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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
├── run_wall_time: 0 seconds
├── run_wall_time / 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 => Callback of stop_time_exceeded on IterationInterval(1)
│   ├── stop_iteration_exceeded => Callback of stop_iteration_exceeded on IterationInterval(1)
│   ├── wall_time_limit_exceeded => Callback of wall_time_limit_exceeded on IterationInterval(1)
│   └── nan_checker => Callback of NaNChecker for u on IterationInterval(100)
└── output_writers: 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: 44.8 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: 44.4 KiB

Running the simulation

This part is easy,

run!(simulation)
[ Info: Initializing simulation...
[ Info: i: 0000, t: 0 seconds, Δt: 49.500 seconds, umax = (1.7e-03, 9.4e-04, 1.7e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (17.934 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.846 seconds).
[ Info: i: 0020, t: 11.995 minutes, Δt: 30.173 seconds, umax = (2.8e-02, 1.1e-02, 2.0e-02) ms⁻¹, wall time: 21.952 seconds
[ Info: i: 0040, t: 20.787 minutes, Δt: 19.567 seconds, umax = (4.1e-02, 1.0e-02, 1.8e-02) ms⁻¹, wall time: 22.532 seconds
[ Info: i: 0060, t: 26.813 minutes, Δt: 16.165 seconds, umax = (4.8e-02, 1.5e-02, 2.0e-02) ms⁻¹, wall time: 22.924 seconds
[ Info: i: 0080, t: 32.204 minutes, Δt: 16.221 seconds, umax = (5.0e-02, 1.6e-02, 2.2e-02) ms⁻¹, wall time: 23.379 seconds
[ Info: i: 0100, t: 37.348 minutes, Δt: 16.191 seconds, umax = (5.1e-02, 1.9e-02, 2.4e-02) ms⁻¹, wall time: 24.023 seconds
[ Info: i: 0120, t: 42.520 minutes, Δt: 14.979 seconds, umax = (5.4e-02, 2.0e-02, 2.5e-02) ms⁻¹, wall time: 25.204 seconds
[ Info: i: 0140, t: 47.490 minutes, Δt: 14.384 seconds, umax = (5.8e-02, 2.3e-02, 2.7e-02) ms⁻¹, wall time: 25.695 seconds
[ Info: i: 0160, t: 52.105 minutes, Δt: 13.994 seconds, umax = (5.9e-02, 2.2e-02, 2.8e-02) ms⁻¹, wall time: 26.183 seconds
[ Info: i: 0180, t: 56.578 minutes, Δt: 13.389 seconds, umax = (6.6e-02, 2.4e-02, 2.9e-02) ms⁻¹, wall time: 26.688 seconds
[ Info: i: 0200, t: 1.015 hours, Δt: 13.049 seconds, umax = (6.1e-02, 2.5e-02, 3.0e-02) ms⁻¹, wall time: 27.217 seconds
[ Info: i: 0220, t: 1.083 hours, Δt: 12.417 seconds, umax = (6.3e-02, 2.7e-02, 3.2e-02) ms⁻¹, wall time: 27.590 seconds
[ Info: i: 0240, t: 1.153 hours, Δt: 12.222 seconds, umax = (6.7e-02, 3.1e-02, 3.2e-02) ms⁻¹, wall time: 28.064 seconds
[ Info: i: 0260, t: 1.217 hours, Δt: 11.510 seconds, umax = (6.7e-02, 3.0e-02, 3.4e-02) ms⁻¹, wall time: 28.520 seconds
[ Info: i: 0280, t: 1.279 hours, Δt: 11.017 seconds, umax = (6.6e-02, 3.3e-02, 3.2e-02) ms⁻¹, wall time: 29.015 seconds
[ Info: i: 0300, t: 1.339 hours, Δt: 10.923 seconds, umax = (7.0e-02, 3.6e-02, 3.5e-02) ms⁻¹, wall time: 29.605 seconds
[ Info: i: 0320, t: 1.399 hours, Δt: 10.482 seconds, umax = (7.0e-02, 3.4e-02, 3.5e-02) ms⁻¹, wall time: 29.944 seconds
[ Info: i: 0340, t: 1.458 hours, Δt: 10.561 seconds, umax = (7.0e-02, 3.4e-02, 3.8e-02) ms⁻¹, wall time: 30.417 seconds
[ Info: i: 0360, t: 1.515 hours, Δt: 10.280 seconds, umax = (7.4e-02, 3.8e-02, 3.9e-02) ms⁻¹, wall time: 30.989 seconds
