240 MALKUS [chap. 4 



The solution to the differential equation under specified boundary conditions 

 is thus an infinite family of logarithmic spirals (modified slightly in the core), 

 each differing from the others by means of a different inflow angle ^ [or more 

 generally by a different parameter C{r)]. We shall indicate later how the non- 

 uniqueness is resolved through thermal constraints and discuss the relation- 

 ship between the parameter C{r) and sea-air exchange. 



With the chosen values of tq, ri and the ratio of CdI^z, inflow angles of 20° 

 give moderate storms (maximum W0~ 100 knots) while inflow angles of about 

 25° give the much more rarely observed intense ones (maximum W0~175- 

 200 knots). We shall be concerned here mainly with the moderate model 

 hurricane and its relations to the ocean. Unlike the trade- wind model, 

 we do not here plug the heat source in to the differential motion equation 

 directly, but rather work first from the pressure field prescribed by its 

 solution to the heat source using the physical modelling comprised in Figs. 68 

 and 69. 



Table XVII presents the calculated dynamic relationships for the moderate 

 case. First u^ is obtained from (59) and (60), u from u^jcos jS and Ur from 

 W0 tan |8. For the purposes of the vertical motion and shearing stress calcula- 

 tions, the depth of the inflow layer, 82, has been chosen as 1.1 km (or S^ = 

 100 mb), thus cd^ 1.5 x 10~3. The ageostrophic mass flow, needed later in the 

 heat budget, is 'l-nrur ^p/g or proportional to UrV. The horizontal velocity 

 divergence is {l/r) d{Ur-r)ldr, yielding the mean vertical motion w at 1.1 km 

 elevation from mass continuity. Irregular values of divergence and vertical 

 motion occur at r= 100 km through rapid reduction in mass flow arising from 

 the assumption that the inflow angle /8 begins to decrease at that radius. This 

 minor difficulty apart, it is seen that all strong convergence is concentrated in 

 the core. An average ascent rate of about 30 cm/sec, or 1 km/h, is required at 

 the top of the inflow layer. Actually, we envisage this as an average over 

 concentrated convective ascent, with 5-10% of the core area covered with up- 

 drafts of 3-6 m/sec at this level. The surface pressure, ps, was calculated from 

 evaluating gradients from (53) and integrating graphically with a boundary 

 pressure of 1011.8 mb at r= 800 km. 



The features of Table XVII are realistic and consistent with presently 

 available observations. Fig. 70 shows some comparisons. Fig. 70a contains 

 wind and pressure profiles of Table XVII together with those of two medium 

 strength hurricanes obtained by the National Hurricane Research Project. 

 Both storms were encountered between 25°-30°N. Fig. 70b compares mass 

 inflow and radial velocity distributions for these same storms with those 

 calculated in Table XVII. Fig. 70c shows good agreement between the shearing 

 stress of Table XVII (calculated from the u column with (17) and Cz)~ 1.5 x 10-3) 

 as a function of radius and those obtained by Palmen and Riehl (1957) from 

 momentum -budget requirements established from mean hurricane data. We 

 see that the momentum budget is readily balanced with exchange relationships 

 no different from the normal trade situation and a cd comparable to that given 

 in Fig. 6. Even so, it is interesting to note that surface stresses of 40 dynes/cm^ 



