In this example, we initialize a random velocity field and observe its viscous, turbulent decay in a two-dimensional domain. This example demonstrates:

• How to use a tuple of turbulence closures
• How to use hyperdiffusivity
• How to implement background velocity and tracer distributions
• How to use ComputedFields for output

## Install dependencies

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

using Pkg

The "Eady problem" simulates the baroclinic instability problem proposed by Eric Eady in the classic paper "Long waves and cyclone waves," Tellus (1949). The Eady problem is a simple, canonical model for the generation of mid-latitude atmospheric storms and the ocean eddies that permeate the world sea.

In the Eady problem, baroclinic motion and turublence is generated by the interaction between turbulent motions and a stationary, geostrophically-balanced basic state that is unstable to baroclinic instability. In this example, the baroclinic generation of turbulence due to extraction of energy from the geostrophic basic state is balanced by a bottom boundary condition that extracts momentum from turbulent motions and serves as a crude model for the drag associated with an unresolved and small-scale turbulent bottom boundary layer.

### The geostrophic basic state

The geostrophic basic state in the Eady problem is represented by the streamfunction,

$$$ψ(y, z) = - α y (z + L_z) \, ,$$$

where $α$ is the geostrophic shear and $L_z$ is the depth of the domain. The background buoyancy includes both the geostrophic flow component, $f ∂_z ψ$, where $f$ is the Coriolis parameter, and a background stable stratification component, $N^2 z$, where $N$ is the buoyancy frequency:

$$$B(y, z) = f ∂_z ψ + N^2 z = - α f y + N^2 z \, .$$$

The background velocity field is related to the geostrophic streamfunction via $U = - ∂_y ψ$ such that

$$$U(z) = α (z + L_z) \, .$$$

### Boundary conditions

All fields are periodic in the horizontal directions. We use "insulating", or zero-flux boundary conditions on the buoyancy perturbation at the top and bottom. We thus implicitly assume that the background vertical density gradient, $N^2 z$, is maintained by a process external to our simulation. We use free-slip, or zero-flux boundary conditions on $u$ and $v$ at the surface where $z=0$. At the bottom, we impose a momentum flux that extracts momentum and energy from the flow.

#### Bottom boundary condition: quadratic bottom drag

We model the effects of a turbulent bottom boundary layer on the eddy momentum budget with quadratic bottom drag. A quadratic cottom drag is introduced by imposing a vertical flux of horizontal momentum that removes momentum from the layer immediately above: in other words, the flux is negative (downwards) when the velocity at the bottom boundary is positive, and positive (upwards) with the velocity at the bottom boundary is negative. This drag term is "quadratic" because the rate at which momentum is removed is proportional to $\boldsymbol{u} |\boldsymbol{u}|$, where $\boldsymbol{u} = u \boldsymbol{\hat{x}} + v \boldsymbol{\hat{y}}$ is the horizontal velocity.

The $x$-component of the quadratic bottom drag is thus

$$$\tau_{xz}(z=L_z) = - c^D u \sqrt{u^2 + v^2} \, ,$$$

while the $y$-component is

$$$\tau_{yz}(z=L_z) = - c^D v \sqrt{u^2 + v^2} \, ,$$$

where $c^D$ is a dimensionless drag coefficient and $\tau_{xz}(z=L_z)$ and $\tau_{yz}(z=L_z)$ denote the flux of $u$ and $v$ momentum at $z = L_z$, the bottom of the domain.

### Vertical and horizontal viscosity and diffusivity

Vertical and horizontal viscosities and diffusivities are required to stabilize the Eady problem and can be idealized as modeling the effect of turbulent mixing below the grid scale. For both tracers and velocities we use a Laplacian vertical diffusivity $κ_z ∂_z^2 c$ and a horizontal hyperdiffusivity $ϰ_h (∂_x^4 + ∂_y^4) c$.

