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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: 44.7 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 = (1.7e-03, 9.5e-04, 1.3e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (17.960 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (3.916 seconds).
[ Info: i: 0020, t: 11.804 minutes, Δt: 29.175 seconds, umax = (2.9e-02, 1.2e-02, 2.1e-02) ms⁻¹, wall time: 22.938 seconds
[ Info: i: 0040, t: 20.409 minutes, Δt: 19.651 seconds, umax = (4.0e-02, 1.0e-02, 2.1e-02) ms⁻¹, wall time: 23.514 seconds
[ Info: i: 0060, t: 26.487 minutes, Δt: 17.058 seconds, umax = (4.7e-02, 1.4e-02, 2.0e-02) ms⁻¹, wall time: 23.922 seconds
[ Info: i: 0080, t: 31.875 minutes, Δt: 15.652 seconds, umax = (5.2e-02, 1.7e-02, 2.1e-02) ms⁻¹, wall time: 24.360 seconds
[ Info: i: 0100, t: 37.143 minutes, Δt: 15.232 seconds, umax = (5.1e-02, 1.9e-02, 2.4e-02) ms⁻¹, wall time: 24.821 seconds
[ Info: i: 0120, t: 42.059 minutes, Δt: 15.515 seconds, umax = (5.3e-02, 1.8e-02, 2.8e-02) ms⁻¹, wall time: 25.309 seconds
[ Info: i: 0140, t: 46.956 minutes, Δt: 14.616 seconds, umax = (5.5e-02, 2.0e-02, 2.6e-02) ms⁻¹, wall time: 25.801 seconds
[ Info: i: 0160, t: 51.689 minutes, Δt: 13.907 seconds, umax = (5.9e-02, 2.1e-02, 2.8e-02) ms⁻¹, wall time: 26.298 seconds
[ Info: i: 0180, t: 56.137 minutes, Δt: 13.314 seconds, umax = (6.0e-02, 2.3e-02, 2.8e-02) ms⁻¹, wall time: 26.825 seconds
[ Info: i: 0200, t: 1.007 hours, Δt: 12.013 seconds, umax = (6.1e-02, 2.7e-02, 3.1e-02) ms⁻¹, wall time: 27.355 seconds
[ Info: i: 0220, t: 1.074 hours, Δt: 12.208 seconds, umax = (6.4e-02, 2.7e-02, 3.1e-02) ms⁻¹, wall time: 27.709 seconds
[ Info: i: 0240, t: 1.141 hours, Δt: 12.353 seconds, umax = (6.3e-02, 2.8e-02, 3.3e-02) ms⁻¹, wall time: 28.177 seconds
[ Info: i: 0260, t: 1.207 hours, Δt: 11.747 seconds, umax = (6.5e-02, 2.9e-02, 3.5e-02) ms⁻¹, wall time: 28.644 seconds
[ Info: i: 0280, t: 1.269 hours, Δt: 11.610 seconds, umax = (6.7e-02, 3.2e-02, 3.4e-02) ms⁻¹, wall time: 29.172 seconds
[ Info: i: 0300, t: 1.333 hours, Δt: 10.943 seconds, umax = (7.5e-02, 3.3e-02, 3.6e-02) ms⁻¹, wall time: 29.592 seconds
[ Info: i: 0320, t: 1.394 hours, Δt: 10.893 seconds, umax = (7.0e-02, 3.3e-02, 3.6e-02) ms⁻¹, wall time: 30.076 seconds
[ Info: i: 0340, t: 1.452 hours, Δt: 10.317 seconds, umax = (6.9e-02, 3.7e-02, 3.5e-02) ms⁻¹, wall time: 30.557 seconds
[ Info: i: 0360, t: 1.509 hours, Δt: 10.638 seconds, umax = (7.1e-02, 3.9e-02, 4.1e-02) ms⁻¹, wall time: 31.144 seconds
[ Info: i: 0380, t: 1.567 hours, Δt: 10.525 seconds, umax = (7.1e-02, 3.8e-02, 3.9e-02) ms⁻¹, wall time: 31.523 seconds
[ Info: i: 0400, t: 1.624 hours, Δt: 10.229 seconds, umax = (7.1e-02, 3.8e-02, 3.8e-02) ms⁻¹, wall time: 32.003 seconds
[ Info: i: 0420, t: 1.681 hours, Δt: 10.144 seconds, umax = (7.3e-02, 4.1e-02, 4.1e-02) ms⁻¹, wall time: 32.572 seconds
[ Info: i: 0440, t: 1.738 hours, Δt: 10.294 seconds, umax = (7.3e-02, 4.2e-02, 3.9e-02) ms⁻¹, wall time: 32.993 seconds
