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Langmuir turbulence example

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

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

This example demonstrates

  • How to run large eddy simulations with surface wave effects via the Craik-Leibovich approximation.

  • How to specify time- and horizontally-averaged output.

Install dependencies

First let's make sure we have all required packages installed.

using Pkg
pkg"add Oceananigans, CairoMakie, CUDA"
using Oceananigans
using Oceananigans.Units: minute, minutes, hours
using CUDA

Model set-up

To build the model, we specify the grid, Stokes drift, boundary conditions, and Coriolis parameter.

Domain and numerical grid specification

We use a modest resolution and the same total extent as Wagner et al. (2021),

grid = RectilinearGrid(GPU(), size=(128, 128, 64), extent=(128, 128, 64))
128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CUDAGPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 128.0) regularly spaced with Δx=1.0
├── Periodic y ∈ [0.0, 128.0) regularly spaced with Δy=1.0
└── Bounded  z ∈ [-64.0, 0.0] regularly spaced with Δz=1.0

The Stokes Drift profile

The surface wave Stokes drift profile prescribed in Wagner et al. (2021), corresponds to a 'monochromatic' (that is, single-frequency) wave field.

A monochromatic wave field is characterized by its wavelength and amplitude (half the distance from wave crest to wave trough), which determine the wave frequency and the vertical scale of the Stokes drift profile.

g = Oceananigans.defaults.gravitational_acceleration

amplitude = 0.8 # m
wavelength = 60  # m
wavenumber = 2π / wavelength # m⁻¹
frequency = sqrt(g * wavenumber) # s⁻¹

# The vertical scale over which the Stokes drift of a monochromatic surface wave
# decays away from the surface is `1/2wavenumber`, or
const vertical_scale = wavelength / 4π

# Stokes drift velocity at the surface
const Uˢ = amplitude^2 * wavenumber * frequency # m s⁻¹
0.06791774197745354

The const declarations ensure that Stokes drift functions compile on the GPU. To run this example on the CPU, replace GPU() with CPU() in the RectilinearGrid constructor above.

The Stokes drift profile is

uˢ(z) = Uˢ * exp(z / vertical_scale)
uˢ (generic function with 1 method)

and its z-derivative is

∂z_uˢ(z, t) = 1 / vertical_scale * Uˢ * exp(z / vertical_scale)
∂z_uˢ (generic function with 1 method)
The Craik-Leibovich equations in Oceananigans

Oceananigans implements the Craik-Leibovich approximation for surface wave effects using the Lagrangian-mean velocity field as its prognostic momentum variable. In other words, model.velocities.u is the Lagrangian-mean $x$-velocity beneath surface waves. This differs from models that use the Eulerian-mean velocity field as a prognostic variable, but has the advantage that $u$ accounts for the total advection of tracers and momentum, and that $u = v = w = 0$ is a steady solution even when Coriolis forces are present. See the physics documentation for more information.

Finally, we note that the time-derivative of the Stokes drift must be provided if the Stokes drift and surface wave field undergoes forced changes in time. In this example, the Stokes drift is constant and thus the time-derivative of the Stokes drift is 0.

Boundary conditions

At the surface $z = 0$, Wagner et al. (2021) impose

τx = -3.72e-5 # m² s⁻², surface kinematic momentum flux
u_boundary_conditions = FieldBoundaryConditions(top = FluxBoundaryCondition(τx))
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── top: FluxBoundaryCondition: -3.72e-5
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)

Wagner et al. (2021) impose a linear buoyancy gradient N² at the bottom along with a weak, destabilizing flux of buoyancy at the surface to faciliate spin-up from rest.

