# Horizontal convection example

In "horizontal convection", a non-uniform buoyancy is imposed on top of an initially resting fluid.

This example demonstrates:

• How to use computed Fields for output.
• How to post-process saved output using FieldTimeSeries.

## Install dependencies

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

using Pkg
pkg"add Oceananigans, CairoMakie"

## Horizontal convection

We consider two-dimensional horizontal convection of an incompressible flow $\boldsymbol{u} = (u, w)$ on the $(x, z)$-plane ($-L_x/2 \le x \le L_x/2$ and $-H \le z \le 0$). The flow evolves under the effect of gravity. The only forcing on the fluid comes from a prescribed, non-uniform buoyancy at the top-surface of the domain.

We start by importing Oceananigans and Printf.

using Oceananigans
using Printf

### The grid

H = 1.0          # vertical domain extent
Lx = 2H          # horizontal domain extent
Nx, Nz = 128, 64 # horizontal, vertical resolution

grid = RectilinearGrid(size = (Nx, Nz),
x = (-Lx/2, Lx/2),
z = (-H, 0),
topology = (Bounded, Flat, Bounded))
128×1×64 RectilinearGrid{Float64, Bounded, Flat, Bounded} on CPU with 3×0×3 halo
├── Bounded  x ∈ [-1.0, 1.0]      regularly spaced with Δx=0.015625
├── Flat y
└── Bounded  z ∈ [-1.0, 0.0]      regularly spaced with Δz=0.015625

### Boundary conditions

At the surface, the imposed buoyancy is

$$$b(x, z = 0, t) = - b_* \cos (2 \pi x / L_x) \, ,$$$

while zero-flux boundary conditions are imposed on all other boundaries. We use free-slip boundary conditions on $u$ and $w$ everywhere.

b★ = 1.0

@inline bˢ(x, y, t, p) = - p.b★ * cos(2π * x / p.Lx)

b_bcs = FieldBoundaryConditions(top = ValueBoundaryCondition(bˢ, parameters=(; b★, Lx)))
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: ValueBoundaryCondition: ContinuousBoundaryFunction bˢ at (Nothing, Nothing, Nothing)
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)

### Non-dimensional control parameters and Turbulence closure

The problem is characterized by three non-dimensional parameters. The first is the domain's aspect ratio, $L_x / H$ and the other two are the Rayleigh ($Ra$) and Prandtl ($Pr$) numbers:

$$$Ra = \frac{b_* L_x^3}{\nu \kappa} \, , \quad \text{and}\, \quad Pr = \frac{\nu}{\kappa} \, .$$$

The Prandtl number expresses the ratio of momentum over heat diffusion; the Rayleigh number is a measure of the relative importance of gravity over viscosity in the momentum equation.

For a domain with a given extent, the nondimensional values of $Ra$ and $Pr$ uniquely determine the viscosity and diffusivity, i.e.,

$$$\nu = \sqrt{\frac{Pr b_* L_x^3}{Ra}} \quad \text{and} \quad \kappa = \sqrt{\frac{b_* L_x^3}{Pr Ra}} \, .$$$

We use isotropic viscosity and diffusivities, ν and κ whose values are obtain from the prescribed $Ra$ and $Pr$ numbers. Here, we use $Pr = 1$ and $Ra = 10^8$:

Pr = 1.0    # Prandtl number
Ra = 1e8    # Rayleigh number

ν = sqrt(Pr * b★ * Lx^3 / Ra)  # Laplacian viscosity
κ = ν * Pr                     # Laplacian diffusivity
nothing # hide

## Model instantiation

We instantiate the model with the fifth-order WENO advection scheme, a 3rd order Runge-Kutta time-stepping scheme, and a BuoyancyTracer.

model = NonhydrostaticModel(; grid,
timestepper = :RungeKutta3,
tracers = :b,
buoyancy = BuoyancyTracer(),
closure = ScalarDiffusivity(; ν, κ),
boundary_conditions = (; b=b_bcs))
NonhydrostaticModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 128×1×64 RectilinearGrid{Float64, Bounded, Flat, Bounded} on CPU with 3×0×3 halo
├── timestepper: RungeKutta3TimeStepper
├── tracers: b
├── closure: ScalarDiffusivity{ExplicitTimeDiscretization}(ν=0.000282843, κ=(b=0.000282843,))
├── buoyancy: BuoyancyTracer with ĝ = NegativeZDirection()
└── coriolis: Nothing

