Wind- and convection-driven mixing in an ocean surface boundary layer

This example simulates mixing by three-dimensional turbulence in an ocean surface boundary layer driven by atmospheric winds and convection. It demonstrates:

• How to set-up a grid with varying spacing in the vertical direction
• How to use the SeawaterBuoyancy model for buoyancy with TEOS10EquationOfState.
• How to use a turbulence closure for large eddy simulation.
• How to use a function to impose a boundary condition.

Install dependencies

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

using Pkg
pkg"add Oceananigans, CairoMakie, SeawaterPolynomials, CUDA"

We start by importing all of the packages and functions that we'll need for this example.

using Oceananigans
using Oceananigans.Units

using CUDA
using Random
using Printf
using CairoMakie
using SeawaterPolynomials.TEOS10: TEOS10EquationOfState

The grid

We use 128²×64 grid points with 1 m grid spacing in the horizontal and varying spacing in the vertical, with higher resolution closer to the surface. Here we use a stretching function for the vertical nodes that maintains relatively constant vertical spacing in the mixed layer, which is desirable from a numerical standpoint:

Nx = Ny = 128    # number of points in each of horizontal directions
Nz = 64          # number of points in the vertical direction

Lx = Ly = 128    # (m) domain horizontal extents
Lz = 64          # (m) domain depth

refinement = 1.2 # controls spacing near surface (higher means finer spaced)
stretching = 12  # controls rate of stretching at bottom

# Normalized height ranging from 0 to 1
h(k) = (k - 1) / Nz

# Linear near-surface generator
ζ₀(k) = 1 + (h(k) - 1) / refinement

# Bottom-intensified stretching function
Σ(k) = (1 - exp(-stretching * h(k))) / (1 - exp(-stretching))

# Generating function
z_interfaces(k) = Lz * (ζ₀(k) * Σ(k) - 1)

grid = RectilinearGrid(GPU(),
                       size = (Nx, Nx, Nz),
                       x = (0, Lx),
                       y = (0, Ly),
                       z = z_interfaces)

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] variably spaced with min(Δz)=0.833413, max(Δz)=1.96618

We plot vertical spacing versus depth to inspect the prescribed grid stretching:

fig = Figure(size=(1200, 800))
ax = Axis(fig[1, 1], ylabel = "z (m)", xlabel = "Vertical spacing (m)")

lines!(ax, zspacings(grid, Center()))
scatter!(ax, zspacings(grid, Center()))

fig

Buoyancy that depends on temperature and salinity

We use the SeawaterBuoyancy model with the TEOS10 equation of state,

ρₒ = 1026 # kg m⁻³, average density at the surface of the world ocean
equation_of_state = TEOS10EquationOfState(reference_density=ρₒ)
buoyancy = SeawaterBuoyancy(; equation_of_state)

SeawaterBuoyancy{Float64}:
├── gravitational_acceleration: 9.80665
└── equation_of_state: BoussinesqEquationOfState{Float64}

Boundary conditions

We calculate the surface temperature flux associated with surface cooling of 200 W m⁻², reference density ρₒ, and heat capacity cᴾ,

Q = 200   # W m⁻², surface _heat_ flux
cᴾ = 3991 # J K⁻¹ kg⁻¹, typical heat capacity for seawater

Jᵀ = Q / (ρₒ * cᴾ) # K m s⁻¹, surface _temperature_ flux

4.884283985946938e-5

Finally, we impose a temperature gradient dTdz both initially (see "Initial conditions" section below) and at the bottom of the domain, culminating in the boundary conditions on temperature,

dTdz = 0.01 # K m⁻¹

T_bcs = FieldBoundaryConditions(top = FluxBoundaryCondition(Jᵀ),
                                bottom = GradientBoundaryCondition(dTdz))

Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: GradientBoundaryCondition: 0.01
├── top: FluxBoundaryCondition: 4.88428e-5
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)

Note that a positive temperature flux at the surface of the ocean implies cooling. This is because a positive temperature flux implies that temperature is fluxed upwards, out of the ocean.

