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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{GPU, 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{GPU, 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 = (1.6e-03, 8.1e-04, 1.3e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (18.298 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.851 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 18.026 seconds, umax = (3.6e-02, 1.3e-02, 2.2e-02) ms⁻¹, wall time: 23.488 seconds
[ Info: i: 0040, t: 16.515 minutes, Δt: 13.510 seconds, umax = (5.2e-02, 1.8e-02, 2.4e-02) ms⁻¹, wall time: 23.977 seconds
[ Info: i: 0060, t: 20.584 minutes, Δt: 10.850 seconds, umax = (6.0e-02, 2.8e-02, 3.4e-02) ms⁻¹, wall time: 24.554 seconds
[ Info: i: 0080, t: 24.212 minutes, Δt: 10.493 seconds, umax = (6.1e-02, 3.0e-02, 3.4e-02) ms⁻¹, wall time: 24.891 seconds
[ Info: i: 0100, t: 27.690 minutes, Δt: 11.080 seconds, umax = (6.1e-02, 2.8e-02, 3.0e-02) ms⁻¹, wall time: 25.398 seconds
[ Info: i: 0120, t: 31.317 minutes, Δt: 11.147 seconds, umax = (6.1e-02, 3.5e-02, 2.8e-02) ms⁻¹, wall time: 25.876 seconds
[ Info: i: 0140, t: 34.958 minutes, Δt: 10.480 seconds, umax = (6.5e-02, 3.2e-02, 3.0e-02) ms⁻¹, wall time: 26.293 seconds
[ Info: i: 0160, t: 38.320 minutes, Δt: 9.791 seconds, umax = (6.8e-02, 3.5e-02, 3.4e-02) ms⁻¹, wall time: 26.738 seconds
[ Info: i: 0180, t: 41.463 minutes, Δt: 9.953 seconds, umax = (7.8e-02, 4.0e-02, 3.2e-02) ms⁻¹, wall time: 27.187 seconds
[ Info: i: 0200, t: 44.727 minutes, Δt: 9.776 seconds, umax = (6.9e-02, 3.6e-02, 3.3e-02) ms⁻¹, wall time: 27.612 seconds
[ Info: i: 0220, t: 47.869 minutes, Δt: 8.984 seconds, umax = (7.3e-02, 4.3e-02, 3.5e-02) ms⁻¹, wall time: 28.045 seconds
[ Info: i: 0240, t: 50.714 minutes, Δt: 8.782 seconds, umax = (7.2e-02, 4.3e-02, 3.6e-02) ms⁻¹, wall time: 28.542 seconds
[ Info: i: 0260, t: 53.730 minutes, Δt: 9.040 seconds, umax = (8.1e-02, 3.9e-02, 3.6e-02) ms⁻¹, wall time: 28.915 seconds
[ Info: i: 0280, t: 56.613 minutes, Δt: 8.915 seconds, umax = (8.5e-02, 4.6e-02, 3.7e-02) ms⁻¹, wall time: 29.352 seconds
[ Info: i: 0300, t: 59.522 minutes, Δt: 8.215 seconds, umax = (7.7e-02, 4.4e-02, 3.7e-02) ms⁻¹, wall time: 29.779 seconds
[ Info: i: 0320, t: 1.037 hours, Δt: 8.340 seconds, umax = (8.4e-02, 4.4e-02, 3.9e-02) ms⁻¹, wall time: 30.214 seconds
[ Info: i: 0340, t: 1.083 hours, Δt: 8.835 seconds, umax = (8.3e-02, 4.2e-02, 3.8e-02) ms⁻¹, wall time: 30.642 seconds
[ Info: i: 0360, t: 1.129 hours, Δt: 7.864 seconds, umax = (8.8e-02, 4.6e-02, 3.5e-02) ms⁻¹, wall time: 31.076 seconds
