GitHub

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.

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 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
├── 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: 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 = (1.4e-03, 8.4e-04, 1.4e-03) ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (33.458 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (7.189 seconds).
[ Info: i: 0020, t: 12.554 minutes, Δt: 27.535 seconds, umax = (3.0e-02, 1.1e-02, 2.0e-02) ms⁻¹, wall time: 54.543 seconds
[ Info: i: 0040, t: 20.809 minutes, Δt: 20.134 seconds, umax = (4.1e-02, 1.1e-02, 1.9e-02) ms⁻¹, wall time: 55.255 seconds
[ Info: i: 0060, t: 27.106 minutes, Δt: 16.757 seconds, umax = (4.9e-02, 1.4e-02, 1.9e-02) ms⁻¹, wall time: 55.740 seconds
[ Info: i: 0080, t: 32.486 minutes, Δt: 16.475 seconds, umax = (5.3e-02, 1.8e-02, 2.3e-02) ms⁻¹, wall time: 56.342 seconds
[ Info: i: 0100, t: 37.654 minutes, Δt: 15.791 seconds, umax = (5.3e-02, 1.8e-02, 2.2e-02) ms⁻¹, wall time: 57.012 seconds
[ Info: i: 0120, t: 42.892 minutes, Δt: 14.913 seconds, umax = (5.3e-02, 2.0e-02, 2.5e-02) ms⁻¹, wall time: 57.544 seconds
[ Info: i: 0140, t: 47.538 minutes, Δt: 14.668 seconds, umax = (5.5e-02, 2.1e-02, 2.7e-02) ms⁻¹, wall time: 58.155 seconds
[ Info: i: 0160, t: 52.140 minutes, Δt: 13.909 seconds, umax = (5.8e-02, 2.2e-02, 2.8e-02) ms⁻¹, wall time: 58.702 seconds
[ Info: i: 0180, t: 56.598 minutes, Δt: 13.233 seconds, umax = (6.0e-02, 2.4e-02, 2.9e-02) ms⁻¹, wall time: 59.256 seconds
[ Info: i: 0200, t: 1.014 hours, Δt: 12.761 seconds, umax = (6.2e-02, 2.7e-02, 3.0e-02) ms⁻¹, wall time: 59.867 seconds
[ Info: i: 0220, t: 1.083 hours, Δt: 11.948 seconds, umax = (6.4e-02, 2.7e-02, 3.1e-02) ms⁻¹, wall time: 1.005 minutes
[ Info: i: 0240, t: 1.150 hours, Δt: 12.091 seconds, umax = (6.4e-02, 3.0e-02, 3.2e-02) ms⁻¹, wall time: 1.013 minutes
[ Info: i: 0260, t: 1.213 hours, Δt: 11.623 seconds, umax = (6.5e-02, 2.9e-02, 3.1e-02) ms⁻¹, wall time: 1.022 minutes
[ Info: i: 0280, t: 1.276 hours, Δt: 10.810 seconds, umax = (6.8e-02, 3.0e-02, 3.4e-02) ms⁻¹, wall time: 1.031 minutes
[ Info: i: 0300, t: 1.333 hours, Δt: 11.262 seconds, umax = (7.1e-02, 3.0e-02, 3.6e-02) ms⁻¹, wall time: 1.039 minutes
[ Info: i: 0320, t: 1.395 hours, Δt: 11.039 seconds, umax = (7.0e-02, 3.3e-02, 3.9e-02) ms⁻¹, wall time: 1.048 minutes
[ Info: i: 0340, t: 1.453 hours, Δt: 10.598 seconds, umax = (7.2e-02, 3.5e-02, 3.7e-02) ms⁻¹, wall time: 1.057 minutes
[ Info: i: 0360, t: 1.512 hours, Δt: 9.831 seconds, umax = (7.2e-02, 3.5e-02, 3.8e-02) ms⁻¹, wall time: 1.068 minutes
[ Info: i: 0380, t: 1.566 hours, Δt: 10.124 seconds, umax = (7.1e-02, 4.0e-02, 3.6e-02) ms⁻¹, wall time: 1.075 minutes
[ Info: i: 0400, t: 1.620 hours, Δt: 9.816 seconds, umax = (7.4e-02, 3.8e-02, 3.8e-02) ms⁻¹, wall time: 1.084 minutes
[ Info: i: 0420, t: 1.672 hours, Δt: 9.896 seconds, umax = (7.6e-02, 3.9e-02, 3.8e-02) ms⁻¹, wall time: 1.095 minutes
