Question

How does relative vorticity vary with height of column and planetary vorticity?

Answer 1

Vorticity is a measure of the local spin of a fluid element given by
ω~ = ∇ ×~v (1)
So, if the flow is two dimensional the vorticity will be a vector in the direction perpendicular to the flow.
• Divergence is the divergence of the velocity field given by
D = ∇.~v (2)
• Circulation around a loop is the integral of the tangential velocity around the loop Γ =I ~v.d~ l (3)
For example, consider the isolated vortex patch in Fig. 1. The circulation around the closed curve C is given by Γ =Ic ~v.d~ l =ZZ (∇ ×~v).d~s =ZZ ω~.d~s =ZZA ads = aA (4)
where we have made use of Stokes’ theorem. The circulation around the loop can also be approximated as the mean tangential velocity times the length of the loop and the length of the loop will be proportional to it’s characteristic length scale r e.g. if the loop were a circle L = 2πr. It therefore follows that the tangential velocity around the loop is proportional to aA/r i.e. it does not decay exponentially with distance from the vortex patch. So regions of vorticity have a remote influence on the flow in analogy with electrostatics or gravitational fields. The circulation is defined to be positive for anti-clockwise integration around a loop.