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.

julia

using Pkg
pkg"add Oceananigans, CairoMakie, CUDA"

julia

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),

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 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.

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 = (1.5e-03, 8.6e-04, 1.3e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (9.817 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (1.932 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 18.714 seconds, umax = (3.6e-02, 1.4e-02, 2.0e-02) ms⁻¹, wall time: 13.346 seconds
[ Info: i: 0040, t: 16.832 minutes, Δt: 12.776 seconds, umax = (5.4e-02, 2.1e-02, 2.2e-02) ms⁻¹, wall time: 14.113 seconds
[ Info: i: 0060, t: 20.825 minutes, Δt: 11.124 seconds, umax = (6.2e-02, 2.7e-02, 3.0e-02) ms⁻¹, wall time: 15.034 seconds
[ Info: i: 0080, t: 24.456 minutes, Δt: 11.082 seconds, umax = (6.3e-02, 3.0e-02, 3.3e-02) ms⁻¹, wall time: 15.680 seconds
[ Info: i: 0100, t: 28.033 minutes, Δt: 10.880 seconds, umax = (5.9e-02, 3.4e-02, 3.1e-02) ms⁻¹, wall time: 16.489 seconds
[ Info: i: 0120, t: 31.635 minutes, Δt: 11.456 seconds, umax = (6.3e-02, 3.4e-02, 3.2e-02) ms⁻¹, wall time: 17.296 seconds
[ Info: i: 0140, t: 35.179 minutes, Δt: 10.049 seconds, umax = (7.1e-02, 3.3e-02, 3.0e-02) ms⁻¹, wall time: 18.256 seconds
[ Info: i: 0160, t: 38.514 minutes, Δt: 9.762 seconds, umax = (6.9e-02, 3.6e-02, 3.3e-02) ms⁻¹, wall time: 18.796 seconds
[ Info: i: 0180, t: 41.626 minutes, Δt: 9.968 seconds, umax = (6.9e-02, 3.9e-02, 3.6e-02) ms⁻¹, wall time: 19.546 seconds
[ Info: i: 0200, t: 44.876 minutes, Δt: 9.653 seconds, umax = (7.0e-02, 3.7e-02, 3.6e-02) ms⁻¹, wall time: 20.288 seconds
[ Info: i: 0220, t: 48.045 minutes, Δt: 9.282 seconds, umax = (7.4e-02, 3.7e-02, 3.4e-02) ms⁻¹, wall time: 21.084 seconds
[ Info: i: 0240, t: 51.079 minutes, Δt: 8.580 seconds, umax = (7.4e-02, 4.0e-02, 4.0e-02) ms⁻¹, wall time: 21.910 seconds
[ Info: i: 0260, t: 53.983 minutes, Δt: 8.059 seconds, umax = (8.1e-02, 4.2e-02, 3.8e-02) ms⁻¹, wall time: 22.670 seconds
[ Info: i: 0280, t: 56.684 minutes, Δt: 7.895 seconds, umax = (8.0e-02, 4.5e-02, 3.6e-02) ms⁻¹, wall time: 23.467 seconds
[ Info: i: 0300, t: 59.429 minutes, Δt: 8.428 seconds, umax = (8.0e-02, 4.6e-02, 3.7e-02) ms⁻¹, wall time: 24.216 seconds
[ Info: i: 0320, t: 1.034 hours, Δt: 7.625 seconds, umax = (8.2e-02, 5.0e-02, 3.7e-02) ms⁻¹, wall time: 25.016 seconds
[ Info: i: 0340, t: 1.076 hours, Δt: 7.153 seconds, umax = (8.8e-02, 4.8e-02, 4.3e-02) ms⁻¹, wall time: 25.677 seconds
[ Info: i: 0360, t: 1.117 hours, Δt: 8.655 seconds, umax = (9.0e-02, 4.7e-02, 4.2e-02) ms⁻¹, wall time: 26.267 seconds
[ Info: i: 0380, t: 1.163 hours, Δt: 8.195 seconds, umax = (9.1e-02, 4.8e-02, 3.8e-02) ms⁻¹, wall time: 26.810 seconds
[ Info: i: 0400, t: 1.208 hours, Δt: 8.034 seconds, umax = (8.7e-02, 4.6e-02, 4.5e-02) ms⁻¹, wall time: 27.529 seconds
[ Info: i: 0420, t: 1.250 hours, Δt: 8.046 seconds, umax = (9.0e-02, 4.8e-02, 4.1e-02) ms⁻¹, wall time: 28.311 seconds
[ Info: i: 0440, t: 1.294 hours, Δt: 7.606 seconds, umax = (9.3e-02, 5.2e-02, 3.9e-02) ms⁻¹, wall time: 29.766 seconds
