Tornadoes - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 360

modeled vortices is sensitive to the type of subgrid-scale parameterization scheme
used.
To simulate idealized laboratory vortices, there is the problem of what
boundary conditions to use. No-slip lower-boundary conditions are appropriate
when the effects of surface friction need to be taken into account, while free-slip
lower-boundary conditions are used to isolate the behavior of the vortex when
surface friction plays no role. What are the boundary conditions at the top of the
domain if the parent storm is not explicitly represented? The reader is referred to
Rich Rotunno's early papers and the review papers by Davies-Jones et al. (2001)
and Rotunno (2012), and elsewhere for a summary of the more technical issues.

6.6.1 Vortex structure

The idealized vortex that is produced may be thought of as an intense (''primary'')
vortex that is intensified by storm-updraft-associated convergence acting on pre-
existing vorticity. In the meantime, the intensifying vortex rubs against the
surface, where friction slows it down. The surface boundary-layer flow may be
laminar or turbulent, depending on the Reynolds number. The degree of smooth-
ness of the surface underneath the vortex can therefore play a role in the nature of
the flow. The Reynolds number, as we have shown, is extremely high and there-
fore indicative of turbulence. It is therefore thought that in nature tornado
boundary layers are turbulent, especially when the surface of the Earth is relatively
rough.
The radial and azimuthal components of the equations of motion for

axisymmetric ( is the azimuthal angle in cylindrical coordinates)

motions in a non-rotating atmosphere, including turbulent friction, are given in
cylindrical coordinates as follows:

@=@ ¼ 0, where

2

Du = Dt ¼ @ u =@ t þ u @ u =@ r þ w @ u =@ z v = r

p 0

2 u 2 r þ 1

r 2

2 u

z 2

¼ 0 @ =@ r þð@ =@ = r @ u =@ r u = þ@ =@ Þ ð 6 : 9 Þ

D v= Dt ¼ @v=@ t þ u @v=@ r þ w @v=@ z þ u v= r

2

r 2 r 2

2

z 2 ¼ ð@ v=@ þ 1 = r @v=@ r v= þ@ v=@ Þ ð 6 : 10 Þ

where u is the radial wind component; v is the azimuthal wind component; r is the
radial coordinate; z is the vertical coordinate; 0 is the specific density at the

surface; and is the kinematic coecient of viscosity for turbulent eddies—not

for molecules. The v

2

r term on the LHS of (6.9) is the centripetal acceleration;
if moved to the RHS so that the reference frame changes from that of an air
parcel to that of the rotating reference frame of the vortex, it (i.e.,

=

2

v = r) is the

p 0 centrifugal acceleration. The 0 @ r term represents the acceleration due to the

radial pressure gradient force. The last term on the RHS of (6.9) is the turbulent
friction term. The most significant contribution to the friction term comes from
the vertical term,

=@

z 2 . The equation for the azimuthal wind component
(6.10) is sometimes expressed in terms of angular momentum

2 u

@ =@

G ¼ r v . Since the