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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: 57.9 MiB

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: 57.9 MiB

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, 8.2e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (18.981 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.764 seconds).
[ Info: i: 0020, t: 11.930 minutes, Δt: 29.708 seconds, umax = (2.8e-02, 1.1e-02, 2.1e-02) ms⁻¹, wall time: 22.928 seconds
[ Info: i: 0040, t: 20.412 minutes, Δt: 20.698 seconds, umax = (4.2e-02, 1.1e-02, 1.7e-02) ms⁻¹, wall time: 23.589 seconds
[ Info: i: 0060, t: 26.848 minutes, Δt: 17.223 seconds, umax = (4.7e-02, 1.5e-02, 2.0e-02) ms⁻¹, wall time: 24.010 seconds
[ Info: i: 0080, t: 32.158 minutes, Δt: 15.474 seconds, umax = (5.4e-02, 1.7e-02, 2.4e-02) ms⁻¹, wall time: 24.478 seconds
[ Info: i: 0100, t: 37.079 minutes, Δt: 15.865 seconds, umax = (5.3e-02, 1.9e-02, 2.5e-02) ms⁻¹, wall time: 24.959 seconds
[ Info: i: 0120, t: 42.008 minutes, Δt: 14.783 seconds, umax = (5.1e-02, 1.8e-02, 2.5e-02) ms⁻¹, wall time: 26.248 seconds
[ Info: i: 0140, t: 46.631 minutes, Δt: 14.383 seconds, umax = (5.6e-02, 2.1e-02, 2.8e-02) ms⁻¹, wall time: 26.855 seconds
[ Info: i: 0160, t: 51.189 minutes, Δt: 13.967 seconds, umax = (5.9e-02, 2.3e-02, 2.8e-02) ms⁻¹, wall time: 27.402 seconds
[ Info: i: 0180, t: 55.672 minutes, Δt: 13.257 seconds, umax = (5.8e-02, 2.6e-02, 2.9e-02) ms⁻¹, wall time: 28.168 seconds
[ Info: i: 0200, t: 59.984 minutes, Δt: 13.020 seconds, umax = (6.1e-02, 2.6e-02, 3.3e-02) ms⁻¹, wall time: 28.546 seconds
[ Info: i: 0220, t: 1.068 hours, Δt: 12.753 seconds, umax = (6.3e-02, 2.8e-02, 3.2e-02) ms⁻¹, wall time: 29.182 seconds
[ Info: i: 0240, t: 1.134 hours, Δt: 11.676 seconds, umax = (6.4e-02, 3.2e-02, 3.2e-02) ms⁻¹, wall time: 29.791 seconds
[ Info: i: 0260, t: 1.200 hours, Δt: 11.744 seconds, umax = (6.6e-02, 3.2e-02, 3.5e-02) ms⁻¹, wall time: 30.275 seconds
[ Info: i: 0280, t: 1.263 hours, Δt: 11.059 seconds, umax = (6.9e-02, 3.1e-02, 3.8e-02) ms⁻¹, wall time: 30.819 seconds
[ Info: i: 0300, t: 1.324 hours, Δt: 11.068 seconds, umax = (6.8e-02, 3.2e-02, 3.7e-02) ms⁻¹, wall time: 31.214 seconds
[ Info: i: 0320, t: 1.385 hours, Δt: 10.899 seconds, umax = (7.1e-02, 3.5e-02, 4.0e-02) ms⁻¹, wall time: 31.692 seconds
[ Info: i: 0340, t: 1.443 hours, Δt: 10.999 seconds, umax = (6.9e-02, 3.4e-02, 3.6e-02) ms⁻¹, wall time: 32.183 seconds
[ Info: i: 0360, t: 1.503 hours, Δt: 10.489 seconds, umax = (7.2e-02, 3.6e-02, 3.5e-02) ms⁻¹, wall time: 32.793 seconds
[ Info: i: 0380, t: 1.561 hours, Δt: 10.569 seconds, umax = (7.2e-02, 3.5e-02, 3.8e-02) ms⁻¹, wall time: 33.136 seconds
[ Info: i: 0400, t: 1.618 hours, Δt: 10.116 seconds, umax = (7.3e-02, 3.9e-02, 3.6e-02) ms⁻¹, wall time: 34.464 seconds
[ Info: i: 0420, t: 1.672 hours, Δt: 9.908 seconds, umax = (7.4e-02, 4.0e-02, 4.0e-02) ms⁻¹, wall time: 35.055 seconds
