Complete Coriolis force in spherical coordinates

[francispoulin]

In Cartesian coordinates, we offer both the traditional and non-traditional Coriolis forces.

In spherical coordinates, we seem to offer only the traditional Coriolis forces. Is this true? This is based on looking at this file that only involves the sin and not the cos of the latitude.

https://github.com/CliMA/Oceananigans.jl/blob/main/src/Coriolis/hydrostatic_spherical_coriolis.jl

To include the non-traditional component, we need to add another force to the zonal momentum equation, which is easy, but we then need to impose quasi-hydrostatic balance. The MITgcm does this. as is talked about here.

Is this done anywhere? If not, how difficult would this be to do?

@glwagner @simone-silvestri @navidcy

[glwagner]

True but we can only do this in a nonhydrostatic model, right? Otherwise we would not conserve kinetic energy. That said, it is possible to use a spherical Coriolis term in a nonhydrostatic model (either if we have LatitudeLongitudeGrid plus a pressure solver, or if we want to somehow compute a latitude from the "y" coordinate, but otherwise pretend we are on a Cartesian tangent plane).

[francispoulin]

Thanks @glwagner .

You don't need nonhydrostatic terms to include the complete Coriolis force. If you look in the original paper for the MITgcm here, you will see there are two changes.

First, for the x-momentum equation you add the following,

du/dt + ... + f^* w = ...

This is easy to do as it's just an additional term in the body force.

Second, you have to modify the vertical mometnum equation so it becomes

-f^* u = - dp/dz + b

This means that when we compute the hydrostatic pressure we don't just integrate the buoyancy but instead integrate the buoyancy + f^* u. The paper calls this Quasi-Hydrostatic, but I'm not sure if that is standard terminology.

But if it's possible to do this more easily in the Nonhydrostatic model in spherical cordinates, then I'm happy to try that. I have not tried using the nonhydrostatic model on a sphere before. Maybe that's the first thing I should try?

[glwagner]

Interesting. It wouldn't be hard to add a coriolis contribution to the pressure integral. Right now the non-hydrostatic and hydrostatic model use the same pressure integral, but the nonhydrostatic model has an explicit z-Coriolis contribution in the vertical momentum equation. So to make this change correctly, you need to add the Coriolis term to the pressure integral and remove the z-Coriolis contribution from the nonhydrostatic model.

[francispoulin]

That does not sound too difficult and I am keen to give it a try.

Just to be clear, if I try the nonhydrostatic model with lat-lon coordinates that should work, right? If yes, I can use that to compare with what I do with the quasi-hydrostatic model.

Also, are there docs that talk about the pressure integral formulation?

[glwagner]

There's no pressure solver for NonhydrostaticModel on a LatitudeLongitudeGrid, though --- you'd have to implement that.

[glwagner]

the pressure is found here: https://github.com/CliMA/Oceananigans.jl/blob/main/src/Models/NonhydrostaticModels/update_hydrostatic_pressure.jl

[francispoulin]

I have created a branch (fjp/complete-Coriolis-sphereical) that I will work on.

My first step is to create a file in Coriolis similar to hydrostatic_spherical_coriolis.jl, but for the quasi-hydrostatic problem. I see that this allows the option to have either energy or enstrophy conserving forms of the Coriolis term. Are these methods from a paper? If yes, I'd be happy and dig it up and see how this generalizes to the complete terms.

@simone-silvestri

[glwagner]

I don't think we should rely on descriptions of other models for documentation.

[glwagner]

We need to write it ourselves, rather than using a link.

[francispoulin]

Good point. I would be happy to put something together for the spherical case (after this PR and I figure out the details) but we don't have a lot of details on how the averaging works in the equations in the docs. I suspect it would be much more technical than the rest of the docs, but would be nice to have.

[glwagner]

No, it's not very complicated, it's just a few lines of code and is easily written up.

[francispoulin]

As long as the averaging and differencing operators are defined, which they are, then I guess you're right.