Jᵇ = 2.307e-8 # m² s⁻³, surface buoyancy flux
N² = 1.936e-5 # s⁻², initial and bottom buoyancy gradient

b_boundary_conditions = FieldBoundaryConditions(top = FluxBoundaryCondition(Jᵇ),
                                                bottom = GradientBoundaryCondition(N²))
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: GradientBoundaryCondition: 1.936e-5
├── top: FluxBoundaryCondition: 2.307e-8
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
The flux convention in Oceananigans

Note that Oceananigans uses "positive upward" conventions for all fluxes. In consequence, a negative flux at the surface drives positive velocities, and a positive flux of buoyancy drives cooling.

Coriolis parameter

Wagner et al. (2021) use

coriolis = FPlane(f=1e-4) # s⁻¹
FPlane{Float64}(f=0.0001)

which is typical for mid-latitudes on Earth.

Model instantiation

We are ready to build the model. We use a fifth-order Weighted Essentially Non-Oscillatory (WENO) advection scheme and the AnisotropicMinimumDissipation model for large eddy simulation. Because our Stokes drift does not vary in $x, y$, we use UniformStokesDrift, which expects Stokes drift functions of $z, t$ only.

model = NonhydrostaticModel(; grid, coriolis,
                            advection = WENO(order=9),
                            tracers = :b,
                            buoyancy = BuoyancyTracer(),
                            stokes_drift = UniformStokesDrift(∂z_uˢ=∂z_uˢ),
                            boundary_conditions = (u=u_boundary_conditions, b=b_boundary_conditions))
NonhydrostaticModel{CUDAGPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CUDAGPU with 5×5×5 halo
├── timestepper: RungeKutta3TimeStepper
├── advection scheme: WENO{5, Float64, Float32}(order=9)
├── tracers: b
├── closure: Nothing
├── buoyancy: BuoyancyTracer with ĝ = NegativeZDirection()
└── coriolis: FPlane{Float64}(f=0.0001)

Initial conditions

We make use of random noise concentrated in the upper 4 meters for buoyancy and velocity initial conditions,

Ξ(z) = randn() * exp(z / 4)

Our initial condition for buoyancy consists of a surface mixed layer 33 m deep, a deep linear stratification, plus noise,

initial_mixed_layer_depth = 33 # m
stratification(z) = z < - initial_mixed_layer_depth ? N² * z : N² * (-initial_mixed_layer_depth)

bᵢ(x, y, z) = stratification(z) + 1e-1 * Ξ(z) * N² * model.grid.Lz
bᵢ (generic function with 1 method)

The simulation we reproduce from Wagner et al. (2021) is zero Lagrangian-mean velocity. This initial condition is consistent with a wavy, quiescent ocean suddenly impacted by winds. To this quiescent state we add noise scaled by the friction velocity to $u$ and $w$.

u★ = sqrt(abs(τx))
uᵢ(x, y, z) = u★ * 1e-1 * Ξ(z)
wᵢ(x, y, z) = u★ * 1e-1 * Ξ(z)

set!(model, u=uᵢ, w=wᵢ, b=bᵢ)

Setting up the simulation

simulation = Simulation(model, Δt=45.0, stop_time=4hours)
Simulation of NonhydrostaticModel{CUDAGPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── Next time step: 45 seconds
├── run_wall_time: 0 seconds
├── run_wall_time / iteration: NaN days
├── stop_time: 4 hours
├── stop_iteration: Inf
├── wall_time_limit: Inf
├── minimum_relative_step: 0.0
├── callbacks: OrderedDict with 4 entries:
│   ├── stop_time_exceeded => Callback of stop_time_exceeded on IterationInterval(1)
│   ├── stop_iteration_exceeded => Callback of stop_iteration_exceeded on IterationInterval(1)
│   ├── wall_time_limit_exceeded => Callback of wall_time_limit_exceeded on IterationInterval(1)
│   └── nan_checker => Callback of NaNChecker for u on IterationInterval(100)
└── output_writers: OrderedDict with no entries

We use the TimeStepWizard for adaptive time-stepping with a Courant-Freidrichs-Lewy (CFL) number of 1.0,

conjure_time_step_wizard!(simulation, cfl=1.0, max_Δt=1minute)

