SECT. 2] I.ARGK-SCALE INTERACTIONS 239 



Dividing out u and utilizing the definition (Hilds= —{duj'dr) sin ^, we derive 

 tlie following first-order differential equation for the velocity u along any 

 trajectory as a function of radial distance, r. from the storm center 



du 

 dr 



CD 



r sin jS 82 



^ (56) 



cos j8 



Since it will prove more convenient to set boundary conditions on u^, the 

 tangential velocity component, we may obtain an equation for it by using the 

 fact that u = m^/cos ^. namely, 



duj, 



- + C(r) 

 r 



-L (57) 



where C{r) = — colsin ^ 8z. 



This simple differential equation is readily solved for u^ when C{r) and a 

 boundary condition are specified. 



The results of a mean hurricane-momentum budget by Palmen and Riehl 

 (1957) suggest that, although cd and 8z may each vary by a factor of two 

 under hurricane conditions (increasing inward), their ratio is constant within 

 20%. The ratio c^/Ss; was chosen as 1.36 x 10^^ cm~i for the analysis, which 

 permits Cd to vary from 1.1 to 2.7 x 10"^ for a range of depth of the inflow layer 

 from 750 m to 2 km. 



We make the simplest choice of ^ for our trajectory calculations, namely 

 ^ = |8i = constant for the outer rain area r > 100 km, and decreasing from there 

 linearly to zero at r = 25 km, the assumed radius of the eye wall. 



Integrating (57) when C(r) = constant, we have 



f 

 u^r = ±-^{l-Cr) + Cie-cr_ (53) 



The constant of integration Ci is evaluated by choosing an outer radius ro 

 where the relative vorticity vanishes, that is du^ldr+{u^lr) = 0. Since Ur = 

 U4, tan jS, Ur also satisfies this relation at ro, which thus separates the region of 

 horizontal convergence (rain area) from that of the surrounding horizontal 

 divergence. Choice of ro is arbitrary, but the computed structure of the rain 

 area is not sensitive to the choice so long as ro> 500 km, a figure suggested by 

 moderate storm data. Under these conditions 



u^r = /[l-(7r-eC(ro-r)]. (59) 



For the inner rain area, namely rE<r<ri where ri is 100 km and te, the eye 

 boundary, is 25 km, we choose sin /S = sin ^^^(r — r£)/(ri — r^). Near the core, 

 Coriolis forces are negligible compared to centrifugal, leaving only the homo- 

 geneous part of (57). Thus, for the inner rain area 



u^r = c2(r-r£)-('-i-'-A)c^ (60) 



where C2 is determined by matching u^ at 100 km with the results of (59). 



