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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{5, Float64, Float32}(order=9)
├── tracers: b
├── closure: Nothing
├── buoyancy: BuoyancyTracer with ĝ = NegativeZDirection()
└── coriolis: FPlane{Float64}(f=0.0001)

Initial conditions

We make use of random noise concentrated in the upper 4 meters for buoyancy and velocity initial conditions,

Ξ(z) = randn() * exp(z / 4)

Our initial condition for buoyancy consists of a surface mixed layer 33 m deep, a deep linear stratification, plus noise,

initial_mixed_layer_depth = 33 # m
stratification(z) = z < - initial_mixed_layer_depth ? N² * z : N² * (-initial_mixed_layer_depth)

bᵢ(x, y, z) = stratification(z) + 1e-1 * Ξ(z) * N² * model.grid.Lz
bᵢ (generic function with 1 method)

The simulation we reproduce from Wagner et al. (2021) is zero Lagrangian-mean velocity. This initial condition is consistent with a wavy, quiescent ocean suddenly impacted by winds. To this quiescent state we add noise scaled by the friction velocity to $u$ and $w$.

u★ = sqrt(abs(τx))
uᵢ(x, y, z) = u★ * 1e-1 * Ξ(z)
wᵢ(x, y, z) = u★ * 1e-1 * Ξ(z)

set!(model, u=uᵢ, w=wᵢ, b=bᵢ)

Setting up the simulation

simulation = Simulation(model, Δt=45.0, stop_time=4hours)
Simulation of NonhydrostaticModel{GPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── Next time step: 45 seconds
├── Elapsed wall time: 0 seconds
├── Wall time per iteration: NaN days
├── Stop time: 4 hours
├── Stop iteration: Inf
├── Wall time limit: Inf
├── Minimum relative step: 0.0
├── Callbacks: OrderedDict with 4 entries:
│   ├── stop_time_exceeded => 4
│   ├── stop_iteration_exceeded => -
│   ├── wall_time_limit_exceeded => e
│   └── nan_checker => }
├── Output writers: OrderedDict with no entries
└── Diagnostics: OrderedDict with no entries

We use the TimeStepWizard for adaptive time-stepping with a Courant-Freidrichs-Lewy (CFL) number of 1.0,

conjure_time_step_wizard!(simulation, cfl=1.0, max_Δt=1minute)

Nice progress messaging

We define a function that prints a helpful message with maximum absolute value of $u, v, w$ and the current wall clock time.

using Printf

function progress(simulation)
    u, v, w = simulation.model.velocities

    # Print a progress message
    msg = @sprintf("i: %04d, t: %s, Δt: %s, umax = (%.1e, %.1e, %.1e) ms⁻¹, wall time: %s\n",
                   iteration(simulation),
                   prettytime(time(simulation)),
                   prettytime(simulation.Δt),
                   maximum(abs, u), maximum(abs, v), maximum(abs, w),
                   prettytime(simulation.run_wall_time))

    @info msg

    return nothing
end

simulation.callbacks[:progress] = Callback(progress, IterationInterval(20))
Callback of progress on IterationInterval(20)

Output

A field writer

We set up an output writer for the simulation that saves all velocity fields, tracer fields, and the subgrid turbulent diffusivity.

output_interval = 5minutes

fields_to_output = merge(model.velocities, model.tracers)

