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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"
using Oceananigans
using Oceananigans.Units: minute, minutes, hours

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(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 => Callback of stop_time_exceeded on IterationInterval(1)
│   ├── stop_iteration_exceeded => Callback of stop_iteration_exceeded on IterationInterval(1)
│   ├── wall_time_limit_exceeded => Callback of wall_time_limit_exceeded on IterationInterval(1)
│   └── nan_checker => Callback of NaNChecker for u on IterationInterval(100)
├── 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, 8.4e-04, 1.3e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (52.006 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (6.488 seconds).
[ Info: i: 0020, t: 12.601 minutes, Δt: 29.112 seconds, umax = (2.9e-02, 1.3e-02, 2.0e-02) ms⁻¹, wall time: 1.010 minutes
[ Info: i: 0040, t: 21.140 minutes, Δt: 19.709 seconds, umax = (4.2e-02, 1.1e-02, 1.9e-02) ms⁻¹, wall time: 1.022 minutes
[ Info: i: 0060, t: 27.003 minutes, Δt: 17.193 seconds, umax = (4.8e-02, 1.5e-02, 1.9e-02) ms⁻¹, wall time: 1.030 minutes
[ Info: i: 0080, t: 32.461 minutes, Δt: 16.184 seconds, umax = (5.2e-02, 1.7e-02, 2.2e-02) ms⁻¹, wall time: 1.039 minutes
[ Info: i: 0100, t: 37.736 minutes, Δt: 16.030 seconds, umax = (5.0e-02, 1.8e-02, 2.2e-02) ms⁻¹, wall time: 1.048 minutes
[ Info: i: 0120, t: 42.794 minutes, Δt: 15.059 seconds, umax = (5.2e-02, 1.9e-02, 2.4e-02) ms⁻¹, wall time: 1.057 minutes
[ Info: i: 0140, t: 47.582 minutes, Δt: 14.225 seconds, umax = (5.4e-02, 2.0e-02, 2.8e-02) ms⁻¹, wall time: 1.067 minutes
[ Info: i: 0160, t: 52.161 minutes, Δt: 13.986 seconds, umax = (5.8e-02, 2.1e-02, 2.9e-02) ms⁻¹, wall time: 1.077 minutes
[ Info: i: 0180, t: 56.560 minutes, Δt: 13.233 seconds, umax = (6.1e-02, 2.4e-02, 2.7e-02) ms⁻¹, wall time: 1.087 minutes
[ Info: i: 0200, t: 1.014 hours, Δt: 12.833 seconds, umax = (6.3e-02, 2.6e-02, 3.0e-02) ms⁻¹, wall time: 1.097 minutes
[ Info: i: 0220, t: 1.083 hours, Δt: 11.819 seconds, umax = (6.4e-02, 2.7e-02, 3.2e-02) ms⁻¹, wall time: 1.105 minutes
[ Info: i: 0240, t: 1.146 hours, Δt: 12.258 seconds, umax = (6.5e-02, 2.8e-02, 3.6e-02) ms⁻¹, wall time: 1.113 minutes
[ Info: i: 0260, t: 1.209 hours, Δt: 11.677 seconds, umax = (6.6e-02, 3.4e-02, 3.1e-02) ms⁻¹, wall time: 1.122 minutes
[ Info: i: 0280, t: 1.272 hours, Δt: 10.731 seconds, umax = (6.7e-02, 3.4e-02, 3.5e-02) ms⁻¹, wall time: 1.132 minutes
[ Info: i: 0300, t: 1.332 hours, Δt: 10.951 seconds, umax = (7.0e-02, 3.2e-02, 3.8e-02) ms⁻¹, wall time: 1.140 minutes
[ Info: i: 0320, t: 1.392 hours, Δt: 10.928 seconds, umax = (7.4e-02, 3.2e-02, 3.4e-02) ms⁻¹, wall time: 1.149 minutes
[ Info: i: 0340, t: 1.449 hours, Δt: 10.363 seconds, umax = (7.4e-02, 3.4e-02, 3.5e-02) ms⁻¹, wall time: 1.158 minutes
[ Info: i: 0360, t: 1.506 hours, Δt: 10.021 seconds, umax = (7.2e-02, 3.6e-02, 3.8e-02) ms⁻¹, wall time: 1.170 minutes
[ Info: i: 0380, t: 1.562 hours, Δt: 10.383 seconds, umax = (7.4e-02, 3.8e-02, 3.9e-02) ms⁻¹, wall time: 1.177 minutes
