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{CUDAGPU, 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{CUDAGPU, 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: 0 bytes (file not yet created)

We're ready:

run!(simulation)

[ Info: Initializing simulation...
Iteration: 0000, time: 0 seconds, Δt: 11 seconds, max(|w|) = 1.4e-05 ms⁻¹, wall time: 0 seconds
[ Info:     ... simulation initialization complete (10.174 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (2.788 seconds).
Iteration: 0040, time: 7.344 minutes, Δt: 7.910 seconds, max(|w|) = 4.6e-05 ms⁻¹, wall time: 13.734 seconds
Iteration: 0080, time: 11.440 minutes, Δt: 4.650 seconds, max(|w|) = 9.0e-03 ms⁻¹, wall time: 14.100 seconds
Iteration: 0120, time: 14.203 minutes, Δt: 4.136 seconds, max(|w|) = 2.4e-02 ms⁻¹, wall time: 14.539 seconds
Iteration: 0160, time: 16.902 minutes, Δt: 4.197 seconds, max(|w|) = 2.5e-02 ms⁻¹, wall time: 14.907 seconds
Iteration: 0200, time: 19.556 minutes, Δt: 4.142 seconds, max(|w|) = 2.5e-02 ms⁻¹, wall time: 15.350 seconds
Iteration: 0240, time: 22.132 minutes, Δt: 3.868 seconds, max(|w|) = 3.2e-02 ms⁻¹, wall time: 15.719 seconds
Iteration: 0280, time: 24.531 minutes, Δt: 3.390 seconds, max(|w|) = 3.2e-02 ms⁻¹, wall time: 16.114 seconds
Iteration: 0320, time: 26.732 minutes, Δt: 3.431 seconds, max(|w|) = 2.8e-02 ms⁻¹, wall time: 16.518 seconds
Iteration: 0360, time: 28.960 minutes, Δt: 3.440 seconds, max(|w|) = 3.0e-02 ms⁻¹, wall time: 16.933 seconds
Iteration: 0400, time: 31.174 minutes, Δt: 3.507 seconds, max(|w|) = 3.8e-02 ms⁻¹, wall time: 17.391 seconds
Iteration: 0440, time: 33.391 minutes, Δt: 3.362 seconds, max(|w|) = 3.5e-02 ms⁻¹, wall time: 17.779 seconds
Iteration: 0480, time: 35.498 minutes, Δt: 3.210 seconds, max(|w|) = 3.1e-02 ms⁻¹, wall time: 18.148 seconds
Iteration: 0520, time: 37.515 minutes, Δt: 3.121 seconds, max(|w|) = 3.4e-02 ms⁻¹, wall time: 18.529 seconds
Iteration: 0560, time: 39.569 minutes, Δt: 3.034 seconds, max(|w|) = 3.3e-02 ms⁻¹, wall time: 18.924 seconds
Iteration: 0600, time: 41.576 minutes, Δt: 3.069 seconds, max(|w|) = 3.1e-02 ms⁻¹, wall time: 19.286 seconds
Iteration: 0640, time: 43.537 minutes, Δt: 2.863 seconds, max(|w|) = 3.5e-02 ms⁻¹, wall time: 19.673 seconds
Iteration: 0680, time: 45.461 minutes, Δt: 2.993 seconds, max(|w|) = 3.7e-02 ms⁻¹, wall time: 20.059 seconds
Iteration: 0720, time: 47.394 minutes, Δt: 2.981 seconds, max(|w|) = 4.1e-02 ms⁻¹, wall time: 20.422 seconds
Iteration: 0760, time: 49.291 minutes, Δt: 2.900 seconds, max(|w|) = 3.8e-02 ms⁻¹, wall time: 20.802 seconds
Iteration: 0800, time: 51.089 minutes, Δt: 2.745 seconds, max(|w|) = 4.0e-02 ms⁻¹, wall time: 21.183 seconds
Iteration: 0840, time: 52.884 minutes, Δt: 2.829 seconds, max(|w|) = 3.6e-02 ms⁻¹, wall time: 21.545 seconds
Iteration: 0880, time: 54.705 minutes, Δt: 2.794 seconds, max(|w|) = 4.2e-02 ms⁻¹, wall time: 21.932 seconds
Iteration: 0920, time: 56.528 minutes, Δt: 2.848 seconds, max(|w|) = 4.8e-02 ms⁻¹, wall time: 22.380 seconds
Iteration: 0960, time: 58.370 minutes, Δt: 2.745 seconds, max(|w|) = 5.4e-02 ms⁻¹, wall time: 22.747 seconds
Iteration: 1000, time: 1.003 hours, Δt: 2.729 seconds, max(|w|) = 5.5e-02 ms⁻¹, wall time: 23.139 seconds
Iteration: 1040, time: 1.033 hours, Δt: 2.834 seconds, max(|w|) = 4.7e-02 ms⁻¹, wall time: 23.558 seconds
Iteration: 1080, time: 1.064 hours, Δt: 2.740 seconds, max(|w|) = 4.6e-02 ms⁻¹, wall time: 23.928 seconds
Iteration: 1120, time: 1.094 hours, Δt: 2.688 seconds, max(|w|) = 5.9e-02 ms⁻¹, wall time: 24.326 seconds
Iteration: 1160, time: 1.123 hours, Δt: 2.512 seconds, max(|w|) = 5.4e-02 ms⁻¹, wall time: 24.719 seconds
Iteration: 1200, time: 1.151 hours, Δt: 2.598 seconds, max(|w|) = 6.2e-02 ms⁻¹, wall time: 25.126 seconds
Iteration: 1240, time: 1.180 hours, Δt: 2.701 seconds, max(|w|) = 5.1e-02 ms⁻¹, wall time: 25.527 seconds
