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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"
using Oceananigans
using Oceananigans.Units: minute, minutes, hours

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.

using Oceananigans.BuoyancyFormulations: g_Earth

 amplitude = 0.8 # m
wavelength = 60  # m
wavenumber = 2π / wavelength # m⁻¹
 frequency = sqrt(g_Earth * 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 GPU, include GPU() 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(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
├── Elapsed wall time: 0 seconds
├── Wall time per 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
└── Diagnostics: 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: 45.0 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: 44.3 KiB

Running the simulation

This part is easy,

run!(simulation)
[ Info: Initializing simulation...
[ Info: i: 0000, t: 0 seconds, Δt: 49.500 seconds, umax = (1.5e-03, 8.3e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (35.201 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (5.307 seconds).
[ Info: i: 0020, t: 11.942 minutes, Δt: 29.089 seconds, umax = (2.9e-02, 1.2e-02, 2.1e-02) ms⁻¹, wall time: 53.689 seconds
[ Info: i: 0040, t: 20.409 minutes, Δt: 19.835 seconds, umax = (4.1e-02, 1.1e-02, 2.2e-02) ms⁻¹, wall time: 54.366 seconds
[ Info: i: 0060, t: 26.473 minutes, Δt: 16.661 seconds, umax = (4.9e-02, 1.5e-02, 2.0e-02) ms⁻¹, wall time: 54.850 seconds
[ Info: i: 0080, t: 31.904 minutes, Δt: 16.417 seconds, umax = (5.0e-02, 1.6e-02, 2.3e-02) ms⁻¹, wall time: 55.351 seconds
[ Info: i: 0100, t: 36.853 minutes, Δt: 15.482 seconds, umax = (5.2e-02, 1.8e-02, 2.5e-02) ms⁻¹, wall time: 55.908 seconds
[ Info: i: 0120, t: 41.720 minutes, Δt: 14.783 seconds, umax = (5.5e-02, 1.9e-02, 2.4e-02) ms⁻¹, wall time: 56.486 seconds
[ Info: i: 0140, t: 46.431 minutes, Δt: 14.496 seconds, umax = (5.6e-02, 2.1e-02, 2.5e-02) ms⁻¹, wall time: 57.021 seconds
[ Info: i: 0160, t: 51.182 minutes, Δt: 13.434 seconds, umax = (5.9e-02, 2.3e-02, 2.6e-02) ms⁻¹, wall time: 57.581 seconds
[ Info: i: 0180, t: 55.436 minutes, Δt: 13.658 seconds, umax = (6.0e-02, 2.6e-02, 3.1e-02) ms⁻¹, wall time: 58.174 seconds
[ Info: i: 0200, t: 59.896 minutes, Δt: 12.476 seconds, umax = (6.5e-02, 2.5e-02, 3.1e-02) ms⁻¹, wall time: 58.532 seconds
[ Info: i: 0220, t: 1.066 hours, Δt: 12.321 seconds, umax = (6.4e-02, 2.7e-02, 3.2e-02) ms⁻¹, wall time: 59.078 seconds
[ Info: i: 0240, t: 1.132 hours, Δt: 11.654 seconds, umax = (6.5e-02, 2.9e-02, 3.3e-02) ms⁻¹, wall time: 59.589 seconds
[ Info: i: 0260, t: 1.196 hours, Δt: 11.615 seconds, umax = (6.6e-02, 3.1e-02, 3.2e-02) ms⁻¹, wall time: 1.003 minutes
[ Info: i: 0280, t: 1.256 hours, Δt: 11.189 seconds, umax = (6.9e-02, 3.1e-02, 3.5e-02) ms⁻¹, wall time: 1.014 minutes
[ Info: i: 0300, t: 1.319 hours, Δt: 11.413 seconds, umax = (6.9e-02, 3.3e-02, 3.6e-02) ms⁻¹, wall time: 1.020 minutes
[ Info: i: 0320, t: 1.377 hours, Δt: 11.172 seconds, umax = (6.9e-02, 3.2e-02, 3.7e-02) ms⁻¹, wall time: 1.028 minutes
[ Info: i: 0340, t: 1.438 hours, Δt: 10.918 seconds, umax = (6.9e-02, 3.4e-02, 3.8e-02) ms⁻¹, wall time: 1.038 minutes
