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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 = (1.6e-03, 8.4e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (12.072 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.837 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 18.416 seconds, umax = (3.5e-02, 1.3e-02, 2.0e-02) ms⁻¹, wall time: 16.043 seconds
[ Info: i: 0040, t: 16.806 minutes, Δt: 13.128 seconds, umax = (5.2e-02, 2.3e-02, 2.2e-02) ms⁻¹, wall time: 16.474 seconds
[ Info: i: 0060, t: 20.802 minutes, Δt: 11.161 seconds, umax = (6.3e-02, 2.8e-02, 3.3e-02) ms⁻¹, wall time: 17.026 seconds
[ Info: i: 0080, t: 24.508 minutes, Δt: 10.901 seconds, umax = (6.7e-02, 3.0e-02, 3.1e-02) ms⁻¹, wall time: 17.387 seconds
[ Info: i: 0100, t: 28.111 minutes, Δt: 10.669 seconds, umax = (6.1e-02, 3.0e-02, 3.2e-02) ms⁻¹, wall time: 17.854 seconds
[ Info: i: 0120, t: 31.554 minutes, Δt: 10.641 seconds, umax = (6.3e-02, 3.4e-02, 3.0e-02) ms⁻¹, wall time: 18.336 seconds
[ Info: i: 0140, t: 35 minutes, Δt: 9.996 seconds, umax = (6.4e-02, 3.7e-02, 2.8e-02) ms⁻¹, wall time: 18.747 seconds
[ Info: i: 0160, t: 38.294 minutes, Δt: 9.801 seconds, umax = (6.6e-02, 3.7e-02, 3.2e-02) ms⁻¹, wall time: 19.229 seconds
[ Info: i: 0180, t: 41.511 minutes, Δt: 9.734 seconds, umax = (7.2e-02, 3.8e-02, 4.0e-02) ms⁻¹, wall time: 19.717 seconds
[ Info: i: 0200, t: 44.602 minutes, Δt: 9.457 seconds, umax = (8.1e-02, 3.9e-02, 3.5e-02) ms⁻¹, wall time: 20.126 seconds
[ Info: i: 0220, t: 47.575 minutes, Δt: 9.025 seconds, umax = (7.6e-02, 4.0e-02, 3.5e-02) ms⁻¹, wall time: 20.602 seconds
[ Info: i: 0240, t: 50.629 minutes, Δt: 9.293 seconds, umax = (7.5e-02, 3.8e-02, 3.6e-02) ms⁻¹, wall time: 21.156 seconds
[ Info: i: 0260, t: 53.747 minutes, Δt: 8.980 seconds, umax = (7.8e-02, 3.8e-02, 3.7e-02) ms⁻¹, wall time: 21.516 seconds
[ Info: i: 0280, t: 56.611 minutes, Δt: 8.531 seconds, umax = (8.0e-02, 4.3e-02, 3.7e-02) ms⁻¹, wall time: 21.988 seconds
[ Info: i: 0300, t: 59.425 minutes, Δt: 8.542 seconds, umax = (8.4e-02, 4.1e-02, 4.6e-02) ms⁻¹, wall time: 22.408 seconds
[ Info: i: 0320, t: 1.035 hours, Δt: 9.096 seconds, umax = (8.3e-02, 4.6e-02, 4.1e-02) ms⁻¹, wall time: 22.850 seconds
[ Info: i: 0340, t: 1.082 hours, Δt: 8.137 seconds, umax = (8.2e-02, 4.3e-02, 3.8e-02) ms⁻¹, wall time: 23.272 seconds
[ Info: i: 0360, t: 1.127 hours, Δt: 8.063 seconds, umax = (8.8e-02, 4.8e-02, 3.8e-02) ms⁻¹, wall time: 23.712 seconds
[ Info: i: 0380, t: 1.169 hours, Δt: 7.658 seconds, umax = (9.0e-02, 4.8e-02, 4.1e-02) ms⁻¹, wall time: 24.278 seconds
