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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: 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 = (1.5e-03, 8.3e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (32.929 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (7.254 seconds).
[ Info: i: 0020, t: 12.591 minutes, Δt: 28.995 seconds, umax = (3.0e-02, 1.1e-02, 2.2e-02) ms⁻¹, wall time: 54.384 seconds
[ Info: i: 0040, t: 21.151 minutes, Δt: 19.657 seconds, umax = (4.2e-02, 1.2e-02, 2.0e-02) ms⁻¹, wall time: 55.060 seconds
[ Info: i: 0060, t: 27.349 minutes, Δt: 16.666 seconds, umax = (5.0e-02, 1.5e-02, 2.2e-02) ms⁻¹, wall time: 55.549 seconds
[ Info: i: 0080, t: 32.744 minutes, Δt: 16.526 seconds, umax = (5.0e-02, 1.6e-02, 2.3e-02) ms⁻¹, wall time: 56.064 seconds
[ Info: i: 0100, t: 37.902 minutes, Δt: 15.170 seconds, umax = (5.0e-02, 1.8e-02, 2.4e-02) ms⁻¹, wall time: 56.629 seconds
[ Info: i: 0120, t: 42.764 minutes, Δt: 14.303 seconds, umax = (5.3e-02, 2.0e-02, 2.4e-02) ms⁻¹, wall time: 57.139 seconds
[ Info: i: 0140, t: 47.423 minutes, Δt: 14.260 seconds, umax = (5.6e-02, 2.1e-02, 2.6e-02) ms⁻¹, wall time: 57.751 seconds
[ Info: i: 0160, t: 51.922 minutes, Δt: 13.938 seconds, umax = (5.6e-02, 2.2e-02, 2.8e-02) ms⁻¹, wall time: 58.283 seconds
[ Info: i: 0180, t: 56.350 minutes, Δt: 13.653 seconds, umax = (6.1e-02, 2.3e-02, 3.1e-02) ms⁻¹, wall time: 58.905 seconds
[ Info: i: 0200, t: 1.011 hours, Δt: 12.802 seconds, umax = (6.2e-02, 2.7e-02, 2.9e-02) ms⁻¹, wall time: 59.467 seconds
[ Info: i: 0220, t: 1.082 hours, Δt: 12.591 seconds, umax = (6.5e-02, 2.7e-02, 3.4e-02) ms⁻¹, wall time: 59.859 seconds
[ Info: i: 0240, t: 1.148 hours, Δt: 11.403 seconds, umax = (6.4e-02, 3.0e-02, 3.1e-02) ms⁻¹, wall time: 1.007 minutes
[ Info: i: 0260, t: 1.211 hours, Δt: 11.775 seconds, umax = (6.3e-02, 2.8e-02, 3.4e-02) ms⁻¹, wall time: 1.016 minutes
[ Info: i: 0280, t: 1.276 hours, Δt: 11.239 seconds, umax = (6.9e-02, 3.3e-02, 3.3e-02) ms⁻¹, wall time: 1.026 minutes
[ Info: i: 0300, t: 1.337 hours, Δt: 10.682 seconds, umax = (7.1e-02, 3.6e-02, 3.7e-02) ms⁻¹, wall time: 1.036 minutes
[ Info: i: 0320, t: 1.396 hours, Δt: 10.147 seconds, umax = (6.9e-02, 3.4e-02, 3.5e-02) ms⁻¹, wall time: 1.042 minutes
[ Info: i: 0340, t: 1.451 hours, Δt: 10.363 seconds, umax = (7.0e-02, 3.4e-02, 3.7e-02) ms⁻¹, wall time: 1.051 minutes
[ Info: i: 0360, t: 1.509 hours, Δt: 10.533 seconds, umax = (7.1e-02, 3.5e-02, 4.4e-02) ms⁻¹, wall time: 1.062 minutes
[ Info: i: 0380, t: 1.567 hours, Δt: 9.877 seconds, umax = (7.5e-02, 3.7e-02, 3.8e-02) ms⁻¹, wall time: 1.068 minutes
[ Info: i: 0400, t: 1.623 hours, Δt: 9.900 seconds, umax = (7.5e-02, 3.9e-02, 3.9e-02) ms⁻¹, wall time: 1.077 minutes
[ Info: i: 0420, t: 1.678 hours, Δt: 9.699 seconds, umax = (7.4e-02, 4.1e-02, 3.8e-02) ms⁻¹, wall time: 1.087 minutes
[ Info: i: 0440, t: 1.732 hours, Δt: 9.833 seconds, umax = (7.6e-02, 4.0e-02, 3.7e-02) ms⁻¹, wall time: 1.093 minutes