simulation.output_writers[:fields] =
    JLD2Writer(model, fields_to_output,
               schedule = TimeInterval(output_interval),
               filename = "langmuir_turbulence_fields.jld2",
               overwrite_existing = true)
JLD2Writer scheduled on TimeInterval(5 minutes):
├── filepath: langmuir_turbulence_fields.jld2
├── 4 outputs: (u, v, w, b)
├── array type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 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 = (1.6e-03, 9.1e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (52.235 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (8.631 seconds).
[ Info: i: 0020, t: 11.240 minutes, Δt: 27.278 seconds, umax = (2.8e-02, 1.2e-02, 2.4e-02) ms⁻¹, wall time: 1.363 minutes
[ Info: i: 0040, t: 19.680 minutes, Δt: 20.311 seconds, umax = (3.9e-02, 1.1e-02, 1.9e-02) ms⁻¹, wall time: 1.372 minutes
[ Info: i: 0060, t: 25.948 minutes, Δt: 16.979 seconds, umax = (4.7e-02, 1.4e-02, 1.9e-02) ms⁻¹, wall time: 1.384 minutes
[ Info: i: 0080, t: 31.390 minutes, Δt: 16.326 seconds, umax = (4.9e-02, 1.7e-02, 2.4e-02) ms⁻¹, wall time: 1.394 minutes
[ Info: i: 0100, t: 36.631 minutes, Δt: 15.869 seconds, umax = (5.1e-02, 1.8e-02, 2.5e-02) ms⁻¹, wall time: 1.404 minutes
[ Info: i: 0120, t: 41.794 minutes, Δt: 15.202 seconds, umax = (5.3e-02, 1.9e-02, 2.5e-02) ms⁻¹, wall time: 1.414 minutes
[ Info: i: 0140, t: 46.742 minutes, Δt: 14.485 seconds, umax = (5.5e-02, 2.0e-02, 2.6e-02) ms⁻¹, wall time: 1.425 minutes
[ Info: i: 0160, t: 51.196 minutes, Δt: 14.001 seconds, umax = (5.9e-02, 2.1e-02, 2.6e-02) ms⁻¹, wall time: 1.435 minutes
[ Info: i: 0180, t: 55.673 minutes, Δt: 13.223 seconds, umax = (6.2e-02, 2.3e-02, 2.8e-02) ms⁻¹, wall time: 1.445 minutes
[ Info: i: 0200, t: 1 hour, Δt: 13.213 seconds, umax = (6.2e-02, 2.5e-02, 2.9e-02) ms⁻¹, wall time: 1.453 minutes
[ Info: i: 0220, t: 1.073 hours, Δt: 12.301 seconds, umax = (6.4e-02, 2.8e-02, 3.3e-02) ms⁻¹, wall time: 1.463 minutes
[ Info: i: 0240, t: 1.138 hours, Δt: 11.962 seconds, umax = (6.8e-02, 2.6e-02, 3.4e-02) ms⁻¹, wall time: 1.471 minutes
[ Info: i: 0260, t: 1.201 hours, Δt: 11.639 seconds, umax = (6.5e-02, 2.9e-02, 3.3e-02) ms⁻¹, wall time: 1.484 minutes
[ Info: i: 0280, t: 1.263 hours, Δt: 11.355 seconds, umax = (6.8e-02, 3.2e-02, 3.8e-02) ms⁻¹, wall time: 1.495 minutes
[ Info: i: 0300, t: 1.326 hours, Δt: 11.074 seconds, umax = (7.0e-02, 3.5e-02, 3.5e-02) ms⁻¹, wall time: 1.503 minutes
[ Info: i: 0320, t: 1.387 hours, Δt: 11.355 seconds, umax = (7.1e-02, 3.3e-02, 3.5e-02) ms⁻¹, wall time: 1.512 minutes
[ Info: i: 0340, t: 1.445 hours, Δt: 11.129 seconds, umax = (7.3e-02, 3.5e-02, 3.6e-02) ms⁻¹, wall time: 1.522 minutes
[ Info: i: 0360, t: 1.506 hours, Δt: 10.484 seconds, umax = (7.2e-02, 3.6e-02, 3.9e-02) ms⁻¹, wall time: 1.536 minutes
[ Info: i: 0380, t: 1.565 hours, Δt: 10.439 seconds, umax = (7.2e-02, 4.0e-02, 4.0e-02) ms⁻¹, wall time: 1.544 minutes
[ Info: i: 0400, t: 1.620 hours, Δt: 10.243 seconds, umax = (7.3e-02, 3.9e-02, 3.8e-02) ms⁻¹, wall time: 1.555 minutes