For the velocity field, we imagine a wind blowing over the ocean surface with an average velocity at 10 meters u₁₀, and use a drag coefficient cᴰ to estimate the kinematic stress (that is, stress divided by density) exerted by the wind on the ocean:

u₁₀ = 10  # m s⁻¹, average wind velocity 10 meters above the ocean
cᴰ = 2e-3 # dimensionless drag coefficient
ρₐ = 1.2  # kg m⁻³, approximate average density of air at sea-level
τx = - ρₐ / ρₒ * cᴰ * u₁₀ * abs(u₁₀) # m² s⁻²

-0.00023391812865497074

The boundary conditions on u are thus

u_bcs = 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: -0.000233918
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)

For salinity, S, we impose an evaporative flux of the form

@inline Jˢ(x, y, t, S, evaporation_rate) = - evaporation_rate * S # [salinity unit] m s⁻¹

where S is salinity. We use an evaporation rate of 1 millimeter per hour,

evaporation_rate = 1e-3 / hour # m s⁻¹

2.7777777777777776e-7

We build the Flux evaporation BoundaryCondition with the function Jˢ, indicating that Jˢ depends on salinity S and passing the parameter evaporation_rate,

evaporation_bc = FluxBoundaryCondition(Jˢ, field_dependencies=:S, parameters=evaporation_rate)

FluxBoundaryCondition: ContinuousBoundaryFunction Jˢ at (Nothing, Nothing, Nothing)

The full salinity boundary conditions are

S_bcs = FieldBoundaryConditions(top=evaporation_bc)

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: ContinuousBoundaryFunction Jˢ at (Nothing, Nothing, Nothing)
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)

Model instantiation

We fill in the final details of the model here, i.e., Coriolis forces, and the AnisotropicMinimumDissipation closure for large eddy simulation to model the effect of turbulent motions at scales smaller than the grid scale that are not explicitly resolved.

model = NonhydrostaticModel(; grid, buoyancy,
                            tracers = (:T, :S),
                            coriolis = FPlane(f=1e-4),
                            closure = AnisotropicMinimumDissipation(),
                            boundary_conditions = (u=u_bcs, T=T_bcs, S=S_bcs))

NonhydrostaticModel{GPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CUDAGPU with 3×3×3 halo
├── timestepper: RungeKutta3TimeStepper
├── advection scheme: Centered(order=2)
├── tracers: (T, S)
├── closure: AnisotropicMinimumDissipation{ExplicitTimeDiscretization, @NamedTuple{T::Float64, S::Float64}, Float64, Nothing}
├── buoyancy: SeawaterBuoyancy with g=9.80665 and BoussinesqEquationOfState{Float64} with ĝ = NegativeZDirection()
└── coriolis: FPlane{Float64}(f=0.0001)

Note: To use the Smagorinsky-Lilly turbulence closure (with a constant model coefficient) rather than AnisotropicMinimumDissipation, use closure = SmagorinskyLilly() in the model constructor.

Initial conditions

Our initial condition for temperature consists of a linear stratification superposed with random noise damped at the walls, while our initial condition for velocity consists only of random noise.

# Random noise damped at top and bottom
Ξ(z) = randn() * z / model.grid.Lz * (1 + z / model.grid.Lz) # noise

# Temperature initial condition: a stable density gradient with random noise superposed.
Tᵢ(x, y, z) = 20 + dTdz * z + dTdz * model.grid.Lz * 1e-6 * Ξ(z)

# Velocity initial condition: random noise scaled by the friction velocity.
uᵢ(x, y, z) = sqrt(abs(τx)) * 1e-3 * Ξ(z)

# `set!` the `model` fields using functions or constants:
set!(model, u=uᵢ, w=uᵢ, T=Tᵢ, S=35)

Setting up a simulation

We set-up a simulation with an initial time-step of 10 seconds that stops at 2 hours, with adaptive time-stepping and progress printing.

simulation = Simulation(model, Δt=10, stop_time=2hours)

Simulation of NonhydrostaticModel{GPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── Next time step: 10 seconds
├── run_wall_time: 0 seconds
├── run_wall_time / iteration: NaN days
├── stop_time: 2 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

