Langmuir turbulence example

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

Wagner et al., "Near-inertial waves and turbulence driven by the growth of swell", Journal of Physical Oceanography (2021)

This example demonstrates

How to run large eddy simulations with surface wave effects via the Craik-Leibovich approximation.
How to specify time- and horizontally-averaged output.

Install dependencies

First let's make sure we have all required packages installed.

using Pkg
pkg"add Oceananigans, CairoMakie, CUDA"

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

Model set-up

To build the model, we specify the grid, Stokes drift, boundary conditions, and Coriolis parameter.

Domain and numerical grid specification

We use a modest resolution and the same total extent as Wagner et al. (2021),

grid = RectilinearGrid(GPU(), size=(128, 128, 64), extent=(128, 128, 64))

128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CUDAGPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 128.0) regularly spaced with Δx=1.0
├── Periodic y ∈ [0.0, 128.0) regularly spaced with Δy=1.0
└── Bounded  z ∈ [-64.0, 0.0] regularly spaced with Δz=1.0

The Stokes Drift profile

The surface wave Stokes drift profile prescribed in Wagner et al. (2021), corresponds to a 'monochromatic' (that is, single-frequency) wave field.

A monochromatic wave field is characterized by its wavelength and amplitude (half the distance from wave crest to wave trough), which determine the wave frequency and the vertical scale of the Stokes drift profile.

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 CPU, replace GPU() with CPU() in the RectilinearGrid constructor above.

The Stokes drift profile is

uˢ(z) = Uˢ * exp(z / vertical_scale)

uˢ (generic function with 1 method)

and its z-derivative is

∂z_uˢ(z, t) = 1 / vertical_scale * Uˢ * exp(z / vertical_scale)

∂z_uˢ (generic function with 1 method)

The Craik-Leibovich equations in Oceananigans

Oceananigans implements the Craik-Leibovich approximation for surface wave effects using the Lagrangian-mean velocity field as its prognostic momentum variable. In other words, model.velocities.u is the Lagrangian-mean $x$-velocity beneath surface waves. This differs from models that use the Eulerian-mean velocity field as a prognostic variable, but has the advantage that $u$ accounts for the total advection of tracers and momentum, and that $u = v = w = 0$ is a steady solution even when Coriolis forces are present. See the physics documentation for more information.

Finally, we note that the time-derivative of the Stokes drift must be provided if the Stokes drift and surface wave field undergoes forced changes in time. In this example, the Stokes drift is constant and thus the time-derivative of the Stokes drift is 0.

Boundary conditions

At the surface $z = 0$, Wagner et al. (2021) impose

τx = -3.72e-5 # m² s⁻², surface kinematic momentum flux
u_boundary_conditions = FieldBoundaryConditions(top = FluxBoundaryCondition(τx))

Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── top: FluxBoundaryCondition: -3.72e-5
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)

Wagner et al. (2021) impose a linear buoyancy gradient N² at the bottom along with a weak, destabilizing flux of buoyancy at the surface to faciliate spin-up from rest.

Jᵇ = 2.307e-8 # m² s⁻³, surface buoyancy flux
N² = 1.936e-5 # s⁻², initial and bottom buoyancy gradient

b_boundary_conditions = FieldBoundaryConditions(top = FluxBoundaryCondition(Jᵇ),
                                                bottom = GradientBoundaryCondition(N²))

Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: GradientBoundaryCondition: 1.936e-5
├── top: FluxBoundaryCondition: 2.307e-8
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)

The flux convention in Oceananigans

Note that Oceananigans uses "positive upward" conventions for all fluxes. In consequence, a negative flux at the surface drives positive velocities, and a positive flux of buoyancy drives cooling.

Coriolis parameter

Wagner et al. (2021) use

coriolis = FPlane(f=1e-4) # s⁻¹

FPlane{Float64}(f=0.0001)

which is typical for mid-latitudes on Earth.