### Eady problem summary and parameters

To summarize, the Eady problem parameters along with the values we use in this example are

Parameter nameDescriptionValueUnits
$f$Coriolis parameter$10^{-4}$$\mathrm{s^{-1}} NBuoyancy frequency (square root of \partial_z B)10^{-3}$$\mathrm{s^{-1}}$
$\alpha$Background vertical shear $\partial_z U$$10^{-3}$$\mathrm{s^{-1}}$
$c^D$Bottom quadratic drag coefficient$10^{-4}$none
$κ_z$Laplacian vertical diffusivity$10^{-2}$$\mathrm{m^2 s^{-1}} ϰ_hBiharmonic horizontal diffusivity10^{-2} \times \Delta x^4 / \mathrm{day}$$\mathrm{m^4 s^{-1}}$

We start off by importing Oceananigans, Printf, and some convenient constants for specifying dimensional units:

using Printf
using Oceananigans
using Oceananigans.Units: hours, day, days

## The grid

We use a three-dimensional grid with a depth of 4000 m and a horizontal extent of 1000 km, appropriate for mesoscale ocean dynamics with characteristic scales of 50-200 km.

grid = RegularRectilinearGrid(size=(48, 48, 16), x=(0, 1e6), y=(0, 1e6), z=(-4e3, 0))
RegularRectilinearGrid{Float64, Periodic, Periodic, Bounded}
domain: x ∈ [0.0, 1.0e6], y ∈ [0.0, 1.0e6], z ∈ [-4000.0, 0.0]
topology: (Periodic, Periodic, Bounded)
resolution (Nx, Ny, Nz): (48, 48, 16)
halo size (Hx, Hy, Hz): (1, 1, 1)
grid spacing (Δx, Δy, Δz): (20833.333333333332, 20833.333333333332, 250.0)

## Rotation

The classical Eady problem is posed on an $f$-plane. We use a Coriolis parameter typical to mid-latitudes on Earth,

coriolis = FPlane(f=1e-4) # [s⁻¹]
FPlane{Float64}: f = 1.00e-04

## The background flow

We build a NamedTuple of parameters that describe the background flow,

basic_state_parameters = ( α = 10 * coriolis.f, # s⁻¹, geostrophic shear
f = coriolis.f,      # s⁻¹, Coriolis parameter
N = 1e-3,            # s⁻¹, buoyancy frequency
Lz = grid.Lz)         # m, ocean depth
(α = 0.001, f = 0.0001, N = 0.001, Lz = 4000.0)

and then construct the background fields $U$ and $B$

# Background fields are defined via functions of x, y, z, t, and optional parameters
U(x, y, z, t, p) = + p.α * (z + p.Lz)
B(x, y, z, t, p) = - p.α * p.f * y + p.N^2 * z

U_field = BackgroundField(U, parameters=basic_state_parameters)
B_field = BackgroundField(B, parameters=basic_state_parameters)
BackgroundField{typeof(Main.B), NamedTuple{(:α, :f, :N, :Lz), NTuple{4, Float64}}}
├── func: B
└── parameters: (α = 0.001, f = 0.0001, N = 0.001, Lz = 4000.0)

## Boundary conditions

The boundary conditions prescribe a quadratic drag at the bottom as a flux condition.

cᴰ = 1e-4 # quadratic drag coefficient

@inline drag_u(x, y, t, u, v, cᴰ) = - cᴰ * u * sqrt(u^2 + v^2)
@inline drag_v(x, y, t, u, v, cᴰ) = - cᴰ * v * sqrt(u^2 + v^2)

drag_bc_u = FluxBoundaryCondition(drag_u, field_dependencies=(:u, :v), parameters=cᴰ)
drag_bc_v = FluxBoundaryCondition(drag_v, field_dependencies=(:u, :v), parameters=cᴰ)

u_bcs = FieldBoundaryConditions(bottom = drag_bc_u)
v_bcs = FieldBoundaryConditions(bottom = drag_bc_v)
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: Oceananigans.BoundaryConditions.DefaultPrognosticFieldBoundaryCondition
├── east: Oceananigans.BoundaryConditions.DefaultPrognosticFieldBoundaryCondition
├── south: Oceananigans.BoundaryConditions.DefaultPrognosticFieldBoundaryCondition
├── north: Oceananigans.BoundaryConditions.DefaultPrognosticFieldBoundaryCondition
├── bottom: BoundaryCondition{Flux, Oceananigans.BoundaryConditions.ContinuousBoundaryFunction{Nothing, Nothing, Nothing, Nothing, typeof(Main.drag_v), Float64, Tuple{Symbol, Symbol}, Nothing, Nothing}}
├── top: Oceananigans.BoundaryConditions.DefaultPrognosticFieldBoundaryCondition
└── immersed: BoundaryCondition{Flux, Nothing}

