Baroclinic adjustment

In this example, we simulate the evolution and equilibration of a baroclinically unstable front.

Install dependencies

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

using Pkg
pkg"add Oceananigans, CairoMakie"

using Oceananigans
using Oceananigans.Units

Grid

We use a three-dimensional channel that is periodic in the x direction:

Lx = 1000kilometers # east-west extent [m]
Ly = 1000kilometers # north-south extent [m]
Lz = 1kilometers    # depth [m]

grid = RectilinearGrid(size = (48, 48, 8),
                       x = (0, Lx),
                       y = (-Ly/2, Ly/2),
                       z = (-Lz, 0),
                       topology = (Periodic, Bounded, Bounded))

48×48×8 RectilinearGrid{Float64, Periodic, Bounded, Bounded} on CPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 1.0e6)          regularly spaced with Δx=20833.3
├── Bounded  y ∈ [-500000.0, 500000.0] regularly spaced with Δy=20833.3
└── Bounded  z ∈ [-1000.0, 0.0]        regularly spaced with Δz=125.0

Model

We built a HydrostaticFreeSurfaceModel with an ImplicitFreeSurface solver. Regarding Coriolis, we use a beta-plane centered at 45° South.

model = HydrostaticFreeSurfaceModel(; grid,
                                    coriolis = BetaPlane(latitude = -45),
                                    buoyancy = BuoyancyTracer(),
                                    tracers = :b,
                                    momentum_advection = WENO(),
                                    tracer_advection = WENO())

HydrostaticFreeSurfaceModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 48×48×8 RectilinearGrid{Float64, Periodic, Bounded, Bounded} on CPU with 3×3×3 halo
├── timestepper: QuasiAdamsBashforth2TimeStepper
├── tracers: b
├── closure: Nothing
├── buoyancy: BuoyancyTracer with ĝ = NegativeZDirection()
├── free surface: ImplicitFreeSurface with gravitational acceleration 9.80665 m s⁻²
│   └── solver: FFTImplicitFreeSurfaceSolver
├── advection scheme: 
│   ├── momentum: WENO{3, Float64, Float32}(order=5)
│   └── b: WENO{3, Float64, Float32}(order=5)
├── vertical_coordinate: ZCoordinate
└── coriolis: BetaPlane{Float64}

We start our simulation from rest with a baroclinically unstable buoyancy distribution. We use ramp(y, Δy), defined below, to specify a front with width Δy and horizontal buoyancy gradient M². We impose the front on top of a vertical buoyancy gradient N² and a bit of noise.

"""
    ramp(y, Δy)

Linear ramp from 0 to 1 between -Δy/2 and +Δy/2.

For example:
```
            y < -Δy/2 => ramp = 0
    -Δy/2 < y < -Δy/2 => ramp = y / Δy
            y >  Δy/2 => ramp = 1
```
"""
ramp(y, Δy) = min(max(0, y/Δy + 1/2), 1)

N² = 1e-5 # [s⁻²] buoyancy frequency / stratification
M² = 1e-7 # [s⁻²] horizontal buoyancy gradient

Δy = 100kilometers # width of the region of the front
Δb = Δy * M²       # buoyancy jump associated with the front
ϵb = 1e-2 * Δb     # noise amplitude

bᵢ(x, y, z) = N² * z + Δb * ramp(y, Δy) + ϵb * randn()

set!(model, b=bᵢ)

Let's visualize the initial buoyancy distribution.

using CairoMakie
set_theme!(Theme(fontsize = 20))

# Build coordinates with units of kilometers
x, y, z = 1e-3 .* nodes(grid, (Center(), Center(), Center()))

b = model.tracers.b

fig, ax, hm = heatmap(view(b, 1, :, :),
                      colormap = :deep,
                      axis = (xlabel = "y [km]",
                              ylabel = "z [km]",
                              title = "b(x=0, y, z, t=0)",
                              titlesize = 24))

