250 MALKUS [chap. 4 



family of logarithmic spiral trajectories for each choice of the boundary condi- 

 tion, ro. Using the language of the new approach, we suggest that the thermal 

 constraints, which in nature operate on the system in addition to the dynamic 

 laws, impose a choice between or a limit upon the range of dynamically possible 

 trajectories, so that most or even all of these may be prevented from occurring 

 in a real situation. The thermal constraints operate through the surface- 

 pressure gradient along the trajectory, s, in the storm core ; the realizable 

 dpics is restricted by the possible heat transfer at the air-sea boundary and 

 the thermodynamics of condensation heat release in the vertical. 



As in the work in Section 6-C (pages 190-201) on the trades, we introduce 

 thermal constraints by means of the first law of thermodynamics. There we 

 wrote a standard meteorological form of this law, namely, 



where H is the rate of sensible heat addition per unit mass. The equation says 

 that sensible heat added may be used to increase the enthalpy (CpT) or to do 

 mechanical work by means of pressure gradients. In the steady-state hurricane 

 situation, for particles moving horizontally and isothermally toward the 

 center, (40) ma}'^ be written 



The well-known kinetic energy equation is obtained by multiplying equation 

 (52) by u, 



du dK u dp u drsz ,„„, 



dt dt p ds p oz 



where K is kinetic energy per unit mass. The term —{ulp)dplds is the rate of 

 production of kinetic energy by pressure forces. It is now apparent that kinetic 

 energy is produced from the oceanic heat source at a maximum rate during 

 isothermal and horizontal motion, since for a prescribed Qsp, dpfds is maximal 

 when dTjdt is zero. 



From equations (61) and (40a), the pressure-gradient force may be related 

 explicitly to the boundary-heat transfers 



Qsp = ksp{To-Ta)u = --^ (66) 



p cs 



so that 



--^ = ksp{To-Ta). (67) 



The pressure gradient along the trajectory is thus limited by the input rate of 

 sensible heat from sea to air. 



A second thermal constraint must be met by the system. Since hydrostatic 

 equilibrium is required, pressures exerted by and on sub-cloud air must be 



