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

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

where $\boldsymbol{u}$ 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"

The grid

We use a two-dimensional grid with 64² points and 1 m grid spacing and assign Flat to the y direction:

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

grid = RegularRectilinearGrid(size=(64, 64), extent=(64, 64), topology=(Periodic, Flat, Bounded))
RegularRectilinearGrid{Float64, Periodic, Flat, Bounded}
                   domain: x ∈ [0.0, 64.0], y ∈ [0.0, 0.0], z ∈ [-64.0, 0.0]
                 topology: (Periodic, Flat, Bounded)
  resolution (Nx, Ny, Nz): (64, 1, 64)
   halo size (Hx, Hy, Hz): (1, 0, 1)
grid spacing (Δx, Δy, Δz): (1.0, 0.0, 1.0)

Boundary conditions

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

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 = FluxBoundaryCondition(buoyancy_flux, parameters = buoyancy_flux_parameters)
BoundaryCondition: type=Flux, condition=buoyancy_flux(x, y, t, p) in Main.ex-convecting_plankton at none:1

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 = GradientBoundaryCondition(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{Nothing,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.

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: RegularRectilinearGrid{Float64, Periodic, Flat, Bounded}(Nx=64, Ny=1, Nz=64)
├── tracers: (:b, :P)
├── closure: IsotropicDiffusivity{Oceananigans.TurbulenceClosures.ExplicitTimeDiscretization,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,typeof(Oceananigans.Utils.cell_advection_timescale),typeof(Oceananigans.Simulations.infinite_diffusion_timescale)}(1.0, Inf, 1.1, 0.5, 120.0, 0.0, 120.0, Oceananigans.Utils.cell_advection_timescale, Oceananigans.Simulations.infinite_diffusion_timescale)

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=24hours,
                        iteration_interval=20, progress=progress)
Simulation{IncompressibleModel{CPU, Float64}}
├── Model clock: time = 0 seconds, iteration = 0 
├── Next time step (TimeStepWizard{Float64,typeof(Oceananigans.Utils.cell_advection_timescale),typeof(Oceananigans.Simulations.infinite_diffusion_timescale)}): 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,

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: 2 hours, Δt: 2 minutes
Iteration: 80, time: 2.393 hours, Δt: 1.191 minutes
Iteration: 100, time: 2.786 hours, Δt: 1.189 minutes
Iteration: 120, time: 3.154 hours, Δt: 1.152 minutes
Iteration: 140, time: 3.518 hours, Δt: 1.110 minutes
Iteration: 160, time: 3.828 hours, Δt: 57.956 seconds
Iteration: 180, time: 4.155 hours, Δt: 1.036 minutes
Iteration: 200, time: 4.452 hours, Δt: 53.381 seconds
Iteration: 220, time: 4.748 hours, Δt: 58.322 seconds
Iteration: 240, time: 5.046 hours, Δt: 55.428 seconds
Iteration: 260, time: 5.384 hours, Δt: 1.016 minutes
Iteration: 280, time: 5.682 hours, Δt: 55.711 seconds
Iteration: 300, time: 6.017 hours, Δt: 1.021 minutes
Iteration: 320, time: 6.369 hours, Δt: 1.066 minutes
Iteration: 340, time: 6.743 hours, Δt: 1.150 minutes
Iteration: 360, time: 7.091 hours, Δt: 1.095 minutes
Iteration: 380, time: 7.418 hours, Δt: 1.019 minutes
Iteration: 400, time: 7.779 hours, Δt: 1.121 minutes
Iteration: 420, time: 8.185 hours, Δt: 1.233 minutes
Iteration: 440, time: 8.627 hours, Δt: 1.356 minutes
Iteration: 460, time: 9.099 hours, Δt: 1.486 minutes
Iteration: 480, time: 9.497 hours, Δt: 1.228 minutes
Iteration: 500, time: 9.937 hours, Δt: 1.350 minutes
Iteration: 520, time: 10.381 hours, Δt: 1.419 minutes
Iteration: 540, time: 10.826 hours, Δt: 1.362 minutes
Iteration: 560, time: 11.325 hours, Δt: 1.498 minutes
Iteration: 580, time: 11.823 hours, Δt: 1.565 minutes
Iteration: 600, time: 12.284 hours, Δt: 1.421 minutes
Iteration: 620, time: 12.797 hours, Δt: 1.563 minutes
Iteration: 640, time: 13.333 hours, Δt: 1.719 minutes
Iteration: 660, time: 13.919 hours, Δt: 1.891 minutes
Iteration: 680, time: 14.478 hours, Δt: 1.732 minutes
Iteration: 700, time: 15.095 hours, Δt: 1.906 minutes
Iteration: 720, time: 15.733 hours, Δt: 2 minutes
Iteration: 740, time: 16.400 hours, Δt: 2 minutes
Iteration: 760, time: 17.067 hours, Δt: 2 minutes
Iteration: 780, time: 17.733 hours, Δt: 2 minutes
Iteration: 800, time: 18.400 hours, Δt: 2 minutes
Iteration: 820, time: 19.067 hours, Δt: 2 minutes
Iteration: 840, time: 19.733 hours, Δt: 2 minutes
Iteration: 860, time: 20.400 hours, Δt: 2 minutes
Iteration: 880, time: 21.067 hours, Δt: 2 minutes
Iteration: 900, time: 21.733 hours, Δt: 2 minutes
Iteration: 920, time: 22.400 hours, Δt: 2 minutes
Iteration: 940, time: 23.067 hours, Δt: 2 minutes
Iteration: 960, time: 23.733 hours, Δt: 2 minutes
Iteration: 968, 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,

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=(1000.5, 1000.5))
end

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