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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.

g = Oceananigans.defaults.gravitational_acceleration

amplitude = 0.8 # m
wavelength = 60  # m
wavenumber = 2π / wavelength # m⁻¹
frequency = sqrt(g * wavenumber) # s⁻¹

# The vertical scale over which the Stokes drift of a monochromatic surface wave
# decays away from the surface is `1/2wavenumber`, or
const vertical_scale = wavelength / 4π

# Stokes drift velocity at the surface
const Uˢ = amplitude^2 * wavenumber * frequency # m s⁻¹
0.06791774197745354

The const declarations ensure that Stokes drift functions compile on the GPU. To run this example on the CPU, replace GPU() with CPU() in the RectilinearGrid constructor above.

The Stokes drift profile is

uˢ(z) = Uˢ * exp(z / vertical_scale)
uˢ (generic function with 1 method)

and its z-derivative is

∂z_uˢ(z, t) = 1 / vertical_scale * Uˢ * exp(z / vertical_scale)
∂z_uˢ (generic function with 1 method)
The Craik-Leibovich equations in Oceananigans

Oceananigans implements the Craik-Leibovich approximation for surface wave effects using the Lagrangian-mean velocity field as its prognostic momentum variable. In other words, model.velocities.u is the Lagrangian-mean $x$-velocity beneath surface waves. This differs from models that use the Eulerian-mean velocity field as a prognostic variable, but has the advantage that $u$ accounts for the total advection of tracers and momentum, and that $u = v = w = 0$ is a steady solution even when Coriolis forces are present. See the physics documentation for more information.

Finally, we note that the time-derivative of the Stokes drift must be provided if the Stokes drift and surface wave field undergoes forced changes in time. In this example, the Stokes drift is constant and thus the time-derivative of the Stokes drift is 0.

Boundary conditions

At the surface $z = 0$, Wagner et al. (2021) impose

τx = -3.72e-5 # m² s⁻², surface kinematic momentum flux
u_boundary_conditions = FieldBoundaryConditions(top = FluxBoundaryCondition(τx))
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── top: FluxBoundaryCondition: -3.72e-5
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)

Wagner et al. (2021) impose a linear buoyancy gradient N² at the bottom along with a weak, destabilizing flux of buoyancy at the surface to faciliate spin-up from rest.

Jᵇ = 2.307e-8 # m² s⁻³, surface buoyancy flux
N² = 1.936e-5 # s⁻², initial and bottom buoyancy gradient

b_boundary_conditions = FieldBoundaryConditions(top = FluxBoundaryCondition(Jᵇ),
                                                bottom = GradientBoundaryCondition(N²))
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: GradientBoundaryCondition: 1.936e-5
├── top: FluxBoundaryCondition: 2.307e-8
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
The flux convention in Oceananigans

Note that Oceananigans uses "positive upward" conventions for all fluxes. In consequence, a negative flux at the surface drives positive velocities, and a positive flux of buoyancy drives cooling.

Coriolis parameter

Wagner et al. (2021) use

coriolis = FPlane(f=1e-4) # s⁻¹
FPlane{Float64}(f=0.0001)

which is typical for mid-latitudes on Earth.

Model instantiation

We are ready to build the model. We use a fifth-order Weighted Essentially Non-Oscillatory (WENO) advection scheme and the AnisotropicMinimumDissipation model for large eddy simulation. Because our Stokes drift does not vary in $x, y$, we use UniformStokesDrift, which expects Stokes drift functions of $z, t$ only.

model = NonhydrostaticModel(; grid, coriolis,
                            advection = WENO(order=9),
                            tracers = :b,
                            buoyancy = BuoyancyTracer(),
                            stokes_drift = UniformStokesDrift(∂z_uˢ=∂z_uˢ),
                            boundary_conditions = (u=u_boundary_conditions, b=b_boundary_conditions))
NonhydrostaticModel{GPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CUDAGPU with 5×5×5 halo
├── timestepper: RungeKutta3TimeStepper
├── advection scheme: WENO{5, Float64, Float32}(order=9)
├── tracers: b
├── closure: Nothing
├── buoyancy: BuoyancyTracer with ĝ = NegativeZDirection()
└── coriolis: FPlane{Float64}(f=0.0001)

