Oceananigans.jl

Appearance

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

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 = (1.5e-03, 1.0e-03, 1.5e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (10.997 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (3.144 seconds).
[ Info: i: 0020, t: 11.238 minutes, Δt: 18.388 seconds, umax = (3.5e-02, 1.4e-02, 2.2e-02) ms⁻¹, wall time: 15.675 seconds
[ Info: i: 0040, t: 16.793 minutes, Δt: 13.209 seconds, umax = (5.3e-02, 2.0e-02, 2.2e-02) ms⁻¹, wall time: 16.428 seconds
[ Info: i: 0060, t: 20.783 minutes, Δt: 11.330 seconds, umax = (6.2e-02, 2.7e-02, 3.2e-02) ms⁻¹, wall time: 17.390 seconds
[ Info: i: 0080, t: 24.425 minutes, Δt: 11.029 seconds, umax = (6.3e-02, 3.0e-02, 3.4e-02) ms⁻¹, wall time: 18.036 seconds
[ Info: i: 0100, t: 28.027 minutes, Δt: 10.881 seconds, umax = (6.2e-02, 3.3e-02, 3.2e-02) ms⁻¹, wall time: 18.748 seconds
[ Info: i: 0120, t: 31.588 minutes, Δt: 11.563 seconds, umax = (6.2e-02, 2.8e-02, 3.0e-02) ms⁻¹, wall time: 19.353 seconds
[ Info: i: 0140, t: 35.389 minutes, Δt: 10.179 seconds, umax = (6.7e-02, 3.5e-02, 2.9e-02) ms⁻¹, wall time: 20.111 seconds
[ Info: i: 0160, t: 38.822 minutes, Δt: 9.922 seconds, umax = (6.7e-02, 3.9e-02, 3.0e-02) ms⁻¹, wall time: 20.495 seconds
[ Info: i: 0180, t: 41.967 minutes, Δt: 9.859 seconds, umax = (6.8e-02, 3.8e-02, 3.3e-02) ms⁻¹, wall time: 21.091 seconds
[ Info: i: 0200, t: 45 minutes, Δt: 9.758 seconds, umax = (7.6e-02, 3.8e-02, 3.3e-02) ms⁻¹, wall time: 21.795 seconds
[ Info: i: 0220, t: 48.232 minutes, Δt: 8.684 seconds, umax = (8.1e-02, 3.7e-02, 3.8e-02) ms⁻¹, wall time: 22.598 seconds
[ Info: i: 0240, t: 51.114 minutes, Δt: 9.304 seconds, umax = (7.3e-02, 3.7e-02, 3.5e-02) ms⁻¹, wall time: 23.412 seconds
[ Info: i: 0260, t: 54.204 minutes, Δt: 8.283 seconds, umax = (8.0e-02, 4.0e-02, 3.5e-02) ms⁻¹, wall time: 24.136 seconds
[ Info: i: 0280, t: 56.912 minutes, Δt: 8.484 seconds, umax = (8.4e-02, 4.2e-02, 3.6e-02) ms⁻¹, wall time: 24.782 seconds
[ Info: i: 0300, t: 59.771 minutes, Δt: 8.699 seconds, umax = (7.9e-02, 4.1e-02, 3.9e-02) ms⁻¹, wall time: 25.285 seconds
[ Info: i: 0320, t: 1.043 hours, Δt: 8.120 seconds, umax = (8.5e-02, 4.4e-02, 4.2e-02) ms⁻¹, wall time: 25.887 seconds
[ Info: i: 0340, t: 1.088 hours, Δt: 8.446 seconds, umax = (8.4e-02, 5.0e-02, 3.7e-02) ms⁻¹, wall time: 26.659 seconds
[ Info: i: 0360, t: 1.133 hours, Δt: 7.537 seconds, umax = (8.3e-02, 4.4e-02, 3.8e-02) ms⁻¹, wall time: 27.208 seconds
