Simplifying notation for difference and interpolation operators

[tomchor]

Now that most operators are 3D we could simplify the nomenclature for the difference and interpolation operators by getting rid of the ᵃ, no? I think it would make code more legible.

For example something like this (from Oceanostics)

@inline function turbulent_kinetic_energy_ccc(i, j, k, grid, u, v, w, U, V, W)
    return (ℑxᶜᵃᵃ(i, j, k, grid, ψ′², u, U) +
            ℑyᵃᶜᵃ(i, j, k, grid, ψ′², v, V) +
            ℑzᵃᵃᶜ(i, j, k, grid, ψ′², w, W)) / 2
end

could become this, which I think is way easier on the eyes:

@inline function turbulent_kinetic_energy_ccc(i, j, k, grid, u, v, w, U, V, W)
    return (ℑxᶜ(i, j, k, grid, ψ′², u, U) +
            ℑyᶜ(i, j, k, grid, ψ′², v, V) +
            ℑzᶜ(i, j, k, grid, ψ′², w, W)) / 2
end

Double interpolators could go from ℑxyᶠᶜᵃ to ℑxyᶠᶜ or ℑxᶠyᶜ.

Thoughts?

This was originally suggested by @simone-silvestri in #2214

[tomchor]

Just saw that @simone-silvestri mentioned that in #2214 (comment), my bad. I'll modify the issue to reflect that

[glwagner]

I prefer the three letter notation because it's more explicit. I'm worried that mistakes might be easier to make, since the current format has a redundancy that in my experience often catches hard to find bugs when coding and formatting long expressions. I guess I also feel it helps me "visualize" terms but maybe that's just me.

Here's an example of such an expression:

Oceananigans.jl/src/TurbulenceClosures/turbulence_closure_implementations/anisotropic_minimum_dissipation.jl

Lines 262 to 301 in b7871d3

	@inline function norm_uᵢₐ_uⱼₐ_Σᵢⱼᶜᶜᶜ(i, j, k, grid, closure, u, v, w)
	ijk = (i, j, k, grid)
	uvw = (u, v, w)
	ijkuvw = (i, j, k, grid, u, v, w)
	uᵢ₁_uⱼ₁_Σ₁ⱼ = (
	norm_Σ₁₁(ijkuvw...) * norm_∂x_u(ijk..., u)^2
	+ norm_Σ₂₂(ijkuvw...) * ℑxyᶜᶜᵃ(ijk..., norm_∂x_v², uvw...)
	+ norm_Σ₃₃(ijkuvw...) * ℑxzᶜᵃᶜ(ijk..., norm_∂x_w², uvw...)
	+ 2 * norm_∂x_u(ijkuvw...) * ℑxyᶜᶜᵃ(ijk..., norm_∂x_v_Σ₁₂, uvw...)
	+ 2 * norm_∂x_u(ijkuvw...) * ℑxzᶜᵃᶜ(ijk..., norm_∂x_w_Σ₁₃, uvw...)
	+ 2 * ℑxyᶜᶜᵃ(ijk..., norm_∂x_v, uvw...) * ℑxzᶜᵃᶜ(ijk..., norm_∂x_w, uvw...)
	* ℑyzᵃᶜᶜ(ijk..., norm_Σ₂₃, uvw...)
	)
	uᵢ₂_uⱼ₂_Σ₂ⱼ = (
	+ norm_Σ₁₁(ijkuvw...) * ℑxyᶜᶜᵃ(ijk..., norm_∂y_u², uvw...)
	+ norm_Σ₂₂(ijkuvw...) * norm_∂y_v(ijk..., v)^2
	+ norm_Σ₃₃(ijkuvw...) * ℑyzᵃᶜᶜ(ijk..., norm_∂y_w², uvw...)
	+ 2 * norm_∂y_v(ijkuvw...) * ℑxyᶜᶜᵃ(ijk..., norm_∂y_u_Σ₁₂, uvw...)
	+ 2 * ℑxyᶜᶜᵃ(ijk..., norm_∂y_u, uvw...) * ℑyzᵃᶜᶜ(ijk..., norm_∂y_w, uvw...)
	* ℑxzᶜᵃᶜ(ijk..., norm_Σ₁₃, uvw...)
	+ 2 * norm_∂y_v(ijkuvw...) * ℑyzᵃᶜᶜ(ijk..., norm_∂y_w_Σ₂₃, uvw...)
	)
	uᵢ₃_uⱼ₃_Σ₃ⱼ = (
	+ norm_Σ₁₁(ijkuvw...) * ℑxzᶜᵃᶜ(ijk..., norm_∂z_u², uvw...)
	+ norm_Σ₂₂(ijkuvw...) * ℑyzᵃᶜᶜ(ijk..., norm_∂z_v², uvw...)
	+ norm_Σ₃₃(ijkuvw...) * norm_∂z_w(ijk..., w)^2
	+ 2 * ℑxzᶜᵃᶜ(ijk..., norm_∂z_u, uvw...) * ℑyzᵃᶜᶜ(ijk..., norm_∂z_v, uvw...)
	* ℑxyᶜᶜᵃ(ijk..., norm_Σ₁₂, uvw...)
	+ 2 * norm_∂z_w(ijkuvw...) * ℑxzᶜᵃᶜ(ijk..., norm_∂z_u_Σ₁₃, uvw...)
	+ 2 * norm_∂z_w(ijkuvw...) * ℑyzᵃᶜᶜ(ijk..., norm_∂z_v_Σ₂₃, uvw...)
	)
	return uᵢ₁_uⱼ₁_Σ₁ⱼ + uᵢ₂_uⱼ₂_Σ₂ⱼ + uᵢ₃_uⱼ₃_Σ₃ⱼ
	end

I agree though that the "minimal" notation is more readable for short expressions like the example above.

[francispoulin]

Even though it takes more effort, because there are three directions, I also prefer having three letters. It seems more explicit, even thought it mght be more information than is necessarily needed.

[glwagner]

@tomchor do you mind if I reopen this? I think it's worth discussing more!

[tomchor]

Not at all

[simone-silvestri]

I am in favour of removing the indices, but not necessarily too much against keeping them either. I can see reasons for both. Since these functions are not user-facing (and would be understandable either way), I guess the best choice is to do what makes the most developers happy.

[glwagner]

I'm closing this issue because I'm judging that it's not of current, timely relevance to Oceananigans development. If you would like to make it a higher priority or if you think the issue was closed in error please feel free to re-open.