Initial conditions

We make use of random noise concentrated in the upper 4 meters for buoyancy and velocity initial conditions,

Ξ(z) = randn() * exp(z / 4)

Our initial condition for buoyancy consists of a surface mixed layer 33 m deep, a deep linear stratification, plus noise,

initial_mixed_layer_depth = 33 # m
stratification(z) = z < - initial_mixed_layer_depth ? N² * z : N² * (-initial_mixed_layer_depth)

bᵢ(x, y, z) = stratification(z) + 1e-1 * Ξ(z) * N² * model.grid.Lz
bᵢ (generic function with 1 method)

The simulation we reproduce from Wagner et al. (2021) is zero Lagrangian-mean velocity. This initial condition is consistent with a wavy, quiescent ocean suddenly impacted by winds. To this quiescent state we add noise scaled by the friction velocity to $u$ and $w$.

u★ = sqrt(abs(τx))
uᵢ(x, y, z) = u★ * 1e-1 * Ξ(z)
wᵢ(x, y, z) = u★ * 1e-1 * Ξ(z)

set!(model, u=uᵢ, w=wᵢ, b=bᵢ)

Setting up the simulation

simulation = Simulation(model, Δt=45.0, stop_time=4hours)
Simulation of NonhydrostaticModel{GPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── Next time step: 45 seconds
├── Elapsed wall time: 0 seconds
├── Wall time per iteration: NaN days
├── Stop time: 4 hours
├── Stop iteration: Inf
├── Wall time limit: Inf
├── Minimum relative step: 0.0
├── Callbacks: OrderedDict with 4 entries:
│   ├── stop_time_exceeded => 4
│   ├── stop_iteration_exceeded => -
│   ├── wall_time_limit_exceeded => e
│   └── nan_checker => }
├── Output writers: OrderedDict with no entries
└── Diagnostics: OrderedDict with no entries

We use the TimeStepWizard for adaptive time-stepping with a Courant-Freidrichs-Lewy (CFL) number of 1.0,

conjure_time_step_wizard!(simulation, cfl=1.0, max_Δt=1minute)

Nice progress messaging

We define a function that prints a helpful message with maximum absolute value of $u, v, w$ and the current wall clock time.

using Printf

function progress(simulation)
    u, v, w = simulation.model.velocities

    # Print a progress message
    msg = @sprintf("i: %04d, t: %s, Δt: %s, umax = (%.1e, %.1e, %.1e) ms⁻¹, wall time: %s\n",
                   iteration(simulation),
                   prettytime(time(simulation)),
                   prettytime(simulation.Δt),
                   maximum(abs, u), maximum(abs, v), maximum(abs, w),
                   prettytime(simulation.run_wall_time))

    @info msg

    return nothing
end

simulation.callbacks[:progress] = Callback(progress, IterationInterval(20))
Callback of progress on IterationInterval(20)

Output

A field writer

We set up an output writer for the simulation that saves all velocity fields, tracer fields, and the subgrid turbulent diffusivity.

output_interval = 5minutes

fields_to_output = merge(model.velocities, model.tracers)

simulation.output_writers[:fields] =
    JLD2Writer(model, fields_to_output,
               schedule = TimeInterval(output_interval),
               filename = "langmuir_turbulence_fields.jld2",
               overwrite_existing = true)
JLD2Writer scheduled on TimeInterval(5 minutes):
├── filepath: langmuir_turbulence_fields.jld2
├── 4 outputs: (u, v, w, b)
├── array_type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 44.7 KiB

An "averages" writer

We also set up output of time- and horizontally-averaged velocity field and momentum fluxes.