The TimeStepWizard helps ensure stable time-stepping with a Courant-Freidrichs-Lewy (CFL) number of 1.0.

wizard = TimeStepWizard(cfl=1, max_change=1.1, max_Δt=1minute)
simulation.callbacks[:wizard] = Callback(wizard, IterationInterval(10))

Callback of TimeStepWizard(cfl=1.0, max_Δt=60.0, min_Δt=0.0) on IterationInterval(10)

Nice progress messaging is helpful:

# Print a progress message
progress_message(sim) = @printf("Iteration: %04d, time: %s, Δt: %s, max(|w|) = %.1e ms⁻¹, wall time: %s\n",
                                iteration(sim), prettytime(sim), prettytime(sim.Δt),
                                maximum(abs, sim.model.velocities.w), prettytime(sim.run_wall_time))

add_callback!(simulation, progress_message, IterationInterval(40))

We then set up the simulation:

Output

We use the JLD2Writer to save $x, z$ slices of the velocity fields, tracer fields, and eddy diffusivities. The prefix keyword argument to JLD2Writer indicates that output will be saved in ocean_wind_mixing_and_convection.jld2.

# Create a NamedTuple with eddy viscosity
eddy_viscosity = (; νₑ = model.closure_fields.νₑ)

filename = "ocean_wind_mixing_and_convection"

simulation.output_writers[:slices] =
    JLD2Writer(model, merge(model.velocities, model.tracers, eddy_viscosity),
               filename = filename * ".jld2",
               indices = (:, grid.Ny/2, :),
               schedule = TimeInterval(1minute),
               overwrite_existing = true)

JLD2Writer scheduled on TimeInterval(1 minute):
├── filepath: ocean_wind_mixing_and_convection.jld2
├── 6 outputs: (u, v, w, T, S, νₑ)
├── array_type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 63.8 KiB

We're ready:

run!(simulation)

