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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{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 = (1.7e-03, 9.0e-04, 1.9e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (49.524 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (6.494 seconds).
[ Info: i: 0020, t: 11.941 minutes, Δt: 29.743 seconds, umax = (2.9e-02, 1.1e-02, 1.9e-02) ms⁻¹, wall time: 1.273 minutes
[ Info: i: 0040, t: 20.396 minutes, Δt: 20.441 seconds, umax = (4.2e-02, 1.1e-02, 1.8e-02) ms⁻¹, wall time: 1.288 minutes
[ Info: i: 0060, t: 26.834 minutes, Δt: 16.842 seconds, umax = (4.8e-02, 1.5e-02, 2.2e-02) ms⁻¹, wall time: 1.298 minutes
[ Info: i: 0080, t: 32.177 minutes, Δt: 16.051 seconds, umax = (5.0e-02, 1.6e-02, 2.3e-02) ms⁻¹, wall time: 1.306 minutes
[ Info: i: 0100, t: 37.407 minutes, Δt: 16.122 seconds, umax = (5.2e-02, 1.7e-02, 2.4e-02) ms⁻¹, wall time: 1.316 minutes
[ Info: i: 0120, t: 42.614 minutes, Δt: 15.149 seconds, umax = (5.2e-02, 1.8e-02, 2.4e-02) ms⁻¹, wall time: 1.325 minutes
[ Info: i: 0140, t: 47.539 minutes, Δt: 14.606 seconds, umax = (5.6e-02, 2.2e-02, 2.5e-02) ms⁻¹, wall time: 1.335 minutes
[ Info: i: 0160, t: 51.879 minutes, Δt: 13.865 seconds, umax = (6.0e-02, 2.2e-02, 3.2e-02) ms⁻¹, wall time: 1.345 minutes
[ Info: i: 0180, t: 56.337 minutes, Δt: 13.253 seconds, umax = (6.0e-02, 2.4e-02, 2.7e-02) ms⁻¹, wall time: 1.354 minutes
[ Info: i: 0200, t: 1.011 hours, Δt: 13.272 seconds, umax = (6.1e-02, 2.5e-02, 3.0e-02) ms⁻¹, wall time: 1.365 minutes
[ Info: i: 0220, t: 1.083 hours, Δt: 11.962 seconds, umax = (6.5e-02, 2.7e-02, 3.2e-02) ms⁻¹, wall time: 1.372 minutes
[ Info: i: 0240, t: 1.150 hours, Δt: 11.959 seconds, umax = (6.5e-02, 2.8e-02, 3.6e-02) ms⁻¹, wall time: 1.379 minutes
[ Info: i: 0260, t: 1.212 hours, Δt: 11.462 seconds, umax = (6.6e-02, 3.2e-02, 3.4e-02) ms⁻¹, wall time: 1.388 minutes
[ Info: i: 0280, t: 1.276 hours, Δt: 11.382 seconds, umax = (6.8e-02, 3.0e-02, 3.3e-02) ms⁻¹, wall time: 1.406 minutes
[ Info: i: 0300, t: 1.336 hours, Δt: 11.047 seconds, umax = (6.8e-02, 2.9e-02, 3.7e-02) ms⁻¹, wall time: 1.417 minutes
[ Info: i: 0320, t: 1.398 hours, Δt: 10.928 seconds, umax = (7.1e-02, 3.1e-02, 3.7e-02) ms⁻¹, wall time: 1.423 minutes
[ Info: i: 0340, t: 1.455 hours, Δt: 10.738 seconds, umax = (7.1e-02, 3.3e-02, 3.5e-02) ms⁻¹, wall time: 1.433 minutes
[ Info: i: 0360, t: 1.512 hours, Δt: 10.208 seconds, umax = (7.3e-02, 3.9e-02, 3.5e-02) ms⁻¹, wall time: 1.442 minutes
[ Info: i: 0380, t: 1.569 hours, Δt: 10.250 seconds, umax = (7.2e-02, 3.5e-02, 3.9e-02) ms⁻¹, wall time: 1.451 minutes
[ Info: i: 0400, t: 1.623 hours, Δt: 10.326 seconds, umax = (7.5e-02, 3.9e-02, 4.1e-02) ms⁻¹, wall time: 1.460 minutes
[ Info: i: 0420, t: 1.678 hours, Δt: 9.740 seconds, umax = (7.5e-02, 3.9e-02, 4.1e-02) ms⁻¹, wall time: 1.471 minutes
[ Info: i: 0440, t: 1.731 hours, Δt: 9.399 seconds, umax = (7.5e-02, 4.1e-02, 3.8e-02) ms⁻¹, wall time: 1.478 minutes