[ Info: i: 0400, t: 1.618 hours, Δt: 10.305 seconds, umax = (7.4e-02, 3.7e-02, 3.8e-02) ms⁻¹, wall time: 1.188 minutes
[ Info: i: 0420, t: 1.672 hours, Δt: 9.977 seconds, umax = (7.3e-02, 3.9e-02, 4.1e-02) ms⁻¹, wall time: 1.200 minutes
[ Info: i: 0440, t: 1.728 hours, Δt: 9.694 seconds, umax = (7.5e-02, 4.2e-02, 3.9e-02) ms⁻¹, wall time: 1.206 minutes
[ Info: i: 0460, t: 1.780 hours, Δt: 10.225 seconds, umax = (7.4e-02, 4.1e-02, 4.2e-02) ms⁻¹, wall time: 1.217 minutes
[ Info: i: 0480, t: 1.836 hours, Δt: 9.789 seconds, umax = (7.3e-02, 4.4e-02, 3.8e-02) ms⁻¹, wall time: 1.228 minutes
[ Info: i: 0500, t: 1.891 hours, Δt: 9.545 seconds, umax = (7.7e-02, 4.4e-02, 4.2e-02) ms⁻¹, wall time: 1.233 minutes
[ Info: i: 0520, t: 1.943 hours, Δt: 9.694 seconds, umax = (7.8e-02, 4.4e-02, 4.4e-02) ms⁻¹, wall time: 1.244 minutes
[ Info: i: 0540, t: 1.998 hours, Δt: 9.843 seconds, umax = (7.9e-02, 4.6e-02, 4.1e-02) ms⁻¹, wall time: 1.253 minutes
[ Info: i: 0560, t: 2.049 hours, Δt: 9.528 seconds, umax = (8.1e-02, 4.4e-02, 4.0e-02) ms⁻¹, wall time: 1.263 minutes
[ Info: i: 0580, t: 2.099 hours, Δt: 9.405 seconds, umax = (8.2e-02, 4.5e-02, 4.0e-02) ms⁻¹, wall time: 1.275 minutes
[ Info: i: 0600, t: 2.150 hours, Δt: 9.274 seconds, umax = (7.9e-02, 4.8e-02, 4.1e-02) ms⁻¹, wall time: 1.283 minutes
[ Info: i: 0620, t: 2.199 hours, Δt: 8.997 seconds, umax = (7.9e-02, 4.7e-02, 4.0e-02) ms⁻¹, wall time: 1.293 minutes
[ Info: i: 0640, t: 2.250 hours, Δt: 9.163 seconds, umax = (8.1e-02, 4.8e-02, 4.4e-02) ms⁻¹, wall time: 1.302 minutes
[ Info: i: 0660, t: 2.296 hours, Δt: 9.306 seconds, umax = (8.8e-02, 4.8e-02, 4.2e-02) ms⁻¹, wall time: 1.312 minutes
[ Info: i: 0680, t: 2.344 hours, Δt: 8.989 seconds, umax = (8.9e-02, 4.5e-02, 4.4e-02) ms⁻¹, wall time: 1.323 minutes
[ Info: i: 0700, t: 2.394 hours, Δt: 8.966 seconds, umax = (8.2e-02, 4.9e-02, 4.2e-02) ms⁻¹, wall time: 1.330 minutes
[ Info: i: 0720, t: 2.441 hours, Δt: 9.059 seconds, umax = (8.1e-02, 5.0e-02, 4.7e-02) ms⁻¹, wall time: 1.341 minutes
[ Info: i: 0740, t: 2.492 hours, Δt: 9.059 seconds, umax = (8.3e-02, 4.9e-02, 4.0e-02) ms⁻¹, wall time: 1.351 minutes
[ Info: i: 0760, t: 2.540 hours, Δt: 8.760 seconds, umax = (8.3e-02, 5.7e-02, 4.5e-02) ms⁻¹, wall time: 1.360 minutes
[ Info: i: 0780, t: 2.588 hours, Δt: 8.454 seconds, umax = (8.0e-02, 5.5e-02, 4.6e-02) ms⁻¹, wall time: 1.372 minutes
[ Info: i: 0800, t: 2.636 hours, Δt: 8.935 seconds, umax = (8.2e-02, 5.1e-02, 4.5e-02) ms⁻¹, wall time: 1.379 minutes
[ Info: i: 0820, t: 2.684 hours, Δt: 8.826 seconds, umax = (7.9e-02, 5.2e-02, 4.5e-02) ms⁻¹, wall time: 1.389 minutes
[ Info: i: 0840, t: 2.732 hours, Δt: 8.574 seconds, umax = (8.9e-02, 5.4e-02, 4.3e-02) ms⁻¹, wall time: 1.397 minutes
[ Info: i: 0860, t: 2.778 hours, Δt: 8.790 seconds, umax = (8.3e-02, 5.1e-02, 4.7e-02) ms⁻¹, wall time: 1.408 minutes
[ Info: i: 0880, t: 2.826 hours, Δt: 8.899 seconds, umax = (8.3e-02, 5.3e-02, 4.4e-02) ms⁻¹, wall time: 1.417 minutes
[ Info: i: 0900, t: 2.874 hours, Δt: 8.387 seconds, umax = (8.4e-02, 5.1e-02, 4.3e-02) ms⁻¹, wall time: 1.427 minutes
[ Info: i: 0920, t: 2.919 hours, Δt: 8.464 seconds, umax = (8.6e-02, 5.1e-02, 4.8e-02) ms⁻¹, wall time: 1.442 minutes
