Oceananigans.jl

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Langmuir turbulence example ​

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

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.

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

Random.seed!(1337) # for reproducible results
Random.TaskLocalRNG()

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),

julia
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 by 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.

julia
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

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

and its z-derivative is

julia
∂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 -velocity beneath surface waves. This differs from models that use the Eulerian-mean velocity field as a prognostic variable, but has the advantage that accounts for the total advection of tracers and momentum, and that     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  , Wagner et al. (2021) impose

julia
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.

julia
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

julia
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 , we use UniformStokesDrift, which expects Stokes drift functions of only.

julia
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,

julia
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,

julia
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 and .

julia
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 ​

julia
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,

julia
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 and the current wall clock time.

julia
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.

julia
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.

julia
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,

julia
run!(simulation)
[ Info: Initializing simulation...
[ Info: i: 0000, t: 0 seconds, Δt: 49.500 seconds, umax = (1.8e-03, 9.5e-04, 1.5e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (10.556 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (3.022 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 19.470 seconds, umax = (3.6e-02, 1.2e-02, 2.1e-02) ms⁻¹, wall time: 15.010 seconds
[ Info: i: 0040, t: 17.083 minutes, Δt: 12.980 seconds, umax = (5.3e-02, 2.1e-02, 2.5e-02) ms⁻¹, wall time: 15.443 seconds
[ Info: i: 0060, t: 21.181 minutes, Δt: 10.514 seconds, umax = (6.4e-02, 2.9e-02, 3.2e-02) ms⁻¹, wall time: 15.968 seconds
[ Info: i: 0080, t: 24.649 minutes, Δt: 10.835 seconds, umax = (6.4e-02, 3.1e-02, 3.4e-02) ms⁻¹, wall time: 16.444 seconds
[ Info: i: 0100, t: 28.387 minutes, Δt: 11.357 seconds, umax = (6.1e-02, 3.0e-02, 3.0e-02) ms⁻¹, wall time: 16.981 seconds
[ Info: i: 0120, t: 32.015 minutes, Δt: 11.254 seconds, umax = (6.1e-02, 2.9e-02, 2.8e-02) ms⁻¹, wall time: 17.548 seconds
[ Info: i: 0140, t: 35.548 minutes, Δt: 10.870 seconds, umax = (6.6e-02, 3.4e-02, 3.0e-02) ms⁻¹, wall time: 18.203 seconds