[ Info: i: 0440, t: 1.727 hours, Δt: 10.047 seconds, umax = (7.6e-02, 4.6e-02, 4.4e-02) ms⁻¹, wall time: 1.102 minutes
[ Info: i: 0460, t: 1.780 hours, Δt: 9.862 seconds, umax = (7.8e-02, 4.3e-02, 4.8e-02) ms⁻¹, wall time: 1.111 minutes
[ Info: i: 0480, t: 1.833 hours, Δt: 9.718 seconds, umax = (7.4e-02, 4.5e-02, 4.5e-02) ms⁻¹, wall time: 1.120 minutes
[ Info: i: 0500, t: 1.888 hours, Δt: 9.720 seconds, umax = (7.9e-02, 4.3e-02, 4.4e-02) ms⁻¹, wall time: 1.130 minutes
[ Info: i: 0520, t: 1.937 hours, Δt: 9.191 seconds, umax = (7.6e-02, 4.8e-02, 4.5e-02) ms⁻¹, wall time: 1.141 minutes
[ Info: i: 0540, t: 1.988 hours, Δt: 8.910 seconds, umax = (7.5e-02, 4.6e-02, 4.3e-02) ms⁻¹, wall time: 1.150 minutes
[ Info: i: 0560, t: 2.037 hours, Δt: 8.641 seconds, umax = (7.8e-02, 4.4e-02, 4.4e-02) ms⁻¹, wall time: 1.159 minutes
[ Info: i: 0580, t: 2.086 hours, Δt: 9.046 seconds, umax = (8.0e-02, 4.7e-02, 4.3e-02) ms⁻¹, wall time: 1.171 minutes
[ Info: i: 0600, t: 2.136 hours, Δt: 8.740 seconds, umax = (7.9e-02, 4.9e-02, 4.6e-02) ms⁻¹, wall time: 1.177 minutes
[ Info: i: 0620, t: 2.183 hours, Δt: 8.413 seconds, umax = (8.0e-02, 4.8e-02, 4.6e-02) ms⁻¹, wall time: 1.187 minutes
[ Info: i: 0640, t: 2.231 hours, Δt: 8.734 seconds, umax = (7.9e-02, 5.0e-02, 5.0e-02) ms⁻¹, wall time: 1.195 minutes
[ Info: i: 0660, t: 2.279 hours, Δt: 8.768 seconds, umax = (8.2e-02, 4.6e-02, 4.2e-02) ms⁻¹, wall time: 1.204 minutes
[ Info: i: 0680, t: 2.326 hours, Δt: 9.027 seconds, umax = (8.1e-02, 4.6e-02, 4.1e-02) ms⁻¹, wall time: 1.213 minutes
[ Info: i: 0700, t: 2.376 hours, Δt: 8.792 seconds, umax = (8.2e-02, 5.5e-02, 4.0e-02) ms⁻¹, wall time: 1.222 minutes
[ Info: i: 0720, t: 2.421 hours, Δt: 8.518 seconds, umax = (8.4e-02, 5.8e-02, 4.1e-02) ms⁻¹, wall time: 1.235 minutes
[ Info: i: 0740, t: 2.468 hours, Δt: 8.825 seconds, umax = (8.1e-02, 5.2e-02, 4.5e-02) ms⁻¹, wall time: 1.242 minutes
[ Info: i: 0760, t: 2.518 hours, Δt: 8.937 seconds, umax = (8.4e-02, 5.2e-02, 4.4e-02) ms⁻¹, wall time: 1.253 minutes
[ Info: i: 0780, t: 2.567 hours, Δt: 8.610 seconds, umax = (8.1e-02, 5.8e-02, 4.5e-02) ms⁻¹, wall time: 1.261 minutes
[ Info: i: 0800, t: 2.612 hours, Δt: 8.627 seconds, umax = (8.2e-02, 5.6e-02, 4.6e-02) ms⁻¹, wall time: 1.270 minutes
[ Info: i: 0820, t: 2.661 hours, Δt: 8.697 seconds, umax = (8.7e-02, 5.2e-02, 4.2e-02) ms⁻¹, wall time: 1.279 minutes
[ Info: i: 0840, t: 2.708 hours, Δt: 8.689 seconds, umax = (8.5e-02, 5.4e-02, 4.1e-02) ms⁻¹, wall time: 1.287 minutes
[ Info: i: 0860, t: 2.755 hours, Δt: 8.351 seconds, umax = (8.4e-02, 5.4e-02, 4.2e-02) ms⁻¹, wall time: 1.299 minutes
[ Info: i: 0880, t: 2.802 hours, Δt: 8.425 seconds, umax = (8.3e-02, 5.6e-02, 4.5e-02) ms⁻¹, wall time: 1.305 minutes
[ Info: i: 0900, t: 2.847 hours, Δt: 8.526 seconds, umax = (8.6e-02, 5.6e-02, 4.9e-02) ms⁻¹, wall time: 1.315 minutes
[ Info: i: 0920, t: 2.894 hours, Δt: 8.846 seconds, umax = (8.5e-02, 6.2e-02, 4.8e-02) ms⁻¹, wall time: 1.323 minutes