Nice progress messaging

We define a function that prints a helpful message with maximum absolute value of $u, v, w$ and the current wall clock time.

using Printf

function progress(simulation)
    u, v, w = simulation.model.velocities

    # Print a progress message
    msg = @sprintf("i: %04d, t: %s, Δt: %s, umax = (%.1e, %.1e, %.1e) ms⁻¹, wall time: %s\n",
                   iteration(simulation),
                   prettytime(time(simulation)),
                   prettytime(simulation.Δt),
                   maximum(abs, u), maximum(abs, v), maximum(abs, w),
                   prettytime(simulation.run_wall_time))

    @info msg

    return nothing
end

simulation.callbacks[:progress] = Callback(progress, IterationInterval(20))
Callback of progress on IterationInterval(20)

Output

A field writer

We set up an output writer for the simulation that saves all velocity fields, tracer fields, and the subgrid turbulent diffusivity.

output_interval = 5minutes

fields_to_output = merge(model.velocities, model.tracers)

simulation.output_writers[:fields] =
    JLD2Writer(model, fields_to_output,
               schedule = TimeInterval(output_interval),
               filename = "langmuir_turbulence_fields.jld2",
               overwrite_existing = true)
JLD2Writer scheduled on TimeInterval(5 minutes):
├── filepath: langmuir_turbulence_fields.jld2
├── 4 outputs: (u, v, w, b)
├── array_type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 0 bytes (file not yet created)

An "averages" writer

We also set up output of time- and horizontally-averaged velocity field and momentum fluxes.

u, v, w = model.velocities
b = model.tracers.b

 U = Average(u, dims=(1, 2))
 V = Average(v, dims=(1, 2))
 B = Average(b, dims=(1, 2))
wu = Average(w * u, dims=(1, 2))
wv = Average(w * v, dims=(1, 2))

simulation.output_writers[:averages] =
    JLD2Writer(model, (; U, V, B, wu, wv),
               schedule = AveragedTimeInterval(output_interval, window=2minutes),
               filename = "langmuir_turbulence_averages.jld2",
               overwrite_existing = true)
JLD2Writer scheduled on TimeInterval(5 minutes):
├── filepath: langmuir_turbulence_averages.jld2
├── 5 outputs: (U, V, B, wu, wv) averaged on AveragedTimeInterval(window=2 minutes, stride=1, interval=5 minutes)
├── array_type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 0 bytes (file not yet created)

