add Field sum reproducer

[juliasloan25]

In CliMA/ClimaLand.jl#970, I encountered that sum(f, field) != sum(f, parent(field)). I'm not sure if this is the expected behavior but it was surprising to me, so I wanted to open a reproducer for it.

@charleskawczynski do you have any insight?

[juliasloan25]

The behavior of sum is described somewhat in this thread, but it would be good to have more complete documentation of the expected behavior

[charleskawczynski]

Yes, I hope I can shine some light on this. Our sum by default uses a weighting by the jacobian. From ClimaCore:

function Base.sum(
    field::Union{Field, Base.Broadcast.Broadcasted{<:FieldStyle}},
    dev::ClimaComms.CUDADevice,
)
    context = ClimaComms.context(axes(field))
    localsum = mapreduce_cuda(identity, +, field, weighting = true)
    ClimaComms.allreduce!(context, parent(localsum), +)
    call_post_op_callback() && post_op_callback(localsum[], field, dev)
    return localsum[]
end

Basically, it's an integral

[charleskawczynski]

That said, there is likely still inconsistencies with how we handle sum when slicing data, as mentioned in #943.

I think we need to assemble a list of requirements / expectations, and perhaps define integral in ClimaCore, and document expected behavior in different situations.