[ Info: i: 0460, t: 1.335 hours, Δt: 7.560 seconds, umax = (9.0e-02, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 30.862 seconds
[ Info: i: 0480, t: 1.377 hours, Δt: 7.576 seconds, umax = (9.0e-02, 4.9e-02, 4.2e-02) ms⁻¹, wall time: 31.399 seconds
[ Info: i: 0500, t: 1.417 hours, Δt: 7.467 seconds, umax = (9.2e-02, 5.0e-02, 4.4e-02) ms⁻¹, wall time: 32.176 seconds
[ Info: i: 0520, t: 1.458 hours, Δt: 7.482 seconds, umax = (9.1e-02, 5.7e-02, 4.2e-02) ms⁻¹, wall time: 33.063 seconds
[ Info: i: 0540, t: 1.500 hours, Δt: 7.430 seconds, umax = (9.4e-02, 5.2e-02, 4.2e-02) ms⁻¹, wall time: 33.843 seconds
[ Info: i: 0560, t: 1.539 hours, Δt: 7.374 seconds, umax = (9.8e-02, 5.3e-02, 4.2e-02) ms⁻¹, wall time: 34.694 seconds
[ Info: i: 0580, t: 1.581 hours, Δt: 7.307 seconds, umax = (9.4e-02, 5.8e-02, 4.4e-02) ms⁻¹, wall time: 35.486 seconds
[ Info: i: 0600, t: 1.620 hours, Δt: 7.292 seconds, umax = (1.0e-01, 5.7e-02, 4.2e-02) ms⁻¹, wall time: 36.339 seconds
[ Info: i: 0620, t: 1.661 hours, Δt: 7.060 seconds, umax = (9.7e-02, 5.3e-02, 4.3e-02) ms⁻¹, wall time: 37.128 seconds
[ Info: i: 0640, t: 1.701 hours, Δt: 6.730 seconds, umax = (9.7e-02, 5.8e-02, 4.5e-02) ms⁻¹, wall time: 37.975 seconds
[ Info: i: 0660, t: 1.739 hours, Δt: 6.842 seconds, umax = (9.6e-02, 5.2e-02, 4.1e-02) ms⁻¹, wall time: 38.776 seconds
[ Info: i: 0680, t: 1.778 hours, Δt: 7.126 seconds, umax = (1.0e-01, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 39.648 seconds
[ Info: i: 0700, t: 1.818 hours, Δt: 6.968 seconds, umax = (9.7e-02, 5.8e-02, 4.0e-02) ms⁻¹, wall time: 40.454 seconds
[ Info: i: 0720, t: 1.853 hours, Δt: 6.815 seconds, umax = (9.8e-02, 6.0e-02, 4.2e-02) ms⁻¹, wall time: 41.325 seconds
[ Info: i: 0740, t: 1.891 hours, Δt: 6.919 seconds, umax = (1.0e-01, 5.4e-02, 4.3e-02) ms⁻¹, wall time: 41.998 seconds
[ Info: i: 0760, t: 1.928 hours, Δt: 6.660 seconds, umax = (1.0e-01, 6.5e-02, 4.8e-02) ms⁻¹, wall time: 42.700 seconds
[ Info: i: 0780, t: 1.965 hours, Δt: 6.866 seconds, umax = (1.0e-01, 6.6e-02, 4.8e-02) ms⁻¹, wall time: 43.243 seconds
[ Info: i: 0800, t: 2.002 hours, Δt: 6.833 seconds, umax = (1.1e-01, 6.0e-02, 4.5e-02) ms⁻¹, wall time: 44.475 seconds
[ Info: i: 0820, t: 2.040 hours, Δt: 6.301 seconds, umax = (1.0e-01, 6.3e-02, 4.1e-02) ms⁻¹, wall time: 45.162 seconds
[ Info: i: 0840, t: 2.075 hours, Δt: 6.316 seconds, umax = (1.0e-01, 6.7e-02, 4.5e-02) ms⁻¹, wall time: 46.124 seconds
[ Info: i: 0860, t: 2.109 hours, Δt: 5.880 seconds, umax = (1.0e-01, 6.8e-02, 4.4e-02) ms⁻¹, wall time: 47.138 seconds
[ Info: i: 0880, t: 2.142 hours, Δt: 5.536 seconds, umax = (1.2e-01, 6.4e-02, 4.6e-02) ms⁻¹, wall time: 48.014 seconds
[ Info: i: 0900, t: 2.173 hours, Δt: 6.236 seconds, umax = (1.1e-01, 6.5e-02, 4.7e-02) ms⁻¹, wall time: 48.982 seconds
[ Info: i: 0920, t: 2.209 hours, Δt: 6.666 seconds, umax = (1.0e-01, 6.4e-02, 4.5e-02) ms⁻¹, wall time: 49.671 seconds
[ Info: i: 0940, t: 2.246 hours, Δt: 6.494 seconds, umax = (1.0e-01, 6.3e-02, 4.4e-02) ms⁻¹, wall time: 50.506 seconds
[ Info: i: 0960, t: 2.279 hours, Δt: 6.612 seconds, umax = (1.1e-01, 6.4e-02, 4.5e-02) ms⁻¹, wall time: 51.364 seconds