[ Info: i: 0160, t: 39.055 minutes, Δt: 10.051 seconds, umax = (6.9e-02, 3.7e-02, 3.0e-02) ms⁻¹, wall time: 18.680 seconds
[ Info: i: 0180, t: 42.196 minutes, Δt: 9.376 seconds, umax = (7.2e-02, 3.6e-02, 3.4e-02) ms⁻¹, wall time: 19.412 seconds
[ Info: i: 0200, t: 45.154 minutes, Δt: 9.332 seconds, umax = (7.0e-02, 3.7e-02, 3.3e-02) ms⁻¹, wall time: 20.354 seconds
[ Info: i: 0220, t: 48.263 minutes, Δt: 8.574 seconds, umax = (7.5e-02, 4.3e-02, 3.5e-02) ms⁻¹, wall time: 20.840 seconds
[ Info: i: 0240, t: 51.238 minutes, Δt: 8.667 seconds, umax = (7.5e-02, 4.1e-02, 3.8e-02) ms⁻¹, wall time: 21.549 seconds
[ Info: i: 0260, t: 54.183 minutes, Δt: 8.782 seconds, umax = (7.8e-02, 3.9e-02, 3.6e-02) ms⁻¹, wall time: 22.044 seconds
[ Info: i: 0280, t: 57.056 minutes, Δt: 8.470 seconds, umax = (7.8e-02, 4.6e-02, 3.7e-02) ms⁻¹, wall time: 22.606 seconds
[ Info: i: 0300, t: 59.841 minutes, Δt: 8.423 seconds, umax = (8.2e-02, 4.1e-02, 4.1e-02) ms⁻¹, wall time: 23.110 seconds
[ Info: i: 0320, t: 1.042 hours, Δt: 7.873 seconds, umax = (8.9e-02, 4.4e-02, 3.6e-02) ms⁻¹, wall time: 23.690 seconds
[ Info: i: 0340, t: 1.086 hours, Δt: 8.292 seconds, umax = (8.2e-02, 4.7e-02, 3.8e-02) ms⁻¹, wall time: 24.426 seconds
[ Info: i: 0360, t: 1.132 hours, Δt: 8.269 seconds, umax = (8.2e-02, 4.3e-02, 4.0e-02) ms⁻¹, wall time: 24.784 seconds
[ Info: i: 0380, t: 1.179 hours, Δt: 8.177 seconds, umax = (8.7e-02, 4.6e-02, 3.6e-02) ms⁻¹, wall time: 25.405 seconds
[ Info: i: 0400, t: 1.224 hours, Δt: 7.848 seconds, umax = (8.3e-02, 4.8e-02, 3.6e-02) ms⁻¹, wall time: 25.861 seconds
[ Info: i: 0420, t: 1.267 hours, Δt: 7.613 seconds, umax = (8.8e-02, 5.3e-02, 4.3e-02) ms⁻¹, wall time: 26.545 seconds
[ Info: i: 0440, t: 1.310 hours, Δt: 7.647 seconds, umax = (9.1e-02, 5.1e-02, 3.9e-02) ms⁻¹, wall time: 27.250 seconds
[ Info: i: 0460, t: 1.350 hours, Δt: 7.444 seconds, umax = (9.2e-02, 5.5e-02, 3.8e-02) ms⁻¹, wall time: 28.013 seconds
[ Info: i: 0480, t: 1.392 hours, Δt: 7.478 seconds, umax = (9.3e-02, 5.2e-02, 4.1e-02) ms⁻¹, wall time: 28.720 seconds
[ Info: i: 0500, t: 1.433 hours, Δt: 7.334 seconds, umax = (9.3e-02, 5.3e-02, 4.0e-02) ms⁻¹, wall time: 29.368 seconds
[ Info: i: 0520, t: 1.474 hours, Δt: 7.630 seconds, umax = (9.0e-02, 5.6e-02, 3.8e-02) ms⁻¹, wall time: 29.880 seconds
[ Info: i: 0540, t: 1.515 hours, Δt: 7.618 seconds, umax = (9.1e-02, 5.6e-02, 4.6e-02) ms⁻¹, wall time: 30.468 seconds
[ Info: i: 0560, t: 1.555 hours, Δt: 7.350 seconds, umax = (9.2e-02, 5.5e-02, 4.1e-02) ms⁻¹, wall time: 30.980 seconds
[ Info: i: 0580, t: 1.596 hours, Δt: 7.264 seconds, umax = (9.8e-02, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 31.631 seconds
[ Info: i: 0600, t: 1.636 hours, Δt: 7.464 seconds, umax = (9.6e-02, 5.5e-02, 4.1e-02) ms⁻¹, wall time: 32.101 seconds
[ Info: i: 0620, t: 1.675 hours, Δt: 6.749 seconds, umax = (1.0e-01, 5.6e-02, 4.2e-02) ms⁻¹, wall time: 32.782 seconds
[ Info: i: 0640, t: 1.712 hours, Δt: 7.013 seconds, umax = (9.8e-02, 5.3e-02, 4.2e-02) ms⁻¹, wall time: 33.367 seconds
[ Info: i: 0660, t: 1.750 hours, Δt: 6.605 seconds, umax = (9.8e-02, 5.4e-02, 4.4e-02) ms⁻¹, wall time: 34.113 seconds