[ Info: i: 0380, t: 1.171 hours, Δt: 7.684 seconds, umax = (8.7e-02, 4.6e-02, 3.9e-02) ms⁻¹, wall time: 31.654 seconds
[ Info: i: 0400, t: 1.215 hours, Δt: 8.050 seconds, umax = (8.5e-02, 5.0e-02, 3.7e-02) ms⁻¹, wall time: 31.957 seconds
[ Info: i: 0420, t: 1.256 hours, Δt: 7.784 seconds, umax = (8.8e-02, 5.9e-02, 3.8e-02) ms⁻¹, wall time: 32.506 seconds
[ Info: i: 0440, t: 1.299 hours, Δt: 7.925 seconds, umax = (9.3e-02, 5.0e-02, 4.0e-02) ms⁻¹, wall time: 32.833 seconds
[ Info: i: 0460, t: 1.340 hours, Δt: 7.778 seconds, umax = (9.8e-02, 4.6e-02, 4.2e-02) ms⁻¹, wall time: 33.383 seconds
[ Info: i: 0480, t: 1.381 hours, Δt: 7.077 seconds, umax = (9.1e-02, 5.8e-02, 3.9e-02) ms⁻¹, wall time: 33.716 seconds
[ Info: i: 0500, t: 1.421 hours, Δt: 7.854 seconds, umax = (9.4e-02, 4.9e-02, 3.8e-02) ms⁻¹, wall time: 34.294 seconds
[ Info: i: 0520, t: 1.464 hours, Δt: 7.167 seconds, umax = (9.0e-02, 5.3e-02, 4.0e-02) ms⁻¹, wall time: 34.607 seconds
[ Info: i: 0540, t: 1.502 hours, Δt: 6.776 seconds, umax = (9.4e-02, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 35.200 seconds
[ Info: i: 0560, t: 1.541 hours, Δt: 7.378 seconds, umax = (9.0e-02, 5.2e-02, 4.3e-02) ms⁻¹, wall time: 35.503 seconds
[ Info: i: 0580, t: 1.581 hours, Δt: 6.831 seconds, umax = (9.4e-02, 5.5e-02, 3.9e-02) ms⁻¹, wall time: 35.966 seconds
[ Info: i: 0600, t: 1.618 hours, Δt: 7.280 seconds, umax = (9.4e-02, 6.3e-02, 4.3e-02) ms⁻¹, wall time: 36.422 seconds
[ Info: i: 0620, t: 1.659 hours, Δt: 7.378 seconds, umax = (9.1e-02, 5.8e-02, 4.4e-02) ms⁻¹, wall time: 36.878 seconds
[ Info: i: 0640, t: 1.699 hours, Δt: 6.995 seconds, umax = (9.2e-02, 5.7e-02, 4.0e-02) ms⁻¹, wall time: 37.348 seconds
[ Info: i: 0660, t: 1.738 hours, Δt: 7.257 seconds, umax = (9.4e-02, 5.5e-02, 4.3e-02) ms⁻¹, wall time: 37.810 seconds
[ Info: i: 0680, t: 1.777 hours, Δt: 6.857 seconds, umax = (9.5e-02, 5.6e-02, 4.4e-02) ms⁻¹, wall time: 38.288 seconds
[ Info: i: 0700, t: 1.815 hours, Δt: 6.628 seconds, umax = (9.4e-02, 5.8e-02, 4.5e-02) ms⁻¹, wall time: 38.751 seconds
[ Info: i: 0720, t: 1.851 hours, Δt: 6.178 seconds, umax = (1.0e-01, 6.2e-02, 4.2e-02) ms⁻¹, wall time: 39.236 seconds
[ Info: i: 0740, t: 1.887 hours, Δt: 6.660 seconds, umax = (9.8e-02, 5.8e-02, 4.1e-02) ms⁻¹, wall time: 39.682 seconds
[ Info: i: 0760, t: 1.924 hours, Δt: 6.382 seconds, umax = (9.8e-02, 6.4e-02, 4.1e-02) ms⁻¹, wall time: 41.091 seconds
[ Info: i: 0780, t: 1.959 hours, Δt: 6.599 seconds, umax = (9.9e-02, 6.1e-02, 4.0e-02) ms⁻¹, wall time: 41.466 seconds
