We seek to identify an accessible observable for anticipating tornadog
enesis in mesocyclones, because such tornadoes are characterized by ex
ceptionally extended lifespan, exceptionally long and wide path, and e
xceptionally high wind speed. We associate tornadogenesis with the tra
nsition from a one-cell vortex to a two-cell vortex. After such a tran
sition, the core (''eye'') of the vortex consists of virtually nonrota
ting, slowly recirculating, relatively dry air. Rapidly swirling air s
wiftly ascends in an annulus (''eyewall''), situated at a small but fi
nite distance from the axis of rotation. The swiftly ascending air is
described by a locus of thermodynamic states well approximated by a mo
ist adiabat. Such a transition from a one-cell vortex to a two-cell vo
rtex, on vastly larger lateral scale, is known to characterize the int
ensification of a tropical storm to a typhoon. We adopt a simplified,
tractable model for our initial analytic efforts. We examine a quasist
eady axisymmetric vortex with a four-part structure, consisting of a b
ulk potential vortex, a near-ground inflow layer, an ''eyewall'', and
an ''eye''. We inquire under what conditions such a four-part intense
vortex, formed in convectively unstable stratified air, is self-sustai
ning. In particular, we inquire whether the vertical profile of the an
gular momentum outside of the eyewall is a discriminant for identifyin
g the conditions for which an intense (two-cell) vortex could be self-
sustaining. Guidance from laboratory experiments would be helpful conc
erning the turnaround (the portion of the flow field in which the near
-ground swirling inflow separates to form the swirling updraft of the
eyewall annulus).