[ Info: i: 0460, t: 1.782 hours, Δt: 9.552 seconds, umax = (7.5e-02, 4.3e-02, 3.6e-02) ms⁻¹, wall time: 1.487 minutes
[ Info: i: 0480, t: 1.836 hours, Δt: 10.029 seconds, umax = (7.9e-02, 3.8e-02, 4.0e-02) ms⁻¹, wall time: 1.498 minutes
[ Info: i: 0500, t: 1.891 hours, Δt: 9.547 seconds, umax = (7.8e-02, 4.4e-02, 4.0e-02) ms⁻¹, wall time: 1.504 minutes
[ Info: i: 0520, t: 1.943 hours, Δt: 9.396 seconds, umax = (7.5e-02, 4.6e-02, 4.5e-02) ms⁻¹, wall time: 1.514 minutes
[ Info: i: 0540, t: 1.995 hours, Δt: 9.142 seconds, umax = (7.5e-02, 4.9e-02, 4.5e-02) ms⁻¹, wall time: 1.524 minutes
[ Info: i: 0560, t: 2.043 hours, Δt: 9.136 seconds, umax = (7.9e-02, 5.1e-02, 4.2e-02) ms⁻¹, wall time: 1.532 minutes
[ Info: i: 0580, t: 2.094 hours, Δt: 9.367 seconds, umax = (7.5e-02, 5.3e-02, 4.3e-02) ms⁻¹, wall time: 1.543 minutes
[ Info: i: 0600, t: 2.145 hours, Δt: 9.099 seconds, umax = (7.9e-02, 5.1e-02, 3.9e-02) ms⁻¹, wall time: 1.551 minutes
[ Info: i: 0620, t: 2.195 hours, Δt: 9.354 seconds, umax = (8.3e-02, 5.0e-02, 4.5e-02) ms⁻¹, wall time: 1.560 minutes
[ Info: i: 0640, t: 2.247 hours, Δt: 9.312 seconds, umax = (7.8e-02, 5.1e-02, 4.4e-02) ms⁻¹, wall time: 1.569 minutes
[ Info: i: 0660, t: 2.297 hours, Δt: 9.191 seconds, umax = (7.7e-02, 5.1e-02, 4.3e-02) ms⁻¹, wall time: 1.579 minutes
[ Info: i: 0680, t: 2.346 hours, Δt: 9.139 seconds, umax = (7.9e-02, 5.0e-02, 4.3e-02) ms⁻¹, wall time: 1.592 minutes
[ Info: i: 0700, t: 2.397 hours, Δt: 8.654 seconds, umax = (8.1e-02, 4.9e-02, 4.1e-02) ms⁻¹, wall time: 1.601 minutes
[ Info: i: 0720, t: 2.442 hours, Δt: 8.516 seconds, umax = (8.0e-02, 5.5e-02, 4.3e-02) ms⁻¹, wall time: 1.610 minutes
[ Info: i: 0740, t: 2.488 hours, Δt: 8.293 seconds, umax = (8.0e-02, 5.7e-02, 4.2e-02) ms⁻¹, wall time: 1.620 minutes
[ Info: i: 0760, t: 2.534 hours, Δt: 8.915 seconds, umax = (8.0e-02, 5.4e-02, 4.7e-02) ms⁻¹, wall time: 1.629 minutes
[ Info: i: 0780, t: 2.582 hours, Δt: 8.839 seconds, umax = (8.2e-02, 5.4e-02, 4.6e-02) ms⁻¹, wall time: 1.637 minutes
[ Info: i: 0800, t: 2.630 hours, Δt: 8.533 seconds, umax = (8.4e-02, 5.7e-02, 5.0e-02) ms⁻¹, wall time: 1.647 minutes
[ Info: i: 0820, t: 2.676 hours, Δt: 8.682 seconds, umax = (8.5e-02, 5.5e-02, 5.1e-02) ms⁻¹, wall time: 1.658 minutes
[ Info: i: 0840, t: 2.725 hours, Δt: 8.765 seconds, umax = (8.1e-02, 5.2e-02, 4.3e-02) ms⁻¹, wall time: 1.666 minutes
[ Info: i: 0860, t: 2.772 hours, Δt: 8.609 seconds, umax = (8.1e-02, 5.4e-02, 4.7e-02) ms⁻¹, wall time: 1.676 minutes
[ Info: i: 0880, t: 2.820 hours, Δt: 8.339 seconds, umax = (8.3e-02, 5.1e-02, 4.4e-02) ms⁻¹, wall time: 1.687 minutes
[ Info: i: 0900, t: 2.866 hours, Δt: 8.613 seconds, umax = (8.3e-02, 5.1e-02, 4.5e-02) ms⁻¹, wall time: 1.699 minutes
[ Info: i: 0920, t: 2.915 hours, Δt: 8.425 seconds, umax = (8.4e-02, 5.3e-02, 4.4e-02) ms⁻¹, wall time: 1.710 minutes
[ Info: i: 0940, t: 2.961 hours, Δt: 8.646 seconds, umax = (8.2e-02, 5.4e-02, 4.0e-02) ms⁻¹, wall time: 1.722 minutes
