Tornadoes - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 398

3
| {z }
CEN

2

Du=Dt ¼ u @u=@r
| {z }
ACCEL

¼ G =r 0 @p=@r

| {z }
PGF

Figure 6.59. Idealized illustrations of force diagrams (vectors) for various swirl ratios and
radii, for steady-state, frictionless flow at a level surface (where w ¼ 0), for a potential vortex
characterized by angular momentum G . The curved dashed streamlines indicate the motion of
the air around the center of a cyclonic vortex. The governing equation of motion in the radial
direction is given at the top. The net acceleration is indicated by ACCEL, centrifugal accel-
eration by CEN, and pressure gradient force by PGF. For low swirl (ratio), at large r, CEN is
very small, and the radial acceleration is determined by the radially inward-directed PGF. As r
decreases, CEN increases, but is still relatively small. At small r, the air parcel must decelerate
before reaching r ¼ 0, so the PGF reverses (especially if the swirl is very low and CEN very
low); as the parcel decelerates it turns upward before reaching r ¼ 0. For high swirl (ratio), at
large r CEN detracts from the inward-directed PGF, so the parcel accelerates radially inward
more slowly. At smaller r, CEN increases so much that it eventually overwhelms the inward-
directed PGF and the parcel decelerates, finally coming to a halt and turning upward at some
relatively large value of r (much larger than that for the low-swirl case). A downward-directed
PGF (not shown here) at r ¼ 0 drives a central downdraft.