[ Info: i: 0360, t: 1.498 hours, Δt: 10.321 seconds, umax = (7.2e-02, 3.7e-02, 3.6e-02) ms⁻¹, wall time: 1.046 minutes
[ Info: i: 0380, t: 1.553 hours, Δt: 9.605 seconds, umax = (7.3e-02, 3.6e-02, 4.1e-02) ms⁻¹, wall time: 1.054 minutes
[ Info: i: 0400, t: 1.604 hours, Δt: 9.475 seconds, umax = (7.3e-02, 4.1e-02, 4.0e-02) ms⁻¹, wall time: 1.064 minutes
[ Info: i: 0420, t: 1.655 hours, Δt: 8.833 seconds, umax = (7.5e-02, 4.2e-02, 3.9e-02) ms⁻¹, wall time: 1.073 minutes
[ Info: i: 0440, t: 1.704 hours, Δt: 9.082 seconds, umax = (7.2e-02, 4.5e-02, 3.7e-02) ms⁻¹, wall time: 1.081 minutes
[ Info: i: 0460, t: 1.753 hours, Δt: 8.933 seconds, umax = (7.8e-02, 4.5e-02, 3.7e-02) ms⁻¹, wall time: 1.092 minutes
[ Info: i: 0480, t: 1.804 hours, Δt: 10.094 seconds, umax = (7.7e-02, 4.0e-02, 4.0e-02) ms⁻¹, wall time: 1.097 minutes
[ Info: i: 0500, t: 1.858 hours, Δt: 9.502 seconds, umax = (7.5e-02, 4.2e-02, 4.1e-02) ms⁻¹, wall time: 1.107 minutes
[ Info: i: 0520, t: 1.909 hours, Δt: 9.224 seconds, umax = (7.8e-02, 4.9e-02, 4.1e-02) ms⁻¹, wall time: 1.116 minutes
[ Info: i: 0540, t: 1.960 hours, Δt: 8.920 seconds, umax = (8.1e-02, 4.8e-02, 5.1e-02) ms⁻¹, wall time: 1.124 minutes
[ Info: i: 0560, t: 2.010 hours, Δt: 9.457 seconds, umax = (8.3e-02, 4.6e-02, 5.0e-02) ms⁻¹, wall time: 1.136 minutes
[ Info: i: 0580, t: 2.063 hours, Δt: 9.466 seconds, umax = (8.0e-02, 4.6e-02, 4.2e-02) ms⁻¹, wall time: 1.142 minutes
[ Info: i: 0600, t: 2.115 hours, Δt: 9.313 seconds, umax = (8.1e-02, 4.4e-02, 4.8e-02) ms⁻¹, wall time: 1.151 minutes
[ Info: i: 0620, t: 2.167 hours, Δt: 9.367 seconds, umax = (7.9e-02, 4.8e-02, 4.3e-02) ms⁻¹, wall time: 1.161 minutes
[ Info: i: 0640, t: 2.218 hours, Δt: 9.356 seconds, umax = (7.8e-02, 5.1e-02, 4.3e-02) ms⁻¹, wall time: 1.169 minutes
[ Info: i: 0660, t: 2.268 hours, Δt: 9.182 seconds, umax = (7.9e-02, 4.7e-02, 4.4e-02) ms⁻¹, wall time: 1.179 minutes
[ Info: i: 0680, t: 2.318 hours, Δt: 9.257 seconds, umax = (7.8e-02, 4.7e-02, 4.5e-02) ms⁻¹, wall time: 1.187 minutes
[ Info: i: 0700, t: 2.369 hours, Δt: 9.209 seconds, umax = (8.1e-02, 5.4e-02, 4.3e-02) ms⁻¹, wall time: 1.197 minutes
[ Info: i: 0720, t: 2.417 hours, Δt: 8.726 seconds, umax = (8.0e-02, 5.5e-02, 4.5e-02) ms⁻¹, wall time: 1.207 minutes
[ Info: i: 0740, t: 2.466 hours, Δt: 8.880 seconds, umax = (8.1e-02, 5.2e-02, 4.4e-02) ms⁻¹, wall time: 1.216 minutes
[ Info: i: 0760, t: 2.515 hours, Δt: 8.760 seconds, umax = (8.2e-02, 5.2e-02, 4.3e-02) ms⁻¹, wall time: 1.228 minutes
[ Info: i: 0780, t: 2.564 hours, Δt: 8.773 seconds, umax = (8.3e-02, 4.9e-02, 5.1e-02) ms⁻¹, wall time: 1.237 minutes
[ Info: i: 0800, t: 2.612 hours, Δt: 8.507 seconds, umax = (8.3e-02, 5.2e-02, 4.4e-02) ms⁻¹, wall time: 1.246 minutes
[ Info: i: 0820, t: 2.659 hours, Δt: 8.454 seconds, umax = (8.7e-02, 5.9e-02, 4.0e-02) ms⁻¹, wall time: 1.257 minutes
[ Info: i: 0840, t: 2.704 hours, Δt: 8.579 seconds, umax = (8.6e-02, 6.0e-02, 4.4e-02) ms⁻¹, wall time: 1.266 minutes
[ Info: i: 0860, t: 2.750 hours, Δt: 8.013 seconds, umax = (8.4e-02, 5.6e-02, 4.4e-02) ms⁻¹, wall time: 1.277 minutes
[ Info: i: 0880, t: 2.796 hours, Δt: 8.417 seconds, umax = (8.5e-02, 5.5e-02, 4.4e-02) ms⁻¹, wall time: 1.286 minutes
[ Info: i: 0900, t: 2.838 hours, Δt: 8.425 seconds, umax = (8.3e-02, 5.6e-02, 4.1e-02) ms⁻¹, wall time: 1.300 minutes
