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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: 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 = (1.6e-03, 1.0e-03, 1.5e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (17.957 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (3.799 seconds).
[ Info: i: 0020, t: 12.586 minutes, Δt: 28.653 seconds, umax = (2.9e-02, 1.1e-02, 2.1e-02) ms⁻¹, wall time: 22.750 seconds
[ Info: i: 0040, t: 20.803 minutes, Δt: 19.683 seconds, umax = (4.2e-02, 1.1e-02, 1.8e-02) ms⁻¹, wall time: 23.329 seconds
[ Info: i: 0060, t: 26.765 minutes, Δt: 15.734 seconds, umax = (4.8e-02, 1.5e-02, 1.9e-02) ms⁻¹, wall time: 23.727 seconds
[ Info: i: 0080, t: 31.899 minutes, Δt: 16.706 seconds, umax = (5.0e-02, 1.6e-02, 2.3e-02) ms⁻¹, wall time: 24.176 seconds
[ Info: i: 0100, t: 37.124 minutes, Δt: 15.754 seconds, umax = (5.0e-02, 1.8e-02, 2.3e-02) ms⁻¹, wall time: 24.634 seconds
[ Info: i: 0120, t: 42.400 minutes, Δt: 15.126 seconds, umax = (5.3e-02, 2.0e-02, 2.2e-02) ms⁻¹, wall time: 25.103 seconds
[ Info: i: 0140, t: 47.240 minutes, Δt: 14.421 seconds, umax = (5.6e-02, 2.0e-02, 2.6e-02) ms⁻¹, wall time: 25.576 seconds
[ Info: i: 0160, t: 51.885 minutes, Δt: 14.068 seconds, umax = (5.8e-02, 2.4e-02, 2.8e-02) ms⁻¹, wall time: 26.049 seconds
[ Info: i: 0180, t: 56.353 minutes, Δt: 13.128 seconds, umax = (6.0e-02, 2.5e-02, 3.0e-02) ms⁻¹, wall time: 26.553 seconds
[ Info: i: 0200, t: 1.010 hours, Δt: 12.889 seconds, umax = (6.1e-02, 2.6e-02, 3.0e-02) ms⁻¹, wall time: 27.087 seconds
[ Info: i: 0220, t: 1.081 hours, Δt: 12.570 seconds, umax = (6.4e-02, 2.8e-02, 3.1e-02) ms⁻¹, wall time: 27.460 seconds
[ Info: i: 0240, t: 1.150 hours, Δt: 12.409 seconds, umax = (6.5e-02, 2.7e-02, 3.3e-02) ms⁻¹, wall time: 27.948 seconds
[ Info: i: 0260, t: 1.216 hours, Δt: 11.194 seconds, umax = (7.2e-02, 2.8e-02, 3.1e-02) ms⁻¹, wall time: 28.419 seconds
[ Info: i: 0280, t: 1.276 hours, Δt: 11.684 seconds, umax = (6.8e-02, 3.1e-02, 3.5e-02) ms⁻¹, wall time: 28.908 seconds
[ Info: i: 0300, t: 1.340 hours, Δt: 11.217 seconds, umax = (6.9e-02, 3.4e-02, 3.5e-02) ms⁻¹, wall time: 29.503 seconds
[ Info: i: 0320, t: 1.401 hours, Δt: 11.187 seconds, umax = (7.1e-02, 3.3e-02, 3.7e-02) ms⁻¹, wall time: 29.851 seconds
[ Info: i: 0340, t: 1.459 hours, Δt: 10.673 seconds, umax = (7.1e-02, 3.7e-02, 3.5e-02) ms⁻¹, wall time: 30.316 seconds
[ Info: i: 0360, t: 1.518 hours, Δt: 10.780 seconds, umax = (7.1e-02, 3.6e-02, 3.6e-02) ms⁻¹, wall time: 30.838 seconds
[ Info: i: 0380, t: 1.577 hours, Δt: 10.433 seconds, umax = (7.3e-02, 3.8e-02, 4.3e-02) ms⁻¹, wall time: 31.271 seconds
[ Info: i: 0400, t: 1.632 hours, Δt: 10.170 seconds, umax = (7.1e-02, 4.3e-02, 3.7e-02) ms⁻¹, wall time: 31.747 seconds
[ Info: i: 0420, t: 1.685 hours, Δt: 9.590 seconds, umax = (7.0e-02, 4.5e-02, 3.7e-02) ms⁻¹, wall time: 32.258 seconds