[ Info: i: 0980, t: 2.317 hours, Δt: 6.345 seconds, umax = (1.1e-01, 6.1e-02, 4.3e-02) ms⁻¹, wall time: 52.233 seconds
[ Info: i: 1000, t: 2.352 hours, Δt: 6.513 seconds, umax = (1.1e-01, 6.5e-02, 4.6e-02) ms⁻¹, wall time: 53.120 seconds
[ Info: i: 1020, t: 2.388 hours, Δt: 6.334 seconds, umax = (1.0e-01, 7.4e-02, 4.6e-02) ms⁻¹, wall time: 53.967 seconds
[ Info: i: 1040, t: 2.422 hours, Δt: 6.230 seconds, umax = (1.0e-01, 6.5e-02, 4.9e-02) ms⁻¹, wall time: 55.016 seconds
[ Info: i: 1060, t: 2.456 hours, Δt: 6.272 seconds, umax = (1.0e-01, 6.7e-02, 4.7e-02) ms⁻¹, wall time: 55.679 seconds
[ Info: i: 1080, t: 2.490 hours, Δt: 6.254 seconds, umax = (1.1e-01, 6.6e-02, 4.2e-02) ms⁻¹, wall time: 56.536 seconds
[ Info: i: 1100, t: 2.524 hours, Δt: 6.085 seconds, umax = (1.1e-01, 7.0e-02, 4.6e-02) ms⁻¹, wall time: 57.418 seconds
[ Info: i: 1120, t: 2.557 hours, Δt: 6.117 seconds, umax = (1.1e-01, 6.6e-02, 4.5e-02) ms⁻¹, wall time: 58.081 seconds
[ Info: i: 1140, t: 2.590 hours, Δt: 6.267 seconds, umax = (1.1e-01, 6.7e-02, 5.2e-02) ms⁻¹, wall time: 58.823 seconds
[ Info: i: 1160, t: 2.626 hours, Δt: 6.166 seconds, umax = (1.0e-01, 7.3e-02, 4.8e-02) ms⁻¹, wall time: 59.358 seconds
[ Info: i: 1180, t: 2.659 hours, Δt: 6.314 seconds, umax = (1.1e-01, 6.7e-02, 4.7e-02) ms⁻¹, wall time: 59.975 seconds
[ Info: i: 1200, t: 2.694 hours, Δt: 6.419 seconds, umax = (1.1e-01, 6.7e-02, 4.6e-02) ms⁻¹, wall time: 1.011 minutes
[ Info: i: 1220, t: 2.729 hours, Δt: 5.736 seconds, umax = (1.1e-01, 6.8e-02, 4.8e-02) ms⁻¹, wall time: 1.021 minutes
[ Info: i: 1240, t: 2.760 hours, Δt: 6.122 seconds, umax = (1.1e-01, 6.9e-02, 4.7e-02) ms⁻¹, wall time: 1.065 minutes
[ Info: i: 1260, t: 2.794 hours, Δt: 5.843 seconds, umax = (1.1e-01, 7.1e-02, 4.5e-02) ms⁻¹, wall time: 1.075 minutes
[ Info: i: 1280, t: 2.826 hours, Δt: 6.290 seconds, umax = (1.0e-01, 6.7e-02, 4.7e-02) ms⁻¹, wall time: 1.085 minutes
[ Info: i: 1300, t: 2.860 hours, Δt: 6.030 seconds, umax = (1.1e-01, 6.8e-02, 4.5e-02) ms⁻¹, wall time: 1.097 minutes
[ Info: i: 1320, t: 2.893 hours, Δt: 5.616 seconds, umax = (1.1e-01, 8.2e-02, 5.0e-02) ms⁻¹, wall time: 1.107 minutes
[ Info: i: 1340, t: 2.923 hours, Δt: 5.934 seconds, umax = (1.0e-01, 6.7e-02, 4.9e-02) ms⁻¹, wall time: 1.121 minutes
[ Info: i: 1360, t: 2.957 hours, Δt: 6.152 seconds, umax = (1.1e-01, 7.0e-02, 4.8e-02) ms⁻¹, wall time: 1.129 minutes
[ Info: i: 1380, t: 2.990 hours, Δt: 6.153 seconds, umax = (1.1e-01, 7.2e-02, 5.2e-02) ms⁻¹, wall time: 1.140 minutes
[ Info: i: 1400, t: 3.024 hours, Δt: 6.110 seconds, umax = (1.1e-01, 7.3e-02, 5.0e-02) ms⁻¹, wall time: 1.152 minutes
[ Info: i: 1420, t: 3.059 hours, Δt: 6.110 seconds, umax = (1.0e-01, 6.8e-02, 4.9e-02) ms⁻¹, wall time: 1.163 minutes
[ Info: i: 1440, t: 3.092 hours, Δt: 6.116 seconds, umax = (1.1e-01, 6.9e-02, 4.4e-02) ms⁻¹, wall time: 1.177 minutes
[ Info: i: 1460, t: 3.126 hours, Δt: 6.061 seconds, umax = (1.0e-01, 7.2e-02, 4.4e-02) ms⁻¹, wall time: 1.186 minutes
[ Info: i: 1480, t: 3.160 hours, Δt: 6.287 seconds, umax = (1.0e-01, 7.6e-02, 4.4e-02) ms⁻¹, wall time: 1.197 minutes
