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: 18.875 seconds, max(u): (0.000e+00, 0.000e+00, 0.000e+00) m/s, next Δt: 20 minutes
[ Info:     ... simulation initialization complete (13.128 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (9.282 seconds).
[06.94%] i: 100, t: 1.389 days, wall time: 19.849 seconds, max(u): (1.315e-01, 1.147e-01, 1.598e-03) m/s, next Δt: 20 minutes
[13.89%] i: 200, t: 2.778 days, wall time: 1.873 seconds, max(u): (2.259e-01, 1.715e-01, 1.803e-03) m/s, next Δt: 20 minutes
[20.83%] i: 300, t: 4.167 days, wall time: 1.767 seconds, max(u): (3.026e-01, 2.198e-01, 1.722e-03) m/s, next Δt: 20 minutes
[27.78%] i: 400, t: 5.556 days, wall time: 1.843 seconds, max(u): (3.758e-01, 2.820e-01, 1.836e-03) m/s, next Δt: 20 minutes
[34.72%] i: 500, t: 6.944 days, wall time: 1.758 seconds, max(u): (4.589e-01, 4.282e-01, 1.783e-03) m/s, next Δt: 20 minutes
[41.67%] i: 600, t: 8.333 days, wall time: 1.856 seconds, max(u): (5.783e-01, 7.636e-01, 2.269e-03) m/s, next Δt: 20 minutes
[48.61%] i: 700, t: 9.722 days, wall time: 1.760 seconds, max(u): (8.526e-01, 1.180e+00, 3.325e-03) m/s, next Δt: 20 minutes
[55.56%] i: 800, t: 11.111 days, wall time: 1.779 seconds, max(u): (1.244e+00, 1.388e+00, 4.452e-03) m/s, next Δt: 20 minutes
[62.50%] i: 900, t: 12.500 days, wall time: 1.869 seconds, max(u): (1.447e+00, 1.319e+00, 5.244e-03) m/s, next Δt: 20 minutes
[69.44%] i: 1000, t: 13.889 days, wall time: 1.765 seconds, max(u): (1.464e+00, 1.476e+00, 4.427e-03) m/s, next Δt: 20 minutes
[76.39%] i: 1100, t: 15.278 days, wall time: 1.828 seconds, max(u): (1.520e+00, 1.547e+00, 3.324e-03) m/s, next Δt: 20 minutes
[83.33%] i: 1200, t: 16.667 days, wall time: 1.794 seconds, max(u): (1.474e+00, 1.365e+00, 3.309e-03) m/s, next Δt: 20 minutes
[90.28%] i: 1300, t: 18.056 days, wall time: 1.868 seconds, max(u): (1.465e+00, 1.283e+00, 4.010e-03) m/s, next Δt: 20 minutes
[97.22%] i: 1400, t: 19.444 days, wall time: 1.823 seconds, max(u): (1.449e+00, 1.381e+00, 3.134e-03) m/s, next Δt: 20 minutes
[ Info: Simulation is stopping after running for 50.867 seconds.
[ Info: Simulation time 20 days equals or exceeds stop time 20 days.