[ Info: i: 0440, t: 1.739 hours, Δt: 9.798 seconds, umax = (7.8e-02, 4.4e-02, 4.2e-02) ms⁻¹, wall time: 32.720 seconds
[ Info: i: 0460, t: 1.791 hours, Δt: 10.068 seconds, umax = (7.6e-02, 4.3e-02, 4.1e-02) ms⁻¹, wall time: 33.205 seconds
[ Info: i: 0480, t: 1.844 hours, Δt: 9.797 seconds, umax = (7.7e-02, 4.3e-02, 4.2e-02) ms⁻¹, wall time: 33.806 seconds
[ Info: i: 0500, t: 1.899 hours, Δt: 9.928 seconds, umax = (8.0e-02, 4.3e-02, 3.7e-02) ms⁻¹, wall time: 34.218 seconds
[ Info: i: 0520, t: 1.953 hours, Δt: 9.933 seconds, umax = (8.1e-02, 4.5e-02, 4.1e-02) ms⁻¹, wall time: 34.703 seconds
[ Info: i: 0540, t: 2.005 hours, Δt: 9.509 seconds, umax = (7.8e-02, 4.7e-02, 4.1e-02) ms⁻¹, wall time: 35.335 seconds
[ Info: i: 0560, t: 2.058 hours, Δt: 9.328 seconds, umax = (8.7e-02, 4.8e-02, 4.2e-02) ms⁻¹, wall time: 35.695 seconds
[ Info: i: 0580, t: 2.110 hours, Δt: 8.979 seconds, umax = (7.9e-02, 4.7e-02, 4.5e-02) ms⁻¹, wall time: 36.220 seconds
[ Info: i: 0600, t: 2.160 hours, Δt: 9.412 seconds, umax = (7.8e-02, 4.6e-02, 3.9e-02) ms⁻¹, wall time: 36.714 seconds
[ Info: i: 0620, t: 2.210 hours, Δt: 9.063 seconds, umax = (7.7e-02, 4.9e-02, 3.9e-02) ms⁻¹, wall time: 37.215 seconds
[ Info: i: 0640, t: 2.260 hours, Δt: 8.834 seconds, umax = (8.0e-02, 4.8e-02, 4.3e-02) ms⁻¹, wall time: 37.817 seconds
[ Info: i: 0660, t: 2.309 hours, Δt: 8.918 seconds, umax = (8.5e-02, 5.2e-02, 4.6e-02) ms⁻¹, wall time: 38.239 seconds
[ Info: i: 0680, t: 2.358 hours, Δt: 8.734 seconds, umax = (8.3e-02, 5.1e-02, 4.2e-02) ms⁻¹, wall time: 38.753 seconds
[ Info: i: 0700, t: 2.407 hours, Δt: 9.013 seconds, umax = (8.5e-02, 5.2e-02, 4.3e-02) ms⁻¹, wall time: 39.248 seconds
[ Info: i: 0720, t: 2.457 hours, Δt: 8.866 seconds, umax = (8.2e-02, 5.2e-02, 4.5e-02) ms⁻¹, wall time: 39.773 seconds
[ Info: i: 0740, t: 2.505 hours, Δt: 8.578 seconds, umax = (8.5e-02, 5.0e-02, 5.1e-02) ms⁻¹, wall time: 40.457 seconds
[ Info: i: 0760, t: 2.553 hours, Δt: 8.909 seconds, umax = (8.7e-02, 5.0e-02, 4.7e-02) ms⁻¹, wall time: 40.820 seconds
[ Info: i: 0780, t: 2.600 hours, Δt: 8.480 seconds, umax = (8.4e-02, 5.1e-02, 4.4e-02) ms⁻¹, wall time: 41.355 seconds
[ Info: i: 0800, t: 2.648 hours, Δt: 8.535 seconds, umax = (8.9e-02, 5.0e-02, 4.7e-02) ms⁻¹, wall time: 41.841 seconds
[ Info: i: 0820, t: 2.693 hours, Δt: 8.858 seconds, umax = (7.9e-02, 5.1e-02, 5.0e-02) ms⁻¹, wall time: 42.347 seconds
[ Info: i: 0840, t: 2.742 hours, Δt: 8.592 seconds, umax = (8.3e-02, 5.5e-02, 4.5e-02) ms⁻¹, wall time: 42.850 seconds
[ Info: i: 0860, t: 2.788 hours, Δt: 8.653 seconds, umax = (8.2e-02, 5.5e-02, 4.6e-02) ms⁻¹, wall time: 43.353 seconds
[ Info: i: 0880, t: 2.833 hours, Δt: 8.314 seconds, umax = (8.4e-02, 5.2e-02, 4.8e-02) ms⁻¹, wall time: 43.915 seconds
[ Info: i: 0900, t: 2.879 hours, Δt: 8.326 seconds, umax = (8.4e-02, 5.2e-02, 4.3e-02) ms⁻¹, wall time: 44.498 seconds
[ Info: i: 0920, t: 2.924 hours, Δt: 8.495 seconds, umax = (8.8e-02, 5.5e-02, 4.5e-02) ms⁻¹, wall time: 45.223 seconds