[ Info: i: 0380, t: 1.175 hours, Δt: 7.894 seconds, umax = (8.6e-02, 4.7e-02, 4.2e-02) ms⁻¹, wall time: 28.134 seconds
[ Info: i: 0400, t: 1.220 hours, Δt: 7.748 seconds, umax = (9.1e-02, 4.7e-02, 4.1e-02) ms⁻¹, wall time: 28.740 seconds
[ Info: i: 0420, t: 1.263 hours, Δt: 7.736 seconds, umax = (8.7e-02, 5.1e-02, 4.5e-02) ms⁻¹, wall time: 29.625 seconds
[ Info: i: 0440, t: 1.305 hours, Δt: 7.373 seconds, umax = (9.0e-02, 5.0e-02, 4.1e-02) ms⁻¹, wall time: 30.117 seconds
[ Info: i: 0460, t: 1.346 hours, Δt: 7.650 seconds, umax = (9.0e-02, 5.1e-02, 3.8e-02) ms⁻¹, wall time: 30.749 seconds
[ Info: i: 0480, t: 1.388 hours, Δt: 7.255 seconds, umax = (9.1e-02, 5.2e-02, 3.8e-02) ms⁻¹, wall time: 31.236 seconds
[ Info: i: 0500, t: 1.426 hours, Δt: 7.122 seconds, umax = (9.6e-02, 5.4e-02, 4.0e-02) ms⁻¹, wall time: 32.151 seconds
[ Info: i: 0520, t: 1.467 hours, Δt: 7.474 seconds, umax = (9.7e-02, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 32.826 seconds
[ Info: i: 0540, t: 1.506 hours, Δt: 7.563 seconds, umax = (9.2e-02, 5.5e-02, 3.9e-02) ms⁻¹, wall time: 33.783 seconds
[ Info: i: 0560, t: 1.548 hours, Δt: 7.294 seconds, umax = (9.4e-02, 5.1e-02, 3.9e-02) ms⁻¹, wall time: 34.392 seconds
[ Info: i: 0580, t: 1.587 hours, Δt: 6.957 seconds, umax = (9.4e-02, 5.8e-02, 3.9e-02) ms⁻¹, wall time: 35.430 seconds
[ Info: i: 0600, t: 1.627 hours, Δt: 6.703 seconds, umax = (9.8e-02, 5.9e-02, 4.1e-02) ms⁻¹, wall time: 35.995 seconds
[ Info: i: 0620, t: 1.664 hours, Δt: 7.045 seconds, umax = (9.7e-02, 5.9e-02, 4.1e-02) ms⁻¹, wall time: 36.757 seconds
[ Info: i: 0640, t: 1.702 hours, Δt: 6.766 seconds, umax = (9.6e-02, 5.4e-02, 3.9e-02) ms⁻¹, wall time: 37.615 seconds
[ Info: i: 0660, t: 1.739 hours, Δt: 6.831 seconds, umax = (1.0e-01, 5.7e-02, 4.4e-02) ms⁻¹, wall time: 38.367 seconds
[ Info: i: 0680, t: 1.776 hours, Δt: 6.133 seconds, umax = (1.1e-01, 5.7e-02, 4.3e-02) ms⁻¹, wall time: 39.342 seconds
[ Info: i: 0700, t: 1.812 hours, Δt: 6.988 seconds, umax = (9.7e-02, 5.7e-02, 4.4e-02) ms⁻¹, wall time: 40.029 seconds
[ Info: i: 0720, t: 1.850 hours, Δt: 6.850 seconds, umax = (9.9e-02, 6.4e-02, 4.3e-02) ms⁻¹, wall time: 40.624 seconds
[ Info: i: 0740, t: 1.888 hours, Δt: 6.590 seconds, umax = (9.8e-02, 6.0e-02, 5.3e-02) ms⁻¹, wall time: 41.155 seconds
[ Info: i: 0760, t: 1.924 hours, Δt: 6.928 seconds, umax = (1.0e-01, 6.0e-02, 5.0e-02) ms⁻¹, wall time: 41.992 seconds
[ Info: i: 0780, t: 1.963 hours, Δt: 6.699 seconds, umax = (1.0e-01, 6.7e-02, 4.8e-02) ms⁻¹, wall time: 42.646 seconds