## Turbulence closures

We use a horizontal hyperdiffusivity and a Laplacian vertical diffusivity to dissipate energy in the Eady problem. To use both of these closures at the same time, we set the keyword argument closure to a tuple of two closures.

κ₂z = 1e-2 # [m² s⁻¹] Laplacian vertical viscosity and diffusivity
κ₄h = 1e-1 / day * grid.Δx^4 # [m⁴ s⁻¹] horizontal hyperviscosity and hyperdiffusivity

Laplacian_vertical_diffusivity = AnisotropicDiffusivity(νh=0, κh=0, νz=κ₂z, κz=κ₂z)
biharmonic_horizontal_diffusivity = AnisotropicBiharmonicDiffusivity(νh=κ₄h, κh=κ₄h)
AnisotropicBiharmonicDiffusivity{Float64, Float64, Float64, Float64}(2.1803253690129166e11, 2.1803253690129166e11, 0.0, 2.1803253690129166e11, 2.1803253690129166e11, 0.0)

## 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(
architecture = CPU(),
grid = grid,
timestepper = :RungeKutta3,
coriolis = coriolis,
tracers = :b,
buoyancy = BuoyancyTracer(),
background_fields = (b=B_field, u=U_field),
closure = (Laplacian_vertical_diffusivity, biharmonic_horizontal_diffusivity),
boundary_conditions = (u=u_bcs, v=v_bcs)
)
NonhydrostaticModel{CPU, Float64}(time = 0 seconds, iteration = 0)
├── grid: RegularRectilinearGrid{Float64, Periodic, Periodic, Bounded}(Nx=48, Ny=48, Nz=16)
├── tracers: (:b,)
├── closure: Tuple{AnisotropicDiffusivity{Oceananigans.TurbulenceClosures.ExplicitTimeDiscretization, Float64, Float64, Float64, NamedTuple{(:b,), Tuple{Float64}}, NamedTuple{(:b,), Tuple{Float64}}, NamedTuple{(:b,), Tuple{Float64}}}, AnisotropicBiharmonicDiffusivity{Float64, NamedTuple{(:b,), Tuple{Float64}}, NamedTuple{(:b,), Tuple{Float64}}, NamedTuple{(:b,), Tuple{Float64}}}}
├── buoyancy: BuoyancyTracer
└── coriolis: FPlane{Float64}

## Initial conditions

We seed our initial conditions with random noise stimulate the growth of baroclinic instability.

# A noise function, damped at the top and bottom
Ξ(z) = randn() * z/grid.Lz * (z/grid.Lz + 1)

# Scales for the initial velocity and buoyancy
Ũ = 1e-1 * basic_state_parameters.α * grid.Lz
B̃ = 1e-2 * basic_state_parameters.α * coriolis.f

uᵢ(x, y, z) = Ũ * Ξ(z)
vᵢ(x, y, z) = Ũ * Ξ(z)
bᵢ(x, y, z) = B̃ * Ξ(z)

set!(model, u=uᵢ, v=vᵢ, b=bᵢ)

We subtract off any residual mean velocity to avoid exciting domain-scale inertial oscillations. We use a sum over the entire parent arrays or data to ensure this operation is efficient on the GPU (set architecture = GPU() in NonhydrostaticModel constructor to run this problem on the GPU if one is available).