[ Info: i: 0800, t: 1.995 hours, Δt: 6.498 seconds, umax = (1.1e-01, 6.3e-02, 4.6e-02) ms⁻¹, wall time: 41.934 seconds
[ Info: i: 0820, t: 2.031 hours, Δt: 6.943 seconds, umax = (1.0e-01, 6.5e-02, 4.7e-02) ms⁻¹, wall time: 42.403 seconds
[ Info: i: 0840, t: 2.069 hours, Δt: 6.591 seconds, umax = (1.0e-01, 6.4e-02, 4.9e-02) ms⁻¹, wall time: 42.876 seconds
[ Info: i: 0860, t: 2.105 hours, Δt: 6.493 seconds, umax = (1.0e-01, 6.5e-02, 4.1e-02) ms⁻¹, wall time: 43.384 seconds
[ Info: i: 0880, t: 2.141 hours, Δt: 6.533 seconds, umax = (1.0e-01, 6.9e-02, 4.3e-02) ms⁻¹, wall time: 43.869 seconds
[ Info: i: 0900, t: 2.178 hours, Δt: 6.926 seconds, umax = (1.1e-01, 6.0e-02, 4.5e-02) ms⁻¹, wall time: 44.388 seconds
[ Info: i: 0920, t: 2.215 hours, Δt: 6.570 seconds, umax = (1.0e-01, 6.5e-02, 4.4e-02) ms⁻¹, wall time: 44.810 seconds
[ Info: i: 0940, t: 2.252 hours, Δt: 6.147 seconds, umax = (1.1e-01, 6.0e-02, 4.5e-02) ms⁻¹, wall time: 45.454 seconds
[ Info: i: 0960, t: 2.284 hours, Δt: 5.747 seconds, umax = (1.0e-01, 6.0e-02, 4.4e-02) ms⁻¹, wall time: 45.773 seconds
[ Info: i: 0980, t: 2.316 hours, Δt: 6.261 seconds, umax = (1.0e-01, 6.3e-02, 4.8e-02) ms⁻¹, wall time: 46.246 seconds
[ Info: i: 1000, t: 2.351 hours, Δt: 6.424 seconds, umax = (1.0e-01, 6.4e-02, 4.3e-02) ms⁻¹, wall time: 46.735 seconds
[ Info: i: 1020, t: 2.386 hours, Δt: 6.544 seconds, umax = (9.9e-02, 6.7e-02, 4.4e-02) ms⁻¹, wall time: 47.197 seconds
[ Info: i: 1040, t: 2.422 hours, Δt: 6.314 seconds, umax = (1.0e-01, 6.4e-02, 4.3e-02) ms⁻¹, wall time: 47.787 seconds
[ Info: i: 1060, t: 2.457 hours, Δt: 6.259 seconds, umax = (1.1e-01, 7.4e-02, 4.5e-02) ms⁻¹, wall time: 48.148 seconds
[ Info: i: 1080, t: 2.493 hours, Δt: 6.557 seconds, umax = (1.0e-01, 6.7e-02, 4.2e-02) ms⁻¹, wall time: 48.625 seconds
[ Info: i: 1100, t: 2.527 hours, Δt: 5.868 seconds, umax = (1.1e-01, 6.8e-02, 4.8e-02) ms⁻¹, wall time: 49.125 seconds
[ Info: i: 1120, t: 2.560 hours, Δt: 6.335 seconds, umax = (1.0e-01, 7.0e-02, 4.7e-02) ms⁻¹, wall time: 49.601 seconds
[ Info: i: 1140, t: 2.594 hours, Δt: 6.322 seconds, umax = (1.0e-01, 6.9e-02, 4.8e-02) ms⁻¹, wall time: 50.120 seconds
[ Info: i: 1160, t: 2.629 hours, Δt: 6.059 seconds, umax = (1.1e-01, 6.6e-02, 4.6e-02) ms⁻¹, wall time: 50.542 seconds
[ Info: i: 1180, t: 2.663 hours, Δt: 6.502 seconds, umax = (1.1e-01, 6.9e-02, 5.0e-02) ms⁻¹, wall time: 51.031 seconds
[ Info: i: 1200, t: 2.699 hours, Δt: 6.501 seconds, umax = (1.0e-01, 6.7e-02, 4.2e-02) ms⁻¹, wall time: 51.522 seconds
[ Info: i: 1220, t: 2.735 hours, Δt: 6.428 seconds, umax = (1.0e-01, 6.5e-02, 4.5e-02) ms⁻¹, wall time: 52.007 seconds
