Plankton mixing and blooming

In this example, we simulate the mixing of phytoplankton by convection that decreases in time and eventually shuts off, thereby precipitating a phytoplankton bloom. A similar scenario was simulated by Taylor and Ferrari (2011), providing evidence that the "critical turbulence hypothesis" explains the explosive bloom of oceanic phytoplankton observed in spring.

The phytoplankton in our model are advected, diffuse, grow, and die according to

\[∂_t P + \boldsymbol{v ⋅ ∇} P - κ ∇²P = (μ₀ \exp(z / λ) - m) \, P \, ,\]

where $\boldsymbol{v}$ is the turbulent velocity field, $κ$ is an isotropic diffusivity, $μ₀$ is the phytoplankton growth rate at the surface, $λ$ is the scale over which sunlight attenuates away from the surface, and $m$ is the mortality rate of phytoplankton due to viruses and grazing by zooplankton. We use Oceananigans' Forcing abstraction to implement the phytoplankton dynamics described by the right side of the phytoplankton equation above.

This example demonstrates

  • How to use a user-defined forcing function to simulate the dynamics of phytoplankton growth in sunlight and grazing by zooplankton.
  • How to set time-dependent boundary conditions.
  • How to use the TimeStepWizard to adapt the simulation time-step.
  • How to use Average to diagnose spatial averages of model fields.

Install dependencies

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

using Pkg
pkg"add Oceananigans, CairoMakie"

The grid

We use a two-dimensional grid with 64² points, 3² halo points for high-order advection, 1 m grid spacing, and a Flat y-direction:

using Oceananigans
using Oceananigans.Units: minutes, hour, hours, day

grid = RectilinearGrid(size=(64, 64), extent=(64, 64), halo=(3, 3), topology=(Periodic, Flat, Bounded))
64×1×64 RectilinearGrid{Float64, Periodic, Flat, Bounded} on CPU with 3×0×3 halo
├── Periodic x ∈ [0.0, 64.0)      regularly spaced with Δx=1.0
├── Flat y
└── Bounded  z ∈ [-64.0, 0.0]     regularly spaced with Δz=1.0

Boundary conditions

We impose a surface buoyancy flux that's initially constant and then decays to zero,

buoyancy_flux(x, y, t, params) = params.initial_buoyancy_flux * exp(-t^4 / (24 * params.shut_off_time^4))

buoyancy_flux_parameters = (initial_buoyancy_flux = 1e-8, # m² s⁻³
                                    shut_off_time = 2hours)

buoyancy_flux_bc = FluxBoundaryCondition(buoyancy_flux, parameters = buoyancy_flux_parameters)
FluxBoundaryCondition: ContinuousBoundaryFunction buoyancy_flux at (Nothing, Nothing, Nothing)

The fourth power in the argument of exp above helps keep the buoyancy flux relatively constant during the first phase of the simulation. We produce a plot of this time-dependent buoyancy flux for the visually-oriented,

using CairoMakie
set_theme!(Theme(fontsize = 24, linewidth=2))

times = range(0, 12hours, length=100)

fig = Figure(resolution = (800, 300))
ax = Axis(fig[1, 1]; xlabel = "Time (hours)", ylabel = "Surface buoyancy flux (m² s⁻³)")

flux_time_series = [buoyancy_flux(0, 0, t, buoyancy_flux_parameters) for t in times]
lines!(ax, times ./ hour, flux_time_series)

current_figure() # hide

The buoyancy flux effectively shuts off after 6 hours of simulation time.

The flux convention in Oceananigans.jl

Fluxes are defined by the direction a quantity is carried: positive velocities produce positive fluxes, while negative velocities produce negative fluxes. Diffusive fluxes are defined with the same convention. A positive flux at the top boundary transports buoyancy upwards, out of the domain. This means that a positive flux of buoyancy at the top boundary reduces the buoyancy of near-surface fluid, causing convection.