Running the simulation

This part is easy,

run!(simulation)
[ Info: Initializing simulation...
[ Info: i: 0000, t: 0 seconds, Δt: 49.500 seconds, umax = (1.4e-03, 9.2e-04, 1.6e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (21.185 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.838 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 18.950 seconds, umax = (3.5e-02, 1.2e-02, 2.1e-02) ms⁻¹, wall time: 25.217 seconds
[ Info: i: 0040, t: 16.807 minutes, Δt: 13.170 seconds, umax = (5.2e-02, 2.1e-02, 2.4e-02) ms⁻¹, wall time: 25.632 seconds
[ Info: i: 0060, t: 20.796 minutes, Δt: 11.082 seconds, umax = (6.3e-02, 2.7e-02, 3.2e-02) ms⁻¹, wall time: 26.173 seconds
[ Info: i: 0080, t: 24.410 minutes, Δt: 11.137 seconds, umax = (6.6e-02, 3.1e-02, 3.3e-02) ms⁻¹, wall time: 26.539 seconds
[ Info: i: 0100, t: 27.995 minutes, Δt: 11.062 seconds, umax = (6.2e-02, 3.1e-02, 3.0e-02) ms⁻¹, wall time: 26.972 seconds
[ Info: i: 0120, t: 31.583 minutes, Δt: 11.402 seconds, umax = (6.4e-02, 3.2e-02, 2.9e-02) ms⁻¹, wall time: 27.433 seconds
[ Info: i: 0140, t: 35 minutes, Δt: 9.708 seconds, umax = (6.3e-02, 3.4e-02, 3.0e-02) ms⁻¹, wall time: 27.863 seconds
[ Info: i: 0160, t: 38.383 minutes, Δt: 9.507 seconds, umax = (7.1e-02, 3.4e-02, 3.0e-02) ms⁻¹, wall time: 28.301 seconds
[ Info: i: 0180, t: 41.415 minutes, Δt: 9.323 seconds, umax = (7.3e-02, 3.8e-02, 3.2e-02) ms⁻¹, wall time: 28.766 seconds
[ Info: i: 0200, t: 44.503 minutes, Δt: 9.597 seconds, umax = (7.3e-02, 3.8e-02, 3.5e-02) ms⁻¹, wall time: 29.193 seconds
[ Info: i: 0220, t: 47.490 minutes, Δt: 9.771 seconds, umax = (7.0e-02, 3.9e-02, 3.6e-02) ms⁻¹, wall time: 29.632 seconds
[ Info: i: 0240, t: 50.605 minutes, Δt: 8.794 seconds, umax = (7.8e-02, 4.0e-02, 3.6e-02) ms⁻¹, wall time: 30.166 seconds
[ Info: i: 0260, t: 53.511 minutes, Δt: 8.779 seconds, umax = (7.8e-02, 4.1e-02, 3.4e-02) ms⁻¹, wall time: 30.515 seconds
[ Info: i: 0280, t: 56.307 minutes, Δt: 8.518 seconds, umax = (7.7e-02, 4.1e-02, 3.5e-02) ms⁻¹, wall time: 30.966 seconds
[ Info: i: 0300, t: 59.177 minutes, Δt: 8.585 seconds, umax = (8.5e-02, 4.5e-02, 3.3e-02) ms⁻¹, wall time: 31.391 seconds
[ Info: i: 0320, t: 1.032 hours, Δt: 7.929 seconds, umax = (8.3e-02, 5.0e-02, 3.9e-02) ms⁻¹, wall time: 31.831 seconds
[ Info: i: 0340, t: 1.076 hours, Δt: 8.122 seconds, umax = (9.5e-02, 4.7e-02, 4.3e-02) ms⁻¹, wall time: 32.263 seconds
[ Info: i: 0360, t: 1.119 hours, Δt: 7.897 seconds, umax = (8.3e-02, 5.2e-02, 3.8e-02) ms⁻¹, wall time: 32.699 seconds