[ Info: i: 0960, t: 3.007 hours, Δt: 8.559 seconds, umax = (8.7e-02, 5.7e-02, 4.5e-02) ms⁻¹, wall time: 1.735 minutes
[ Info: i: 0980, t: 3.055 hours, Δt: 8.684 seconds, umax = (8.7e-02, 5.6e-02, 4.6e-02) ms⁻¹, wall time: 1.743 minutes
[ Info: i: 1000, t: 3.103 hours, Δt: 8.833 seconds, umax = (8.8e-02, 5.7e-02, 5.1e-02) ms⁻¹, wall time: 1.755 minutes
[ Info: i: 1020, t: 3.152 hours, Δt: 8.162 seconds, umax = (8.5e-02, 6.0e-02, 4.4e-02) ms⁻¹, wall time: 1.764 minutes
[ Info: i: 1040, t: 3.196 hours, Δt: 8.100 seconds, umax = (8.9e-02, 5.9e-02, 4.9e-02) ms⁻¹, wall time: 1.775 minutes
[ Info: i: 1060, t: 3.242 hours, Δt: 8.287 seconds, umax = (9.0e-02, 6.0e-02, 4.4e-02) ms⁻¹, wall time: 1.785 minutes
[ Info: i: 1080, t: 3.284 hours, Δt: 8.032 seconds, umax = (8.9e-02, 6.1e-02, 4.5e-02) ms⁻¹, wall time: 1.795 minutes
[ Info: i: 1100, t: 3.329 hours, Δt: 8.441 seconds, umax = (8.5e-02, 6.0e-02, 5.5e-02) ms⁻¹, wall time: 1.805 minutes
[ Info: i: 1120, t: 3.376 hours, Δt: 8.519 seconds, umax = (8.2e-02, 6.2e-02, 4.7e-02) ms⁻¹, wall time: 1.813 minutes
[ Info: i: 1140, t: 3.419 hours, Δt: 8.549 seconds, umax = (8.3e-02, 6.1e-02, 5.0e-02) ms⁻¹, wall time: 1.827 minutes
[ Info: i: 1160, t: 3.466 hours, Δt: 8.258 seconds, umax = (8.7e-02, 6.4e-02, 4.8e-02) ms⁻¹, wall time: 1.832 minutes
[ Info: i: 1180, t: 3.509 hours, Δt: 7.719 seconds, umax = (8.8e-02, 6.0e-02, 4.3e-02) ms⁻¹, wall time: 1.845 minutes
[ Info: i: 1200, t: 3.553 hours, Δt: 8.521 seconds, umax = (8.4e-02, 6.1e-02, 4.8e-02) ms⁻¹, wall time: 1.851 minutes
[ Info: i: 1220, t: 3.600 hours, Δt: 8.234 seconds, umax = (8.9e-02, 5.9e-02, 4.2e-02) ms⁻¹, wall time: 1.863 minutes
[ Info: i: 1240, t: 3.645 hours, Δt: 7.989 seconds, umax = (8.4e-02, 6.1e-02, 4.3e-02) ms⁻¹, wall time: 1.871 minutes
[ Info: i: 1260, t: 3.689 hours, Δt: 7.268 seconds, umax = (8.7e-02, 6.4e-02, 5.0e-02) ms⁻¹, wall time: 1.881 minutes
[ Info: i: 1280, t: 3.731 hours, Δt: 7.830 seconds, umax = (8.9e-02, 6.3e-02, 5.4e-02) ms⁻¹, wall time: 1.890 minutes
[ Info: i: 1300, t: 3.775 hours, Δt: 8.316 seconds, umax = (8.5e-02, 6.8e-02, 4.8e-02) ms⁻¹, wall time: 1.899 minutes
[ Info: i: 1320, t: 3.820 hours, Δt: 7.657 seconds, umax = (8.8e-02, 6.7e-02, 4.4e-02) ms⁻¹, wall time: 1.910 minutes
[ Info: i: 1340, t: 3.862 hours, Δt: 8.272 seconds, umax = (8.8e-02, 6.5e-02, 4.8e-02) ms⁻¹, wall time: 1.920 minutes
[ Info: i: 1360, t: 3.907 hours, Δt: 7.617 seconds, umax = (8.9e-02, 6.4e-02, 5.0e-02) ms⁻¹, wall time: 1.931 minutes
[ Info: i: 1380, t: 3.948 hours, Δt: 7.740 seconds, umax = (8.6e-02, 6.7e-02, 4.7e-02) ms⁻¹, wall time: 1.940 minutes
[ Info: i: 1400, t: 3.991 hours, Δt: 7.387 seconds, umax = (8.3e-02, 7.3e-02, 4.6e-02) ms⁻¹, wall time: 1.949 minutes
[ Info: Simulation is stopping after running for 1.952 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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