## Simulation set-up

We set up a simulation that runs up to $t = 40$ with a JLD2OutputWriter that saves the flow speed, $\sqrt{u^2 + w^2}$, the buoyancy, $b$, and the vorticity, $\partial_z u - \partial_x w$.

simulation = Simulation(model, Δt=1e-2, stop_time=40.0)
Simulation of NonhydrostaticModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── Next time step: 10 ms
├── Elapsed wall time: 0 seconds
├── Wall time per iteration: NaN days
├── Stop time: 40 seconds
├── 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 entries

### The TimeStepWizard

The TimeStepWizard manages the time-step adaptively, keeping the Courant-Freidrichs-Lewy (CFL) number close to 0.75 while ensuring the time-step does not increase beyond the maximum allowable value for numerical stability.

wizard = TimeStepWizard(cfl=0.75, max_change=1.2, max_Δt=1e-1)

simulation.callbacks[:wizard] = Callback(wizard, IterationInterval(50))
Callback of TimeStepWizard(cfl=0.75, max_Δt=0.1, min_Δt=0.0) on IterationInterval(50)

### A progress messenger

We write a function that prints out a helpful progress message while the simulation runs.

progress(sim) = @printf("i: % 6d, sim time: % 1.3f, wall time: % 10s, Δt: % 1.4f, advective CFL: %.2e, diffusive CFL: %.2e\n",
iteration(sim), time(sim), prettytime(sim.run_wall_time),

simulation.callbacks[:progress] = Callback(progress, IterationInterval(50))
Callback of progress on IterationInterval(50)

### Output

We use computed Fields to diagnose and output the total flow speed, the vorticity, $\zeta$, and the buoyancy, $b$. Note that computed Fields take "AbstractOperations" on Fields as input:

u, v, w = model.velocities # unpack velocity Fields
b = model.tracers.b        # unpack buoyancy Field

# total flow speed
s = @at (Center, Center, Center) sqrt(u^2 + w^2)

# y-component of vorticity
ζ = ∂z(u) - ∂x(w)
nothing # hide

We create a JLD2OutputWriter that saves the speed, and the vorticity. Because we want to post-process buoyancy and compute the buoyancy variance dissipation (which is proportional to $|\boldsymbol{\nabla} b|^2$) we use the with_halos = true. This way, the halos for the fields are saved and thus when we load them as fields they will come with the proper boundary conditions.

We then add the JLD2OutputWriter to the simulation.

saved_output_filename = "horizontal_convection.jld2"

simulation.output_writers[:fields] = JLD2OutputWriter(model, (; s, b, ζ),
schedule = TimeInterval(0.5),
filename = saved_output_filename,
with_halos = true,
overwrite_existing = true)
nothing # hide

Ready to press the big red button:

run!(simulation)
[ Info: Initializing simulation...
i:      0, sim time:  0.000, wall time:  0 seconds, Δt:  0.0120, advective CFL: 0.00e+00, diffusive CFL: 1.39e-02
[ Info:     ... simulation initialization complete (10.456 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (22.500 seconds).
i:     50, sim time:  0.596, wall time: 36.109 seconds, Δt:  0.0144, advective CFL: 6.09e-03, diffusive CFL: 1.67e-02
i:    100, sim time:  1.302, wall time: 39.464 seconds, Δt:  0.0173, advective CFL: 2.64e-02, diffusive CFL: 2.00e-02
i:    150, sim time:  2.156, wall time: 42.446 seconds, Δt:  0.0207, advective CFL: 6.96e-02, diffusive CFL: 2.40e-02
i:    200, sim time:  3.166, wall time: 45.382 seconds, Δt:  0.0249, advective CFL: 1.53e-01, diffusive CFL: 2.88e-02
i:    250, sim time:  4.373, wall time: 48.334 seconds, Δt:  0.0299, advective CFL: 3.69e-01, diffusive CFL: 3.46e-02
i:    300, sim time:  5.828, wall time: 51.282 seconds, Δt:  0.0188, advective CFL: 7.50e-01, diffusive CFL: 2.17e-02
i:    350, sim time:  6.744, wall time: 54.243 seconds, Δt:  0.0170, advective CFL: 7.50e-01, diffusive CFL: 1.97e-02
i:    400, sim time:  7.568, wall time: 57.161 seconds, Δt:  0.0168, advective CFL: 7.50e-01, diffusive CFL: 1.95e-02
i:    450, sim time:  8.403, wall time: 1.001 minutes, Δt:  0.0164, advective CFL: 7.50e-01, diffusive CFL: 1.90e-02
i:    500, sim time:  9.213, wall time: 1.050 minutes, Δt:  0.0178, advective CFL: 7.50e-01, diffusive CFL: 2.06e-02
i:    550, sim time:  10.071, wall time: 1.100 minutes, Δt:  0.0165, advective CFL: 7.50e-01, diffusive CFL: 1.91e-02
i:    600, sim time:  10.878, wall time: 1.151 minutes, Δt:  0.0169, advective CFL: 7.50e-01, diffusive CFL: 1.96e-02
i:    650, sim time:  11.703, wall time: 1.202 minutes, Δt:  0.0188, advective CFL: 7.50e-01, diffusive CFL: 2.17e-02
i:    700, sim time:  12.631, wall time: 1.254 minutes, Δt:  0.0189, advective CFL: 7.50e-01, diffusive CFL: 2.19e-02
i:    750, sim time:  13.557, wall time: 1.305 minutes, Δt:  0.0193, advective CFL: 7.50e-01, diffusive CFL: 2.24e-02
i:    800, sim time:  14.519, wall time: 1.353 minutes, Δt:  0.0199, advective CFL: 7.50e-01, diffusive CFL: 2.31e-02
i:    850, sim time:  15.498, wall time: 1.403 minutes, Δt:  0.0211, advective CFL: 7.50e-01, diffusive CFL: 2.45e-02
i:    900, sim time:  16.521, wall time: 1.452 minutes, Δt:  0.0210, advective CFL: 7.50e-01, diffusive CFL: 2.44e-02
i:    950, sim time:  17.563, wall time: 1.502 minutes, Δt:  0.0213, advective CFL: 7.50e-01, diffusive CFL: 2.46e-02
i:   1000, sim time:  18.606, wall time: 1.551 minutes, Δt:  0.0220, advective CFL: 7.50e-01, diffusive CFL: 2.54e-02
i:   1050, sim time:  19.698, wall time: 1.600 minutes, Δt:  0.0231, advective CFL: 7.50e-01, diffusive CFL: 2.68e-02
i:   1100, sim time:  20.824, wall time: 1.649 minutes, Δt:  0.0237, advective CFL: 7.50e-01, diffusive CFL: 2.74e-02
i:   1150, sim time:  21.950, wall time: 1.698 minutes, Δt:  0.0208, advective CFL: 7.50e-01, diffusive CFL: 2.41e-02
i:   1200, sim time:  22.957, wall time: 1.747 minutes, Δt:  0.0211, advective CFL: 7.50e-01, diffusive CFL: 2.44e-02
i:   1250, sim time:  23.985, wall time: 1.796 minutes, Δt:  0.0232, advective CFL: 7.50e-01, diffusive CFL: 2.69e-02
i:   1300, sim time:  25.116, wall time: 1.845 minutes, Δt:  0.0251, advective CFL: 7.50e-01, diffusive CFL: 2.91e-02
i:   1350, sim time:  26.352, wall time: 1.894 minutes, Δt:  0.0278, advective CFL: 7.50e-01, diffusive CFL: 3.22e-02
i:   1400, sim time:  27.695, wall time: 1.946 minutes, Δt:  0.0220, advective CFL: 7.50e-01, diffusive CFL: 2.55e-02
i:   1450, sim time:  28.786, wall time: 1.996 minutes, Δt:  0.0171, advective CFL: 7.50e-01, diffusive CFL: 1.99e-02
i:   1500, sim time:  29.620, wall time: 2.048 minutes, Δt:  0.0159, advective CFL: 7.50e-01, diffusive CFL: 1.84e-02
i:   1550, sim time:  30.413, wall time: 2.099 minutes, Δt:  0.0155, advective CFL: 7.50e-01, diffusive CFL: 1.80e-02
i:   1600, sim time:  31.171, wall time: 2.151 minutes, Δt:  0.0187, advective CFL: 7.15e-01, diffusive CFL: 2.16e-02
i:   1650, sim time:  32.093, wall time: 2.202 minutes, Δt:  0.0207, advective CFL: 7.50e-01, diffusive CFL: 2.39e-02
i:   1700, sim time:  33.103, wall time: 2.253 minutes, Δt:  0.0221, advective CFL: 7.50e-01, diffusive CFL: 2.57e-02
i:   1750, sim time:  34.199, wall time: 2.303 minutes, Δt:  0.0230, advective CFL: 7.50e-01, diffusive CFL: 2.67e-02
i:   1800, sim time:  35.323, wall time: 2.353 minutes, Δt:  0.0236, advective CFL: 7.50e-01, diffusive CFL: 2.74e-02
i:   1850, sim time:  36.473, wall time: 2.404 minutes, Δt:  0.0240, advective CFL: 7.50e-01, diffusive CFL: 2.78e-02
i:   1900, sim time:  37.644, wall time: 2.456 minutes, Δt:  0.0246, advective CFL: 7.50e-01, diffusive CFL: 2.85e-02
i:   1950, sim time:  38.845, wall time: 2.507 minutes, Δt:  0.0255, advective CFL: 7.50e-01, diffusive CFL: 2.96e-02
[ Info: Simulation is stopping after running for 2.553 minutes.
[ Info: Simulation time 40 seconds equals or exceeds stop time 40 seconds.


