100 seconds
[ Info: i: 0580, t: 1.589 hours, Δt: 6.467 seconds, umax = (1.0e-01, 5.2e-02, 4.2e-02) ms⁻¹, wall time: 35.690 seconds
[ Info: i: 0600, t: 1.626 hours, Δt: 7.396 seconds, umax = (9.1e-02, 5.8e-02, 4.5e-02) ms⁻¹, wall time: 36.081 seconds
[ Info: i: 0620, t: 1.666 hours, Δt: 7.350 seconds, umax = (9.3e-02, 5.3e-02, 4.0e-02) ms⁻¹, wall time: 36.613 seconds
[ Info: i: 0640, t: 1.704 hours, Δt: 6.680 seconds, umax = (9.4e-02, 6.1e-02, 4.2e-02) ms⁻¹, wall time: 37.142 seconds
[ Info: i: 0660, t: 1.741 hours, Δt: 7.150 seconds, umax = (9.7e-02, 5.7e-02, 4.3e-02) ms⁻¹, wall time: 37.693 seconds
[ Info: i: 0680, t: 1.779 hours, Δt: 6.811 seconds, umax = (9.6e-02, 6.0e-02, 4.3e-02) ms⁻¹, wall time: 38.216 seconds
[ Info: i: 0700, t: 1.817 hours, Δt: 6.918 seconds, umax = (9.5e-02, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 38.702 seconds
[ Info: i: 0720, t: 1.855 hours, Δt: 6.657 seconds, umax = (9.7e-02, 5.7e-02, 4.2e-02) ms⁻¹, wall time: 39.200 seconds
[ Info: i: 0740, t: 1.891 hours, Δt: 6.091 seconds, umax = (9.7e-02, 5.8e-02, 4.1e-02) ms⁻¹, wall time: 39.667 seconds
[ Info: i: 0760, t: 1.926 hours, Δt: 6.591 seconds, umax = (9.9e-02, 6.2e-02, 4.3e-02) ms⁻¹, wall time: 40.228 seconds
[ Info: i: 0780, t: 1.962 hours, Δt: 6.557 seconds, umax = (9.7e-02, 6.4e-02, 5.0e-02) ms⁻¹, wall time: 40.628 seconds
[ Info: i: 0800, t: 1.999 hours, Δt: 6.601 seconds, umax = (1.0e-01, 6.3e-02, 4.8e-02) ms⁻¹, wall time: 41.101 seconds
[ Info: i: 0820, t: 2.036 hours, Δt: 6.857 seconds, umax = (1.1e-01, 5.8e-02, 4.1e-02) ms⁻¹, wall time: 41.605 seconds
[ Info: i: 0840, t: 2.074 hours, Δt: 6.554 seconds, umax = (9.9e-02, 6.1e-02, 4.5e-02) ms⁻¹, wall time: 42.081 seconds
[ Info: i: 0860, t: 2.111 hours, Δt: 6.873 seconds, umax = (1.0e-01, 5.8e-02, 4.3e-02) ms⁻¹, wall time: 42.585 seconds
[ Info: i: 0880, t: 2.150 hours, Δt: 6.808 seconds, umax = (9.9e-02, 6.9e-02, 4.2e-02) ms⁻¹, wall time: 43.075 seconds
[ Info: i: 0900, t: 2.185 hours, Δt: 6.675 seconds, umax = (1.0e-01, 6.5e-02, 4.5e-02) ms⁻¹, wall time: 43.573 seconds
[ Info: i: 0920, t: 2.223 hours, Δt: 6.473 seconds, umax = (1.0e-01, 5.8e-02, 4.5e-02) ms⁻¹, wall time: 44.043 seconds
[ Info: i: 0940, t: 2.259 hours, Δt: 6.663 seconds, umax = (1.0e-01, 5.9e-02, 4.3e-02) ms⁻¹, wall time: 44.627 seconds
[ Info: i: 0960, t: 2.295 hours, Δt: 6.384 seconds, umax = (1.0e-01, 6.1e-02, 5.2e-02) ms⁻¹, wall time: 45.033 seconds
[ Info: i: 0980, t: 2.331 hours, Δt: 6.452 seconds, umax = (1.0e-01, 6.7e-02, 5.4e-02) ms⁻¹, wall time: 45.514 seconds
[ Info: i: 1000, t: 2.366 hours, Δt: 6.496 seconds, umax = (1.0e-01, 7.2e-02, 5.5e-02) ms⁻¹, wall time: 46.050 seconds
[ Info: i: 1020, t: 2.401 hours, Δt: 6.555 seconds, umax = (1.0e-01, 6.3e-02, 5.2e-02) ms⁻¹, wall time: 46.537 seconds
[ Info: i: 1040, t: 2.436 hours, Δt: 5.976 seconds, umax = (1.0e-01, 6.6e-02, 4.7e-02) ms⁻¹, wall time: 48.015 seconds
[ Info: i: 1060, t: 2.471 hours, Δt: 6.178 seconds, umax = (9.9e-02, 6.6e-02, 4.4e-02) ms⁻¹, wall time: 48.481 seconds
