# 2D forced-dissipative turbulence

A simulation of forced-dissipative two-dimensional turbulence. We solve the two-dimensional vorticity equation with stochastic excitation and dissipation in the form of linear drag and hyperviscosity.

## Install dependencies

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

```
using Pkg
pkg"add GeophysicalFlows, CUDA, CairoMakie"
```

## Let's begin

Let's load `GeophysicalFlows.jl`

and some other packages we need.

```
using GeophysicalFlows, CUDA, Random, Printf, CairoMakie
parsevalsum = FourierFlows.parsevalsum
record = CairoMakie.record # disambiguate between CairoMakie.record and CUDA.record
```

## Choosing a device: CPU or GPU

`dev = CPU() # Device (CPU/GPU)`

## Numerical, domain, and simulation parameters

First, we pick some numerical and physical parameters for our model.

```
n, L = 256, 2π # grid resolution and domain length
ν, nν = 2e-7, 2 # hyperviscosity coefficient and hyperviscosity order
μ, nμ = 1e-1, 0 # linear drag coefficient
dt = 0.005 # timestep
nsteps = 4000 # total number of steps
nsubs = 20 # number of steps between each plot
```

## Forcing

We force the vorticity equation with stochastic excitation that is delta-correlated in time and while spatially homogeneously and isotropically correlated. The forcing has a spectrum with power in a ring in wavenumber space of radius $k_f$ (`forcing_wavenumber`

) and width $δ_f$ (`forcing_bandwidth`

), and it injects energy per unit area and per unit time equal to $\varepsilon$. That is, the forcing covariance spectrum is proportional to $\exp{[-(|\bm{k}| - k_f)^2 / (2 δ_f^2)]}$.

```
forcing_wavenumber = 14.0 * 2π/L # the forcing wavenumber, `k_f`, for a spectrum that is a ring in wavenumber space
forcing_bandwidth = 1.5 * 2π/L # the width of the forcing spectrum, `δ_f`
ε = 0.1 # energy input rate by the forcing
grid = TwoDGrid(dev; nx=n, Lx=L)
K = @. sqrt(grid.Krsq) # a 2D array with the total wavenumber
forcing_spectrum = @. exp(-(K - forcing_wavenumber)^2 / (2 * forcing_bandwidth^2))
@CUDA.allowscalar forcing_spectrum[grid.Krsq .== 0] .= 0 # ensure forcing has zero domain-average
ε0 = parsevalsum(forcing_spectrum .* grid.invKrsq / 2, grid) / (grid.Lx * grid.Ly)
@. forcing_spectrum *= ε/ε0 # normalize forcing to inject energy at rate ε
```

We reset of the random number generator for reproducibility

`if dev==CPU(); Random.seed!(1234); else; CUDA.seed!(1234); end`

Next we construct function `calcF!`

that computes a forcing realization every timestep. For that, we call `randn!`

to obtain complex numbers whose real and imaginary part are normally-distributed with zero mean and variance 1/2.

```
function calcF!(Fh, sol, t, clock, vars, params, grid)
randn!(Fh)
@. Fh *= sqrt(forcing_spectrum) / sqrt(clock.dt)
return nothing
end
```

## Problem setup

We initialize a `Problem`

by providing a set of keyword arguments. The `stepper`

keyword defines the time-stepper to be used.

```
prob = TwoDNavierStokes.Problem(dev; nx=n, Lx=L, ν, nν, μ, nμ, dt, stepper="ETDRK4",
calcF=calcF!, stochastic=true)
```

Define some shortcuts for convenience.

```
sol, clock, vars, params, grid = prob.sol, prob.clock, prob.vars, prob.params, prob.grid
x, y = grid.x, grid.y
```

First let's see how a forcing realization looks like. Function `calcF!()`

computes the forcing in Fourier space and saves it into variable `vars.Fh`

, so we first need to go back to physical space.