[ Info: i: 0920, t: 2.884 hours, Δt: 8.247 seconds, umax = (8.0e-02, 5.6e-02, 4.5e-02) ms⁻¹, wall time: 1.306 minutes
[ Info: i: 0940, t: 2.928 hours, Δt: 8.443 seconds, umax = (8.3e-02, 5.4e-02, 4.9e-02) ms⁻¹, wall time: 1.319 minutes
[ Info: i: 0960, t: 2.975 hours, Δt: 8.486 seconds, umax = (8.4e-02, 5.8e-02, 4.4e-02) ms⁻¹, wall time: 1.327 minutes
[ Info: i: 0980, t: 3.019 hours, Δt: 8.342 seconds, umax = (8.4e-02, 5.6e-02, 4.4e-02) ms⁻¹, wall time: 1.337 minutes
[ Info: i: 1000, t: 3.065 hours, Δt: 8.560 seconds, umax = (8.4e-02, 6.1e-02, 4.4e-02) ms⁻¹, wall time: 1.347 minutes
[ Info: i: 1020, t: 3.111 hours, Δt: 8.453 seconds, umax = (8.6e-02, 5.9e-02, 5.1e-02) ms⁻¹, wall time: 1.356 minutes
[ Info: i: 1040, t: 3.158 hours, Δt: 8.448 seconds, umax = (8.2e-02, 6.1e-02, 4.8e-02) ms⁻¹, wall time: 1.366 minutes
[ Info: i: 1060, t: 3.204 hours, Δt: 8.468 seconds, umax = (8.3e-02, 5.9e-02, 4.6e-02) ms⁻¹, wall time: 1.376 minutes
[ Info: i: 1080, t: 3.250 hours, Δt: 8.697 seconds, umax = (8.6e-02, 5.8e-02, 4.6e-02) ms⁻¹, wall time: 1.386 minutes
[ Info: i: 1100, t: 3.297 hours, Δt: 8.229 seconds, umax = (8.9e-02, 6.1e-02, 4.2e-02) ms⁻¹, wall time: 1.395 minutes
[ Info: i: 1120, t: 3.343 hours, Δt: 8.311 seconds, umax = (8.8e-02, 6.6e-02, 4.3e-02) ms⁻¹, wall time: 1.408 minutes
[ Info: i: 1140, t: 3.389 hours, Δt: 8.186 seconds, umax = (8.7e-02, 5.7e-02, 4.2e-02) ms⁻¹, wall time: 1.415 minutes
[ Info: i: 1160, t: 3.433 hours, Δt: 8.371 seconds, umax = (8.7e-02, 5.9e-02, 4.4e-02) ms⁻¹, wall time: 1.427 minutes
[ Info: i: 1180, t: 3.478 hours, Δt: 7.723 seconds, umax = (8.9e-02, 6.6e-02, 4.5e-02) ms⁻¹, wall time: 1.436 minutes
[ Info: i: 1200, t: 3.520 hours, Δt: 7.997 seconds, umax = (8.9e-02, 5.8e-02, 5.4e-02) ms⁻¹, wall time: 1.446 minutes
[ Info: i: 1220, t: 3.565 hours, Δt: 8.365 seconds, umax = (9.0e-02, 5.7e-02, 4.9e-02) ms⁻¹, wall time: 1.456 minutes
[ Info: i: 1240, t: 3.611 hours, Δt: 8.287 seconds, umax = (8.4e-02, 5.9e-02, 4.9e-02) ms⁻¹, wall time: 1.467 minutes
[ Info: i: 1260, t: 3.658 hours, Δt: 8.445 seconds, umax = (8.8e-02, 5.5e-02, 4.9e-02) ms⁻¹, wall time: 1.476 minutes
[ Info: i: 1280, t: 3.704 hours, Δt: 8.457 seconds, umax = (8.7e-02, 5.9e-02, 4.7e-02) ms⁻¹, wall time: 1.486 minutes
[ Info: i: 1300, t: 3.750 hours, Δt: 7.858 seconds, umax = (8.6e-02, 6.1e-02, 4.4e-02) ms⁻¹, wall time: 1.496 minutes
[ Info: i: 1320, t: 3.793 hours, Δt: 7.888 seconds, umax = (8.9e-02, 5.8e-02, 4.5e-02) ms⁻¹, wall time: 1.506 minutes
[ Info: i: 1340, t: 3.836 hours, Δt: 8.128 seconds, umax = (8.9e-02, 6.6e-02, 4.8e-02) ms⁻¹, wall time: 1.519 minutes
[ Info: i: 1360, t: 3.881 hours, Δt: 7.962 seconds, umax = (8.5e-02, 6.0e-02, 4.6e-02) ms⁻¹, wall time: 1.525 minutes
[ Info: i: 1380, t: 3.923 hours, Δt: 8.098 seconds, umax = (8.7e-02, 5.9e-02, 4.4e-02) ms⁻¹, wall time: 1.538 minutes
[ Info: i: 1400, t: 3.968 hours, Δt: 8.226 seconds, umax = (8.7e-02, 6.3e-02, 4.5e-02) ms⁻¹, wall time: 1.545 minutes
[ Info: Simulation is stopping after running for 1.554 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)

record(fig, "langmuir_turbulence.mp4", frames, framerate=8) do i
    n[] = i
end

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