[ Info: i: 1500, t: 3.194 hours, Δt: 6.266 seconds, umax = (1.1e-01, 7.2e-02, 4.5e-02) ms⁻¹, wall time: 1.210 minutes
[ Info: i: 1520, t: 3.229 hours, Δt: 6.230 seconds, umax = (1.1e-01, 6.9e-02, 4.9e-02) ms⁻¹, wall time: 1.220 minutes
[ Info: i: 1540, t: 3.262 hours, Δt: 6.218 seconds, umax = (1.2e-01, 7.1e-02, 4.9e-02) ms⁻¹, wall time: 1.232 minutes
[ Info: i: 1560, t: 3.296 hours, Δt: 6.022 seconds, umax = (1.1e-01, 7.0e-02, 4.4e-02) ms⁻¹, wall time: 1.242 minutes
[ Info: i: 1580, t: 3.330 hours, Δt: 5.953 seconds, umax = (1.3e-01, 7.8e-02, 4.5e-02) ms⁻¹, wall time: 1.253 minutes
[ Info: i: 1600, t: 3.362 hours, Δt: 5.987 seconds, umax = (1.1e-01, 8.7e-02, 4.6e-02) ms⁻¹, wall time: 1.264 minutes
[ Info: i: 1620, t: 3.394 hours, Δt: 5.493 seconds, umax = (1.1e-01, 7.9e-02, 5.0e-02) ms⁻¹, wall time: 1.275 minutes
[ Info: i: 1640, t: 3.425 hours, Δt: 5.597 seconds, umax = (1.1e-01, 6.8e-02, 5.0e-02) ms⁻¹, wall time: 1.288 minutes
[ Info: i: 1660, t: 3.455 hours, Δt: 5.860 seconds, umax = (1.2e-01, 7.6e-02, 4.8e-02) ms⁻¹, wall time: 1.297 minutes
[ Info: i: 1680, t: 3.487 hours, Δt: 5.648 seconds, umax = (1.2e-01, 7.6e-02, 4.9e-02) ms⁻¹, wall time: 1.307 minutes
[ Info: i: 1700, t: 3.518 hours, Δt: 6.275 seconds, umax = (1.1e-01, 7.0e-02, 4.9e-02) ms⁻¹, wall time: 1.319 minutes
[ Info: i: 1720, t: 3.552 hours, Δt: 6.074 seconds, umax = (1.1e-01, 7.7e-02, 5.3e-02) ms⁻¹, wall time: 1.329 minutes
[ Info: i: 1740, t: 3.585 hours, Δt: 6.140 seconds, umax = (1.1e-01, 7.5e-02, 4.6e-02) ms⁻¹, wall time: 1.343 minutes
[ Info: i: 1760, t: 3.619 hours, Δt: 6.175 seconds, umax = (1.1e-01, 7.5e-02, 4.5e-02) ms⁻¹, wall time: 1.351 minutes
[ Info: i: 1780, t: 3.653 hours, Δt: 5.937 seconds, umax = (1.2e-01, 7.9e-02, 4.8e-02) ms⁻¹, wall time: 1.365 minutes
[ Info: i: 1800, t: 3.685 hours, Δt: 5.643 seconds, umax = (1.2e-01, 7.7e-02, 4.6e-02) ms⁻¹, wall time: 1.385 minutes
[ Info: i: 1820, t: 3.718 hours, Δt: 5.996 seconds, umax = (1.1e-01, 7.4e-02, 4.9e-02) ms⁻¹, wall time: 1.403 minutes
[ Info: i: 1840, t: 3.750 hours, Δt: 5.299 seconds, umax = (1.1e-01, 7.8e-02, 5.3e-02) ms⁻¹, wall time: 1.419 minutes
[ Info: i: 1860, t: 3.779 hours, Δt: 5.546 seconds, umax = (1.1e-01, 7.9e-02, 4.8e-02) ms⁻¹, wall time: 1.513 minutes
[ Info: i: 1880, t: 3.810 hours, Δt: 5.673 seconds, umax = (1.1e-01, 7.6e-02, 5.1e-02) ms⁻¹, wall time: 1.532 minutes
[ Info: i: 1900, t: 3.841 hours, Δt: 6.017 seconds, umax = (1.1e-01, 7.8e-02, 4.9e-02) ms⁻¹, wall time: 1.554 minutes
[ Info: i: 1920, t: 3.874 hours, Δt: 5.865 seconds, umax = (1.1e-01, 7.7e-02, 4.9e-02) ms⁻¹, wall time: 1.571 minutes
[ Info: i: 1940, t: 3.906 hours, Δt: 6.018 seconds, umax = (1.2e-01, 7.6e-02, 5.1e-02) ms⁻¹, wall time: 1.587 minutes
[ Info: i: 1960, t: 3.938 hours, Δt: 5.974 seconds, umax = (1.1e-01, 9.4e-02, 4.8e-02) ms⁻¹, wall time: 1.690 minutes
[ Info: i: 1980, t: 3.971 hours, Δt: 5.413 seconds, umax = (1.0e-01, 8.0e-02, 4.7e-02) ms⁻¹, wall time: 1.708 minutes
[ Info: Simulation is stopping after running for 1.722 minutes.
[ Info: Simulation time 4 hours equals or exceeds stop time 4 hours.
[ Info: i: 2000, t: 4 hours, Δt: 5.865 seconds, umax = (1.2e-01, 7.8e-02, 4.5e-02) ms⁻¹, wall time: 1.722 minutes