[ Info: i: 0460, t: 1.792 hours, Δt: 9.792 seconds, umax = (7.6e-02, 4.3e-02, 3.9e-02) ms⁻¹, wall time: 33.487 seconds
[ Info: i: 0480, t: 1.842 hours, Δt: 9.983 seconds, umax = (7.3e-02, 4.4e-02, 4.2e-02) ms⁻¹, wall time: 34.087 seconds
[ Info: i: 0500, t: 1.896 hours, Δt: 9.811 seconds, umax = (7.9e-02, 4.2e-02, 4.2e-02) ms⁻¹, wall time: 34.462 seconds
[ Info: i: 0520, t: 1.949 hours, Δt: 9.531 seconds, umax = (7.9e-02, 4.2e-02, 4.0e-02) ms⁻¹, wall time: 34.946 seconds
[ Info: i: 0540, t: 2 hours, Δt: 9.164 seconds, umax = (7.7e-02, 4.4e-02, 4.3e-02) ms⁻¹, wall time: 35.425 seconds
[ Info: i: 0560, t: 2.052 hours, Δt: 9.356 seconds, umax = (7.9e-02, 4.8e-02, 4.5e-02) ms⁻¹, wall time: 35.909 seconds
[ Info: i: 0580, t: 2.101 hours, Δt: 9.027 seconds, umax = (7.8e-02, 4.5e-02, 4.6e-02) ms⁻¹, wall time: 36.451 seconds
[ Info: i: 0600, t: 2.150 hours, Δt: 8.410 seconds, umax = (7.8e-02, 4.7e-02, 4.2e-02) ms⁻¹, wall time: 36.915 seconds
[ Info: i: 0620, t: 2.198 hours, Δt: 9.124 seconds, umax = (8.2e-02, 4.4e-02, 4.0e-02) ms⁻¹, wall time: 37.436 seconds
[ Info: i: 0640, t: 2.249 hours, Δt: 9.208 seconds, umax = (8.0e-02, 4.6e-02, 4.3e-02) ms⁻¹, wall time: 37.923 seconds
[ Info: i: 0660, t: 2.298 hours, Δt: 8.876 seconds, umax = (8.0e-02, 4.8e-02, 3.9e-02) ms⁻¹, wall time: 38.403 seconds
[ Info: i: 0680, t: 2.345 hours, Δt: 9.067 seconds, umax = (7.9e-02, 5.1e-02, 4.0e-02) ms⁻¹, wall time: 39.014 seconds
[ Info: i: 0700, t: 2.396 hours, Δt: 9.256 seconds, umax = (7.8e-02, 5.1e-02, 4.0e-02) ms⁻¹, wall time: 39.452 seconds
[ Info: i: 0720, t: 2.447 hours, Δt: 8.686 seconds, umax = (7.9e-02, 5.2e-02, 4.3e-02) ms⁻¹, wall time: 40.004 seconds
[ Info: i: 0740, t: 2.495 hours, Δt: 8.351 seconds, umax = (8.0e-02, 5.1e-02, 4.3e-02) ms⁻¹, wall time: 40.571 seconds
[ Info: i: 0760, t: 2.541 hours, Δt: 8.862 seconds, umax = (7.9e-02, 5.1e-02, 4.1e-02) ms⁻¹, wall time: 41.133 seconds
[ Info: i: 0780, t: 2.588 hours, Δt: 8.998 seconds, umax = (8.2e-02, 5.0e-02, 4.2e-02) ms⁻¹, wall time: 41.790 seconds
[ Info: i: 0800, t: 2.638 hours, Δt: 8.756 seconds, umax = (8.6e-02, 5.0e-02, 4.8e-02) ms⁻¹, wall time: 42.149 seconds
[ Info: i: 0820, t: 2.686 hours, Δt: 8.336 seconds, umax = (8.8e-02, 4.9e-02, 5.0e-02) ms⁻¹, wall time: 42.724 seconds
[ Info: i: 0840, t: 2.733 hours, Δt: 8.389 seconds, umax = (8.2e-02, 5.2e-02, 4.4e-02) ms⁻¹, wall time: 43.299 seconds
[ Info: i: 0860, t: 2.779 hours, Δt: 8.527 seconds, umax = (8.5e-02, 5.2e-02, 4.4e-02) ms⁻¹, wall time: 43.876 seconds
[ Info: i: 0880, t: 2.826 hours, Δt: 8.344 seconds, umax = (8.5e-02, 5.4e-02, 4.5e-02) ms⁻¹, wall time: 44.442 seconds
[ Info: i: 0900, t: 2.870 hours, Δt: 8.050 seconds, umax = (8.6e-02, 5.2e-02, 4.6e-02) ms⁻¹, wall time: 45.017 seconds
[ Info: i: 0920, t: 2.916 hours, Δt: 8.198 seconds, umax = (8.6e-02, 5.3e-02, 5.0e-02) ms⁻¹, wall time: 45.597 seconds
[ Info: i: 0940, t: 2.960 hours, Δt: 8.380 seconds, umax = (8.4e-02, 5.2e-02, 4.6e-02) ms⁻¹, wall time: 46.175 seconds
