Potential missing factor in MultiLayerQG.energies

[apaloczy]

I think there may be a missing 1 / sum(params.H) factor in the potential energies part of MultiLayerQG.energies.

This is because in the n-layer case, multiplying the PV equations by $\psi_n H_n/H$ (with $H = \Sigma H_n$) and adding them together to form the energy equation gives potential energy terms that look like

$$\mathrm{PE_{j+1/2}} = \frac{1}{2} \frac{f_0^2}{g_{j+1/2}'H} \iint (\psi_j - \psi_{j+1})^2 \frac{dx dy}{Lx Ly}.$$

The total energy is nearly conserved as expected in two-layer decaying turbulence simulations if the 1 / sum(params.H) is added.

Does this check? If so, I can open a pull request.

[navidcy]

This checks! Even dimensionally the KE and PE expressions in the docs don't match.

(https://fourierflows.github.io/GeophysicalFlowsDocumentation/v0.14.0/modules/multilayerqg/#Diagnostics)

Open a PR please!