Model instantiation

We are ready to build the model. We use a fifth-order Weighted Essentially Non-Oscillatory (WENO) advection scheme and the AnisotropicMinimumDissipation model for large eddy simulation. Because our Stokes drift does not vary in $x, y$, we use UniformStokesDrift, which expects Stokes drift functions of $z, t$ only.

model = NonhydrostaticModel(; grid, coriolis,
                            advection = WENO(order=9),
                            tracers = :b,
                            buoyancy = BuoyancyTracer(),
                            stokes_drift = UniformStokesDrift(∂z_uˢ=∂z_uˢ),
                            boundary_conditions = (u=u_boundary_conditions, b=b_boundary_conditions))

NonhydrostaticModel{GPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CUDAGPU with 5×5×5 halo
├── timestepper: RungeKutta3TimeStepper
├── advection scheme: WENO{5, Float64, Float32}(order=9)
├── tracers: b
├── closure: Nothing
├── buoyancy: BuoyancyTracer with ĝ = NegativeZDirection()
└── coriolis: FPlane{Float64}(f=0.0001)

Initial conditions

We make use of random noise concentrated in the upper 4 meters for buoyancy and velocity initial conditions,

Ξ(z) = randn() * exp(z / 4)

Our initial condition for buoyancy consists of a surface mixed layer 33 m deep, a deep linear stratification, plus noise,

initial_mixed_layer_depth = 33 # m
stratification(z) = z < - initial_mixed_layer_depth ? N² * z : N² * (-initial_mixed_layer_depth)

bᵢ(x, y, z) = stratification(z) + 1e-1 * Ξ(z) * N² * model.grid.Lz

bᵢ (generic function with 1 method)

The simulation we reproduce from Wagner et al. (2021) is zero Lagrangian-mean velocity. This initial condition is consistent with a wavy, quiescent ocean suddenly impacted by winds. To this quiescent state we add noise scaled by the friction velocity to $u$ and $w$.

u★ = sqrt(abs(τx))
uᵢ(x, y, z) = u★ * 1e-1 * Ξ(z)
wᵢ(x, y, z) = u★ * 1e-1 * Ξ(z)

set!(model, u=uᵢ, w=wᵢ, b=bᵢ)

Setting up the simulation

simulation = Simulation(model, Δt=45.0, stop_time=4hours)

