In this paper, we propose a form for potential vorticity (PV), rescaled using the Lorenz's rearranged density profile, the novelty being that we here take into account its time evolution. We argue this rescaled PV is more representative of the dynamics, in particular to evaluate the respective impact of mixing and friction on the generation of geostrophic circulation. The impact of mixing at global scale, which only modifies the global stratification at rest, is taken into account in the evolution equation of this "objective" definition of PV, in the sense that it scales the PV changes with respect to its effect on the circulation. Numerically, we show that all terms can be calculated coherently using a single computation cell. We illustrate our purpose by studying the instability of coastal upwelling currents, using a numerical model at high resolution. The configuration is a periodic flat channel on the f-plane with vertical walls at the southern and northern boundaries. A constant wind is applied over a fluid at rest with an initial linear stratification. An upwelling current forms at the northern coast. After a few days, instabilities develop and vortices eventually emerge with surface intensified cyclones and subsurface anticyclones. We show that these instabilities and eddies are associated with (rescaled) PV anomalies,