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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, 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 = (1.7e-03, 8.5e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (34.728 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (5.147 seconds).
[ Info: i: 0020, t: 11.906 minutes, Δt: 30.380 seconds, umax = (2.8e-02, 1.2e-02, 2.0e-02) ms⁻¹, wall time: 53.312 seconds
[ Info: i: 0040, t: 20.404 minutes, Δt: 20.147 seconds, umax = (4.1e-02, 1.1e-02, 1.8e-02) ms⁻¹, wall time: 54.151 seconds
[ Info: i: 0060, t: 26.481 minutes, Δt: 16.420 seconds, umax = (4.8e-02, 1.6e-02, 1.9e-02) ms⁻¹, wall time: 54.604 seconds
[ Info: i: 0080, t: 31.898 minutes, Δt: 16.090 seconds, umax = (5.0e-02, 1.7e-02, 2.5e-02) ms⁻¹, wall time: 55.090 seconds
[ Info: i: 0100, t: 37.100 minutes, Δt: 15.327 seconds, umax = (5.3e-02, 1.8e-02, 2.3e-02) ms⁻¹, wall time: 55.636 seconds
[ Info: i: 0120, t: 42.030 minutes, Δt: 14.883 seconds, umax = (5.3e-02, 1.8e-02, 2.5e-02) ms⁻¹, wall time: 56.163 seconds
[ Info: i: 0140, t: 46.998 minutes, Δt: 14.785 seconds, umax = (5.4e-02, 2.1e-02, 2.6e-02) ms⁻¹, wall time: 56.696 seconds
[ Info: i: 0160, t: 51.690 minutes, Δt: 14.189 seconds, umax = (5.8e-02, 2.2e-02, 2.9e-02) ms⁻¹, wall time: 57.266 seconds
[ Info: i: 0180, t: 56.169 minutes, Δt: 13.420 seconds, umax = (5.9e-02, 2.3e-02, 2.8e-02) ms⁻¹, wall time: 57.833 seconds
[ Info: i: 0200, t: 1.007 hours, Δt: 12.751 seconds, umax = (6.2e-02, 2.5e-02, 3.2e-02) ms⁻¹, wall time: 58.424 seconds
[ Info: i: 0220, t: 1.078 hours, Δt: 12.201 seconds, umax = (6.3e-02, 2.7e-02, 3.2e-02) ms⁻¹, wall time: 58.781 seconds
[ Info: i: 0240, t: 1.144 hours, Δt: 12.008 seconds, umax = (6.5e-02, 2.7e-02, 3.6e-02) ms⁻¹, wall time: 59.271 seconds
[ Info: i: 0260, t: 1.209 hours, Δt: 11.351 seconds, umax = (6.6e-02, 2.8e-02, 3.3e-02) ms⁻¹, wall time: 59.808 seconds
[ Info: i: 0280, t: 1.273 hours, Δt: 11.495 seconds, umax = (7.1e-02, 3.2e-02, 3.7e-02) ms⁻¹, wall time: 1.008 minutes
[ Info: i: 0300, t: 1.333 hours, Δt: 10.473 seconds, umax = (7.0e-02, 3.3e-02, 3.7e-02) ms⁻¹, wall time: 1.015 minutes
[ Info: i: 0320, t: 1.392 hours, Δt: 10.859 seconds, umax = (6.7e-02, 3.4e-02, 3.7e-02) ms⁻¹, wall time: 1.025 minutes
[ Info: i: 0340, t: 1.450 hours, Δt: 10.689 seconds, umax = (6.9e-02, 3.6e-02, 3.6e-02) ms⁻¹, wall time: 1.035 minutes
[ Info: i: 0360, t: 1.506 hours, Δt: 10.673 seconds, umax = (7.5e-02, 3.6e-02, 3.6e-02) ms⁻¹, wall time: 1.048 minutes
[ Info: i: 0380, t: 1.565 hours, Δt: 10.571 seconds, umax = (7.3e-02, 3.8e-02, 3.7e-02) ms⁻¹, wall time: 1.054 minutes
[ Info: i: 0400, t: 1.621 hours, Δt: 9.896 seconds, umax = (7.4e-02, 3.9e-02, 4.0e-02) ms⁻¹, wall time: 1.062 minutes
[ Info: i: 0420, t: 1.675 hours, Δt: 10.037 seconds, umax = (7.3e-02, 3.7e-02, 3.7e-02) ms⁻¹, wall time: 1.074 minutes
[ Info: i: 0440, t: 1.730 hours, Δt: 10.036 seconds, umax = (7.5e-02, 4.1e-02, 3.8e-02) ms⁻¹, wall time: 1.080 minutes