Colorbar(fig[1, 2], hm, label = "[m s⁻²]")

fig

Simulation

Now let's build a Simulation.

simulation = Simulation(model, Δt=20minutes, stop_time=20days)

Simulation of HydrostaticFreeSurfaceModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── Next time step: 20 minutes
├── Elapsed wall time: 0 seconds
├── Wall time per iteration: NaN days
├── Stop time: 20 days
├── Stop iteration: Inf
├── Wall time limit: Inf
├── Minimum relative step: 0.0
├── Callbacks: OrderedDict with 4 entries:
│   ├── stop_time_exceeded => 4
│   ├── stop_iteration_exceeded => -
│   ├── wall_time_limit_exceeded => e
│   └── nan_checker => }
├── Output writers: OrderedDict with no entries
└── Diagnostics: OrderedDict with no entries

We add a TimeStepWizard callback to adapt the simulation's time-step,

conjure_time_step_wizard!(simulation, IterationInterval(20), cfl=0.2, max_Δt=20minutes)

Also, we add a callback to print a message about how the simulation is going,

using Printf

wall_clock = Ref(time_ns())

function print_progress(sim)
    u, v, w = model.velocities
    progress = 100 * (time(sim) / sim.stop_time)
    elapsed = (time_ns() - wall_clock[]) / 1e9

    @printf("[%05.2f%%] i: %d, t: %s, wall time: %s, max(u): (%6.3e, %6.3e, %6.3e) m/s, next Δt: %s\n",
            progress, iteration(sim), prettytime(sim), prettytime(elapsed),
            maximum(abs, u), maximum(abs, v), maximum(abs, w), prettytime(sim.Δt))

    wall_clock[] = time_ns()

    return nothing
end

add_callback!(simulation, print_progress, IterationInterval(100))

Diagnostics/Output

Here, we save the buoyancy, $b$, at the edges of our domain as well as the zonal ($x$) average of buoyancy.

u, v, w = model.velocities
ζ = ∂x(v) - ∂y(u)
B = Average(b, dims=1)
U = Average(u, dims=1)
V = Average(v, dims=1)

filename = "baroclinic_adjustment"
save_fields_interval = 0.5day

slicers = (east = (grid.Nx, :, :),
           north = (:, grid.Ny, :),
           bottom = (:, :, 1),
           top = (:, :, grid.Nz))

for side in keys(slicers)
    indices = slicers[side]

    simulation.output_writers[side] = JLD2Writer(model, (; b, ζ);
                                                 filename = filename * "_$(side)_slice",
                                                 schedule = TimeInterval(save_fields_interval),
                                                 overwrite_existing = true,
                                                 indices)
end

simulation.output_writers[:zonal] = JLD2Writer(model, (; b=B, u=U, v=V);
                                               filename = filename * "_zonal_average",
                                               schedule = TimeInterval(save_fields_interval),
                                               overwrite_existing = true)

JLD2Writer scheduled on TimeInterval(12 hours):
├── filepath: baroclinic_adjustment_zonal_average.jld2
├── 3 outputs: (b, u, v)
├── array_type: Array{Float32}
├── including: [:grid, :coriolis, :buoyancy, :closure]
├── file_splitting: NoFileSplitting
└── file size: 32.5 KiB

Now we're ready to run.

@info "Running the simulation..."

run!(simulation)

@info "Simulation completed in " * prettytime(simulation.run_wall_time)