[ Info: i: 0420, t: 1.675 hours, Δt: 9.858 seconds, umax = (7.6e-02, 4.1e-02, 3.7e-02) ms⁻¹, wall time: 1.569 minutes
[ Info: i: 0440, t: 1.729 hours, Δt: 9.694 seconds, umax = (7.5e-02, 3.9e-02, 3.8e-02) ms⁻¹, wall time: 1.576 minutes
[ Info: i: 0460, t: 1.783 hours, Δt: 9.804 seconds, umax = (8.0e-02, 4.2e-02, 4.0e-02) ms⁻¹, wall time: 1.587 minutes
[ Info: i: 0480, t: 1.836 hours, Δt: 9.463 seconds, umax = (7.8e-02, 4.4e-02, 4.1e-02) ms⁻¹, wall time: 1.601 minutes
[ Info: i: 0500, t: 1.888 hours, Δt: 8.919 seconds, umax = (7.7e-02, 4.2e-02, 4.3e-02) ms⁻¹, wall time: 1.609 minutes
[ Info: i: 0520, t: 1.936 hours, Δt: 9.085 seconds, umax = (7.7e-02, 4.6e-02, 4.5e-02) ms⁻¹, wall time: 1.621 minutes
[ Info: i: 0540, t: 1.989 hours, Δt: 9.512 seconds, umax = (8.0e-02, 4.3e-02, 4.1e-02) ms⁻¹, wall time: 1.631 minutes
[ Info: i: 0560, t: 2.037 hours, Δt: 9.650 seconds, umax = (7.8e-02, 4.6e-02, 4.2e-02) ms⁻¹, wall time: 1.641 minutes
[ Info: i: 0580, t: 2.089 hours, Δt: 9.658 seconds, umax = (7.7e-02, 4.7e-02, 4.7e-02) ms⁻¹, wall time: 1.653 minutes
[ Info: i: 0600, t: 2.142 hours, Δt: 9.351 seconds, umax = (7.9e-02, 4.9e-02, 4.8e-02) ms⁻¹, wall time: 1.659 minutes
[ Info: i: 0620, t: 2.191 hours, Δt: 8.885 seconds, umax = (8.1e-02, 4.7e-02, 4.5e-02) ms⁻¹, wall time: 1.670 minutes
[ Info: i: 0640, t: 2.241 hours, Δt: 8.876 seconds, umax = (8.0e-02, 5.5e-02, 4.5e-02) ms⁻¹, wall time: 1.679 minutes
[ Info: i: 0660, t: 2.290 hours, Δt: 9.270 seconds, umax = (8.3e-02, 4.7e-02, 4.3e-02) ms⁻¹, wall time: 1.690 minutes
[ Info: i: 0680, t: 2.341 hours, Δt: 9.193 seconds, umax = (7.8e-02, 5.0e-02, 4.2e-02) ms⁻¹, wall time: 1.703 minutes
[ Info: i: 0700, t: 2.391 hours, Δt: 8.726 seconds, umax = (7.8e-02, 4.8e-02, 4.1e-02) ms⁻¹, wall time: 1.710 minutes
[ Info: i: 0720, t: 2.439 hours, Δt: 8.932 seconds, umax = (7.9e-02, 5.4e-02, 4.2e-02) ms⁻¹, wall time: 1.722 minutes
[ Info: i: 0740, t: 2.488 hours, Δt: 8.776 seconds, umax = (8.3e-02, 5.3e-02, 4.4e-02) ms⁻¹, wall time: 1.730 minutes
[ Info: i: 0760, t: 2.537 hours, Δt: 8.563 seconds, umax = (8.3e-02, 5.1e-02, 4.1e-02) ms⁻¹, wall time: 1.740 minutes
[ Info: i: 0780, t: 2.583 hours, Δt: 8.661 seconds, umax = (8.3e-02, 5.2e-02, 4.4e-02) ms⁻¹, wall time: 1.750 minutes
[ Info: i: 0800, t: 2.632 hours, Δt: 9.081 seconds, umax = (8.2e-02, 5.5e-02, 4.1e-02) ms⁻¹, wall time: 1.759 minutes
[ Info: i: 0820, t: 2.682 hours, Δt: 8.976 seconds, umax = (8.2e-02, 5.4e-02, 4.2e-02) ms⁻¹, wall time: 1.770 minutes
[ Info: i: 0840, t: 2.731 hours, Δt: 8.650 seconds, umax = (8.1e-02, 5.3e-02, 4.8e-02) ms⁻¹, wall time: 1.778 minutes
[ Info: i: 0860, t: 2.779 hours, Δt: 8.798 seconds, umax = (8.1e-02, 5.1e-02, 4.8e-02) ms⁻¹, wall time: 1.789 minutes
[ Info: i: 0880, t: 2.828 hours, Δt: 8.606 seconds, umax = (8.2e-02, 5.1e-02, 4.6e-02) ms⁻¹, wall time: 1.800 minutes
[ Info: i: 0900, t: 2.873 hours, Δt: 8.115 seconds, umax = (8.4e-02, 5.2e-02, 4.2e-02) ms⁻¹, wall time: 1.810 minutes
[ Info: i: 0920, t: 2.917 hours, Δt: 8.143 seconds, umax = (8.2e-02, 5.4e-02, 4.3e-02) ms⁻¹, wall time: 1.820 minutes