[ Info: i: 1080, t: 2.505 hours, Δt: 6.332 seconds, umax = (1.0e-01, 6.9e-02, 4.6e-02) ms⁻¹, wall time: 49.099 seconds
[ Info: i: 1100, t: 2.540 hours, Δt: 6.521 seconds, umax = (1.1e-01, 7.0e-02, 5.0e-02) ms⁻¹, wall time: 49.472 seconds
[ Info: i: 1120, t: 2.576 hours, Δt: 6.538 seconds, umax = (1.1e-01, 6.6e-02, 4.9e-02) ms⁻¹, wall time: 49.955 seconds
[ Info: i: 1140, t: 2.612 hours, Δt: 6.380 seconds, umax = (1.0e-01, 6.7e-02, 5.1e-02) ms⁻¹, wall time: 50.476 seconds
[ Info: i: 1160, t: 2.647 hours, Δt: 6.399 seconds, umax = (1.1e-01, 7.5e-02, 4.8e-02) ms⁻¹, wall time: 50.939 seconds
[ Info: i: 1180, t: 2.680 hours, Δt: 6.436 seconds, umax = (1.1e-01, 7.6e-02, 4.7e-02) ms⁻¹, wall time: 51.460 seconds
[ Info: i: 1200, t: 2.715 hours, Δt: 5.949 seconds, umax = (1.1e-01, 7.5e-02, 5.1e-02) ms⁻¹, wall time: 51.923 seconds
[ Info: i: 1220, t: 2.749 hours, Δt: 5.977 seconds, umax = (1.0e-01, 7.4e-02, 4.9e-02) ms⁻¹, wall time: 52.414 seconds
[ Info: i: 1240, t: 2.782 hours, Δt: 6.027 seconds, umax = (1.1e-01, 7.4e-02, 4.6e-02) ms⁻¹, wall time: 52.902 seconds
[ Info: i: 1260, t: 2.815 hours, Δt: 6.380 seconds, umax = (1.1e-01, 7.3e-02, 4.6e-02) ms⁻¹, wall time: 53.389 seconds
[ Info: i: 1280, t: 2.849 hours, Δt: 6.248 seconds, umax = (1.1e-01, 7.0e-02, 5.0e-02) ms⁻¹, wall time: 53.877 seconds
[ Info: i: 1300, t: 2.882 hours, Δt: 6.132 seconds, umax = (1.0e-01, 7.1e-02, 4.8e-02) ms⁻¹, wall time: 54.331 seconds
[ Info: i: 1320, t: 2.916 hours, Δt: 5.917 seconds, umax = (1.1e-01, 7.6e-02, 5.4e-02) ms⁻¹, wall time: 54.821 seconds
[ Info: i: 1340, t: 2.948 hours, Δt: 5.901 seconds, umax = (1.1e-01, 7.5e-02, 4.9e-02) ms⁻¹, wall time: 55.345 seconds
[ Info: i: 1360, t: 2.981 hours, Δt: 6.004 seconds, umax = (1.1e-01, 7.5e-02, 4.7e-02) ms⁻¹, wall time: 55.906 seconds
[ Info: i: 1380, t: 3.014 hours, Δt: 5.684 seconds, umax = (1.1e-01, 8.2e-02, 5.0e-02) ms⁻¹, wall time: 56.485 seconds
[ Info: i: 1400, t: 3.045 hours, Δt: 5.883 seconds, umax = (1.1e-01, 7.4e-02, 4.9e-02) ms⁻¹, wall time: 57.011 seconds
[ Info: i: 1420, t: 3.077 hours, Δt: 5.413 seconds, umax = (1.2e-01, 7.8e-02, 4.7e-02) ms⁻¹, wall time: 57.573 seconds
[ Info: i: 1440, t: 3.105 hours, Δt: 5.437 seconds, umax = (1.1e-01, 8.3e-02, 5.6e-02) ms⁻¹, wall time: 58.100 seconds
[ Info: i: 1460, t: 3.136 hours, Δt: 5.664 seconds, umax = (1.1e-01, 7.5e-02, 5.6e-02) ms⁻¹, wall time: 58.571 seconds
[ Info: i: 1480, t: 3.167 hours, Δt: 5.913 seconds, umax = (1.0e-01, 6.9e-02, 5.6e-02) ms⁻¹, wall time: 59.051 seconds
[ Info: i: 1500, t: 3.199 hours, Δt: 5.992 seconds, umax = (1.1e-01, 7.0e-02, 4.8e-02) ms⁻¹, wall time: 59.569 seconds
[ Info: i: 1520, t: 3.233 hours, Δt: 5.984 seconds, umax = (1.1e-01, 7.2e-02, 4.8e-02) ms⁻¹, wall time: 1.001 minutes
[ Info: i: 1540, t: 3.264 hours, Δt: 5.791 seconds, umax = (1.1e-01, 7.9e-02, 5.1e-02) ms⁻¹, wall time: 1.009 minutes
[ Info: i: 1560, t: 3.296 hours, Δt: 6.002 seconds, umax = (1.1e-01, 7.0e-02, 4.8e-02) ms⁻¹, wall time: 1.017 minutes
