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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 = (1.6e-03, 9.6e-04, 1.3e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (44.288 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (11.104 seconds).
[ Info: i: 0020, t: 11.511 minutes, Δt: 30.146 seconds, umax = (2.7e-02, 1.2e-02, 2.4e-02) ms⁻¹, wall time: 1.245 minutes
[ Info: i: 0040, t: 20.407 minutes, Δt: 19.979 seconds, umax = (4.1e-02, 1.1e-02, 1.9e-02) ms⁻¹, wall time: 1.259 minutes
[ Info: i: 0060, t: 26.517 minutes, Δt: 17.679 seconds, umax = (4.9e-02, 1.4e-02, 2.0e-02) ms⁻¹, wall time: 1.270 minutes
[ Info: i: 0080, t: 32.149 minutes, Δt: 16.018 seconds, umax = (5.1e-02, 1.7e-02, 2.3e-02) ms⁻¹, wall time: 1.278 minutes
[ Info: i: 0100, t: 37.341 minutes, Δt: 15.404 seconds, umax = (5.1e-02, 1.9e-02, 2.6e-02) ms⁻¹, wall time: 1.287 minutes
[ Info: i: 0120, t: 42.077 minutes, Δt: 15.261 seconds, umax = (5.2e-02, 1.9e-02, 2.5e-02) ms⁻¹, wall time: 1.295 minutes
[ Info: i: 0140, t: 46.917 minutes, Δt: 14.000 seconds, umax = (5.4e-02, 2.1e-02, 2.6e-02) ms⁻¹, wall time: 1.305 minutes
[ Info: i: 0160, t: 51.433 minutes, Δt: 13.901 seconds, umax = (5.7e-02, 2.3e-02, 2.8e-02) ms⁻¹, wall time: 1.316 minutes
[ Info: i: 0180, t: 55.910 minutes, Δt: 13.067 seconds, umax = (5.9e-02, 2.5e-02, 2.8e-02) ms⁻¹, wall time: 1.326 minutes
[ Info: i: 0200, t: 1 hour, Δt: 12.622 seconds, umax = (6.2e-02, 2.4e-02, 3.0e-02) ms⁻¹, wall time: 1.333 minutes
[ Info: i: 0220, t: 1.071 hours, Δt: 12.600 seconds, umax = (6.3e-02, 2.7e-02, 3.1e-02) ms⁻¹, wall time: 1.341 minutes
[ Info: i: 0240, t: 1.138 hours, Δt: 11.719 seconds, umax = (6.6e-02, 2.8e-02, 3.1e-02) ms⁻¹, wall time: 1.352 minutes
[ Info: i: 0260, t: 1.203 hours, Δt: 12.090 seconds, umax = (6.6e-02, 2.9e-02, 3.8e-02) ms⁻¹, wall time: 1.363 minutes
[ Info: i: 0280, t: 1.266 hours, Δt: 11.607 seconds, umax = (6.8e-02, 3.0e-02, 3.4e-02) ms⁻¹, wall time: 1.374 minutes
[ Info: i: 0300, t: 1.329 hours, Δt: 11.214 seconds, umax = (7.1e-02, 2.9e-02, 3.3e-02) ms⁻¹, wall time: 1.383 minutes
[ Info: i: 0320, t: 1.389 hours, Δt: 10.783 seconds, umax = (7.2e-02, 3.3e-02, 3.7e-02) ms⁻¹, wall time: 1.393 minutes
[ Info: i: 0340, t: 1.447 hours, Δt: 10.390 seconds, umax = (7.0e-02, 3.8e-02, 3.8e-02) ms⁻¹, wall time: 1.406 minutes
[ Info: i: 0360, t: 1.503 hours, Δt: 10.446 seconds, umax = (7.0e-02, 3.8e-02, 3.8e-02) ms⁻¹, wall time: 1.420 minutes
[ Info: i: 0380, t: 1.560 hours, Δt: 10.131 seconds, umax = (7.2e-02, 4.0e-02, 3.8e-02) ms⁻¹, wall time: 1.426 minutes
[ Info: i: 0400, t: 1.615 hours, Δt: 10.135 seconds, umax = (7.4e-02, 3.7e-02, 4.4e-02) ms⁻¹, wall time: 1.438 minutes
[ Info: i: 0420, t: 1.670 hours, Δt: 10.264 seconds, umax = (7.4e-02, 4.0e-02, 4.1e-02) ms⁻¹, wall time: 1.452 minutes
[ Info: i: 0440, t: 1.726 hours, Δt: 10.241 seconds, umax = (7.7e-02, 3.8e-02, 3.9e-02) ms⁻¹, wall time: 1.461 minutes
