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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 = (1.4e-03, 8.5e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (49.931 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (8.674 seconds).
[ Info: i: 0020, t: 11.987 minutes, Δt: 30.022 seconds, umax = (2.8e-02, 1.2e-02, 2.1e-02) ms⁻¹, wall time: 1.324 minutes
[ Info: i: 0040, t: 20.401 minutes, Δt: 20.419 seconds, umax = (4.1e-02, 1.1e-02, 2.2e-02) ms⁻¹, wall time: 1.336 minutes
[ Info: i: 0060, t: 26.459 minutes, Δt: 17.371 seconds, umax = (4.7e-02, 1.5e-02, 1.9e-02) ms⁻¹, wall time: 1.344 minutes
[ Info: i: 0080, t: 31.918 minutes, Δt: 16.288 seconds, umax = (4.9e-02, 1.8e-02, 2.4e-02) ms⁻¹, wall time: 1.353 minutes
[ Info: i: 0100, t: 36.985 minutes, Δt: 15.439 seconds, umax = (5.4e-02, 1.7e-02, 2.3e-02) ms⁻¹, wall time: 1.362 minutes
[ Info: i: 0120, t: 42.097 minutes, Δt: 14.756 seconds, umax = (5.5e-02, 1.8e-02, 2.4e-02) ms⁻¹, wall time: 1.370 minutes
[ Info: i: 0140, t: 46.956 minutes, Δt: 13.994 seconds, umax = (5.5e-02, 2.0e-02, 2.9e-02) ms⁻¹, wall time: 1.379 minutes
[ Info: i: 0160, t: 51.393 minutes, Δt: 13.375 seconds, umax = (5.8e-02, 2.1e-02, 2.6e-02) ms⁻¹, wall time: 1.389 minutes
[ Info: i: 0180, t: 55.630 minutes, Δt: 12.173 seconds, umax = (6.1e-02, 2.6e-02, 2.9e-02) ms⁻¹, wall time: 1.399 minutes
[ Info: i: 0200, t: 59.738 minutes, Δt: 13.010 seconds, umax = (6.1e-02, 2.8e-02, 3.0e-02) ms⁻¹, wall time: 1.406 minutes
[ Info: i: 0220, t: 1.063 hours, Δt: 12.432 seconds, umax = (6.3e-02, 2.6e-02, 3.4e-02) ms⁻¹, wall time: 1.414 minutes
[ Info: i: 0240, t: 1.131 hours, Δt: 11.301 seconds, umax = (6.3e-02, 3.0e-02, 3.3e-02) ms⁻¹, wall time: 1.423 minutes
[ Info: i: 0260, t: 1.193 hours, Δt: 11.618 seconds, umax = (6.6e-02, 2.8e-02, 3.2e-02) ms⁻¹, wall time: 1.433 minutes
[ Info: i: 0280, t: 1.256 hours, Δt: 11.589 seconds, umax = (6.6e-02, 3.1e-02, 3.5e-02) ms⁻¹, wall time: 1.443 minutes
[ Info: i: 0300, t: 1.320 hours, Δt: 11.249 seconds, umax = (6.9e-02, 3.1e-02, 3.5e-02) ms⁻¹, wall time: 1.450 minutes
[ Info: i: 0320, t: 1.380 hours, Δt: 11.025 seconds, umax = (7.0e-02, 3.2e-02, 4.0e-02) ms⁻¹, wall time: 1.460 minutes
[ Info: i: 0340, t: 1.441 hours, Δt: 10.629 seconds, umax = (6.8e-02, 3.3e-02, 3.7e-02) ms⁻¹, wall time: 1.471 minutes
[ Info: i: 0360, t: 1.500 hours, Δt: 10.209 seconds, umax = (7.3e-02, 3.5e-02, 3.9e-02) ms⁻¹, wall time: 1.479 minutes
[ Info: i: 0380, t: 1.558 hours, Δt: 10.359 seconds, umax = (7.6e-02, 3.4e-02, 3.8e-02) ms⁻¹, wall time: 1.488 minutes
[ Info: i: 0400, t: 1.615 hours, Δt: 9.844 seconds, umax = (7.3e-02, 3.8e-02, 3.9e-02) ms⁻¹, wall time: 1.497 minutes
[ Info: i: 0420, t: 1.667 hours, Δt: 10.101 seconds, umax = (7.3e-02, 4.1e-02, 3.7e-02) ms⁻¹, wall time: 1.506 minutes
[ Info: i: 0440, t: 1.724 hours, Δt: 10.080 seconds, umax = (7.6e-02, 3.9e-02, 3.8e-02) ms⁻¹, wall time: 1.515 minutes