[ Info: Running the simulation...
[ Info: Initializing simulation...
[00.00%] i: 0, t: 0 seconds, wall time: 20.430 seconds, max(u): (0.000e+00, 0.000e+00, 0.000e+00) m/s, next Δt: 20 minutes
[ Info:     ... simulation initialization complete (14.376 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (9.291 seconds).
[06.94%] i: 100, t: 1.389 days, wall time: 20.462 seconds, max(u): (1.284e-01, 1.227e-01, 1.563e-03) m/s, next Δt: 20 minutes
[13.89%] i: 200, t: 2.778 days, wall time: 1.851 seconds, max(u): (2.155e-01, 1.674e-01, 1.764e-03) m/s, next Δt: 20 minutes
[20.83%] i: 300, t: 4.167 days, wall time: 2.244 seconds, max(u): (2.813e-01, 2.280e-01, 1.809e-03) m/s, next Δt: 20 minutes
[27.78%] i: 400, t: 5.556 days, wall time: 1.970 seconds, max(u): (3.593e-01, 3.025e-01, 1.842e-03) m/s, next Δt: 20 minutes
[34.72%] i: 500, t: 6.944 days, wall time: 1.907 seconds, max(u): (4.373e-01, 4.624e-01, 1.951e-03) m/s, next Δt: 20 minutes
[41.67%] i: 600, t: 8.333 days, wall time: 1.923 seconds, max(u): (5.818e-01, 7.891e-01, 2.814e-03) m/s, next Δt: 20 minutes
[48.61%] i: 700, t: 9.722 days, wall time: 1.998 seconds, max(u): (8.124e-01, 1.235e+00, 3.451e-03) m/s, next Δt: 20 minutes
[55.56%] i: 800, t: 11.111 days, wall time: 1.997 seconds, max(u): (1.222e+00, 1.180e+00, 4.798e-03) m/s, next Δt: 20 minutes
[62.50%] i: 900, t: 12.500 days, wall time: 2.146 seconds, max(u): (1.312e+00, 1.229e+00, 5.190e-03) m/s, next Δt: 20 minutes
[69.44%] i: 1000, t: 13.889 days, wall time: 2.205 seconds, max(u): (1.333e+00, 1.121e+00, 5.026e-03) m/s, next Δt: 20 minutes
[76.39%] i: 1100, t: 15.278 days, wall time: 2.240 seconds, max(u): (1.326e+00, 1.144e+00, 4.886e-03) m/s, next Δt: 20 minutes
[83.33%] i: 1200, t: 16.667 days, wall time: 2.255 seconds, max(u): (1.223e+00, 1.245e+00, 3.073e-03) m/s, next Δt: 20 minutes
[90.28%] i: 1300, t: 18.056 days, wall time: 2.265 seconds, max(u): (1.304e+00, 1.091e+00, 3.441e-03) m/s, next Δt: 20 minutes
[97.22%] i: 1400, t: 19.444 days, wall time: 2.283 seconds, max(u): (1.439e+00, 1.228e+00, 3.533e-03) m/s, next Δt: 20 minutes
[ Info: Simulation is stopping after running for 56.031 seconds.
[ Info: Simulation time 20 days equals or exceeds stop time 20 days.
[ Info: Simulation completed in 56.050 seconds

Visualization

All that's left is to make a pretty movie. Actually, we make two visualizations here. First, we illustrate how to make a 3D visualization with Makie's Axis3 and Makie.surface. Then we make a movie in 2D. We use CairoMakie in this example, but note that using GLMakie is more convenient on a system with OpenGL, as figures will be displayed on the screen.

using CairoMakie

Three-dimensional visualization

We load the saved buoyancy output on the top, north, and east surface as FieldTimeSerieses.

filename = "baroclinic_adjustment"

sides = keys(slicers)

slice_filenames = NamedTuple(side => filename * "_$(side)_slice.jld2" for side in sides)

b_timeserieses = (east   = FieldTimeSeries(slice_filenames.east, "b"),
                  north  = FieldTimeSeries(slice_filenames.north, "b"),
                  top    = FieldTimeSeries(slice_filenames.top, "b"))

B_timeseries = FieldTimeSeries(filename * "_zonal_average.jld2", "b")

times = B_timeseries.times
grid = B_timeseries.grid

48×48×8 RectilinearGrid{Float64, Periodic, Bounded, Bounded} on CPU with 3×3×3 halo
├── Periodic x ∈ [0.0, 1.0e6)          regularly spaced with Δx=20833.3
├── Bounded  y ∈ [-500000.0, 500000.0] regularly spaced with Δy=20833.3
└── Bounded  z ∈ [-1000.0, 0.0]        regularly spaced with Δz=125.0