[ Info: Initializing simulation...
Iteration: 0000, time: 0 seconds, Δt: 11 seconds, max(|w|) = 1.3e-05 ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (8.254 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (3.285 seconds).
Iteration: 0040, time: 7.344 minutes, Δt: 7.911 seconds, max(|w|) = 4.7e-05 ms⁻¹, wall time: 12.415 seconds
Iteration: 0080, time: 11.441 minutes, Δt: 4.746 seconds, max(|w|) = 9.3e-03 ms⁻¹, wall time: 13.154 seconds
Iteration: 0120, time: 14.201 minutes, Δt: 4.111 seconds, max(|w|) = 2.4e-02 ms⁻¹, wall time: 13.573 seconds
Iteration: 0160, time: 16.901 minutes, Δt: 4.108 seconds, max(|w|) = 2.2e-02 ms⁻¹, wall time: 13.965 seconds
Iteration: 0200, time: 19.541 minutes, Δt: 4.079 seconds, max(|w|) = 3.1e-02 ms⁻¹, wall time: 14.489 seconds
Iteration: 0240, time: 22.131 minutes, Δt: 3.788 seconds, max(|w|) = 3.3e-02 ms⁻¹, wall time: 14.867 seconds
Iteration: 0280, time: 24.627 minutes, Δt: 3.608 seconds, max(|w|) = 2.8e-02 ms⁻¹, wall time: 15.326 seconds
Iteration: 0320, time: 26.892 minutes, Δt: 3.570 seconds, max(|w|) = 3.3e-02 ms⁻¹, wall time: 15.730 seconds
Iteration: 0360, time: 29.235 minutes, Δt: 3.399 seconds, max(|w|) = 3.1e-02 ms⁻¹, wall time: 16.201 seconds
Iteration: 0400, time: 31.465 minutes, Δt: 3.366 seconds, max(|w|) = 3.1e-02 ms⁻¹, wall time: 16.609 seconds
Iteration: 0440, time: 33.681 minutes, Δt: 3.434 seconds, max(|w|) = 3.0e-02 ms⁻¹, wall time: 17.012 seconds
Iteration: 0480, time: 35.863 minutes, Δt: 3.258 seconds, max(|w|) = 2.8e-02 ms⁻¹, wall time: 17.416 seconds
Iteration: 0520, time: 37.975 minutes, Δt: 3.167 seconds, max(|w|) = 3.5e-02 ms⁻¹, wall time: 17.840 seconds
Iteration: 0560, time: 39.965 minutes, Δt: 3.006 seconds, max(|w|) = 3.5e-02 ms⁻¹, wall time: 18.232 seconds
Iteration: 0600, time: 41.893 minutes, Δt: 2.926 seconds, max(|w|) = 3.1e-02 ms⁻¹, wall time: 18.647 seconds
Iteration: 0640, time: 43.888 minutes, Δt: 3.190 seconds, max(|w|) = 3.6e-02 ms⁻¹, wall time: 19.020 seconds
Iteration: 0680, time: 45.860 minutes, Δt: 2.997 seconds, max(|w|) = 3.7e-02 ms⁻¹, wall time: 19.434 seconds
Iteration: 0720, time: 47.798 minutes, Δt: 2.945 seconds, max(|w|) = 3.5e-02 ms⁻¹, wall time: 19.840 seconds
Iteration: 0760, time: 49.692 minutes, Δt: 2.850 seconds, max(|w|) = 3.1e-02 ms⁻¹, wall time: 20.217 seconds
Iteration: 0800, time: 51.552 minutes, Δt: 2.742 seconds, max(|w|) = 4.3e-02 ms⁻¹, wall time: 20.626 seconds
Iteration: 0840, time: 53.371 minutes, Δt: 2.729 seconds, max(|w|) = 5.0e-02 ms⁻¹, wall time: 21.064 seconds
Iteration: 0880, time: 55.187 minutes, Δt: 2.865 seconds, max(|w|) = 5.3e-02 ms⁻¹, wall time: 21.482 seconds
Iteration: 0920, time: 57 minutes, Δt: 2.766 seconds, max(|w|) = 5.1e-02 ms⁻¹, wall time: 21.938 seconds
Iteration: 0960, time: 58.807 minutes, Δt: 2.677 seconds, max(|w|) = 4.9e-02 ms⁻¹, wall time: 22.381 seconds
Iteration: 1000, time: 1.008 hours, Δt: 2.657 seconds, max(|w|) = 4.3e-02 ms⁻¹, wall time: 22.780 seconds
Iteration: 1040, time: 1.036 hours, Δt: 2.564 seconds, max(|w|) = 4.3e-02 ms⁻¹, wall time: 23.178 seconds
Iteration: 1080, time: 1.064 hours, Δt: 2.479 seconds, max(|w|) = 4.0e-02 ms⁻¹, wall time: 23.546 seconds
Iteration: 1120, time: 1.091 hours, Δt: 2.577 seconds, max(|w|) = 4.0e-02 ms⁻¹, wall time: 23.924 seconds
Iteration: 1160, time: 1.119 hours, Δt: 2.564 seconds, max(|w|) = 4.0e-02 ms⁻¹, wall time: 24.363 seconds
Iteration: 1200, time: 1.147 hours, Δt: 2.528 seconds, max(|w|) = 4.6e-02 ms⁻¹, wall time: 24.790 seconds
