238 MALKus [chap. 4 



or the angle at which they cross circles of constant r ; / is the Coriolis parameter, 

 and the remaining symbols have the same meaning as in the trade case. Addi- 

 tional assumptions made so far are as follows : 



(1) The pressure field is radially nearly symmetrical as observed, i.e., 

 (1/r) dpl8ip<^ Bpjdr. Since all other quantities may vary from one trajectory to 

 the next, and thence with azimuth angle ip, this choice does not restrict the 

 applicability of the model to symmetrical circulations. 



(2) The vertical transport of s-momentum by the mean motion w dujdz 

 {w is the vertical velocity) is small compared to dulds. This assumption is valid 

 because the vertical motion is zero at the ground and dujdz is small through 

 the inflow layer except quite close to the ground. Vertical momentum transport 

 by convective-scale elements is included in the shearing stress term. 



(3) Lateral turbulent transport of s-momentum is neglected compared to 

 vertical transport, with the hypothesis that momentum from the air inside 

 the hurricane is abstracted by the ocean and not diffused laterally outward 

 by small-scale eddies. This assumption is probably weak if single trajectories 

 or only small segments of a storm are considered, but, due to the difficulty of 

 prescribing vertical eddy transport accurately, is not critical at present. 



(4) The wind direction is nearly constant with height through the inflow 

 layer so that the shearing stress term drmldz may be omitted. 



We shall now substitute r/cos j8 for R, the radius of trajectory curvature in 

 (53). This is exact for the logarithmic spiral, which is closely followed by 

 observed hurricane trajectories (Senn, Hiser and Bourret, 1957); here cosj3 = 

 constant. It is nearly true when the inflow angle varies only slowly along the 

 trajectory. When |8:^20° or less, R differs from r by only about 6% for any 

 trajectory. Let equation (52) be multiplied by cos j8 and equation (53) by sin j8. 

 The pressure-gradient force is then eliminated by combining these equations 

 and we have, after dividing again by cos {3, 



— sm B + fu tan B — u-;— = — (54) 



r ^ '' ^ ds p dz ^ 



Upon averaging vertically through the inflow layer of height Sz we obtain 



du 



p hz 



— sin B + fu tan B — u^ 



r ds 



= Tso = CdpqUo" (55) 



introducing (17) for surface stress. The symbol ~ denotes vertical averaging. 

 We neglect the slight difference between po and p, also between uq and u since 

 the vertical shear above anemometer level is very weak in hurricane interiors, 

 especially over water. The important assumption in going from the right side 

 of equation (54) to that of (55) is that the shearing stress vanishes at the top 

 of the inflow layer, which is near the level of maximum wind. Since we shall 

 consider only quantities averaged through the inflow layer, the symbol ~ is 

 henceforth omitted. 