Simulation of NonhydrostaticModel{GPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── Next time step: 45 seconds
├── 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 => 4
│   ├── stop_iteration_exceeded => -
│   ├── wall_time_limit_exceeded => e
│   └── nan_checker => }
├── 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 = (1.4e-03, 8.7e-04, 1.5e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (35.855 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (5.570 seconds).
[ Info: i: 0020, t: 12.583 minutes, Δt: 29.420 seconds, umax = (3.0e-02, 1.1e-02, 2.1e-02) ms⁻¹, wall time: 54.730 seconds
[ Info: i: 0040, t: 21.133 minutes, Δt: 20.004 seconds, umax = (4.2e-02, 1.1e-02, 2.0e-02) ms⁻¹, wall time: 55.389 seconds
[ Info: i: 0060, t: 27.352 minutes, Δt: 16.073 seconds, umax = (4.9e-02, 1.6e-02, 2.1e-02) ms⁻¹, wall time: 55.850 seconds
[ Info: i: 0080, t: 32.764 minutes, Δt: 15.812 seconds, umax = (5.0e-02, 1.6e-02, 2.4e-02) ms⁻¹, wall time: 56.329 seconds
[ Info: i: 0100, t: 37.932 minutes, Δt: 16.229 seconds, umax = (5.3e-02, 1.7e-02, 2.4e-02) ms⁻¹, wall time: 56.847 seconds
[ Info: i: 0120, t: 43.127 minutes, Δt: 14.486 seconds, umax = (5.3e-02, 2.0e-02, 2.4e-02) ms⁻¹, wall time: 57.388 seconds
[ Info: i: 0140, t: 47.867 minutes, Δt: 14.508 seconds, umax = (5.5e-02, 2.2e-02, 2.6e-02) ms⁻¹, wall time: 57.941 seconds
[ Info: i: 0160, t: 52.650 minutes, Δt: 14.032 seconds, umax = (5.6e-02, 2.3e-02, 2.7e-02) ms⁻¹, wall time: 58.432 seconds
[ Info: i: 0180, t: 57.046 minutes, Δt: 13.511 seconds, umax = (6.0e-02, 2.4e-02, 3.2e-02) ms⁻¹, wall time: 58.967 seconds
[ Info: i: 0200, t: 1.022 hours, Δt: 12.378 seconds, umax = (6.3e-02, 2.7e-02, 3.0e-02) ms⁻¹, wall time: 59.653 seconds
[ Info: i: 0220, t: 1.087 hours, Δt: 12.617 seconds, umax = (6.4e-02, 2.8e-02, 3.2e-02) ms⁻¹, wall time: 1.004 minutes
[ Info: i: 0240, t: 1.156 hours, Δt: 11.723 seconds, umax = (6.6e-02, 2.9e-02, 3.5e-02) ms⁻¹, wall time: 1.010 minutes
[ Info: i: 0260, t: 1.215 hours, Δt: 11.639 seconds, umax = (6.8e-02, 3.1e-02, 3.7e-02) ms⁻¹, wall time: 1.020 minutes
[ Info: i: 0280, t: 1.278 hours, Δt: 10.783 seconds, umax = (6.8e-02, 3.5e-02, 3.3e-02) ms⁻¹, wall time: 1.030 minutes
[ Info: i: 0300, t: 1.336 hours, Δt: 10.481 seconds, umax = (6.9e-02, 3.4e-02, 4.0e-02) ms⁻¹, wall time: 1.041 minutes
[ Info: i: 0320, t: 1.395 hours, Δt: 10.842 seconds, umax = (7.2e-02, 3.5e-02, 3.9e-02) ms⁻¹, wall time: 1.047 minutes
[ Info: i: 0340, t: 1.453 hours, Δt: 10.946 seconds, umax = (7.1e-02, 3.6e-02, 4.3e-02) ms⁻¹, wall time: 1.056 minutes
[ Info: i: 0360, t: 1.512 hours, Δt: 10.367 seconds, umax = (7.2e-02, 3.8e-02, 3.8e-02) ms⁻¹, wall time: 1.066 minutes
[ Info: i: 0380, t: 1.568 hours, Δt: 10.693 seconds, umax = (7.2e-02, 4.0e-02, 3.9e-02) ms⁻¹, wall time: 1.073 minutes
[ Info: i: 0400, t: 1.624 hours, Δt: 10.065 seconds, umax = (7.8e-02, 4.0e-02, 4.3e-02) ms⁻¹, wall time: 1.082 minutes
[ Info: i: 0420, t: 1.678 hours, Δt: 10.052 seconds, umax = (7.2e-02, 4.1e-02, 3.9e-02) ms⁻¹, wall time: 1.093 minutes
