252 MALKTjs [chap. 4 



Adhere A and B are held fixed. We now substitute u from the sohition to the 

 differential equation (57) as a function of C{r) and find (after some labor) the 

 value of sin /3 which maximizes dKjdt. In the case where ksp{To-Ta) was 

 fixed as in Table XIX. we get the now obvious result that /S must be 20"" I The 

 method, however, permitted a rapid extension to exchange and release efficiency 

 requirements for intense and record storms, which will only be briefly sum- 

 marized by a physical discussion here. 



As we mentioned earlier, the dynamic calculation gave an intense storm 

 with an inflow angle of about 2o\ Higher inflow angles permit stronger central 

 winds due to reduction of trajectory distance and thereby of frictional dissipa- 

 tion of kinetic energy. The larger the inflow angle, the closer the velocity 

 profile with r approaches that of the constant angular momentum vortex 

 which, without friction, would arise from any inflow angle. Ver\- high inflow 

 angles are prevented in real situations due to thermodynamic restrictions on 

 the reahzable pressure gradients. AATien the trajectory distance is reduced too 

 much, the air cannot pick up and release by condensation enough extra heat 

 energy to achieve the required gradient of Q and thus of surface pressure. To 

 achieve record storms, with 200-knot winds, inflow angles of 25'' and central 

 pressure at or below 900 mb. the analysis shows that .4. or core pressure gradient 

 cpjcs, must be held at a value about twice that for the moderate storm studied. 

 Since values of To— Ta of 5°-6T are probably excessive, it appears marginal 

 whether normal heat-energy exchange mechanisms are adequate ; it may prove 

 necessary to call upon slanting eye walls to sustain the pressure fields in these 

 extreme situations. 



In any case, maximum development is almost surely restricted by boundary 

 exchange since, as pursued further in the paper by Malkus and Riehl (1960), 

 dissipation increases as the wind speed cubed, and input and condensation only 

 linearly. It is interesting to note that their extreme storm released condensation 

 energy more than twice as efficiently as did the moderate storm. Attempts are 

 now being made to apply the energetic principles of relative stability to cumu- 

 lonimbus convection in hurricanes, the other facet of pressure-field maintenance. 

 It is hoped to relate thereby the sizes of the convection elements to large-scale 

 dynamics, although the degree of rigor achievable is not yet clear ; the problem 

 hes, as it did in the exchange aspect, in specification of the constraints upon the 

 system. 



We have now completed our presentation of the efforts to relate air-sea 

 exchange to the dynamics of large-scale circulations in terms of models which 

 are at least, to a degree, analytic and begin from the basic laws governing fluid 

 motions. These endeavors date only from the 1950's. They are plainly in the 

 rudimentary beginning stages and are conflned to the relatively simplest and 

 best-documented aspects of planetary flows, such as the lower trades and the 

 mature hurricane. It is nevertheless significant that such models have been 

 attempted, and their inquiries framed in such a way that the new developments 

 in the fundamental mechanics of turbulent heated fluids may be used, if 

 crudely, to guide the question-asking of those facing real geophysical problems. 



