Potential vorticity

Simplified approach for understanding fluid motions in a rotating system

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.

Equation (3) is the atmospheric equivalent to angular momentum. For example, a spinning ice skater with her arms spread out laterally can accelerate her rate of spin by contracting her arms. Similarly, when a vortex of air is broadened, it in turn spins more slowly. When the air converges horizontally, the air speed increases to maintain potential vorticity, and the vertical extent increases to conserve mass. On the other hand, divergence causes the vortex to spread, slowing down the rate of spin.

The Ertel PV conservation theorem, equation (12), states that for a dry atmosphere, if an air parcel conserves its potential temperature, its potential vorticity is also conserved following its full three dimensional motions. In other words, in adiabatic motion, air parcels conserve Ertel PV on an isentropic surface. Remarkably, this quantity can serve as a Lagrangian tracer that links the wind and temperature fields. Using the Ertel PV conservation theorem has led to various advances in understanding the general circulation. One of them was "tropopause folding" process described in Reed et al., (1950). For the upper troposphere and stratosphere, air parcels follow adiabatic movements during a synoptic period of time. In the extratropical region, isentropic surfaces in the stratosphere can penetrate into the tropopause, and thus air parcels can move between stratosphere and troposphere, although the strong gradient of PV near the tropopause usually prevents this motion. However, in frontal region near jet streaks, which is a concentrated region within a jet stream where the wind speeds are the strongest, the PV contour can extend substantially downward into the troposphere, which is similar to the isentropic surfaces. Therefore, stratospheric air can be advected, following both constant PV and isentropic surfaces, downwards deep into the troposphere. The use of PV maps was also proved to be accurate in distinguishing air parcels of recent stratospheric origin even under sub synoptic scale disturbances. (An illustration can be found in Holton, 2004, figure 6.4)

The Ertel PV also acts as a flow tracer in the ocean, and can be used to explain how a range of mountains, such as the Andes, can make the upper westerly winds swerve towards the equator and back. Maps depicting Ertel PV are usually used In meteorological analysis in which the potential vorticity unit (PVU) defined as .

One of the simplest but nevertheless insightful balancing conditions is in the form of quasi geostrophic equations. This approximation basically states that for three dimensional atmospheric motions that are nearly hydrostatic and geostrophic, their geostrophic part can be determined approximately by the pressure field, whereas the ageostrophic part governs the evolution of the geostrophic flow. The potential vorticity in the quasi geostrophic limit (QGPV) was first formulated by Charney and Stern in 1960. Similar to Chapter 6.3 in Holton 2004,

Despite theoretical elegance of Ertel's potential vorticity, early applications of Ertel PV are limited to tracer studies using special isentropic maps. It is generally insufficient to deduce other variables from the knowledge of Ertel PV fields only, since it is a product of wind and temperature fields ( and ). However, large scale atmospheric motions are inherently quasi static; wind and mass fields are adjusted and balanced against each other (e. g., gradient balance, geostrophic balance). Therefore, other assumptions can be made to form a closure and deduce the complete structure of the flow in question:(1) introduce balancing conditions of certain form. These conditions must be physically realizable and stable without instabilities such as static instability. Also, the space and time scales of the motion must be compatible with the assumed balance;(2) specify a certain reference state, such as distribution of temperature, potential temperature, or geopotential height;(3) assert proper boundary conditions and invert the PV field globally. The first and second assumptions are expressed explicitly in the derivation of quasi geostrophic PV. Leading order geostrophic balance is used as the balancing condition. The second order terms such as ageostrophic winds, perturbations of potential temperature and perturbations of geostrophic height should have consistent magnitude, i. e., of the order of Rossby number. The reference state is zonally averaged potential temperature and geopotential height. The third assumption is apparent even for 2 dimensional vorticity inversion because inverting the Laplace operator in equation (21), which is a second order elliptic operator, requires knowledge of the boundary conditions. For example, in equation (20), invertibility implies that given the knowledge of , the Laplace like operator can be inverted to yield geopotential height is also proportional to the QG streamfunction under the quasi geostrophic assumption. The geostrophic wind field can then be readily deduced from Lastly, the temperature field is given by substituting into the hydrostatic equation (17).