[ Info: i: 0460, t: 1.786 hours, Δt: 9.895 seconds, umax = (7.9e-02, 4.1e-02, 3.7e-02) ms⁻¹, wall time: 1.102 minutes
[ Info: i: 0480, t: 1.839 hours, Δt: 9.784 seconds, umax = (8.2e-02, 4.4e-02, 4.3e-02) ms⁻¹, wall time: 1.114 minutes
[ Info: i: 0500, t: 1.892 hours, Δt: 9.307 seconds, umax = (7.9e-02, 4.3e-02, 4.1e-02) ms⁻¹, wall time: 1.120 minutes
[ Info: i: 0520, t: 1.940 hours, Δt: 9.494 seconds, umax = (7.7e-02, 4.4e-02, 4.5e-02) ms⁻¹, wall time: 1.129 minutes
[ Info: i: 0540, t: 1.992 hours, Δt: 9.371 seconds, umax = (8.0e-02, 4.6e-02, 4.6e-02) ms⁻¹, wall time: 1.138 minutes
[ Info: i: 0560, t: 2.041 hours, Δt: 9.375 seconds, umax = (7.7e-02, 4.7e-02, 4.1e-02) ms⁻¹, wall time: 1.147 minutes
[ Info: i: 0580, t: 2.091 hours, Δt: 8.938 seconds, umax = (7.7e-02, 4.5e-02, 4.3e-02) ms⁻¹, wall time: 1.159 minutes
[ Info: i: 0600, t: 2.141 hours, Δt: 8.984 seconds, umax = (7.6e-02, 4.7e-02, 4.6e-02) ms⁻¹, wall time: 1.165 minutes
[ Info: i: 0620, t: 2.190 hours, Δt: 9.489 seconds, umax = (7.7e-02, 4.8e-02, 4.4e-02) ms⁻¹, wall time: 1.174 minutes
[ Info: i: 0640, t: 2.243 hours, Δt: 9.474 seconds, umax = (8.0e-02, 4.8e-02, 4.5e-02) ms⁻¹, wall time: 1.183 minutes
[ Info: i: 0660, t: 2.294 hours, Δt: 9.142 seconds, umax = (8.1e-02, 4.8e-02, 4.5e-02) ms⁻¹, wall time: 1.191 minutes
[ Info: i: 0680, t: 2.343 hours, Δt: 9.037 seconds, umax = (8.5e-02, 4.8e-02, 4.6e-02) ms⁻¹, wall time: 1.203 minutes
[ Info: i: 0700, t: 2.393 hours, Δt: 8.814 seconds, umax = (8.1e-02, 4.9e-02, 4.3e-02) ms⁻¹, wall time: 1.209 minutes
[ Info: i: 0720, t: 2.441 hours, Δt: 8.787 seconds, umax = (8.2e-02, 4.8e-02, 4.7e-02) ms⁻¹, wall time: 1.218 minutes
[ Info: i: 0740, t: 2.489 hours, Δt: 8.406 seconds, umax = (8.4e-02, 5.5e-02, 5.0e-02) ms⁻¹, wall time: 1.227 minutes
[ Info: i: 0760, t: 2.536 hours, Δt: 8.970 seconds, umax = (7.9e-02, 5.5e-02, 4.8e-02) ms⁻¹, wall time: 1.236 minutes
[ Info: i: 0780, t: 2.583 hours, Δt: 8.648 seconds, umax = (8.4e-02, 5.2e-02, 4.8e-02) ms⁻¹, wall time: 1.245 minutes
[ Info: i: 0800, t: 2.632 hours, Δt: 8.894 seconds, umax = (8.4e-02, 5.2e-02, 4.6e-02) ms⁻¹, wall time: 1.254 minutes
[ Info: i: 0820, t: 2.676 hours, Δt: 8.430 seconds, umax = (8.2e-02, 5.4e-02, 4.5e-02) ms⁻¹, wall time: 1.267 minutes
[ Info: i: 0840, t: 2.723 hours, Δt: 8.675 seconds, umax = (8.3e-02, 5.0e-02, 4.6e-02) ms⁻¹, wall time: 1.273 minutes
[ Info: i: 0860, t: 2.769 hours, Δt: 8.763 seconds, umax = (8.6e-02, 5.1e-02, 4.9e-02) ms⁻¹, wall time: 1.284 minutes
[ Info: i: 0880, t: 2.818 hours, Δt: 8.890 seconds, umax = (8.3e-02, 5.5e-02, 4.9e-02) ms⁻¹, wall time: 1.292 minutes
[ Info: i: 0900, t: 2.865 hours, Δt: 8.766 seconds, umax = (8.3e-02, 5.4e-02, 4.6e-02) ms⁻¹, wall time: 1.301 minutes
[ Info: i: 0920, t: 2.912 hours, Δt: 8.192 seconds, umax = (8.4e-02, 5.5e-02, 4.4e-02) ms⁻¹, wall time: 1.310 minutes
[ Info: i: 0940, t: 2.959 hours, Δt: 8.584 seconds, umax = (8.5e-02, 5.0e-02, 4.6e-02) ms⁻¹, wall time: 1.319 minutes