[ Info: i: 0460, t: 1.782 hours, Δt: 10.359 seconds, umax = (7.7e-02, 3.7e-02, 3.9e-02) ms⁻¹, wall time: 1.472 minutes
[ Info: i: 0480, t: 1.836 hours, Δt: 9.930 seconds, umax = (7.5e-02, 4.3e-02, 4.3e-02) ms⁻¹, wall time: 1.485 minutes
[ Info: i: 0500, t: 1.891 hours, Δt: 9.824 seconds, umax = (7.7e-02, 4.3e-02, 4.2e-02) ms⁻¹, wall time: 1.494 minutes
[ Info: i: 0520, t: 1.944 hours, Δt: 9.733 seconds, umax = (7.7e-02, 4.2e-02, 4.5e-02) ms⁻¹, wall time: 1.503 minutes
[ Info: i: 0540, t: 1.997 hours, Δt: 9.340 seconds, umax = (7.8e-02, 4.5e-02, 4.4e-02) ms⁻¹, wall time: 1.514 minutes
[ Info: i: 0560, t: 2.046 hours, Δt: 8.771 seconds, umax = (8.2e-02, 4.7e-02, 4.1e-02) ms⁻¹, wall time: 1.522 minutes
[ Info: i: 0580, t: 2.093 hours, Δt: 8.627 seconds, umax = (8.1e-02, 4.8e-02, 4.1e-02) ms⁻¹, wall time: 1.535 minutes
[ Info: i: 0600, t: 2.142 hours, Δt: 9.427 seconds, umax = (8.0e-02, 4.5e-02, 4.2e-02) ms⁻¹, wall time: 1.545 minutes
[ Info: i: 0620, t: 2.192 hours, Δt: 8.657 seconds, umax = (8.2e-02, 5.0e-02, 4.8e-02) ms⁻¹, wall time: 1.560 minutes
[ Info: i: 0640, t: 2.240 hours, Δt: 9.189 seconds, umax = (8.1e-02, 4.9e-02, 4.3e-02) ms⁻¹, wall time: 1.569 minutes
[ Info: i: 0660, t: 2.291 hours, Δt: 8.846 seconds, umax = (8.3e-02, 5.1e-02, 4.6e-02) ms⁻¹, wall time: 1.587 minutes
[ Info: i: 0680, t: 2.338 hours, Δt: 8.930 seconds, umax = (8.1e-02, 5.1e-02, 4.3e-02) ms⁻¹, wall time: 1.598 minutes
[ Info: i: 0700, t: 2.388 hours, Δt: 8.399 seconds, umax = (8.1e-02, 5.5e-02, 4.0e-02) ms⁻¹, wall time: 1.605 minutes
[ Info: i: 0720, t: 2.433 hours, Δt: 8.701 seconds, umax = (8.4e-02, 5.0e-02, 4.4e-02) ms⁻¹, wall time: 1.622 minutes
[ Info: i: 0740, t: 2.482 hours, Δt: 8.803 seconds, umax = (8.5e-02, 4.9e-02, 4.7e-02) ms⁻¹, wall time: 1.630 minutes
[ Info: i: 0760, t: 2.529 hours, Δt: 8.772 seconds, umax = (8.2e-02, 4.9e-02, 4.7e-02) ms⁻¹, wall time: 1.639 minutes
[ Info: i: 0780, t: 2.578 hours, Δt: 9.010 seconds, umax = (8.1e-02, 5.4e-02, 4.6e-02) ms⁻¹, wall time: 1.648 minutes
[ Info: i: 0800, t: 2.625 hours, Δt: 8.918 seconds, umax = (8.2e-02, 5.1e-02, 4.6e-02) ms⁻¹, wall time: 1.657 minutes
[ Info: i: 0820, t: 2.674 hours, Δt: 8.842 seconds, umax = (8.2e-02, 4.9e-02, 5.0e-02) ms⁻¹, wall time: 1.670 minutes
[ Info: i: 0840, t: 2.723 hours, Δt: 8.567 seconds, umax = (7.9e-02, 5.1e-02, 4.4e-02) ms⁻¹, wall time: 1.677 minutes
[ Info: i: 0860, t: 2.768 hours, Δt: 8.527 seconds, umax = (8.0e-02, 5.2e-02, 5.0e-02) ms⁻¹, wall time: 1.689 minutes
[ Info: i: 0880, t: 2.816 hours, Δt: 8.648 seconds, umax = (8.2e-02, 5.3e-02, 4.8e-02) ms⁻¹, wall time: 1.699 minutes
[ Info: i: 0900, t: 2.863 hours, Δt: 8.789 seconds, umax = (8.5e-02, 5.1e-02, 5.2e-02) ms⁻¹, wall time: 1.710 minutes
[ Info: i: 0920, t: 2.911 hours, Δt: 8.326 seconds, umax = (8.4e-02, 4.9e-02, 4.6e-02) ms⁻¹, wall time: 1.722 minutes
[ Info: i: 0940, t: 2.955 hours, Δt: 8.394 seconds, umax = (8.6e-02, 5.2e-02, 4.3e-02) ms⁻¹, wall time: 1.732 minutes
