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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, 8.9e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (37.310 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (5.371 seconds).
[ Info: i: 0020, t: 11.914 minutes, Δt: 29.167 seconds, umax = (2.9e-02, 1.2e-02, 2.1e-02) ms⁻¹, wall time: 56.687 seconds
[ Info: i: 0040, t: 20.408 minutes, Δt: 20.544 seconds, umax = (4.1e-02, 1.1e-02, 1.9e-02) ms⁻¹, wall time: 57.485 seconds
[ Info: i: 0060, t: 26.423 minutes, Δt: 17.757 seconds, umax = (4.7e-02, 1.5e-02, 2.0e-02) ms⁻¹, wall time: 58.002 seconds
[ Info: i: 0080, t: 31.892 minutes, Δt: 16.592 seconds, umax = (5.0e-02, 1.6e-02, 2.2e-02) ms⁻¹, wall time: 58.491 seconds
[ Info: i: 0100, t: 36.923 minutes, Δt: 15.562 seconds, umax = (5.1e-02, 1.7e-02, 2.6e-02) ms⁻¹, wall time: 59.028 seconds
[ Info: i: 0120, t: 42.015 minutes, Δt: 15.052 seconds, umax = (5.5e-02, 1.9e-02, 2.5e-02) ms⁻¹, wall time: 59.590 seconds
[ Info: i: 0140, t: 46.956 minutes, Δt: 14.417 seconds, umax = (5.6e-02, 2.0e-02, 2.4e-02) ms⁻¹, wall time: 1.002 minutes
[ Info: i: 0160, t: 51.682 minutes, Δt: 13.261 seconds, umax = (5.7e-02, 2.4e-02, 2.7e-02) ms⁻¹, wall time: 1.011 minutes
[ Info: i: 0180, t: 56.108 minutes, Δt: 13.368 seconds, umax = (6.1e-02, 2.5e-02, 2.8e-02) ms⁻¹, wall time: 1.022 minutes
[ Info: i: 0200, t: 1.007 hours, Δt: 13.002 seconds, umax = (6.6e-02, 2.8e-02, 3.0e-02) ms⁻¹, wall time: 1.031 minutes
[ Info: i: 0220, t: 1.077 hours, Δt: 11.367 seconds, umax = (6.9e-02, 2.8e-02, 3.6e-02) ms⁻¹, wall time: 1.038 minutes
[ Info: i: 0240, t: 1.141 hours, Δt: 12.244 seconds, umax = (6.2e-02, 2.9e-02, 3.6e-02) ms⁻¹, wall time: 1.046 minutes
[ Info: i: 0260, t: 1.206 hours, Δt: 11.930 seconds, umax = (6.4e-02, 3.2e-02, 3.6e-02) ms⁻¹, wall time: 1.057 minutes
[ Info: i: 0280, t: 1.270 hours, Δt: 11.573 seconds, umax = (7.1e-02, 2.9e-02, 3.4e-02) ms⁻¹, wall time: 1.069 minutes
[ Info: i: 0300, t: 1.333 hours, Δt: 11.157 seconds, umax = (6.9e-02, 3.1e-02, 3.7e-02) ms⁻¹, wall time: 1.076 minutes
[ Info: i: 0320, t: 1.394 hours, Δt: 10.924 seconds, umax = (6.9e-02, 3.4e-02, 3.6e-02) ms⁻¹, wall time: 1.084 minutes
[ Info: i: 0340, t: 1.453 hours, Δt: 10.962 seconds, umax = (6.8e-02, 3.4e-02, 3.7e-02) ms⁻¹, wall time: 1.093 minutes
[ Info: i: 0360, t: 1.512 hours, Δt: 10.314 seconds, umax = (7.1e-02, 3.6e-02, 3.8e-02) ms⁻¹, wall time: 1.104 minutes
[ Info: i: 0380, t: 1.567 hours, Δt: 9.105 seconds, umax = (7.4e-02, 3.6e-02, 3.5e-02) ms⁻¹, wall time: 1.111 minutes
[ Info: i: 0400, t: 1.618 hours, Δt: 10.565 seconds, umax = (7.7e-02, 3.7e-02, 3.7e-02) ms⁻¹, wall time: 1.120 minutes
[ Info: i: 0420, t: 1.675 hours, Δt: 10.356 seconds, umax = (7.4e-02, 3.9e-02, 3.9e-02) ms⁻¹, wall time: 1.131 minutes
[ Info: i: 0440, t: 1.733 hours, Δt: 10.199 seconds, umax = (7.4e-02, 4.0e-02, 4.0e-02) ms⁻¹, wall time: 1.138 minutes
