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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 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{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, 8.3e-04, 1.5e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (35.536 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (5.472 seconds).
[ Info: i: 0020, t: 12.005 minutes, Δt: 29.041 seconds, umax = (2.9e-02, 1.2e-02, 2.0e-02) ms⁻¹, wall time: 56.638 seconds
[ Info: i: 0040, t: 20.401 minutes, Δt: 20.351 seconds, umax = (4.1e-02, 1.1e-02, 1.8e-02) ms⁻¹, wall time: 57.378 seconds
[ Info: i: 0060, t: 26.828 minutes, Δt: 16.480 seconds, umax = (4.7e-02, 1.5e-02, 1.9e-02) ms⁻¹, wall time: 57.814 seconds
[ Info: i: 0080, t: 32.205 minutes, Δt: 15.855 seconds, umax = (5.3e-02, 1.8e-02, 2.4e-02) ms⁻¹, wall time: 58.318 seconds
[ Info: i: 0100, t: 37.366 minutes, Δt: 15.416 seconds, umax = (5.2e-02, 1.7e-02, 2.2e-02) ms⁻¹, wall time: 58.824 seconds
[ Info: i: 0120, t: 42.293 minutes, Δt: 14.372 seconds, umax = (5.3e-02, 1.8e-02, 2.4e-02) ms⁻¹, wall time: 59.361 seconds
[ Info: i: 0140, t: 46.912 minutes, Δt: 13.822 seconds, umax = (5.3e-02, 2.1e-02, 2.6e-02) ms⁻¹, wall time: 59.966 seconds
[ Info: i: 0160, t: 51.424 minutes, Δt: 14.161 seconds, umax = (5.8e-02, 2.3e-02, 3.0e-02) ms⁻¹, wall time: 1.010 minutes
[ Info: i: 0180, t: 55.689 minutes, Δt: 13.520 seconds, umax = (6.1e-02, 2.3e-02, 3.2e-02) ms⁻¹, wall time: 1.020 minutes
[ Info: i: 0200, t: 1 hour, Δt: 12.807 seconds, umax = (6.4e-02, 2.7e-02, 3.2e-02) ms⁻¹, wall time: 1.028 minutes
[ Info: i: 0220, t: 1.070 hours, Δt: 12.369 seconds, umax = (6.8e-02, 3.1e-02, 3.5e-02) ms⁻¹, wall time: 1.038 minutes
[ Info: i: 0240, t: 1.138 hours, Δt: 11.574 seconds, umax = (6.5e-02, 2.9e-02, 3.4e-02) ms⁻¹, wall time: 1.048 minutes
[ Info: i: 0260, t: 1.199 hours, Δt: 12.060 seconds, umax = (6.6e-02, 3.5e-02, 3.3e-02) ms⁻¹, wall time: 1.059 minutes
[ Info: i: 0280, t: 1.263 hours, Δt: 11.333 seconds, umax = (6.8e-02, 3.2e-02, 3.2e-02) ms⁻¹, wall time: 1.072 minutes
[ Info: i: 0300, t: 1.326 hours, Δt: 11.547 seconds, umax = (6.8e-02, 3.4e-02, 3.5e-02) ms⁻¹, wall time: 1.080 minutes
[ Info: i: 0320, t: 1.387 hours, Δt: 10.903 seconds, umax = (6.8e-02, 3.5e-02, 3.7e-02) ms⁻¹, wall time: 1.091 minutes
[ Info: i: 0340, t: 1.448 hours, Δt: 10.975 seconds, umax = (7.2e-02, 3.3e-02, 3.7e-02) ms⁻¹, wall time: 1.101 minutes
[ Info: i: 0360, t: 1.509 hours, Δt: 10.885 seconds, umax = (7.3e-02, 3.6e-02, 3.7e-02) ms⁻¹, wall time: 1.115 minutes
[ Info: i: 0380, t: 1.568 hours, Δt: 9.946 seconds, umax = (7.5e-02, 3.3e-02, 3.9e-02) ms⁻¹, wall time: 1.122 minutes
[ Info: i: 0400, t: 1.623 hours, Δt: 10.623 seconds, umax = (7.6e-02, 3.7e-02, 4.0e-02) ms⁻¹, wall time: 1.134 minutes
[ Info: i: 0420, t: 1.681 hours, Δt: 10.398 seconds, umax = (7.9e-02, 4.0e-02, 4.0e-02) ms⁻¹, wall time: 1.146 minutes
