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Langmuir turbulence example

This example implements a Langmuir turbulence simulation similar to the one reported in section 4 of

Wagner et al., "Near-inertial waves and turbulence driven by the growth of swell", Journal of Physical Oceanography (2021)

This example demonstrates

  • How to run large eddy simulations with surface wave effects via the Craik-Leibovich approximation.

  • How to specify time- and horizontally-averaged output.

Install dependencies

First let's make sure we have all required packages installed.

using Pkg
pkg"add Oceananigans, CairoMakie"
using Oceananigans
using Oceananigans.Units: minute, minutes, hours

Model set-up

To build the model, we specify the grid, Stokes drift, boundary conditions, and Coriolis parameter.

Domain and numerical grid specification

We use a modest resolution and the same total extent as Wagner et al. (2021),

grid = RectilinearGrid(GPU(), size=(128, 128, 64), extent=(128, 128, 64))
128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CUDAGPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 128.0) regularly spaced with Δx=1.0
├── Periodic y ∈ [0.0, 128.0) regularly spaced with Δy=1.0
└── Bounded  z ∈ [-64.0, 0.0] regularly spaced with Δz=1.0

The Stokes Drift profile

The surface wave Stokes drift profile prescribed in Wagner et al. (2021), corresponds to a 'monochromatic' (that is, single-frequency) wave field.

A monochromatic wave field is characterized by its wavelength and amplitude (half the distance from wave crest to wave trough), which determine the wave frequency and the vertical scale of the Stokes drift profile.

using Oceananigans.BuoyancyFormulations: g_Earth

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

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

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

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

The Stokes drift profile is

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

and its z-derivative is

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

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

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

Boundary conditions

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

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

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

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

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

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

Coriolis parameter

Wagner et al. (2021) use

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

which is typical for mid-latitudes on Earth.

