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
SeawaterBuoyancymodel for buoyancy with a linear equation of state. - 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"We start by importing all of the packages and functions that we'll need for this example.
using Random
using Printf
using CairoMakie
using Oceananigans
using Oceananigans.Units: minute, minutes, hourThe grid
We use 32²×24 grid points with 2 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 = 32 # number of points in each of horizontal directions
Nz = 24 # number of points in the vertical direction
Lx = Ly = 64 # (m) domain horizontal extents
Lz = 32 # (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_faces(k) = Lz * (ζ₀(k) * Σ(k) - 1)
grid = RectilinearGrid(size = (Nx, Nx, Nz),
x = (0, Lx),
y = (0, Ly),
z = z_faces)32×32×24 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 64.0) regularly spaced with Δx=2.0
├── Periodic y ∈ [0.0, 64.0) regularly spaced with Δy=2.0
└── Bounded z ∈ [-32.0, 0.0] variably spaced with min(Δz)=1.11123, max(Δz)=2.53571We plot vertical spacing versus depth to inspect the prescribed grid stretching:
fig = Figure(size=(1200, 800))
ax = Axis(fig[1, 1], ylabel = "Depth (m)", xlabel = "Vertical spacing (m)")
lines!(ax, zspacings(grid, Center()), znodes(grid, Center()))
scatter!(ax, zspacings(grid, Center()), znodes(grid, Center()))
fig
Buoyancy that depends on temperature and salinity
We use the SeawaterBuoyancy model with a linear equation of state,
buoyancy = SeawaterBuoyancy(equation_of_state=LinearEquationOfState(thermal_expansion = 2e-4,
haline_contraction = 8e-4))SeawaterBuoyancy{Float64}:
├── gravitational_acceleration: 9.80665
└── equation_of_state: LinearEquationOfState(thermal_expansion=0.0002, haline_contraction=0.0008)Boundary conditions
We calculate the surface temperature flux associated with surface cooling of 200 W m⁻², reference density ρₒ, and heat capacity cᴾ,
Qʰ = 200.0 # W m⁻², surface _heat_ flux
ρₒ = 1026.0 # kg m⁻³, average density at the surface of the world ocean
cᴾ = 3991.0 # J K⁻¹ kg⁻¹, typical heat capacity for seawater
Qᵀ = Qʰ / (ρₒ * cᴾ) # K m s⁻¹, surface _temperature_ flux4.884283985946938e-5Finally, we impose a temperature gradient dTdz both initially 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(Qᵀ),
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 = 2.5e-3 # dimensionless drag coefficient
ρₐ = 1.225 # kg m⁻³, average density of air at sea-level
Qᵘ = - ρₐ / ρₒ * cᴰ * u₁₀ * abs(u₁₀) # m² s⁻²-0.0002984892787524367The boundary conditions on u are thus
u_bcs = FieldBoundaryConditions(top = FluxBoundaryCondition(Qᵘ))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: -0.000298489
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)For salinity, S, we impose an evaporative flux of the form
@inline Qˢ(x, y, t, S, evaporation_rate) = - evaporation_rate * S # [salinity unit] m s⁻¹where S is salinity. We use an evporation rate of 1 millimeter per hour,
evaporation_rate = 1e-3 / hour # m s⁻¹2.7777777777777776e-7We build the Flux evaporation BoundaryCondition with the function Qˢ, indicating that Qˢ depends on salinity S and passing the parameter evaporation_rate,
evaporation_bc = FluxBoundaryCondition(Qˢ, field_dependencies=:S, parameters=evaporation_rate)FluxBoundaryCondition: ContinuousBoundaryFunction Qˢ 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 Qˢ at (Nothing, Nothing, Nothing)
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)Model instantiation
We fill in the final details of the model here: upwind-biased 5th-order advection for momentum and tracers, 3rd-order Runge-Kutta time-stepping, 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 we cannot explicitly resolve.
