Tornadoes - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 361

Coriolis force is much less than the centrifugal force in a tornado, the Coriolis
force is neglected.
The vertical equation of motion is

Dw = Dt ¼ @ w =@ t þ u @ w =@ r þ w @ w =@ z

p 0

2 w

r 2

2 w 2 z Þ ð 6

¼ 0 @ =@ z þ B þð@ =@ þ 1 = r @ w =@ r þ@ =@ : 11 Þ

where B ¼ g T 0 = T. The reader is reminded that B ¼ 0 in laboratory models, which

are driven by exhaust fans—not by positive buoyancy in a cloud overhead—as a
result of latent heat release.
The components of vorticity in an axisymmetric vortex in the radial,
azimuthal, and vertical directions respectively are

¼@v=@ z ð 6 : 12 Þ

¼ @ u =@ z @ w =@ r ð 6 : 13 Þ

¼ 1 = r @=@ r ð r vÞ ð 6 : 14 Þ

The reader should note that these three components of vorticity in cylindrical
coordinates are given the same names (Greek letters) as the components of
vorticity in Cartesian coordinates ((4.36)-(4.38)), and should not be confused with
them. Vorticity in cylindrical coordinates will be used later but are provided now
for reference.
The equation of continuity in a Boussinesq atmosphere for axisymmetric
motions is

1 = r @=@ r ð ru Þþ@ w =@ z ¼ 0 ð 6 : 15 Þ

The adiabatic form of the thermodynamic equation is
DT 0 @ z Þ T 0 þ w

= Dt ¼ð@=@ t þ u @=@ r þ w @=@ T =@ z

2 2 r 2 2

z 2 Þ T 0

¼ ð@ =@ þ 1 = r @=@ r þ@ =@ ð 6 : 16 Þ

where is the eddy coecient of turbulent diffusivity. The simplest models of

tornadoes are those in an atmosphere that is adiabatic and one in which density is
constant. It follows that no thermodynamic equation is needed to determine
vortex structure in these models.
Beyond a certain radius, the effects of surface friction are felt in the boundary
layer ( Figure 6.41 ). Al Barcilon in 1967 was perhaps the first to analyze the
problem of having a tornado-like vortex interact with the ground and infer that
there are several dynamically different regions. The boundary layer is divided up
into the ''friction layer'' (2b) and the ''inertial layer'' (2a). Above the boundary
layer, in the free atmosphere, away from the axis of rotation (beyond r ¼ r c ), the
region is typically referred to as the ''outer flow'' (1). In the outer flow region, a
radially inward-directed pressure gradient force (the acceleration is 0 @

=@

r)is
counterbalanced by a radially outward-directed centrifugal force (the acceleration
is v

p

2

=

r), the condition of ''cyclostrophic balance'', if the flow is steady state. It is
assumed that the outer flow is characterized by ''constant angular momentum'' G ,