Model instantiation

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

model = NonhydrostaticModel(; grid, coriolis,
                            advection = WENO(order=9),
                            tracers = :b,
                            buoyancy = BuoyancyTracer(),
                            stokes_drift = UniformStokesDrift(∂z_uˢ=∂z_uˢ),
                            boundary_conditions = (u=u_boundary_conditions, b=b_boundary_conditions))
NonhydrostaticModel{GPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CUDAGPU with 5×5×5 halo
├── timestepper: RungeKutta3TimeStepper
├── advection scheme: WENO(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 => 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
└── 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 = (1.4e-03, 8.4e-04, 1.6e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (37.228 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (5.314 seconds).
[ Info: i: 0020, t: 11.838 minutes, Δt: 29.155 seconds, umax = (2.9e-02, 1.2e-02, 2.3e-02) ms⁻¹, wall time: 56.723 seconds
[ Info: i: 0040, t: 20.396 minutes, Δt: 20.238 seconds, umax = (4.2e-02, 1.1e-02, 1.9e-02) ms⁻¹, wall time: 57.409 seconds
[ Info: i: 0060, t: 26.452 minutes, Δt: 16.717 seconds, umax = (4.8e-02, 1.6e-02, 2.0e-02) ms⁻¹, wall time: 57.873 seconds
[ Info: i: 0080, t: 31.863 minutes, Δt: 16.099 seconds, umax = (4.9e-02, 1.7e-02, 2.4e-02) ms⁻¹, wall time: 58.379 seconds
[ Info: i: 0100, t: 37.162 minutes, Δt: 14.715 seconds, umax = (5.2e-02, 1.7e-02, 2.3e-02) ms⁻¹, wall time: 58.910 seconds
[ Info: i: 0120, t: 42.094 minutes, Δt: 15.355 seconds, umax = (5.3e-02, 1.9e-02, 2.4e-02) ms⁻¹, wall time: 59.474 seconds
[ Info: i: 0140, t: 46.974 minutes, Δt: 14.661 seconds, umax = (5.8e-02, 2.1e-02, 2.7e-02) ms⁻¹, wall time: 1.000 minutes
[ Info: i: 0160, t: 51.603 minutes, Δt: 14.058 seconds, umax = (5.8e-02, 2.1e-02, 2.6e-02) ms⁻¹, wall time: 1.009 minutes
[ Info: i: 0180, t: 55.908 minutes, Δt: 13.102 seconds, umax = (6.1e-02, 2.4e-02, 2.8e-02) ms⁻¹, wall time: 1.021 minutes
[ Info: i: 0200, t: 1 hour, Δt: 12.281 seconds, umax = (6.1e-02, 2.6e-02, 3.0e-02) ms⁻¹, wall time: 1.029 minutes
[ Info: i: 0220, t: 1.069 hours, Δt: 12.273 seconds, umax = (6.4e-02, 2.9e-02, 3.5e-02) ms⁻¹, wall time: 1.039 minutes
[ Info: i: 0240, t: 1.135 hours, Δt: 12.017 seconds, umax = (6.3e-02, 2.9e-02, 3.3e-02) ms⁻¹, wall time: 1.047 minutes
[ Info: i: 0260, t: 1.200 hours, Δt: 11.837 seconds, umax = (6.4e-02, 3.1e-02, 3.2e-02) ms⁻¹, wall time: 1.056 minutes
[ Info: i: 0280, t: 1.260 hours, Δt: 11.826 seconds, umax = (6.8e-02, 3.5e-02, 3.3e-02) ms⁻¹, wall time: 1.067 minutes
[ Info: i: 0300, t: 1.324 hours, Δt: 11.098 seconds, umax = (7.1e-02, 3.4e-02, 3.5e-02) ms⁻¹, wall time: 1.074 minutes
[ Info: i: 0320, t: 1.386 hours, Δt: 11.228 seconds, umax = (6.9e-02, 3.4e-02, 3.6e-02) ms⁻¹, wall time: 1.085 minutes
[ Info: i: 0340, t: 1.444 hours, Δt: 10.514 seconds, umax = (7.2e-02, 3.3e-02, 3.7e-02) ms⁻¹, wall time: 1.096 minutes
[ Info: i: 0360, t: 1.500 hours, Δt: 10.198 seconds, umax = (7.2e-02, 3.4e-02, 3.7e-02) ms⁻¹, wall time: 1.106 minutes