[ Info: i: 0460, t: 1.783 hours, Δt: 9.790 seconds, umax = (7.6e-02, 4.3e-02, 4.1e-02) ms⁻¹, wall time: 1.089 minutes
[ Info: i: 0480, t: 1.836 hours, Δt: 9.974 seconds, umax = (7.5e-02, 4.1e-02, 4.3e-02) ms⁻¹, wall time: 1.101 minutes
[ Info: i: 0500, t: 1.891 hours, Δt: 9.912 seconds, umax = (7.7e-02, 4.3e-02, 4.3e-02) ms⁻¹, wall time: 1.107 minutes
[ Info: i: 0520, t: 1.944 hours, Δt: 9.429 seconds, umax = (7.7e-02, 4.4e-02, 4.1e-02) ms⁻¹, wall time: 1.116 minutes
[ Info: i: 0540, t: 1.996 hours, Δt: 9.434 seconds, umax = (7.5e-02, 4.3e-02, 4.2e-02) ms⁻¹, wall time: 1.125 minutes
[ Info: i: 0560, t: 2.048 hours, Δt: 9.499 seconds, umax = (7.7e-02, 4.5e-02, 4.2e-02) ms⁻¹, wall time: 1.134 minutes
[ Info: i: 0580, t: 2.099 hours, Δt: 9.289 seconds, umax = (7.6e-02, 4.6e-02, 4.5e-02) ms⁻¹, wall time: 1.145 minutes
[ Info: i: 0600, t: 2.151 hours, Δt: 9.408 seconds, umax = (7.8e-02, 4.5e-02, 4.3e-02) ms⁻¹, wall time: 1.152 minutes
[ Info: i: 0620, t: 2.200 hours, Δt: 9.221 seconds, umax = (7.8e-02, 4.5e-02, 4.1e-02) ms⁻¹, wall time: 1.161 minutes
[ Info: i: 0640, t: 2.250 hours, Δt: 8.997 seconds, umax = (7.9e-02, 4.7e-02, 3.9e-02) ms⁻¹, wall time: 1.170 minutes
[ Info: i: 0660, t: 2.301 hours, Δt: 9.171 seconds, umax = (7.7e-02, 4.8e-02, 3.8e-02) ms⁻¹, wall time: 1.178 minutes
[ Info: i: 0680, t: 2.351 hours, Δt: 8.656 seconds, umax = (7.9e-02, 5.0e-02, 4.5e-02) ms⁻¹, wall time: 1.189 minutes
[ Info: i: 0700, t: 2.399 hours, Δt: 8.994 seconds, umax = (8.1e-02, 4.9e-02, 4.3e-02) ms⁻¹, wall time: 1.197 minutes
[ Info: i: 0720, t: 2.446 hours, Δt: 8.935 seconds, umax = (8.2e-02, 5.3e-02, 4.3e-02) ms⁻¹, wall time: 1.206 minutes
[ Info: i: 0740, t: 2.495 hours, Δt: 8.340 seconds, umax = (8.3e-02, 5.2e-02, 3.9e-02) ms⁻¹, wall time: 1.215 minutes
[ Info: i: 0760, t: 2.543 hours, Δt: 8.724 seconds, umax = (8.2e-02, 5.2e-02, 4.3e-02) ms⁻¹, wall time: 1.224 minutes
[ Info: i: 0780, t: 2.590 hours, Δt: 8.649 seconds, umax = (8.7e-02, 5.3e-02, 4.1e-02) ms⁻¹, wall time: 1.236 minutes
[ Info: i: 0800, t: 2.638 hours, Δt: 8.674 seconds, umax = (8.3e-02, 5.3e-02, 4.7e-02) ms⁻¹, wall time: 1.243 minutes
[ Info: i: 0820, t: 2.686 hours, Δt: 8.558 seconds, umax = (8.0e-02, 5.5e-02, 4.4e-02) ms⁻¹, wall time: 1.253 minutes
[ Info: i: 0840, t: 2.733 hours, Δt: 8.370 seconds, umax = (8.2e-02, 5.4e-02, 4.2e-02) ms⁻¹, wall time: 1.262 minutes
[ Info: i: 0860, t: 2.777 hours, Δt: 8.615 seconds, umax = (8.2e-02, 5.8e-02, 4.9e-02) ms⁻¹, wall time: 1.271 minutes
[ Info: i: 0880, t: 2.824 hours, Δt: 8.631 seconds, umax = (8.4e-02, 5.9e-02, 4.9e-02) ms⁻¹, wall time: 1.280 minutes
[ Info: i: 0900, t: 2.869 hours, Δt: 8.300 seconds, umax = (8.5e-02, 5.7e-02, 4.4e-02) ms⁻¹, wall time: 1.289 minutes
[ Info: i: 0920, t: 2.916 hours, Δt: 8.131 seconds, umax = (8.7e-02, 6.3e-02, 4.6e-02) ms⁻¹, wall time: 1.299 minutes
[ Info: i: 0940, t: 2.958 hours, Δt: 8.489 seconds, umax = (8.5e-02, 5.6e-02, 4.9e-02) ms⁻¹, wall time: 1.307 minutes