u, v, w = model.velocities
b = model.tracers.b

 U = Average(u, dims=(1, 2))
 V = Average(v, dims=(1, 2))
 B = Average(b, dims=(1, 2))
wu = Average(w * u, dims=(1, 2))
wv = Average(w * v, dims=(1, 2))

simulation.output_writers[:averages] =
    JLD2Writer(model, (; U, V, B, wu, wv),
               schedule = AveragedTimeInterval(output_interval, window=2minutes),
               filename = "langmuir_turbulence_averages.jld2",
               overwrite_existing = true)
JLD2Writer scheduled on TimeInterval(5 minutes):
├── filepath: langmuir_turbulence_averages.jld2
├── 5 outputs: (U, V, B, wu, wv) averaged on AveragedTimeInterval(window=2 minutes, stride=1, interval=5 minutes)
├── array_type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 44.3 KiB

Running the simulation

This part is easy,

run!(simulation)
[ Info: Initializing simulation...
[ Info: i: 0000, t: 0 seconds, Δt: 49.500 seconds, umax = (1.7e-03, 1.0e-03, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (19.857 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (4.090 seconds).
[ Info: i: 0020, t: 11.774 minutes, Δt: 27.957 seconds, umax = (2.9e-02, 1.1e-02, 2.3e-02) ms⁻¹, wall time: 25.229 seconds
[ Info: i: 0040, t: 20 minutes, Δt: 21.192 seconds, umax = (4.1e-02, 1.0e-02, 2.1e-02) ms⁻¹, wall time: 25.795 seconds
[ Info: i: 0060, t: 26.507 minutes, Δt: 17.408 seconds, umax = (4.7e-02, 1.5e-02, 2.3e-02) ms⁻¹, wall time: 26.420 seconds
[ Info: i: 0080, t: 31.859 minutes, Δt: 15.887 seconds, umax = (5.1e-02, 1.8e-02, 2.6e-02) ms⁻¹, wall time: 26.966 seconds
[ Info: i: 0100, t: 37.088 minutes, Δt: 16.151 seconds, umax = (5.0e-02, 1.8e-02, 2.3e-02) ms⁻¹, wall time: 27.469 seconds
[ Info: i: 0120, t: 42.364 minutes, Δt: 15.144 seconds, umax = (5.4e-02, 1.9e-02, 2.5e-02) ms⁻¹, wall time: 28.017 seconds
[ Info: i: 0140, t: 47.231 minutes, Δt: 14.653 seconds, umax = (5.6e-02, 2.1e-02, 2.5e-02) ms⁻¹, wall time: 28.542 seconds
[ Info: i: 0160, t: 51.938 minutes, Δt: 13.484 seconds, umax = (5.9e-02, 2.2e-02, 2.7e-02) ms⁻¹, wall time: 29.119 seconds
[ Info: i: 0180, t: 56.401 minutes, Δt: 13.495 seconds, umax = (5.9e-02, 2.4e-02, 3.0e-02) ms⁻¹, wall time: 29.711 seconds
[ Info: i: 0200, t: 1.007 hours, Δt: 12.795 seconds, umax = (6.3e-02, 2.5e-02, 2.9e-02) ms⁻¹, wall time: 30.311 seconds
[ Info: i: 0220, t: 1.078 hours, Δt: 12.029 seconds, umax = (6.7e-02, 2.6e-02, 2.9e-02) ms⁻¹, wall time: 30.682 seconds