## Load saved output, process, visualize

We animate the results by loading the saved output, extracting data for the iterations we ended up saving at, and ploting the saved fields. From the saved buoyancy field we compute the buoyancy dissipation, $\chi = \kappa |\boldsymbol{\nabla} b|^2$, and plot that also.

To start we load the saved fields are FieldTimeSeries and prepare for animating the flow by creating coordinate arrays that each field lives on.

using CairoMakie
using Oceananigans
using Oceananigans.Fields
using Oceananigans.AbstractOperations: volume

saved_output_filename = "horizontal_convection.jld2"

# Open the file with our data
s_timeseries = FieldTimeSeries(saved_output_filename, "s")
b_timeseries = FieldTimeSeries(saved_output_filename, "b")
ζ_timeseries = FieldTimeSeries(saved_output_filename, "ζ")

times = b_timeseries.times

# Coordinate arrays
xc, yc, zc = nodes(b_timeseries)
xζ, yζ, zζ = nodes(ζ_timeseries)
nothing # hide

χ_timeseries = deepcopy(b_timeseries)

for i in 1:length(times)
bᵢ = b_timeseries[i]
χ_timeseries[i] .= @at (Center, Center, Center) κ * (∂x(bᵢ)^2 + ∂z(bᵢ)^2)
end

Now we're ready to animate using Makie.

@info "Making an animation from saved data..."

n = Observable(1)

title = @lift @sprintf("t=%1.2f", times[$n]) sₙ = @lift interior(s_timeseries[$n], :, 1, :)
ζₙ = @lift interior(ζ_timeseries[$n], :, 1, :) bₙ = @lift interior(b_timeseries[$n], :, 1, :)
χₙ = @lift interior(χ_timeseries[$n], :, 1, :) slim = 0.6 blim = 0.6 ζlim = 9 χlim = 0.025 axis_kwargs = (xlabel = L"x / H", ylabel = L"z / H", limits = ((-Lx/2, Lx/2), (-H, 0)), aspect = Lx / H, titlesize = 20) fig = Figure(resolution = (600, 1100)) ax_s = Axis(fig[2, 1]; title = L"speed,$(u^2+w^2)^{1/2} / (L_x b_*) ^{1/2}", axis_kwargs...)

ax_b = Axis(fig[3, 1];
title = L"buoyancy, $b / b_*$", axis_kwargs...)

ax_ζ = Axis(fig[4, 1];
title = L"vorticity, $(∂u/∂z - ∂w/∂x) \, (L_x / b_*)^{1/2}$", axis_kwargs...)

ax_χ = Axis(fig[5, 1];
title = L"buoyancy dissipation, $κ |\mathbf{\nabla}b|^2 \, (L_x / {b_*}^5)^{1/2}$", axis_kwargs...)