The initial condition and bottom boundary condition impose the constant buoyancy gradient

N² = 1e-4 # s⁻²

buoyancy_gradient_bc = GradientBoundaryCondition(N²)
GradientBoundaryCondition: 0.0001

In summary, the buoyancy boundary conditions impose a destabilizing flux at the top and a stable buoyancy gradient at the bottom:

buoyancy_bcs = FieldBoundaryConditions(top = buoyancy_flux_bc, bottom = buoyancy_gradient_bc)
Oceananigans.FieldBoundaryConditions, with boundary conditions
├── west: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── east: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── south: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── north: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)
├── bottom: GradientBoundaryCondition: 0.0001
├── top: FluxBoundaryCondition: ContinuousBoundaryFunction buoyancy_flux at (Nothing, Nothing, Nothing)
└── immersed: DefaultBoundaryCondition (FluxBoundaryCondition: Nothing)

Phytoplankton dynamics: light-dependent growth and uniform mortality

We use a simple model for the growth of phytoplankton in sunlight and decay due to viruses and grazing by zooplankton,

growing_and_grazing(x, y, z, t, P, params) = (params.μ₀ * exp(z / params.λ) - params.m) * P
nothing # hide

with parameters

plankton_dynamics_parameters = (μ₀ = 1/day,   # surface growth rate
                                 λ = 5,       # sunlight attenuation length scale (m)
                                 m = 0.1/day) # mortality rate due to virus and zooplankton grazing
(μ₀ = 1.1574074074074073e-5, λ = 5, m = 1.1574074074074074e-6)

We tell Forcing that our plankton model depends on the plankton concentration P and the chosen parameters,

plankton_dynamics = Forcing(growing_and_grazing, field_dependencies = :P,
                            parameters = plankton_dynamics_parameters)
ContinuousForcing{NamedTuple{(:μ₀, :λ, :m), Tuple{Float64, Int64, Float64}}}
├── func: growing_and_grazing (generic function with 1 method)
├── parameters: (μ₀ = 1.1574074074074073e-5, λ = 5, m = 1.1574074074074074e-6)
└── field dependencies: (:P,)

The model

The name "P" for phytoplankton is specified in the constructor for NonhydrostaticModel. We additionally specify a fifth-order advection scheme, third-order Runge-Kutta time-stepping, isotropic viscosity and diffusivities, and Coriolis forces appropriate for planktonic convection at mid-latitudes on Earth.

model = NonhydrostaticModel(; grid,
                            advection = UpwindBiasedFifthOrder(),
                            timestepper = :RungeKutta3,
                            closure = ScalarDiffusivity(ν=1e-4, κ=1e-4),
                            coriolis = FPlane(f=1e-4),
                            tracers = (:b, :P), # P for Plankton
                            buoyancy = BuoyancyTracer(),
                            forcing = (; P=plankton_dynamics),
                            boundary_conditions = (; b=buoyancy_bcs))
NonhydrostaticModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── grid: 64×1×64 RectilinearGrid{Float64, Periodic, Flat, Bounded} on CPU with 3×0×3 halo
├── timestepper: RungeKutta3TimeStepper
├── tracers: (b, P)
├── closure: ScalarDiffusivity{ExplicitTimeDiscretization}(ν=0.0001, κ=(b=0.0001, P=0.0001))
├── buoyancy: BuoyancyTracer with ĝ = NegativeZDirection()
└── coriolis: FPlane{Float64}(f=0.0001)

Initial condition

We set the initial phytoplankton at $P = 1 \, \rm{μM}$. For buoyancy, we use a stratification that's mixed near the surface and linearly stratified below, superposed with surface-concentrated random noise.

mixed_layer_depth = 32 # m

stratification(z) = z < -mixed_layer_depth ? N² * z : - N² * mixed_layer_depth
noise(z) = 1e-4 * N² * grid.Lz * randn() * exp(z / 4)
initial_buoyancy(x, y, z) = stratification(z) + noise(z)

set!(model, b=initial_buoyancy, P=1)