[ Info: i: 0960, t: 3.005 hours, Δt: 8.518 seconds, umax = (8.1e-02, 5.5e-02, 4.6e-02) ms⁻¹, wall time: 1.319 minutes
[ Info: i: 0980, t: 3.052 hours, Δt: 8.408 seconds, umax = (8.2e-02, 5.4e-02, 4.6e-02) ms⁻¹, wall time: 1.325 minutes
[ Info: i: 1000, t: 3.097 hours, Δt: 8.093 seconds, umax = (8.3e-02, 5.8e-02, 5.0e-02) ms⁻¹, wall time: 1.336 minutes
[ Info: i: 1020, t: 3.143 hours, Δt: 8.350 seconds, umax = (8.4e-02, 6.3e-02, 4.9e-02) ms⁻¹, wall time: 1.345 minutes
[ Info: i: 1040, t: 3.188 hours, Δt: 8.166 seconds, umax = (8.4e-02, 5.7e-02, 4.6e-02) ms⁻¹, wall time: 1.356 minutes
[ Info: i: 1060, t: 3.233 hours, Δt: 8.212 seconds, umax = (8.7e-02, 5.6e-02, 4.5e-02) ms⁻¹, wall time: 1.364 minutes
[ Info: i: 1080, t: 3.277 hours, Δt: 7.984 seconds, umax = (8.8e-02, 5.8e-02, 4.8e-02) ms⁻¹, wall time: 1.373 minutes
[ Info: i: 1100, t: 3.322 hours, Δt: 8.039 seconds, umax = (8.9e-02, 5.7e-02, 4.9e-02) ms⁻¹, wall time: 1.383 minutes
[ Info: i: 1120, t: 3.367 hours, Δt: 8.094 seconds, umax = (9.8e-02, 5.6e-02, 4.3e-02) ms⁻¹, wall time: 1.391 minutes
[ Info: i: 1140, t: 3.412 hours, Δt: 8.172 seconds, umax = (8.7e-02, 5.6e-02, 4.5e-02) ms⁻¹, wall time: 1.400 minutes
[ Info: i: 1160, t: 3.455 hours, Δt: 8.060 seconds, umax = (8.6e-02, 5.7e-02, 4.8e-02) ms⁻¹, wall time: 1.409 minutes
[ Info: i: 1180, t: 3.500 hours, Δt: 7.993 seconds, umax = (8.8e-02, 5.7e-02, 4.8e-02) ms⁻¹, wall time: 1.418 minutes
[ Info: i: 1200, t: 3.543 hours, Δt: 8.187 seconds, umax = (8.9e-02, 6.0e-02, 4.5e-02) ms⁻¹, wall time: 1.427 minutes
[ Info: i: 1220, t: 3.588 hours, Δt: 8.106 seconds, umax = (8.4e-02, 5.9e-02, 4.6e-02) ms⁻¹, wall time: 1.440 minutes
[ Info: i: 1240, t: 3.632 hours, Δt: 7.771 seconds, umax = (8.6e-02, 6.2e-02, 4.6e-02) ms⁻¹, wall time: 1.446 minutes
[ Info: i: 1260, t: 3.673 hours, Δt: 7.440 seconds, umax = (8.5e-02, 6.8e-02, 4.7e-02) ms⁻¹, wall time: 1.458 minutes
[ Info: i: 1280, t: 3.715 hours, Δt: 8.289 seconds, umax = (8.4e-02, 7.0e-02, 4.7e-02) ms⁻¹, wall time: 1.464 minutes
[ Info: i: 1300, t: 3.759 hours, Δt: 8.225 seconds, umax = (8.4e-02, 6.3e-02, 4.6e-02) ms⁻¹, wall time: 1.476 minutes
[ Info: i: 1320, t: 3.805 hours, Δt: 7.840 seconds, umax = (8.6e-02, 5.8e-02, 4.4e-02) ms⁻¹, wall time: 1.482 minutes
[ Info: i: 1340, t: 3.847 hours, Δt: 8.258 seconds, umax = (9.1e-02, 6.2e-02, 4.6e-02) ms⁻¹, wall time: 1.493 minutes
[ Info: i: 1360, t: 3.893 hours, Δt: 7.861 seconds, umax = (8.6e-02, 6.1e-02, 4.5e-02) ms⁻¹, wall time: 1.500 minutes
[ Info: i: 1380, t: 3.931 hours, Δt: 7.519 seconds, umax = (8.5e-02, 6.3e-02, 4.4e-02) ms⁻¹, wall time: 1.512 minutes
[ Info: i: 1400, t: 3.974 hours, Δt: 8.112 seconds, umax = (8.8e-02, 6.5e-02, 4.7e-02) ms⁻¹, wall time: 1.520 minutes
[ Info: Simulation is stopping after running for 1.527 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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