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 = (1.5e-03, 9.2e-04, 1.3e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (34.347 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (7.408 seconds).
[ Info: i: 0020, t: 11.711 minutes, Δt: 28.238 seconds, umax = (2.8e-02, 1.1e-02, 2.3e-02) ms⁻¹, wall time: 56.984 seconds
[ Info: i: 0040, t: 20 minutes, Δt: 20.591 seconds, umax = (4.1e-02, 1.2e-02, 2.0e-02) ms⁻¹, wall time: 57.502 seconds
[ Info: i: 0060, t: 26.220 minutes, Δt: 17.356 seconds, umax = (4.7e-02, 1.5e-02, 1.9e-02) ms⁻¹, wall time: 58.194 seconds
[ Info: i: 0080, t: 31.666 minutes, Δt: 15.191 seconds, umax = (5.0e-02, 1.7e-02, 2.3e-02) ms⁻¹, wall time: 58.728 seconds
[ Info: i: 0100, t: 36.896 minutes, Δt: 15.619 seconds, umax = (4.9e-02, 1.7e-02, 2.3e-02) ms⁻¹, wall time: 59.242 seconds
[ Info: i: 0120, t: 41.739 minutes, Δt: 14.726 seconds, umax = (5.2e-02, 1.8e-02, 2.4e-02) ms⁻¹, wall time: 59.880 seconds
[ Info: i: 0140, t: 46.447 minutes, Δt: 14.744 seconds, umax = (5.5e-02, 2.0e-02, 2.7e-02) ms⁻¹, wall time: 1.009 minutes
[ Info: i: 0160, t: 51.217 minutes, Δt: 14.294 seconds, umax = (5.7e-02, 2.1e-02, 2.5e-02) ms⁻¹, wall time: 1.020 minutes
[ Info: i: 0180, t: 55.698 minutes, Δt: 13.586 seconds, umax = (5.9e-02, 2.3e-02, 2.7e-02) ms⁻¹, wall time: 1.031 minutes
[ Info: i: 0200, t: 1 hour, Δt: 12.456 seconds, umax = (6.1e-02, 2.4e-02, 2.9e-02) ms⁻¹, wall time: 1.037 minutes
[ Info: i: 0220, t: 1.069 hours, Δt: 12.650 seconds, umax = (6.4e-02, 2.7e-02, 2.9e-02) ms⁻¹, wall time: 1.046 minutes
[ Info: i: 0240, t: 1.139 hours, Δt: 12.203 seconds, umax = (6.4e-02, 2.7e-02, 3.1e-02) ms⁻¹, wall time: 1.054 minutes
[ Info: i: 0260, t: 1.202 hours, Δt: 11.684 seconds, umax = (6.5e-02, 2.9e-02, 3.3e-02) ms⁻¹, wall time: 1.063 minutes
[ Info: i: 0280, t: 1.262 hours, Δt: 11.436 seconds, umax = (6.8e-02, 3.0e-02, 3.2e-02) ms⁻¹, wall time: 1.074 minutes
[ Info: i: 0300, t: 1.327 hours, Δt: 11.643 seconds, umax = (6.8e-02, 3.2e-02, 3.6e-02) ms⁻¹, wall time: 1.080 minutes
[ Info: i: 0320, t: 1.383 hours, Δt: 10.606 seconds, umax = (6.9e-02, 3.3e-02, 3.3e-02) ms⁻¹, wall time: 1.090 minutes
[ Info: i: 0340, t: 1.440 hours, Δt: 10.830 seconds, umax = (7.0e-02, 3.6e-02, 3.5e-02) ms⁻¹, wall time: 1.100 minutes
[ Info: i: 0360, t: 1.500 hours, Δt: 10.773 seconds, umax = (7.2e-02, 3.6e-02, 3.9e-02) ms⁻¹, wall time: 1.109 minutes
[ Info: i: 0380, t: 1.560 hours, Δt: 10.542 seconds, umax = (7.1e-02, 3.6e-02, 3.8e-02) ms⁻¹, wall time: 1.118 minutes
[ Info: i: 0400, t: 1.616 hours, Δt: 10.523 seconds, umax = (7.3e-02, 3.7e-02, 3.7e-02) ms⁻¹, wall time: 1.127 minutes
[ Info: i: 0420, t: 1.672 hours, Δt: 10.080 seconds, umax = (7.2e-02, 3.8e-02, 3.8e-02) ms⁻¹, wall time: 1.138 minutes
[ Info: i: 0440, t: 1.728 hours, Δt: 10.149 seconds, umax = (7.5e-02, 4.1e-02, 3.8e-02) ms⁻¹, wall time: 1.144 minutes
