Directional Recording of Swell from Distant Storms - Department of ...

576 W. H. MUNK, G. R. MILLER, F. E. SNODGRASS AND N. F. BARBERassociated with the energy-containing components of the turbulence. The total scatteredenergy is of the ordery(k) = 2 f s(8, k) dO 30g-1k0J. (15.3)On the basis of some diffusion experiments by Stommel, Phillips suggests e -0-5 cm2 S-3,and this leads to y = 0-O2 per day for waves of frequency 50 c/ks. These waves have a groupvelocity of 120 latitude per day, and so cross the trade wind belt in 2 days. Total traveltimes are 10 days. Thus 400 or 200 of the energy is scattered, depending upon whether weconsider scattering in the trades only, or along the entire transmission path. In the lattercase, 80 % of the energy comes along the great circle route, 20 % is scattered and most of thisthrough small angles, with a root-mean-square value estimated at 60?. The correspondingangle for the total field, direct plus scattered, is then 12?, and the associated delay, t- t',is 1 h according to (15.1). This amount of forward scattering would then go unnoticed.But the waves are scattered in the surface layers where the value of 6 might be an order ofmagnitude above average, and the scattering would then bejust below the limits of detectionwith the existing angular resolution of our array.(d) Island scatteringtNumerous island groups are in the path of the swell (figures 49 and 50). Typically theseare atolls many times larger than the wave lengths here under consideration. A representativeslope from the surface to several hundred fathoms is 260. With this slope the waterdepth reaches half the wavelength at a distance of 1P03 wavelengths from the shore line. Thesteep slope suggests effective reflexion of the low frequencies, and forward scattering at thesteep sides. Where the islands are dense, multiple reflexion may contribute to small anglescattering. Forward scattering leads to a broadening of the beam and a spreading of thedispersive spectral peaks.The simplest theoretical construction is to compute the fractional shadow cast by theislands for waves from various directions (figure 51). On this basis the arrival of swell fromthe Indian Ocean is confined to two narrow windows. The window south of New Zealandis also limited by pack ice.The shadow method assumes total absorption (by breaking) whenever the wave frontsintersect land, and total penetration otherwise. Reflexion and diffraction are then neglected.We shall consider these processes.Geometric optics. In the case of specular reflexion from a smooth, cylindrical island ofradius a > A, the ratio of reflected to incident power (as in equation 15.2) is given bys(O, k) == ar2(k) sin 101, (15.4)where r2(k) is the energy reflectivity (assumed independent of incidence angle). There isno forward reflexion, s(0, k) 0, and a maximum backward reflexion s(ir, k) =-ar2. Thetotal reflected and absorbed power per unit frequency band arerespectively. 2ar2, 2a(1-r2) (15.5)t We are indebted to George Hess for the calculations in this section.