Closing Momentum Budget using WENO advection scheme

[weigwu]

Hi all,

I am trying to close the momentum/divergence/vorticity budgets using nonhydrostatic model outputs. The model uses a WENO fifth-order scheme for advection. I have been using the following approach to calculate the advection terms for momentum. However, I suspect it may be an approximation of the "true" momentum flux. Is there a way to obtain the advection terms that align with WENO scheme?

Many thanks,
Roger

u, v, w = simulation.model.velocities

uu_x=@at((Center, Center, Center), u*∂x(u)),
vu_y=@at((Center, Center, Center), v*∂y(u)),
wu_z=@at((Center, Center, Center), w*∂z(u)),

[glwagner]

Your suspicions are correct!

The tendency kernel functions are here:

https://github.com/CliMA/Oceananigans.jl/blob/cc51ff0ee0a1326cebcbfa207ec0ed29ae768648/src/Models/NonhydrostaticModels/nonhydrostatic_tendency_kernel_functions.jl

for example advection is

Oceananigans.jl/src/Models/NonhydrostaticModels/nonhydrostatic_tendency_kernel_functions.jl

Line 71 in cc51ff0

return ( - div_𝐯u(i, j, k, grid, advection, total_velocities, velocities.u)

which you can compute via KernelFunctionOperation by writing

using Oceananigans.Advection: div_𝐯u

advection = model.advection
u, v, w = velocities = model.velocities
uu_x = KernelFunctionOperation{Face, Center, Center}(div_𝐯u, grid, advection, velocities, u)

the other components are

using Oceananigans.Advection: div_𝐯v, div_𝐯w
vu_y = KernelFunctionOperation{Center, Face, Center}(div_𝐯v, grid, advection, velocities, v)
wu_z = KernelFunctionOperation{Center, Center, Face}(div_𝐯w, grid, advection, velocities, w)

that will work provided your model doesn't have background velocity fields. If it does, you need to use total_velocities as in the kernel function.

Once you have the KernelFunctionOperations you can pass them directly to output writers, or wrap them in Field to compute somewhere else.

Note that these advection flux divergences are located at the same place as the variables to which they refer (not at Center, Center, Center, unless it is the tracer advective flux divergence).

[zhihua-zheng]

Thank you! That's helpful! But ∂x(U*u) is not the same as U*∂x(u), right? What if one wants to calculate the latter?

[glwagner]

You're free to compute U*∂x(u) of course! The question posed was how to calculate momentum advection using the "native" advection scheme, ie the same numerical scheme that's used to integrate the momentum equation. With NonhydrostaticModel we're always using the "flux form", ie something like ∂x(U*u), with appropriate reconstructions of U and u depending on the scheme. (Note that the HydrostaticFreeSurfaceModel also supports a "vector invariant" scheme, but this still does not use the advective form U*∂x(u).)

[glwagner]

So just to complete the thought --- while one could try to come up with a scheme to compute U*∂x(u) based on WENO reconstruction, that scheme wouldn't be the one used natively by the model.

Are you interested in using (some kind of) WENO reconstruction to compute U*∂x(u) even if NonhydrostaticModel doesn't use such a scheme to integrate the momentum equation? (If so, why?)

[zhihua-zheng]

I see, the native flux form is what I should use to close the budget, but each component of the flux divergence is not exactly the advection in each direction, which makes it a bit unclear when I want to compare individual advection terms.

Since diagnosing terms in advective form would involve a scheme that was not used by the model anyway, I guess that may be unnecessary.

[glwagner]

Yes, that's true. Though terms like u*u are indeed the "momentum flux", and there are three of them in the x-momentum equation: u*u, v*u, and w*u. So you can indeed decompose the flux divergence into three contributions if that is what you would like to do...