[ Info: i: 0440, t: 1.739 hours, Δt: 10.191 seconds, umax = (7.5e-02, 3.9e-02, 3.8e-02) ms⁻¹, wall time: 1.155 minutes
[ Info: i: 0460, t: 1.795 hours, Δt: 10.166 seconds, umax = (7.5e-02, 4.0e-02, 4.1e-02) ms⁻¹, wall time: 1.165 minutes
[ Info: i: 0480, t: 1.850 hours, Δt: 9.933 seconds, umax = (7.6e-02, 4.0e-02, 3.8e-02) ms⁻¹, wall time: 1.178 minutes
[ Info: i: 0500, t: 1.905 hours, Δt: 9.597 seconds, umax = (7.8e-02, 4.3e-02, 4.2e-02) ms⁻¹, wall time: 1.187 minutes
[ Info: i: 0520, t: 1.956 hours, Δt: 9.386 seconds, umax = (7.5e-02, 4.4e-02, 4.0e-02) ms⁻¹, wall time: 1.197 minutes
[ Info: i: 0540, t: 2.008 hours, Δt: 9.517 seconds, umax = (7.7e-02, 4.2e-02, 4.2e-02) ms⁻¹, wall time: 1.210 minutes
[ Info: i: 0560, t: 2.061 hours, Δt: 9.207 seconds, umax = (7.9e-02, 4.4e-02, 3.7e-02) ms⁻¹, wall time: 1.218 minutes
[ Info: i: 0580, t: 2.112 hours, Δt: 9.583 seconds, umax = (8.0e-02, 4.6e-02, 3.9e-02) ms⁻¹, wall time: 1.229 minutes
[ Info: i: 0600, t: 2.165 hours, Δt: 9.064 seconds, umax = (8.0e-02, 4.7e-02, 4.4e-02) ms⁻¹, wall time: 1.240 minutes
[ Info: i: 0620, t: 2.215 hours, Δt: 9.275 seconds, umax = (7.9e-02, 4.9e-02, 4.5e-02) ms⁻¹, wall time: 1.250 minutes
[ Info: i: 0640, t: 2.263 hours, Δt: 9.260 seconds, umax = (7.9e-02, 4.7e-02, 4.0e-02) ms⁻¹, wall time: 1.260 minutes
[ Info: i: 0660, t: 2.314 hours, Δt: 9.283 seconds, umax = (7.9e-02, 4.5e-02, 4.4e-02) ms⁻¹, wall time: 1.267 minutes
[ Info: i: 0680, t: 2.364 hours, Δt: 9.113 seconds, umax = (7.9e-02, 4.6e-02, 4.5e-02) ms⁻¹, wall time: 1.277 minutes
[ Info: i: 0700, t: 2.414 hours, Δt: 9.055 seconds, umax = (8.1e-02, 5.0e-02, 4.7e-02) ms⁻¹, wall time: 1.287 minutes
[ Info: i: 0720, t: 2.461 hours, Δt: 8.605 seconds, umax = (8.3e-02, 5.0e-02, 4.2e-02) ms⁻¹, wall time: 1.296 minutes
[ Info: i: 0740, t: 2.507 hours, Δt: 8.836 seconds, umax = (8.2e-02, 4.9e-02, 4.3e-02) ms⁻¹, wall time: 1.308 minutes
[ Info: i: 0760, t: 2.556 hours, Δt: 8.674 seconds, umax = (8.1e-02, 4.9e-02, 4.2e-02) ms⁻¹, wall time: 1.315 minutes
[ Info: i: 0780, t: 2.602 hours, Δt: 8.382 seconds, umax = (8.2e-02, 5.2e-02, 4.6e-02) ms⁻¹, wall time: 1.324 minutes
[ Info: i: 0800, t: 2.648 hours, Δt: 8.118 seconds, umax = (8.2e-02, 5.7e-02, 4.5e-02) ms⁻¹, wall time: 1.333 minutes
[ Info: i: 0820, t: 2.692 hours, Δt: 8.345 seconds, umax = (8.3e-02, 5.7e-02, 4.7e-02) ms⁻¹, wall time: 1.343 minutes
[ Info: i: 0840, t: 2.738 hours, Δt: 8.385 seconds, umax = (8.0e-02, 5.8e-02, 4.5e-02) ms⁻¹, wall time: 1.352 minutes
[ Info: i: 0860, t: 2.783 hours, Δt: 8.678 seconds, umax = (8.1e-02, 5.8e-02, 4.3e-02) ms⁻¹, wall time: 1.362 minutes
[ Info: i: 0880, t: 2.832 hours, Δt: 8.524 seconds, umax = (8.5e-02, 5.5e-02, 4.7e-02) ms⁻¹, wall time: 1.371 minutes
[ Info: i: 0900, t: 2.878 hours, Δt: 8.330 seconds, umax = (8.6e-02, 5.7e-02, 4.6e-02) ms⁻¹, wall time: 1.379 minutes
[ Info: i: 0920, t: 2.924 hours, Δt: 8.278 seconds, umax = (8.8e-02, 6.1e-02, 4.7e-02) ms⁻¹, wall time: 1.391 minutes