[ Info: i: 0380, t: 1.162 hours, Δt: 7.733 seconds, umax = (9.2e-02, 5.6e-02, 4.4e-02) ms⁻¹, wall time: 33.135 seconds
[ Info: i: 0400, t: 1.206 hours, Δt: 8.196 seconds, umax = (9.1e-02, 4.7e-02, 3.9e-02) ms⁻¹, wall time: 33.592 seconds
[ Info: i: 0420, t: 1.250 hours, Δt: 7.313 seconds, umax = (8.6e-02, 5.0e-02, 3.8e-02) ms⁻¹, wall time: 34.027 seconds
[ Info: i: 0440, t: 1.291 hours, Δt: 7.686 seconds, umax = (8.8e-02, 4.7e-02, 3.8e-02) ms⁻¹, wall time: 34.476 seconds
[ Info: i: 0460, t: 1.333 hours, Δt: 7.898 seconds, umax = (9.3e-02, 5.3e-02, 3.9e-02) ms⁻¹, wall time: 34.920 seconds
[ Info: i: 0480, t: 1.376 hours, Δt: 7.225 seconds, umax = (1.0e-01, 5.2e-02, 3.8e-02) ms⁻¹, wall time: 35.374 seconds
[ Info: i: 0500, t: 1.417 hours, Δt: 7.249 seconds, umax = (9.6e-02, 4.8e-02, 4.2e-02) ms⁻¹, wall time: 35.824 seconds
[ Info: i: 0520, t: 1.456 hours, Δt: 7.523 seconds, umax = (1.0e-01, 5.4e-02, 4.0e-02) ms⁻¹, wall time: 36.319 seconds
[ Info: i: 0540, t: 1.498 hours, Δt: 7.258 seconds, umax = (9.8e-02, 5.6e-02, 4.0e-02) ms⁻¹, wall time: 36.817 seconds
[ Info: i: 0560, t: 1.537 hours, Δt: 7.114 seconds, umax = (9.3e-02, 5.4e-02, 4.2e-02) ms⁻¹, wall time: 37.289 seconds
[ Info: i: 0580, t: 1.577 hours, Δt: 6.775 seconds, umax = (9.3e-02, 5.9e-02, 4.0e-02) ms⁻¹, wall time: 37.742 seconds
[ Info: i: 0600, t: 1.613 hours, Δt: 7.268 seconds, umax = (9.4e-02, 5.7e-02, 4.2e-02) ms⁻¹, wall time: 38.210 seconds
[ Info: i: 0620, t: 1.654 hours, Δt: 7.081 seconds, umax = (1.0e-01, 5.4e-02, 4.5e-02) ms⁻¹, wall time: 38.668 seconds
[ Info: i: 0640, t: 1.693 hours, Δt: 6.845 seconds, umax = (9.3e-02, 6.5e-02, 4.5e-02) ms⁻¹, wall time: 39.138 seconds
[ Info: i: 0660, t: 1.731 hours, Δt: 6.863 seconds, umax = (9.6e-02, 5.7e-02, 4.8e-02) ms⁻¹, wall time: 39.604 seconds
[ Info: i: 0680, t: 1.767 hours, Δt: 6.748 seconds, umax = (1.0e-01, 5.4e-02, 5.2e-02) ms⁻¹, wall time: 40.087 seconds
[ Info: i: 0700, t: 1.806 hours, Δt: 7.102 seconds, umax = (9.7e-02, 5.5e-02, 4.5e-02) ms⁻¹, wall time: 40.539 seconds
[ Info: i: 0720, t: 1.843 hours, Δt: 6.812 seconds, umax = (1.0e-01, 6.2e-02, 4.3e-02) ms⁻¹, wall time: 41.090 seconds
[ Info: i: 0740, t: 1.879 hours, Δt: 6.798 seconds, umax = (9.5e-02, 5.8e-02, 4.5e-02) ms⁻¹, wall time: 41.487 seconds
[ Info: i: 0760, t: 1.917 hours, Δt: 7.087 seconds, umax = (9.8e-02, 5.7e-02, 4.2e-02) ms⁻¹, wall time: 41.957 seconds
[ Info: i: 0780, t: 1.955 hours, Δt: 6.770 seconds, umax = (9.6e-02, 6.0e-02, 4.0e-02) ms⁻¹, wall time: 42.440 seconds