[ Info: i: 0960, t: 3.005 hours, Δt: 8.675 seconds, umax = (8.5e-02, 5.3e-02, 4.9e-02) ms⁻¹, wall time: 46.923 seconds
[ Info: i: 0980, t: 3.052 hours, Δt: 8.394 seconds, umax = (9.0e-02, 5.5e-02, 4.4e-02) ms⁻¹, wall time: 47.341 seconds
[ Info: i: 1000, t: 3.097 hours, Δt: 8.522 seconds, umax = (8.7e-02, 5.6e-02, 4.7e-02) ms⁻¹, wall time: 47.985 seconds
[ Info: i: 1020, t: 3.144 hours, Δt: 8.444 seconds, umax = (8.5e-02, 5.9e-02, 4.8e-02) ms⁻¹, wall time: 48.515 seconds
[ Info: i: 1040, t: 3.191 hours, Δt: 8.439 seconds, umax = (8.6e-02, 5.5e-02, 5.1e-02) ms⁻¹, wall time: 49.094 seconds
[ Info: i: 1060, t: 3.236 hours, Δt: 8.149 seconds, umax = (9.0e-02, 5.5e-02, 5.1e-02) ms⁻¹, wall time: 49.676 seconds
[ Info: i: 1080, t: 3.280 hours, Δt: 8.296 seconds, umax = (8.8e-02, 5.4e-02, 5.0e-02) ms⁻¹, wall time: 50.254 seconds
[ Info: i: 1100, t: 3.326 hours, Δt: 8.510 seconds, umax = (8.7e-02, 5.2e-02, 4.7e-02) ms⁻¹, wall time: 50.838 seconds
[ Info: i: 1120, t: 3.373 hours, Δt: 8.425 seconds, umax = (9.0e-02, 5.3e-02, 5.2e-02) ms⁻¹, wall time: 51.415 seconds
[ Info: i: 1140, t: 3.417 hours, Δt: 8.178 seconds, umax = (8.7e-02, 5.7e-02, 4.8e-02) ms⁻¹, wall time: 51.998 seconds
[ Info: i: 1160, t: 3.462 hours, Δt: 8.109 seconds, umax = (8.5e-02, 6.0e-02, 4.9e-02) ms⁻¹, wall time: 52.576 seconds
[ Info: i: 1180, t: 3.507 hours, Δt: 8.359 seconds, umax = (8.5e-02, 6.7e-02, 4.8e-02) ms⁻¹, wall time: 53.205 seconds
[ Info: i: 1200, t: 3.552 hours, Δt: 7.905 seconds, umax = (8.5e-02, 6.6e-02, 4.6e-02) ms⁻¹, wall time: 53.603 seconds
[ Info: i: 1220, t: 3.594 hours, Δt: 7.624 seconds, umax = (8.5e-02, 6.7e-02, 4.8e-02) ms⁻¹, wall time: 54.189 seconds
[ Info: i: 1240, t: 3.637 hours, Δt: 7.789 seconds, umax = (8.7e-02, 6.5e-02, 4.7e-02) ms⁻¹, wall time: 54.622 seconds
[ Info: i: 1260, t: 3.680 hours, Δt: 7.979 seconds, umax = (8.7e-02, 6.8e-02, 5.1e-02) ms⁻¹, wall time: 55.184 seconds
[ Info: i: 1280, t: 3.724 hours, Δt: 7.905 seconds, umax = (8.9e-02, 6.9e-02, 5.2e-02) ms⁻¹, wall time: 55.637 seconds
[ Info: i: 1300, t: 3.767 hours, Δt: 7.300 seconds, umax = (8.8e-02, 7.1e-02, 4.8e-02) ms⁻¹, wall time: 56.139 seconds
[ Info: i: 1320, t: 3.809 hours, Δt: 8.178 seconds, umax = (8.5e-02, 6.6e-02, 4.8e-02) ms⁻¹, wall time: 56.636 seconds
[ Info: i: 1340, t: 3.854 hours, Δt: 8.151 seconds, umax = (8.6e-02, 6.2e-02, 4.5e-02) ms⁻¹, wall time: 57.137 seconds
[ Info: i: 1360, t: 3.899 hours, Δt: 8.093 seconds, umax = (8.8e-02, 6.5e-02, 4.7e-02) ms⁻¹, wall time: 57.716 seconds
[ Info: i: 1380, t: 3.943 hours, Δt: 7.861 seconds, umax = (9.2e-02, 6.6e-02, 5.0e-02) ms⁻¹, wall time: 58.295 seconds
[ Info: i: 1400, t: 3.988 hours, Δt: 8.187 seconds, umax = (8.8e-02, 7.1e-02, 5.1e-02) ms⁻¹, wall time: 58.891 seconds
[ Info: Simulation is stopping after running for 59.078 seconds.
[ 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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