Omega V1 Governing Equations Design Document

[mark-petersen]

Derivation of non-Boussinesq primitive equations for Omega Version 1.

See compiled document at
https://portal.nersc.gov/project/e3sm/lvroekel/omega_docs/html/design/OmegaV1GoverningEqns.html

[xylar]

Some comments, mostly about generalizing the gravitational potential and adding it a few places I think it's missing.

Also, a request to remove the commented-out text to make review easier.

[mark-petersen]

I have added substantial updates based on feedback and conversations this past week. In particular, the document now uses layer mass-thickness $h$ in kg/m^2. Squashing commits, rebasing and pushing a single commit.

[vanroekel]

the buoyancy term is missing here rho g khat

[mark-petersen]

The first equation set is the most general, so gravity would be within ${\bf F}^u$ but is not yet specified. A few paragraphs down I begin "Geophysical fluids such as the ocean and atmosphere are rotating and stratified..." and start to make assumptions and call out specific terms. I have the sentence "The gravitational force $g {\bf k}$ is included in ${\bf F}^u$".

[vanroekel]

oh sorry, totally missed that. Thanks for the explanation

[xylar]

I realize that one could include gravity in ${\bf F}^u$ but I would much prefer that we not do so. In my view, it makes a lot of the subsequent derivation clearer if we include $\nabla_{3D} \Phi$ here from the start. I have been pushing for this the whole time and will continue to do so.

[sbrus89]

Thanks for putting this together @mark-petersen, I just have a couple comments.

[sbrus89]

Suggested change

	where conservative temperature, $\Theta$, and salinity, $S$, are examples of tracers $\varphi$.
	where conservative temperature, $\Theta$, and absolute salinity, $S_A$, are examples of tracers $\varphi$.

[mark-petersen]

Is there a need to work with other types of salinity? If not, I would prefer to just use $S$ for absolute salinity, rather than $S_A$. Likewise, we work with conservative rather than potential temperature, and conservative temperature simply uses $\Theta$.

[mark-petersen]

rebased on head. Squashed small commits about typos, delinting, white space etc, to reduce number of commits. Current compiled version is here:
https://portal.nersc.gov/project/e3sm/mpeterse/omega/design-doc-v1-eqns-Aug6/design/OmegaV1GoverningEqns.html

[mark-petersen]

A negative sign must be included for the pressure terms derived from the surface force terms. According to Leishman 2025, Chapter 21, above equation 2, "the negative sign indicates that the force is inward and in the opposite direction to the unit normal vector area, which always points outward by convention". Also see discussion in Kundu et al. 2016 page 104 and equation 4.26.

I added that text to the derivation and changed the sign on the pressure gradient throughout, as well as the related grad(z) terms.

[sbrus89]

Approved. Thanks for all the work on this everyone. This is a great resource.