[ Info: i: 1240, t: 2.769 hours, Δt: 5.659 seconds, umax = (1.0e-01, 7.5e-02, 4.5e-02) ms⁻¹, wall time: 52.499 seconds
[ Info: i: 1260, t: 2.801 hours, Δt: 6.028 seconds, umax = (1.0e-01, 6.6e-02, 4.3e-02) ms⁻¹, wall time: 52.958 seconds
[ Info: i: 1280, t: 2.833 hours, Δt: 5.908 seconds, umax = (1.0e-01, 7.1e-02, 4.6e-02) ms⁻¹, wall time: 53.448 seconds
[ Info: i: 1300, t: 2.868 hours, Δt: 6.801 seconds, umax = (1.2e-01, 7.1e-02, 5.0e-02) ms⁻¹, wall time: 53.933 seconds
[ Info: i: 1320, t: 2.905 hours, Δt: 6.361 seconds, umax = (1.1e-01, 7.1e-02, 4.9e-02) ms⁻¹, wall time: 54.415 seconds
[ Info: i: 1340, t: 2.938 hours, Δt: 6.215 seconds, umax = (1.1e-01, 6.9e-02, 4.7e-02) ms⁻¹, wall time: 54.904 seconds
[ Info: i: 1360, t: 2.971 hours, Δt: 5.926 seconds, umax = (1.1e-01, 7.6e-02, 5.0e-02) ms⁻¹, wall time: 55.383 seconds
[ Info: i: 1380, t: 3.003 hours, Δt: 5.920 seconds, umax = (1.1e-01, 6.7e-02, 4.6e-02) ms⁻¹, wall time: 56.003 seconds
[ Info: i: 1400, t: 3.037 hours, Δt: 5.968 seconds, umax = (1.1e-01, 7.5e-02, 5.0e-02) ms⁻¹, wall time: 56.347 seconds
[ Info: i: 1420, t: 3.070 hours, Δt: 5.993 seconds, umax = (1.1e-01, 7.0e-02, 4.6e-02) ms⁻¹, wall time: 56.830 seconds
[ Info: i: 1440, t: 3.101 hours, Δt: 5.748 seconds, umax = (1.1e-01, 7.7e-02, 4.6e-02) ms⁻¹, wall time: 57.341 seconds
[ Info: i: 1460, t: 3.134 hours, Δt: 5.787 seconds, umax = (1.1e-01, 7.4e-02, 4.4e-02) ms⁻¹, wall time: 57.805 seconds
[ Info: i: 1480, t: 3.166 hours, Δt: 5.669 seconds, umax = (1.1e-01, 7.5e-02, 4.8e-02) ms⁻¹, wall time: 58.295 seconds
[ Info: i: 1500, t: 3.197 hours, Δt: 6.282 seconds, umax = (1.0e-01, 7.3e-02, 4.5e-02) ms⁻¹, wall time: 58.780 seconds
[ Info: i: 1520, t: 3.231 hours, Δt: 5.782 seconds, umax = (1.1e-01, 8.1e-02, 4.6e-02) ms⁻¹, wall time: 59.262 seconds
[ Info: i: 1540, t: 3.263 hours, Δt: 5.891 seconds, umax = (1.1e-01, 8.2e-02, 5.0e-02) ms⁻¹, wall time: 59.756 seconds
[ Info: i: 1560, t: 3.296 hours, Δt: 5.833 seconds, umax = (1.1e-01, 7.7e-02, 4.5e-02) ms⁻¹, wall time: 1.004 minutes
[ Info: i: 1580, t: 3.327 hours, Δt: 5.718 seconds, umax = (1.1e-01, 8.0e-02, 4.4e-02) ms⁻¹, wall time: 1.012 minutes
[ Info: i: 1600, t: 3.358 hours, Δt: 5.522 seconds, umax = (1.0e-01, 8.4e-02, 4.3e-02) ms⁻¹, wall time: 1.035 minutes
[ Info: i: 1620, t: 3.389 hours, Δt: 5.919 seconds, umax = (1.1e-01, 7.7e-02, 4.2e-02) ms⁻¹, wall time: 1.043 minutes
[ Info: i: 1640, t: 3.422 hours, Δt: 6.170 seconds, umax = (1.1e-01, 7.8e-02, 4.6e-02) ms⁻¹, wall time: 1.053 minutes