[ Info: i: 0940, t: 2.971 hours, Δt: 8.836 seconds, umax = (8.7e-02, 5.5e-02, 4.7e-02) ms⁻¹, wall time: 45.675 seconds
[ Info: i: 0960, t: 3.019 hours, Δt: 8.507 seconds, umax = (8.6e-02, 5.5e-02, 4.8e-02) ms⁻¹, wall time: 46.279 seconds
[ Info: i: 0980, t: 3.066 hours, Δt: 8.335 seconds, umax = (8.6e-02, 5.5e-02, 4.9e-02) ms⁻¹, wall time: 46.865 seconds
[ Info: i: 1000, t: 3.112 hours, Δt: 8.505 seconds, umax = (8.9e-02, 5.4e-02, 4.8e-02) ms⁻¹, wall time: 47.376 seconds
[ Info: i: 1020, t: 3.159 hours, Δt: 7.970 seconds, umax = (8.7e-02, 6.2e-02, 4.7e-02) ms⁻¹, wall time: 47.889 seconds
[ Info: i: 1040, t: 3.203 hours, Δt: 8.199 seconds, umax = (8.7e-02, 5.8e-02, 4.9e-02) ms⁻¹, wall time: 48.398 seconds
[ Info: i: 1060, t: 3.248 hours, Δt: 7.954 seconds, umax = (9.0e-02, 6.1e-02, 4.8e-02) ms⁻¹, wall time: 48.917 seconds
[ Info: i: 1080, t: 3.289 hours, Δt: 7.550 seconds, umax = (8.8e-02, 6.1e-02, 5.4e-02) ms⁻¹, wall time: 49.433 seconds
[ Info: i: 1100, t: 3.331 hours, Δt: 7.586 seconds, umax = (9.3e-02, 6.1e-02, 4.9e-02) ms⁻¹, wall time: 49.948 seconds
[ Info: i: 1120, t: 3.371 hours, Δt: 7.597 seconds, umax = (9.1e-02, 6.2e-02, 5.0e-02) ms⁻¹, wall time: 50.458 seconds
[ Info: i: 1140, t: 3.415 hours, Δt: 8.062 seconds, umax = (9.1e-02, 6.7e-02, 4.6e-02) ms⁻¹, wall time: 50.986 seconds
[ Info: i: 1160, t: 3.460 hours, Δt: 8.285 seconds, umax = (8.8e-02, 6.3e-02, 4.5e-02) ms⁻¹, wall time: 51.489 seconds
[ Info: i: 1180, t: 3.505 hours, Δt: 8.459 seconds, umax = (8.9e-02, 6.7e-02, 4.5e-02) ms⁻¹, wall time: 52.138 seconds
[ Info: i: 1200, t: 3.551 hours, Δt: 8.227 seconds, umax = (8.9e-02, 5.9e-02, 4.6e-02) ms⁻¹, wall time: 52.489 seconds
[ Info: i: 1220, t: 3.595 hours, Δt: 8.568 seconds, umax = (8.4e-02, 5.9e-02, 5.0e-02) ms⁻¹, wall time: 53.067 seconds
[ Info: i: 1240, t: 3.642 hours, Δt: 8.391 seconds, umax = (8.4e-02, 5.9e-02, 5.0e-02) ms⁻¹, wall time: 53.503 seconds
[ Info: i: 1260, t: 3.685 hours, Δt: 8.283 seconds, umax = (8.5e-02, 6.0e-02, 4.8e-02) ms⁻¹, wall time: 54.016 seconds
[ Info: i: 1280, t: 3.731 hours, Δt: 8.301 seconds, umax = (8.9e-02, 6.3e-02, 4.4e-02) ms⁻¹, wall time: 54.530 seconds
[ Info: i: 1300, t: 3.775 hours, Δt: 8.024 seconds, umax = (8.9e-02, 6.0e-02, 4.7e-02) ms⁻¹, wall time: 55.038 seconds
[ Info: i: 1320, t: 3.820 hours, Δt: 8.537 seconds, umax = (8.5e-02, 6.2e-02, 4.6e-02) ms⁻¹, wall time: 55.557 seconds
[ Info: i: 1340, t: 3.866 hours, Δt: 8.239 seconds, umax = (8.4e-02, 6.3e-02, 4.6e-02) ms⁻¹, wall time: 56.068 seconds
[ Info: i: 1360, t: 3.912 hours, Δt: 8.388 seconds, umax = (8.6e-02, 6.1e-02, 4.6e-02) ms⁻¹, wall time: 56.572 seconds
[ Info: i: 1380, t: 3.958 hours, Δt: 8.132 seconds, umax = (8.7e-02, 6.1e-02, 5.2e-02) ms⁻¹, wall time: 57.077 seconds
[ Info: Simulation is stopping after running for 57.560 seconds.
[ 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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