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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: 45.1 KiB

An "averages" writer

We also set up output of time- and horizontally-averaged velocity field and momentum fluxes.

u, v, w = model.velocities
b = model.tracers.b

 U = Average(u, dims=(1, 2))
 V = Average(v, dims=(1, 2))
 B = Average(b, dims=(1, 2))
wu = Average(w * u, dims=(1, 2))
wv = Average(w * v, dims=(1, 2))

simulation.output_writers[:averages] =
    JLD2Writer(model, (; U, V, B, wu, wv),
               schedule = AveragedTimeInterval(output_interval, window=2minutes),
               filename = "langmuir_turbulence_averages.jld2",
               overwrite_existing = true)
JLD2Writer scheduled on TimeInterval(5 minutes):
├── filepath: langmuir_turbulence_averages.jld2
├── 5 outputs: (U, V, B, wu, wv) averaged on AveragedTimeInterval(window=2 minutes, stride=1, interval=5 minutes)
├── array_type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 44.3 KiB

Running the simulation

This part is easy,

run!(simulation)
[ Info: Initializing simulation...
[ Info: i: 0000, t: 0 seconds, Δt: 49.500 seconds, umax = (1.5e-03, 8.4e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (32.920 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (7.336 seconds).
[ Info: i: 0020, t: 12.001 minutes, Δt: 28.825 seconds, umax = (2.9e-02, 1.3e-02, 2.0e-02) ms⁻¹, wall time: 53.787 seconds
[ Info: i: 0040, t: 20.409 minutes, Δt: 19.411 seconds, umax = (4.1e-02, 1.1e-02, 1.9e-02) ms⁻¹, wall time: 54.499 seconds
[ Info: i: 0060, t: 26.556 minutes, Δt: 17.179 seconds, umax = (4.9e-02, 1.5e-02, 2.0e-02) ms⁻¹, wall time: 55.021 seconds
[ Info: i: 0080, t: 31.864 minutes, Δt: 16.127 seconds, umax = (4.9e-02, 1.6e-02, 2.3e-02) ms⁻¹, wall time: 55.508 seconds
[ Info: i: 0100, t: 37.124 minutes, Δt: 14.952 seconds, umax = (5.4e-02, 1.9e-02, 2.5e-02) ms⁻¹, wall time: 56.034 seconds
[ Info: i: 0120, t: 42.053 minutes, Δt: 14.884 seconds, umax = (5.4e-02, 2.1e-02, 2.4e-02) ms⁻¹, wall time: 56.601 seconds
[ Info: i: 0140, t: 46.934 minutes, Δt: 13.252 seconds, umax = (5.6e-02, 2.1e-02, 2.6e-02) ms⁻¹, wall time: 57.174 seconds
[ Info: i: 0160, t: 51.398 minutes, Δt: 14.055 seconds, umax = (5.8e-02, 2.2e-02, 2.8e-02) ms⁻¹, wall time: 57.837 seconds
[ Info: i: 0180, t: 55.930 minutes, Δt: 13.649 seconds, umax = (6.0e-02, 2.3e-02, 3.1e-02) ms⁻¹, wall time: 58.422 seconds
