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

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

where $\bm{u}$ 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 AveragedField 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, Plots, JLD2, Measures"

   Updating registry at `/storage7/buildkite-agent/.julia-865/registries/General`

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  Resolving package versions...
  Installed Oceananigans ─ v0.45.2
Updating `/storage7/buildkite-agent/builds/tartarus-mit-edu-6/clima/oceananigans/docs/Project.toml`
  [9e8cae18] ~ Oceananigans v0.45.2 `/storage7/buildkite-agent/builds/tartarus-mit-edu-6/clima/oceananigans` ⇒ v0.45.2
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  [9e8cae18] ~ Oceananigans v0.45.2 `/storage7/buildkite-agent/builds/tartarus-mit-edu-6/clima/oceananigans` ⇒ v0.45.2

The grid

We use a two-dimensional grid with 64² points and 1 m grid spacing:

using Oceananigans

grid = RegularCartesianGrid(size=(64, 1, 64), extent=(64, 1, 64))

RegularCartesianGrid{Float64, Periodic, Periodic, Bounded}
                   domain: x ∈ [0.0, 64.0], y ∈ [0.0, 1.0], z ∈ [-64.0, 0.0]
                 topology: (Periodic, Periodic, Bounded)
  resolution (Nx, Ny, Nz): (64, 1, 64)
   halo size (Hx, Hy, Hz): (1, 1, 1)
grid spacing (Δx, Δy, Δz): (1.0, 1.0, 1.0)

Boundary conditions

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

using Oceananigans.Utils

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

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

buoyancy_flux_bc = BoundaryCondition(Flux, buoyancy_flux, parameters = buoyancy_flux_parameters)

BoundaryCondition: type=Flux, condition=buoyancy_flux(x, y, t, p) in Main.ex-convecting_plankton at none:2

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 Plots, Measures

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

flux_plot = plot(time ./ hour, [buoyancy_flux(0, 0, t, buoyancy_flux_parameters) for t in time],
                 linewidth = 2, xlabel = "Time (hours)", ylabel = "Surface buoyancy flux (m² s⁻³)",
                 size = (800, 300), margin = 5mm, label = nothing)

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 = BoundaryCondition(Gradient, N²)

BoundaryCondition: type=Gradient, condition=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 = TracerBoundaryConditions(grid, top = buoyancy_flux_bc, bottom = buoyancy_gradient_bc)

Oceananigans.FieldBoundaryConditions (NamedTuple{(:x, :y, :z)}), with boundary conditions
├── x: CoordinateBoundaryConditions{BoundaryCondition{Oceananigans.BoundaryConditions.Periodic,Nothing},BoundaryCondition{Oceananigans.BoundaryConditions.Periodic,Nothing}}
├── y: CoordinateBoundaryConditions{BoundaryCondition{Oceananigans.BoundaryConditions.Periodic,Nothing},BoundaryCondition{Oceananigans.BoundaryConditions.Periodic,Nothing}}
└── z: CoordinateBoundaryConditions{BoundaryCondition{Gradient,Float64},BoundaryCondition{Flux,Oceananigans.BoundaryConditions.ContinuousBoundaryFunction{Nothing,Nothing,Nothing,Nothing,typeof(Main.ex-convecting_plankton.buoyancy_flux),NamedTuple{(:initial_buoyancy_flux, :shut_off_time),Tuple{Float64,Float64}},Tuple{},Nothing,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, p) = (p.μ₀ * exp(z / p.λ) - p.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, :λ, :m),Tuple{Float64,Int64,Float64}}}
├── func: growing_and_grazing
├── 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 IncompressibleModel. 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.

using Oceananigans.Advection

model = IncompressibleModel(
                   grid = grid,
              advection = UpwindBiasedFifthOrder(),
            timestepper = :RungeKutta3,
                closure = IsotropicDiffusivity(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,)
)

IncompressibleModel{CPU, Float64}(time = 0 seconds, iteration = 0) 
├── grid: RegularCartesianGrid{Float64, Periodic, Periodic, Bounded}(Nx=64, Ny=1, Nz=64)
├── tracers: (:b, :P)
├── closure: IsotropicDiffusivity{Float64,NamedTuple{(:b, :P),Tuple{Float64,Float64}}}
├── buoyancy: BuoyancyTracer
└── coriolis: FPlane{Float64}

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)

Adaptive time-stepping, logging, output and simulation setup

We use 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, Δt=2minutes, max_change=1.1, max_Δt=2minutes)

TimeStepWizard{Float64}(1.0, Inf, 1.1, 0.5, 120.0, 0.0, 120.0)

We also write a function that prints the progress of the simulation

using Printf

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

simulation = Simulation(model, Δt=wizard, stop_time=24hour,
                        iteration_interval=20, progress=progress)

