Directional recording of swell from distant storms

We have measured the distribution of wave energy with frequency and direction for several months, and attempted to interpret the resulting field E(f, $\theta $, t) in terms of pertinent geophysical processes. The fluctuating pressure on the sea bottom was measured with a triangular array of sensitive transducers located 2 miles off shore from San Clemente Island, California, at a depth of 100 m. The transducer output was brought ashore by undersea cable and telemetered to the laboratory at La Jolla, California, where it was digitally recorded. The analysis by means of high-speed computers consists of two distinct steps: (i) computing the spectral matrix S$_{ijk}$ = C$_{ijk}$ + iQ$_{ijk}$ between the records x$_{i}$(t) and x$_{j}$(t) (i, j = 1, 2, 3) at frequency k (k = 1, 2,..., 100); C$_{ijk}$, Q$_{ijk}$ are the co- and quadrature spectra, and S$_{iik}$ = C$_{iik}$ the ordinary power spectra of the three records; (ii) interpreting S$_{ijk}$ in terms of the energy spectrum in frequency and direction (or equivalently in horizontal wave-number space). The problem is discussed in some detail. The method adapted is to find the energy and direction of a point source which can best account (in the least-square sense) for S$_{ijk}$; to compute the matrix S$_{ijk}^{\prime}$ associated with this optimum point source; then to find the optimum point source of the 'residual matrix' S$_{ijk}$ - $\gamma $S$_{ijk}^{\prime}$ (we chose $\gamma $ = 0$\cdot $1), etc, for 100 iterations, or until the residual matrix vanishes. The calculation is made independently for each frequency. The iterative point-source method is satisfactory if the incoming radiation is associated with one or two strong sources. From daily measurements over several months we obtain E(f, $\theta $, t). The non-directional spectra, E(f, t) = $\int $E(f, $\theta $, t) d$\theta $ are contoured, and show pronounced slanting ridges associated with the dispersive arrivals from individual storms. Their slope and intercept give the distance and time of the source, respectively; the energy density along the ridge is associated with the wind speed, the width of the ridge with the storm duration, and the variation of width with the rate at which the storm approaches the station. The distribution of energy with direction along these ridge lines give the direction of the source (after correction for local refraction), some upper limit to source aperture and an indication of its motion normal to the line-of-site. With distance, direction, time and five other 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. From June to September most sources lie in the New Zealand-Australia-Antarctic section or in the Ross Sea. In three instances the swell was generated in the Indian Ocean near the antipole and entered the Pacific along the great circle route between New Zealand and Antarctica. By October the southern winter has 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 New Zealand or by impairing storm fetches in the Ross Sea. The northward travelling swell is further impaired by dense island groups in the South Pacific, and an attempt is made to estimate the resulting scattering. The recorded spectrum at frequencies below 0$\cdot $05 c/s is in rough accord with what would be expected from typical storms after due allowance for geometric spreading; at higher frequencies the spectrum fails to rise in the expected manner. A resonant interaction between the swell and the trade wind sea may be responsible for back-scattering frequencies above 0$\cdot $05 c/s. The observed widths of the beam (< 0$\cdot $1 radian) and of the spectral peak ($\Delta $f/f < 0$\cdot $1) put an upper limit on forward scattering over the transmission path of 10$^{4}$ wavelengths, and thus place some restriction on the energy of turbulent eddies with dimensions of the order of 1 km in the shallow layers of the sea. Finally, a comparison of mean monthly wave spectra with those from individual storm permits us to estimate the absorption time of the North Pacific basin. The resultant value of $\frac{1}{2}$ week is not inconsistent 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 predominantly absorbing boundary as the frequency increases from 0$\cdot $03 to 0$\cdot $05 c/s.