Iteration: 1240, time: 1.174 hours, Δt: 2.536 seconds, max(|w|) = 5.6e-02 ms⁻¹, wall time: 25.247 seconds
Iteration: 1280, time: 1.202 hours, Δt: 2.584 seconds, max(|w|) = 5.4e-02 ms⁻¹, wall time: 25.615 seconds
Iteration: 1320, time: 1.231 hours, Δt: 2.677 seconds, max(|w|) = 5.6e-02 ms⁻¹, wall time: 25.994 seconds
Iteration: 1360, time: 1.259 hours, Δt: 2.609 seconds, max(|w|) = 5.8e-02 ms⁻¹, wall time: 26.376 seconds
Iteration: 1400, time: 1.288 hours, Δt: 2.631 seconds, max(|w|) = 5.6e-02 ms⁻¹, wall time: 26.786 seconds
Iteration: 1440, time: 1.317 hours, Δt: 2.573 seconds, max(|w|) = 5.4e-02 ms⁻¹, wall time: 27.181 seconds
Iteration: 1480, time: 1.344 hours, Δt: 2.531 seconds, max(|w|) = 5.5e-02 ms⁻¹, wall time: 27.557 seconds
Iteration: 1520, time: 1.371 hours, Δt: 2.474 seconds, max(|w|) = 5.4e-02 ms⁻¹, wall time: 27.951 seconds
Iteration: 1560, time: 1.398 hours, Δt: 2.624 seconds, max(|w|) = 5.0e-02 ms⁻¹, wall time: 28.374 seconds
Iteration: 1600, time: 1.427 hours, Δt: 2.653 seconds, max(|w|) = 5.1e-02 ms⁻¹, wall time: 28.817 seconds
Iteration: 1640, time: 1.456 hours, Δt: 2.626 seconds, max(|w|) = 5.6e-02 ms⁻¹, wall time: 29.212 seconds
Iteration: 1680, time: 1.485 hours, Δt: 2.583 seconds, max(|w|) = 5.7e-02 ms⁻¹, wall time: 29.586 seconds
Iteration: 1720, time: 1.513 hours, Δt: 2.541 seconds, max(|w|) = 6.1e-02 ms⁻¹, wall time: 29.967 seconds
Iteration: 1760, time: 1.539 hours, Δt: 2.378 seconds, max(|w|) = 6.0e-02 ms⁻¹, wall time: 30.367 seconds
Iteration: 1800, time: 1.565 hours, Δt: 2.475 seconds, max(|w|) = 5.5e-02 ms⁻¹, wall time: 30.738 seconds
Iteration: 1840, time: 1.592 hours, Δt: 2.379 seconds, max(|w|) = 5.2e-02 ms⁻¹, wall time: 31.145 seconds
Iteration: 1880, time: 1.617 hours, Δt: 2.365 seconds, max(|w|) = 5.7e-02 ms⁻¹, wall time: 31.588 seconds
Iteration: 1920, time: 1.643 hours, Δt: 2.398 seconds, max(|w|) = 5.7e-02 ms⁻¹, wall time: 32.061 seconds
Iteration: 1960, time: 1.669 hours, Δt: 2.395 seconds, max(|w|) = 6.6e-02 ms⁻¹, wall time: 32.536 seconds
Iteration: 2000, time: 1.696 hours, Δt: 2.431 seconds, max(|w|) = 7.2e-02 ms⁻¹, wall time: 33.008 seconds
Iteration: 2040, time: 1.724 hours, Δt: 2.488 seconds, max(|w|) = 6.7e-02 ms⁻¹, wall time: 33.471 seconds
Iteration: 2080, time: 1.750 hours, Δt: 2.436 seconds, max(|w|) = 5.5e-02 ms⁻¹, wall time: 33.876 seconds
Iteration: 2120, time: 1.777 hours, Δt: 2.486 seconds, max(|w|) = 5.9e-02 ms⁻¹, wall time: 34.311 seconds
Iteration: 2160, time: 1.803 hours, Δt: 2.414 seconds, max(|w|) = 6.2e-02 ms⁻¹, wall time: 34.736 seconds
Iteration: 2200, time: 1.831 hours, Δt: 2.538 seconds, max(|w|) = 6.5e-02 ms⁻¹, wall time: 35.129 seconds
Iteration: 2240, time: 1.857 hours, Δt: 2.434 seconds, max(|w|) = 5.9e-02 ms⁻¹, wall time: 35.550 seconds
Iteration: 2280, time: 1.884 hours, Δt: 2.544 seconds, max(|w|) = 6.8e-02 ms⁻¹, wall time: 35.998 seconds
Iteration: 2320, time: 1.912 hours, Δt: 2.508 seconds, max(|w|) = 7.5e-02 ms⁻¹, wall time: 36.412 seconds
Iteration: 2360, time: 1.939 hours, Δt: 2.505 seconds, max(|w|) = 6.6e-02 ms⁻¹, wall time: 36.854 seconds
Iteration: 2400, time: 1.967 hours, Δt: 2.436 seconds, max(|w|) = 6.0e-02 ms⁻¹, wall time: 37.344 seconds
Iteration: 2440, time: 1.993 hours, Δt: 2.378 seconds, max(|w|) = 5.4e-02 ms⁻¹, wall time: 37.754 seconds
[ Info: Simulation is stopping after running for 37.855 seconds.
[ Info: Simulation time 2 hours equals or exceeds stop time 2 hours.

