Entropic Balance Theory and Radar Observation for Prospective Tornado Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 424

The entropic balance theory hypothesizes that changes in entropy are a quasi-
adiabatic process, that is, the microphysical phase change of a small ensemble
of hydrometeor molecules is instantaneous, creating a new entropy level, with
adiabatic conditions before and after the phase change. It is hypothesized that this
phase change timescale is significantly shorter than the time-scales of convective
storms and tornadoes (Hypothesis 1), schematically shown in Fig. 18.15 ,

t phase change << t supercell, tornado (18.2)

Variations of the initial entropy levels are small enough and allow us to approximate
them by their ensemble means (Hypothesis 2). These hypotheses are further
discussed in Sect. 18.10 .
The Lagrangian density L is thus formulated as

v 2 U L WD .1=2 .; S / ˆ/ ˛.@ t Cr . v // ˇ.@ t . S / Cr . v S //;

(18.3)

where ,S,and v are density of the air, internal energy, gravitational

potential energy, entropy, and flow velocity respectively, and

,U, ˆ

are the
Lagrange multipliers to satisfy the constraints of conservation of mass and entropy,
respectively. Then, the Lagrangian (action) denoted by L is defined as

˛ and ˇ

Z

L WD I L d ; (18.4)

where represents the temporal and spatial integration domain, and the ensemble

of air molecules is represented by the spatial integration.
The first variation of L leads to the Euler-Lagrange (E-L) equations, which, after
mathematical manipulation, lead to a full set of dynamical and thermodynamical,
nonlinear, equations of the ideal flow ( Lamb 1932 ; Bateman , 1932 ; Sasaki , 1955 ;
Dutton 1976 ). The E-L equations are all prognostic except for one that is diagnostic,
so-called by Lamb as the Clebsch's transformation ( Clebsch 1859 ) of flow velocity,

v Dr ˛ Sr ˇ: (18.5)

Then, the vorticity, ! , equation becomes

! D .1= S / rS . Sr ˇ/: (18.6)

The vector relation ( 18.6 ) is found to be extremely important to gain clear insight
into the development mechanisms of supercells and tornadogenesis. The diagnostic
velocity ( 18.5 ) is universal for the ideal flow. The vorticity ( 18.6 ), derived from
( 18.5 ), is demonstrated in convenience by the mutually orthogonal vector relation,
similar to the so-called Fleming's right hand rule of electromagnetic fields, called
by the author as “entropic right-hand rule”, among the orthogonal variables of
the spatial three dimensions, the vorticity

r

! , the entropy gradient (1/S) S, and