Maximum potential intensity
The maximum potential intensity of a tropical cyclone is the theoretical limit of the strength of a tropical cyclone. Maximum potential intensity. Due to surface friction, the inflow only partially conserves angular momentum. Thus, the sea surface lower boundary acts as both a source (evaporation) and sink (friction) of energy for the system. This fact leads to the existence of a theoretical upper bound on the strongest wind speed that a tropical cyclone can attain. Because evaporation increases linearly with wind speed (just as climbing out of a pool feels much colder on a windy day), there  is a positive feedback on energy input into the system known as the Wind-Induced Surface Heat Exchange (WISHE) feedback. This feedback is offset when frictional dissipation, which increases with the cube of the wind speed, becomes sufficiently large. This upper bound is called the "maximum potential intensity", formula_1, and is given by where formula_3 is the temperature of the sea surface, formula_4 is the temperature of the outflow ([K]), formula_5 is the enthalpy difference between the surface and the overlying air ([J/kg]), and formula_6 and formula_7 are the surface exchange coefficients (dimensionless) of enthalpy and momentum, respectively. The  surface-air enthalpy difference is taken as formula_8, where formula_9 is the saturation enthalpy of air at sea surface temperature and sea-level pressure and formula_10 is the enthalpy of boundary layer air overlying the surface. The maximum potential intensity is predominantly a function of the background environment alone (i.e. without a tropical cyclone), and thus this quantity can be used to determine which regions on Earth can support tropical cyclones of a given intensity, and how these regions may evolve in time. Specifically, the maximum potential intensity has three components, but its variability in space and time is due  predominantly to the variability in the surface-air enthalpy difference component formula_5. Derivation. A tropical cyclone may be viewed as a heat engine that converts input heat energy from the surface into mechanical energy that can be used to do mechanical work against surface friction. At equilibrium, the rate of net energy production in the system must equal the rate of energy loss due to frictional dissipation at the surface, i.e. The rate of energy loss per unit surface area from surface friction, formula_13, is given by where formula_15 is the density of near-surface air ([kg/m]) and formula_16  is the near surface wind speed ([m/s]). The rate of energy production per unit surface area, formula_17 is given by where formula_19 is the heat engine efficiency and formula_20 is the total rate of heat input into the system per unit surface area. Given that a tropical cyclone may be idealized as a , the Carnot heat engine efficiency is given by Heat (enthalpy) per unit mass is given by where formula_23 is the heat capacity of air, formula_24 is air temperature, formula_25 is the latent heat of vaporization, and formula_26 is the concentration of water vapor. The first component  corresponds to sensible heat and the second to latent heat. There are two sources of heat input. The dominant source is the input of heat at the surface, primarily due to evaporation. The bulk aerodynamic formula for the rate of heat input per unit area at the surface, formula_27, is given by where formula_8 represents the enthalpy difference between the ocean surface and the overlying air. The second source is the internal sensible heat generated from frictional dissipation (equal to formula_13), which occurs near the surface within the tropical cyclone and is recycled to the system. Thus, the total rate  of net energy production per unit surface area is given by Setting formula_12 and taking formula_34 (i.e. the rotational wind speed is dominant) leads to the solution for formula_1 given above. This derivation assumes that total energy input and loss within the system can be approximated by their values at the radius of maximum wind. The inclusion of formula_36 acts to multiply the total heat input rate by the factor formula_37. Mathematically, this has the effect of replacing formula_3 with formula_4 in the denominator of the Carnot efficiency. An alternative definition for the maximum potential intensity, which is mathematically equivalent  to the above formulation, is where CAPE stands for the Convective Available Potential Energy, formula_41 is the CAPE of an air parcel lifted from saturation at sea level in reference to the environmental sounding, formula_42 is the CAPE of the boundary layer air, and both quantities are calculated at the radius of maximum wind. Characteristic values and variability on Earth. On Earth, a characteristic temperature for formula_3 is 300 K and for formula_4 is 200 K, corresponding to a Carnot efficiency of formula_45. The ratio of the surface exchange coefficients, formula_46, is typically taken to be 1. However, observations suggest  that the drag coefficient formula_7 varies with wind speed and may decrease at high wind speeds within the boundary layer of a mature hurricane. Additionally, formula_6 may vary at high wind speeds due to the effect of sea spray on evaporation within the boundary layer. A characteristic value of the maximum potential intensity, formula_1, is . However, this quantity varies significantly across space and time, particularly within the seasonal cycle, spanning a range of . This variability is primarily due to variability in the surface enthalpy disequilibrium ( formula_5 ) as well as in the thermodynamic structure of the troposphere, which are controlled by  the large-scale dynamics of the tropical climate. These processes are modulated by factors including the sea surface temperature (and underlying ocean dynamics), background near-surface wind speed, and the vertical structure of atmospheric radiative heating. The nature of this modulation is complex, particularly on climate time-scales (decades or longer). On shorter time-scales, variability in the maximum potential intensity is commonly linked to sea surface temperature perturbations from the tropical mean, as regions with relatively warm water have thermodynamic states much more capable of sustaining a tropical cyclone than regions with relatively cold water. However, this relationship is  indirect via the large-scale dynamics of the tropics; the direct influence of the absolute sea surface temperature on formula_1 is weak in comparison. Empirical Limit on Intensity. An empirical limit on tropical cyclone intensity can also be computed using the following formula: formula_52 Where formula_53 is the maximum potential velocity in meters per second; formula_24 is the sea surface temperature underneath the center of the tropical cyclone, formula_55 is a reference temperature (30 ˚C) and formula_56, formula_57 and formula_58 are curve-fit constants. When formula_59, formula_60, and formula_61, the graph generated by this function corresponds to the 99th percentile  of empirical tropical cyclone intensity data.