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

hm_s = heatmap!(ax_s, xc, zc, sₙ;
colorrange = (0, slim),
colormap = :speed)
Colorbar(fig[2, 2], hm_s)

hm_b = heatmap!(ax_b, xc, zc, bₙ;
colorrange = (-blim, blim),
colormap = :thermal)
Colorbar(fig[3, 2], hm_b)

hm_ζ = heatmap!(ax_ζ, xζ, zζ, ζₙ;
colorrange = (-ζlim, ζlim),
colormap = :balance)
Colorbar(fig[4, 2], hm_ζ)

hm_χ = heatmap!(ax_χ, xc, zc, χₙ;
colorrange = (0, χlim),
colormap = :dense)
Colorbar(fig[5, 2], hm_χ)
Colorbar()

And, finally, we record a movie.

frames = 1:length(times)

record(fig, "horizontal_convection.mp4", frames, framerate=8) do i
msg = string("Plotting frame ", i, " of ", frames[end])
print(msg * " \r")
n[] = i
end
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At higher Rayleigh numbers the flow becomes much more vigorous. See, for example, an animation of the voricity of the fluid at $Ra = 10^{12}$ on vimeo.

### The Nusselt number

Often we are interested on how much the flow enhances mixing. This is quantified by the Nusselt number, which measures how much the flow enhances mixing compared if only diffusion was in operation. The Nusselt number is given by

$$$Nu = \frac{\langle \chi \rangle}{\langle \chi_{\rm diff} \rangle} \, ,$$$

where angle brackets above denote both a volume and time average and $\chi_{\rm diff}$ is the buoyancy dissipation that we get without any flow, i.e., the buoyancy dissipation associated with the buoyancy distribution that satisfies

$$$\kappa \nabla^2 b_{\rm diff} = 0 \, ,$$$

with the same boundary conditions same as our setup. In this case, we can readily find that

$$$b_{\rm diff}(x, z) = b_s(x) \frac{\cosh \left [2 \pi (H + z) / L_x \right ]}{\cosh(2 \pi H / L_x)} \, ,$$$

where $b_s(x)$ is the surface boundary condition. The diffusive solution implies $\langle \chi_{\rm diff} \rangle = \kappa b_*^2 \pi \tanh(2 \pi Η /Lx) / (L_x H)$.

We use the loaded FieldTimeSeries to compute the Nusselt number from buoyancy and the volume average kinetic energy of the fluid.

First we compute the diffusive buoyancy dissipation, $\chi_{\rm diff}$ (which is just a scalar):

χ_diff = κ * b★^2 * π * tanh(2π * H / Lx) / (Lx * H)
nothing # hide

We recover the time from the saved FieldTimeSeries and construct two empty arrays to store the volume-averaged kinetic energy and the instantaneous Nusselt number,

t = b_timeseries.times

kinetic_energy, Nu = zeros(length(t)), zeros(length(t))
nothing # hide

Now we can loop over the fields in the FieldTimeSeries, compute kinetic energy and $Nu$, and plot. We make use of Integral to compute the volume integral of fields over our domain.

for i = 1:length(t)
ke = Field(Integral(1/2 * s_timeseries[i]^2 / (Lx * H)))
compute!(ke)
kinetic_energy[i] = ke[1, 1, 1]

χ = Field(Integral(χ_timeseries[i] / (Lx * H)))
compute!(χ)

Nu[i] = χ[1, 1, 1] / χ_diff
end

fig = Figure(resolution = (850, 450))

ax_KE = Axis(fig[1, 1], xlabel = L"t \, (b_* / L_x)^{1/2}", ylabel = L"KE $/ (L_x b_*)$")
lines!(ax_KE, t, kinetic_energy; linewidth = 3)

ax_Nu = Axis(fig[2, 1], xlabel = L"t \, (b_* / L_x)^{1/2}", ylabel = L"Nu")
lines!(ax_Nu, t, Nu; linewidth = 3)

current_figure() # hide
fig 