[ Info: i: 0400, t: 1.213 hours, Δt: 7.827 seconds, umax = (8.8e-02, 4.6e-02, 3.8e-02) ms⁻¹, wall time: 24.582 seconds
[ Info: i: 0420, t: 1.254 hours, Δt: 8.158 seconds, umax = (9.0e-02, 4.9e-02, 3.9e-02) ms⁻¹, wall time: 25.130 seconds
[ Info: i: 0440, t: 1.300 hours, Δt: 7.801 seconds, umax = (8.8e-02, 4.9e-02, 4.7e-02) ms⁻¹, wall time: 25.449 seconds
[ Info: i: 0460, t: 1.342 hours, Δt: 7.704 seconds, umax = (8.4e-02, 4.9e-02, 3.9e-02) ms⁻¹, wall time: 25.964 seconds
[ Info: i: 0480, t: 1.386 hours, Δt: 7.388 seconds, umax = (9.1e-02, 5.4e-02, 4.2e-02) ms⁻¹, wall time: 26.325 seconds
[ Info: i: 0500, t: 1.424 hours, Δt: 7.566 seconds, umax = (8.9e-02, 5.6e-02, 4.1e-02) ms⁻¹, wall time: 26.848 seconds
[ Info: i: 0520, t: 1.467 hours, Δt: 7.751 seconds, umax = (9.5e-02, 5.1e-02, 4.3e-02) ms⁻¹, wall time: 27.213 seconds
[ Info: i: 0540, t: 1.508 hours, Δt: 7.512 seconds, umax = (9.6e-02, 6.1e-02, 4.2e-02) ms⁻¹, wall time: 27.743 seconds
[ Info: i: 0560, t: 1.549 hours, Δt: 7.502 seconds, umax = (9.5e-02, 5.4e-02, 4.2e-02) ms⁻¹, wall time: 28.105 seconds
[ Info: i: 0580, t: 1.589 hours, Δt: 7.618 seconds, umax = (9.9e-02, 5.2e-02, 4.8e-02) ms⁻¹, wall time: 28.649 seconds
[ Info: i: 0600, t: 1.631 hours, Δt: 7.163 seconds, umax = (9.8e-02, 5.7e-02, 4.1e-02) ms⁻¹, wall time: 28.997 seconds
[ Info: i: 0620, t: 1.669 hours, Δt: 6.953 seconds, umax = (9.4e-02, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 29.580 seconds
[ Info: i: 0640, t: 1.707 hours, Δt: 7.066 seconds, umax = (9.5e-02, 6.0e-02, 4.3e-02) ms⁻¹, wall time: 29.893 seconds
[ Info: i: 0660, t: 1.745 hours, Δt: 7.409 seconds, umax = (9.8e-02, 5.2e-02, 4.3e-02) ms⁻¹, wall time: 30.334 seconds
[ Info: i: 0680, t: 1.784 hours, Δt: 7.289 seconds, umax = (1.0e-01, 5.4e-02, 4.5e-02) ms⁻¹, wall time: 30.798 seconds
[ Info: i: 0700, t: 1.825 hours, Δt: 7.083 seconds, umax = (9.9e-02, 5.6e-02, 4.0e-02) ms⁻¹, wall time: 31.241 seconds
[ Info: i: 0720, t: 1.862 hours, Δt: 6.455 seconds, umax = (9.7e-02, 5.9e-02, 4.0e-02) ms⁻¹, wall time: 31.708 seconds
[ Info: i: 0740, t: 1.900 hours, Δt: 6.924 seconds, umax = (1.0e-01, 5.3e-02, 4.0e-02) ms⁻¹, wall time: 32.160 seconds
[ Info: i: 0760, t: 1.938 hours, Δt: 6.569 seconds, umax = (1.0e-01, 5.9e-02, 4.2e-02) ms⁻¹, wall time: 32.630 seconds
[ Info: i: 0780, t: 1.975 hours, Δt: 6.849 seconds, umax = (1.1e-01, 6.0e-02, 4.7e-02) ms⁻¹, wall time: 33.075 seconds
[ Info: i: 0800, t: 2.011 hours, Δt: 6.539 seconds, umax = (1.2e-01, 6.2e-02, 4.3e-02) ms⁻¹, wall time: 33.569 seconds
[ Info: i: 0820, t: 2.048 hours, Δt: 6.387 seconds, umax = (1.1e-01, 6.7e-02, 4.2e-02) ms⁻¹, wall time: 33.988 seconds