Simulation{IncompressibleModel{CPU, Float64}}
├── Model clock: time = 0 seconds, iteration = 0 
├── Next time step (TimeStepWizard{Float64}): 2 minutes 
├── Iteration interval: 20
├── Stop criteria: Any[Oceananigans.Simulations.iteration_limit_exceeded, Oceananigans.Simulations.stop_time_exceeded, Oceananigans.Simulations.wall_time_limit_exceeded]
├── Run time: 0 seconds, wall time limit: Inf
├── Stop time: 1 day, stop iteration: Inf
├── Diagnostics: OrderedCollections.OrderedDict with 1 entry:
│   └── nan_checker => NaNChecker
└── Output writers: OrderedCollections.OrderedDict with no entries

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

using Oceananigans.OutputWriters, Oceananigans.Fields

averaged_plankton = AveragedField(model.tracers.P, dims=(1, 2))

outputs = (w = model.velocities.w,
           plankton = model.tracers.P,
           averaged_plankton = averaged_plankton)

simulation.output_writers[:simple_output] =
    JLD2OutputWriter(model, outputs,
                     schedule = TimeInterval(20minutes),
                     prefix = "convecting_plankton",
                     force = true)

JLD2OutputWriter scheduled on TimeInterval(20 minutes):
├── filepath: ./convecting_plankton.jld2
├── 3 outputs: (:w, :plankton, :averaged_plankton)
├── field slicer: FieldSlicer(:, :, :, with_halos=false)
├── array type: Array{Float32}
├── 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, reduced fields like AveragedField usually consume less disk space than two- or three-dimensional fields, and can thus be output more frequently without blowing up 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:

run!(simulation)

Iteration: 20, time: 40 minutes, Δt: 2 minutes
Iteration: 40, time: 1.333 hours, Δt: 2 minutes
Iteration: 60, time: 1.913 hours, Δt: 1.848 minutes
Iteration: 80, time: 2.305 hours, Δt: 1.218 minutes
Iteration: 100, time: 2.607 hours, Δt: 54.755 seconds
Iteration: 120, time: 2.856 hours, Δt: 45.398 seconds
Iteration: 140, time: 3.125 hours, Δt: 49.938 seconds
Iteration: 160, time: 3.404 hours, Δt: 51.217 seconds
Iteration: 180, time: 3.714 hours, Δt: 56.339 seconds
Iteration: 200, time: 4.016 hours, Δt: 55.848 seconds
Iteration: 220, time: 4.319 hours, Δt: 54.602 seconds
Iteration: 240, time: 4.605 hours, Δt: 54.389 seconds
Iteration: 260, time: 4.933 hours, Δt: 59.828 seconds
Iteration: 280, time: 5.225 hours, Δt: 53.983 seconds
Iteration: 300, time: 5.535 hours, Δt: 55.812 seconds
Iteration: 320, time: 5.843 hours, Δt: 57.848 seconds
Iteration: 340, time: 6.165 hours, Δt: 59.527 seconds
Iteration: 360, time: 6.454 hours, Δt: 54.081 seconds
Iteration: 380, time: 6.782 hours, Δt: 59.489 seconds
Iteration: 400, time: 7.123 hours, Δt: 1.058 minutes
Iteration: 420, time: 7.482 hours, Δt: 1.118 minutes
Iteration: 440, time: 7.828 hours, Δt: 1.073 minutes
Iteration: 460, time: 8.173 hours, Δt: 1.039 minutes
Iteration: 480, time: 8.535 hours, Δt: 1.098 minutes
Iteration: 500, time: 8.890 hours, Δt: 1.117 minutes
Iteration: 520, time: 9.226 hours, Δt: 1.042 minutes
Iteration: 540, time: 9.593 hours, Δt: 1.114 minutes
Iteration: 560, time: 9.994 hours, Δt: 1.226 minutes
Iteration: 580, time: 10.423 hours, Δt: 1.348 minutes
Iteration: 600, time: 10.912 hours, Δt: 1.473 minutes
Iteration: 620, time: 11.414 hours, Δt: 1.621 minutes
Iteration: 640, time: 11.921 hours, Δt: 1.527 minutes
Iteration: 660, time: 12.473 hours, Δt: 1.680 minutes
Iteration: 680, time: 12.979 hours, Δt: 1.560 minutes
Iteration: 700, time: 13.530 hours, Δt: 1.686 minutes
Iteration: 720, time: 14.124 hours, Δt: 1.855 minutes
Iteration: 740, time: 14.767 hours, Δt: 2 minutes
Iteration: 760, time: 15.365 hours, Δt: 1.929 minutes
Iteration: 780, time: 16 hours, Δt: 2 minutes
Iteration: 800, time: 16.667 hours, Δt: 2 minutes
Iteration: 820, time: 17.333 hours, Δt: 2 minutes
Iteration: 840, time: 18 hours, Δt: 2 minutes
Iteration: 860, time: 18.667 hours, Δt: 2 minutes
Iteration: 880, time: 19.333 hours, Δt: 2 minutes
Iteration: 900, time: 20 hours, Δt: 2 minutes
Iteration: 920, time: 20.667 hours, Δt: 2 minutes
Iteration: 940, time: 21.333 hours, Δt: 2 minutes
Iteration: 960, time: 22 hours, Δt: 2 minutes
Iteration: 980, time: 22.667 hours, Δt: 2 minutes
Iteration: 1000, time: 23.333 hours, Δt: 2 minutes
Iteration: 1020, time: 1 day, Δt: 2 minutes
[ Info: Simulation is stopping. Model time 1 day has hit or exceeded simulation 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,

