Cubed sphere tracer advection script

[glwagner]

Some TODOs:

• Make set! work for Face, Face, Center fields (eg streamfunctions)
• Make broadcasting work for cubed sphere fields
• Use plotting utilities from Imaginocean

With the above and fixing the tests then I think we are ready to merge.

Supersedes #3221; closes #3204

[glwagner]

multi_region_tracer_advection_latlon.mp4

[glwagner]

However, it doesn't work when the tracer passes over a corner:

multi_region_tracer_advection_latlon.mp4

[glwagner]

(Note that the two videos above are with tracer_advection = WENO().)

[glwagner]

It might've just been a CFL issue (here at coarser resolution, with a time-step estimated based on the minimum grid spacing):

multi_region_tracer_advection_latlon.mp4

[glwagner]

Now with WENO(order=9)

multi_region_tracer_advection_latlon.mp4

[navidcy]

Let's leave the set! for another PR because there is still work to be done, in particular to fill halos for Face, Face fields...

[navidcy]

There seems to be some artifacty things introduced by the corners. See the below which starts with a tracer:

θ₀ = 1
Δφ = 20
θᵢ(λ, φ, z) = - θ₀ * cosd(4λ) * exp(-φ^2 / 2Δφ^2)

advected by streamfunction that corresponds to solid-body rotation about axis that passes via (lat, lon) = (0, 0).

φʳ = 0        # Latitude pierced by the axis of rotation
α  = 90 - φʳ  # Angle between axis of rotation and north pole (degrees)
ψᵣ(λ, φ, z) = - U * R * (sind(φ) * cosd(α) - cosd(λ) * cosd(φ) * sind(α))

cubed_sphere_tracer_advection.mp4

[simone-silvestri]

I think we need stretched WENO with 2D coefficients. Now WENO(grid) assumes 1D coefficients which is okay for latitude-longitude and rectilinear but not for cubed sphere.

It will be straightforward to update WENO once everything is in place

[navidcy]

I think we need stretched WENO with 2D coefficients. Now WENO(grid) assumes 1D coefficients which is okay for latitude-longitude and rectilinear but not for cubed sphere.

It will be straightforward to update WENO once everything is in place

Good point. So if I fall back to the default scheme for tracers then it should be OK?

[simone-silvestri]

A centered second order advection should be okay but you have to pair it with some diffusion otherwise oscillatory errors will break your solution

[navidcy]

A centered second order advection should be okay but you have to pair it with some diffusion otherwise oscillatory errors will break your solution

True! At least we know what's going on. Thanks @simone-silvestri!

Here it is without any diffusion:

cubed_sphere_tracer_advection.mp4

And here it is with ScalarDiffusivity(; κ=1e-2):

cubed_sphere_tracer_advection.mp4

You can still "see" the corner.

[navidcy]

Some remarks:

• Make set! work for Face, Face, Center fields (eg streamfunctions)

It seems like there is a topology problem. For Face, Face, Center fields it seems that there are two corner points which do not correspond to any point in the interior. So even if set! fills everything in the interior (e.g., for 1:Nx) then fill_halo_regions! won't be able to assign values on those two special corner points. Is the FullyConnected topology a bit problematic when panels are rotated to one another? Or are we not thinking something straight? We were discussing this over zoom w @glwagner, @simone-silvestri, and @siddharthabishnu and decided that it belongs to a new PR.

• Make broadcasting work for cubed sphere fields

We also decided to leave making broadcasting work for MultiRegionFields part of another PR.

So, with this and that, we should merge this PR at its current state.