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Piecewise potential vorticity inversion (PPVI) seeks to determine the impact of observed potential vorticity (PV) anomalies on the surrounding flow. This widely used technique is based on dividing a flow domain D into subdomains D-1 and D-2 = D - D-1. The influence of PV in D-1 on the flow in D-2 is assessed by removing all PV anomalies in D-2 and then inverting the modified PV in D. The resulting flow with streamfunction psi(1) is attributed to the PV anomalies in D-1. The relation of PV in D-1 to psi(1) in D-2 is not unique, because there are many PV distributions in D-1 that induce the same psi(1). There is, however, a unique solution if the ageostrophic circulation is included in the inversion procedure.

The superposition principle requires that the sum of inverted flows with PV = 0 in D-2 and the complementary ones with PV = 0 in D-1 equal the inverted flow for the complete observed PV in D. It is demonstrated, using two isolated PV balls as a paradigmatic example, that the superposition principle is violated if the ageostrophic circulation is included in PPVI, because the ageostrophic circulation cannot be associated with only one of the anomalies.

Inversions of Ertel's PV are carried out using Charney's balance condition. PPVI is not unique in that case, because many different PV fields can be specified in D-1, which all lead to the same inverted flow in D-2. The balance condition assumes vanishing vertical velocity w so that uniqueness cannot be established by including w in the inversion, as was possible in the quasigeostrophic case.