using JLD2

file = jldopen(simulation.output_writers[:simple_output].filepath)

iterations = parse.(Int, keys(file["timeseries/t"]))

times = [file["timeseries/t/$iter"] for iter in iterations]

buoyancy_flux_time_series = [buoyancy_flux(0, 0, t, buoyancy_flux_parameters) for t in times]

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

using Oceananigans.Grids: nodes

xw, yw, zw = nodes(model.velocities.w)
xp, yp, zp = nodes(model.tracers.P)

Finally, we animate plankton mixing and blooming,

using Plots

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

w_lim = 0   # the maximum(abs(w)) across the whole timeseries

for (i, iteration) in enumerate(iterations)
    w = file["timeseries/w/$iteration"][:, 1, :]

    global w_lim = maximum([w_lim, maximum(abs.(w))])
end

anim = @animate for (i, iteration) in enumerate(iterations)

    @info "Plotting frame $i from iteration $iteration..."

    t = file["timeseries/t/$iteration"]
    w = file["timeseries/w/$iteration"][:, 1, :]
    P = file["timeseries/plankton/$iteration"][:, 1, :]
    averaged_P = file["timeseries/averaged_plankton/$iteration"][1, 1, :]

    P_min = minimum(P) - 1e-9
    P_max = maximum(P) + 1e-9
    P_lims = (0.95, 1.1)

    w_levels = range(-w_lim, stop=w_lim, length=20)

    P_levels = collect(range(P_lims[1], stop=P_lims[2], length=20))
    P_lims[1] > P_min && pushfirst!(P_levels, P_min)
    P_lims[2] < P_max && push!(P_levels, P_max)

    kwargs = (xlabel="x (m)", ylabel="y (m)", aspectratio=1, linewidth=0, colorbar=true,
              xlims=(0, model.grid.Lx), ylims=(-model.grid.Lz, 0))

    w_contours = contourf(xw, zw, w';
                          color = :balance,
                          levels = w_levels,
                          clims = (-w_lim, w_lim),
                          kwargs...)

    P_contours = contourf(xp, zp, clamp.(P, P_lims[1], P_lims[2])';
                          color = :matter,
                          levels = P_levels,
                          clims = P_lims,
                          kwargs...)

    P_profile = plot(averaged_P, zp,
                     linewidth = 2,
                     label = nothing,
                     xlims = (0.9, 1.3),
                     ylabel = "z (m)",
                     xlabel = "Plankton concentration (μM)")

    flux_plot = plot(times ./ hour, buoyancy_flux_time_series,
                     linewidth = 1,
                     label = "Buoyancy flux time series",
                     color = :black,
                     alpha = 0.4,
                     legend = :topright,
                     xlabel = "Time (hours)",
                     ylabel = "Buoyancy flux (m² s⁻³)",
                     ylims = (0.0, 1.1 * buoyancy_flux_parameters.initial_buoyancy_flux))

    plot!(flux_plot, times[1:i] ./ hour, buoyancy_flux_time_series[1:i],
          color = :steelblue,
          linewidth = 6,
          label = nothing)

    scatter!(flux_plot, times[i:i] / hour, buoyancy_flux_time_series[i:i],
             markershape = :circle,
             color = :steelblue,
             markerstrokewidth = 0,
             markersize = 15,
             label = "Current buoyancy flux")

    layout = Plots.grid(2, 2, widths=(0.7, 0.3))

    w_title = @sprintf("Vertical velocity (m s⁻¹) at %s", prettytime(t))
    P_title = @sprintf("Plankton concentration (μM) at %s", prettytime(t))

    plot(w_contours, flux_plot, P_contours, P_profile,
         title=[w_title "" P_title ""],
         layout=layout, size=(1000, 1000))
end

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