Simulation with adaptive time-stepping, logging, and output

We build a simulation

simulation = Simulation(model, Δt=2minutes, stop_time=24hours)
Simulation of NonhydrostaticModel{CPU, RectilinearGrid}(time = 0 seconds, iteration = 0)
├── Next time step: 2 minutes
├── Elapsed wall time: 0 seconds
├── Wall time per iteration: NaN days
├── Stop time: 1 day
├── Stop iteration : Inf
├── Wall time limit: Inf
├── Callbacks: OrderedDict with 4 entries:
│   ├── stop_time_exceeded => Callback of stop_time_exceeded on IterationInterval(1)
│   ├── stop_iteration_exceeded => Callback of stop_iteration_exceeded on IterationInterval(1)
│   ├── wall_time_limit_exceeded => Callback of wall_time_limit_exceeded on IterationInterval(1)
│   └── nan_checker => Callback of NaNChecker for u on IterationInterval(100)
├── Output writers: OrderedDict with no entries
└── Diagnostics: OrderedDict with no entries

with a TimeStepWizard that limits the time-step to 2 minutes, and adapts the time-step such that CFL (Courant-Freidrichs-Lewy) number hovers around 1.0,

wizard = TimeStepWizard(cfl=1.0, max_change=1.1, max_Δt=2minutes)

simulation.callbacks[:wizard] = Callback(wizard, IterationInterval(10))
Callback of TimeStepWizard(cfl=1.0, max_Δt=120.0, min_Δt=0.0) on IterationInterval(10)

We also add a callback that prints the progress of the simulation,

using Printf

progress(sim) = @printf("Iteration: %d, time: %s, Δt: %s\n",
                        iteration(sim), prettytime(sim), prettytime(sim.Δt))

simulation.callbacks[:progress] = Callback(progress, IterationInterval(100))
Callback of progress on IterationInterval(100)

and a basic JLD2OutputWriter that writes velocities and both the two-dimensional and horizontally-averaged plankton concentration,

outputs = (w = model.velocities.w,
           P = model.tracers.P,
           avg_P = Average(model.tracers.P, dims=(1, 2)))

simulation.output_writers[:simple_output] =
    JLD2OutputWriter(model, outputs,
                     schedule = TimeInterval(20minutes),
                     filename = "convecting_plankton.jld2",
                     overwrite_existing = true)
JLD2OutputWriter scheduled on TimeInterval(20 minutes):
├── filepath: ./convecting_plankton.jld2
├── 3 outputs: (w, P, avg_P)
├── array type: Array{Float64}
├── including: [:grid, :coriolis, :buoyancy, :closure]
└── max filesize: Inf YiB
Using multiple output writers

Because each output writer is associated with a single output schedule, it often makes sense to use different output writers for different types of output. For example, smaller outputs that consume less disk space may be written more frequently without threatening the capacity of your hard drive. An arbitrary number of output writers may be added to simulation.output_writers.

The simulation is set up. Let there be plankton:

[ Info: Initializing simulation...
Iteration: 0, time: 0 seconds, Δt: 2 minutes
[ Info:     ... simulation initialization complete (4.995 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (13.510 seconds).
Iteration: 100, time: 2.348 hours, Δt: 52.750 seconds
Iteration: 200, time: 3.454 hours, Δt: 38.014 seconds
Iteration: 300, time: 4.490 hours, Δt: 40.264 seconds
Iteration: 400, time: 5.461 hours, Δt: 39.416 seconds
Iteration: 500, time: 6.632 hours, Δt: 44.463 seconds
Iteration: 600, time: 7.907 hours, Δt: 49.649 seconds
Iteration: 700, time: 9.329 hours, Δt: 51.970 seconds
Iteration: 800, time: 10.822 hours, Δt: 55.082 seconds
Iteration: 900, time: 12.295 hours, Δt: 50.594 seconds
Iteration: 1000, time: 13.771 hours, Δt: 54.444 seconds
Iteration: 1100, time: 15.348 hours, Δt: 52.339 seconds
Iteration: 1200, time: 16.783 hours, Δt: 51.740 seconds
Iteration: 1300, time: 18.213 hours, Δt: 56.683 seconds
Iteration: 1400, time: 19.871 hours, Δt: 1.123 minutes
Iteration: 1500, time: 21.576 hours, Δt: 1.119 minutes
Iteration: 1600, time: 23.308 hours, Δt: 1.153 minutes
[ Info: Simulation is stopping after running for 1.246 minutes.
[ Info: Simulation time 1 day equals or exceeds stop time 1 day.