[ Info: i: 0200, t: 1.004 hours, Δt: 12.818 seconds, umax = (6.3e-02, 2.4e-02, 2.9e-02) ms⁻¹, wall time: 58.991 seconds
[ Info: i: 0220, t: 1.074 hours, Δt: 12.294 seconds, umax = (6.6e-02, 2.6e-02, 3.3e-02) ms⁻¹, wall time: 59.365 seconds
[ Info: i: 0240, t: 1.141 hours, Δt: 11.991 seconds, umax = (6.5e-02, 3.0e-02, 3.3e-02) ms⁻¹, wall time: 59.862 seconds
[ Info: i: 0260, t: 1.206 hours, Δt: 11.727 seconds, umax = (6.6e-02, 3.0e-02, 3.3e-02) ms⁻¹, wall time: 1.007 minutes
[ Info: i: 0280, t: 1.269 hours, Δt: 11.318 seconds, umax = (6.7e-02, 3.3e-02, 3.4e-02) ms⁻¹, wall time: 1.016 minutes
[ Info: i: 0300, t: 1.332 hours, Δt: 11.049 seconds, umax = (6.8e-02, 3.4e-02, 3.7e-02) ms⁻¹, wall time: 1.024 minutes
[ Info: i: 0320, t: 1.390 hours, Δt: 10.799 seconds, umax = (7.0e-02, 3.5e-02, 3.6e-02) ms⁻¹, wall time: 1.032 minutes
[ Info: i: 0340, t: 1.451 hours, Δt: 10.968 seconds, umax = (7.1e-02, 3.5e-02, 4.0e-02) ms⁻¹, wall time: 1.041 minutes
[ Info: i: 0360, t: 1.509 hours, Δt: 10.225 seconds, umax = (7.2e-02, 3.6e-02, 3.7e-02) ms⁻¹, wall time: 1.053 minutes
[ Info: i: 0380, t: 1.566 hours, Δt: 10.237 seconds, umax = (7.1e-02, 3.5e-02, 3.6e-02) ms⁻¹, wall time: 1.060 minutes
[ Info: i: 0400, t: 1.621 hours, Δt: 10.657 seconds, umax = (7.4e-02, 3.9e-02, 3.7e-02) ms⁻¹, wall time: 1.070 minutes
[ Info: i: 0420, t: 1.678 hours, Δt: 10.045 seconds, umax = (7.6e-02, 3.7e-02, 3.9e-02) ms⁻¹, wall time: 1.080 minutes
[ Info: i: 0440, t: 1.734 hours, Δt: 9.928 seconds, umax = (7.5e-02, 3.7e-02, 4.4e-02) ms⁻¹, wall time: 1.087 minutes
[ Info: i: 0460, t: 1.789 hours, Δt: 10.162 seconds, umax = (7.5e-02, 4.2e-02, 4.3e-02) ms⁻¹, wall time: 1.096 minutes
[ Info: i: 0480, t: 1.842 hours, Δt: 9.783 seconds, umax = (7.8e-02, 4.4e-02, 4.1e-02) ms⁻¹, wall time: 1.107 minutes
[ Info: i: 0500, t: 1.896 hours, Δt: 9.556 seconds, umax = (8.0e-02, 4.2e-02, 4.3e-02) ms⁻¹, wall time: 1.114 minutes
[ Info: i: 0520, t: 1.949 hours, Δt: 9.375 seconds, umax = (7.9e-02, 4.5e-02, 4.3e-02) ms⁻¹, wall time: 1.124 minutes
[ Info: i: 0540, t: 2 hours, Δt: 9.512 seconds, umax = (7.5e-02, 4.7e-02, 4.4e-02) ms⁻¹, wall time: 1.134 minutes
[ Info: i: 0560, t: 2.052 hours, Δt: 9.192 seconds, umax = (7.7e-02, 4.5e-02, 4.2e-02) ms⁻¹, wall time: 1.142 minutes
[ Info: i: 0580, t: 2.101 hours, Δt: 9.244 seconds, umax = (7.8e-02, 4.5e-02, 4.6e-02) ms⁻¹, wall time: 1.152 minutes
[ Info: i: 0600, t: 2.152 hours, Δt: 9.407 seconds, umax = (8.2e-02, 4.5e-02, 4.6e-02) ms⁻¹, wall time: 1.160 minutes
