Improve implementation of imposed `U(y)` in `SingleLayerQG`

[navidcy]

#360 implemented the ability of U(y) in SingleLayerQG module.

• If imposed zonal flow is constant ($U_0$) then term $- i k_x U_0$ is included in the L operator & the $U_0 \partial_x \eta$ term in N.
• If imposed zonal flow varies with y ($U(y)$) then all extra advection terms are added in N.

A potential improvement is that we should take any prescribed imposed zonal flow $U(y)$ and decompose it into its y-mean and deviation from that, e.g.,

$$ U(y) = U_0 + u(y)$$

with $\int u(y) \mathrm{d} y = 0 $. Then we add the $U_0$ via L (as 1 above) and the rest via N (as 2 above).

cc @mncrowe

[mncrowe]

I did consider it as an option as you can just define $U_0$ as the zero Fourier mode of $U$. I started implementing it but in the end used the current approach as it was simpler to write down the Nonlinear term this way.

If we're going to do this, it might be worth implementing @glwagner's suggestion of including support for U as a function of $y$ and $t$ at the same time.

[navidcy]

If we're going to do this, it might be worth implementing @glwagner's suggestion of including support for U as a function of y and t at the same time.

Good point.

[mncrowe]

I'm going to have a look at doing this and then ensure consistency with the SQG case by implementing there too. Our options for U are:

• constant U, added to L with one additional nonlinear term,
• array U = U(y), subtract off average and add to L, rest in nonlinear term,
• function U = U(y, t), all in nonlinear term, evaluated at each step so may be slow
• function U = U(y), calculate array straight away and proceed as with array case?

The first three are easy enough to separate, is there a good way to use multiple dispatch to differentiate between a function of 1 variable and a function of 2 variables? And is it even worth bothering with the fourth case when we can just tell people to use an array instead? Thoughts?