PDF《H. E. WILLOUGHBY》

2006 W I L L O U G H B Y E T A L .

1103 and zero elsewhere. The coefficient C is chosen to make w(1) ? 1, and the exponent k is the order of the bell and ramp curves, even though the resulting polynomi- als are of order 2k and 2k ? 1, respectively. We have coined the term bellramp functions to denote this family of polynomials. Since the shapes of fitted pro- files are insensitive to the order of the polynomials, we selected fourth-order ramp functions for smoothness and differentiability. Based upon parameters Rmax, Vmax, X1, and n, the full wind profile is constructed as follows. First, the width of the transition R1 ? R2 is specified a priori at a value between

10 and

25 km. Then, the location of the transition zone is determined by requiring the radial derivative of (1b) to vanish at r ? Rmax, recognizing that Vi(Rmax) ? Vo(Rmax) ? Vmax. This condition yields the value of w at the wind maximum: w?Rmax ? R1 R2 ? R1 ?? ?Vi ?r ?Vi ?r ? ?Vo ?r ? nX1 nX1 ? Rmax , ?3? which may be solved for R1 through numerical inver- sion of (2). As shown subsequently, in many situations the pro- file described by (1a)C(1c) suffers from the problem that vexed the Holland profile in Part I. Relatively large values of X1 chosen to generate profiles that match the outer part of the vortex may fail to capture the rapid decrease of wind just outside the eyewall;

conversely, smaller values of X1 generate profiles that match the steep gradient outside the eyewall and decrease too rapidly farther from the center........