Zero PV checks in the unbalanced transform

The unbalanced part of the U-transform imposes zero PV. PV here is linearised Ertel PV (with just vertical component of vorticity) in the bulk. This PV cannot be computed at the top and bottom boundaries and so two normal-mode-like PV values are used instead of the top and bottom Ertel PV.

Given a field of 'unbalanced pressure', the unbalanced transform compues a field of 'unbalanced vorticity'. This is the vorticity that, together with the unbalanced pressures, gives rise to zero PV. Since unbalanced pressure is on p-points, the transform has been written on p-points and so the unbalanced vorticity emerges on p-points. The poisson solver is used to compue the unbalanced streamfunction (on p-points). This streamfunction is interpolated to psi-points which is then used to compute unbalanced u and v winds.

This transform has been written. In order to check that it works, I perform two tests.

PV test 1: To check that PV is zero on p-points. Input: unbalanced pressure and unbalanced vorticity, both on p-points. Output: Bulk and normal-mode PV on p-points.
PV test 2: To check that PV is zero on input variable's natural points. Input: unbalanced pressure on p-points and unbalanced winds on u/v-points. Output: Bulk and normal-mode PV on psi-points.
PV test 3: To gauge typical values of PV (PV that is not meant to be zero). Input: pressure on p-points and winds on u/v-points. Output: Bulk and normal-mode PV on psi-points.

PV Test 1: To check that PV is zero on p-points.
Bulk PV (eta=0.056)	PV of normal mode 1	PV of normal mode 2


PV Test 2: To check that PV is zero on input variable's natural points.
Bulk PV (eta=0.056), all points	PV of normal mode 1, all points	PV of normal mode 2, all points

PV Test 2, Bulk PV (eta=0.056), points closest to pole excluded	PV Test 2, PV of normal mode 1, points closest to pole excluded	PV Test 2, PV of normal mode 2, points closest to pole excluded


PV Test 3: To gauge typical values of PV.
Bulk PV	PV of normal mode 1	PV of normal mode 2

Notes:

The PV calculated from the unbalanced fields (first row), computed on the same grid that the unbalanced transform was used (p-points) gives values that are essentially zero.
The PV in rows 2 and 3 has been calculated from unbalanced pressure and unbalanced u and v. All of these variables are located at their correct grid positions. Unbalanced u and v are found by transforming unbalanced vorticity to unbalanced streamfunction (on -points), interpolating this to psi-points and then finding u and v. Values (rows 2 and 3) are smaller than 'typical' values that are not meant to be zero (row 4), but they are not as small as I would like.
There are strange values near the poles in row 2. I have not yet been able to trace the cause of this.

Back to PV page

Zero PV checks in the unbalanced transform

PV Test 1: To check that PV is zero on p-points.

PV Test 2: To check that PV is zero on input variable's natural points.

PV Test 3: To gauge typical values of PV.