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Eady turbulence example

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

The Eady problem

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 $\bm{u}_h |\bm{u}_h|$, where $\bm{u}_h = u \bm{\hat{x}} + v \bm{\hat{y}}$ 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 viscosties 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}$$\mathrm{s^{-1}}$
$N$Buoyancy frequency (square root of $\partial_z B$)$10^{-3}$$\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}$$\mathrm{m^2 s^{-1}}$
$ϰ_h$Biharmonic horizontal diffusivity$10^{-2} \times \Delta x^4 / \mathrm{day}$$\mathrm{m^4 s^{-1}}$

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

using Oceananigans, Oceananigans.Utils, Printf

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 = RegularCartesianGrid(size=(48, 48, 16), x=(0, 1e6), y=(0, 1e6), z=(-4e3, 0))
RegularCartesianGrid{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,

background_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 $B$

using Oceananigans.Fields: BackgroundField

# 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=background_parameters)
B_field = BackgroundField(B, parameters=background_parameters)
BackgroundField{typeof(Main.ex-eady_turbulence.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.

drag_coefficient = 1e-4

@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 = BoundaryCondition(Flux, drag_u, field_dependencies=(:u, :v), parameters=drag_coefficient)
drag_bc_v = BoundaryCondition(Flux, drag_v, field_dependencies=(:u, :v), parameters=drag_coefficient)

u_bcs = UVelocityBoundaryConditions(grid, bottom = drag_bc_u)
v_bcs = VVelocityBoundaryConditions(grid, bottom = drag_bc_v)
Oceananigans.FieldBoundaryConditions (NamedTuple{(:x, :y, :z)}), with boundary conditions
├── x: CoordinateBoundaryConditions{BoundaryCondition{Oceananigans.BoundaryConditions.Periodic,Nothing},BoundaryCondition{Oceananigans.BoundaryConditions.Periodic,Nothing}}
├── y: CoordinateBoundaryConditions{BoundaryCondition{Oceananigans.BoundaryConditions.Periodic,Nothing},BoundaryCondition{Oceananigans.BoundaryConditions.Periodic,Nothing}}
└── z: CoordinateBoundaryConditions{BoundaryCondition{Flux,Oceananigans.BoundaryConditions.ContinuousBoundaryFunction{Nothing,Nothing,Nothing,Nothing,typeof(Main.ex-eady_turbulence.drag_v),Float64,Tuple{Symbol,Symbol},Nothing,Nothing}},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.

using Oceananigans.Advection: WENO5

model = IncompressibleModel(
           architecture = CPU(),
                   grid = grid,
              advection = WENO5(),
            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)
)
IncompressibleModel{CPU, Float64}(time = 0 seconds, iteration = 0) 
├── grid: RegularCartesianGrid{Float64, Periodic, Periodic, Bounded}(Nx=48, Ny=48, Nz=16)
├── tracers: (:b,)
├── closure: Tuple{AnisotropicDiffusivity{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 * background_parameters.α * grid.Lz
B̃ = 1e-2 * background_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 IncompressibleModel 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 = background_parameters.α * grid.Lz

max_Δt = min(grid.Δx / Ū, grid.Δx^4 / κ₄h, grid.Δz^2 / κ₂z, 0.2/coriolis.f)

wizard = TimeStepWizard(cfl=1.0, Δt=max_Δt, max_change=1.1, max_Δt=max_Δt)
TimeStepWizard{Float64}(1.0, Inf, 1.1, 0.5, 2000.0, 2000.0)

A progress messenger

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

using Oceananigans.Diagnostics: AdvectiveCFL

CFL = AdvectiveCFL(wizard)

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{IncompressibleModel{CPU, Float64}}
├── Model clock: time = 0 seconds, iteration = 0 
├── Next time step (TimeStepWizard{Float64}): 33.333 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: OrderedCollections.OrderedDict with no entries
└── Output writers: 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:

using Oceananigans.AbstractOperations
using Oceananigans.Fields: ComputedField

u, v, w = model.velocities # unpack velocity `Field`s

# 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 (Cell, Cell, Cell) of UnaryOperation at (Cell, Cell, Cell)
├── data: OffsetArrays.OffsetArray{Float64,3,Array{Float64,3}}, size: (54, 54, 22)
├── grid: RegularCartesianGrid{Float64, Periodic, Periodic, Bounded}(Nx=48, Ny=48, Nz=16)
├── operand: UnaryOperation at (Cell, Cell, Cell)
└── 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.

