Entropic Balance Theory and Radar Observation for Prospective Tornado Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 422

found by Clebsch ( 1859 ) and called the Clebsch transformation ( Lamb 1932 ),
shows indeed a new balance condition different from the known hydrostatic,
(quasi-) geostrophic, cyclostrophic, and anelastic balance conditions, as well as
the Boussinesq approximation. It is named as entropic balance because of its
analogy to other balance conditions, and found it essential to explain tornadogenesis
mechanism ( Sasaki 1999 , 2009 , 2010 ). The entropic balance theory is based on the
variational principle, which is proved valid for the tornado-producing, atmospheric
fluid of Reynolds number R e D 10 8 12 and Rossby number R o D 10 2 4 .The
Euler-Lagrange equations are all prognostic except one that is diagnostic, which
was found to play key role in the long-lasting steady state of a tornado. The state
is analogous to the attractor of nonlinear dynamics. Based on the entropic balance
theory, the wrap-around mechanism is introduced to explain explicitly the nonlinear
process of tornadogenesis. The results are consistent with advanced observations
and successful tornado simulations of phenomena in tornadic storms, such as over-
shooting hydrometeors against the upper-level westerlies, the mesocyclone, hook
echo, discontinuous transition from supercell to tornadic stages as a transition from
baroclinic to barotropic stages, an increase of the relative helicity to one (its maxi-
mum value), and the tornado touching the ground in the perpendicular direction.
The wrap-around mechanism is analogous to a nonlinear process, the so-
called “baker's transformation,” and the transition is discontinuous from baroclinic
to barotropic stages by trapping entropic sink core inside the vortex, like a
nonlinear attractor (Fig. 18.10 ). Note that the entropic source and sink are of
larger magnitudes, it is baroclinic while they are of smaller magnitudes, it is
barotropic. Note also that the wrap-around mechanism is two-dimensional while the
baker's transformation is one-dimensional. In the entropic balance theory, the sole
diagnostic Euler-Lagrange equation is the key equation of the steady state, long-
lasting mesocyclonic and tornadic states, where the entropy anomaly is an essential
term. Consequently, to estimate the entropy anomaly from radar reflectivity, dual-
polarization radar data, or other observational means is a new challenge for data
assimilation and prediction of tornadoes.
The entropic balance theory predicts that the supercell and tornado stages
correspond well to the cases of high values of helicity. Helicity has been known
as an important index for conservation certain rotation to determine the spin and
velocity of air particle or parcel, the swirling direction and magnitude of streamlines
in fluid mechanics, and the earth magnetism in the dynamo theory of the earth's
core magma convection. It is also an important index of mature tornadoes from data
analyses, numerical simulations, and theoretical analyses over recent decades. In
meteorology, a positive relation between vertical velocity and vertical vorticity was
found by Klemp et al. ( 1981 )and We i s m a n a n d K l e m p ( 1982 ) in their analyses
of observations and numerical simulations, which seems to support higher values
of helicity in supercell thunderstorms. Theories were developed for the helicity to
reach nearly a maximum value at the mature stage of rotating convective storms and
tornadoes from dynamical analyses by Lilly ( 1982 , 1986 ), Davies-Jones ( 1984 ), and
Davies-Jones et al. ( 2001 ). Their theoretical analyses suggested the upward tilting
of the horizontal vortex tube for tornadogenesis. Note that fluid mechanics allows