[ Info: i: 0840, t: 2.083 hours, Δt: 6.350 seconds, umax = (1.0e-01, 5.8e-02, 4.4e-02) ms⁻¹, wall time: 34.443 seconds
[ Info: i: 0860, t: 2.119 hours, Δt: 6.780 seconds, umax = (9.8e-02, 6.2e-02, 4.5e-02) ms⁻¹, wall time: 34.919 seconds
[ Info: i: 0880, t: 2.157 hours, Δt: 6.887 seconds, umax = (1.0e-01, 5.7e-02, 4.5e-02) ms⁻¹, wall time: 35.378 seconds
[ Info: i: 0900, t: 2.195 hours, Δt: 6.038 seconds, umax = (1.1e-01, 6.0e-02, 4.3e-02) ms⁻¹, wall time: 35.858 seconds
[ Info: i: 0920, t: 2.230 hours, Δt: 6.610 seconds, umax = (1.2e-01, 6.0e-02, 4.2e-02) ms⁻¹, wall time: 36.315 seconds
[ Info: i: 0940, t: 2.265 hours, Δt: 6.514 seconds, umax = (1.1e-01, 6.2e-02, 4.7e-02) ms⁻¹, wall time: 36.801 seconds
[ Info: i: 0960, t: 2.302 hours, Δt: 6.704 seconds, umax = (1.1e-01, 6.2e-02, 5.5e-02) ms⁻¹, wall time: 37.245 seconds
[ Info: i: 0980, t: 2.337 hours, Δt: 6.604 seconds, umax = (1.1e-01, 5.9e-02, 5.0e-02) ms⁻¹, wall time: 37.841 seconds
[ Info: i: 1000, t: 2.373 hours, Δt: 6.315 seconds, umax = (1.0e-01, 6.4e-02, 4.6e-02) ms⁻¹, wall time: 38.184 seconds
[ Info: i: 1020, t: 2.406 hours, Δt: 5.642 seconds, umax = (1.0e-01, 6.8e-02, 4.1e-02) ms⁻¹, wall time: 38.643 seconds
[ Info: i: 1040, t: 2.439 hours, Δt: 6.063 seconds, umax = (1.0e-01, 6.8e-02, 4.2e-02) ms⁻¹, wall time: 39.127 seconds
[ Info: i: 1060, t: 2.474 hours, Δt: 6.592 seconds, umax = (1.0e-01, 6.7e-02, 4.3e-02) ms⁻¹, wall time: 39.579 seconds
[ Info: i: 1080, t: 2.509 hours, Δt: 6.483 seconds, umax = (1.1e-01, 6.6e-02, 4.7e-02) ms⁻¹, wall time: 40.109 seconds
[ Info: i: 1100, t: 2.545 hours, Δt: 6.136 seconds, umax = (1.0e-01, 6.6e-02, 5.0e-02) ms⁻¹, wall time: 40.512 seconds
[ Info: i: 1120, t: 2.579 hours, Δt: 6.163 seconds, umax = (1.0e-01, 6.6e-02, 4.9e-02) ms⁻¹, wall time: 40.978 seconds
[ Info: i: 1140, t: 2.613 hours, Δt: 6.208 seconds, umax = (1.0e-01, 7.1e-02, 4.9e-02) ms⁻¹, wall time: 41.459 seconds
[ Info: i: 1160, t: 2.647 hours, Δt: 5.990 seconds, umax = (1.0e-01, 7.3e-02, 4.8e-02) ms⁻¹, wall time: 41.918 seconds
[ Info: i: 1180, t: 2.680 hours, Δt: 6.063 seconds, umax = (1.2e-01, 8.1e-02, 4.6e-02) ms⁻¹, wall time: 42.401 seconds
[ Info: i: 1200, t: 2.715 hours, Δt: 6.152 seconds, umax = (1.1e-01, 7.0e-02, 4.6e-02) ms⁻¹, wall time: 42.846 seconds
[ Info: i: 1220, t: 2.749 hours, Δt: 5.822 seconds, umax = (1.1e-01, 7.0e-02, 4.7e-02) ms⁻¹, wall time: 43.312 seconds
[ Info: i: 1240, t: 2.782 hours, Δt: 6.405 seconds, umax = (1.0e-01, 7.0e-02, 4.9e-02) ms⁻¹, wall time: 43.793 seconds
[ Info: i: 1260, t: 2.817 hours, Δt: 5.764 seconds, umax = (1.1e-01, 7.5e-02, 4.5e-02) ms⁻¹, wall time: 44.255 seconds