using Oceananigans.OutputWriters: JLD2OutputWriter, TimeInterval

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

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

run!(simulation)
i:     20, sim time: 11.111 hours, wall time: 1.213 minutes, Δt: 33.333 minutes, CFL: 9.76e-02
i:     40, sim time: 22.222 hours, wall time: 1.279 minutes, Δt: 33.333 minutes, CFL: 1.28e-01
i:     60, sim time: 1.389 days, wall time: 1.342 minutes, Δt: 33.333 minutes, CFL: 2.64e-01
i:     80, sim time: 1.852 days, wall time: 1.405 minutes, Δt: 33.333 minutes, CFL: 3.48e-01
i:    100, sim time: 2.315 days, wall time: 1.468 minutes, Δt: 33.333 minutes, CFL: 5.05e-01
i:    120, sim time: 2.778 days, wall time: 1.531 minutes, Δt: 33.333 minutes, CFL: 6.71e-01
i:    140, sim time: 3.241 days, wall time: 1.595 minutes, Δt: 33.333 minutes, CFL: 8.12e-01
i:    160, sim time: 3.704 days, wall time: 1.658 minutes, Δt: 33.333 minutes, CFL: 1.18e+00
i:    180, sim time: 4.095 days, wall time: 1.722 minutes, Δt: 28.179 minutes, CFL: 9.44e-01
i:    200, sim time: 4.510 days, wall time: 1.787 minutes, Δt: 29.860 minutes, CFL: 1.01e+00
i:    220, sim time: 4.922 days, wall time: 1.850 minutes, Δt: 29.702 minutes, CFL: 9.37e-01
i:    240, sim time: 5.363 days, wall time: 1.913 minutes, Δt: 31.710 minutes, CFL: 1.13e+00
i:    260, sim time: 5.752 days, wall time: 1.978 minutes, Δt: 28.015 minutes, CFL: 9.21e-01
i:    280, sim time: 6.174 days, wall time: 2.041 minutes, Δt: 30.428 minutes, CFL: 1.88e+00
i:    300, sim time: 6.400 days, wall time: 2.104 minutes, Δt: 16.221 minutes, CFL: 1.30e+00
i:    320, sim time: 6.573 days, wall time: 2.167 minutes, Δt: 12.509 minutes, CFL: 8.30e-01
i:    340, sim time: 6.765 days, wall time: 2.230 minutes, Δt: 13.760 minutes, CFL: 7.08e-01
i:    360, sim time: 6.975 days, wall time: 2.294 minutes, Δt: 15.136 minutes, CFL: 5.99e-01
i:    380, sim time: 7.206 days, wall time: 2.357 minutes, Δt: 16.650 minutes, CFL: 7.04e-01
i:    400, sim time: 7.460 days, wall time: 2.420 minutes, Δt: 18.315 minutes, CFL: 7.00e-01
i:    420, sim time: 7.740 days, wall time: 2.484 minutes, Δt: 20.147 minutes, CFL: 7.69e-01
i:    440, sim time: 8.048 days, wall time: 2.547 minutes, Δt: 22.161 minutes, CFL: 9.97e-01
[ Info: Simulation is stopping. Model time 8.048 days has hit or exceeded simulation stop time 8 days.

Visualizing Eady turbulence

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

using Oceananigans.Grids: nodes

# 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 Array{Int64,1}:
   0
   8
  15
  22
  29
  37
  44
  51
  58
  65
   ⋮
 330
 347
 363
 377
 391
 403
 415
 427
 437

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

Now we're ready to animate.

@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),
           link = :x,
         layout = Plots.grid(2, 2, heights=[0.5, 0.5, 0.2, 0.2]),
          title = [@sprintf("ζ(t=%s) / f", prettytime(t)) @sprintf("δ(t=%s) (s⁻¹)", prettytime(t)) "" ""])

    iter == iterations[end] && close(file)
end

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