[ Info: i: 0940, t: 2.963 hours, Δt: 8.476 seconds, umax = (8.1e-02, 5.1e-02, 4.6e-02) ms⁻¹, wall time: 1.828 minutes
[ Info: i: 0960, t: 3.007 hours, Δt: 8.608 seconds, umax = (8.1e-02, 5.7e-02, 4.3e-02) ms⁻¹, wall time: 1.841 minutes
[ Info: i: 0980, t: 3.055 hours, Δt: 8.457 seconds, umax = (8.4e-02, 5.5e-02, 4.2e-02) ms⁻¹, wall time: 1.848 minutes
[ Info: i: 1000, t: 3.102 hours, Δt: 8.513 seconds, umax = (8.3e-02, 5.4e-02, 4.7e-02) ms⁻¹, wall time: 1.858 minutes
[ Info: i: 1020, t: 3.149 hours, Δt: 8.409 seconds, umax = (8.5e-02, 5.2e-02, 4.5e-02) ms⁻¹, wall time: 1.867 minutes
[ Info: i: 1040, t: 3.194 hours, Δt: 8.287 seconds, umax = (8.4e-02, 5.4e-02, 4.5e-02) ms⁻¹, wall time: 1.878 minutes
[ Info: i: 1060, t: 3.240 hours, Δt: 8.692 seconds, umax = (8.2e-02, 5.5e-02, 4.8e-02) ms⁻¹, wall time: 1.887 minutes
[ Info: i: 1080, t: 3.286 hours, Δt: 8.506 seconds, umax = (8.4e-02, 6.1e-02, 5.3e-02) ms⁻¹, wall time: 1.896 minutes
[ Info: i: 1100, t: 3.333 hours, Δt: 8.494 seconds, umax = (8.4e-02, 6.3e-02, 5.0e-02) ms⁻¹, wall time: 1.905 minutes
[ Info: i: 1120, t: 3.375 hours, Δt: 8.326 seconds, umax = (8.3e-02, 5.9e-02, 4.6e-02) ms⁻¹, wall time: 1.914 minutes
[ Info: i: 1140, t: 3.421 hours, Δt: 8.225 seconds, umax = (8.4e-02, 5.7e-02, 4.8e-02) ms⁻¹, wall time: 1.926 minutes
[ Info: i: 1160, t: 3.467 hours, Δt: 8.208 seconds, umax = (8.4e-02, 5.6e-02, 4.3e-02) ms⁻¹, wall time: 1.934 minutes
[ Info: i: 1180, t: 3.511 hours, Δt: 8.192 seconds, umax = (8.2e-02, 6.2e-02, 4.5e-02) ms⁻¹, wall time: 1.948 minutes
[ Info: i: 1200, t: 3.557 hours, Δt: 8.332 seconds, umax = (8.1e-02, 6.7e-02, 4.6e-02) ms⁻¹, wall time: 1.955 minutes
[ Info: i: 1220, t: 3.601 hours, Δt: 8.050 seconds, umax = (8.4e-02, 6.4e-02, 4.4e-02) ms⁻¹, wall time: 1.966 minutes
[ Info: i: 1240, t: 3.646 hours, Δt: 7.891 seconds, umax = (8.5e-02, 5.9e-02, 4.5e-02) ms⁻¹, wall time: 1.975 minutes
[ Info: i: 1260, t: 3.689 hours, Δt: 8.001 seconds, umax = (8.6e-02, 5.7e-02, 4.3e-02) ms⁻¹, wall time: 1.985 minutes
[ Info: i: 1280, t: 3.734 hours, Δt: 8.093 seconds, umax = (8.4e-02, 6.0e-02, 4.3e-02) ms⁻¹, wall time: 1.994 minutes
[ Info: i: 1300, t: 3.776 hours, Δt: 7.611 seconds, umax = (8.7e-02, 6.1e-02, 4.6e-02) ms⁻¹, wall time: 2.003 minutes
[ Info: i: 1320, t: 3.819 hours, Δt: 7.993 seconds, umax = (8.6e-02, 6.4e-02, 5.1e-02) ms⁻¹, wall time: 2.012 minutes
[ Info: i: 1340, t: 3.862 hours, Δt: 7.897 seconds, umax = (8.4e-02, 6.2e-02, 4.6e-02) ms⁻¹, wall time: 2.021 minutes
[ Info: i: 1360, t: 3.907 hours, Δt: 8.187 seconds, umax = (8.6e-02, 6.4e-02, 5.0e-02) ms⁻¹, wall time: 2.031 minutes
[ Info: i: 1380, t: 3.950 hours, Δt: 8.183 seconds, umax = (8.7e-02, 7.0e-02, 4.9e-02) ms⁻¹, wall time: 2.041 minutes
[ Info: i: 1400, t: 3.996 hours, Δt: 7.970 seconds, umax = (8.9e-02, 6.6e-02, 4.5e-02) ms⁻¹, wall time: 2.052 minutes
[ Info: Simulation is stopping after running for 2.053 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)

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

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