Entropic Balance Theory and Radar Observation for Prospective Tornado Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 427

/ 2 C .. / 2 D 1; .. v !/= D v !/= D (18.10)

where D 2 WD v 2 ! 2 ,and(v !/

/D is called relative helicity or normalized helicity, or
simply helicity. When the relative helicity approaches unity, ( 18.10 ) imposes that

. v !/ ! 0: (18.11)

Then, from ( 18.9 )to( 18.10 ), we get

@ t ! ! 0: (18.12)

This means that a steady state of vorticity will be reached when the magnitude
of relative helicity increases to unity. Also, it means that the mature stage of a
tornado is a long-lasting system, which is similar to the attractor of a nonlinear
system (e.g., Lorenz strange attractor of Rayleigh convection). This result agrees
with the solution classification in the solution space of the entropic balance theory,
as shown as the steady state attractor in Fig. 18.2 . It is also clear that ( 18.11 ) will be
satisfied if the vector v is parallel to the vector ! , and the helicity ( 18.8 ) becomes a

maximum.

18.5 A Form of Helicity Based on Entropic Balance theory

The entropic balance theory gives further new insight into helicity and entropy. The
following E-L equation is the only diagnostic one among all E-L equations obtained
from the Lagrangian density of the flow of high Reynolds and Rossby numbers
shown by ( 18.5 )as

v Dr ˛ Sr ˇ: same as ( 18.5 ) (18.13)

In ( 18.13 ), S is entropy, are the Lagrange multipliers of mass conservation

and thermodynamics of quasi-adiabatic process, adiabatic with instantaneous phase-
change, then entropy change, of microphysics in the Lagrangian density. The
Lagrange multipliers

˛ and ˇ

are potentials, and they are analogous to the well-
known velocity potential usually designated by

˛ and ˇ

˛

as follows

˛ D ˛ C S 0 ˇ; (18.14)

where S 0 is a constant along each molecular trajectory and may be determined from
the initial condition. Note that S 0 is S 0 (x, y, z) at t D t 0 . Determination of S 0 is

discussed in Sect. 18.6 .
The vorticity is computed from ( 18.5 ,or 18.13 )andshownby( 18.6 )as

!. WD r v / D .1= S / rS . Sr ˇ/: same as ( 18.6 ) (18.15)