[ Info: i: 0940, t: 2.940 hours, Δt: 8.393 seconds, umax = (8.7e-02, 6.3e-02, 4.6e-02) ms⁻¹, wall time: 1.333 minutes
[ Info: i: 0960, t: 2.987 hours, Δt: 8.301 seconds, umax = (8.8e-02, 6.0e-02, 5.3e-02) ms⁻¹, wall time: 1.342 minutes
[ Info: i: 0980, t: 3.032 hours, Δt: 7.690 seconds, umax = (8.9e-02, 6.2e-02, 4.8e-02) ms⁻¹, wall time: 1.351 minutes
[ Info: i: 1000, t: 3.075 hours, Δt: 8.419 seconds, umax = (8.9e-02, 5.9e-02, 4.8e-02) ms⁻¹, wall time: 1.360 minutes
[ Info: i: 1020, t: 3.121 hours, Δt: 8.449 seconds, umax = (8.8e-02, 5.9e-02, 4.7e-02) ms⁻¹, wall time: 1.370 minutes
[ Info: i: 1040, t: 3.167 hours, Δt: 8.094 seconds, umax = (8.9e-02, 5.4e-02, 4.6e-02) ms⁻¹, wall time: 1.379 minutes
[ Info: i: 1060, t: 3.211 hours, Δt: 7.949 seconds, umax = (8.6e-02, 5.6e-02, 4.8e-02) ms⁻¹, wall time: 1.389 minutes
[ Info: i: 1080, t: 3.255 hours, Δt: 8.495 seconds, umax = (8.9e-02, 5.8e-02, 5.5e-02) ms⁻¹, wall time: 1.400 minutes
[ Info: i: 1100, t: 3.301 hours, Δt: 8.337 seconds, umax = (9.2e-02, 5.7e-02, 4.6e-02) ms⁻¹, wall time: 1.407 minutes
[ Info: i: 1120, t: 3.347 hours, Δt: 8.123 seconds, umax = (9.2e-02, 5.8e-02, 4.9e-02) ms⁻¹, wall time: 1.417 minutes
[ Info: i: 1140, t: 3.392 hours, Δt: 8.348 seconds, umax = (8.8e-02, 6.1e-02, 5.0e-02) ms⁻¹, wall time: 1.425 minutes
[ Info: i: 1160, t: 3.438 hours, Δt: 8.675 seconds, umax = (8.6e-02, 5.9e-02, 5.0e-02) ms⁻¹, wall time: 1.434 minutes
[ Info: i: 1180, t: 3.486 hours, Δt: 8.392 seconds, umax = (8.7e-02, 6.0e-02, 4.8e-02) ms⁻¹, wall time: 1.444 minutes
[ Info: i: 1200, t: 3.531 hours, Δt: 8.420 seconds, umax = (8.7e-02, 6.0e-02, 4.5e-02) ms⁻¹, wall time: 1.454 minutes
[ Info: i: 1220, t: 3.578 hours, Δt: 8.444 seconds, umax = (8.8e-02, 5.9e-02, 4.5e-02) ms⁻¹, wall time: 1.463 minutes
[ Info: i: 1240, t: 3.623 hours, Δt: 8.211 seconds, umax = (8.3e-02, 5.3e-02, 4.9e-02) ms⁻¹, wall time: 1.472 minutes
[ Info: i: 1260, t: 3.667 hours, Δt: 8.373 seconds, umax = (8.5e-02, 5.3e-02, 4.6e-02) ms⁻¹, wall time: 1.481 minutes
[ Info: i: 1280, t: 3.713 hours, Δt: 8.054 seconds, umax = (8.6e-02, 5.5e-02, 5.0e-02) ms⁻¹, wall time: 1.490 minutes
[ Info: i: 1300, t: 3.757 hours, Δt: 7.941 seconds, umax = (8.7e-02, 5.7e-02, 4.4e-02) ms⁻¹, wall time: 1.502 minutes
[ Info: i: 1320, t: 3.800 hours, Δt: 7.922 seconds, umax = (8.8e-02, 6.0e-02, 4.7e-02) ms⁻¹, wall time: 1.508 minutes
[ Info: i: 1340, t: 3.845 hours, Δt: 8.346 seconds, umax = (8.5e-02, 6.1e-02, 5.0e-02) ms⁻¹, wall time: 1.519 minutes
[ Info: i: 1360, t: 3.891 hours, Δt: 8.156 seconds, umax = (8.6e-02, 6.0e-02, 5.1e-02) ms⁻¹, wall time: 1.527 minutes
[ Info: i: 1380, t: 3.934 hours, Δt: 7.975 seconds, umax = (9.2e-02, 5.8e-02, 5.1e-02) ms⁻¹, wall time: 1.537 minutes
[ Info: i: 1400, t: 3.978 hours, Δt: 7.599 seconds, umax = (8.7e-02, 6.0e-02, 4.9e-02) ms⁻¹, wall time: 1.545 minutes
[ Info: Simulation is stopping after running for 1.552 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

This page was generated using Literate.jl.