[ Info: i: 0960, t: 3 hours, Δt: 8.428 seconds, umax = (8.4e-02, 5.8e-02, 4.3e-02) ms⁻¹, wall time: 1.740 minutes
[ Info: i: 0980, t: 3.046 hours, Δt: 7.936 seconds, umax = (8.4e-02, 5.2e-02, 4.8e-02) ms⁻¹, wall time: 1.749 minutes
[ Info: i: 1000, t: 3.090 hours, Δt: 8.554 seconds, umax = (8.4e-02, 5.5e-02, 4.8e-02) ms⁻¹, wall time: 1.760 minutes
[ Info: i: 1020, t: 3.137 hours, Δt: 8.186 seconds, umax = (8.5e-02, 5.7e-02, 4.8e-02) ms⁻¹, wall time: 1.767 minutes
[ Info: i: 1040, t: 3.184 hours, Δt: 8.591 seconds, umax = (8.9e-02, 5.4e-02, 4.9e-02) ms⁻¹, wall time: 1.777 minutes
[ Info: i: 1060, t: 3.230 hours, Δt: 7.967 seconds, umax = (8.7e-02, 6.2e-02, 5.0e-02) ms⁻¹, wall time: 1.785 minutes
[ Info: i: 1080, t: 3.275 hours, Δt: 8.238 seconds, umax = (8.5e-02, 5.8e-02, 4.9e-02) ms⁻¹, wall time: 1.795 minutes
[ Info: i: 1100, t: 3.320 hours, Δt: 7.724 seconds, umax = (8.6e-02, 5.9e-02, 5.2e-02) ms⁻¹, wall time: 1.803 minutes
[ Info: i: 1120, t: 3.361 hours, Δt: 8.122 seconds, umax = (8.4e-02, 5.9e-02, 5.3e-02) ms⁻¹, wall time: 1.814 minutes
[ Info: i: 1140, t: 3.406 hours, Δt: 7.764 seconds, umax = (8.9e-02, 6.0e-02, 4.8e-02) ms⁻¹, wall time: 1.822 minutes
[ Info: i: 1160, t: 3.448 hours, Δt: 7.708 seconds, umax = (8.6e-02, 6.1e-02, 4.8e-02) ms⁻¹, wall time: 1.832 minutes
[ Info: i: 1180, t: 3.491 hours, Δt: 8.127 seconds, umax = (8.5e-02, 6.0e-02, 5.4e-02) ms⁻¹, wall time: 1.841 minutes
[ Info: i: 1200, t: 3.536 hours, Δt: 7.926 seconds, umax = (8.9e-02, 6.6e-02, 4.9e-02) ms⁻¹, wall time: 1.850 minutes
[ Info: i: 1220, t: 3.580 hours, Δt: 8.009 seconds, umax = (8.8e-02, 6.5e-02, 5.3e-02) ms⁻¹, wall time: 1.860 minutes
[ Info: i: 1240, t: 3.623 hours, Δt: 8.097 seconds, umax = (8.6e-02, 6.0e-02, 4.8e-02) ms⁻¹, wall time: 1.871 minutes
[ Info: i: 1260, t: 3.667 hours, Δt: 8.199 seconds, umax = (8.8e-02, 5.8e-02, 4.7e-02) ms⁻¹, wall time: 1.880 minutes
[ Info: i: 1280, t: 3.711 hours, Δt: 7.889 seconds, umax = (8.3e-02, 6.2e-02, 4.9e-02) ms⁻¹, wall time: 1.890 minutes
[ Info: i: 1300, t: 3.754 hours, Δt: 8.089 seconds, umax = (8.4e-02, 6.1e-02, 4.9e-02) ms⁻¹, wall time: 1.904 minutes
[ Info: i: 1320, t: 3.799 hours, Δt: 8.120 seconds, umax = (8.6e-02, 6.2e-02, 4.3e-02) ms⁻¹, wall time: 1.910 minutes
[ Info: i: 1340, t: 3.842 hours, Δt: 7.958 seconds, umax = (8.3e-02, 6.5e-02, 5.3e-02) ms⁻¹, wall time: 1.922 minutes
[ Info: i: 1360, t: 3.887 hours, Δt: 8.199 seconds, umax = (8.6e-02, 6.5e-02, 4.7e-02) ms⁻¹, wall time: 1.928 minutes
[ Info: i: 1380, t: 3.930 hours, Δt: 7.957 seconds, umax = (8.8e-02, 6.6e-02, 4.8e-02) ms⁻¹, wall time: 1.940 minutes
[ Info: i: 1400, t: 3.974 hours, Δt: 7.902 seconds, umax = (8.6e-02, 6.1e-02, 4.6e-02) ms⁻¹, wall time: 1.948 minutes
[ Info: Simulation is stopping after running for 1.955 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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