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

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

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

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

fig

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

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, z, t, P, params) = (params.μ₀ * exp(z / params.λ) - params.m) * P

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::Float64, λ::Int64, m::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
├── advection scheme: Upwind Biased reconstruction order 5
├── 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, 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,

conjure_time_step_wizard!(simulation, cfl=1.0, max_Δt=2minutes)

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))

add_callback!(simulation, progress, 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]
├── file_splitting: NoFileSplitting
└── file size: 31.5 KiB

The simulation is set up. Let there be plankton:

run!(simulation)

[ Info: Initializing simulation...
Iteration: 0, time: 0 seconds, Δt: 2 minutes
[ Info:     ... simulation initialization complete (8.236 seconds)
[ Info: Executing initial time step...
[ Info:     ... initial time step complete (12.840 seconds).
Iteration: 100, time: 2.461 hours, Δt: 59.459 seconds
Iteration: 200, time: 3.647 hours, Δt: 40.801 seconds
Iteration: 300, time: 4.698 hours, Δt: 34.846 seconds
Iteration: 400, time: 5.705 hours, Δt: 33.642 seconds
Iteration: 500, time: 6.892 hours, Δt: 51.247 seconds
Iteration: 600, time: 8.278 hours, Δt: 49.406 seconds
Iteration: 700, time: 9.715 hours, Δt: 1.063 minutes
Iteration: 800, time: 11.524 hours, Δt: 1.031 minutes
Iteration: 900, time: 13.232 hours, Δt: 1.017 minutes
Iteration: 1000, time: 14.879 hours, Δt: 58.669 seconds
Iteration: 1100, time: 16.733 hours, Δt: 1.328 minutes
Iteration: 1200, time: 19 hours, Δt: 1.385 minutes
Iteration: 1300, time: 21.202 hours, Δt: 1.357 minutes
Iteration: 1400, time: 23.614 hours, Δt: 1.777 minutes
[ Info: Simulation is stopping after running for 1.389 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, t, buoyancy_flux_parameters) for t in times]

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

xw, yw, zw = nodes(w_timeseries)
xp, yp, zp = nodes(P_timeseries)

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(size = (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)

fig

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
end

[ Info: Making an animation of convecting plankton...

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