Inverse Laplace ∇⁻²! via kernel in ∇²!

[milankl]

fixes #141

[maximilian-gelbrecht]

Looks good to me!

[milankl]

I just realised that we shouldn't replace eigenvalues⁻¹ by 1/eigenvalues as that makes the integration constant infinity. We currently don't test for this (and it's not used by the dynamical core atm) hence the tests still pass

julia> alms = LowerTriangularMatrix(randn(ComplexF32,33,32));
julia> alms[:,1] .= real.(alms[:,1]);
julia> ∇⁻²alms = copy(alms)
julia> S = SpectralTransform(alms);
julia> SpeedyWeather.∇²!(∇⁻²alms,alms,S,inverse=true)
33×32 LowerTriangularMatrix{ComplexF32}:
   -Inf+NaN*im                0.0+0.0im          …          0.0+0.0im
    -0.369622-0.0im      0.199208-0.408254im                0.0+0.0im
...

[maximilian-gelbrecht]

I am on vacation for a week. I can look into it when I am back, if you don’t do it until then.

[milankl]

Don't worry, I'll resolve that, just don't merge it yet. Enjoy your vacation 🌴

[milankl]

Also @maximilian-gelbrecht, a little change I made in ∇²! and ∇⁻²! is that previously it excluded the the last row = highest degree l. This is because scalar quantities have lmax==mmax, but vector quantities use one more degree due to the recursion in the meridional gradient. But all LowerTriangularMatrices are of size lmax+1,mmax (as if they were all vector quantities). I changed ∇²! and ∇⁻²! to include all degrees and orders, to have a generally valid (inverse) Laplace operator (the higest degree l otherwise wouldn't be touched, and values may carry over which I don't like). I believe this does not have any unforeseen consequences but in case it does, let me know.

[maximilian-gelbrecht]

Great, thanks @milankl