[ Info: i: 0460, t: 1.777 hours, Δt: 9.649 seconds, umax = (7.9e-02, 4.3e-02, 4.1e-02) ms⁻¹, wall time: 1.525 minutes
[ Info: i: 0480, t: 1.830 hours, Δt: 9.404 seconds, umax = (7.9e-02, 5.1e-02, 3.8e-02) ms⁻¹, wall time: 1.534 minutes
[ Info: i: 0500, t: 1.880 hours, Δt: 9.047 seconds, umax = (7.8e-02, 4.6e-02, 3.9e-02) ms⁻¹, wall time: 1.543 minutes
[ Info: i: 0520, t: 1.929 hours, Δt: 8.979 seconds, umax = (7.7e-02, 4.4e-02, 3.8e-02) ms⁻¹, wall time: 1.553 minutes
[ Info: i: 0540, t: 1.979 hours, Δt: 9.106 seconds, umax = (7.7e-02, 4.6e-02, 3.9e-02) ms⁻¹, wall time: 1.561 minutes
[ Info: i: 0560, t: 2.028 hours, Δt: 9.243 seconds, umax = (7.7e-02, 4.7e-02, 3.8e-02) ms⁻¹, wall time: 1.569 minutes
[ Info: i: 0580, t: 2.079 hours, Δt: 9.553 seconds, umax = (8.0e-02, 4.5e-02, 4.3e-02) ms⁻¹, wall time: 1.579 minutes
[ Info: i: 0600, t: 2.131 hours, Δt: 9.180 seconds, umax = (8.2e-02, 4.5e-02, 4.1e-02) ms⁻¹, wall time: 1.589 minutes
[ Info: i: 0620, t: 2.180 hours, Δt: 9.301 seconds, umax = (8.3e-02, 4.4e-02, 4.5e-02) ms⁻¹, wall time: 1.602 minutes
[ Info: i: 0640, t: 2.231 hours, Δt: 8.924 seconds, umax = (8.1e-02, 4.5e-02, 4.3e-02) ms⁻¹, wall time: 1.611 minutes
[ Info: i: 0660, t: 2.280 hours, Δt: 9.082 seconds, umax = (8.0e-02, 4.4e-02, 4.5e-02) ms⁻¹, wall time: 1.620 minutes
[ Info: i: 0680, t: 2.331 hours, Δt: 9.005 seconds, umax = (7.8e-02, 4.6e-02, 4.5e-02) ms⁻¹, wall time: 1.630 minutes
[ Info: i: 0700, t: 2.379 hours, Δt: 8.918 seconds, umax = (8.0e-02, 4.6e-02, 4.4e-02) ms⁻¹, wall time: 1.639 minutes
[ Info: i: 0720, t: 2.427 hours, Δt: 8.994 seconds, umax = (7.7e-02, 4.8e-02, 4.3e-02) ms⁻¹, wall time: 1.652 minutes
[ Info: i: 0740, t: 2.476 hours, Δt: 8.790 seconds, umax = (8.1e-02, 4.7e-02, 5.2e-02) ms⁻¹, wall time: 1.660 minutes
[ Info: i: 0760, t: 2.525 hours, Δt: 9.077 seconds, umax = (8.0e-02, 5.4e-02, 6.5e-02) ms⁻¹, wall time: 1.672 minutes
[ Info: i: 0780, t: 2.574 hours, Δt: 8.799 seconds, umax = (8.3e-02, 5.5e-02, 5.4e-02) ms⁻¹, wall time: 1.682 minutes
[ Info: i: 0800, t: 2.622 hours, Δt: 8.633 seconds, umax = (8.1e-02, 5.0e-02, 4.6e-02) ms⁻¹, wall time: 1.692 minutes
[ Info: i: 0820, t: 2.667 hours, Δt: 8.583 seconds, umax = (8.0e-02, 5.4e-02, 4.6e-02) ms⁻¹, wall time: 1.702 minutes
[ Info: i: 0840, t: 2.714 hours, Δt: 8.426 seconds, umax = (8.3e-02, 6.0e-02, 5.0e-02) ms⁻¹, wall time: 1.712 minutes
[ Info: i: 0860, t: 2.759 hours, Δt: 8.396 seconds, umax = (8.3e-02, 5.6e-02, 5.1e-02) ms⁻¹, wall time: 1.724 minutes
[ Info: i: 0880, t: 2.806 hours, Δt: 8.624 seconds, umax = (8.4e-02, 5.3e-02, 5.0e-02) ms⁻¹, wall time: 1.732 minutes
[ Info: i: 0900, t: 2.853 hours, Δt: 8.802 seconds, umax = (8.7e-02, 5.6e-02, 4.6e-02) ms⁻¹, wall time: 1.743 minutes
[ Info: i: 0920, t: 2.901 hours, Δt: 8.382 seconds, umax = (8.5e-02, 5.8e-02, 4.8e-02) ms⁻¹, wall time: 1.752 minutes
[ Info: i: 0940, t: 2.947 hours, Δt: 8.738 seconds, umax = (8.4e-02, 6.0e-02, 4.9e-02) ms⁻¹, wall time: 1.763 minutes