[ Info: i: 0800, t: 1.999 hours, Δt: 6.624 seconds, umax = (9.3e-02, 6.4e-02, 4.1e-02) ms⁻¹, wall time: 43.414 seconds
[ Info: i: 0820, t: 2.035 hours, Δt: 6.416 seconds, umax = (9.8e-02, 5.8e-02, 4.7e-02) ms⁻¹, wall time: 44.227 seconds
[ Info: i: 0840, t: 2.072 hours, Δt: 6.369 seconds, umax = (9.6e-02, 6.6e-02, 4.4e-02) ms⁻¹, wall time: 45.001 seconds
[ Info: i: 0860, t: 2.107 hours, Δt: 6.566 seconds, umax = (1.0e-01, 6.2e-02, 4.3e-02) ms⁻¹, wall time: 45.830 seconds
[ Info: i: 0880, t: 2.143 hours, Δt: 6.062 seconds, umax = (1.1e-01, 6.8e-02, 4.3e-02) ms⁻¹, wall time: 46.589 seconds
[ Info: i: 0900, t: 2.177 hours, Δt: 6.725 seconds, umax = (1.1e-01, 6.1e-02, 4.4e-02) ms⁻¹, wall time: 47.435 seconds
[ Info: i: 0920, t: 2.213 hours, Δt: 6.417 seconds, umax = (1.1e-01, 6.5e-02, 4.5e-02) ms⁻¹, wall time: 48.143 seconds
[ Info: i: 0940, t: 2.248 hours, Δt: 6.281 seconds, umax = (1.1e-01, 6.2e-02, 4.3e-02) ms⁻¹, wall time: 48.928 seconds
[ Info: i: 0960, t: 2.283 hours, Δt: 6.188 seconds, umax = (1.1e-01, 7.1e-02, 4.5e-02) ms⁻¹, wall time: 49.710 seconds
[ Info: i: 0980, t: 2.318 hours, Δt: 6.097 seconds, umax = (1.1e-01, 7.9e-02, 4.8e-02) ms⁻¹, wall time: 50.467 seconds
[ Info: i: 1000, t: 2.351 hours, Δt: 6.342 seconds, umax = (1.1e-01, 7.0e-02, 4.9e-02) ms⁻¹, wall time: 51.100 seconds
[ Info: i: 1020, t: 2.387 hours, Δt: 6.442 seconds, umax = (1.1e-01, 6.5e-02, 4.3e-02) ms⁻¹, wall time: 51.630 seconds
[ Info: i: 1040, t: 2.422 hours, Δt: 6.287 seconds, umax = (1.0e-01, 6.9e-02, 4.3e-02) ms⁻¹, wall time: 52.361 seconds
[ Info: i: 1060, t: 2.457 hours, Δt: 6.177 seconds, umax = (1.1e-01, 6.9e-02, 4.5e-02) ms⁻¹, wall time: 52.992 seconds
[ Info: i: 1080, t: 2.492 hours, Δt: 6.326 seconds, umax = (1.1e-01, 7.7e-02, 4.2e-02) ms⁻¹, wall time: 53.759 seconds
[ Info: i: 1100, t: 2.526 hours, Δt: 6.500 seconds, umax = (1.1e-01, 7.0e-02, 4.4e-02) ms⁻¹, wall time: 54.582 seconds
[ Info: i: 1120, t: 2.562 hours, Δt: 6.371 seconds, umax = (1.1e-01, 7.2e-02, 4.1e-02) ms⁻¹, wall time: 55.332 seconds
[ Info: i: 1140, t: 2.595 hours, Δt: 6.386 seconds, umax = (1.0e-01, 6.9e-02, 4.3e-02) ms⁻¹, wall time: 56.189 seconds
[ Info: i: 1160, t: 2.631 hours, Δt: 6.328 seconds, umax = (1.1e-01, 6.9e-02, 4.4e-02) ms⁻¹, wall time: 56.940 seconds
[ Info: i: 1180, t: 2.665 hours, Δt: 5.983 seconds, umax = (1.1e-01, 7.4e-02, 4.6e-02) ms⁻¹, wall time: 57.722 seconds
[ Info: i: 1200, t: 2.697 hours, Δt: 6.044 seconds, umax = (1.1e-01, 7.9e-02, 4.8e-02) ms⁻¹, wall time: 58.559 seconds
