Missing (-1)^{n+1} factor from hyperdiffusion term

[navidcy]

I believe that the hyperdiffusion term is missing a $(-1)^{n+1}$ factor? For example, for $n=1$ it's plain-old viscosity so it should be $+\nu\nabla^2\zeta$ but for $n=2$ it should be $-\nu\nabla^4\zeta$, otherwise the hyperdiffusion is not dissipative.

Am I right? If so, perhaps this mistake propagates elsewhere in the docs?

closes #286

[navidcy]

Seems like in the docs where the the eigenvalues of the Laplacian/hyperLaplacian are written down, the $(-1)^{n+1}$ factor has been taken into account.

(Btw, docs report the eigenvalues of $-\nabla^2$, right?)

[milankl]

This is a good point, I missed that. And it's not just in the documentation we also need to address that here

SpeedyWeather.jl/src/dynamics/define_diffusion.jl

Line 78 in 4b5f106

∇²ⁿ[l+1] = norm_eigenvalue^power/(3600*time_scale)*radius

Because theoretically one should be able to choose HyperDiffusion(power=3) but as it's written now it should blow up if I'm not wrong.

Update: No changes to the code needed as we normalise by the largest eigenvalue (which changes sign depending on the power!) anyway, so that the dissipative nature is retained for power=odd and even.

[milankl]

@navidcy I've changed the documentation to (hopefully) be clearer and consistent with the sign of diffusion and also adapted the dynamics to actually match the documentation. Feel free to review!

[navidcy]

nice!

I can't approve because I opened the PR ;)

[milankl]

Let me quickly check that I can reproduce the figures in #286 with the signs changed in the dynamical core