Turbulence visualization

We animate the data saved in ocean_wind_mixing_and_convection.jld2. We prepare for animating the flow by loading the data into FieldTimeSeries and defining functions for computing colorbar limits.

filepath = filename * ".jld2"

time_series = (w = FieldTimeSeries(filepath, "w"),
               T = FieldTimeSeries(filepath, "T"),
               S = FieldTimeSeries(filepath, "S"),
               νₑ = FieldTimeSeries(filepath, "νₑ"))

(w = 128×1×65×121 FieldTimeSeries{InMemory} located at (Center, Center, Face) of w at ocean_wind_mixing_and_convection.jld2
├── grid: 128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── indices: (:, 64:64, :)
├── time_indexing: Linear()
├── backend: InMemory()
├── path: ocean_wind_mixing_and_convection.jld2
├── name: w
└── data: 134×1×71×121 OffsetArray(::Array{Float64, 4}, -2:131, 64:64, -2:68, 1:121) with eltype Float64 with indices -2:131×64:64×-2:68×1:121
    └── max=0.0421237, min=-0.0473499, mean=-2.05376e-5, T = 128×1×64×121 FieldTimeSeries{InMemory} located at (Center, Center, Center) of T at ocean_wind_mixing_and_convection.jld2
├── grid: 128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── indices: (:, 64:64, :)
├── time_indexing: Linear()
├── backend: InMemory()
├── path: ocean_wind_mixing_and_convection.jld2
├── name: T
└── data: 134×1×70×121 OffsetArray(::Array{Float64, 4}, -2:131, 64:64, -2:67, 1:121) with eltype Float64 with indices -2:131×64:64×-2:67×1:121
    └── max=19.9958, min=0.0, mean=18.5884, S = 128×1×64×121 FieldTimeSeries{InMemory} located at (Center, Center, Center) of S at ocean_wind_mixing_and_convection.jld2
├── grid: 128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── indices: (:, 64:64, :)
├── time_indexing: Linear()
├── backend: InMemory()
├── path: ocean_wind_mixing_and_convection.jld2
├── name: S
└── data: 134×1×70×121 OffsetArray(::Array{Float64, 4}, -2:131, 64:64, -2:67, 1:121) with eltype Float64 with indices -2:131×64:64×-2:67×1:121
    └── max=35.013, min=0.0, mean=33.0007, νₑ = 128×1×64×121 FieldTimeSeries{InMemory} located at (Center, Center, Center) of νₑ at ocean_wind_mixing_and_convection.jld2
├── grid: 128×128×64 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── indices: (:, 64:64, :)
├── time_indexing: Linear()
├── backend: InMemory()
├── path: ocean_wind_mixing_and_convection.jld2
├── name: νₑ
└── data: 134×1×70×121 OffsetArray(::Array{Float64, 4}, -2:131, 64:64, -2:67, 1:121) with eltype Float64 with indices -2:131×64:64×-2:67×1:121
    └── max=0.0190295, min=0.0, mean=0.000412483)

We start the animation at $t = 10$ minutes since things are pretty boring till then:

times = time_series.w.times
intro = searchsortedfirst(times, 10minutes)

