Vorticity equation

The final concept needed to understand the general horizontal motions in giant planet atmospheres is the concept of vorticity ra, which is defined simply enough as the curl or of the velocity vector ra = VxV. (5.27)

Vorticity is important since it (and properties derived from it) is conserved under certain conditions. The physical meaning of vorticity is that it is a measure of the spin of a fluid flow at any particular point and as such tells us something about the angular momentum of the fluid flow. For velocities measured in a frame rotating with constant angular velocity H we know that

Taking the curl of both sides we find ra/ = raR + 2H. (5.29)

For the special case of horizontal flow only, which very well approximates to the conditions in giant planet atmospheres where zonal and meridional velocities greatly exceed vertical velocities V — (u, v, 0) and uR — (0,0, £), where (, known as relative vorticity, is equal to

Substituting for raR in Equation (5.29) we then find raj =(0,2Q cos + 2Q sin (5.31)

A vorticity equation may be derived from the x and y components of the momentum equations expressed in whatever coordinates are most convenient and taking into account any suitable approximations. For example, differentiating Equation (5.11) with respect to y and differentiating Equation (5.12) with respect to x and subtracting (ignoring frictional forces) we find that, assuming pressure and density do not vary with horizontal position and that there are no vertical motions (Houghton, 1986),

where

If the flow can be considered to be two-dimensional and non-divergent (i.e., vertical velocities are negligible) it can be seen from Equation (5.33) that the quantity ( + f, known as absolute vorticity, is conserved. We will see in Section 5.3.2 that this conservation law is the central restoring force that supports Rossby or planetary waves. A less stringent approximation is to assume that the air has roughly constant density and temperature for which the mass continuity equation ft + V-(PV) = 0 (5.35)

becomes simply V- V = 0. This is an example of the Boussinesq approximation, where density and temperature variations are neglected, except where they appear in buoyancy terms. Substituting Equation (5.35) into Equation (5.33) we find

from which it may be deduced that (Houghton, 1986)

where h is the separation between material levels in the fluid. The quantity (C + f )/h is a simplified form of potential vorticity, and its conservation is analogous to the conservation of angular momentum. When a parcel is vertically stretched (and hence made narrower by conservation of mass) it rotates faster to maintain its angular momentum in the same way as ice skaters spin faster when they draw their arms inwards. The most general form of potential vorticity is Ertel's potential vorticity which is defined as

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