[ Info: i: 0460, t: 1.781 hours, Δt: 9.788 seconds, umax = (7.5e-02, 4.3e-02, 4.1e-02) ms⁻¹, wall time: 1.153 minutes
[ Info: i: 0480, t: 1.833 hours, Δt: 9.259 seconds, umax = (7.8e-02, 4.4e-02, 3.8e-02) ms⁻¹, wall time: 1.162 minutes
[ Info: i: 0500, t: 1.884 hours, Δt: 9.749 seconds, umax = (7.7e-02, 4.7e-02, 3.9e-02) ms⁻¹, wall time: 1.171 minutes
[ Info: i: 0520, t: 1.938 hours, Δt: 9.763 seconds, umax = (8.1e-02, 4.4e-02, 4.0e-02) ms⁻¹, wall time: 1.181 minutes
[ Info: i: 0540, t: 1.992 hours, Δt: 8.894 seconds, umax = (7.5e-02, 4.5e-02, 4.2e-02) ms⁻¹, wall time: 1.189 minutes
[ Info: i: 0560, t: 2.038 hours, Δt: 9.013 seconds, umax = (7.5e-02, 4.7e-02, 4.6e-02) ms⁻¹, wall time: 1.198 minutes
[ Info: i: 0580, t: 2.088 hours, Δt: 9.212 seconds, umax = (7.9e-02, 4.8e-02, 4.2e-02) ms⁻¹, wall time: 1.210 minutes
[ Info: i: 0600, t: 2.140 hours, Δt: 9.047 seconds, umax = (8.0e-02, 4.7e-02, 4.1e-02) ms⁻¹, wall time: 1.216 minutes
[ Info: i: 0620, t: 2.189 hours, Δt: 8.741 seconds, umax = (8.0e-02, 4.5e-02, 4.3e-02) ms⁻¹, wall time: 1.225 minutes
[ Info: i: 0640, t: 2.238 hours, Δt: 9.340 seconds, umax = (8.4e-02, 4.7e-02, 4.0e-02) ms⁻¹, wall time: 1.234 minutes
[ Info: i: 0660, t: 2.289 hours, Δt: 9.075 seconds, umax = (7.9e-02, 4.5e-02, 4.4e-02) ms⁻¹, wall time: 1.244 minutes
[ Info: i: 0680, t: 2.336 hours, Δt: 8.843 seconds, umax = (8.0e-02, 5.3e-02, 4.8e-02) ms⁻¹, wall time: 1.256 minutes
[ Info: i: 0700, t: 2.384 hours, Δt: 8.698 seconds, umax = (8.0e-02, 4.8e-02, 4.7e-02) ms⁻¹, wall time: 1.263 minutes
[ Info: i: 0720, t: 2.432 hours, Δt: 8.849 seconds, umax = (7.9e-02, 4.9e-02, 4.5e-02) ms⁻¹, wall time: 1.273 minutes
[ Info: i: 0740, t: 2.479 hours, Δt: 8.277 seconds, umax = (8.4e-02, 5.5e-02, 4.7e-02) ms⁻¹, wall time: 1.280 minutes
[ Info: i: 0760, t: 2.526 hours, Δt: 8.285 seconds, umax = (8.3e-02, 6.0e-02, 4.2e-02) ms⁻¹, wall time: 1.290 minutes
[ Info: i: 0780, t: 2.572 hours, Δt: 8.648 seconds, umax = (8.2e-02, 5.4e-02, 4.8e-02) ms⁻¹, wall time: 1.298 minutes
[ Info: i: 0800, t: 2.619 hours, Δt: 8.325 seconds, umax = (8.3e-02, 5.5e-02, 4.4e-02) ms⁻¹, wall time: 1.308 minutes
[ Info: i: 0820, t: 2.666 hours, Δt: 8.659 seconds, umax = (8.3e-02, 5.5e-02, 4.3e-02) ms⁻¹, wall time: 1.319 minutes
[ Info: i: 0840, t: 2.713 hours, Δt: 8.930 seconds, umax = (8.4e-02, 5.4e-02, 4.8e-02) ms⁻¹, wall time: 1.328 minutes
[ Info: i: 0860, t: 2.760 hours, Δt: 8.606 seconds, umax = (8.5e-02, 5.6e-02, 4.5e-02) ms⁻¹, wall time: 1.340 minutes
[ Info: i: 0880, t: 2.808 hours, Δt: 8.693 seconds, umax = (8.4e-02, 5.6e-02, 4.7e-02) ms⁻¹, wall time: 1.347 minutes
[ Info: i: 0900, t: 2.855 hours, Δt: 8.864 seconds, umax = (8.8e-02, 5.2e-02, 5.1e-02) ms⁻¹, wall time: 1.357 minutes
[ Info: i: 0920, t: 2.904 hours, Δt: 8.678 seconds, umax = (9.1e-02, 5.3e-02, 4.5e-02) ms⁻¹, wall time: 1.366 minutes
[ Info: i: 0940, t: 2.950 hours, Δt: 8.491 seconds, umax = (8.6e-02, 5.1e-02, 4.3e-02) ms⁻¹, wall time: 1.375 minutes