ū = sum(model.velocities.u.data.parent) / (grid.Nx * grid.Ny * grid.Nz)
v̄ = sum(model.velocities.v.data.parent) / (grid.Nx * grid.Ny * grid.Nz)

model.velocities.u.data.parent .-= ū
model.velocities.v.data.parent .-= v̄

## Simulation set-up

We set up a simulation that runs for 10 days with a JLD2OutputWriter that saves the vertical vorticity and divergence every 2 hours.

### The TimeStepWizard

The TimeStepWizard manages the time-step adaptively, keeping the Courant-Freidrichs-Lewy (CFL) number close to 1.0 while ensuring the time-step does not increase beyond the maximum allowable value for numerical stability given the specified background flow, Coriolis time scales, and diffusion time scales.

# Calculate absolute limit on time-step using diffusivities and
# background velocity.
Ū = basic_state_parameters.α * grid.Lz

max_Δt = min(grid.Δx / Ū, grid.Δx^4 / κ₄h, grid.Δz^2 / κ₂z, 1/basic_state_parameters.N)

wizard = TimeStepWizard(cfl=0.85, Δt=max_Δt, max_change=1.1, max_Δt=max_Δt)

### A progress messenger

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

start_time = time_ns()

progress(sim) = @printf("i: % 6d, sim time: % 10s, wall time: % 10s, Δt: % 10s, CFL: %.2e\n",
sim.model.clock.iteration,
prettytime(sim.model.clock.time),
prettytime(1e-9 * (time_ns() - start_time)),
prettytime(sim.Δt.Δt),
CFL(sim.model))

### Build the simulation

We're ready to build and run the simulation. We ask for a progress message and time-step update every 20 iterations,

simulation = Simulation(model, Δt = wizard, iteration_interval = 20,
stop_time = 8days,
progress = progress)
Simulation{typename(NonhydrostaticModel){typename(CPU), Float64}}
├── Model clock: time = 0 seconds, iteration = 0
├── Next time step (TimeStepWizard{Float64, typeof(Oceananigans.Utils.cell_advection_timescale), typeof(Oceananigans.Simulations.infinite_diffusion_timescale)}): 16.667 minutes
├── Iteration interval: 20
├── Stop criteria: Any[Oceananigans.Simulations.iteration_limit_exceeded, Oceananigans.Simulations.stop_time_exceeded, Oceananigans.Simulations.wall_time_limit_exceeded]
├── Run time: 0 seconds, wall time limit: Inf
├── Stop time: 8 days, stop iteration: Inf
├── Diagnostics: typename(OrderedCollections.OrderedDict) with 1 entry:
│   └── nan_checker => typename(NaNChecker)
└── Output writers: typename(OrderedCollections.OrderedDict) with no entries

### Output

To visualize the baroclinic turbulence ensuing in the Eady problem, we use ComputedFields to diagnose and output vertical vorticity and divergence. Note that ComputedFields take "AbstractOperations" on Fields as input:

u, v, w = model.velocities # unpack velocity Fields

# Vertical vorticity [s⁻¹]
ζ = ComputedField(∂x(v) - ∂y(u))

# Horizontal divergence, or ∂x(u) + ∂y(v) [s⁻¹]
δ = ComputedField(-∂z(w))
ComputedField located at (Center, Center, Center) of UnaryOperation at (Center, Center, Center)
├── data: OffsetArrays.OffsetArray{Float64, 3, Array{Float64, 3}}, size: (48, 48, 16)
├── grid: RegularRectilinearGrid{Float64, Periodic, Periodic, Bounded}(Nx=48, Ny=48, Nz=16)
├── operand: UnaryOperation at (Center, Center, Center)
└── status: time=0.0

With the vertical vorticity, ζ, and the horizontal divergence, δ in hand, we create a JLD2OutputWriter that saves ζ and δ and add them to simulation.

simulation.output_writers[:fields] = JLD2OutputWriter(model, (ζ=ζ, δ=δ),
schedule = TimeInterval(4hours),
force = true)