[ Info: i: 0940, t: 2.966 hours, Δt: 8.464 seconds, umax = (8.7e-02, 5.0e-02, 5.1e-02) ms⁻¹, wall time: 1.448 minutes
[ Info: i: 0960, t: 3.011 hours, Δt: 8.164 seconds, umax = (8.8e-02, 5.2e-02, 4.6e-02) ms⁻¹, wall time: 1.460 minutes
[ Info: i: 0980, t: 3.056 hours, Δt: 7.922 seconds, umax = (8.6e-02, 5.8e-02, 4.5e-02) ms⁻¹, wall time: 1.467 minutes
[ Info: i: 1000, t: 3.099 hours, Δt: 8.503 seconds, umax = (8.8e-02, 5.4e-02, 4.4e-02) ms⁻¹, wall time: 1.479 minutes
[ Info: i: 1020, t: 3.146 hours, Δt: 8.541 seconds, umax = (8.5e-02, 5.6e-02, 4.6e-02) ms⁻¹, wall time: 1.487 minutes
[ Info: i: 1040, t: 3.193 hours, Δt: 8.396 seconds, umax = (8.4e-02, 5.7e-02, 4.4e-02) ms⁻¹, wall time: 1.496 minutes
[ Info: i: 1060, t: 3.239 hours, Δt: 8.408 seconds, umax = (9.0e-02, 6.4e-02, 4.5e-02) ms⁻¹, wall time: 1.505 minutes
[ Info: i: 1080, t: 3.285 hours, Δt: 8.256 seconds, umax = (8.8e-02, 6.2e-02, 4.3e-02) ms⁻¹, wall time: 1.515 minutes
[ Info: i: 1100, t: 3.330 hours, Δt: 8.003 seconds, umax = (8.7e-02, 5.6e-02, 4.5e-02) ms⁻¹, wall time: 1.524 minutes
[ Info: i: 1120, t: 3.373 hours, Δt: 7.992 seconds, umax = (8.7e-02, 5.8e-02, 4.5e-02) ms⁻¹, wall time: 1.533 minutes
[ Info: i: 1140, t: 3.417 hours, Δt: 8.043 seconds, umax = (8.8e-02, 5.7e-02, 4.5e-02) ms⁻¹, wall time: 1.543 minutes
[ Info: i: 1160, t: 3.462 hours, Δt: 8.236 seconds, umax = (9.1e-02, 6.0e-02, 4.5e-02) ms⁻¹, wall time: 1.553 minutes
[ Info: i: 1180, t: 3.507 hours, Δt: 8.165 seconds, umax = (8.6e-02, 6.1e-02, 5.1e-02) ms⁻¹, wall time: 1.566 minutes
[ Info: i: 1200, t: 3.552 hours, Δt: 8.005 seconds, umax = (8.7e-02, 6.7e-02, 4.9e-02) ms⁻¹, wall time: 1.573 minutes
[ Info: i: 1220, t: 3.594 hours, Δt: 7.973 seconds, umax = (8.5e-02, 7.1e-02, 4.7e-02) ms⁻¹, wall time: 1.585 minutes
[ Info: i: 1240, t: 3.639 hours, Δt: 7.916 seconds, umax = (9.2e-02, 6.9e-02, 4.4e-02) ms⁻¹, wall time: 1.593 minutes
[ Info: i: 1260, t: 3.682 hours, Δt: 8.032 seconds, umax = (8.9e-02, 7.1e-02, 4.7e-02) ms⁻¹, wall time: 1.603 minutes
[ Info: i: 1280, t: 3.727 hours, Δt: 7.980 seconds, umax = (9.2e-02, 6.4e-02, 4.6e-02) ms⁻¹, wall time: 1.612 minutes
[ Info: i: 1300, t: 3.770 hours, Δt: 7.647 seconds, umax = (8.7e-02, 6.6e-02, 4.6e-02) ms⁻¹, wall time: 1.622 minutes
[ Info: i: 1320, t: 3.812 hours, Δt: 7.696 seconds, umax = (8.7e-02, 6.6e-02, 5.2e-02) ms⁻¹, wall time: 1.632 minutes
[ Info: i: 1340, t: 3.856 hours, Δt: 8.155 seconds, umax = (8.9e-02, 6.3e-02, 4.9e-02) ms⁻¹, wall time: 1.642 minutes
[ Info: i: 1360, t: 3.901 hours, Δt: 8.050 seconds, umax = (8.9e-02, 5.8e-02, 4.9e-02) ms⁻¹, wall time: 1.651 minutes
[ Info: i: 1380, t: 3.945 hours, Δt: 7.556 seconds, umax = (8.8e-02, 6.1e-02, 4.8e-02) ms⁻¹, wall time: 1.664 minutes
[ Info: i: 1400, t: 3.987 hours, Δt: 7.341 seconds, umax = (8.8e-02, 6.1e-02, 5.4e-02) ms⁻¹, wall time: 1.673 minutes
[ Info: Simulation is stopping after running for 1.678 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)

record(fig, "langmuir_turbulence.mp4", frames, framerate=8) do i
    n[] = i
end

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