Potential vorticity

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.

It is a simplified approach for understanding fluid motions in a rotating system such as the Earth's atmosphere and ocean.

Starting from Hoskins et al., 1985,[2] PV has been more commonly used in operational weather diagnosis such as tracing dynamics of air parcels and inverting for the full flow field.

[3] 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,[1] i.e., fluids of varying density in a rotational frame which has a constant angular speed.

However, in the context of atmospheric dynamics, such conditions are not a good approximation: if the fluid circuit moves from the equatorial region to the extratropics,

Furthermore, the complex geometry of the material circuit approach is not ideal for making an argument about fluid motions.

His later paper in 1940[5] relaxed this theory from 2D flow to quasi-2D shallow water equations on a beta plane.

In this system, the atmosphere is separated into several incompressible layers stacked upon each other, and the vertical velocity can be deduced from integrating the convergence of horizontal flow.

For a one-layer shallow water system without external forces or diabatic heating, Rossby showed that where

The conserved quantity, in parenthesis in equation (3), was later named the shallow water potential vorticity.

is a measure of the weight of unit cross-section of an individual air column inside the layer.

Specifically for the atmosphere, potential temperature is chosen as the invariant for frictionless and adiabatic motions.

Then, if we start the derivation from the horizontal momentum equation in isentropic coordinates, Ertel PV takes a much simpler form[8] where

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.

[9] For the upper-troposphere and stratosphere, air parcels follow adiabatic movements during a synoptic period of time.

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

With some manipulation (see Quasi-geostrophic equations or Holton 2004, Chapter 6 for details), one can arrive at a conservation law where

Apart from the diabatic heating term on the right-hand-side of equation(19), it can also be shown that QGPV can be changed by frictional forces.

QGPV has been widely used in depicting large-scale atmospheric flow structures, as discussed in the section PV invertibility principle; Apart from being a Lagrangian tracer, the potential vorticity also gives dynamical implications via the invertibility principle.

For a 2-dimensional ideal fluid, the vorticity distribution controls the stream function by a Laplace operator, where

A similar principle was originally introduced for the potential vorticity in three-dimensional fluid in the 1940s by Kleinschmit, and was developed by Charney and Stern in their quasi-geostrophic theory.

It is generally insufficient to deduce other variables from the knowledge of Ertel PV fields only, since it is a product of wind (

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.