[ Info: i: 0240, t: 1.142 hours, Δt: 12.083 seconds, umax = (6.5e-02, 2.8e-02, 3.1e-02) ms⁻¹, wall time: 31.228 seconds
[ Info: i: 0260, t: 1.207 hours, Δt: 11.744 seconds, umax = (6.7e-02, 3.1e-02, 3.8e-02) ms⁻¹, wall time: 31.726 seconds
[ Info: i: 0280, t: 1.269 hours, Δt: 11.321 seconds, umax = (6.7e-02, 3.3e-02, 3.6e-02) ms⁻¹, wall time: 32.267 seconds
[ Info: i: 0300, t: 1.331 hours, Δt: 10.868 seconds, umax = (6.9e-02, 3.4e-02, 3.7e-02) ms⁻¹, wall time: 32.697 seconds
[ Info: i: 0320, t: 1.391 hours, Δt: 10.858 seconds, umax = (7.0e-02, 3.6e-02, 3.7e-02) ms⁻¹, wall time: 33.209 seconds
[ Info: i: 0340, t: 1.449 hours, Δt: 10.867 seconds, umax = (7.2e-02, 3.5e-02, 3.6e-02) ms⁻¹, wall time: 33.718 seconds
[ Info: i: 0360, t: 1.506 hours, Δt: 10.198 seconds, umax = (7.7e-02, 4.0e-02, 3.9e-02) ms⁻¹, wall time: 34.359 seconds
[ Info: i: 0380, t: 1.562 hours, Δt: 10.354 seconds, umax = (7.1e-02, 3.9e-02, 4.0e-02) ms⁻¹, wall time: 34.714 seconds
[ Info: i: 0400, t: 1.618 hours, Δt: 10.376 seconds, umax = (7.2e-02, 3.7e-02, 4.1e-02) ms⁻¹, wall time: 35.231 seconds
[ Info: i: 0420, t: 1.672 hours, Δt: 9.688 seconds, umax = (7.3e-02, 3.8e-02, 4.0e-02) ms⁻¹, wall time: 35.896 seconds
[ Info: i: 0440, t: 1.725 hours, Δt: 10.082 seconds, umax = (7.2e-02, 4.3e-02, 3.6e-02) ms⁻¹, wall time: 36.252 seconds
[ Info: i: 0460, t: 1.781 hours, Δt: 9.747 seconds, umax = (7.5e-02, 4.0e-02, 4.1e-02) ms⁻¹, wall time: 36.775 seconds
[ Info: i: 0480, t: 1.833 hours, Δt: 9.235 seconds, umax = (7.7e-02, 4.3e-02, 4.5e-02) ms⁻¹, wall time: 37.254 seconds
[ Info: i: 0500, t: 1.885 hours, Δt: 9.221 seconds, umax = (7.8e-02, 4.3e-02, 4.0e-02) ms⁻¹, wall time: 37.753 seconds
[ Info: i: 0520, t: 1.935 hours, Δt: 9.586 seconds, umax = (8.2e-02, 4.3e-02, 4.0e-02) ms⁻¹, wall time: 38.306 seconds
[ Info: i: 0540, t: 1.987 hours, Δt: 8.963 seconds, umax = (7.8e-02, 4.5e-02, 4.4e-02) ms⁻¹, wall time: 38.872 seconds
[ Info: i: 0560, t: 2.036 hours, Δt: 9.216 seconds, umax = (7.9e-02, 4.5e-02, 3.9e-02) ms⁻¹, wall time: 39.450 seconds
[ Info: i: 0580, t: 2.086 hours, Δt: 9.180 seconds, umax = (7.7e-02, 4.7e-02, 4.4e-02) ms⁻¹, wall time: 40.232 seconds
[ Info: i: 0600, t: 2.138 hours, Δt: 9.050 seconds, umax = (7.8e-02, 5.0e-02, 4.3e-02) ms⁻¹, wall time: 40.635 seconds
[ Info: i: 0620, t: 2.187 hours, Δt: 9.184 seconds, umax = (7.8e-02, 5.0e-02, 4.2e-02) ms⁻¹, wall time: 41.217 seconds