[ Info: i: 1220, t: 2.731 hours, Δt: 5.879 seconds, umax = (1.0e-01, 7.1e-02, 4.9e-02) ms⁻¹, wall time: 59.294 seconds
[ Info: i: 1240, t: 2.764 hours, Δt: 6.316 seconds, umax = (1.2e-01, 7.0e-02, 4.8e-02) ms⁻¹, wall time: 1.002 minutes
[ Info: i: 1260, t: 2.799 hours, Δt: 6.354 seconds, umax = (1.1e-01, 7.4e-02, 5.0e-02) ms⁻¹, wall time: 1.014 minutes
[ Info: i: 1280, t: 2.833 hours, Δt: 6.168 seconds, umax = (1.0e-01, 7.1e-02, 4.9e-02) ms⁻¹, wall time: 1.026 minutes
[ Info: i: 1300, t: 2.866 hours, Δt: 5.839 seconds, umax = (1.1e-01, 7.2e-02, 4.7e-02) ms⁻¹, wall time: 1.037 minutes
[ Info: i: 1320, t: 2.899 hours, Δt: 5.649 seconds, umax = (1.1e-01, 7.6e-02, 4.8e-02) ms⁻¹, wall time: 1.047 minutes
[ Info: i: 1340, t: 2.930 hours, Δt: 6.042 seconds, umax = (1.1e-01, 7.2e-02, 5.3e-02) ms⁻¹, wall time: 1.060 minutes
[ Info: i: 1360, t: 2.964 hours, Δt: 6.307 seconds, umax = (1.1e-01, 7.1e-02, 5.1e-02) ms⁻¹, wall time: 1.072 minutes
[ Info: i: 1380, t: 2.999 hours, Δt: 6.533 seconds, umax = (1.0e-01, 7.2e-02, 5.0e-02) ms⁻¹, wall time: 1.085 minutes
[ Info: i: 1400, t: 3.034 hours, Δt: 6.019 seconds, umax = (1.1e-01, 7.1e-02, 4.7e-02) ms⁻¹, wall time: 1.097 minutes
[ Info: i: 1420, t: 3.068 hours, Δt: 6.175 seconds, umax = (1.0e-01, 7.1e-02, 5.0e-02) ms⁻¹, wall time: 1.110 minutes
[ Info: i: 1440, t: 3.101 hours, Δt: 6.003 seconds, umax = (1.1e-01, 7.7e-02, 5.3e-02) ms⁻¹, wall time: 1.123 minutes
[ Info: i: 1460, t: 3.134 hours, Δt: 5.540 seconds, umax = (1.1e-01, 7.5e-02, 4.8e-02) ms⁻¹, wall time: 1.135 minutes
[ Info: i: 1480, t: 3.165 hours, Δt: 5.929 seconds, umax = (1.1e-01, 7.3e-02, 5.0e-02) ms⁻¹, wall time: 1.147 minutes
[ Info: i: 1500, t: 3.196 hours, Δt: 5.949 seconds, umax = (1.1e-01, 7.3e-02, 4.9e-02) ms⁻¹, wall time: 1.161 minutes
[ Info: i: 1520, t: 3.230 hours, Δt: 6.089 seconds, umax = (1.2e-01, 7.4e-02, 5.1e-02) ms⁻¹, wall time: 1.173 minutes
[ Info: i: 1540, t: 3.261 hours, Δt: 5.737 seconds, umax = (1.1e-01, 7.1e-02, 5.0e-02) ms⁻¹, wall time: 1.276 minutes
[ Info: i: 1560, t: 3.293 hours, Δt: 5.777 seconds, umax = (1.1e-01, 7.3e-02, 4.7e-02) ms⁻¹, wall time: 1.288 minutes
[ Info: i: 1580, t: 3.325 hours, Δt: 5.473 seconds, umax = (1.1e-01, 7.3e-02, 4.5e-02) ms⁻¹, wall time: 1.296 minutes
[ Info: i: 1600, t: 3.355 hours, Δt: 5.564 seconds, umax = (1.1e-01, 8.1e-02, 4.6e-02) ms⁻¹, wall time: 1.309 minutes
[ Info: i: 1620, t: 3.387 hours, Δt: 5.971 seconds, umax = (1.1e-01, 7.3e-02, 5.1e-02) ms⁻¹, wall time: 1.317 minutes
