Tornadoes - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 364

Figure 6.43. Balance of forces in the friction layer.

pressure explicitly expressed as

0 ¼ V 2

2 u

z 2

= r þ@ =@ ð 6 : 18 Þ

where V is the azimuthal wind speed if there were cyclostrophic balance

(V 2 p 0

r). In the friction layer, the flow is sub-cyclostrophic.
Since the turbulent friction term on the RHS of (6.18) is the same order of
magnitude as the V 2

= r ¼ 0 @ =@

= r term, the kinematic coecient of turbulent viscosity

ð V 2

2 u

z 2 ½ð 10 m s 1

2

=ð 100 m Þ=½ 10 m s 1

2

= r Þ=@ =@ Þ =ð 100 m Þ

10 3 m 2 s 1

ð 6 : 19 Þ

In the friction layer, the vertical derivatives of wind components are much greater
than radial gradients.
Just above the friction layer, where the effects of friction are negligible, it

follows from (6.9) that

2

V 2 p 0

2

u @ u =@ r þ w @ u =@ z ðv Þ= r ¼ 0 @ =@ r þv = r ð 6 : 20 Þ

( Figure 6.42 ). This layer ( just above the friction layer) is called the ''inertial layer''
because inertial accelerations—the LHS of (6.20)—are significant: the flow in the
reference frame of the vortex is unbalanced. It is noteworthy that while turbulent
mixing does not appear explicitly in this equation, the effects of surface friction
have been communicated to this layer via deviation of the azimuthal component
of the wind from its cyclostrophic value. Since the flow is sub-cyclostrophic, the
term on the RHS of (6.20) is 0. In the reference frame of a rotating air parcel,

the LHS of (6.20) is just the parcel acceleration in the radial direction, so that air
parcels accelerate radially inward toward the center. Air parcels become less
sub-cyclostrophic with height until they are exactly cyclostrophic at the top of the
inertial layer. The vertical derivatives of wind components are much larger than
radial derivatives, as they are in the friction layer. At the surface, where w ¼ 0,
u

<

@ u =@ r < 0, since u < 0 and @ u =@ r > 0; this pattern is consistent with the sign of

the RHS of (6.20).
We now consider the depth of the friction and inertial layers. From analytic
solutions O. Burggraf and co-authors in 1971 showed in a seminal paper that the
depth of the friction layer for a vortex having potential flow is ð=GÞ

1 = 2 r:In