11

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(intro)

 wₙ = @lift time_series.w[$n]
 Tₙ = @lift time_series.T[$n]
 Sₙ = @lift time_series.S[$n]
νₑₙ = @lift time_series.νₑ[$n]

fig = Figure(size = (1800, 900))

axis_kwargs = (xlabel="x (m)",
               ylabel="z (m)",
               aspect = AxisAspect(grid.Lx/grid.Lz),
               limits = ((0, grid.Lx), (-grid.Lz, 0)))

ax_w  = Axis(fig[2, 1]; title = "Vertical velocity", axis_kwargs...)
ax_T  = Axis(fig[2, 3]; title = "Temperature", axis_kwargs...)
ax_S  = Axis(fig[3, 1]; title = "Salinity", axis_kwargs...)
ax_νₑ = Axis(fig[3, 3]; title = "Eddy viscocity", axis_kwargs...)

title = @lift @sprintf("t = %s", prettytime(times[$n]))

 wlims = (-0.05, 0.05)
 Tlims = (19.7, 19.99)
 Slims = (35, 35.005)
νₑlims = (1e-6, 5e-3)

hm_w = heatmap!(ax_w, wₙ; colormap = :balance, colorrange = wlims)
Colorbar(fig[2, 2], hm_w; label = "m s⁻¹")

hm_T = heatmap!(ax_T, Tₙ; colormap = :thermal, colorrange = Tlims)
Colorbar(fig[2, 4], hm_T; label = "ᵒC")

hm_S = heatmap!(ax_S, Sₙ; colormap = :haline, colorrange = Slims)
Colorbar(fig[3, 2], hm_S; label = "g / kg")

hm_νₑ = heatmap!(ax_νₑ, νₑₙ; colormap = :thermal, colorrange = νₑlims)
Colorbar(fig[3, 4], hm_νₑ; label = "m s⁻²")

fig[1, 1:4] = Label(fig, title, fontsize=24, tellwidth=false)

fig

And now record a movie.

frames = intro:length(times)

@info "Making a motion picture of ocean wind mixing and convection..."

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

[ Info: Making a motion picture of ocean wind mixing and convection...

Julia version and environment information

This example was executed with the following version of Julia:

using InteractiveUtils: versioninfo
versioninfo()

Julia Version 1.12.2
Commit ca9b6662be4 (2025-11-20 16:25 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 9374F 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver4)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 128 virtual cores)
Environment:
  LD_LIBRARY_PATH = 
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
  JULIA_DEPOT_PATH = /var/lib/buildkite-agent/.julia-oceananigans
  JULIA_PROJECT = /var/lib/buildkite-agent/Oceananigans.jl-27559/docs/
  JULIA_VERSION = 1.12.2
  JULIA_LOAD_PATH = @:@v#.#:@stdlib
  JULIA_VERSION_ENZYME = 1.10.10
  JULIA_PYTHONCALL_EXE = /var/lib/buildkite-agent/Oceananigans.jl-27559/docs/.CondaPkg/.pixi/envs/default/bin/python
  JULIA_DEBUG = Literate

These were the top-level packages installed in the environment:

import Pkg
Pkg.status()

Status `~/Oceananigans.jl-27559/docs/Project.toml`
  [79e6a3ab] Adapt v4.4.0
  [052768ef] CUDA v5.9.5
  [13f3f980] CairoMakie v0.15.8
  [e30172f5] Documenter v1.16.1
  [daee34ce] DocumenterCitations v1.4.1
  [033835bb] JLD2 v0.6.3
  [98b081ad] Literate v2.21.0
  [da04e1cc] MPI v0.20.23
  [85f8d34a] NCDatasets v0.14.10
  [9e8cae18] Oceananigans v0.102.5 `~/Oceananigans.jl-27559`
  [f27b6e38] Polynomials v4.1.0
  [6038ab10] Rotations v1.7.1
  [d496a93d] SeawaterPolynomials v0.3.10
  [09ab397b] StructArrays v0.7.2
  [bdfc003b] TimesDates v0.3.3
  [2e0b0046] XESMF v0.1.6
  [b77e0a4c] InteractiveUtils v1.11.0
  [37e2e46d] LinearAlgebra v1.12.0
  [44cfe95a] Pkg v1.12.0

This page was generated using Literate.jl.