The connection between Carnot and CAPE formulations of TC potential
intensity

Tropical cyclone (TC) potential intensity (PI) theory has a well known
form, consistent with a Carnot cycle interpretation of TC energetics,
which relates PI to mean environmental conditions: the difference
between surface and TC outflow temperatures and the air–sea enthalpy
disequilibrium. PI has also been defined as a difference in convective
available potential energy (CAPE) between two parcels, and quantitative
assessments of future changes make use of a numerical algorithm based on
this definition. Here, an analysis shows the conditions under which
these Carnot and CAPE-based PI definitions are equivalent. There are
multiple conditions, not previously enumerated, which in particular
reveal a role for irreversible entropy production from surface
evaporation. This mathematical analysis is verified by numerical
calculations of PI’s sensitivity to large changes in surface-air
relative humidity. To gain physical insight into the connection between
the CAPE and Carnot formulations of PI, we use a recently developed
analytic theory for CAPE to derive, starting from the CAPE-based
definition, a new approximate formula for PI which nearly recovers the
previous Carnot PI formula. The derivation shows that the difference in
undilute buoyancies of saturated and environmental parcels which
determines CAPE PI can in fact be expressed as a difference in the
parcels’ surface moist static energy, providing a physical link between
the Carnot and CAPE formulations of PI. This combination of analysis and
physical interpretation builds confidence in previous numerical
CAPE-based PI calculations that use climate model projections of the
future tropical environment.