[ Info: i: 0800, t: 1.992 hours, Δt: 6.087 seconds, umax = (1.0e-01, 6.3e-02, 4.1e-02) ms⁻¹, wall time: 42.908 seconds
[ Info: i: 0820, t: 2.026 hours, Δt: 6.039 seconds, umax = (1.1e-01, 7.3e-02, 4.5e-02) ms⁻¹, wall time: 43.408 seconds
[ Info: i: 0840, t: 2.060 hours, Δt: 6.683 seconds, umax = (1.1e-01, 6.6e-02, 4.4e-02) ms⁻¹, wall time: 43.877 seconds
[ Info: i: 0860, t: 2.097 hours, Δt: 6.518 seconds, umax = (1.0e-01, 6.2e-02, 4.1e-02) ms⁻¹, wall time: 44.383 seconds
[ Info: i: 0880, t: 2.132 hours, Δt: 6.493 seconds, umax = (1.0e-01, 6.1e-02, 4.4e-02) ms⁻¹, wall time: 44.829 seconds
[ Info: i: 0900, t: 2.169 hours, Δt: 6.628 seconds, umax = (1.0e-01, 7.0e-02, 4.6e-02) ms⁻¹, wall time: 45.485 seconds
[ Info: i: 0920, t: 2.203 hours, Δt: 6.496 seconds, umax = (1.0e-01, 6.9e-02, 4.4e-02) ms⁻¹, wall time: 45.808 seconds
[ Info: i: 0940, t: 2.238 hours, Δt: 6.534 seconds, umax = (9.7e-02, 6.4e-02, 4.4e-02) ms⁻¹, wall time: 46.286 seconds
[ Info: i: 0960, t: 2.274 hours, Δt: 6.138 seconds, umax = (1.0e-01, 6.3e-02, 4.6e-02) ms⁻¹, wall time: 46.772 seconds
[ Info: i: 0980, t: 2.309 hours, Δt: 6.608 seconds, umax = (1.0e-01, 6.3e-02, 4.4e-02) ms⁻¹, wall time: 47.251 seconds
[ Info: i: 1000, t: 2.343 hours, Δt: 6.463 seconds, umax = (1.1e-01, 7.0e-02, 4.5e-02) ms⁻¹, wall time: 47.780 seconds
[ Info: i: 1020, t: 2.379 hours, Δt: 6.221 seconds, umax = (1.0e-01, 6.4e-02, 4.6e-02) ms⁻¹, wall time: 48.202 seconds
[ Info: i: 1040, t: 2.414 hours, Δt: 6.131 seconds, umax = (1.1e-01, 6.4e-02, 4.5e-02) ms⁻¹, wall time: 48.688 seconds
[ Info: i: 1060, t: 2.447 hours, Δt: 6.299 seconds, umax = (1.1e-01, 7.2e-02, 4.6e-02) ms⁻¹, wall time: 49.180 seconds
[ Info: i: 1080, t: 2.482 hours, Δt: 6.022 seconds, umax = (1.0e-01, 7.8e-02, 4.7e-02) ms⁻¹, wall time: 49.661 seconds
[ Info: i: 1100, t: 2.516 hours, Δt: 5.863 seconds, umax = (1.0e-01, 6.4e-02, 5.8e-02) ms⁻¹, wall time: 50.163 seconds
[ Info: i: 1120, t: 2.549 hours, Δt: 6.161 seconds, umax = (1.1e-01, 7.0e-02, 4.8e-02) ms⁻¹, wall time: 50.623 seconds
[ Info: i: 1140, t: 2.583 hours, Δt: 6.215 seconds, umax = (1.1e-01, 6.8e-02, 4.4e-02) ms⁻¹, wall time: 51.116 seconds
[ Info: i: 1160, t: 2.618 hours, Δt: 6.502 seconds, umax = (1.0e-01, 6.9e-02, 4.4e-02) ms⁻¹, wall time: 51.612 seconds
[ Info: i: 1180, t: 2.653 hours, Δt: 6.128 seconds, umax = (1.1e-01, 7.2e-02, 4.8e-02) ms⁻¹, wall time: 52.156 seconds
[ Info: i: 1200, t: 2.687 hours, Δt: 5.606 seconds, umax = (1.1e-01, 6.6e-02, 5.0e-02) ms⁻¹, wall time: 52.648 seconds
[ Info: i: 1220, t: 2.717 hours, Δt: 5.636 seconds, umax = (1.1e-01, 7.2e-02, 4.6e-02) ms⁻¹, wall time: 53.120 seconds
