Barotropic conversion in MKE and EKE budgets do not cancel each other

[chen315]

Hi,

We got a problem about barotropic energy conversion. Any suggestions would be greatly appreciated.

1. Goal: To calculate mean kinetic energy (MKE) and eddy kinetic energy (EKE) budgets from ROMS 3D momentum diagnostics

2. Model problem: Eady problem of baroclinic instability with surface wind forcing

3. Formulations
MKE budget (x-component) = mean( u_mean * (x-momentum equation) )
EKE budget (x-component) = mean( u_prime * (x-momentum equation) )

4. Problem:
After volume integration (over a domain that is periodic in x and closed in y), the barotropic energy conversion comes out of the advection terms.
We expect the barotropic conversion terms to cancel each other when we add EKE and MKE budget. However, we cannot get them to cancel out.
The details and derivations are in the attached text. We’ve been really struggling with this problem for a while. We will appreciate if someone can provide us with some hints or suggestions. Thanks

CJ