[ Info: i: 0620, t: 2.203 hours, Δt: 8.876 seconds, umax = (8.0e-02, 4.7e-02, 4.6e-02) ms⁻¹, wall time: 1.169 minutes
[ Info: i: 0640, t: 2.250 hours, Δt: 8.551 seconds, umax = (7.9e-02, 4.4e-02, 4.5e-02) ms⁻¹, wall time: 1.179 minutes
[ Info: i: 0660, t: 2.297 hours, Δt: 8.695 seconds, umax = (8.1e-02, 4.6e-02, 4.3e-02) ms⁻¹, wall time: 1.189 minutes
[ Info: i: 0680, t: 2.346 hours, Δt: 8.984 seconds, umax = (8.1e-02, 4.9e-02, 4.4e-02) ms⁻¹, wall time: 1.199 minutes
[ Info: i: 0700, t: 2.395 hours, Δt: 8.660 seconds, umax = (7.9e-02, 5.4e-02, 4.2e-02) ms⁻¹, wall time: 1.207 minutes
[ Info: i: 0720, t: 2.440 hours, Δt: 8.782 seconds, umax = (8.2e-02, 6.0e-02, 4.3e-02) ms⁻¹, wall time: 1.216 minutes
[ Info: i: 0740, t: 2.489 hours, Δt: 8.866 seconds, umax = (8.2e-02, 5.9e-02, 4.4e-02) ms⁻¹, wall time: 1.225 minutes
[ Info: i: 0760, t: 2.537 hours, Δt: 9.045 seconds, umax = (8.2e-02, 5.1e-02, 4.5e-02) ms⁻¹, wall time: 1.234 minutes
[ Info: i: 0780, t: 2.586 hours, Δt: 8.727 seconds, umax = (8.6e-02, 5.1e-02, 4.0e-02) ms⁻¹, wall time: 1.246 minutes
[ Info: i: 0800, t: 2.634 hours, Δt: 8.594 seconds, umax = (8.5e-02, 5.0e-02, 5.0e-02) ms⁻¹, wall time: 1.251 minutes
[ Info: i: 0820, t: 2.681 hours, Δt: 8.431 seconds, umax = (8.4e-02, 5.0e-02, 4.8e-02) ms⁻¹, wall time: 1.262 minutes
[ Info: i: 0840, t: 2.728 hours, Δt: 8.218 seconds, umax = (8.4e-02, 5.5e-02, 5.0e-02) ms⁻¹, wall time: 1.271 minutes
[ Info: i: 0860, t: 2.773 hours, Δt: 7.993 seconds, umax = (8.3e-02, 6.2e-02, 4.5e-02) ms⁻¹, wall time: 1.281 minutes
[ Info: i: 0880, t: 2.817 hours, Δt: 7.877 seconds, umax = (8.4e-02, 5.6e-02, 4.4e-02) ms⁻¹, wall time: 1.291 minutes
[ Info: i: 0900, t: 2.859 hours, Δt: 8.096 seconds, umax = (8.4e-02, 5.5e-02, 5.1e-02) ms⁻¹, wall time: 1.299 minutes
[ Info: i: 0920, t: 2.904 hours, Δt: 8.229 seconds, umax = (8.2e-02, 6.3e-02, 4.8e-02) ms⁻¹, wall time: 1.308 minutes
[ Info: i: 0940, t: 2.948 hours, Δt: 8.170 seconds, umax = (8.3e-02, 5.9e-02, 4.8e-02) ms⁻¹, wall time: 1.317 minutes
[ Info: i: 0960, t: 2.994 hours, Δt: 8.620 seconds, umax = (8.6e-02, 5.4e-02, 4.6e-02) ms⁻¹, wall time: 1.326 minutes
[ Info: i: 0980, t: 3.041 hours, Δt: 8.471 seconds, umax = (8.6e-02, 5.9e-02, 4.6e-02) ms⁻¹, wall time: 1.335 minutes
[ Info: i: 1000, t: 3.086 hours, Δt: 8.161 seconds, umax = (8.7e-02, 5.9e-02, 4.3e-02) ms⁻¹, wall time: 1.347 minutes
[ Info: i: 1020, t: 3.131 hours, Δt: 8.151 seconds, umax = (8.7e-02, 6.3e-02, 5.8e-02) ms⁻¹, wall time: 1.353 minutes