[ Info: i: 0380, t: 1.557 hours, Δt: 10.322 seconds, umax = (7.6e-02, 4.1e-02, 3.8e-02) ms⁻¹, wall time: 1.116 minutes
[ Info: i: 0400, t: 1.612 hours, Δt: 10.280 seconds, umax = (7.3e-02, 4.2e-02, 4.0e-02) ms⁻¹, wall time: 1.127 minutes
[ Info: i: 0420, t: 1.667 hours, Δt: 9.800 seconds, umax = (7.4e-02, 4.0e-02, 3.8e-02) ms⁻¹, wall time: 1.139 minutes
[ Info: i: 0440, t: 1.722 hours, Δt: 10.155 seconds, umax = (7.5e-02, 3.9e-02, 3.8e-02) ms⁻¹, wall time: 1.149 minutes
[ Info: i: 0460, t: 1.775 hours, Δt: 10.141 seconds, umax = (7.5e-02, 3.9e-02, 4.0e-02) ms⁻¹, wall time: 1.161 minutes
[ Info: i: 0480, t: 1.831 hours, Δt: 9.726 seconds, umax = (7.5e-02, 4.2e-02, 4.3e-02) ms⁻¹, wall time: 1.172 minutes
[ Info: i: 0500, t: 1.885 hours, Δt: 9.763 seconds, umax = (7.5e-02, 4.3e-02, 4.6e-02) ms⁻¹, wall time: 1.181 minutes
[ Info: i: 0520, t: 1.938 hours, Δt: 9.604 seconds, umax = (7.7e-02, 4.4e-02, 4.1e-02) ms⁻¹, wall time: 1.192 minutes
[ Info: i: 0540, t: 1.991 hours, Δt: 8.831 seconds, umax = (8.2e-02, 4.6e-02, 4.0e-02) ms⁻¹, wall time: 1.200 minutes
[ Info: i: 0560, t: 2.040 hours, Δt: 9.170 seconds, umax = (8.2e-02, 4.4e-02, 4.1e-02) ms⁻¹, wall time: 1.209 minutes
[ Info: i: 0580, t: 2.088 hours, Δt: 9.021 seconds, umax = (7.9e-02, 4.6e-02, 4.4e-02) ms⁻¹, wall time: 1.220 minutes
[ Info: i: 0600, t: 2.139 hours, Δt: 9.630 seconds, umax = (8.3e-02, 4.8e-02, 4.2e-02) ms⁻¹, wall time: 1.226 minutes
[ Info: i: 0620, t: 2.191 hours, Δt: 9.390 seconds, umax = (7.9e-02, 4.7e-02, 4.4e-02) ms⁻¹, wall time: 1.235 minutes
[ Info: i: 0640, t: 2.242 hours, Δt: 9.088 seconds, umax = (8.0e-02, 5.1e-02, 4.2e-02) ms⁻¹, wall time: 1.244 minutes
[ Info: i: 0660, t: 2.290 hours, Δt: 9.206 seconds, umax = (7.9e-02, 5.0e-02, 4.2e-02) ms⁻¹, wall time: 1.253 minutes
[ Info: i: 0680, t: 2.341 hours, Δt: 8.942 seconds, umax = (7.9e-02, 5.3e-02, 4.0e-02) ms⁻¹, wall time: 1.265 minutes
[ Info: i: 0700, t: 2.389 hours, Δt: 8.408 seconds, umax = (8.2e-02, 5.9e-02, 4.2e-02) ms⁻¹, wall time: 1.271 minutes
[ Info: i: 0720, t: 2.435 hours, Δt: 8.519 seconds, umax = (8.1e-02, 5.4e-02, 4.3e-02) ms⁻¹, wall time: 1.281 minutes
[ Info: i: 0740, t: 2.482 hours, Δt: 8.209 seconds, umax = (8.4e-02, 5.5e-02, 4.7e-02) ms⁻¹, wall time: 1.290 minutes
[ Info: i: 0760, t: 2.525 hours, Δt: 8.028 seconds, umax = (8.3e-02, 5.2e-02, 5.0e-02) ms⁻¹, wall time: 1.300 minutes
[ Info: i: 0780, t: 2.570 hours, Δt: 8.515 seconds, umax = (8.2e-02, 6.4e-02, 4.6e-02) ms⁻¹, wall time: 1.310 minutes
[ Info: i: 0800, t: 2.616 hours, Δt: 8.410 seconds, umax = (8.1e-02, 6.2e-02, 4.6e-02) ms⁻¹, wall time: 1.319 minutes
[ Info: i: 0820, t: 2.663 hours, Δt: 8.841 seconds, umax = (8.2e-02, 5.1e-02, 4.2e-02) ms⁻¹, wall time: 1.329 minutes
[ Info: i: 0840, t: 2.711 hours, Δt: 8.511 seconds, umax = (8.2e-02, 5.3e-02, 4.5e-02) ms⁻¹, wall time: 1.337 minutes
[ Info: i: 0860, t: 2.757 hours, Δt: 8.410 seconds, umax = (8.3e-02, 5.4e-02, 4.4e-02) ms⁻¹, wall time: 1.349 minutes
[ Info: i: 0880, t: 2.804 hours, Δt: 8.840 seconds, umax = (8.2e-02, 5.7e-02, 5.0e-02) ms⁻¹, wall time: 1.355 minutes
[ Info: i: 0900, t: 2.853 hours, Δt: 8.800 seconds, umax = (8.7e-02, 5.5e-02, 5.1e-02) ms⁻¹, wall time: 1.365 minutes
