What is Vorticity
Vorticity (which is expressed as ζ) is a measure of the rotation of fluids about a vertical axis relative to the earth's surface. In the northern hemisphere it is defined as positive or cyclonic in the counterclockwise direction. The unit of measurement of vorticity is inverse seconds (s-1). The following two equations are identical:
Where U and V are the east and northward wind components of the wind velocity. R is the radius or curvature travelled by a moving air parcel, M is the tangential speed along that curve in a cyclonic direction, and n is the direction pointing inward toward the centre of curvature.
In the above equation you see the small subset r, which stands for relative. In these equations we do not take into account the rotation of the earth and therefore consider relative vorticity. A physical interpretation of the first equation can be seen in this animation.
Cyclonic vorticity induced on a small paddle as a result of windshear.
If fluid flows along a straight channel, and is bound by shear, then it also has vorticity because a tiny paddle would be seen to rotate.
The second equation is interpreted below, where fluid along a curved path also has vorticity, so long as radial shear of the tangential velocity does not cancel it.
Vorticity for flow around a curved path where both the curvature and shear contribute to positive vorticity.
A special case of the last equation is where the radial shear is just great enough so that winds at different radii sweep out identical angular velocities about the centre of curvature. In other words, the fluid rotates as a solid body. For this the last equation reduces to:
Below we see a maximum of cyclonic vorticity north of the wind maximum and a maximum of anticyclonic vorticity south of the wind maximum, as is recognised by the rotating paddle wheels. The lower of the two paddles will turn clockwise (anticyclonic) since the wind force on the northern side of the blades is stronger than the force on the blades south of the axes.
Two dimensional flow with linear shear resulting in cyclonic and anticyclonic vorticity centers.
For this fact the poleward and equatorward sides of a westerly jet are always referred to as the cyclonic and anticyclonic shear sides, respectively.
Absolute Vorticity
The absolute vorticity is the sum of the relative vorticity and the earths' rotation.
where fc is the coriolis parameter fc = 2 Ω sin (Φ) and is a measure of the velocity of the earths' rotation.
Potential Vorticity
Potential vorticity ζp is defined as the absolute vorticity divided by the depth Δz of the column of air that is rotating.
It has units of m-1 s-1. In the absence of turbulent mixing and heating (latent radiation), potential vorticity is conserved (i.e. remains constant).
Isentropic Potential Vorticity
The Isentropic Potential Vorticity (IPV) can be defined as:
Where ζp is the potential vorticity measured on an isentropic surface (i.e. a surface connecting points of equal potential temperature θ) and where ρ is the air density (e.g., 1.2 kg m-3)
By rewriting the above equation as:
we can derive that larger IPVs exist where the air is less dense and where the static stability Δθ / Δz is greater. For this reason, the IPV is two orders if magnitude greater in the stratosphere than the troposphere. IPV is measured in potential vorticity units (PVU) defined by 1 PVU = 10-6 K m2 s-1 Kg-1. On average the air in the troposphere ζIPV <1.5, while in the troposphere this is greater.
Isentropic Potential Vorticity is conserved for air moving adiabatically and without friction along an isentropic surface (i.e., a surface of constant potential temperature). Thus, it can be used as tracer of air. Stratospheric air entrained in the troposphere retains its IPV>1.5 PVU for a while before losing its intensity due to turbulent mixing.
Vorticity Advection
One parameter that is often presented on weather maps is positive vorticity advection, or PVA. This is a parameter widely used to explain the concept of cyclogenesis but can be a bit harder to understand. The following animation is designed to simplify the processes leading up to PVA and what effect it has on the weather.