[ Info: i: 0440, t: 1.734 hours, Δt: 9.869 seconds, umax = (7.6e-02, 4.0e-02, 3.6e-02) ms⁻¹, wall time: 1.100 minutes
[ Info: i: 0460, t: 1.788 hours, Δt: 9.661 seconds, umax = (7.7e-02, 4.3e-02, 3.9e-02) ms⁻¹, wall time: 1.109 minutes
[ Info: i: 0480, t: 1.839 hours, Δt: 9.806 seconds, umax = (7.9e-02, 4.1e-02, 4.1e-02) ms⁻¹, wall time: 1.123 minutes
[ Info: i: 0500, t: 1.893 hours, Δt: 9.186 seconds, umax = (8.0e-02, 4.5e-02, 3.9e-02) ms⁻¹, wall time: 1.129 minutes
[ Info: i: 0520, t: 1.941 hours, Δt: 9.617 seconds, umax = (7.8e-02, 4.7e-02, 4.2e-02) ms⁻¹, wall time: 1.138 minutes
[ Info: i: 0540, t: 1.995 hours, Δt: 9.584 seconds, umax = (8.2e-02, 4.6e-02, 4.5e-02) ms⁻¹, wall time: 1.147 minutes
[ Info: i: 0560, t: 2.047 hours, Δt: 9.270 seconds, umax = (8.1e-02, 4.6e-02, 4.5e-02) ms⁻¹, wall time: 1.156 minutes
[ Info: i: 0580, t: 2.096 hours, Δt: 9.369 seconds, umax = (8.2e-02, 4.4e-02, 4.3e-02) ms⁻¹, wall time: 1.167 minutes
[ Info: i: 0600, t: 2.148 hours, Δt: 9.359 seconds, umax = (7.9e-02, 4.6e-02, 4.4e-02) ms⁻¹, wall time: 1.174 minutes
[ Info: i: 0620, t: 2.197 hours, Δt: 8.916 seconds, umax = (8.1e-02, 4.8e-02, 4.7e-02) ms⁻¹, wall time: 1.183 minutes
[ Info: i: 0640, t: 2.247 hours, Δt: 8.994 seconds, umax = (8.3e-02, 4.8e-02, 4.4e-02) ms⁻¹, wall time: 1.192 minutes
[ Info: i: 0660, t: 2.295 hours, Δt: 8.859 seconds, umax = (8.3e-02, 5.2e-02, 4.3e-02) ms⁻¹, wall time: 1.200 minutes
[ Info: i: 0680, t: 2.343 hours, Δt: 8.576 seconds, umax = (8.5e-02, 5.6e-02, 4.1e-02) ms⁻¹, wall time: 1.212 minutes
[ Info: i: 0700, t: 2.391 hours, Δt: 8.774 seconds, umax = (8.2e-02, 5.0e-02, 4.2e-02) ms⁻¹, wall time: 1.219 minutes
[ Info: i: 0720, t: 2.439 hours, Δt: 9.068 seconds, umax = (8.2e-02, 4.9e-02, 4.2e-02) ms⁻¹, wall time: 1.228 minutes
[ Info: i: 0740, t: 2.488 hours, Δt: 8.691 seconds, umax = (8.3e-02, 5.3e-02, 4.1e-02) ms⁻¹, wall time: 1.238 minutes
[ Info: i: 0760, t: 2.537 hours, Δt: 8.697 seconds, umax = (8.3e-02, 5.1e-02, 4.0e-02) ms⁻¹, wall time: 1.248 minutes
[ Info: i: 0780, t: 2.583 hours, Δt: 8.517 seconds, umax = (8.4e-02, 5.1e-02, 4.2e-02) ms⁻¹, wall time: 1.258 minutes
[ Info: i: 0800, t: 2.630 hours, Δt: 8.273 seconds, umax = (8.7e-02, 5.5e-02, 4.3e-02) ms⁻¹, wall time: 1.266 minutes
[ Info: i: 0820, t: 2.676 hours, Δt: 8.091 seconds, umax = (8.4e-02, 6.1e-02, 4.2e-02) ms⁻¹, wall time: 1.277 minutes
[ Info: i: 0840, t: 2.721 hours, Δt: 8.607 seconds, umax = (8.2e-02, 5.3e-02, 4.3e-02) ms⁻¹, wall time: 1.284 minutes
[ Info: i: 0860, t: 2.767 hours, Δt: 8.330 seconds, umax = (8.2e-02, 6.0e-02, 4.2e-02) ms⁻¹, wall time: 1.294 minutes
[ Info: i: 0880, t: 2.813 hours, Δt: 8.496 seconds, umax = (8.4e-02, 5.6e-02, 4.4e-02) ms⁻¹, wall time: 1.302 minutes
[ Info: i: 0900, t: 2.860 hours, Δt: 8.606 seconds, umax = (8.4e-02, 5.1e-02, 4.3e-02) ms⁻¹, wall time: 1.311 minutes
[ Info: i: 0920, t: 2.908 hours, Δt: 8.587 seconds, umax = (8.7e-02, 5.4e-02, 4.8e-02) ms⁻¹, wall time: 1.320 minutes
