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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: 44.7 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 = (1.5e-03, 1.0e-03, 1.5e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (21.310 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (4.336 seconds).
[ Info: i: 0020, t: 11.977 minutes, Δt: 29.935 seconds, umax = (2.8e-02, 1.2e-02, 2.0e-02) ms⁻¹, wall time: 27.014 seconds
[ Info: i: 0040, t: 20.408 minutes, Δt: 20.434 seconds, umax = (4.1e-02, 1.1e-02, 2.3e-02) ms⁻¹, wall time: 27.613 seconds
[ Info: i: 0060, t: 26.800 minutes, Δt: 16.990 seconds, umax = (4.8e-02, 1.4e-02, 2.0e-02) ms⁻¹, wall time: 28.016 seconds
[ Info: i: 0080, t: 32.193 minutes, Δt: 15.809 seconds, umax = (4.9e-02, 1.6e-02, 2.6e-02) ms⁻¹, wall time: 28.473 seconds
[ Info: i: 0100, t: 37.364 minutes, Δt: 15.484 seconds, umax = (5.1e-02, 1.8e-02, 2.3e-02) ms⁻¹, wall time: 28.985 seconds
[ Info: i: 0120, t: 42.302 minutes, Δt: 14.559 seconds, umax = (5.4e-02, 2.0e-02, 2.4e-02) ms⁻¹, wall time: 29.509 seconds
[ Info: i: 0140, t: 46.924 minutes, Δt: 14.572 seconds, umax = (5.7e-02, 2.1e-02, 2.5e-02) ms⁻¹, wall time: 30.017 seconds
[ Info: i: 0160, t: 51.673 minutes, Δt: 13.368 seconds, umax = (5.9e-02, 2.2e-02, 2.7e-02) ms⁻¹, wall time: 30.518 seconds
[ Info: i: 0180, t: 56.112 minutes, Δt: 13.094 seconds, umax = (6.0e-02, 2.5e-02, 2.8e-02) ms⁻¹, wall time: 31.205 seconds
[ Info: i: 0200, t: 1.007 hours, Δt: 12.408 seconds, umax = (6.1e-02, 2.7e-02, 3.0e-02) ms⁻¹, wall time: 31.850 seconds
[ Info: i: 0220, t: 1.077 hours, Δt: 12.397 seconds, umax = (6.3e-02, 2.8e-02, 3.2e-02) ms⁻¹, wall time: 32.201 seconds
[ Info: i: 0240, t: 1.145 hours, Δt: 11.991 seconds, umax = (6.4e-02, 2.9e-02, 3.1e-02) ms⁻¹, wall time: 32.690 seconds
[ Info: i: 0260, t: 1.210 hours, Δt: 11.603 seconds, umax = (6.6e-02, 3.0e-02, 3.2e-02) ms⁻¹, wall time: 33.178 seconds
[ Info: i: 0280, t: 1.273 hours, Δt: 11.118 seconds, umax = (6.8e-02, 3.1e-02, 3.7e-02) ms⁻¹, wall time: 33.685 seconds
[ Info: i: 0300, t: 1.333 hours, Δt: 11.005 seconds, umax = (7.1e-02, 3.6e-02, 3.5e-02) ms⁻¹, wall time: 34.133 seconds
[ Info: i: 0320, t: 1.394 hours, Δt: 10.976 seconds, umax = (7.1e-02, 3.5e-02, 3.6e-02) ms⁻¹, wall time: 34.650 seconds
[ Info: i: 0340, t: 1.453 hours, Δt: 10.623 seconds, umax = (7.1e-02, 4.1e-02, 3.5e-02) ms⁻¹, wall time: 35.152 seconds
[ Info: i: 0360, t: 1.512 hours, Δt: 10.332 seconds, umax = (7.3e-02, 3.6e-02, 3.8e-02) ms⁻¹, wall time: 35.744 seconds
[ Info: i: 0380, t: 1.570 hours, Δt: 10.579 seconds, umax = (7.1e-02, 3.9e-02, 3.6e-02) ms⁻¹, wall time: 36.142 seconds
[ Info: i: 0400, t: 1.627 hours, Δt: 10.095 seconds, umax = (7.7e-02, 4.4e-02, 3.8e-02) ms⁻¹, wall time: 36.642 seconds
[ Info: i: 0420, t: 1.680 hours, Δt: 9.597 seconds, umax = (7.8e-02, 4.1e-02, 3.8e-02) ms⁻¹, wall time: 37.219 seconds
[ Info: i: 0440, t: 1.735 hours, Δt: 9.746 seconds, umax = (7.4e-02, 4.0e-02, 3.8e-02) ms⁻¹, wall time: 37.642 seconds
