Tornadoes - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 365

other words, the depth of the friction layer decreases as one goes radially inward
toward the center of the vortex if the angular momentum is constant. The reader
should appreciate the diculties in finding analytic solutions for the wind field in
a tornado boundary layer since as r ! 0 there are singularities when r is the
denominator of terms and asymptotic solutions must therefore be found. One can
also appreciate how interesting a tornado boundary layer might be dynamically
owing to the circular symmetry of a vortex, something that will be made more
explicit soon. For a kinematic coecient of molecular viscosity of 2 10 5 m 2 s 1
and G (75m s 1 ) (100m), the depth at r c 100m is only 0.5 cm. The depth of

the friction layer for the kinematic coecient of turbulent viscosity of 10 3 m 2 s 1
(6.19) at r c 100m is 10m, which is much deeper than the depth of the friction
layer when not taking turbulent eddies into account. Most of the flow in the
friction layer is in the radial direction.
From both vortex chamber studies and analytic solutions for vortex chamber
flow, the depth of the tornado boundary layer (friction layer þ inertial layer) is
ð=GÞ

1 = 2 r b , where r b is approximately the radius of the rotating bottom of a
vortex chamber, or in the atmosphere the radius at which a boundary layer for
rotating (''swirling'') begins to form. To estimate r b for the real atmosphere, we
make use of vortex chamber data from which it is inferred that the depth of the
inertial layer is approximately the same as the ''core'' radius

1 = 2 r b

r c ð=GÞ ð 6 : 21 Þ

G (75m s 1 )(100m), r b 300m. It is
noted that unlike the depth of the friction layer the depth of the inertial layer
does not vary significantly with radius. Note that the depth of the friction layer at
the core radius ( 10m) is much shallower than the depth of the inertial layer at
the core radius ( 100m).
In the inertial layer air parcels accelerate radially inward, so the time they rub
against the ground in the friction layer decreases as they move radially inward,
limiting the effects of turbulent friction (vertical exchange of turbulent eddies from
aloft with eddies from the surface). Also, it will be shown later that there is a
dynamically induced downdraft that can also act to suppress the height of the
friction layer: thus, the depth of the friction layer decreases radially inward.
Now, consider what happens at low levels inside the core radius. Owing to
circular symmetry, accelerating radial inflow in the inertial region must decelerate
near the origin, since u must vanish at r ¼ 0. Air rushing in from all directions
abruptly slows down and, from continuity considerations, turns and flows
upward. This is perhaps the most interesting aspect of boundary-layer flow in a
tornado alluded to earlier. Paraphrasing comments Rich Rotunno once made to
the author, the radially inward accelerating flow is on a direct ''collision course''
with air coming in from the opposite direction (and others) and something
''catastrophic'' must happen. This region, where the flow abruptly ''turns the
corner'', is called the ''corner'' region ( Figure 6.41 ). In the corner region, both ver-
tical and radial variations are significant. Owing to the mass continuity constraint,
there is an intense frictionally induced updraft jet. When the vortex is accom-

10 3 m 2 s 1 , and So, for r c 100m,