[ Info: i: 0920, t: 2.901 hours, Δt: 8.333 seconds, umax = (8.7e-02, 5.6e-02, 4.7e-02) ms⁻¹, wall time: 1.373 minutes
[ Info: i: 0940, t: 2.947 hours, Δt: 8.235 seconds, umax = (8.2e-02, 6.0e-02, 4.8e-02) ms⁻¹, wall time: 1.382 minutes
[ Info: i: 0960, t: 2.993 hours, Δt: 8.454 seconds, umax = (8.1e-02, 5.9e-02, 4.6e-02) ms⁻¹, wall time: 1.391 minutes
[ Info: i: 0980, t: 3.037 hours, Δt: 8.144 seconds, umax = (8.6e-02, 5.6e-02, 4.6e-02) ms⁻¹, wall time: 1.401 minutes
[ Info: i: 1000, t: 3.083 hours, Δt: 8.309 seconds, umax = (8.7e-02, 5.7e-02, 4.8e-02) ms⁻¹, wall time: 1.412 minutes
[ Info: i: 1020, t: 3.128 hours, Δt: 8.459 seconds, umax = (8.3e-02, 5.6e-02, 4.8e-02) ms⁻¹, wall time: 1.421 minutes
[ Info: i: 1040, t: 3.174 hours, Δt: 7.744 seconds, umax = (8.5e-02, 6.0e-02, 5.2e-02) ms⁻¹, wall time: 1.434 minutes
[ Info: i: 1060, t: 3.218 hours, Δt: 8.022 seconds, umax = (8.6e-02, 5.8e-02, 4.7e-02) ms⁻¹, wall time: 1.440 minutes
[ Info: i: 1080, t: 3.261 hours, Δt: 7.726 seconds, umax = (8.5e-02, 5.8e-02, 4.7e-02) ms⁻¹, wall time: 1.451 minutes
[ Info: i: 1100, t: 3.304 hours, Δt: 8.008 seconds, umax = (8.4e-02, 6.2e-02, 4.8e-02) ms⁻¹, wall time: 1.458 minutes
[ Info: i: 1120, t: 3.349 hours, Δt: 8.382 seconds, umax = (8.7e-02, 6.0e-02, 5.3e-02) ms⁻¹, wall time: 1.469 minutes
[ Info: i: 1140, t: 3.396 hours, Δt: 7.806 seconds, umax = (9.0e-02, 6.2e-02, 4.8e-02) ms⁻¹, wall time: 1.477 minutes
[ Info: i: 1160, t: 3.439 hours, Δt: 7.931 seconds, umax = (9.0e-02, 5.7e-02, 4.8e-02) ms⁻¹, wall time: 1.487 minutes
[ Info: i: 1180, t: 3.482 hours, Δt: 8.107 seconds, umax = (8.6e-02, 6.2e-02, 4.9e-02) ms⁻¹, wall time: 1.496 minutes
[ Info: i: 1200, t: 3.527 hours, Δt: 8.113 seconds, umax = (8.7e-02, 6.0e-02, 4.4e-02) ms⁻¹, wall time: 1.505 minutes
[ Info: i: 1220, t: 3.572 hours, Δt: 8.180 seconds, umax = (8.7e-02, 6.5e-02, 4.8e-02) ms⁻¹, wall time: 1.514 minutes
[ Info: i: 1240, t: 3.618 hours, Δt: 8.254 seconds, umax = (8.4e-02, 6.1e-02, 5.2e-02) ms⁻¹, wall time: 1.525 minutes
[ Info: i: 1260, t: 3.663 hours, Δt: 7.814 seconds, umax = (8.7e-02, 5.7e-02, 4.7e-02) ms⁻¹, wall time: 1.536 minutes
[ Info: i: 1280, t: 3.705 hours, Δt: 7.537 seconds, umax = (9.1e-02, 6.0e-02, 5.2e-02) ms⁻¹, wall time: 1.545 minutes
[ Info: i: 1300, t: 3.747 hours, Δt: 7.835 seconds, umax = (8.9e-02, 5.6e-02, 5.1e-02) ms⁻¹, wall time: 1.555 minutes
[ Info: i: 1320, t: 3.789 hours, Δt: 8.039 seconds, umax = (8.9e-02, 5.7e-02, 4.9e-02) ms⁻¹, wall time: 1.565 minutes
[ Info: i: 1340, t: 3.833 hours, Δt: 8.097 seconds, umax = (8.7e-02, 5.7e-02, 5.0e-02) ms⁻¹, wall time: 1.576 minutes
[ Info: i: 1360, t: 3.878 hours, Δt: 8.113 seconds, umax = (8.3e-02, 5.6e-02, 5.0e-02) ms⁻¹, wall time: 1.585 minutes
[ Info: i: 1380, t: 3.921 hours, Δt: 8.196 seconds, umax = (8.3e-02, 6.4e-02, 4.5e-02) ms⁻¹, wall time: 1.597 minutes
[ Info: i: 1400, t: 3.967 hours, Δt: 8.070 seconds, umax = (8.5e-02, 6.2e-02, 4.6e-02) ms⁻¹, wall time: 1.603 minutes
[ Info: Simulation is stopping after running for 1.612 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)

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

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