[ Info: i: 1280, t: 2.849 hours, Δt: 6.166 seconds, umax = (1.1e-01, 7.7e-02, 4.6e-02) ms⁻¹, wall time: 44.740 seconds
[ Info: i: 1300, t: 2.882 hours, Δt: 6.434 seconds, umax = (1.1e-01, 7.6e-02, 4.6e-02) ms⁻¹, wall time: 45.183 seconds
[ Info: i: 1320, t: 2.917 hours, Δt: 5.744 seconds, umax = (1.1e-01, 6.6e-02, 4.7e-02) ms⁻¹, wall time: 45.649 seconds
[ Info: i: 1340, t: 2.948 hours, Δt: 6.175 seconds, umax = (1.1e-01, 7.0e-02, 4.4e-02) ms⁻¹, wall time: 46.128 seconds
[ Info: i: 1360, t: 2.982 hours, Δt: 6.111 seconds, umax = (1.1e-01, 6.9e-02, 4.6e-02) ms⁻¹, wall time: 46.592 seconds
[ Info: i: 1380, t: 3.015 hours, Δt: 6.090 seconds, umax = (1.1e-01, 6.8e-02, 5.1e-02) ms⁻¹, wall time: 47.076 seconds
[ Info: i: 1400, t: 3.050 hours, Δt: 5.798 seconds, umax = (1.1e-01, 8.5e-02, 5.3e-02) ms⁻¹, wall time: 47.522 seconds
[ Info: i: 1420, t: 3.083 hours, Δt: 6.332 seconds, umax = (1.1e-01, 8.0e-02, 5.2e-02) ms⁻¹, wall time: 47.986 seconds
[ Info: i: 1440, t: 3.116 hours, Δt: 5.638 seconds, umax = (1.1e-01, 7.2e-02, 4.8e-02) ms⁻¹, wall time: 48.465 seconds
[ Info: i: 1460, t: 3.147 hours, Δt: 5.814 seconds, umax = (1.1e-01, 7.9e-02, 4.9e-02) ms⁻¹, wall time: 48.926 seconds
[ Info: i: 1480, t: 3.178 hours, Δt: 5.940 seconds, umax = (1.1e-01, 7.3e-02, 4.9e-02) ms⁻¹, wall time: 49.408 seconds
[ Info: i: 1500, t: 3.211 hours, Δt: 5.905 seconds, umax = (1.1e-01, 7.6e-02, 4.7e-02) ms⁻¹, wall time: 49.853 seconds
[ Info: i: 1520, t: 3.243 hours, Δt: 5.884 seconds, umax = (1.1e-01, 7.7e-02, 4.9e-02) ms⁻¹, wall time: 50.319 seconds
[ Info: i: 1540, t: 3.274 hours, Δt: 5.634 seconds, umax = (1.1e-01, 8.0e-02, 4.9e-02) ms⁻¹, wall time: 50.812 seconds
[ Info: i: 1560, t: 3.306 hours, Δt: 5.562 seconds, umax = (1.2e-01, 7.7e-02, 4.7e-02) ms⁻¹, wall time: 51.270 seconds
[ Info: i: 1580, t: 3.336 hours, Δt: 5.263 seconds, umax = (1.2e-01, 7.4e-02, 5.4e-02) ms⁻¹, wall time: 51.864 seconds
[ Info: i: 1600, t: 3.367 hours, Δt: 5.596 seconds, umax = (1.1e-01, 7.7e-02, 5.2e-02) ms⁻¹, wall time: 52.210 seconds
[ Info: i: 1620, t: 3.400 hours, Δt: 5.979 seconds, umax = (1.1e-01, 7.9e-02, 4.7e-02) ms⁻¹, wall time: 52.675 seconds
[ Info: i: 1640, t: 3.430 hours, Δt: 5.962 seconds, umax = (1.1e-01, 7.5e-02, 4.7e-02) ms⁻¹, wall time: 53.165 seconds
[ Info: i: 1660, t: 3.463 hours, Δt: 5.746 seconds, umax = (1.1e-01, 7.8e-02, 5.1e-02) ms⁻¹, wall time: 53.611 seconds
[ Info: i: 1680, t: 3.496 hours, Δt: 5.876 seconds, umax = (1.1e-01, 7.3e-02, 4.5e-02) ms⁻¹, wall time: 54.077 seconds
[ Info: i: 1700, t: 3.527 hours, Δt: 5.050 seconds, umax = (1.0e-01, 8.8e-02, 4.7e-02) ms⁻¹, wall time: 54.564 seconds