[ Info: i: 0680, t: 1.786 hours, Δt: 6.770 seconds, umax = (1.0e-01, 6.0e-02, 4.3e-02) ms⁻¹, wall time: 34.918 seconds
[ Info: i: 0700, t: 1.825 hours, Δt: 6.998 seconds, umax = (9.9e-02, 5.6e-02, 4.1e-02) ms⁻¹, wall time: 35.664 seconds
[ Info: i: 0720, t: 1.863 hours, Δt: 6.928 seconds, umax = (1.0e-01, 5.8e-02, 4.2e-02) ms⁻¹, wall time: 36.298 seconds
[ Info: i: 0740, t: 1.900 hours, Δt: 6.836 seconds, umax = (1.0e-01, 5.9e-02, 4.6e-02) ms⁻¹, wall time: 36.833 seconds
[ Info: i: 0760, t: 1.936 hours, Δt: 6.637 seconds, umax = (9.7e-02, 5.9e-02, 4.3e-02) ms⁻¹, wall time: 37.483 seconds
[ Info: i: 0780, t: 1.974 hours, Δt: 6.493 seconds, umax = (1.1e-01, 6.3e-02, 4.6e-02) ms⁻¹, wall time: 38.013 seconds
[ Info: i: 0800, t: 2.009 hours, Δt: 6.961 seconds, umax = (1.0e-01, 6.0e-02, 4.5e-02) ms⁻¹, wall time: 38.714 seconds
[ Info: i: 0820, t: 2.047 hours, Δt: 6.288 seconds, umax = (1.1e-01, 6.9e-02, 4.8e-02) ms⁻¹, wall time: 39.191 seconds
[ Info: i: 0840, t: 2.082 hours, Δt: 6.690 seconds, umax = (1.1e-01, 6.8e-02, 4.6e-02) ms⁻¹, wall time: 39.924 seconds
[ Info: i: 0860, t: 2.118 hours, Δt: 6.533 seconds, umax = (1.0e-01, 6.2e-02, 4.2e-02) ms⁻¹, wall time: 40.756 seconds
[ Info: i: 0880, t: 2.156 hours, Δt: 6.926 seconds, umax = (1.0e-01, 6.1e-02, 4.6e-02) ms⁻¹, wall time: 41.507 seconds
[ Info: i: 0900, t: 2.193 hours, Δt: 6.537 seconds, umax = (1.1e-01, 6.3e-02, 4.6e-02) ms⁻¹, wall time: 42.301 seconds
[ Info: i: 0920, t: 2.229 hours, Δt: 6.552 seconds, umax = (1.0e-01, 6.6e-02, 4.7e-02) ms⁻¹, wall time: 43.066 seconds
[ Info: i: 0940, t: 2.264 hours, Δt: 6.617 seconds, umax = (1.0e-01, 6.4e-02, 4.1e-02) ms⁻¹, wall time: 43.865 seconds
[ Info: i: 0960, t: 2.299 hours, Δt: 6.300 seconds, umax = (1.1e-01, 7.3e-02, 4.3e-02) ms⁻¹, wall time: 44.598 seconds
[ Info: i: 0980, t: 2.333 hours, Δt: 5.928 seconds, umax = (1.1e-01, 7.0e-02, 4.4e-02) ms⁻¹, wall time: 45.374 seconds
[ Info: i: 1000, t: 2.367 hours, Δt: 6.263 seconds, umax = (1.0e-01, 6.6e-02, 4.0e-02) ms⁻¹, wall time: 46.206 seconds
[ Info: i: 1020, t: 2.402 hours, Δt: 6.542 seconds, umax = (1.1e-01, 6.4e-02, 4.0e-02) ms⁻¹, wall time: 46.921 seconds
[ Info: i: 1040, t: 2.437 hours, Δt: 6.306 seconds, umax = (1.1e-01, 6.9e-02, 4.4e-02) ms⁻¹, wall time: 47.567 seconds
[ Info: i: 1060, t: 2.471 hours, Δt: 5.965 seconds, umax = (1.1e-01, 6.3e-02, 4.8e-02) ms⁻¹, wall time: 48.173 seconds
[ Info: i: 1080, t: 2.503 hours, Δt: 6.205 seconds, umax = (1.1e-01, 6.8e-02, 4.2e-02) ms⁻¹, wall time: 49.170 seconds
[ Info: i: 1100, t: 2.539 hours, Δt: 6.346 seconds, umax = (1.1e-01, 6.8e-02, 4.2e-02) ms⁻¹, wall time: 49.739 seconds
[ Info: i: 1120, t: 2.573 hours, Δt: 6.143 seconds, umax = (1.1e-01, 6.6e-02, 4.4e-02) ms⁻¹, wall time: 50.531 seconds
[ Info: i: 1140, t: 2.608 hours, Δt: 6.454 seconds, umax = (1.1e-01, 6.4e-02, 4.5e-02) ms⁻¹, wall time: 51.339 seconds
[ Info: i: 1160, t: 2.643 hours, Δt: 6.511 seconds, umax = (1.0e-01, 6.9e-02, 4.7e-02) ms⁻¹, wall time: 52.093 seconds