[ Info: i: 1040, t: 3.174 hours, Δt: 8.053 seconds, umax = (8.7e-02, 6.1e-02, 5.4e-02) ms⁻¹, wall time: 1.364 minutes
[ Info: i: 1060, t: 3.219 hours, Δt: 8.406 seconds, umax = (8.5e-02, 6.3e-02, 5.5e-02) ms⁻¹, wall time: 1.371 minutes
[ Info: i: 1080, t: 3.264 hours, Δt: 8.456 seconds, umax = (8.5e-02, 6.3e-02, 4.9e-02) ms⁻¹, wall time: 1.381 minutes
[ Info: i: 1100, t: 3.311 hours, Δt: 8.232 seconds, umax = (8.5e-02, 6.3e-02, 4.5e-02) ms⁻¹, wall time: 1.389 minutes
[ Info: i: 1120, t: 3.354 hours, Δt: 8.453 seconds, umax = (8.4e-02, 6.0e-02, 5.3e-02) ms⁻¹, wall time: 1.398 minutes
[ Info: i: 1140, t: 3.401 hours, Δt: 8.474 seconds, umax = (8.5e-02, 6.2e-02, 4.5e-02) ms⁻¹, wall time: 1.407 minutes
[ Info: i: 1160, t: 3.447 hours, Δt: 8.441 seconds, umax = (8.6e-02, 6.4e-02, 4.7e-02) ms⁻¹, wall time: 1.416 minutes
[ Info: i: 1180, t: 3.494 hours, Δt: 8.436 seconds, umax = (8.7e-02, 6.7e-02, 4.9e-02) ms⁻¹, wall time: 1.425 minutes
[ Info: i: 1200, t: 3.539 hours, Δt: 8.031 seconds, umax = (8.5e-02, 6.0e-02, 5.4e-02) ms⁻¹, wall time: 1.434 minutes
[ Info: i: 1220, t: 3.583 hours, Δt: 8.166 seconds, umax = (8.9e-02, 6.0e-02, 5.6e-02) ms⁻¹, wall time: 1.443 minutes
[ Info: i: 1240, t: 3.624 hours, Δt: 7.992 seconds, umax = (8.6e-02, 6.4e-02, 5.4e-02) ms⁻¹, wall time: 1.452 minutes
[ Info: i: 1260, t: 3.669 hours, Δt: 8.181 seconds, umax = (8.6e-02, 6.0e-02, 4.8e-02) ms⁻¹, wall time: 1.464 minutes
[ Info: i: 1280, t: 3.714 hours, Δt: 8.284 seconds, umax = (8.6e-02, 6.2e-02, 5.0e-02) ms⁻¹, wall time: 1.469 minutes
[ Info: i: 1300, t: 3.759 hours, Δt: 8.142 seconds, umax = (8.5e-02, 6.2e-02, 4.9e-02) ms⁻¹, wall time: 1.481 minutes
[ Info: i: 1320, t: 3.804 hours, Δt: 8.149 seconds, umax = (8.7e-02, 6.1e-02, 5.5e-02) ms⁻¹, wall time: 1.490 minutes
[ Info: i: 1340, t: 3.849 hours, Δt: 8.297 seconds, umax = (8.7e-02, 5.8e-02, 5.8e-02) ms⁻¹, wall time: 1.501 minutes
[ Info: i: 1360, t: 3.895 hours, Δt: 8.091 seconds, umax = (8.2e-02, 6.4e-02, 6.0e-02) ms⁻¹, wall time: 1.510 minutes
[ Info: i: 1380, t: 3.937 hours, Δt: 8.114 seconds, umax = (8.2e-02, 6.2e-02, 4.6e-02) ms⁻¹, wall time: 1.519 minutes
[ Info: i: 1400, t: 3.981 hours, Δt: 8.180 seconds, umax = (8.8e-02, 6.1e-02, 4.8e-02) ms⁻¹, wall time: 1.528 minutes
[ Info: Simulation is stopping after running for 1.531 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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