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

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[ 505 ]DIRECTIONAL RECORDING OF SWELL FROM DISTANT STORMSBY W. H. MUNK, G. R. MILLER, F. E. SNODGRASS AND N. F. BARBERInstitute of Geophysics and Planetary Physics, University of California, La Jolla(Communicated by G. E. R. Deacon, FER.S.-Received 19 April 1962)PAGE1. INTRODUCTION 5062. INSTALLATION 507(a) Site 507(b) Method of installation 509(c) The Vibrotron pressure transducer509(d) Recording 5103. SAMPLE RECORDS 5124. DIGITAL ANALYSIS 512(a) The detection of errors 512(b) 'Aliasing' and hydrodynamicfiltering 512(c) Suppression of tides 513(d) The spectral matrix 513(e) The reliability of the computedspectra 5145. FIRST INFERENCE 515(a) Sample spectrum 515(b) Power spectra 515(c) Direction finding 515(d) Beam width 517(e) Dispersion 518(f) Effect of finite depth on ridgelines 5196. GENERAL THEORY OF ARRAYS 520(a) Elementary wave trains 520(b) The spectrum 5217. INTERPRETATION OF THE CROSS-SPECTRALMATRIX 523(a) Conventional treatment 523(b) A best-fitting single wave train 524(c) A best-fitting pair of wave trains 526(d) Experimental test of the dispersionlaw 527[Plate 7]CONTENTSPAGE(e) Analysis by iteration 528(f) Fourier-Bessel analysis 531(g) Other methods 5338. RESULTS 5409. REFRACTION 54310. REFLEXION 54911. IDENTIFICATION OF INDIVIDUAL EVENTS 551(a) The hurricane of 23 October 556(b) Antipodal swell 557(c) The quickly moving storm of 12-4May 558(d) The end of the storm season 561(e) Narrow beams and sharp ridges 561(f) A discrepancy in wave direction? 56212. AN EXPERIMENT IN HETERODYNING 562(a) Removal of frequency shift 562(b) Heterodyne theory 563(c) The results 56413. GENERATION AND DECAY 565(a) Ridge cuts 565(b) Geometric spreading 566(c) Attenuation 56714. MEAN MONTHLY SPECTRA 56715. SCATTER AND ABSORPTION 571(a) The evidence 571(b) Scattering by waves 572(c) Scattering by turbulence 575(d) Island scattering 576(e) Absorption 58016. THE SOUTHERN SWELL 581REFERENCES 583VOL. 255 A. io62. (Price $x 5s.) 62 [Published i8 April I963
506 W. H. MUNK, G. R. MILLER, F. E. SNODGRASS AND N. F. BARBERWe have measured the distribution of wave energy with frequency and direction for several months,and attempted to interpret the resulting field E(f, 0, t) in terms of pertinent geophysical processes. Thefluctuating pressure on the sea bottom was measured with a triangular array of sensitive transducerslocated 2 miles off shore from San Clemente Island, California, at a depth of 100 m. The transduceroutput was brought ashore by undersea cable and telemetered to the laboratory at La Jolla,California, where it was digitally recorded. The analysis by means of high-speed computers consistsof two distinct steps: (i) computing the spectral matrix Sijk = Cijk+iQijk between the records xi(t)and xj(t) (i,j = 1, 2, 3) at frequency k (k- 1, 2, ..., 1]00); Cijk, Qijk are the co- and quadraturespectra, and -Siik CLiik the ordinary power spectra of' the three records; (ii) interpreting Sijk interms of the energy spectruim in frequency and direction (or equivalently in horizontal wave-numberspace). 'T'he problem is discussed in some detail. The method adapted is to find the energy anddirection of a point source which can best account (in the least-square sense) for Sijk; to computethe matrix S'k associated with this optimum point source; then to find the optimum point source ofthe 'residual matrix' Sijk.- YS'J (we chose y = 0-1), etc, for 100 iterations, or until the residualmatrix vanishes. The calculation is made independently for each frequency. The iterativepoint-source method is satisfactory if the incoming radiation is associated with one or two strongsources.From daily measurements over several months we obtain E( f, 0, t). The non-directional spectra,E( f, t) =fE(f, 0, t) dO are contoured, and show pronounced slanting ridges associated with thedispersive arrivals from individual storms. Their slope and intercept give the distance and time of thesource, respectively; the energy density along the ridge is associated with the wind speed, the widthof the ridge with the storm duration, and the variation of width with the rate at which the stormapproaches the station. The distribution of energy with direction along these ridge lines give thedirection of the source (after correction for local refraction), some upper limit to source apertureand an indication of its motion normal to the line-of-site. With distance, direction, time and fiveother source characteristics thus derived from the wave records, the sources are fairly well specified.Comparison with weather maps leads to fair agreement in most instances. FromJune to Septembermost sources lie in the New Zealand-Australia-Antarctic section or in the Ross Sea. In threeinstances the swell was generated in the Indian Ocean near the antipole and entered the Pacificalong the great circle route between New Zealand and Antarctica. By October the southern winterhas come to an end and the northern hemisphere takes over as the chief region of swell generation.Antarctic pack ice may be a factor, by narrowing the 'window' between Antarctica and NewZealand or by impairing storm fetches in the Ross Sea. The northward travelling swell is furtherimpaired by densc island groups in the South Pacific, and an attempt is made to estimate theresulting scattering. The recorded spectrum at frequencies below 0 05 c/s is in rough accord withwhat would be expected from typical storms after due allowance for geometric spreading; at higherfrequencies the spectrum fails to rise in the expected manner. A resonant interaction between theswell and the trade wind sea may be responsible for back-scattering frequencies above 0 05 c/s. Theobserved widths of the beam (< 0-1 radian) ancd of the spectral peak (tAf/f < 0.1) put an upper limiton forward scattering over the transmission path of 104 wavelengths, and thus place some restrictionon the energy of turbulent eddies with dimensions of the order of 1 km in the shallow layers of thesea. Finally, a comparison of mean monthly wave spectra with those from individual storm permitsus to estimate the absorption time of the North Pacific basin. The resultant value of j week is notinconsistent with the assumption that most of the swell energy is absorbed along the boundaries.The inferred coastal absorption is roughly in accord with what we found at San Clemente Island(beach slope 1:30): the coast line changes from a predominantly reflecting to a predominantlyabsorbing boundary as the frequency increases from 0 03 to 0 05 c/s.1. INTRODUCTIONAlmost fifteen years ago in the pages of this Journal, one of us presented power spectra ofocean waves and swell off Pendeen and Perranporth in north Cornwall (Barber & Ursell1948). The outstanding feature of these spectra is the successive shift of peaks towardhigher frequencies. This is the expected behaviour of dispersive wave trains from rather
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