**An Expanded Tropical Cyclone Potential Intensity Framework**

Alex M. Kowaleski and Jenni L. Evans

An accurate evaluation of the maximum intensity that a tropical cyclone (TC) can attain from given oceanic and environmental atmospheric conditions is critical to intensity forecasting and for quantifying the effects of climate change on maximum TC intensity.

Sea-surface temperature (SST) has long been known to modulate maximum intensity. Emanuel (1986, 2012) posited a dynamic-thermodynamic potential intensity theory (E-PI), in which the TC is modeled as a Carnot engine. According to E-PI theory potential intensity is highly sensitive to the difference between the SST saturation enthalpy and the actual enthalpy of near-surface air at the radius of maximum winds (RMW). This enthalpy disequilibrium, outflow temperature, and the enthalpy transfer coefficient-drag coefficient ratio (*C _{K}*/

*C*) combine to determine the E-PI value.

_{D}

In E-PI theory energy production from ocean-air enthalpy fluxes is in balance with frictional dissipation. Therefore:

where *r* is the radius from the TC center; *ρ* is the near-surface air density; *V _{REF}* is the wind at the flux reference level;

*k*is the SST saturation enthalpy;

^{*}*k*is the enthalpy of unsaturated near-surface air; and ε is a thermal efficiency term that depends on SST, boundary layer temperature, and outflow temperature.

_{b}

In E-PI calculations winds near the RMW are assumed to dominate the Eqn. (1) balance. The maximum gradient wind speed is assumed to be close to the maximum wind speed at the flux reference level (Emanuel and Rotunno 2011). Therefore, Eqn. (1) is evaluated at the RMW (indicated by the subscript *M*), giving

where *V _{M}* indicates the maximum gradient wind (Emanuel and Rotunno 2011).

Despite E-PI's widespread use, observational and numerical modeling studies have shown that some TCs can exceed their E-PI values. Some modeling and observational studies suggest that ocean-air energy input at radii well beyond the eyewall can contribute to maximum intensity. Here we derive and examine an expanded E-PI calculation that accounts for energy production and frictional dissipation at multiple radii (therefore accumulating over a range of wind speeds). We test the sensitivity of this PI calculation to variations in radial energy production profiles.

** **

**Derivation of Expanded E-PI calculation**

Starting with Eqn. (1), the aggregate balance between energy production and frictional dissipation is evaluated at two radii: the RMW and Radius 2 (subscript R2), where wind speed is λV_{MAX}. Assuming that fractional energy production (*a*) occurs at the RMW, and *b* (1-*a*) energy production occurs at Radius 2, and integrating, Eqn. 1 becomes:

The λ in the *b* energy production at R2 regardless of λ at R2.

If (*k*^{*}-*k _{b}*),

*C*, and

_{K}*C*are radially-constant, Eqn. (3) simplifies to

_{D}Eqns. (3) and (4) can be extended to any number of radii.

The PI equation can also be written by defining a weighting function *A*=*A*(*R*^{*}), where *A* defines the weighted energy input as a function of R/RMW (*R ^{*}*). We can also write

*V*/

*V*(

_{M}*V*

^{*}),

*C*,

_{K}*C*, and (

_{D}*k*

^{*}-

*k*) as functions of R

_{b}^{*}. The equation for maximum gradient wind speed then becomes

Eqn. (5) permits a PI calculation for any radial wind profile and energy production profile between the RMW and the environment.

**Effect of radial energy production profiles on calculated PI**

** **

We calculate expanded PI (Eqn. 5) using various hypothetical energy production profiles *A*(*R*^{*}) (Fig. 1) and compare these results to PI from the standard E-PI calculation. For all calculations presented SST is set to 27° C; outflow temperature is set to 215 K; *C _{K}*/

*C*is set to 0.5; the temperature (relative humidity) at which ocean-air fluxes occur is set to 26° C (80%); and dissipative heating is ignored. The wind profile of the TC is a modified Rankine Vortex in which wind speed decreases outside the RMW with

_{D}*R*

^{-0.5}.

Figure 1. Radial energy production profiles examined in this study.

The set of hypothetical energy production profiles examined here comprise radially-constant energy production beyond the RMW (Fig. 1a), linearly decreasing production beyond the RMW (Fig. 1b), exponentially decreasing production beyond the RMW (Fig. 1c), and production beyond the RMW governed by a negative arctangent function (Fig. 1d). Each energy production profile terminates at 2, 4, or 6 RMW. All profiles account for the area increase with increasing radius; for example, the area under a curve between 2 RMW and 4 RMW reflects the energy production in the annulus between those radii.

For all expanded PI values calculated using the energy production profiles in Fig. 1, the maximum gradient wind speed is between 8 and 42 m s^{-1} greater than that calculated using standard E-PI theory (Table 1). While some energy production profiles likely yield unreasonable maximum intensities, other profiles give more reasonable values. For example, when energy production varies radially with *R*^{*-3} and ocean-air energy fluxes out to 2 RMW contribute to maximum TC intensity, maximum gradient wind speed is 8.4 m s^{-1} greater than using standard E-PI theory (Table 1).

Table 1. Maximum gradient wind speed and increase over standard E-PI values calculated using each energy production profile in Fig. 1.

These results demonstrate that if much of the energy production relevant to maximum intensity occurs beyond the eyewall, a TC can attain a substantially higher intensity than predicted by E-PI theory. We propose that this new PI formulation serves as a bridge between E-PI theory and frameworks in which TC intensity is governed by the total entropy acquired by inflow air at all radii.