[ Info: i: 0960, t: 2.994 hours, Δt: 8.236 seconds, umax = (8.7e-02, 5.7e-02, 4.2e-02) ms⁻¹, wall time: 1.772 minutes
[ Info: i: 0980, t: 3.038 hours, Δt: 8.255 seconds, umax = (8.3e-02, 6.0e-02, 4.8e-02) ms⁻¹, wall time: 1.782 minutes
[ Info: i: 1000, t: 3.083 hours, Δt: 8.265 seconds, umax = (8.6e-02, 5.7e-02, 5.3e-02) ms⁻¹, wall time: 1.792 minutes
[ Info: i: 1020, t: 3.130 hours, Δt: 8.222 seconds, umax = (8.4e-02, 5.8e-02, 4.4e-02) ms⁻¹, wall time: 1.803 minutes
[ Info: i: 1040, t: 3.173 hours, Δt: 8.239 seconds, umax = (8.5e-02, 5.6e-02, 5.2e-02) ms⁻¹, wall time: 1.815 minutes
[ Info: i: 1060, t: 3.220 hours, Δt: 8.490 seconds, umax = (8.6e-02, 5.4e-02, 5.4e-02) ms⁻¹, wall time: 1.822 minutes
[ Info: i: 1080, t: 3.267 hours, Δt: 8.500 seconds, umax = (8.5e-02, 5.7e-02, 4.8e-02) ms⁻¹, wall time: 1.834 minutes
[ Info: i: 1100, t: 3.315 hours, Δt: 8.461 seconds, umax = (8.5e-02, 5.8e-02, 4.6e-02) ms⁻¹, wall time: 1.843 minutes
[ Info: i: 1120, t: 3.358 hours, Δt: 7.698 seconds, umax = (8.5e-02, 6.8e-02, 4.4e-02) ms⁻¹, wall time: 1.853 minutes
[ Info: i: 1140, t: 3.401 hours, Δt: 8.132 seconds, umax = (8.4e-02, 6.8e-02, 4.6e-02) ms⁻¹, wall time: 1.862 minutes
[ Info: i: 1160, t: 3.446 hours, Δt: 8.257 seconds, umax = (8.6e-02, 6.6e-02, 4.7e-02) ms⁻¹, wall time: 1.873 minutes
[ Info: i: 1180, t: 3.492 hours, Δt: 8.381 seconds, umax = (8.7e-02, 6.2e-02, 4.6e-02) ms⁻¹, wall time: 1.883 minutes
[ Info: i: 1200, t: 3.536 hours, Δt: 7.857 seconds, umax = (8.3e-02, 6.5e-02, 4.5e-02) ms⁻¹, wall time: 1.893 minutes
[ Info: i: 1220, t: 3.580 hours, Δt: 7.903 seconds, umax = (8.6e-02, 6.4e-02, 4.5e-02) ms⁻¹, wall time: 1.903 minutes
[ Info: i: 1240, t: 3.621 hours, Δt: 7.790 seconds, umax = (8.9e-02, 6.5e-02, 4.6e-02) ms⁻¹, wall time: 1.913 minutes
[ Info: i: 1260, t: 3.664 hours, Δt: 7.752 seconds, umax = (8.9e-02, 6.8e-02, 4.6e-02) ms⁻¹, wall time: 1.922 minutes
[ Info: i: 1280, t: 3.706 hours, Δt: 7.989 seconds, umax = (9.0e-02, 6.2e-02, 5.1e-02) ms⁻¹, wall time: 1.934 minutes
[ Info: i: 1300, t: 3.750 hours, Δt: 7.924 seconds, umax = (9.1e-02, 6.0e-02, 5.3e-02) ms⁻¹, wall time: 1.945 minutes
[ Info: i: 1320, t: 3.793 hours, Δt: 8.325 seconds, umax = (8.5e-02, 5.7e-02, 5.6e-02) ms⁻¹, wall time: 1.954 minutes
[ Info: i: 1340, t: 3.838 hours, Δt: 7.937 seconds, umax = (8.7e-02, 6.6e-02, 5.0e-02) ms⁻¹, wall time: 1.967 minutes
[ Info: i: 1360, t: 3.882 hours, Δt: 8.160 seconds, umax = (8.7e-02, 6.3e-02, 5.2e-02) ms⁻¹, wall time: 1.975 minutes
[ Info: i: 1380, t: 3.926 hours, Δt: 7.993 seconds, umax = (8.7e-02, 6.4e-02, 5.2e-02) ms⁻¹, wall time: 1.986 minutes
[ Info: i: 1400, t: 3.969 hours, Δt: 7.918 seconds, umax = (8.8e-02, 6.2e-02, 4.9e-02) ms⁻¹, wall time: 1.993 minutes
[ Info: Simulation is stopping after running for 2.002 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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