Rossby-Haurwitz wave initial conditions +docs

[milankl]

Tobias Ehlert @rivwgk had implemented the Rossby-Haurwitz wave as initial conditions for SpeedyWeather, see Williamson et al. 1992, i.e. initial conditions for vorticity are (eq. 145)

With our set! function this is now some easy three lines of code that take care of the transform between grid and spectral space for you

using SpeedyWeather
spectral_grid = SpectralGrid(trunc=127, nlayers=1)
model = BarotropicModel(;spectral_grid)
simulation = initialize!(model)

# Rossby-Haurwitz wave initial conditions for vorticity
R, K, ω = 4, 7.848e-6, 7.848e-6
ζ(λ, θ, σ) = 2ω*sind(θ) - K*sind(θ)*cosd(θ)^R*(R^2 + 3R + 2)*cosd(R*λ)
set!(simulation, vor=ζ)

So only differences to the mathematical version from above is that we use degrees (sind, cosd) as coordinates, and σ is our vertical coordinate that needs to be specified for compatibility with our 3D primitive equations models (yes, it's exactly the same syntax in PrimitiveDryModel and PrimitiveWetModel!). Looks like

@rivwgk can you provide a reference for the cut-off amplitude? It's not discussed in Williamson et al., 1992. You call it $$\sigma$$ but we use that for the vertical coordinate, so it's c here. Anyway, you can do that as easy as

# filter low values below cutoff amplitude c
vor = simulation.prognostic_variables.vor[1]  # 1 = first leapfrog timestep
low_values = abs.(vor) .< c
vor[low_values] .= 0

no need for the loop you wrote, very similar to numpy if you ask me ;)

This pull request

• introduces and exports RossbyHaurwitzWave <: AbstractInitialConditions to have these initial conditions predefined
• adds a section on "Initial conditions" to the documentation outlining how to set the initial conditions using the Rossby-Haurwitz wave as an example

@clessig you wanted to be kept updated ☝🏼

[rivwgk]

Sadly, I don't have a reference for the cut-off amplitude and don't exactly remember why I added it. I think it might be that I added it because at first I had the Rossby-Haurwitz wave given by three spherical harmonics coefficients for $Y_1^0$, $Y_5^4$ and $Y_5^{-4}$ and wanted to cut other numerical noise off?

[milankl]

Sadly, I don't have a reference for the cut-off amplitude and don't exactly remember why I added it. I think it might be that I added it because at first I had the Rossby-Haurwitz wave given by three spherical harmonics coefficients for Y 1 0 , Y 5 4 and Y 5 − 4 and wanted to cut other numerical noise off?

Well it acts now as an example even if it's just you who came up with it ;) I hope that's okay. If you want to make any changes feel free to create a pull request, otherwise you find the Rossby-Haurwitz wave test case now in the documentation:

https://speedyweather.github.io/SpeedyWeather.jl/dev/initial_conditions/#Rossby-Haurwitz-wave-in-a-BarotropicModel