[ Info: i: 1180, t: 2.680 hours, Δt: 6.286 seconds, umax = (1.1e-01, 7.2e-02, 4.6e-02) ms⁻¹, wall time: 52.897 seconds
[ Info: i: 1200, t: 2.715 hours, Δt: 6.212 seconds, umax = (1.1e-01, 7.0e-02, 4.5e-02) ms⁻¹, wall time: 53.611 seconds
[ Info: i: 1220, t: 2.750 hours, Δt: 6.329 seconds, umax = (1.0e-01, 6.5e-02, 5.1e-02) ms⁻¹, wall time: 54.368 seconds
[ Info: i: 1240, t: 2.783 hours, Δt: 6.354 seconds, umax = (1.1e-01, 7.8e-02, 4.8e-02) ms⁻¹, wall time: 55.178 seconds
[ Info: i: 1260, t: 2.818 hours, Δt: 5.757 seconds, umax = (1.1e-01, 7.0e-02, 4.5e-02) ms⁻¹, wall time: 55.931 seconds
[ Info: i: 1280, t: 2.850 hours, Δt: 5.890 seconds, umax = (1.1e-01, 7.0e-02, 4.3e-02) ms⁻¹, wall time: 56.602 seconds
[ Info: i: 1300, t: 2.883 hours, Δt: 6.342 seconds, umax = (1.1e-01, 7.4e-02, 5.3e-02) ms⁻¹, wall time: 57.136 seconds
[ Info: i: 1320, t: 2.917 hours, Δt: 6.080 seconds, umax = (1.2e-01, 6.5e-02, 4.9e-02) ms⁻¹, wall time: 57.698 seconds
[ Info: i: 1340, t: 2.950 hours, Δt: 6.120 seconds, umax = (1.1e-01, 7.2e-02, 4.5e-02) ms⁻¹, wall time: 58.509 seconds
[ Info: i: 1360, t: 2.983 hours, Δt: 6.298 seconds, umax = (1.1e-01, 7.2e-02, 4.9e-02) ms⁻¹, wall time: 59.331 seconds
[ Info: i: 1380, t: 3.017 hours, Δt: 6.017 seconds, umax = (1.1e-01, 7.2e-02, 4.4e-02) ms⁻¹, wall time: 1.004 minutes
[ Info: i: 1400, t: 3.050 hours, Δt: 6.147 seconds, umax = (1.1e-01, 7.1e-02, 4.5e-02) ms⁻¹, wall time: 1.017 minutes
[ Info: i: 1420, t: 3.085 hours, Δt: 6.240 seconds, umax = (1.1e-01, 6.9e-02, 4.3e-02) ms⁻¹, wall time: 1.038 minutes
[ Info: i: 1440, t: 3.120 hours, Δt: 6.203 seconds, umax = (1.1e-01, 7.1e-02, 4.3e-02) ms⁻¹, wall time: 1.047 minutes
[ Info: i: 1460, t: 3.154 hours, Δt: 5.846 seconds, umax = (1.1e-01, 7.7e-02, 4.8e-02) ms⁻¹, wall time: 1.059 minutes
[ Info: i: 1480, t: 3.186 hours, Δt: 6.198 seconds, umax = (1.1e-01, 7.2e-02, 4.8e-02) ms⁻¹, wall time: 1.076 minutes
[ Info: i: 1500, t: 3.220 hours, Δt: 6.136 seconds, umax = (1.1e-01, 6.7e-02, 5.0e-02) ms⁻¹, wall time: 1.088 minutes
[ Info: i: 1520, t: 3.253 hours, Δt: 6.218 seconds, umax = (1.1e-01, 6.5e-02, 4.3e-02) ms⁻¹, wall time: 1.104 minutes
[ Info: i: 1540, t: 3.287 hours, Δt: 5.827 seconds, umax = (1.1e-01, 7.9e-02, 4.7e-02) ms⁻¹, wall time: 1.111 minutes
[ Info: i: 1560, t: 3.319 hours, Δt: 5.929 seconds, umax = (1.1e-01, 7.5e-02, 4.7e-02) ms⁻¹, wall time: 1.122 minutes
[ Info: i: 1580, t: 3.351 hours, Δt: 6.166 seconds, umax = (1.1e-01, 7.1e-02, 5.6e-02) ms⁻¹, wall time: 1.136 minutes
[ Info: i: 1600, t: 3.384 hours, Δt: 6.044 seconds, umax = (1.1e-01, 7.9e-02, 5.4e-02) ms⁻¹, wall time: 1.148 minutes
[ Info: i: 1620, t: 3.417 hours, Δt: 5.845 seconds, umax = (1.1e-01, 7.6e-02, 4.8e-02) ms⁻¹, wall time: 1.161 minutes
[ Info: i: 1640, t: 3.449 hours, Δt: 5.986 seconds, umax = (1.1e-01, 7.2e-02, 4.9e-02) ms⁻¹, wall time: 1.184 minutes
[ Info: i: 1660, t: 3.483 hours, Δt: 5.913 seconds, umax = (1.1e-01, 7.5e-02, 5.1e-02) ms⁻¹, wall time: 1.197 minutes
[ Info: i: 1680, t: 3.515 hours, Δt: 5.832 seconds, umax = (1.1e-01, 7.9e-02, 5.6e-02) ms⁻¹, wall time: 1.257 minutes