model = NonhydrostaticModel(; grid, buoyancy,
advection = UpwindBiasedFifthOrder(),
timestepper = :RungeKutta3,
tracers = (:T, :S),
coriolis = FPlane(f=1e-4),
closure = AnisotropicMinimumDissipation(),
boundary_conditions = (u=u_bcs, T=T_bcs, S=S_bcs))NonhydrostaticModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 32×32×24 RectilinearGrid{Float64, Periodic, Periodic, Bounded} on CPU with 3×3×3 halo
├── timestepper: RungeKutta3TimeStepper
├── advection scheme: Upwind Biased reconstruction order 5
├── tracers: (T, S)
├── closure: AnisotropicMinimumDissipation{ExplicitTimeDiscretization, @NamedTuple{T::Float64, S::Float64}, Float64, Nothing}
├── buoyancy: SeawaterBuoyancy with g=9.80665 and LinearEquationOfState(thermal_expansion=0.0002, haline_contraction=0.0008) with ĝ = NegativeZDirection()
└── coriolis: FPlane{Float64}(f=0.0001)Notes:
To use the Smagorinsky-Lilly turbulence closure (with a constant model coefficient) rather than
AnisotropicMinimumDissipation, useclosure = SmagorinskyLilly()in the model constructor.To change the architecture to
GPU, replaceCPU()withGPU()inside thegridconstructor.
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(Qᵘ)) * 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 40 minutes, with adaptive time-stepping and progress printing.
simulation = Simulation(model, Δt=10.0, stop_time=40minutes)Simulation of NonhydrostaticModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── Next time step: 10 seconds
├── Elapsed wall time: 0 seconds
├── Wall time per iteration: NaN days
├── Stop time: 40 minutes
├── Stop iteration : Inf
├── Wall time limit: Inf
├── 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
└── Diagnostics: OrderedDict with no entriesThe TimeStepWizard helps ensure stable time-stepping with a Courant-Freidrichs-Lewy (CFL) number of 1.0.
wizard = TimeStepWizard(cfl=1.0, 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(20))We then set up the simulation:
Output
We use the JLD2OutputWriter to save $x, z$ slices of the velocity fields, tracer fields, and eddy diffusivities. The prefix keyword argument to JLD2OutputWriter indicates that output will be saved in ocean_wind_mixing_and_convection.jld2.
# Create a NamedTuple with eddy viscosity
eddy_viscosity = (; νₑ = model.diffusivity_fields.νₑ)
filename = "ocean_wind_mixing_and_convection"
simulation.output_writers[:slices] =
JLD2OutputWriter(model, merge(model.velocities, model.tracers, eddy_viscosity),
filename = filename * ".jld2",
indices = (:, grid.Ny/2, :),
schedule = TimeInterval(1minute),
overwrite_existing = true)JLD2OutputWriter scheduled on TimeInterval(1 minute):
├── filepath: ./ocean_wind_mixing_and_convection.jld2
├── 6 outputs: (u, v, w, T, S, νₑ)
├── array type: Array{Float64}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 47.8 KiBWe're ready:
run!(simulation)[ Info: Initializing simulation...
Iteration: 0000, time: 0 seconds, Δt: 11 seconds, max(|w|) = 1.1e-05 ms⁻¹, wall time: 0 seconds
[ Info: ... simulation initialization complete (5.768 seconds)
[ Info: Executing initial time step...
[ Info: ... initial time step complete (11.884 seconds).
Iteration: 0020, time: 3.403 minutes, Δt: 13.310 seconds, max(|w|) = 8.2e-06 ms⁻¹, wall time: 19.531 seconds
Iteration: 0040, time: 7.488 minutes, Δt: 16.105 seconds, max(|w|) = 5.9e-06 ms⁻¹, wall time: 21.193 seconds
Iteration: 0060, time: 12 minutes, Δt: 9.990 seconds, max(|w|) = 1.6e-05 ms⁻¹, wall time: 23.857 seconds
Iteration: 0080, time: 14.884 minutes, Δt: 7.991 seconds, max(|w|) = 4.4e-05 ms⁻¹, wall time: 26.779 seconds