[ Info: i: 0640, t: 2.238 hours, Δt: 9.238 seconds, umax = (8.2e-02, 4.8e-02, 4.5e-02) ms⁻¹, wall time: 41.725 seconds
[ Info: i: 0660, t: 2.284 hours, Δt: 8.954 seconds, umax = (8.2e-02, 4.8e-02, 4.5e-02) ms⁻¹, wall time: 42.257 seconds
[ Info: i: 0680, t: 2.333 hours, Δt: 9.162 seconds, umax = (7.8e-02, 4.8e-02, 4.6e-02) ms⁻¹, wall time: 42.760 seconds
[ Info: i: 0700, t: 2.384 hours, Δt: 8.366 seconds, umax = (8.1e-02, 4.9e-02, 4.6e-02) ms⁻¹, wall time: 43.278 seconds
[ Info: i: 0720, t: 2.429 hours, Δt: 8.886 seconds, umax = (8.2e-02, 5.2e-02, 4.7e-02) ms⁻¹, wall time: 43.904 seconds
[ Info: i: 0740, t: 2.479 hours, Δt: 9.113 seconds, umax = (8.3e-02, 5.2e-02, 4.5e-02) ms⁻¹, wall time: 44.342 seconds
[ Info: i: 0760, t: 2.527 hours, Δt: 8.505 seconds, umax = (8.5e-02, 5.5e-02, 4.5e-02) ms⁻¹, wall time: 44.879 seconds
[ Info: i: 0780, t: 2.575 hours, Δt: 8.428 seconds, umax = (8.2e-02, 6.0e-02, 4.0e-02) ms⁻¹, wall time: 45.394 seconds
[ Info: i: 0800, t: 2.619 hours, Δt: 8.737 seconds, umax = (8.0e-02, 5.8e-02, 4.5e-02) ms⁻¹, wall time: 45.929 seconds
[ Info: i: 0820, t: 2.667 hours, Δt: 8.699 seconds, umax = (8.0e-02, 5.7e-02, 4.4e-02) ms⁻¹, wall time: 46.436 seconds
[ Info: i: 0840, t: 2.715 hours, Δt: 8.800 seconds, umax = (8.3e-02, 5.6e-02, 4.5e-02) ms⁻¹, wall time: 46.947 seconds
[ Info: i: 0860, t: 2.762 hours, Δt: 8.247 seconds, umax = (8.4e-02, 5.8e-02, 4.7e-02) ms⁻¹, wall time: 47.573 seconds
[ Info: i: 0880, t: 2.807 hours, Δt: 8.371 seconds, umax = (8.2e-02, 5.5e-02, 5.3e-02) ms⁻¹, wall time: 48.018 seconds
[ Info: i: 0900, t: 2.852 hours, Δt: 8.360 seconds, umax = (8.6e-02, 5.9e-02, 4.8e-02) ms⁻¹, wall time: 48.557 seconds
[ Info: i: 0920, t: 2.898 hours, Δt: 7.703 seconds, umax = (8.9e-02, 6.2e-02, 4.8e-02) ms⁻¹, wall time: 49.061 seconds
[ Info: i: 0940, t: 2.942 hours, Δt: 8.525 seconds, umax = (8.3e-02, 5.9e-02, 4.9e-02) ms⁻¹, wall time: 49.605 seconds
[ Info: i: 0960, t: 2.989 hours, Δt: 8.349 seconds, umax = (8.4e-02, 5.8e-02, 4.9e-02) ms⁻¹, wall time: 50.125 seconds
[ Info: i: 0980, t: 3.034 hours, Δt: 7.756 seconds, umax = (8.5e-02, 5.9e-02, 4.5e-02) ms⁻¹, wall time: 50.649 seconds
[ Info: i: 1000, t: 3.077 hours, Δt: 7.812 seconds, umax = (8.6e-02, 5.7e-02, 4.7e-02) ms⁻¹, wall time: 51.155 seconds
[ Info: i: 1020, t: 3.120 hours, Δt: 7.790 seconds, umax = (8.8e-02, 5.7e-02, 4.5e-02) ms⁻¹, wall time: 51.699 seconds