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-oceananigans
  JULIA_PROJECT = /var/lib/buildkite-agent/Oceananigans.jl-29586/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-29586/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-29586/docs/Project.toml`
  [79e6a3ab] Adapt v4.4.0
  [052768ef] CUDA v5.9.6
  [13f3f980] CairoMakie v0.15.8
  [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 v0.14.11
  [9e8cae18] Oceananigans v0.105.0 `..`
  [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

This page was generated using Literate.jl.

`NonhydrostaticModel`

`HydrostaticFreeSurfaceModel`

`ShallowWaterModel`

Langmuir turbulence example

Install dependencies

Model set-up

Domain and numerical grid specification

The Stokes Drift profile

Boundary conditions

Coriolis parameter

Model instantiation

Initial conditions

Setting up the simulation

Nice progress messaging

Output

A field writer

An "averages" writer

Running the simulation

Making a neat movie

Julia version and environment information

Langmuir turbulence example ​

Install dependencies ​

Model set-up ​

Domain and numerical grid specification ​

The Stokes Drift profile ​

Boundary conditions ​

Coriolis parameter ​

Model instantiation ​

Initial conditions ​

Setting up the simulation ​

Nice progress messaging ​

Output ​

A field writer ​

An "averages" writer ​

Running the simulation ​

Making a neat movie ​

Julia version and environment information ​

Langmuir turbulence example

Install dependencies

Model set-up

Domain and numerical grid specification

The Stokes Drift profile

Boundary conditions

Coriolis parameter

Model instantiation

Initial conditions

Setting up the simulation

Nice progress messaging

Output

A field writer

An "averages" writer

Running the simulation

Making a neat movie

Julia version and environment information