All that's left is to press the big red button:

run!(simulation)
[ Info: Updating model auxiliary state before the first time step...
[ Info:     ... updated in 3.766 ms.
[ Info: Executing first time step...
i:     20, sim time: 5.389 hours, wall time: 14.268 minutes, Δt: 16.667 minutes, CFL: 2.13e-02
i:     40, sim time: 10.778 hours, wall time: 14.338 minutes, Δt: 16.667 minutes, CFL: 1.92e-02
i:     60, sim time:   16 hours, wall time: 14.403 minutes, Δt: 16.667 minutes, CFL: 2.76e-02
i:     80, sim time: 21.389 hours, wall time: 14.468 minutes, Δt: 16.667 minutes, CFL: 3.39e-02
i:    100, sim time: 1.116 days, wall time: 14.534 minutes, Δt: 16.667 minutes, CFL: 5.00e-02
i:    120, sim time: 1.333 days, wall time: 14.601 minutes, Δt: 16.667 minutes, CFL: 5.39e-02
i:    140, sim time: 1.558 days, wall time: 14.668 minutes, Δt: 16.667 minutes, CFL: 6.22e-02
i:    160, sim time: 1.782 days, wall time: 14.735 minutes, Δt: 16.667 minutes, CFL: 6.47e-02
i:    180, sim time:     2 days, wall time: 14.801 minutes, Δt: 16.667 minutes, CFL: 7.23e-02
i:    200, sim time: 2.225 days, wall time: 14.865 minutes, Δt: 16.667 minutes, CFL: 9.59e-02
i:    220, sim time: 2.449 days, wall time: 14.931 minutes, Δt: 16.667 minutes, CFL: 1.40e-01
i:    240, sim time: 2.667 days, wall time: 14.998 minutes, Δt: 16.667 minutes, CFL: 1.73e-01
i:    260, sim time: 2.891 days, wall time: 15.063 minutes, Δt: 16.667 minutes, CFL: 2.34e-01
i:    280, sim time: 3.116 days, wall time: 15.129 minutes, Δt: 16.667 minutes, CFL: 3.14e-01
i:    300, sim time: 3.333 days, wall time: 15.196 minutes, Δt: 16.667 minutes, CFL: 4.10e-01
i:    320, sim time: 3.558 days, wall time: 15.264 minutes, Δt: 16.667 minutes, CFL: 4.46e-01
i:    340, sim time: 3.782 days, wall time: 15.337 minutes, Δt: 16.667 minutes, CFL: 5.53e-01
i:    360, sim time:     4 days, wall time: 15.403 minutes, Δt: 16.667 minutes, CFL: 5.62e-01
i:    380, sim time: 4.225 days, wall time: 15.470 minutes, Δt: 16.667 minutes, CFL: 7.12e-01
i:    400, sim time: 4.449 days, wall time: 15.536 minutes, Δt: 16.667 minutes, CFL: 7.48e-01
i:    420, sim time: 4.667 days, wall time: 15.600 minutes, Δt: 16.667 minutes, CFL: 5.71e-01
i:    440, sim time: 4.891 days, wall time: 15.666 minutes, Δt: 16.667 minutes, CFL: 6.04e-01
i:    460, sim time: 5.116 days, wall time: 15.731 minutes, Δt: 16.667 minutes, CFL: 6.10e-01
i:    480, sim time: 5.333 days, wall time: 15.799 minutes, Δt: 16.667 minutes, CFL: 4.37e-01
i:    500, sim time: 5.558 days, wall time: 15.865 minutes, Δt: 16.667 minutes, CFL: 3.83e-01
i:    520, sim time: 5.782 days, wall time: 15.931 minutes, Δt: 16.667 minutes, CFL: 4.99e-01
i:    540, sim time:     6 days, wall time: 15.996 minutes, Δt: 16.667 minutes, CFL: 5.12e-01
i:    560, sim time: 6.225 days, wall time: 16.062 minutes, Δt: 16.667 minutes, CFL: 5.26e-01
i:    580, sim time: 6.449 days, wall time: 16.127 minutes, Δt: 16.667 minutes, CFL: 5.61e-01
i:    600, sim time: 6.667 days, wall time: 16.194 minutes, Δt: 16.667 minutes, CFL: 7.29e-01
i:    620, sim time: 6.891 days, wall time: 16.258 minutes, Δt: 16.667 minutes, CFL: 6.94e-01
i:    640, sim time: 7.116 days, wall time: 16.326 minutes, Δt: 16.667 minutes, CFL: 6.02e-01
i:    660, sim time: 7.333 days, wall time: 16.396 minutes, Δt: 16.667 minutes, CFL: 6.76e-01
i:    680, sim time: 7.558 days, wall time: 16.462 minutes, Δt: 16.667 minutes, CFL: 6.20e-01
i:    700, sim time: 7.782 days, wall time: 16.526 minutes, Δt: 16.667 minutes, CFL: 6.80e-01
i:    720, sim time:     8 days, wall time: 16.592 minutes, Δt: 16.667 minutes, CFL: 7.13e-01
[ Info: Simulation is stopping. Model time 8 days has hit or exceeded simulation stop time 8 days.

