PV of vertical normal modes (spherical modes)

We wish to show that the extra PV-like modes have a structure that is similar to the PV of the external and first internal vertical modes.

What is the PV of these normal modes? We try two approaches below. Both involve decomposing the fields into the external and first internal vertical modes.

Attempt 1 ("real-space" grid method): involves estimating PV while the horizontal fields remain in grid-space. The expression used is

(one expression for each vertical mode). Starred quantities are the projections onto each vertical mode and C is the eigenvalue of the vertical structure equation in each case.
Attempt 2 involves decomposing the fields into their spectral components, calculating PV and then transforming back to grid space. The PV expression used is

(one expression for each vertical mode). The notations is different. The fields relate to a given spectral component (n,m) in vertical mode n (unfortunate clash of use of n). B'n is the vertical eigenvalue now ( = C above). The F operator couples different total wavenumbers.

In this document we summarise the two PV values that arise from each method and compare to the two PV-like quantities using orthogonalization to see if we get a match.

The spectral transform is not exact, probably owing to the fact that the grid is regular and not gaussian. Near the bottom of the page I show some tests of acting with the spectral transform and then with its inverse to see if the original fields are recovered.

.	Mode 1	Mode 2
PV - "real-space" grid method
PV - "spectral-space" grid method
.	Missing PV mode-1	Missing PV mode-2
With orthogonalization zonal mean ref. state
With orthogonalization global mean ref. state
.	Before transforms	After transforms
Spec-trans test with p increment
Spec-trans test with vort increment