Note that when plotting, we decorate the variable to be plotted with `Array()`

to make sure it is brought back on the CPU when the variable lives on the GPU.

```
calcF!(vars.Fh, sol, 0.0, clock, vars, params, grid)
fig = Figure()
ax = Axis(fig[1, 1],
xlabel = "x",
ylabel = "y",
aspect = 1,
title = "a forcing realization",
limits = ((-L/2, L/2), (-L/2, L/2)))
heatmap!(ax, x, y, Array(irfft(vars.Fh, grid.nx));
colormap = :balance, colorrange = (-200, 200))
fig
```

## Setting initial conditions

Our initial condition is a fluid at rest.

`TwoDNavierStokes.set_ζ!(prob, device_array(dev)(zeros(grid.nx, grid.ny)))`

## Diagnostics

Create Diagnostics; the diagnostics are aimed to probe the energy budget.

```
E = Diagnostic(TwoDNavierStokes.energy, prob; nsteps) # energy
Z = Diagnostic(TwoDNavierStokes.enstrophy, prob; nsteps) # enstrophy
diags = [E, Z] # a list of Diagnostics passed to `stepforward!` will be updated every timestep.
```

## Visualizing the simulation

We initialize a plot with the vorticity field and the time-series of energy and enstrophy diagnostics. To plot energy and enstrophy on the same axes we scale enstrophy with $k_f^2$.

```
ζ = Observable(Array(vars.ζ))
title_ζ = Observable("vorticity, μ t=" * @sprintf("%.2f", μ * clock.t))
energy = Observable(Point2f[(μ * E.t[1], E.data[1])])
enstrophy = Observable(Point2f[(μ * Z.t[1], Z.data[1] / forcing_wavenumber^2)])
fig = Figure(resolution = (800, 360))
axζ = Axis(fig[1, 1];
xlabel = "x",
ylabel = "y",
title = title_ζ,
aspect = 1,
limits = ((-L/2, L/2), (-L/2, L/2)))
ax2 = Axis(fig[1, 2],
xlabel = "μ t",
limits = ((0, 1.1 * μ * nsteps * dt), (0, 0.55)))
heatmap!(axζ, x, y, ζ;
colormap = :balance, colorrange = (-40, 40))
hE = lines!(ax2, energy; linewidth = 3)
hZ = lines!(ax2, enstrophy; linewidth = 3, color = :red)
Legend(fig[1, 3], [hE, hZ], ["energy E(t)", "enstrophy Z(t) / k_f²"])
fig
```

## Time-stepping the `Problem`

forward

We step the `Problem`

forward in time.

```
startwalltime = time()
record(fig, "twodturb_forced.mp4", 0:round(Int, nsteps / nsubs), framerate = 18) do j
if j % (1000/nsubs) == 0
cfl = clock.dt * maximum([maximum(vars.u) / grid.dx, maximum(vars.v) / grid.dy])
log = @sprintf("step: %04d, t: %d, cfl: %.2f, E: %.4f, Z: %.4f, walltime: %.2f min",
clock.step, clock.t, cfl, E.data[E.i], Z.data[Z.i], (time()-startwalltime)/60)
println(log)
end
ζ[] = vars.ζ
energy[] = push!(energy[], Point2f(μ * E.t[E.i], E.data[E.i]))
enstrophy[] = push!(enstrophy[], Point2f(μ * Z.t[E.i], Z.data[Z.i] / forcing_wavenumber^2))
title_ζ[] = "vorticity, μ t=" * @sprintf("%.2f", μ * clock.t)
stepforward!(prob, diags, nsubs)
TwoDNavierStokes.updatevars!(prob)
end
```

```
step: 0000, t: 0, cfl: 0.00, E: 0.0000, Z: 0.0000, walltime: 0.00 min
step: 1000, t: 5, cfl: 0.43, E: 0.2881, Z: 34.9014, walltime: 0.13 min
step: 2000, t: 10, cfl: 0.53, E: 0.3803, Z: 33.7438, walltime: 0.23 min
step: 3000, t: 15, cfl: 0.48, E: 0.4326, Z: 36.7366, walltime: 0.32 min
step: 4000, t: 20, cfl: 0.53, E: 0.4520, Z: 34.9454, walltime: 0.42 min
```

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