[ Info: i: 1660, t: 3.455 hours, Δt: 5.575 seconds, umax = (1.1e-01, 8.2e-02, 4.9e-02) ms⁻¹, wall time: 1.060 minutes
[ Info: i: 1680, t: 3.486 hours, Δt: 5.651 seconds, umax = (1.1e-01, 7.5e-02, 5.0e-02) ms⁻¹, wall time: 1.068 minutes
[ Info: i: 1700, t: 3.518 hours, Δt: 6.233 seconds, umax = (1.1e-01, 8.2e-02, 4.8e-02) ms⁻¹, wall time: 1.076 minutes
[ Info: i: 1720, t: 3.552 hours, Δt: 6.101 seconds, umax = (1.1e-01, 8.1e-02, 4.8e-02) ms⁻¹, wall time: 1.084 minutes
[ Info: i: 1740, t: 3.585 hours, Δt: 5.645 seconds, umax = (1.1e-01, 8.9e-02, 4.9e-02) ms⁻¹, wall time: 1.095 minutes
[ Info: i: 1760, t: 3.616 hours, Δt: 5.997 seconds, umax = (1.1e-01, 7.6e-02, 4.7e-02) ms⁻¹, wall time: 1.100 minutes
[ Info: i: 1780, t: 3.649 hours, Δt: 5.574 seconds, umax = (1.1e-01, 8.3e-02, 4.7e-02) ms⁻¹, wall time: 1.108 minutes
[ Info: i: 1800, t: 3.680 hours, Δt: 5.714 seconds, umax = (1.1e-01, 8.5e-02, 4.8e-02) ms⁻¹, wall time: 1.116 minutes
[ Info: i: 1820, t: 3.711 hours, Δt: 5.313 seconds, umax = (1.1e-01, 8.5e-02, 4.7e-02) ms⁻¹, wall time: 1.124 minutes
[ Info: i: 1840, t: 3.741 hours, Δt: 5.526 seconds, umax = (1.1e-01, 8.5e-02, 5.8e-02) ms⁻¹, wall time: 1.132 minutes
[ Info: i: 1860, t: 3.771 hours, Δt: 5.641 seconds, umax = (1.1e-01, 8.1e-02, 5.0e-02) ms⁻¹, wall time: 1.141 minutes
[ Info: i: 1880, t: 3.802 hours, Δt: 5.788 seconds, umax = (1.1e-01, 8.6e-02, 4.5e-02) ms⁻¹, wall time: 1.149 minutes
[ Info: i: 1900, t: 3.833 hours, Δt: 6.055 seconds, umax = (1.1e-01, 8.0e-02, 4.9e-02) ms⁻¹, wall time: 1.157 minutes
[ Info: i: 1920, t: 3.867 hours, Δt: 5.886 seconds, umax = (1.1e-01, 8.5e-02, 4.8e-02) ms⁻¹, wall time: 1.165 minutes
[ Info: i: 1940, t: 3.900 hours, Δt: 5.894 seconds, umax = (1.1e-01, 8.0e-02, 5.0e-02) ms⁻¹, wall time: 1.174 minutes
[ Info: i: 1960, t: 3.931 hours, Δt: 5.832 seconds, umax = (1.1e-01, 7.6e-02, 5.4e-02) ms⁻¹, wall time: 1.182 minutes
[ Info: i: 1980, t: 3.963 hours, Δt: 5.676 seconds, umax = (1.1e-01, 8.0e-02, 5.0e-02) ms⁻¹, wall time: 1.189 minutes
[ Info: i: 2000, t: 3.995 hours, Δt: 5.818 seconds, umax = (1.1e-01, 7.9e-02, 4.7e-02) ms⁻¹, wall time: 1.198 minutes
[ Info: Simulation is stopping after running for 1.200 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-27500/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-27500/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-27500/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 v0.102.4 `~/Oceananigans.jl-27500`
  [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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