We animate the results by opening the JLD2 file, extracting data for the iterations we ended up saving at, and ploting slices of the saved fields. We prepare for animating the flow by creating coordinate arrays, opening the file, building a vector of the iterations that we saved data at, and defining a function for computing colorbar limits:

using JLD2, Plots

# Coordinate arrays
xζ, yζ, zζ = nodes(ζ)
xδ, yδ, zδ = nodes(δ)

# Open the file with our data
file = jldopen(simulation.output_writers[:fields].filepath)

# Extract a vector of iterations
iterations = parse.(Int, keys(file["timeseries/t"]))
49-element Vector{Int64}:
0
15
30
45
60
75
90
105
120
135
⋮
600
615
630
645
660
675
690
705
720

This utility is handy for calculating nice contour intervals:

function nice_divergent_levels(c, clim, nlevels=31)
levels = range(-clim, stop=clim, length=nlevels)
cmax = maximum(abs, c)
clim < cmax && (levels = vcat([-cmax], levels, [cmax]))
return levels
end

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

anim = @animate for (i, iter) in enumerate(iterations)

# Load 3D fields from file
t = file["timeseries/t/$iter"] R_snapshot = file["timeseries/ζ/$iter"] ./ coriolis.f
δ_snapshot = file["timeseries/δ/\$iter"]

surface_R = R_snapshot[:, :, grid.Nz]
surface_δ = δ_snapshot[:, :, grid.Nz]

slice_R = R_snapshot[:, 1, :]
slice_δ = δ_snapshot[:, 1, :]

Rlim = 0.5 * maximum(abs, R_snapshot) + 1e-9
δlim = 0.5 * maximum(abs, δ_snapshot) + 1e-9

Rlevels = nice_divergent_levels(R_snapshot, Rlim)
δlevels = nice_divergent_levels(δ_snapshot, δlim)

@info @sprintf("Drawing frame %d from iteration %d: max(ζ̃ / f) = %.3f \n",
i, iter, maximum(abs, surface_R))

xy_kwargs = (xlims = (0, grid.Lx), ylims = (0, grid.Lx),
xlabel = "x (m)", ylabel = "y (m)",
aspectratio = 1,
linewidth = 0, color = :balance, legend = false)

xz_kwargs = (xlims = (0, grid.Lx), ylims = (-grid.Lz, 0),
xlabel = "x (m)", ylabel = "z (m)",
aspectratio = grid.Lx / grid.Lz * 0.5,
linewidth = 0, color = :balance, legend = false)

R_xy = contourf(xζ, yζ, surface_R'; clims=(-Rlim, Rlim), levels=Rlevels, xy_kwargs...)
δ_xy = contourf(xδ, yδ, surface_δ'; clims=(-δlim, δlim), levels=δlevels, xy_kwargs...)
R_xz = contourf(xζ, zζ, slice_R'; clims=(-Rlim, Rlim), levels=Rlevels, xz_kwargs...)
δ_xz = contourf(xδ, zδ, slice_δ'; clims=(-δlim, δlim), levels=δlevels, xz_kwargs...)

plot(R_xy, δ_xy, R_xz, δ_xz,
size = (1000, 800),