[ Info: i: 1580, t: 3.330 hours, Δt: 5.944 seconds, umax = (1.1e-01, 7.5e-02, 4.9e-02) ms⁻¹, wall time: 1.025 minutes
[ Info: i: 1600, t: 3.361 hours, Δt: 6.048 seconds, umax = (1.1e-01, 7.0e-02, 4.7e-02) ms⁻¹, wall time: 1.033 minutes
[ Info: i: 1620, t: 3.394 hours, Δt: 6.061 seconds, umax = (1.1e-01, 7.6e-02, 4.4e-02) ms⁻¹, wall time: 1.041 minutes
[ Info: i: 1640, t: 3.426 hours, Δt: 6.135 seconds, umax = (1.1e-01, 7.2e-02, 4.8e-02) ms⁻¹, wall time: 1.050 minutes
[ Info: i: 1660, t: 3.460 hours, Δt: 6.040 seconds, umax = (1.1e-01, 7.1e-02, 4.9e-02) ms⁻¹, wall time: 1.057 minutes
[ Info: i: 1680, t: 3.494 hours, Δt: 6.174 seconds, umax = (1.1e-01, 7.2e-02, 5.3e-02) ms⁻¹, wall time: 1.065 minutes
[ Info: i: 1700, t: 3.528 hours, Δt: 6.148 seconds, umax = (1.1e-01, 7.4e-02, 4.5e-02) ms⁻¹, wall time: 1.089 minutes
[ Info: i: 1720, t: 3.561 hours, Δt: 6.138 seconds, umax = (1.1e-01, 7.5e-02, 4.8e-02) ms⁻¹, wall time: 1.097 minutes
[ Info: i: 1740, t: 3.593 hours, Δt: 5.503 seconds, umax = (1.1e-01, 8.0e-02, 4.9e-02) ms⁻¹, wall time: 1.106 minutes
[ Info: i: 1760, t: 3.625 hours, Δt: 6.022 seconds, umax = (1.2e-01, 7.2e-02, 4.9e-02) ms⁻¹, wall time: 1.113 minutes
[ Info: i: 1780, t: 3.659 hours, Δt: 5.858 seconds, umax = (1.1e-01, 7.8e-02, 4.8e-02) ms⁻¹, wall time: 1.122 minutes
[ Info: i: 1800, t: 3.691 hours, Δt: 6.207 seconds, umax = (1.1e-01, 8.1e-02, 4.5e-02) ms⁻¹, wall time: 1.130 minutes
[ Info: i: 1820, t: 3.726 hours, Δt: 5.984 seconds, umax = (1.2e-01, 7.9e-02, 4.8e-02) ms⁻¹, wall time: 1.138 minutes
[ Info: i: 1840, t: 3.758 hours, Δt: 5.824 seconds, umax = (1.1e-01, 8.1e-02, 5.5e-02) ms⁻¹, wall time: 1.147 minutes
[ Info: i: 1860, t: 3.790 hours, Δt: 5.750 seconds, umax = (1.1e-01, 8.2e-02, 5.4e-02) ms⁻¹, wall time: 1.154 minutes
[ Info: i: 1880, t: 3.822 hours, Δt: 5.928 seconds, umax = (1.1e-01, 7.7e-02, 5.7e-02) ms⁻¹, wall time: 1.162 minutes
[ Info: i: 1900, t: 3.854 hours, Δt: 6.088 seconds, umax = (1.1e-01, 8.0e-02, 4.9e-02) ms⁻¹, wall time: 1.172 minutes
[ Info: i: 1920, t: 3.887 hours, Δt: 5.372 seconds, umax = (1.2e-01, 8.0e-02, 5.0e-02) ms⁻¹, wall time: 1.181 minutes
[ Info: i: 1940, t: 3.917 hours, Δt: 5.920 seconds, umax = (1.1e-01, 8.3e-02, 5.4e-02) ms⁻¹, wall time: 1.191 minutes
[ Info: i: 1960, t: 3.950 hours, Δt: 5.506 seconds, umax = (1.0e-01, 8.9e-02, 5.9e-02) ms⁻¹, wall time: 1.200 minutes
[ Info: i: 1980, t: 3.982 hours, Δt: 5.959 seconds, umax = (1.1e-01, 7.8e-02, 5.0e-02) ms⁻¹, wall time: 1.210 minutes
[ Info: Simulation is stopping after running for 1.216 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-27724/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-27724/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-27724/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 v0.103.0 `~/Oceananigans.jl-27724`
  [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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