[ Info: i: 1040, t: 3.163 hours, Δt: 7.902 seconds, umax = (8.4e-02, 6.1e-02, 4.6e-02) ms⁻¹, wall time: 52.216 seconds
[ Info: i: 1060, t: 3.207 hours, Δt: 7.946 seconds, umax = (8.6e-02, 6.3e-02, 4.7e-02) ms⁻¹, wall time: 52.748 seconds
[ Info: i: 1080, t: 3.250 hours, Δt: 8.215 seconds, umax = (8.6e-02, 5.4e-02, 5.5e-02) ms⁻¹, wall time: 53.262 seconds
[ Info: i: 1100, t: 3.295 hours, Δt: 8.029 seconds, umax = (8.6e-02, 5.9e-02, 5.4e-02) ms⁻¹, wall time: 53.786 seconds
[ Info: i: 1120, t: 3.340 hours, Δt: 7.979 seconds, umax = (8.7e-02, 5.9e-02, 4.5e-02) ms⁻¹, wall time: 54.437 seconds
[ Info: i: 1140, t: 3.385 hours, Δt: 8.178 seconds, umax = (8.4e-02, 6.2e-02, 4.5e-02) ms⁻¹, wall time: 54.813 seconds
[ Info: i: 1160, t: 3.431 hours, Δt: 7.863 seconds, umax = (8.3e-02, 5.8e-02, 4.8e-02) ms⁻¹, wall time: 55.404 seconds
[ Info: i: 1180, t: 3.475 hours, Δt: 8.396 seconds, umax = (8.4e-02, 6.1e-02, 5.5e-02) ms⁻¹, wall time: 55.872 seconds
[ Info: i: 1200, t: 3.521 hours, Δt: 8.226 seconds, umax = (8.3e-02, 6.0e-02, 4.8e-02) ms⁻¹, wall time: 56.395 seconds
[ Info: i: 1220, t: 3.567 hours, Δt: 8.250 seconds, umax = (8.4e-02, 5.7e-02, 4.8e-02) ms⁻¹, wall time: 56.919 seconds
[ Info: i: 1240, t: 3.611 hours, Δt: 8.055 seconds, umax = (8.8e-02, 5.7e-02, 4.5e-02) ms⁻¹, wall time: 57.429 seconds
[ Info: i: 1260, t: 3.655 hours, Δt: 8.039 seconds, umax = (8.5e-02, 5.4e-02, 4.7e-02) ms⁻¹, wall time: 57.938 seconds
[ Info: i: 1280, t: 3.698 hours, Δt: 7.797 seconds, umax = (8.4e-02, 6.2e-02, 4.5e-02) ms⁻¹, wall time: 58.483 seconds
[ Info: i: 1300, t: 3.741 hours, Δt: 8.014 seconds, umax = (8.4e-02, 6.2e-02, 4.6e-02) ms⁻¹, wall time: 58.988 seconds
[ Info: i: 1320, t: 3.786 hours, Δt: 8.288 seconds, umax = (8.9e-02, 6.3e-02, 4.6e-02) ms⁻¹, wall time: 59.505 seconds
[ Info: i: 1340, t: 3.832 hours, Δt: 8.111 seconds, umax = (8.8e-02, 6.0e-02, 5.3e-02) ms⁻¹, wall time: 1.001 minutes
[ Info: i: 1360, t: 3.875 hours, Δt: 8.034 seconds, umax = (8.8e-02, 6.0e-02, 5.3e-02) ms⁻¹, wall time: 1.010 minutes
[ Info: i: 1380, t: 3.919 hours, Δt: 8.003 seconds, umax = (8.4e-02, 6.2e-02, 4.9e-02) ms⁻¹, wall time: 1.023 minutes
[ Info: i: 1400, t: 3.964 hours, Δt: 8.017 seconds, umax = (8.4e-02, 6.1e-02, 4.9e-02) ms⁻¹, wall time: 1.029 minutes
[ Info: Simulation is stopping after running for 1.038 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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