We build the coordinates. We rescale horizontal coordinates to kilometers.

xb, yb, zb = nodes(b_timeserieses.east)

xb = xb ./ 1e3 # convert m -> km
yb = yb ./ 1e3 # convert m -> km

Nx, Ny, Nz = size(grid)

x_xz = repeat(x, 1, Nz)
y_xz_north = y[end] * ones(Nx, Nz)
z_xz = repeat(reshape(z, 1, Nz), Nx, 1)

x_yz_east = x[end] * ones(Ny, Nz)
y_yz = repeat(y, 1, Nz)
z_yz = repeat(reshape(z, 1, Nz), grid.Ny, 1)

x_xy = x
y_xy = y
z_xy_top = z[end] * ones(grid.Nx, grid.Ny)

Then we create a 3D axis. We use zonal_slice_displacement to control where the plot of the instantaneous zonal average flow is located.

fig = Figure(size = (1600, 800))

zonal_slice_displacement = 1.2

ax = Axis3(fig[2, 1],
           aspect=(1, 1, 1/5),
           xlabel = "x (km)",
           ylabel = "y (km)",
           zlabel = "z (m)",
           xlabeloffset = 100,
           ylabeloffset = 100,
           zlabeloffset = 100,
           limits = ((x[1], zonal_slice_displacement * x[end]), (y[1], y[end]), (z[1], z[end])),
           elevation = 0.45,
           azimuth = 6.8,
           xspinesvisible = false,
           zgridvisible = false,
           protrusions = 40,
           perspectiveness = 0.7)

Axis3()

We use data from the final savepoint for the 3D plot. Note that this plot can easily be animated by using Makie's Observable. To dive into Observables, check out Makie.jl's Documentation.

n = length(times)

41

Now let's make a 3D plot of the buoyancy and in front of it we'll use the zonally-averaged output to plot the instantaneous zonal-average of the buoyancy.

b_slices = (east   = interior(b_timeserieses.east[n], 1, :, :),
            north  = interior(b_timeserieses.north[n], :, 1, :),
            top    = interior(b_timeserieses.top[n], :, :, 1))

# Zonally-averaged buoyancy
B = interior(B_timeseries[n], 1, :, :)

clims = 1.1 .* extrema(b_timeserieses.top[n][:])

kwargs = (colorrange=clims, colormap=:deep, shading=NoShading)

surface!(ax, x_yz_east, y_yz, z_yz;  color = b_slices.east, kwargs...)
surface!(ax, x_xz, y_xz_north, z_xz; color = b_slices.north, kwargs...)
surface!(ax, x_xy, y_xy, z_xy_top;   color = b_slices.top, kwargs...)

sf = surface!(ax, zonal_slice_displacement .* x_yz_east, y_yz, z_yz; color = B, kwargs...)

contour!(ax, y, z, B; transformation = (:yz, zonal_slice_displacement * x[end]),
         levels = 15, linewidth = 2, color = :black)

Colorbar(fig[2, 2], sf, label = "m s⁻²", height = Relative(0.4), tellheight=false)

title = "Buoyancy at t = " * string(round(times[n] / day, digits=1)) * " days"
fig[1, 1:2] = Label(fig, title; fontsize = 24, tellwidth = false, padding = (0, 0, -120, 0))

rowgap!(fig.layout, 1, Relative(-0.2))
colgap!(fig.layout, 1, Relative(-0.1))

save("baroclinic_adjustment_3d.png", fig)

Two-dimensional movie

We make a 2D movie that shows buoyancy $b$ and vertical vorticity $ζ$ at the surface, as well as the zonally-averaged zonal and meridional velocities $U$ and $V$ in the $(y, z)$ plane. First we load the FieldTimeSeries and extract the additional coordinates we'll need for plotting

