Potential vorticity

In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It is a useful concept for understanding the generation of vorticity in cyclogenesis (the birth and development of a cyclone), especially along the polar front, and in analyzing flow in the ocean. Potential vorticity (PV) is seen as one of the important theoretical successes of modern meteorology. It is a simplified approach for understanding fluid motions in a rotating system such as the Earth's atmosphere and ocean. Its development traces back to the circulation theorem by Bjerknes in 1898, which is a specialized form of Kelvin's circulation theorem. Starting from Hoskins et al., 1985, PV has been more commonly used in operational weather diagnosis such as tracing dynamics of air parcels and inverting for the full flow field. Even after detailed numerical weather forecasts on finer scales were made possible by increases in computational power, the PV view is still used in academia and routine weather forecasts, shedding light on the synoptic scale features for forecasters and researchers. Baroclinic instability requires the presence of a potential vorticity gradient along which waves amplify during cyclogenesis. Vilhelm Bjerknes generalized Helmholtz's vorticity equation (1858) and Kelvin's circulation theorem (1869) to inviscid, geostrophic, and baroclinic fluids, i.e., fluids of varying density in a rotational frame which has a constant angular speed. If we define circulation as the integral of the tangent component of velocity around a closed fluid loop and take the integral of a closed chain of fluid parcels, we obtain (1) where is the time derivative in the rotational frame (not inertial frame), is the relative circulation, is projection of the area surrounded by the fluid loop on the equatorial plane, is density, is pressure, and is the frame's angular speed.