[ Info: i: 0440, t: 1.728 hours, Δt: 10.180 seconds, umax = (7.6e-02, 4.0e-02, 3.6e-02) ms⁻¹, wall time: 35.429 seconds
[ Info: i: 0460, t: 1.784 hours, Δt: 10.138 seconds, umax = (7.6e-02, 4.0e-02, 3.7e-02) ms⁻¹, wall time: 36.121 seconds
[ Info: i: 0480, t: 1.839 hours, Δt: 9.816 seconds, umax = (7.6e-02, 4.3e-02, 4.1e-02) ms⁻¹, wall time: 36.771 seconds
[ Info: i: 0500, t: 1.894 hours, Δt: 9.903 seconds, umax = (7.7e-02, 4.3e-02, 3.8e-02) ms⁻¹, wall time: 37.170 seconds
[ Info: i: 0520, t: 1.946 hours, Δt: 9.875 seconds, umax = (7.7e-02, 4.5e-02, 3.9e-02) ms⁻¹, wall time: 37.938 seconds
[ Info: i: 0540, t: 2 hours, Δt: 9.678 seconds, umax = (7.7e-02, 4.4e-02, 3.9e-02) ms⁻¹, wall time: 38.704 seconds
[ Info: i: 0560, t: 2.053 hours, Δt: 9.474 seconds, umax = (7.5e-02, 4.6e-02, 4.1e-02) ms⁻¹, wall time: 39.604 seconds
[ Info: i: 0580, t: 2.104 hours, Δt: 9.183 seconds, umax = (7.6e-02, 4.6e-02, 4.1e-02) ms⁻¹, wall time: 40.141 seconds
[ Info: i: 0600, t: 2.155 hours, Δt: 8.956 seconds, umax = (7.8e-02, 4.9e-02, 4.1e-02) ms⁻¹, wall time: 41.142 seconds
[ Info: i: 0620, t: 2.204 hours, Δt: 8.786 seconds, umax = (7.8e-02, 4.7e-02, 4.1e-02) ms⁻¹, wall time: 41.850 seconds
[ Info: i: 0640, t: 2.250 hours, Δt: 8.874 seconds, umax = (7.8e-02, 4.4e-02, 4.2e-02) ms⁻¹, wall time: 42.685 seconds
[ Info: i: 0660, t: 2.298 hours, Δt: 8.084 seconds, umax = (8.1e-02, 5.0e-02, 3.9e-02) ms⁻¹, wall time: 43.543 seconds
[ Info: i: 0680, t: 2.342 hours, Δt: 8.068 seconds, umax = (7.9e-02, 5.4e-02, 3.8e-02) ms⁻¹, wall time: 44.124 seconds
[ Info: i: 0700, t: 2.387 hours, Δt: 8.000 seconds, umax = (8.0e-02, 5.2e-02, 4.1e-02) ms⁻¹, wall time: 44.543 seconds
[ Info: i: 0720, t: 2.431 hours, Δt: 9.162 seconds, umax = (8.3e-02, 5.0e-02, 4.1e-02) ms⁻¹, wall time: 45.086 seconds
[ Info: i: 0740, t: 2.481 hours, Δt: 8.769 seconds, umax = (8.6e-02, 5.2e-02, 4.1e-02) ms⁻¹, wall time: 45.554 seconds
[ Info: i: 0760, t: 2.529 hours, Δt: 8.353 seconds, umax = (8.1e-02, 4.9e-02, 4.2e-02) ms⁻¹, wall time: 46.759 seconds
[ Info: i: 0780, t: 2.575 hours, Δt: 8.646 seconds, umax = (7.9e-02, 4.9e-02, 4.6e-02) ms⁻¹, wall time: 47.245 seconds
[ Info: i: 0800, t: 2.622 hours, Δt: 8.614 seconds, umax = (8.0e-02, 5.1e-02, 4.8e-02) ms⁻¹, wall time: 47.763 seconds
[ Info: i: 0820, t: 2.669 hours, Δt: 8.990 seconds, umax = (8.3e-02, 5.1e-02, 4.3e-02) ms⁻¹, wall time: 48.406 seconds
[ Info: i: 0840, t: 2.718 hours, Δt: 8.688 seconds, umax = (8.2e-02, 5.0e-02, 4.5e-02) ms⁻¹, wall time: 48.799 seconds
[ Info: i: 0860, t: 2.764 hours, Δt: 8.253 seconds, umax = (9.3e-02, 5.3e-02, 4.1e-02) ms⁻¹, wall time: 49.370 seconds
[ Info: i: 0880, t: 2.811 hours, Δt: 8.456 seconds, umax = (8.3e-02, 5.6e-02, 4.7e-02) ms⁻¹, wall time: 49.829 seconds
[ Info: i: 0900, t: 2.858 hours, Δt: 8.788 seconds, umax = (8.2e-02, 5.2e-02, 4.6e-02) ms⁻¹, wall time: 50.362 seconds
[ Info: i: 0920, t: 2.906 hours, Δt: 8.454 seconds, umax = (8.1e-02, 5.2e-02, 4.5e-02) ms⁻¹, wall time: 50.936 seconds