[ Info: i: 0940, t: 2.954 hours, Δt: 8.315 seconds, umax = (8.5e-02, 5.4e-02, 4.8e-02) ms⁻¹, wall time: 1.329 minutes
[ Info: i: 0960, t: 3.000 hours, Δt: 7.898 seconds, umax = (8.5e-02, 5.5e-02, 4.3e-02) ms⁻¹, wall time: 1.339 minutes
[ Info: i: 0980, t: 3.044 hours, Δt: 8.595 seconds, umax = (8.6e-02, 5.3e-02, 5.0e-02) ms⁻¹, wall time: 1.349 minutes
[ Info: i: 1000, t: 3.090 hours, Δt: 8.485 seconds, umax = (8.7e-02, 5.3e-02, 6.2e-02) ms⁻¹, wall time: 1.362 minutes
[ Info: i: 1020, t: 3.137 hours, Δt: 8.478 seconds, umax = (8.5e-02, 5.7e-02, 5.4e-02) ms⁻¹, wall time: 1.368 minutes
[ Info: i: 1040, t: 3.183 hours, Δt: 8.390 seconds, umax = (8.8e-02, 5.3e-02, 4.5e-02) ms⁻¹, wall time: 1.378 minutes
[ Info: i: 1060, t: 3.230 hours, Δt: 8.485 seconds, umax = (8.9e-02, 5.6e-02, 4.4e-02) ms⁻¹, wall time: 1.386 minutes
[ Info: i: 1080, t: 3.276 hours, Δt: 8.399 seconds, umax = (8.6e-02, 5.6e-02, 4.6e-02) ms⁻¹, wall time: 1.396 minutes
[ Info: i: 1100, t: 3.323 hours, Δt: 8.414 seconds, umax = (8.5e-02, 5.6e-02, 5.1e-02) ms⁻¹, wall time: 1.404 minutes
[ Info: i: 1120, t: 3.368 hours, Δt: 8.346 seconds, umax = (9.0e-02, 5.7e-02, 4.8e-02) ms⁻¹, wall time: 1.414 minutes
[ Info: i: 1140, t: 3.414 hours, Δt: 8.500 seconds, umax = (8.4e-02, 5.9e-02, 4.8e-02) ms⁻¹, wall time: 1.422 minutes
[ Info: i: 1160, t: 3.459 hours, Δt: 8.113 seconds, umax = (8.6e-02, 6.2e-02, 5.1e-02) ms⁻¹, wall time: 1.431 minutes
[ Info: i: 1180, t: 3.502 hours, Δt: 8.514 seconds, umax = (8.8e-02, 6.2e-02, 4.7e-02) ms⁻¹, wall time: 1.443 minutes
[ Info: i: 1200, t: 3.549 hours, Δt: 8.300 seconds, umax = (8.4e-02, 6.1e-02, 4.9e-02) ms⁻¹, wall time: 1.449 minutes
[ Info: i: 1220, t: 3.592 hours, Δt: 8.249 seconds, umax = (8.6e-02, 6.0e-02, 4.6e-02) ms⁻¹, wall time: 1.460 minutes
[ Info: i: 1240, t: 3.637 hours, Δt: 7.678 seconds, umax = (8.8e-02, 6.2e-02, 4.8e-02) ms⁻¹, wall time: 1.467 minutes
[ Info: i: 1260, t: 3.679 hours, Δt: 7.249 seconds, umax = (9.0e-02, 6.3e-02, 5.0e-02) ms⁻¹, wall time: 1.477 minutes
[ Info: i: 1280, t: 3.720 hours, Δt: 7.554 seconds, umax = (8.4e-02, 6.4e-02, 4.8e-02) ms⁻¹, wall time: 1.484 minutes
[ Info: i: 1300, t: 3.761 hours, Δt: 7.921 seconds, umax = (8.5e-02, 5.9e-02, 4.5e-02) ms⁻¹, wall time: 1.497 minutes
[ Info: i: 1320, t: 3.805 hours, Δt: 8.090 seconds, umax = (8.4e-02, 5.9e-02, 5.5e-02) ms⁻¹, wall time: 1.506 minutes
[ Info: i: 1340, t: 3.849 hours, Δt: 8.225 seconds, umax = (9.0e-02, 5.8e-02, 4.5e-02) ms⁻¹, wall time: 1.518 minutes
[ Info: i: 1360, t: 3.894 hours, Δt: 7.912 seconds, umax = (9.1e-02, 5.7e-02, 4.4e-02) ms⁻¹, wall time: 1.528 minutes
[ Info: i: 1380, t: 3.937 hours, Δt: 7.842 seconds, umax = (8.7e-02, 6.2e-02, 4.4e-02) ms⁻¹, wall time: 1.540 minutes
[ Info: i: 1400, t: 3.980 hours, Δt: 7.980 seconds, umax = (8.8e-02, 5.9e-02, 4.3e-02) ms⁻¹, wall time: 1.552 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 $u$ and $w$, and a horizontal slice of $w$ to look for Langmuir cells.