Notice how the time-step is reduced at early times, when turbulence is strong, and increases again towards the end of the simulation when turbulence fades.

Visualizing the solution

We'd like to a make a plankton movie. First we load the output file and build a time-series of the buoyancy flux,

filepath = simulation.output_writers[:simple_output].filepath

w_timeseries = FieldTimeSeries(filepath, "w")
P_timeseries = FieldTimeSeries(filepath, "P")
avg_P_timeseries = FieldTimeSeries(filepath, "avg_P")

times = w_timeseries.times
buoyancy_flux_time_series = [buoyancy_flux(0, 0, t, buoyancy_flux_parameters) for t in times]
nothing # hide

and then we construct the $x, z$ grid,

xw, yw, zw = nodes(w_timeseries)
xp, yp, zp = nodes(P_timeseries)
nothing # hide

Finally, we animate plankton mixing and blooming,

using CairoMakie

@info "Making a movie about plankton..."

n = Observable(1)

title = @lift @sprintf("t = %s", prettytime(times[$n]))

wₙ = @lift interior(w_timeseries[$n], :, 1, :)
Pₙ = @lift interior(P_timeseries[$n], :, 1, :)
avg_Pₙ = @lift interior(avg_P_timeseries[$n], 1, 1, :)

w_lim = maximum(abs, interior(w_timeseries))
w_lims = (-w_lim, w_lim)

P_lims = (0.95, 1.1)

fig = Figure(resolution = (1200, 1000))

ax_w = Axis(fig[2, 2]; xlabel = "x (m)", ylabel = "z (m)", aspect = 1)
ax_P = Axis(fig[3, 2]; xlabel = "x (m)", ylabel = "z (m)", aspect = 1)
ax_b = Axis(fig[2, 3]; xlabel = "Time (hours)", ylabel = "Buoyancy flux (m² s⁻³)", yaxisposition = :right)

ax_avg_P = Axis(fig[3, 3]; xlabel = "Plankton concentration (μM)", ylabel = "z (m)", yaxisposition = :right)
xlims!(ax_avg_P, 0.85, 1.3)

fig[1, 1:3] = Label(fig, title, tellwidth=false)

hm_w = heatmap!(ax_w, xw, zw, wₙ; colormap = :balance, colorrange = w_lims)
Colorbar(fig[2, 1], hm_w; label = "Vertical velocity (m s⁻¹)", flipaxis = false)

hm_P = heatmap!(ax_P, xp, zp, Pₙ; colormap = :matter, colorrange = P_lims)
Colorbar(fig[3, 1], hm_P; label = "Plankton 'concentration'", flipaxis = false)

lines!(ax_b, times ./ hour, buoyancy_flux_time_series; linewidth = 1, color = :black, alpha = 0.4)

b_flux_point = @lift Point2(times[$n] / hour, buoyancy_flux_time_series[$n])
scatter!(ax_b, b_flux_point; marker = :circle, markersize = 16, color = :black)
lines!(ax_avg_P, avg_Pₙ, zp)

current_figure() # hide

And, finally, we record a movie.

frames = 1:length(times)

@info "Making an animation of convecting plankton..."

record(fig, "convecting_plankton.mp4", frames, framerate=8) do i
    n[] = i
[ Info: Making an animation of convecting plankton...

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