Iteration: 1280, time: 1.209 hours, Δt: 2.491 seconds, max(|w|) = 4.9e-02 ms⁻¹, wall time: 25.898 seconds
Iteration: 1320, time: 1.236 hours, Δt: 2.481 seconds, max(|w|) = 4.7e-02 ms⁻¹, wall time: 26.315 seconds
Iteration: 1360, time: 1.265 hours, Δt: 2.615 seconds, max(|w|) = 4.4e-02 ms⁻¹, wall time: 26.714 seconds
Iteration: 1400, time: 1.293 hours, Δt: 2.546 seconds, max(|w|) = 5.7e-02 ms⁻¹, wall time: 27.144 seconds
Iteration: 1440, time: 1.319 hours, Δt: 2.474 seconds, max(|w|) = 6.1e-02 ms⁻¹, wall time: 27.595 seconds
Iteration: 1480, time: 1.346 hours, Δt: 2.489 seconds, max(|w|) = 6.0e-02 ms⁻¹, wall time: 27.984 seconds
Iteration: 1520, time: 1.374 hours, Δt: 2.537 seconds, max(|w|) = 6.0e-02 ms⁻¹, wall time: 28.374 seconds
Iteration: 1560, time: 1.401 hours, Δt: 2.511 seconds, max(|w|) = 5.1e-02 ms⁻¹, wall time: 28.767 seconds
Iteration: 1600, time: 1.429 hours, Δt: 2.548 seconds, max(|w|) = 5.1e-02 ms⁻¹, wall time: 29.157 seconds
Iteration: 1640, time: 1.457 hours, Δt: 2.493 seconds, max(|w|) = 5.6e-02 ms⁻¹, wall time: 29.582 seconds
Iteration: 1680, time: 1.483 hours, Δt: 2.434 seconds, max(|w|) = 5.9e-02 ms⁻¹, wall time: 30.029 seconds
Iteration: 1720, time: 1.510 hours, Δt: 2.446 seconds, max(|w|) = 5.7e-02 ms⁻¹, wall time: 30.457 seconds
Iteration: 1760, time: 1.537 hours, Δt: 2.507 seconds, max(|w|) = 6.3e-02 ms⁻¹, wall time: 30.892 seconds
Iteration: 1800, time: 1.564 hours, Δt: 2.437 seconds, max(|w|) = 5.7e-02 ms⁻¹, wall time: 31.360 seconds
Iteration: 1840, time: 1.590 hours, Δt: 2.514 seconds, max(|w|) = 6.4e-02 ms⁻¹, wall time: 31.835 seconds
Iteration: 1880, time: 1.617 hours, Δt: 2.493 seconds, max(|w|) = 6.4e-02 ms⁻¹, wall time: 32.402 seconds
Iteration: 1920, time: 1.644 hours, Δt: 2.525 seconds, max(|w|) = 5.8e-02 ms⁻¹, wall time: 33.206 seconds
Iteration: 1960, time: 1.670 hours, Δt: 2.474 seconds, max(|w|) = 5.8e-02 ms⁻¹, wall time: 34.235 seconds
Iteration: 2000, time: 1.697 hours, Δt: 2.332 seconds, max(|w|) = 5.5e-02 ms⁻¹, wall time: 35.168 seconds
Iteration: 2040, time: 1.724 hours, Δt: 2.447 seconds, max(|w|) = 5.3e-02 ms⁻¹, wall time: 36.191 seconds
Iteration: 2080, time: 1.750 hours, Δt: 2.394 seconds, max(|w|) = 5.9e-02 ms⁻¹, wall time: 37.146 seconds
Iteration: 2120, time: 1.776 hours, Δt: 2.413 seconds, max(|w|) = 5.3e-02 ms⁻¹, wall time: 38.099 seconds
Iteration: 2160, time: 1.803 hours, Δt: 2.459 seconds, max(|w|) = 5.4e-02 ms⁻¹, wall time: 39.063 seconds
Iteration: 2200, time: 1.829 hours, Δt: 2.289 seconds, max(|w|) = 5.5e-02 ms⁻¹, wall time: 40.022 seconds
Iteration: 2240, time: 1.854 hours, Δt: 2.350 seconds, max(|w|) = 5.9e-02 ms⁻¹, wall time: 40.965 seconds
Iteration: 2280, time: 1.880 hours, Δt: 2.459 seconds, max(|w|) = 6.3e-02 ms⁻¹, wall time: 41.953 seconds
Iteration: 2320, time: 1.906 hours, Δt: 2.515 seconds, max(|w|) = 6.5e-02 ms⁻¹, wall time: 42.885 seconds
Iteration: 2360, time: 1.934 hours, Δt: 2.501 seconds, max(|w|) = 6.7e-02 ms⁻¹, wall time: 43.835 seconds
Iteration: 2400, time: 1.962 hours, Δt: 2.516 seconds, max(|w|) = 6.5e-02 ms⁻¹, wall time: 44.818 seconds
Iteration: 2440, time: 1.989 hours, Δt: 2.493 seconds, max(|w|) = 6.3e-02 ms⁻¹, wall time: 45.785 seconds
[ Info: Simulation is stopping after running for 46.198 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=0.0417438, min=-0.0580294, mean=-4.44501e-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=35.0134, 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=0.018593, min=0.0, mean=0.000427895)

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-28050/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-28050/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-28050/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 v0.104.0 `~/Oceananigans.jl-28050`
  [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.