PV of vertical normal modes

The formula for PV involves vertical derivatives (first and second) of model quantities, making it impossible to compute PV accurately at the top and bottom of the model's domain. Instead of computing PV at these two levels, we use two surrogate PV-like quantities (see Eqs. (24) and (26) of document - reproduced below). These have three contributions: (i) vorticity, (ii) pressure and (iii) vertical derivative of exner pressure. The total quantities and each contribution are shown below for the two PV-like quantities.

The equations for PV1 and PV2 are,

where the hat on the Q inducates vertical interpolation from theta to rho levels.

The two modes are almost identical in structure even though PV1 involves single vertical integrals and PV2 involves double vertical integrals. I presume that the two quantities are similar because the bottom of the domain contributes in the integrals very much more than the other levels, making the integrals - double or single - dominated by the fields at the bottom level. Even in this case, the two PV-like quantities should differ in latitude due to extra zonal-mean factors present only the PV2 quantity. Such factors are only weak functions of latitude at the ground. To show that there is a slight difference between the structures of PV1 and PV2, plotted at the bottom of this page is PV1/PV2 using a scale that highlights the latitudinal dependence.

Also shown below is the full PV field projected onto the external and first internal vertical normal modes in the manner described in the following ps or pdf document. Note that the PV field in this calculation is estimated at the top and bottom due to the problem described above.

The problems with these results are:

The PV1 and PV2 modes have very similar structures.
The PV of the two normal modes have similar structures.
The PV of mode 1 has a different structure to PV1, and similarly for mode 2.

.	PV-like mode 1	PV-like mode 2
Complete term
Term 1 (vorticity integral)
Term 2 (pressure integral)
Term 3 (Q, deriv. of exner p integral)
.	External mode	First internal mode
Normal mode projections