We investigate bulk air-sea enthalpy fluxes in the two-parameter hyperbolic space (surface wind-dependence and thermodynamic-dependence) in which the fluxes take place, for two category 1 hurricanes (Isaac of 2010 and Nate of 2017), and ten major hurricanes (Isidore of 2002; Lili of 2002; Ivan of 2004; Emily of 2005; Dean of 2007; Felix of 2007; Gustav of 2008; Ike of 2008; Earl of 2010; and Edouard of 2014). The goal is identifying the relative contribution of surface wind speeds and thermodynamic disequilibrium to heat extraction from the ocean during steady intensification, rapid intensification, steady-state, and secondary eyewall formation in major hurricanes. Preliminary results indicate that intense latent heat fluxes of more than 1200 W m-2 and 800 W m-2 were supported by moisture disequilibrium of ~9 g Kg-1 during rapid intensification and secondary eyewall formation, respectively, in major Hurricane Earl of 2010, under relatively moderate surface wind speeds. By contrast, moderate latent heat fluxes of ~500 W m-2 were supported by moisture disequilibrium of ~6 g Kg-1 during the moderate intensification of tropical storm Isaac into a hurricane. Both during Earl and Isaac, peak values of moisture disequilibrium and enthalpy fluxes occurred over warmer oceanic features. These results indicate that acquiring accurate measurements of 10-m wind speeds, as well as air-sea moisture and temperature differences, is critical to improve our scientific understanding of intensity change in TCs, as well as to provide the optimal forcing at the sea surface in forecasting models of TC intensity.