using CairoMakie

time_series = (;
     w = FieldTimeSeries("langmuir_turbulence_fields.jld2", "w"),
     u = FieldTimeSeries("langmuir_turbulence_fields.jld2", "u"),
     B = FieldTimeSeries("langmuir_turbulence_averages.jld2", "B"),
     U = FieldTimeSeries("langmuir_turbulence_averages.jld2", "U"),
     V = FieldTimeSeries("langmuir_turbulence_averages.jld2", "V"),
    wu = FieldTimeSeries("langmuir_turbulence_averages.jld2", "wu"),
    wv = FieldTimeSeries("langmuir_turbulence_averages.jld2", "wv"))

times = time_series.w.times

We are now ready to animate using Makie. We use Makie's Observable to animate the data. To dive into how Observables work we refer to Makie.jl's Documentation.

n = Observable(1)

wxy_title = @lift string("w(x, y, t) at z=-8 m and t = ", prettytime(times[$n]))
wxz_title = @lift string("w(x, z, t) at y=0 m and t = ", prettytime(times[$n]))
uxz_title = @lift string("u(x, z, t) at y=0 m and t = ", prettytime(times[$n]))

fig = Figure(size = (850, 850))

ax_B = Axis(fig[1, 4];
            xlabel = "Buoyancy (m s⁻²)",
            ylabel = "z (m)")

ax_U = Axis(fig[2, 4];
            xlabel = "Velocities (m s⁻¹)",
            ylabel = "z (m)",
            limits = ((-0.07, 0.07), nothing))

ax_fluxes = Axis(fig[3, 4];
                 xlabel = "Momentum fluxes (m² s⁻²)",
                 ylabel = "z (m)",
                 limits = ((-3.5e-5, 3.5e-5), nothing))

ax_wxy = Axis(fig[1, 1:2];
              xlabel = "x (m)",
              ylabel = "y (m)",
              aspect = DataAspect(),
              limits = ((0, grid.Lx), (0, grid.Ly)),
              title = wxy_title)

ax_wxz = Axis(fig[2, 1:2];
              xlabel = "x (m)",
              ylabel = "z (m)",
              aspect = AxisAspect(2),
              limits = ((0, grid.Lx), (-grid.Lz, 0)),
              title = wxz_title)

ax_uxz = Axis(fig[3, 1:2];
              xlabel = "x (m)",
              ylabel = "z (m)",
              aspect = AxisAspect(2),
              limits = ((0, grid.Lx), (-grid.Lz, 0)),
              title = uxz_title)


wₙ = @lift time_series.w[$n]
uₙ = @lift time_series.u[$n]
Bₙ = @lift view(time_series.B[$n], 1, 1, :)
Uₙ = @lift view(time_series.U[$n], 1, 1, :)
Vₙ = @lift view(time_series.V[$n], 1, 1, :)
wuₙ = @lift view(time_series.wu[$n], 1, 1, :)
wvₙ = @lift view(time_series.wv[$n], 1, 1, :)

k = searchsortedfirst(znodes(grid, Face(); with_halos=true), -8)
wxyₙ = @lift view(time_series.w[$n], :, :, k)
wxzₙ = @lift view(time_series.w[$n], :, 1, :)
uxzₙ = @lift view(time_series.u[$n], :, 1, :)

wlims = (-0.03, 0.03)
ulims = (-0.05, 0.05)

lines!(ax_B, Bₙ)

lines!(ax_U, Uₙ; label = L"\bar{u}")
lines!(ax_U, Vₙ; label = L"\bar{v}")
axislegend(ax_U; position = :rb)

lines!(ax_fluxes, wuₙ; label = L"mean $wu$")
lines!(ax_fluxes, wvₙ; label = L"mean $wv$")
axislegend(ax_fluxes; position = :rb)

hm_wxy = heatmap!(ax_wxy, wxyₙ;
                  colorrange = wlims,
                  colormap = :balance)

Colorbar(fig[1, 3], hm_wxy; label = "m s⁻¹")

hm_wxz = heatmap!(ax_wxz, wxzₙ;
                  colorrange = wlims,
                  colormap = :balance)

Colorbar(fig[2, 3], hm_wxz; label = "m s⁻¹")

ax_uxz = heatmap!(ax_uxz, uxzₙ;
                  colorrange = ulims,
                  colormap = :balance)

Colorbar(fig[3, 3], ax_uxz; label = "m s⁻¹")

fig

And, finally, we record a movie.

frames = 1:length(times)

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

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