ζ_timeseries = FieldTimeSeries(slice_filenames.top, "ζ")
U_timeseries = FieldTimeSeries(filename * "_zonal_average.jld2", "u")
B_timeseries = FieldTimeSeries(filename * "_zonal_average.jld2", "b")
V_timeseries = FieldTimeSeries(filename * "_zonal_average.jld2", "v")

xζ, yζ, zζ = nodes(ζ_timeseries)
yv = ynodes(V_timeseries)

xζ = xζ ./ 1e3 # convert m -> km
yζ = yζ ./ 1e3 # convert m -> km
yv = yv ./ 1e3 # convert m -> km

-500.0:20.833333333333332:500.0

Next, we set up a plot with 4 panels. The top panels are large and square, while the bottom panels get a reduced aspect ratio through rowsize!.

fig = Figure(size=(1800, 1000))

axb = Axis(fig[1, 2], xlabel="x (km)", ylabel="y (km)", aspect=1)
axζ = Axis(fig[1, 3], xlabel="x (km)", ylabel="y (km)", aspect=1, yaxisposition=:right)

axu = Axis(fig[2, 2], xlabel="y (km)", ylabel="z (m)")
axv = Axis(fig[2, 3], xlabel="y (km)", ylabel="z (m)", yaxisposition=:right)

rowsize!(fig.layout, 2, Relative(0.3))

To prepare a plot for animation, we index the timeseries with an Observable,

n = Observable(1)

b_top = @lift interior(b_timeserieses.top[$n], :, :, 1)
ζ_top = @lift interior(ζ_timeseries[$n], :, :, 1)
U = @lift interior(U_timeseries[$n], 1, :, :)
V = @lift interior(V_timeseries[$n], 1, :, :)
B = @lift interior(B_timeseries[$n], 1, :, :)