[ Info: i: 1640, t: 3.420 hours, Δt: 5.709 seconds, umax = (1.2e-01, 7.2e-02, 4.8e-02) ms⁻¹, wall time: 1.332 minutes
[ Info: i: 1660, t: 3.452 hours, Δt: 5.738 seconds, umax = (1.1e-01, 7.1e-02, 4.6e-02) ms⁻¹, wall time: 1.342 minutes
[ Info: i: 1680, t: 3.484 hours, Δt: 6.159 seconds, umax = (1.2e-01, 7.6e-02, 4.9e-02) ms⁻¹, wall time: 1.354 minutes
[ Info: i: 1700, t: 3.516 hours, Δt: 5.435 seconds, umax = (1.1e-01, 9.3e-02, 4.7e-02) ms⁻¹, wall time: 1.367 minutes
[ Info: i: 1720, t: 3.547 hours, Δt: 6.218 seconds, umax = (1.0e-01, 7.5e-02, 4.8e-02) ms⁻¹, wall time: 1.379 minutes
[ Info: i: 1740, t: 3.581 hours, Δt: 6.265 seconds, umax = (1.1e-01, 7.2e-02, 4.6e-02) ms⁻¹, wall time: 1.391 minutes
[ Info: i: 1760, t: 3.614 hours, Δt: 6.303 seconds, umax = (1.2e-01, 7.9e-02, 4.6e-02) ms⁻¹, wall time: 1.405 minutes
[ Info: i: 1780, t: 3.647 hours, Δt: 5.845 seconds, umax = (1.2e-01, 8.6e-02, 5.1e-02) ms⁻¹, wall time: 1.418 minutes
[ Info: i: 1800, t: 3.678 hours, Δt: 5.919 seconds, umax = (1.3e-01, 7.7e-02, 4.9e-02) ms⁻¹, wall time: 1.432 minutes
[ Info: i: 1820, t: 3.711 hours, Δt: 5.573 seconds, umax = (1.1e-01, 8.7e-02, 4.9e-02) ms⁻¹, wall time: 1.444 minutes
[ Info: i: 1840, t: 3.742 hours, Δt: 6.002 seconds, umax = (1.1e-01, 8.0e-02, 5.1e-02) ms⁻¹, wall time: 1.457 minutes
[ Info: i: 1860, t: 3.775 hours, Δt: 5.831 seconds, umax = (1.1e-01, 8.7e-02, 4.9e-02) ms⁻¹, wall time: 1.471 minutes
[ Info: i: 1880, t: 3.807 hours, Δt: 6.156 seconds, umax = (1.0e-01, 8.8e-02, 4.6e-02) ms⁻¹, wall time: 1.483 minutes
[ Info: i: 1900, t: 3.840 hours, Δt: 6.043 seconds, umax = (1.1e-01, 8.0e-02, 4.5e-02) ms⁻¹, wall time: 1.498 minutes
[ Info: i: 1920, t: 3.873 hours, Δt: 6.159 seconds, umax = (1.1e-01, 7.5e-02, 4.5e-02) ms⁻¹, wall time: 1.509 minutes
[ Info: i: 1940, t: 3.906 hours, Δt: 6.016 seconds, umax = (1.1e-01, 8.5e-02, 5.3e-02) ms⁻¹, wall time: 1.521 minutes
[ Info: i: 1960, t: 3.938 hours, Δt: 5.570 seconds, umax = (1.0e-01, 8.3e-02, 4.6e-02) ms⁻¹, wall time: 1.535 minutes
[ Info: i: 1980, t: 3.969 hours, Δt: 6.068 seconds, umax = (1.1e-01, 7.6e-02, 4.3e-02) ms⁻¹, wall time: 1.546 minutes
[ Info: Simulation is stopping after running for 1.557 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-oceananigans
  JULIA_PROJECT = /var/lib/buildkite-agent/Oceananigans.jl-29683/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-29683/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-29683/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 v0.105.1 `..`
  [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.