[ Info: i: 1720, t: 3.557 hours, Δt: 6.051 seconds, umax = (1.1e-01, 7.8e-02, 4.9e-02) ms⁻¹, wall time: 55.028 seconds
[ Info: i: 1740, t: 3.588 hours, Δt: 5.383 seconds, umax = (1.1e-01, 7.3e-02, 4.9e-02) ms⁻¹, wall time: 55.597 seconds
[ Info: i: 1760, t: 3.619 hours, Δt: 5.798 seconds, umax = (1.1e-01, 7.7e-02, 4.9e-02) ms⁻¹, wall time: 55.963 seconds
[ Info: i: 1780, t: 3.652 hours, Δt: 5.722 seconds, umax = (1.1e-01, 7.5e-02, 5.4e-02) ms⁻¹, wall time: 56.425 seconds
[ Info: i: 1800, t: 3.684 hours, Δt: 6.036 seconds, umax = (1.1e-01, 7.8e-02, 4.9e-02) ms⁻¹, wall time: 56.909 seconds
[ Info: i: 1820, t: 3.718 hours, Δt: 6.048 seconds, umax = (1.1e-01, 7.2e-02, 4.5e-02) ms⁻¹, wall time: 57.355 seconds
[ Info: i: 1840, t: 3.750 hours, Δt: 5.673 seconds, umax = (1.1e-01, 7.6e-02, 4.7e-02) ms⁻¹, wall time: 57.823 seconds
[ Info: i: 1860, t: 3.781 hours, Δt: 5.773 seconds, umax = (1.1e-01, 8.1e-02, 4.5e-02) ms⁻¹, wall time: 58.301 seconds
[ Info: i: 1880, t: 3.813 hours, Δt: 6.032 seconds, umax = (1.2e-01, 7.2e-02, 4.5e-02) ms⁻¹, wall time: 58.760 seconds
[ Info: i: 1900, t: 3.847 hours, Δt: 5.814 seconds, umax = (1.1e-01, 7.2e-02, 4.9e-02) ms⁻¹, wall time: 59.244 seconds
[ Info: i: 1920, t: 3.878 hours, Δt: 5.988 seconds, umax = (1.1e-01, 7.2e-02, 4.5e-02) ms⁻¹, wall time: 59.686 seconds
[ Info: i: 1940, t: 3.911 hours, Δt: 5.649 seconds, umax = (1.1e-01, 7.6e-02, 4.4e-02) ms⁻¹, wall time: 1.002 minutes
[ Info: i: 1960, t: 3.941 hours, Δt: 5.674 seconds, umax = (1.1e-01, 7.4e-02, 5.6e-02) ms⁻¹, wall time: 1.012 minutes
[ Info: i: 1980, t: 3.974 hours, Δt: 6.033 seconds, umax = (1.1e-01, 7.9e-02, 4.5e-02) ms⁻¹, wall time: 1.019 minutes
[ Info: Simulation is stopping after running for 1.026 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-28811/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-28811/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-28811/docs/Project.toml`
  [79e6a3ab] Adapt v4.4.0
  [052768ef] CUDA v5.9.6
  [13f3f980] CairoMakie v0.15.8
  [e30172f5] Documenter v1.16.1
  [daee34ce] DocumenterCitations v1.4.1
  [033835bb] JLD2 v0.6.3
  [63c18a36] KernelAbstractions v0.9.39
  [98b081ad] Literate v2.21.0
  [da04e1cc] MPI v0.20.23
  [85f8d34a] NCDatasets v0.14.10
  [9e8cae18] Oceananigans v0.104.2 `..`
  [f27b6e38] Polynomials v4.1.0
  [6038ab10] Rotations v1.7.1
  [d496a93d] SeawaterPolynomials v0.3.10
  [09ab397b] StructArrays v0.7.2
  [bdfc003b] TimesDates v0.3.3
  [2e0b0046] XESMF v0.1.6
  [b77e0a4c] InteractiveUtils v1.11.0
  [37e2e46d] LinearAlgebra v1.12.0
  [44cfe95a] Pkg v1.12.1

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