[ Info: i: 0380, t: 1.570 hours, Δt: 9.857 seconds, umax = (7.2e-02, 4.1e-02, 3.9e-02) ms⁻¹, wall time: 31.464 seconds
[ Info: i: 0400, t: 1.625 hours, Δt: 10.286 seconds, umax = (7.4e-02, 4.5e-02, 3.5e-02) ms⁻¹, wall time: 31.945 seconds
[ Info: i: 0420, t: 1.681 hours, Δt: 10.253 seconds, umax = (7.4e-02, 4.5e-02, 4.0e-02) ms⁻¹, wall time: 32.497 seconds
[ Info: i: 0440, t: 1.738 hours, Δt: 10.085 seconds, umax = (7.5e-02, 4.8e-02, 3.9e-02) ms⁻¹, wall time: 32.911 seconds
[ Info: i: 0460, t: 1.792 hours, Δt: 10.286 seconds, umax = (7.5e-02, 4.9e-02, 4.5e-02) ms⁻¹, wall time: 33.398 seconds
[ Info: i: 0480, t: 1.847 hours, Δt: 9.769 seconds, umax = (7.6e-02, 4.5e-02, 4.2e-02) ms⁻¹, wall time: 34.044 seconds
[ Info: i: 0500, t: 1.901 hours, Δt: 9.697 seconds, umax = (7.7e-02, 4.6e-02, 3.9e-02) ms⁻¹, wall time: 34.475 seconds
[ Info: i: 0520, t: 1.954 hours, Δt: 9.821 seconds, umax = (7.6e-02, 4.4e-02, 3.9e-02) ms⁻¹, wall time: 35.866 seconds
[ Info: i: 0540, t: 2.008 hours, Δt: 9.681 seconds, umax = (8.8e-02, 4.6e-02, 4.2e-02) ms⁻¹, wall time: 36.561 seconds
[ Info: i: 0560, t: 2.062 hours, Δt: 9.617 seconds, umax = (8.1e-02, 4.8e-02, 4.2e-02) ms⁻¹, wall time: 36.936 seconds
[ Info: i: 0580, t: 2.112 hours, Δt: 9.185 seconds, umax = (7.7e-02, 4.9e-02, 4.1e-02) ms⁻¹, wall time: 37.438 seconds
[ Info: i: 0600, t: 2.164 hours, Δt: 8.998 seconds, umax = (7.9e-02, 4.9e-02, 4.7e-02) ms⁻¹, wall time: 37.922 seconds
[ Info: i: 0620, t: 2.212 hours, Δt: 9.159 seconds, umax = (8.1e-02, 4.9e-02, 4.9e-02) ms⁻¹, wall time: 38.417 seconds
[ Info: i: 0640, t: 2.260 hours, Δt: 8.772 seconds, umax = (7.9e-02, 5.2e-02, 4.6e-02) ms⁻¹, wall time: 39.264 seconds
[ Info: i: 0660, t: 2.310 hours, Δt: 9.184 seconds, umax = (8.4e-02, 4.9e-02, 4.5e-02) ms⁻¹, wall time: 39.668 seconds
[ Info: i: 0680, t: 2.358 hours, Δt: 9.043 seconds, umax = (8.3e-02, 4.7e-02, 4.6e-02) ms⁻¹, wall time: 40.305 seconds
[ Info: i: 0700, t: 2.408 hours, Δt: 8.961 seconds, umax = (7.8e-02, 4.5e-02, 4.5e-02) ms⁻¹, wall time: 40.931 seconds
[ Info: i: 0720, t: 2.457 hours, Δt: 9.057 seconds, umax = (7.9e-02, 4.6e-02, 4.5e-02) ms⁻¹, wall time: 41.456 seconds
[ Info: i: 0740, t: 2.505 hours, Δt: 8.737 seconds, umax = (8.4e-02, 5.2e-02, 4.3e-02) ms⁻¹, wall time: 42.327 seconds
[ Info: i: 0760, t: 2.553 hours, Δt: 8.419 seconds, umax = (8.3e-02, 4.9e-02, 4.5e-02) ms⁻¹, wall time: 42.681 seconds
[ Info: i: 0780, t: 2.600 hours, Δt: 8.285 seconds, umax = (8.5e-02, 5.5e-02, 4.9e-02) ms⁻¹, wall time: 43.215 seconds
[ Info: i: 0800, t: 2.646 hours, Δt: 8.753 seconds, umax = (8.3e-02, 4.9e-02, 4.5e-02) ms⁻¹, wall time: 43.688 seconds
[ Info: i: 0820, t: 2.694 hours, Δt: 8.644 seconds, umax = (8.4e-02, 5.2e-02, 4.9e-02) ms⁻¹, wall time: 44.183 seconds
[ Info: i: 0840, t: 2.741 hours, Δt: 8.589 seconds, umax = (8.4e-02, 5.2e-02, 4.5e-02) ms⁻¹, wall time: 44.668 seconds
[ Info: i: 0860, t: 2.787 hours, Δt: 8.245 seconds, umax = (8.7e-02, 5.4e-02, 4.3e-02) ms⁻¹, wall time: 45.187 seconds
[ Info: i: 0880, t: 2.833 hours, Δt: 8.651 seconds, umax = (8.7e-02, 5.5e-02, 4.3e-02) ms⁻¹, wall time: 45.685 seconds