[ Info: i: 0460, t: 1.788 hours, Δt: 9.380 seconds, umax = (7.4e-02, 4.3e-02, 3.9e-02) ms⁻¹, wall time: 38.147 seconds
[ Info: i: 0480, t: 1.839 hours, Δt: 9.934 seconds, umax = (7.5e-02, 4.0e-02, 4.0e-02) ms⁻¹, wall time: 38.795 seconds
[ Info: i: 0500, t: 1.893 hours, Δt: 9.713 seconds, umax = (7.8e-02, 4.2e-02, 4.0e-02) ms⁻¹, wall time: 39.159 seconds
[ Info: i: 0520, t: 1.947 hours, Δt: 9.661 seconds, umax = (7.8e-02, 4.4e-02, 4.0e-02) ms⁻¹, wall time: 39.664 seconds
[ Info: i: 0540, t: 2 hours, Δt: 9.174 seconds, umax = (7.7e-02, 4.4e-02, 3.9e-02) ms⁻¹, wall time: 40.158 seconds
[ Info: i: 0560, t: 2.051 hours, Δt: 9.409 seconds, umax = (7.8e-02, 4.6e-02, 3.9e-02) ms⁻¹, wall time: 40.685 seconds
[ Info: i: 0580, t: 2.102 hours, Δt: 9.384 seconds, umax = (7.8e-02, 4.5e-02, 3.8e-02) ms⁻¹, wall time: 41.242 seconds
[ Info: i: 0600, t: 2.154 hours, Δt: 8.914 seconds, umax = (7.6e-02, 4.8e-02, 4.4e-02) ms⁻¹, wall time: 41.726 seconds
[ Info: i: 0620, t: 2.201 hours, Δt: 8.668 seconds, umax = (7.9e-02, 4.7e-02, 4.1e-02) ms⁻¹, wall time: 42.243 seconds
[ Info: i: 0640, t: 2.249 hours, Δt: 8.743 seconds, umax = (8.0e-02, 4.9e-02, 4.2e-02) ms⁻¹, wall time: 42.749 seconds
[ Info: i: 0660, t: 2.296 hours, Δt: 8.801 seconds, umax = (8.3e-02, 5.0e-02, 4.4e-02) ms⁻¹, wall time: 43.265 seconds
[ Info: i: 0680, t: 2.343 hours, Δt: 8.544 seconds, umax = (8.1e-02, 4.7e-02, 4.4e-02) ms⁻¹, wall time: 43.877 seconds
[ Info: i: 0700, t: 2.391 hours, Δt: 8.709 seconds, umax = (8.2e-02, 5.4e-02, 4.4e-02) ms⁻¹, wall time: 44.290 seconds
[ Info: i: 0720, t: 2.439 hours, Δt: 8.715 seconds, umax = (8.2e-02, 5.5e-02, 4.6e-02) ms⁻¹, wall time: 44.813 seconds
[ Info: i: 0740, t: 2.488 hours, Δt: 8.815 seconds, umax = (8.3e-02, 5.2e-02, 4.7e-02) ms⁻¹, wall time: 45.318 seconds
[ Info: i: 0760, t: 2.537 hours, Δt: 8.702 seconds, umax = (8.2e-02, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 45.824 seconds
[ Info: i: 0780, t: 2.583 hours, Δt: 8.774 seconds, umax = (8.9e-02, 5.7e-02, 4.6e-02) ms⁻¹, wall time: 46.333 seconds
[ Info: i: 0800, t: 2.632 hours, Δt: 8.814 seconds, umax = (8.3e-02, 5.3e-02, 4.5e-02) ms⁻¹, wall time: 46.846 seconds
[ Info: i: 0820, t: 2.679 hours, Δt: 8.731 seconds, umax = (8.3e-02, 5.1e-02, 4.2e-02) ms⁻¹, wall time: 47.456 seconds
[ Info: i: 0840, t: 2.726 hours, Δt: 8.275 seconds, umax = (8.2e-02, 5.2e-02, 4.4e-02) ms⁻¹, wall time: 47.900 seconds
[ Info: i: 0860, t: 2.771 hours, Δt: 8.495 seconds, umax = (8.5e-02, 5.6e-02, 4.7e-02) ms⁻¹, wall time: 48.435 seconds
[ Info: i: 0880, t: 2.818 hours, Δt: 7.846 seconds, umax = (8.4e-02, 6.0e-02, 5.1e-02) ms⁻¹, wall time: 48.948 seconds
[ Info: i: 0900, t: 2.863 hours, Δt: 8.368 seconds, umax = (8.5e-02, 5.6e-02, 5.5e-02) ms⁻¹, wall time: 49.479 seconds
[ Info: i: 0920, t: 2.909 hours, Δt: 8.261 seconds, umax = (8.5e-02, 5.5e-02, 4.5e-02) ms⁻¹, wall time: 49.993 seconds