Observable([-0.009359532035887241 -0.008101513609290123 -0.006863062269985676 -0.00561874732375145 -0.004381408914923668 -0.0031316138338297606 -0.001891045249067247 -0.0006315717473626137; -0.00939029548317194 -0.008123303763568401 -0.006869222037494183 -0.00562577648088336 -0.004358251113444567 -0.00313223316334188 -0.001874998095445335 -0.00064238061895594; -0.009372878819704056 -0.008128659799695015 -0.006877056322991848 -0.0056500560604035854 -0.004379544407129288 -0.003121863817796111 -0.001886257086880505 -0.0006406665197573602; -0.009386779740452766 -0.00814926065504551 -0.006868869066238403 -0.005628490354865789 -0.004368131514638662 -0.0031174025498330593 -0.0018550818786025047 -0.0006141873309388757; -0.009398477151989937 -0.008104234002530575 -0.006877472624182701 -0.00562161672860384 -0.0044114915654063225 -0.0031414092518389225 -0.0018766727298498154 -0.0006285614217631519; -0.009380477480590343 -0.00813201442360878 -0.0068948292173445225 -0.005619887728244066 -0.004377733450382948 -0.0031223988626152277 -0.0018749033333733678 -0.0006256545311771333; -0.009344516322016716 -0.008134948089718819 -0.006878961343318224 -0.005620104726403952 -0.004355887416750193 -0.0031088150572031736 -0.0018993954872712493 -0.000636213575489819; -0.009376521222293377 -0.008137703873217106 -0.006876274477690458 -0.005627898965030909 -0.004359574057161808 -0.0031290831975638866 -0.0018759211525321007 -0.0006307516014203429; -0.009406140074133873 -0.008122733794152737 -0.00688971858471632 -0.005621298216283321 -0.004385485779494047 -0.003141903318464756 -0.0018932801904156804 -0.0006193392910063267; -0.009369262494146824 -0.00810937862843275 -0.006884425412863493 -0.005643214099109173 -0.004378374200314283 -0.0031342455185949802 -0.0018738068174570799 -0.0006117667653597891; -0.009395601227879524 -0.008109654299914837 -0.006880421191453934 -0.0056300899013876915 -0.004397653508931398 -0.0031099573243409395 -0.0018667463446035981 -0.000649328576400876; -0.009339183568954468 -0.008133879862725735 -0.006886505521833897 -0.005595856346189976 -0.0043735262006521225 -0.003130786120891571 -0.0018742148531600833 -0.0006024560425430536; -0.00936803873628378 -0.008140364661812782 -0.006887377239763737 -0.005634474102407694 -0.00437832111492753 -0.0031197897624224424 -0.0018988576484844089 -0.0006142727797850966; -0.00936865247786045 -0.008129401132464409 -0.006864346098154783 -0.005643409676849842 -0.004368585534393787 -0.0031313621439039707 -0.0018790604081004858 -0.0006361797568388283; -0.009369753301143646 -0.008123183622956276 -0.006849075201898813 -0.005603473633527756 -0.004402667284011841 -0.0031195031479001045 -0.0018606066005304456 -0.0006373743526637554; -0.009373041801154613 -0.00810928363353014 -0.0068895695731043816 -0.005627333652228117 -0.0043799616396427155 -0.0031324794981628656 -0.0018758100923150778 -0.000645565043669194; -0.00938807800412178 -0.008119145408272743 -0.006887809373438358 -0.005624744575470686 -0.004389675334095955 -0.00313111231662333 -0.0018936680862680078 -0.0006186607643030584; -0.009384317323565483 -0.008132114075124264 -0.006866136100143194 -0.00563832325860858 -0.004376459401100874 -0.0031129252165555954 -0.0019058339530602098 -0.0006224632961675525; 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-0.0033301839139312506 -0.0020871483720839024 -0.0008549640187993646 0.0003832023066934198 0.001686501782387495 0.0029335126746445894 0.004134793765842915 0.005393906030803919; -0.0012390721822157502 1.3235008736955933e-5 0.001254584058187902 0.0024868538603186607 0.003745787311345339 0.005007773172110319 0.006235376466065645 0.007483731489628553; 0.0006208490231074393 0.0018817153759300709 0.0031262545380741358 0.004393790382891893 0.00561087392270565 0.006874429527670145 0.008131032809615135 0.00935783889144659; 0.0006471374654211104 0.0018698236672207713 0.0031159238424152136 0.004390180576592684 0.005647376179695129 0.006868797354400158 0.008136725053191185 0.009367212653160095; 0.000622146762907505 0.0018716638442128897 0.0030927779152989388 0.004377598874270916 0.005617094226181507 0.0068785762414336205 0.008119015023112297 0.009367387741804123; 0.0006494564586319029 0.0018703717505559325 0.00313582899980247 0.004380052909255028 0.005617443006485701 0.006906361319124699 0.008124662563204765 0.009363740682601929; 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and then build our plot:

hm = heatmap!(axb, xb, yb, b_top, colorrange=(0, Δb), colormap=:thermal)
Colorbar(fig[1, 1], hm, flipaxis=false, label="Surface b(x, y) (m s⁻²)")

hm = heatmap!(axζ, xζ, yζ, ζ_top, colorrange=(-5e-5, 5e-5), colormap=:balance)
Colorbar(fig[1, 4], hm, label="Surface ζ(x, y) (s⁻¹)")

hm = heatmap!(axu, yb, zb, U; colorrange=(-5e-1, 5e-1), colormap=:balance)
Colorbar(fig[2, 1], hm, flipaxis=false, label="Zonally-averaged U(y, z) (m s⁻¹)")
contour!(axu, yb, zb, B; levels=15, color=:black)

hm = heatmap!(axv, yv, zb, V; colorrange=(-1e-1, 1e-1), colormap=:balance)
Colorbar(fig[2, 4], hm, label="Zonally-averaged V(y, z) (m s⁻¹)")
contour!(axv, yb, zb, B; levels=15, color=:black)

Finally, we're ready to record the movie.

frames = 1:length(times)

record(fig, filename * ".mp4", frames, framerate=8) do i
    n[] = i
end

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