[ Info: i: 0900, t: 2.881 hours, Δt: 8.608 seconds, umax = (8.5e-02, 5.6e-02, 4.4e-02) ms⁻¹, wall time: 46.180 seconds
[ Info: i: 0920, t: 2.926 hours, Δt: 8.193 seconds, umax = (9.0e-02, 5.8e-02, 4.9e-02) ms⁻¹, wall time: 46.964 seconds
[ Info: i: 0940, t: 2.971 hours, Δt: 8.168 seconds, umax = (9.0e-02, 5.3e-02, 4.5e-02) ms⁻¹, wall time: 47.413 seconds
[ Info: i: 0960, t: 3.016 hours, Δt: 8.604 seconds, umax = (8.5e-02, 5.6e-02, 4.7e-02) ms⁻¹, wall time: 47.952 seconds
[ Info: i: 0980, t: 3.064 hours, Δt: 7.960 seconds, umax = (8.5e-02, 6.4e-02, 4.7e-02) ms⁻¹, wall time: 48.530 seconds
[ Info: i: 1000, t: 3.110 hours, Δt: 8.525 seconds, umax = (8.3e-02, 5.6e-02, 4.4e-02) ms⁻¹, wall time: 49.208 seconds
[ Info: i: 1020, t: 3.157 hours, Δt: 8.351 seconds, umax = (8.5e-02, 5.7e-02, 4.6e-02) ms⁻¹, wall time: 49.707 seconds
[ Info: i: 1040, t: 3.201 hours, Δt: 8.232 seconds, umax = (8.8e-02, 6.1e-02, 4.7e-02) ms⁻¹, wall time: 50.355 seconds
[ Info: i: 1060, t: 3.247 hours, Δt: 8.027 seconds, umax = (8.8e-02, 5.6e-02, 4.7e-02) ms⁻¹, wall time: 50.974 seconds
[ Info: i: 1080, t: 3.291 hours, Δt: 8.373 seconds, umax = (8.9e-02, 5.7e-02, 4.8e-02) ms⁻¹, wall time: 51.748 seconds
[ Info: i: 1100, t: 3.336 hours, Δt: 7.739 seconds, umax = (8.3e-02, 6.3e-02, 4.8e-02) ms⁻¹, wall time: 52.920 seconds
[ Info: i: 1120, t: 3.378 hours, Δt: 8.086 seconds, umax = (8.5e-02, 6.5e-02, 5.4e-02) ms⁻¹, wall time: 53.358 seconds
[ Info: i: 1140, t: 3.424 hours, Δt: 8.309 seconds, umax = (8.5e-02, 6.1e-02, 4.7e-02) ms⁻¹, wall time: 54.625 seconds
[ Info: i: 1160, t: 3.470 hours, Δt: 8.455 seconds, umax = (8.5e-02, 5.7e-02, 4.9e-02) ms⁻¹, wall time: 55.010 seconds
[ Info: i: 1180, t: 3.516 hours, Δt: 8.504 seconds, umax = (8.5e-02, 5.9e-02, 4.6e-02) ms⁻¹, wall time: 55.979 seconds
[ Info: i: 1200, t: 3.562 hours, Δt: 7.981 seconds, umax = (8.6e-02, 6.4e-02, 4.9e-02) ms⁻¹, wall time: 56.704 seconds
[ Info: i: 1220, t: 3.605 hours, Δt: 7.850 seconds, umax = (8.8e-02, 6.4e-02, 4.7e-02) ms⁻¹, wall time: 57.216 seconds
[ Info: i: 1240, t: 3.649 hours, Δt: 8.180 seconds, umax = (8.8e-02, 6.1e-02, 5.0e-02) ms⁻¹, wall time: 57.708 seconds
[ Info: i: 1260, t: 3.694 hours, Δt: 7.737 seconds, umax = (9.1e-02, 6.6e-02, 5.1e-02) ms⁻¹, wall time: 58.211 seconds
[ Info: i: 1280, t: 3.737 hours, Δt: 8.215 seconds, umax = (8.7e-02, 7.0e-02, 4.7e-02) ms⁻¹, wall time: 58.699 seconds
[ Info: i: 1300, t: 3.782 hours, Δt: 8.205 seconds, umax = (8.6e-02, 6.4e-02, 5.4e-02) ms⁻¹, wall time: 59.189 seconds
[ Info: i: 1320, t: 3.828 hours, Δt: 8.106 seconds, umax = (8.6e-02, 6.6e-02, 5.2e-02) ms⁻¹, wall time: 59.680 seconds
[ Info: i: 1340, t: 3.872 hours, Δt: 8.243 seconds, umax = (9.0e-02, 6.6e-02, 4.7e-02) ms⁻¹, wall time: 1.003 minutes
[ Info: i: 1360, t: 3.917 hours, Δt: 8.171 seconds, umax = (8.8e-02, 6.2e-02, 5.1e-02) ms⁻¹, wall time: 1.011 minutes
[ Info: i: 1380, t: 3.962 hours, Δt: 7.879 seconds, umax = (8.6e-02, 6.6e-02, 5.0e-02) ms⁻¹, wall time: 1.020 minutes
[ Info: Simulation is stopping after running for 1.027 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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