[ Info: i: 0940, t: 2.971 hours, Δt: 8.623 seconds, umax = (8.3e-02, 5.3e-02, 4.4e-02) ms⁻¹, wall time: 1.397 minutes
[ Info: i: 0960, t: 3.017 hours, Δt: 8.368 seconds, umax = (8.2e-02, 5.2e-02, 4.6e-02) ms⁻¹, wall time: 1.408 minutes
[ Info: i: 0980, t: 3.064 hours, Δt: 8.435 seconds, umax = (8.3e-02, 5.5e-02, 4.7e-02) ms⁻¹, wall time: 1.416 minutes
[ Info: i: 1000, t: 3.109 hours, Δt: 8.589 seconds, umax = (8.2e-02, 5.6e-02, 4.6e-02) ms⁻¹, wall time: 1.426 minutes
[ Info: i: 1020, t: 3.157 hours, Δt: 8.383 seconds, umax = (8.3e-02, 5.4e-02, 4.2e-02) ms⁻¹, wall time: 1.436 minutes
[ Info: i: 1040, t: 3.201 hours, Δt: 8.080 seconds, umax = (8.3e-02, 5.6e-02, 4.3e-02) ms⁻¹, wall time: 1.445 minutes
[ Info: i: 1060, t: 3.247 hours, Δt: 8.694 seconds, umax = (8.4e-02, 5.7e-02, 4.1e-02) ms⁻¹, wall time: 1.453 minutes
[ Info: i: 1080, t: 3.293 hours, Δt: 8.383 seconds, umax = (8.3e-02, 5.7e-02, 4.0e-02) ms⁻¹, wall time: 1.463 minutes
[ Info: i: 1100, t: 3.338 hours, Δt: 7.990 seconds, umax = (8.3e-02, 5.7e-02, 4.4e-02) ms⁻¹, wall time: 1.474 minutes
[ Info: i: 1120, t: 3.382 hours, Δt: 7.877 seconds, umax = (8.3e-02, 5.5e-02, 4.1e-02) ms⁻¹, wall time: 1.480 minutes
[ Info: i: 1140, t: 3.425 hours, Δt: 7.758 seconds, umax = (8.7e-02, 6.0e-02, 4.3e-02) ms⁻¹, wall time: 1.493 minutes
[ Info: i: 1160, t: 3.469 hours, Δt: 7.952 seconds, umax = (8.6e-02, 6.2e-02, 4.4e-02) ms⁻¹, wall time: 1.500 minutes
[ Info: i: 1180, t: 3.514 hours, Δt: 8.449 seconds, umax = (8.4e-02, 6.3e-02, 4.3e-02) ms⁻¹, wall time: 1.512 minutes
[ Info: i: 1200, t: 3.561 hours, Δt: 8.005 seconds, umax = (8.7e-02, 5.8e-02, 5.1e-02) ms⁻¹, wall time: 1.520 minutes
[ Info: i: 1220, t: 3.604 hours, Δt: 7.872 seconds, umax = (8.6e-02, 6.3e-02, 4.9e-02) ms⁻¹, wall time: 1.530 minutes
[ Info: i: 1240, t: 3.647 hours, Δt: 7.937 seconds, umax = (8.8e-02, 6.8e-02, 4.8e-02) ms⁻¹, wall time: 1.539 minutes
[ Info: i: 1260, t: 3.690 hours, Δt: 7.844 seconds, umax = (8.7e-02, 7.2e-02, 4.6e-02) ms⁻¹, wall time: 1.548 minutes
[ Info: i: 1280, t: 3.734 hours, Δt: 7.719 seconds, umax = (9.0e-02, 7.0e-02, 5.0e-02) ms⁻¹, wall time: 1.557 minutes
[ Info: i: 1300, t: 3.777 hours, Δt: 7.665 seconds, umax = (9.3e-02, 6.0e-02, 5.3e-02) ms⁻¹, wall time: 1.566 minutes
[ Info: i: 1320, t: 3.818 hours, Δt: 7.501 seconds, umax = (9.2e-02, 6.1e-02, 4.4e-02) ms⁻¹, wall time: 1.576 minutes
[ Info: i: 1340, t: 3.858 hours, Δt: 7.555 seconds, umax = (9.4e-02, 6.4e-02, 4.5e-02) ms⁻¹, wall time: 1.585 minutes
[ Info: i: 1360, t: 3.901 hours, Δt: 8.004 seconds, umax = (9.0e-02, 6.9e-02, 4.9e-02) ms⁻¹, wall time: 1.594 minutes
[ Info: i: 1380, t: 3.942 hours, Δt: 7.978 seconds, umax = (8.8e-02, 6.7e-02, 4.6e-02) ms⁻¹, wall time: 1.604 minutes
[ Info: i: 1400, t: 3.986 hours, Δt: 7.801 seconds, umax = (8.4e-02, 6.4e-02, 4.8e-02) ms⁻¹, wall time: 1.613 minutes
[ Info: Simulation is stopping after running for 1.616 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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