Iteration: 0100, time: 17.245 minutes, Δt: 6.850 seconds, max(|w|) = 1.4e-04 ms⁻¹, wall time: 29.147 seconds
Iteration: 0120, time: 19.321 minutes, Δt: 6.068 seconds, max(|w|) = 5.4e-04 ms⁻¹, wall time: 30.888 seconds
Iteration: 0140, time: 21.191 minutes, Δt: 5.431 seconds, max(|w|) = 2.6e-03 ms⁻¹, wall time: 32.701 seconds
Iteration: 0160, time: 22.951 minutes, Δt: 4.936 seconds, max(|w|) = 1.0e-02 ms⁻¹, wall time: 34.191 seconds
Iteration: 0180, time: 24.470 minutes, Δt: 4.736 seconds, max(|w|) = 2.1e-02 ms⁻¹, wall time: 35.693 seconds
Iteration: 0200, time: 25.972 minutes, Δt: 4.295 seconds, max(|w|) = 4.8e-02 ms⁻¹, wall time: 37.298 seconds
Iteration: 0220, time: 27.383 minutes, Δt: 4.655 seconds, max(|w|) = 7.5e-02 ms⁻¹, wall time: 39.006 seconds
Iteration: 0240, time: 29 minutes, Δt: 5.539 seconds, max(|w|) = 7.7e-02 ms⁻¹, wall time: 40.686 seconds
Iteration: 0260, time: 30.876 minutes, Δt: 6.231 seconds, max(|w|) = 7.5e-02 ms⁻¹, wall time: 42.404 seconds
Iteration: 0280, time: 32.830 minutes, Δt: 5.968 seconds, max(|w|) = 6.3e-02 ms⁻¹, wall time: 44.978 seconds
Iteration: 0300, time: 34.701 minutes, Δt: 6.237 seconds, max(|w|) = 7.4e-02 ms⁻¹, wall time: 47.656 seconds
Iteration: 0320, time: 36.731 minutes, Δt: 6.895 seconds, max(|w|) = 7.5e-02 ms⁻¹, wall time: 50.345 seconds
Iteration: 0340, time: 38.989 minutes, Δt: 7.199 seconds, max(|w|) = 5.5e-02 ms⁻¹, wall time: 52.972 seconds
[ Info: Simulation is stopping after running for 54.156 seconds.
[ Info: Simulation time 40 minutes equals or exceeds stop time 40 minutes.
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"))
# Coordinate arrays
xw, yw, zw = nodes(time_series.w)
xT, yT, zT = nodes(time_series.T)([1.0, 3.0, 5.0, 7.0, 9.0, 11.0, 13.0, 15.0, 17.0, 19.0, 21.0, 23.0, 25.0, 27.0, 29.0, 31.0, 33.0, 35.0, 37.0, 39.0, 41.0, 43.0, 45.0, 47.0, 49.0, 51.0, 53.0, 55.0, 57.0, 59.0, 61.0, 63.0], [1.0, 3.0, 5.0, 7.0, 9.0, 11.0, 13.0, 15.0, 17.0, 19.0, 21.0, 23.0, 25.0, 27.0, 29.0, 31.0, 33.0, 35.0, 37.0, 39.0, 41.0, 43.0, 45.0, 47.0, 49.0, 51.0, 53.0, 55.0, 57.0, 59.0, 61.0, 63.0], [-30.73214655800215, -28.344120885446316, -26.245517670529452, -24.406267643406927, -22.77515636448867, -21.301119444298777, -19.9410533865803, -18.66145654635607, -17.43754626408272, -16.251584220508228, -15.09116977761991, -13.947785764082138, -12.815662268579008, -11.690933327108011, -10.57103192002591, -9.45426630377412, -8.339528643431164, -7.226097424980429, -6.1135049573687645, -5.001449332843377, -3.889736372710594, -2.7782415919634484, -1.666885417004453, -0.5556171156552487])We start the animation at $t = 10$ minutes since things are pretty boring till then:
times = time_series.w.times
intro = searchsortedfirst(times, 10minutes)11We 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 interior(time_series.w[$n], :, 1, :)
Tₙ = @lift interior(time_series.T[$n], :, 1, :)
Sₙ = @lift interior(time_series.S[$n], :, 1, :)
νₑₙ = @lift interior(time_series.νₑ[$n], :, 1, :)
fig = Figure(size = (1000, 500))
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, xw, zw, wₙ; colormap = :balance, colorrange = wlims)
Colorbar(fig[2, 2], hm_w; label = "m s⁻¹")
hm_T = heatmap!(ax_T, xT, zT, Tₙ; colormap = :thermal, colorrange = Tlims)
Colorbar(fig[2, 4], hm_T; label = "ᵒC")
hm_S = heatmap!(ax_S, xT, zT, Sₙ; colormap = :haline, colorrange = Slims)
Colorbar(fig[3, 2], hm_S; label = "g / kg")
hm_νₑ = heatmap!(ax_νₑ, xT, zT, 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..."
record(fig, filename * ".mp4", frames, framerate=8) do i
n[] = i
end[ Info: Making a motion picture of ocean wind mixing and convection...
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