[ Info: i: 0960, t: 2.997 hours, Δt: 8.300 seconds, umax = (8.2e-02, 5.6e-02, 4.7e-02) ms⁻¹, wall time: 1.384 minutes
[ Info: i: 0980, t: 3.041 hours, Δt: 7.939 seconds, umax = (8.6e-02, 5.9e-02, 4.6e-02) ms⁻¹, wall time: 1.393 minutes
[ Info: i: 1000, t: 3.086 hours, Δt: 8.221 seconds, umax = (8.4e-02, 5.9e-02, 4.8e-02) ms⁻¹, wall time: 1.405 minutes
[ Info: i: 1020, t: 3.131 hours, Δt: 8.437 seconds, umax = (8.4e-02, 5.8e-02, 4.5e-02) ms⁻¹, wall time: 1.411 minutes
[ Info: i: 1040, t: 3.176 hours, Δt: 8.619 seconds, umax = (8.2e-02, 5.6e-02, 5.0e-02) ms⁻¹, wall time: 1.423 minutes
[ Info: i: 1060, t: 3.224 hours, Δt: 8.600 seconds, umax = (8.2e-02, 5.4e-02, 4.8e-02) ms⁻¹, wall time: 1.430 minutes
[ Info: i: 1080, t: 3.269 hours, Δt: 8.298 seconds, umax = (8.3e-02, 5.6e-02, 4.4e-02) ms⁻¹, wall time: 1.440 minutes
[ Info: i: 1100, t: 3.315 hours, Δt: 8.573 seconds, umax = (8.5e-02, 6.1e-02, 4.6e-02) ms⁻¹, wall time: 1.449 minutes
[ Info: i: 1120, t: 3.362 hours, Δt: 8.472 seconds, umax = (8.3e-02, 5.7e-02, 4.5e-02) ms⁻¹, wall time: 1.458 minutes
[ Info: i: 1140, t: 3.408 hours, Δt: 8.401 seconds, umax = (8.6e-02, 5.7e-02, 5.0e-02) ms⁻¹, wall time: 1.467 minutes
[ Info: i: 1160, t: 3.453 hours, Δt: 7.996 seconds, umax = (8.5e-02, 5.7e-02, 4.4e-02) ms⁻¹, wall time: 1.476 minutes
[ Info: i: 1180, t: 3.498 hours, Δt: 8.223 seconds, umax = (8.5e-02, 6.1e-02, 4.2e-02) ms⁻¹, wall time: 1.485 minutes
[ Info: i: 1200, t: 3.542 hours, Δt: 7.917 seconds, umax = (8.5e-02, 6.3e-02, 5.1e-02) ms⁻¹, wall time: 1.496 minutes
[ Info: i: 1220, t: 3.586 hours, Δt: 8.134 seconds, umax = (8.4e-02, 6.3e-02, 4.3e-02) ms⁻¹, wall time: 1.509 minutes
[ Info: i: 1240, t: 3.631 hours, Δt: 7.761 seconds, umax = (8.4e-02, 6.7e-02, 4.5e-02) ms⁻¹, wall time: 1.516 minutes
[ Info: i: 1260, t: 3.673 hours, Δt: 8.263 seconds, umax = (8.7e-02, 6.2e-02, 4.5e-02) ms⁻¹, wall time: 1.527 minutes
[ Info: i: 1280, t: 3.719 hours, Δt: 7.996 seconds, umax = (8.9e-02, 6.5e-02, 4.4e-02) ms⁻¹, wall time: 1.533 minutes
[ Info: i: 1300, t: 3.763 hours, Δt: 8.044 seconds, umax = (8.8e-02, 6.5e-02, 4.5e-02) ms⁻¹, wall time: 1.544 minutes
[ Info: i: 1320, t: 3.808 hours, Δt: 7.656 seconds, umax = (8.7e-02, 6.4e-02, 4.5e-02) ms⁻¹, wall time: 1.551 minutes
[ Info: i: 1340, t: 3.851 hours, Δt: 7.964 seconds, umax = (8.6e-02, 7.0e-02, 4.9e-02) ms⁻¹, wall time: 1.563 minutes
[ Info: i: 1360, t: 3.895 hours, Δt: 7.855 seconds, umax = (9.1e-02, 6.8e-02, 5.2e-02) ms⁻¹, wall time: 1.571 minutes
[ Info: i: 1380, t: 3.936 hours, Δt: 7.797 seconds, umax = (8.9e-02, 6.6e-02, 5.6e-02) ms⁻¹, wall time: 1.581 minutes
[ Info: i: 1400, t: 3.980 hours, Δt: 7.981 seconds, umax = (9.3e-02, 6.6e-02, 5.4e-02) ms⁻¹, wall time: 1.589 minutes
[ Info: Simulation is stopping after running for 1.594 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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