[ Info: i: 0460, t: 1.787 hours, Δt: 10.066 seconds, umax = (7.8e-02, 4.0e-02, 3.8e-02) ms⁻¹, wall time: 1.146 minutes
[ Info: i: 0480, t: 1.841 hours, Δt: 9.750 seconds, umax = (7.9e-02, 4.2e-02, 4.2e-02) ms⁻¹, wall time: 1.158 minutes
[ Info: i: 0500, t: 1.896 hours, Δt: 9.528 seconds, umax = (7.8e-02, 4.5e-02, 3.7e-02) ms⁻¹, wall time: 1.164 minutes
[ Info: i: 0520, t: 1.946 hours, Δt: 9.639 seconds, umax = (7.8e-02, 4.3e-02, 3.9e-02) ms⁻¹, wall time: 1.173 minutes
[ Info: i: 0540, t: 1.998 hours, Δt: 9.188 seconds, umax = (7.8e-02, 4.2e-02, 4.0e-02) ms⁻¹, wall time: 1.184 minutes
[ Info: i: 0560, t: 2.050 hours, Δt: 9.608 seconds, umax = (7.8e-02, 4.3e-02, 3.8e-02) ms⁻¹, wall time: 1.193 minutes
[ Info: i: 0580, t: 2.102 hours, Δt: 8.942 seconds, umax = (7.9e-02, 4.3e-02, 4.3e-02) ms⁻¹, wall time: 1.204 minutes
[ Info: i: 0600, t: 2.151 hours, Δt: 9.108 seconds, umax = (8.2e-02, 4.4e-02, 4.3e-02) ms⁻¹, wall time: 1.212 minutes
[ Info: i: 0620, t: 2.200 hours, Δt: 9.014 seconds, umax = (8.1e-02, 4.8e-02, 4.0e-02) ms⁻¹, wall time: 1.221 minutes
[ Info: i: 0640, t: 2.250 hours, Δt: 8.954 seconds, umax = (8.0e-02, 4.6e-02, 4.1e-02) ms⁻¹, wall time: 1.230 minutes
[ Info: i: 0660, t: 2.297 hours, Δt: 9.141 seconds, umax = (7.9e-02, 4.6e-02, 4.3e-02) ms⁻¹, wall time: 1.239 minutes
[ Info: i: 0680, t: 2.346 hours, Δt: 9.129 seconds, umax = (8.2e-02, 5.1e-02, 4.2e-02) ms⁻¹, wall time: 1.250 minutes
[ Info: i: 0700, t: 2.396 hours, Δt: 8.817 seconds, umax = (7.9e-02, 5.6e-02, 4.5e-02) ms⁻¹, wall time: 1.257 minutes
[ Info: i: 0720, t: 2.444 hours, Δt: 8.704 seconds, umax = (8.1e-02, 5.7e-02, 4.3e-02) ms⁻¹, wall time: 1.267 minutes
[ Info: i: 0740, t: 2.491 hours, Δt: 8.168 seconds, umax = (8.2e-02, 5.2e-02, 4.4e-02) ms⁻¹, wall time: 1.276 minutes
[ Info: i: 0760, t: 2.533 hours, Δt: 7.941 seconds, umax = (8.4e-02, 5.3e-02, 4.5e-02) ms⁻¹, wall time: 1.286 minutes
[ Info: i: 0780, t: 2.579 hours, Δt: 8.869 seconds, umax = (8.4e-02, 5.0e-02, 4.4e-02) ms⁻¹, wall time: 1.297 minutes
[ Info: i: 0800, t: 2.628 hours, Δt: 8.526 seconds, umax = (8.6e-02, 5.5e-02, 4.3e-02) ms⁻¹, wall time: 1.306 minutes
[ Info: i: 0820, t: 2.673 hours, Δt: 8.448 seconds, umax = (8.2e-02, 5.9e-02, 4.5e-02) ms⁻¹, wall time: 1.317 minutes
[ Info: i: 0840, t: 2.720 hours, Δt: 8.900 seconds, umax = (8.0e-02, 5.3e-02, 5.0e-02) ms⁻¹, wall time: 1.324 minutes
[ Info: i: 0860, t: 2.767 hours, Δt: 8.350 seconds, umax = (8.2e-02, 5.6e-02, 5.0e-02) ms⁻¹, wall time: 1.334 minutes
[ Info: i: 0880, t: 2.813 hours, Δt: 8.694 seconds, umax = (8.2e-02, 5.3e-02, 5.3e-02) ms⁻¹, wall time: 1.343 minutes
[ Info: i: 0900, t: 2.860 hours, Δt: 8.324 seconds, umax = (8.3e-02, 5.3e-02, 5.1e-02) ms⁻¹, wall time: 1.351 minutes
[ Info: i: 0920, t: 2.907 hours, Δt: 8.567 seconds, umax = (8.8e-02, 5.3e-02, 4.8e-02) ms⁻¹, wall time: 1.360 minutes
[ Info: i: 0940, t: 2.952 hours, Δt: 8.798 seconds, umax = (8.9e-02, 5.3e-02, 5.1e-02) ms⁻¹, wall time: 1.369 minutes
