Entropic Balance Theory and Radar Observation for Prospective Tornado Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 426

Fig. 18.2 Solution space.
The domain of full solution of
the Euler Lagrange (E-L)
equations is schematically
shown in the solution space
by the heavy solid line .It
includes non-stationary state
(NSS), stationary state (SS),
and the solution of diagnostic
E-L equation (DS). The
solution in the domain
covered by DS and SS has
long-lasting property
mathematically similar to the
attractor

The relationships expressed by ( 18.7 ) emphasize the importance of the diagnostic
E-L equation ( 18.5 ); that is, the transition to a steady state SS or DS from non-
steady state NSS must satisfy ( 18.5 ). In other words, we can find the necessary
conditions for the tornadogenesis and transition among different stages from the
entropic balance theory as discussed further in the next sections. The diagnostic
balance ( 18.6 ) provides insight to a long-lived tornado, presumably by DS and SS
steady states, as expressed by ( 18.7 ). It is important to note that ( 18.7 ) is reached
indirectly by a high value of helicity as shown by ( 18.11 )and( 18.12 ) in the next
sections.

18.4 Helicity and Tornadogenesis

The helicity, H, is defined as a scalar (inner) product of flow velocity and vorticity,

H WD v !; (18.8)

where v is flow velocity and represents vorticity of the flow r v. For fluids of

high Reynolds number and high Rossby number, the fluid motion is assumed as an
ideal fluid. Without solenoidal effects, the vorticity equation is given by

!

@ t ! Dr .

v !/: (18.9)

The case with solenoidal effects will be shown by ( 18.25 ) in the next section.
Because of the normal relationship, sin 2 C cos 2 D 1 , between the scalar product

!

and the vector product, where is the angle between two vectors v and ,weget

( Yoshizawa 2001 ),