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: 37.482 seconds, max(u): (0.000e+00, 0.000e+00, 0.000e+00) m/s, next Δt: 20 minutes
[ Info:     ... simulation initialization complete (31.868 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (18.812 seconds).
[06.94%] i: 100, t: 1.389 days, wall time: 43.281 seconds, max(u): (1.265e-01, 1.200e-01, 1.519e-03) m/s, next Δt: 20 minutes
[13.89%] i: 200, t: 2.778 days, wall time: 602.162 ms, max(u): (2.110e-01, 1.728e-01, 1.722e-03) m/s, next Δt: 20 minutes
[20.83%] i: 300, t: 4.167 days, wall time: 626.915 ms, max(u): (2.927e-01, 2.297e-01, 1.828e-03) m/s, next Δt: 20 minutes
[27.78%] i: 400, t: 5.556 days, wall time: 560.054 ms, max(u): (3.654e-01, 2.788e-01, 1.859e-03) m/s, next Δt: 20 minutes
[34.72%] i: 500, t: 6.944 days, wall time: 570.977 ms, max(u): (4.332e-01, 3.556e-01, 1.741e-03) m/s, next Δt: 20 minutes
[41.67%] i: 600, t: 8.333 days, wall time: 489.253 ms, max(u): (5.616e-01, 5.288e-01, 1.774e-03) m/s, next Δt: 20 minutes
[48.61%] i: 700, t: 9.722 days, wall time: 599.186 ms, max(u): (6.715e-01, 8.010e-01, 2.317e-03) m/s, next Δt: 20 minutes
[55.56%] i: 800, t: 11.111 days, wall time: 558.191 ms, max(u): (8.457e-01, 1.133e+00, 3.311e-03) m/s, next Δt: 20 minutes
[62.50%] i: 900, t: 12.500 days, wall time: 485.822 ms, max(u): (1.068e+00, 1.494e+00, 4.646e-03) m/s, next Δt: 20 minutes
[69.44%] i: 1000, t: 13.889 days, wall time: 523.873 ms, max(u): (1.359e+00, 1.414e+00, 4.780e-03) m/s, next Δt: 20 minutes
[76.39%] i: 1100, t: 15.278 days, wall time: 629.849 ms, max(u): (1.446e+00, 1.330e+00, 4.575e-03) m/s, next Δt: 20 minutes
[83.33%] i: 1200, t: 16.667 days, wall time: 520.088 ms, max(u): (1.457e+00, 1.538e+00, 4.197e-03) m/s, next Δt: 20 minutes
[90.28%] i: 1300, t: 18.056 days, wall time: 505.029 ms, max(u): (1.530e+00, 1.438e+00, 4.346e-03) m/s, next Δt: 20 minutes
[97.22%] i: 1400, t: 19.444 days, wall time: 482.446 ms, max(u): (1.368e+00, 1.340e+00, 3.452e-03) m/s, next Δt: 20 minutes
[ Info: Simulation is stopping after running for 1.034 minutes.
[ Info: Simulation time 20 days equals or exceeds stop time 20 days.