[ Info: i: 0960, t: 3 hours, Δt: 8.648 seconds, umax = (8.7e-02, 5.1e-02, 4.6e-02) ms⁻¹, wall time: 1.378 minutes
[ Info: i: 0980, t: 3.048 hours, Δt: 8.603 seconds, umax = (8.9e-02, 5.5e-02, 4.1e-02) ms⁻¹, wall time: 1.386 minutes
[ Info: i: 1000, t: 3.093 hours, Δt: 8.397 seconds, umax = (8.5e-02, 5.4e-02, 4.5e-02) ms⁻¹, wall time: 1.398 minutes
[ Info: i: 1020, t: 3.140 hours, Δt: 8.384 seconds, umax = (8.6e-02, 5.6e-02, 4.5e-02) ms⁻¹, wall time: 1.405 minutes
[ Info: i: 1040, t: 3.185 hours, Δt: 8.121 seconds, umax = (8.5e-02, 5.7e-02, 4.6e-02) ms⁻¹, wall time: 1.415 minutes
[ Info: i: 1060, t: 3.231 hours, Δt: 8.209 seconds, umax = (8.3e-02, 5.5e-02, 4.3e-02) ms⁻¹, wall time: 1.423 minutes
[ Info: i: 1080, t: 3.275 hours, Δt: 8.269 seconds, umax = (8.6e-02, 5.3e-02, 4.2e-02) ms⁻¹, wall time: 1.432 minutes
[ Info: i: 1100, t: 3.322 hours, Δt: 8.155 seconds, umax = (8.4e-02, 5.6e-02, 4.4e-02) ms⁻¹, wall time: 1.442 minutes
[ Info: i: 1120, t: 3.366 hours, Δt: 8.403 seconds, umax = (8.2e-02, 5.8e-02, 4.4e-02) ms⁻¹, wall time: 1.451 minutes
[ Info: i: 1140, t: 3.412 hours, Δt: 8.061 seconds, umax = (8.7e-02, 6.0e-02, 5.1e-02) ms⁻¹, wall time: 1.462 minutes
[ Info: i: 1160, t: 3.455 hours, Δt: 8.224 seconds, umax = (8.6e-02, 6.3e-02, 4.8e-02) ms⁻¹, wall time: 1.471 minutes
[ Info: i: 1180, t: 3.500 hours, Δt: 8.261 seconds, umax = (8.6e-02, 6.0e-02, 5.1e-02) ms⁻¹, wall time: 1.481 minutes
[ Info: i: 1200, t: 3.546 hours, Δt: 8.230 seconds, umax = (8.4e-02, 5.9e-02, 5.1e-02) ms⁻¹, wall time: 1.492 minutes
[ Info: i: 1220, t: 3.590 hours, Δt: 8.272 seconds, umax = (8.2e-02, 5.8e-02, 4.8e-02) ms⁻¹, wall time: 1.506 minutes
[ Info: i: 1240, t: 3.637 hours, Δt: 8.361 seconds, umax = (8.3e-02, 5.9e-02, 5.2e-02) ms⁻¹, wall time: 1.512 minutes
[ Info: i: 1260, t: 3.682 hours, Δt: 8.337 seconds, umax = (8.5e-02, 6.2e-02, 4.9e-02) ms⁻¹, wall time: 1.523 minutes
[ Info: i: 1280, t: 3.728 hours, Δt: 7.904 seconds, umax = (8.6e-02, 6.1e-02, 4.9e-02) ms⁻¹, wall time: 1.531 minutes
[ Info: i: 1300, t: 3.770 hours, Δt: 7.944 seconds, umax = (8.7e-02, 6.1e-02, 4.9e-02) ms⁻¹, wall time: 1.540 minutes
[ Info: i: 1320, t: 3.814 hours, Δt: 7.838 seconds, umax = (8.6e-02, 5.8e-02, 4.6e-02) ms⁻¹, wall time: 1.549 minutes
[ Info: i: 1340, t: 3.858 hours, Δt: 8.107 seconds, umax = (8.9e-02, 6.1e-02, 4.7e-02) ms⁻¹, wall time: 1.559 minutes
[ Info: i: 1360, t: 3.903 hours, Δt: 8.174 seconds, umax = (8.9e-02, 6.5e-02, 5.0e-02) ms⁻¹, wall time: 1.567 minutes
[ Info: i: 1380, t: 3.947 hours, Δt: 8.284 seconds, umax = (8.5e-02, 6.2e-02, 5.0e-02) ms⁻¹, wall time: 1.577 minutes
[ Info: i: 1400, t: 3.991 hours, Δt: 7.835 seconds, umax = (8.6e-02, 6.0e-02, 5.6e-02) ms⁻¹, wall time: 1.586 minutes
[ Info: Simulation is stopping after running for 1.589 minutes.
[ Info: Simulation time 4.000 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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