[ Info: i: 0960, t: 3.005 hours, Δt: 8.622 seconds, umax = (8.8e-02, 5.4e-02, 4.6e-02) ms⁻¹, wall time: 1.331 minutes
[ Info: i: 0980, t: 3.053 hours, Δt: 8.527 seconds, umax = (8.7e-02, 5.3e-02, 4.6e-02) ms⁻¹, wall time: 1.337 minutes
[ Info: i: 1000, t: 3.100 hours, Δt: 8.333 seconds, umax = (8.5e-02, 5.7e-02, 4.8e-02) ms⁻¹, wall time: 1.347 minutes
[ Info: i: 1020, t: 3.146 hours, Δt: 8.434 seconds, umax = (8.6e-02, 5.5e-02, 4.6e-02) ms⁻¹, wall time: 1.355 minutes
[ Info: i: 1040, t: 3.192 hours, Δt: 8.085 seconds, umax = (8.7e-02, 5.8e-02, 4.4e-02) ms⁻¹, wall time: 1.364 minutes
[ Info: i: 1060, t: 3.236 hours, Δt: 8.016 seconds, umax = (8.8e-02, 5.7e-02, 4.6e-02) ms⁻¹, wall time: 1.374 minutes
[ Info: i: 1080, t: 3.279 hours, Δt: 8.556 seconds, umax = (8.8e-02, 6.0e-02, 4.4e-02) ms⁻¹, wall time: 1.383 minutes
[ Info: i: 1100, t: 3.326 hours, Δt: 8.267 seconds, umax = (8.6e-02, 6.0e-02, 4.5e-02) ms⁻¹, wall time: 1.392 minutes
[ Info: i: 1120, t: 3.367 hours, Δt: 8.019 seconds, umax = (8.8e-02, 6.1e-02, 5.3e-02) ms⁻¹, wall time: 1.401 minutes
[ Info: i: 1140, t: 3.412 hours, Δt: 8.376 seconds, umax = (8.5e-02, 6.0e-02, 5.0e-02) ms⁻¹, wall time: 1.411 minutes
[ Info: i: 1160, t: 3.456 hours, Δt: 8.137 seconds, umax = (8.7e-02, 5.9e-02, 4.9e-02) ms⁻¹, wall time: 1.420 minutes
[ Info: i: 1180, t: 3.500 hours, Δt: 8.099 seconds, umax = (8.8e-02, 5.9e-02, 4.8e-02) ms⁻¹, wall time: 1.428 minutes
[ Info: i: 1200, t: 3.545 hours, Δt: 8.259 seconds, umax = (8.5e-02, 6.2e-02, 4.6e-02) ms⁻¹, wall time: 1.438 minutes
[ Info: i: 1220, t: 3.590 hours, Δt: 8.261 seconds, umax = (8.4e-02, 5.5e-02, 5.0e-02) ms⁻¹, wall time: 1.450 minutes
[ Info: i: 1240, t: 3.637 hours, Δt: 8.189 seconds, umax = (8.7e-02, 6.2e-02, 4.9e-02) ms⁻¹, wall time: 1.456 minutes
[ Info: i: 1260, t: 3.680 hours, Δt: 8.084 seconds, umax = (8.5e-02, 5.9e-02, 5.3e-02) ms⁻¹, wall time: 1.467 minutes
[ Info: i: 1280, t: 3.725 hours, Δt: 8.312 seconds, umax = (8.6e-02, 6.1e-02, 5.9e-02) ms⁻¹, wall time: 1.474 minutes
[ Info: i: 1300, t: 3.769 hours, Δt: 8.116 seconds, umax = (8.4e-02, 6.0e-02, 6.1e-02) ms⁻¹, wall time: 1.484 minutes
[ Info: i: 1320, t: 3.813 hours, Δt: 7.892 seconds, umax = (8.5e-02, 5.9e-02, 5.3e-02) ms⁻¹, wall time: 1.493 minutes
[ Info: i: 1340, t: 3.855 hours, Δt: 7.937 seconds, umax = (8.7e-02, 5.5e-02, 4.6e-02) ms⁻¹, wall time: 1.503 minutes
[ Info: i: 1360, t: 3.900 hours, Δt: 8.150 seconds, umax = (8.9e-02, 5.8e-02, 4.7e-02) ms⁻¹, wall time: 1.513 minutes
[ Info: i: 1380, t: 3.944 hours, Δt: 8.348 seconds, umax = (8.5e-02, 6.4e-02, 5.3e-02) ms⁻¹, wall time: 1.523 minutes
[ Info: i: 1400, t: 3.990 hours, Δt: 8.396 seconds, umax = (8.3e-02, 6.4e-02, 4.9e-02) ms⁻¹, wall time: 1.534 minutes
[ Info: Simulation is stopping after running for 1.538 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)

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

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