Add finite depth parameter to SQG model (surfaceqg.jl)

[mncrowe]

I suggest adding a new depth parameter, say H, which can either be $\infty$, giving the current model, or finite.

In the finite case, the only change is that the Fourier space operator linking b and ψ changes from

$$ \hat{b} = |\textbf{k}|\hat{\psi}, $$

to

$$ \hat{b} = |\textbf{k}| \tanh(|\textbf{k}| H)\hat{\psi}. $$

From looking at the code, I think this just means changing the nonlinear function calcN_advection!, though some of the diagnostics might also need minor changes depending on how they're calculated.

Happy to help implement this.

[glwagner]

Nice! It might make sense to build this feature as an option to precompute this function and store it in an array. That would be more flexible and also may save computational cost since tanh are relatively expensive.

[mncrowe]

I've made a working version of this here. A few notes:

• This probably isn't how you want to handle types in Params. I'm guessing the way to go is similar to SingleLayerQG and have different structures for the finite and infinite parameter cases. I've also fudged the array type slightly in a way that isn't consistent with what you've got. I can sort this out if you let me how how you want to proceed with this.
• I haven't implemented any testing or examples currently.
• There doesn't seem to be any speed difference between the finite and infinite H cases with this implementation.
• No pull request currently, will do so once I'm a bit closer to a final version.

[navidcy]

Cool! It's much easier to see the differences you suggested once you open a PR.

But yes, the if-else statement

https://github.com/mncrowe/GeophysicalFlows.jl/blob/22e2bd4aad11647360ae02d79966bb2caee9aefe/src/surfaceqg.jl#L373-L377

should be replaced with multiple dispatch on the type of Params.

[mncrowe]

PR here.

[navidcy]

Can we combine the two definitions (finite and infinite depth) given that:

julia> tanh(Inf)
1.0

?

What if there is a depth parameter with default value Inf?

[mncrowe]

Can we combine the two definitions (finite and infinite depth) given that:

julia> tanh(Inf)
1.0
?

Yes, I think I've tried this and the only issue is that for the k = 0 mode, you get tanh(0 * Inf) so you have to set that mode manually.

What if there is a depth parameter with default value Inf?

My thinking was that you should only save the values of DirNeu if it was slow to calculate, as Greg hinted might be the case for tanh. Hence the two cases being separated. If you think this isn't necessary, we can revert and either pre-compute or calculate DirNeu? Though the option to pre-save might be useful?

[navidcy]

Fair point about tanh being slow; let's keep it as is.