[ Info: i: 1700, t: 3.548 hours, Δt: 5.905 seconds, umax = (1.1e-01, 7.6e-02, 5.0e-02) ms⁻¹, wall time: 1.269 minutes
[ Info: i: 1720, t: 3.580 hours, Δt: 6.077 seconds, umax = (1.1e-01, 7.4e-02, 5.0e-02) ms⁻¹, wall time: 1.282 minutes
[ Info: i: 1740, t: 3.614 hours, Δt: 6.039 seconds, umax = (1.1e-01, 7.6e-02, 5.0e-02) ms⁻¹, wall time: 1.296 minutes
[ Info: i: 1760, t: 3.647 hours, Δt: 5.865 seconds, umax = (1.1e-01, 8.2e-02, 5.2e-02) ms⁻¹, wall time: 1.309 minutes
[ Info: i: 1780, t: 3.677 hours, Δt: 5.803 seconds, umax = (1.1e-01, 8.8e-02, 4.7e-02) ms⁻¹, wall time: 1.320 minutes
[ Info: i: 1800, t: 3.710 hours, Δt: 5.869 seconds, umax = (1.1e-01, 8.2e-02, 5.0e-02) ms⁻¹, wall time: 1.329 minutes
[ Info: i: 1820, t: 3.743 hours, Δt: 5.915 seconds, umax = (1.1e-01, 7.9e-02, 5.4e-02) ms⁻¹, wall time: 1.340 minutes
[ Info: i: 1840, t: 3.775 hours, Δt: 6.332 seconds, umax = (1.1e-01, 7.8e-02, 5.5e-02) ms⁻¹, wall time: 1.354 minutes
[ Info: i: 1860, t: 3.809 hours, Δt: 5.811 seconds, umax = (1.1e-01, 7.5e-02, 5.0e-02) ms⁻¹, wall time: 1.366 minutes
[ Info: i: 1880, t: 3.842 hours, Δt: 5.871 seconds, umax = (1.1e-01, 8.2e-02, 5.5e-02) ms⁻¹, wall time: 1.381 minutes
[ Info: i: 1900, t: 3.874 hours, Δt: 5.834 seconds, umax = (1.1e-01, 8.3e-02, 5.0e-02) ms⁻¹, wall time: 1.392 minutes
[ Info: i: 1920, t: 3.905 hours, Δt: 5.918 seconds, umax = (1.1e-01, 7.9e-02, 5.3e-02) ms⁻¹, wall time: 1.404 minutes
[ Info: i: 1940, t: 3.939 hours, Δt: 5.855 seconds, umax = (1.2e-01, 7.9e-02, 5.2e-02) ms⁻¹, wall time: 1.420 minutes
[ Info: i: 1960, t: 3.970 hours, Δt: 5.020 seconds, umax = (1.1e-01, 8.5e-02, 4.8e-02) ms⁻¹, wall time: 1.432 minutes
[ Info: i: 1980, t: 3.999 hours, Δt: 5.896 seconds, umax = (1.1e-01, 8.0e-02, 5.5e-02) ms⁻¹, wall time: 1.444 minutes
[ Info: Simulation is stopping after running for 1.448 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 and , and a horizontal slice of to look for Langmuir cells.

julia
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.

julia
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.

julia
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:

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
  JULIA_PROJECT = /var/lib/buildkite-agent/Oceananigans.jl-29902/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-29902/docs/.CondaPkg/.pixi/envs/default/bin/python
  JULIA_DEBUG = Literate

These were the top-level packages installed in the environment:

julia
import Pkg
Pkg.status()
Status `~/Oceananigans.jl-29902/docs/Project.toml`
  [79e6a3ab] Adapt v4.5.0
  [052768ef] CUDA v5.10.0
  [13f3f980] CairoMakie v0.15.9
  [e30172f5] Documenter v1.17.0
  [daee34ce] DocumenterCitations v1.4.1
  [4710194d] DocumenterVitepress v0.3.2
  [033835bb] JLD2 v0.6.3
  [63c18a36] KernelAbstractions v0.9.40
  [98b081ad] Literate v2.21.0
  [da04e1cc] MPI v0.20.23
  [85f8d34a] NCDatasets v0.14.12
  [9e8cae18] Oceananigans v0.105.4 `..`
  [f27b6e38] Polynomials v4.1.1
  [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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