[ Info: i: 0940, t: 2.952 hours, Δt: 8.615 seconds, umax = (8.3e-02, 5.6e-02, 4.6e-02) ms⁻¹, wall time: 51.546 seconds
[ Info: i: 0960, t: 3 hours, Δt: 8.232 seconds, umax = (8.4e-02, 6.1e-02, 4.2e-02) ms⁻¹, wall time: 52.075 seconds
[ Info: i: 0980, t: 3.045 hours, Δt: 8.315 seconds, umax = (8.2e-02, 5.9e-02, 4.9e-02) ms⁻¹, wall time: 52.577 seconds
[ Info: i: 1000, t: 3.090 hours, Δt: 8.400 seconds, umax = (8.4e-02, 5.3e-02, 4.3e-02) ms⁻¹, wall time: 53.175 seconds
[ Info: i: 1020, t: 3.137 hours, Δt: 8.310 seconds, umax = (8.4e-02, 5.2e-02, 4.4e-02) ms⁻¹, wall time: 53.562 seconds
[ Info: i: 1040, t: 3.180 hours, Δt: 8.176 seconds, umax = (8.2e-02, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 54.099 seconds
[ Info: i: 1060, t: 3.226 hours, Δt: 8.008 seconds, umax = (8.6e-02, 5.8e-02, 4.2e-02) ms⁻¹, wall time: 54.572 seconds
[ Info: i: 1080, t: 3.270 hours, Δt: 8.024 seconds, umax = (8.6e-02, 5.8e-02, 4.3e-02) ms⁻¹, wall time: 55.096 seconds
[ Info: i: 1100, t: 3.314 hours, Δt: 8.296 seconds, umax = (8.4e-02, 5.6e-02, 4.5e-02) ms⁻¹, wall time: 55.590 seconds
[ Info: i: 1120, t: 3.359 hours, Δt: 7.868 seconds, umax = (8.5e-02, 5.7e-02, 4.7e-02) ms⁻¹, wall time: 56.120 seconds
[ Info: i: 1140, t: 3.402 hours, Δt: 7.904 seconds, umax = (8.3e-02, 5.7e-02, 4.4e-02) ms⁻¹, wall time: 56.628 seconds
[ Info: i: 1160, t: 3.446 hours, Δt: 8.487 seconds, umax = (8.5e-02, 5.8e-02, 4.4e-02) ms⁻¹, wall time: 57.938 seconds
[ Info: i: 1180, t: 3.493 hours, Δt: 8.312 seconds, umax = (8.2e-02, 5.5e-02, 4.4e-02) ms⁻¹, wall time: 58.425 seconds
[ Info: i: 1200, t: 3.537 hours, Δt: 8.403 seconds, umax = (8.4e-02, 5.9e-02, 4.3e-02) ms⁻¹, wall time: 58.946 seconds
[ Info: i: 1220, t: 3.583 hours, Δt: 8.079 seconds, umax = (8.4e-02, 6.0e-02, 4.5e-02) ms⁻¹, wall time: 59.447 seconds
[ Info: i: 1240, t: 3.626 hours, Δt: 8.033 seconds, umax = (8.7e-02, 6.2e-02, 4.7e-02) ms⁻¹, wall time: 59.971 seconds
[ Info: i: 1260, t: 3.669 hours, Δt: 7.748 seconds, umax = (8.5e-02, 6.0e-02, 4.6e-02) ms⁻¹, wall time: 1.011 minutes
[ Info: i: 1280, t: 3.712 hours, Δt: 8.078 seconds, umax = (8.5e-02, 5.7e-02, 4.4e-02) ms⁻¹, wall time: 1.017 minutes
[ Info: i: 1300, t: 3.757 hours, Δt: 7.795 seconds, umax = (8.9e-02, 5.3e-02, 4.3e-02) ms⁻¹, wall time: 1.027 minutes
[ Info: i: 1320, t: 3.799 hours, Δt: 7.660 seconds, umax = (8.8e-02, 5.5e-02, 4.5e-02) ms⁻¹, wall time: 1.034 minutes
[ Info: i: 1340, t: 3.842 hours, Δt: 7.829 seconds, umax = (8.8e-02, 5.3e-02, 4.6e-02) ms⁻¹, wall time: 1.043 minutes
[ Info: i: 1360, t: 3.887 hours, Δt: 7.911 seconds, umax = (8.5e-02, 6.0e-02, 4.9e-02) ms⁻¹, wall time: 1.051 minutes
[ Info: i: 1380, t: 3.930 hours, Δt: 8.197 seconds, umax = (8.5e-02, 5.7e-02, 4.7e-02) ms⁻¹, wall time: 1.060 minutes
[ Info: i: 1400, t: 3.975 hours, Δt: 7.936 seconds, umax = (8.3e-02, 5.6e-02, 4.7e-02) ms⁻¹, wall time: 1.068 minutes
[ Info: Simulation is stopping after running for 1.074 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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