[ Info: i: 1240, t: 2.750 hours, Δt: 6.228 seconds, umax = (1.0e-01, 6.5e-02, 4.7e-02) ms⁻¹, wall time: 53.606 seconds
[ Info: i: 1260, t: 2.784 hours, Δt: 5.530 seconds, umax = (1.1e-01, 7.5e-02, 5.1e-02) ms⁻¹, wall time: 54.096 seconds
[ Info: i: 1280, t: 2.814 hours, Δt: 5.921 seconds, umax = (1.1e-01, 7.8e-02, 4.8e-02) ms⁻¹, wall time: 54.581 seconds
[ Info: i: 1300, t: 2.847 hours, Δt: 6.359 seconds, umax = (1.1e-01, 7.1e-02, 4.7e-02) ms⁻¹, wall time: 55.081 seconds
[ Info: i: 1320, t: 2.880 hours, Δt: 6.185 seconds, umax = (1.1e-01, 7.3e-02, 5.0e-02) ms⁻¹, wall time: 55.543 seconds
[ Info: i: 1340, t: 2.914 hours, Δt: 5.859 seconds, umax = (1.1e-01, 7.8e-02, 4.6e-02) ms⁻¹, wall time: 56.030 seconds
[ Info: i: 1360, t: 2.946 hours, Δt: 6.435 seconds, umax = (1.1e-01, 7.4e-02, 4.7e-02) ms⁻¹, wall time: 56.532 seconds
[ Info: i: 1380, t: 2.982 hours, Δt: 6.260 seconds, umax = (1.2e-01, 7.3e-02, 4.7e-02) ms⁻¹, wall time: 57.018 seconds
[ Info: i: 1400, t: 3.015 hours, Δt: 6.056 seconds, umax = (1.0e-01, 7.2e-02, 4.5e-02) ms⁻¹, wall time: 57.520 seconds
[ Info: i: 1420, t: 3.049 hours, Δt: 6.190 seconds, umax = (1.1e-01, 7.2e-02, 4.5e-02) ms⁻¹, wall time: 57.998 seconds
[ Info: i: 1440, t: 3.083 hours, Δt: 5.985 seconds, umax = (1.1e-01, 7.1e-02, 4.5e-02) ms⁻¹, wall time: 58.492 seconds
[ Info: i: 1460, t: 3.117 hours, Δt: 5.825 seconds, umax = (1.0e-01, 7.4e-02, 4.4e-02) ms⁻¹, wall time: 58.981 seconds
[ Info: i: 1480, t: 3.150 hours, Δt: 6.004 seconds, umax = (1.1e-01, 7.1e-02, 4.7e-02) ms⁻¹, wall time: 59.466 seconds
[ Info: i: 1500, t: 3.182 hours, Δt: 5.968 seconds, umax = (1.1e-01, 6.8e-02, 4.5e-02) ms⁻¹, wall time: 59.966 seconds
[ Info: i: 1520, t: 3.215 hours, Δt: 5.585 seconds, umax = (1.2e-01, 7.2e-02, 4.9e-02) ms⁻¹, wall time: 1.007 minutes
[ Info: i: 1540, t: 3.247 hours, Δt: 5.715 seconds, umax = (1.1e-01, 7.0e-02, 4.8e-02) ms⁻¹, wall time: 1.015 minutes
[ Info: i: 1560, t: 3.278 hours, Δt: 5.960 seconds, umax = (1.2e-01, 7.5e-02, 4.8e-02) ms⁻¹, wall time: 1.024 minutes
[ Info: i: 1580, t: 3.311 hours, Δt: 6.145 seconds, umax = (1.1e-01, 7.5e-02, 4.9e-02) ms⁻¹, wall time: 1.032 minutes
[ Info: i: 1600, t: 3.345 hours, Δt: 5.825 seconds, umax = (1.1e-01, 7.4e-02, 4.8e-02) ms⁻¹, wall time: 1.041 minutes
[ Info: i: 1620, t: 3.378 hours, Δt: 5.759 seconds, umax = (1.1e-01, 7.6e-02, 4.7e-02) ms⁻¹, wall time: 1.048 minutes
[ Info: i: 1640, t: 3.410 hours, Δt: 6.004 seconds, umax = (1.1e-01, 8.4e-02, 4.4e-02) ms⁻¹, wall time: 1.056 minutes