[ Info: Simulation completed in 50.886 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([-0.009374166838824749 -0.008131640963256359 -0.006879489403218031 -0.005617518909275532 -0.004374624229967594 -0.003113987622782588 -0.0019052387215197086 -0.0006326503353193402; -0.009386367164552212 -0.00812625139951706 -0.006879901979118586 -0.005645452067255974 -0.004385656677186489 -0.0031336715910583735 -0.0018836569506675005 -0.000624393462203443; -0.00936147477477789 -0.008101440966129303 -0.006855867337435484 -0.005637503229081631 -0.004360043443739414 -0.0031256049405783415 -0.001878630486316979 -0.0006320104002952576; -0.009365231730043888 -0.008103038184344769 -0.006868222262710333 -0.005607191473245621 -0.0043926117941737175 -0.0031271055340766907 -0.0018865537131205201 -0.0006601595086976886; -0.00935586541891098 -0.008142529986798763 -0.006867262534797192 -0.005622495897114277 -0.004387888591736555 -0.003143695183098316 -0.0018679858185350895 -0.0006146186497062445; -0.009348494932055473 -0.00812837015837431 -0.006888482253998518 -0.0056322976015508175 -0.004373000003397465 -0.0031285386066883802 -0.0018664660165086389 -0.000612900301348418; -0.009383178316056728 -0.008145365864038467 -0.0068659307435154915 -0.005620866548269987 -0.004386524669826031 -0.003139769658446312 -0.0018808633321896195 -0.0006169165135361254; -0.009396701119840145 -0.008126984350383282 -0.006880280561745167 -0.005622399505227804 -0.0043642534874379635 -0.0031137920450419188 -0.0018384772120043635 -0.0006143167847767472; -0.009380638599395752 -0.00812329351902008 -0.0068784914910793304 -0.00564185855910182 -0.004379226826131344 -0.0031215555500239134 -0.0018467407207936049 -0.0006352211930789053; -0.00936202798038721 -0.00811144057661295 -0.0068764956668019295 -0.005631295498460531 -0.004361295606940985 -0.003118801862001419 -0.0018887667683884501 -0.0006087764631956816; -0.009367299266159534 -0.008108708076179028 -0.006864093244075775 -0.005612303968518972 -0.004415658302605152 -0.0031252994667738676 -0.0018769397865980864 -0.0006319451495073736; -0.009355805814266205 -0.008141838945448399 -0.006880889181047678 -0.0056176199577748775 -0.004371547140181065 -0.003105585929006338 -0.0018504823092371225 -0.0006414502277038991; -0.009360198862850666 -0.008098474703729153 -0.006878195330500603 -0.005626617465168238 -0.00437190430238843 -0.003158821724355221 -0.0018598086899146438 -0.0006122299237176776; -0.009376985020935535 -0.00813269056379795 -0.006880344357341528 -0.005651595536619425 -0.004367688670754433 -0.003104712814092636 -0.0018713973695412278 -0.0006033838144503534; -0.009379496797919273 -0.008114004507660866 -0.006891321856528521 -0.005633784458041191 -0.004361767787486315 -0.0031253304332494736 -0.0018947250209748745 -0.0006086917128413916; -0.009375014342367649 -0.008121117949485779 -0.006870744284242392 -0.005634600296616554 -0.004359634127467871 -0.0031189657747745514 -0.0018490342190489173 -0.0006285829585976899; -0.009383631870150566 -0.008125761523842812 -0.006862250156700611 -0.0056153847835958 -0.004370336886495352 -0.0031144041568040848 -0.0018733860924839973 -0.0006320441025309265; -0.009370693005621433 -0.00812474638223648 -0.006860949099063873 -0.0056246560998260975 -0.004375323187559843 -0.0031507823150604963 -0.0018888113554567099 -0.0006470414227806032; 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-0.003355031367391348 -0.002084931591525674 -0.0008497378439642489 0.0003972278500441462 0.0016754294047132134 0.0029343620408326387 0.004165041726082563 0.00542740523815155; -0.0012411668431013823 -2.2263857317739166e-6 0.0012497982243075967 0.0025045040529221296 0.0037614440079778433 0.005018384661525488 0.006263857241719961 0.007495144847780466; 0.000605622015427798 0.0018712389282882214 0.003127285512164235 0.004383389372378588 0.0056022098287940025 0.006876947823911905 0.008118859492242336 0.009374977089464664; 0.0006229372229427099 0.0018524377373978496 0.003105445299297571 0.004393149167299271 0.00561628257855773 0.006890659686177969 0.00813538022339344 0.009366320446133614; 0.0006059124716557562 0.0018678148044273257 0.0031283029820770025 0.0043731918558478355 0.005628969985991716 0.006849286612123251 0.008132902905344963 0.009356205351650715; 0.0006221050280146301 0.0018547329818829894 0.0031141063664108515 0.004369692411273718 0.005646787583827972 0.006866460666060448 0.008115386590361595 0.00936156790703535; 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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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