[ Info: Simulation completed in 1.034 minutes

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([-0.009396064095199108 -0.008128022775053978 -0.006899877917021513 -0.0056279366835951805 -0.004367631394416094 -0.0031300573609769344 -0.001885316101834178 -0.0006185163510963321; -0.009378359653055668 -0.00812554731965065 -0.006885076407343149 -0.005598502233624458 -0.004394738469272852 -0.003105625044554472 -0.0018552523106336594 -0.0006134114810265601; -0.009374141693115234 -0.008110037073493004 -0.006880635861307383 -0.005617028567939997 -0.004354689735919237 -0.0031339810229837894 -0.0018781980033963919 -0.0006275248597376049; -0.009358996525406837 -0.008111957460641861 -0.0068573434837162495 -0.005597221665084362 -0.00441479217261076 -0.0031295036897063255 -0.0018751627067103982 -0.0006466614431701601; -0.009379809722304344 -0.008113560266792774 -0.006882720626890659 -0.005623043980449438 -0.004390964284539223 -0.0031249653548002243 -0.0018916779663413763 -0.0006263465038500726; -0.009385539218783379 -0.008134022355079651 -0.0068993158638477325 -0.005619140807539225 -0.004391082096844912 -0.003118099644780159 -0.0018646309617906809 -0.0006194387096911669; -0.009388540871441364 -0.008126878179609776 -0.006875893566757441 -0.0056181359104812145 -0.004342980682849884 -0.0031006967183202505 -0.0018677476327866316 -0.000636701297480613; -0.009378195740282536 -0.008155171759426594 -0.0068832142278552055 -0.005611761473119259 -0.0043507046066224575 -0.0031314317602664232 -0.0018720438238233328 -0.0006062278989702463; -0.009371048770844936 -0.008139681071043015 -0.006863285321742296 -0.005634969100356102 -0.0043879179283976555 -0.0031263066921383142 -0.0018819617107510567 -0.0006259153014980257; -0.009386427700519562 -0.008140543475747108 -0.00687278900295496 -0.0056031993590295315 -0.004378690384328365 -0.003120447974652052 -0.0018742447718977928 -0.0006245335098356009; -0.009394255466759205 -0.008112606592476368 -0.006873466074466705 -0.005635794717818499 -0.0044040014035999775 -0.0031259285751730204 -0.0018820733530446887 -0.0006335099460557103; -0.009365552105009556 -0.008135085925459862 -0.00686392979696393 -0.005665382370352745 -0.004403871949762106 -0.0031013996340334415 -0.0018848150502890348 -0.0006217971094883978; -0.00935211218893528 -0.00811131950467825 -0.006874732207506895 -0.005620164796710014 -0.004373582545667887 -0.0031172693707048893 -0.0018494862597435713 -0.000661238213069737; -0.009369590319693089 -0.008117830380797386 -0.006875847466289997 -0.005617386195808649 -0.004383239429444075 -0.0031277695670723915 -0.001869312603957951 -0.0006160768680274487; -0.00934512447565794 -0.008120506070554256 -0.006887894589453936 -0.005616120062768459 -0.004410892724990845 -0.003107166849076748 -0.0018542010802775621 -0.0006106920773163438; -0.009395111352205276 -0.008116126991808414 -0.006892004515975714 -0.005609380081295967 -0.004396507516503334 -0.0031232822220772505 -0.001887973863631487 -0.0006254365434870124; -0.00936704222112894 -0.008142397738993168 -0.006876497995108366 -0.005636271089315414 -0.004371448419988155 -0.0031338268890976906 -0.0018858101684600115 -0.0006228071870282292; -0.009372695349156857 -0.008145096711814404 -0.006891288794577122 -0.005631698295474052 -0.004373540170490742 -0.0031385612674057484 -0.001863485318608582 -0.0006472706445492804; 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-0.0033516031689941883 -0.0020789990667253733 -0.0008389005670323968 0.0004345755442045629 0.0016644441056996584 0.002906976267695427 0.0041806395165622234 0.005413713864982128; -0.0012681422522291541 2.3479831725126132e-5 0.0012766921427100897 0.0025118719786405563 0.0037488762754946947 0.004980985540896654 0.006249838974326849 0.007524255197495222; 0.0006349037284962833 0.0018761279061436653 0.0030923611484467983 0.0043941643089056015 0.005613051820546389 0.006891758181154728 0.008145419880747795 0.009370310232043266; 0.0006346917944028974 0.0018689470598474145 0.0031210011802613735 0.004379511810839176 0.005631830543279648 0.006882459856569767 0.00812148954719305 0.009372596628963947; 0.0006531988619826734 0.0018540280871093273 0.003143155016005039 0.004371972288936377 0.005619246978312731 0.006876147352159023 0.00810480397194624 0.00936903152614832; 0.0006207312690094113 0.0018817808013409376 0.0030995486304163933 0.004365107975900173 0.0056374575942754745 0.006889896467328072 0.008127054199576378 0.009387117810547352; 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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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