[ Info: i: 0940, t: 2.954 hours, Δt: 8.333 seconds, umax = (8.1e-02, 5.7e-02, 5.2e-02) ms⁻¹, wall time: 50.553 seconds
[ Info: i: 0960, t: 3.000 hours, Δt: 8.008 seconds, umax = (8.3e-02, 5.7e-02, 4.6e-02) ms⁻¹, wall time: 51.057 seconds
[ Info: i: 0980, t: 3.042 hours, Δt: 7.957 seconds, umax = (8.1e-02, 6.1e-02, 4.8e-02) ms⁻¹, wall time: 51.582 seconds
[ Info: i: 1000, t: 3.086 hours, Δt: 7.965 seconds, umax = (8.2e-02, 6.0e-02, 4.8e-02) ms⁻¹, wall time: 52.277 seconds
[ Info: i: 1020, t: 3.130 hours, Δt: 8.188 seconds, umax = (8.2e-02, 6.1e-02, 4.8e-02) ms⁻¹, wall time: 52.621 seconds
[ Info: i: 1040, t: 3.173 hours, Δt: 7.954 seconds, umax = (8.3e-02, 5.7e-02, 4.6e-02) ms⁻¹, wall time: 53.267 seconds
[ Info: i: 1060, t: 3.218 hours, Δt: 8.337 seconds, umax = (8.5e-02, 5.6e-02, 4.5e-02) ms⁻¹, wall time: 53.648 seconds
[ Info: i: 1080, t: 3.264 hours, Δt: 8.352 seconds, umax = (8.7e-02, 5.9e-02, 4.4e-02) ms⁻¹, wall time: 54.239 seconds
[ Info: i: 1100, t: 3.310 hours, Δt: 8.396 seconds, umax = (8.5e-02, 5.8e-02, 4.7e-02) ms⁻¹, wall time: 54.704 seconds
[ Info: i: 1120, t: 3.354 hours, Δt: 8.396 seconds, umax = (8.5e-02, 6.2e-02, 4.8e-02) ms⁻¹, wall time: 55.259 seconds
[ Info: i: 1140, t: 3.401 hours, Δt: 8.355 seconds, umax = (8.7e-02, 6.4e-02, 4.9e-02) ms⁻¹, wall time: 55.768 seconds
[ Info: i: 1160, t: 3.447 hours, Δt: 8.356 seconds, umax = (8.5e-02, 6.1e-02, 4.7e-02) ms⁻¹, wall time: 56.306 seconds
[ Info: i: 1180, t: 3.493 hours, Δt: 8.193 seconds, umax = (8.6e-02, 5.6e-02, 4.5e-02) ms⁻¹, wall time: 56.813 seconds
[ Info: i: 1200, t: 3.536 hours, Δt: 8.046 seconds, umax = (8.7e-02, 5.6e-02, 4.6e-02) ms⁻¹, wall time: 57.343 seconds
[ Info: i: 1220, t: 3.582 hours, Δt: 8.309 seconds, umax = (8.5e-02, 5.6e-02, 4.4e-02) ms⁻¹, wall time: 57.871 seconds
[ Info: i: 1240, t: 3.627 hours, Δt: 7.988 seconds, umax = (8.7e-02, 5.7e-02, 4.2e-02) ms⁻¹, wall time: 58.414 seconds
[ Info: i: 1260, t: 3.671 hours, Δt: 8.294 seconds, umax = (8.8e-02, 6.2e-02, 4.5e-02) ms⁻¹, wall time: 59.087 seconds
[ Info: i: 1280, t: 3.716 hours, Δt: 8.323 seconds, umax = (8.9e-02, 6.1e-02, 4.8e-02) ms⁻¹, wall time: 59.460 seconds
[ Info: i: 1300, t: 3.761 hours, Δt: 7.281 seconds, umax = (9.0e-02, 6.1e-02, 4.4e-02) ms⁻¹, wall time: 1.001 minutes
[ Info: i: 1320, t: 3.803 hours, Δt: 7.965 seconds, umax = (8.6e-02, 5.9e-02, 4.7e-02) ms⁻¹, wall time: 1.010 minutes
[ Info: i: 1340, t: 3.846 hours, Δt: 8.013 seconds, umax = (8.6e-02, 5.5e-02, 5.0e-02) ms⁻¹, wall time: 1.020 minutes
[ Info: i: 1360, t: 3.890 hours, Δt: 7.742 seconds, umax = (8.8e-02, 5.9e-02, 5.1e-02) ms⁻¹, wall time: 1.027 minutes
[ Info: i: 1380, t: 3.932 hours, Δt: 7.806 seconds, umax = (9.1e-02, 6.0e-02, 5.0e-02) ms⁻¹, wall time: 1.037 minutes
[ Info: i: 1400, t: 3.975 hours, Δt: 7.941 seconds, umax = (9.1e-02, 5.6e-02, 5.2e-02) ms⁻¹, wall time: 1.045 minutes
[ Info: Simulation is stopping after running for 1.053 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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