[ Info: i: 1660, t: 3.443 hours, Δt: 6.456 seconds, umax = (1.0e-01, 7.5e-02, 4.7e-02) ms⁻¹, wall time: 1.065 minutes
[ Info: i: 1680, t: 3.478 hours, Δt: 5.529 seconds, umax = (1.1e-01, 8.4e-02, 4.6e-02) ms⁻¹, wall time: 1.073 minutes
[ Info: i: 1700, t: 3.508 hours, Δt: 5.883 seconds, umax = (1.1e-01, 8.3e-02, 4.8e-02) ms⁻¹, wall time: 1.082 minutes
[ Info: i: 1720, t: 3.540 hours, Δt: 5.409 seconds, umax = (1.1e-01, 8.3e-02, 5.0e-02) ms⁻¹, wall time: 1.089 minutes
[ Info: i: 1740, t: 3.571 hours, Δt: 6.151 seconds, umax = (1.1e-01, 7.4e-02, 4.5e-02) ms⁻¹, wall time: 1.097 minutes
[ Info: i: 1760, t: 3.603 hours, Δt: 6.017 seconds, umax = (1.0e-01, 7.4e-02, 4.8e-02) ms⁻¹, wall time: 1.106 minutes
[ Info: i: 1780, t: 3.636 hours, Δt: 5.801 seconds, umax = (1.1e-01, 7.5e-02, 5.0e-02) ms⁻¹, wall time: 1.113 minutes
[ Info: i: 1800, t: 3.668 hours, Δt: 5.951 seconds, umax = (1.1e-01, 7.4e-02, 4.7e-02) ms⁻¹, wall time: 1.124 minutes
[ Info: i: 1820, t: 3.702 hours, Δt: 5.980 seconds, umax = (1.1e-01, 7.1e-02, 4.7e-02) ms⁻¹, wall time: 1.130 minutes
[ Info: i: 1840, t: 3.736 hours, Δt: 5.841 seconds, umax = (1.1e-01, 7.3e-02, 5.6e-02) ms⁻¹, wall time: 1.138 minutes
[ Info: i: 1860, t: 3.769 hours, Δt: 5.975 seconds, umax = (1.1e-01, 7.8e-02, 5.0e-02) ms⁻¹, wall time: 1.146 minutes
[ Info: i: 1880, t: 3.802 hours, Δt: 6.136 seconds, umax = (1.1e-01, 7.4e-02, 4.7e-02) ms⁻¹, wall time: 1.154 minutes
[ Info: i: 1900, t: 3.833 hours, Δt: 5.270 seconds, umax = (1.1e-01, 7.9e-02, 5.2e-02) ms⁻¹, wall time: 1.162 minutes
[ Info: i: 1920, t: 3.864 hours, Δt: 6.266 seconds, umax = (1.1e-01, 7.6e-02, 4.8e-02) ms⁻¹, wall time: 1.170 minutes
[ Info: i: 1940, t: 3.897 hours, Δt: 5.388 seconds, umax = (1.1e-01, 7.8e-02, 4.6e-02) ms⁻¹, wall time: 1.178 minutes
[ Info: i: 1960, t: 3.927 hours, Δt: 5.407 seconds, umax = (1.1e-01, 8.0e-02, 5.2e-02) ms⁻¹, wall time: 1.187 minutes
[ Info: i: 1980, t: 3.958 hours, Δt: 5.617 seconds, umax = (1.1e-01, 7.2e-02, 4.5e-02) ms⁻¹, wall time: 1.194 minutes
[ Info: i: 2000, t: 3.990 hours, Δt: 5.721 seconds, umax = (1.1e-01, 7.1e-02, 5.2e-02) ms⁻¹, wall time: 1.202 minutes
[ Info: Simulation is stopping after running for 1.205 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-28008/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-28008/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-28008/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-28008`
  [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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