Mathematics – Logic
Scientific paper
May 1966
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1966rspta.259..533m&link_type=abstract
Philosophical Transactions for the Royal Society of London. Series A, Mathematical and Physical Sciences, Volume 259, Issue 1105
Mathematics
Logic
115
Scientific paper
Nineteen years of hourly tide readings at Honolulu, Hawaii, and Newlyn, England, are analysed without astronomical prejudice as to what frequencies are present, and what are not, thus allowing for background noise. The method consists of generating various complex input functions ci(t) for the same time interval as the recorded tide zeta (t), and of determining the associated lag weights w in the convolutions hat{zeta}(t) = \underset ito{sum}\underset sto{sum}wisci(t - τ s) + \underset ijto{sum}\underset ss'to{sum}wijss'ci(t - τ s)cj(t - τ s') + ldots by the condition < (zeta - hat{zeta})2> = minimum. The two expansions represent linear and bilinear processes; the Fourier transforms of w for any chosen i (or ij) are the linear (or bilinear) admittances. Input functions are the (time variable) spherical harmonics of the gravitational potential and of radiant flux on the Earth's surface; these functions are numerically generated hour by hour, directly from the Kepler-Newton laws and the known orbital constants of Moon and Sun, without time-harmonic expansions (unlike the harmonic method of Kelvin-Darwin-Doodson). The radiative input is required to predict non-gravitational tides, and it allows for the essential distinction that the Earth is opaque to radiation and transparent to gravitation. The input functions are confined to bands, centred at 0, 1, 2,... c/d, which occupy roughly one-fourth the frequency space (less for radiational inputs) at the -60 dB level. Within these bands the admittances turn out to be reasonably smooth, as expected. Subsequently we force the admittances to be smooth by truncating the expansion in s. Subject to this `credo of smoothness' the overlapping gravitational, radiational and nonlinear admittances can be disentangled. The procedure consists of computing the lag weights by inverting a correlation matrix of input functions, and the admittances by subsequent Fourier inversion; the varying uncertainties in the tidal components are automatically allowed for. The residual record, zeta (t) - hat{zeta}(t), is associated with the irregular oscillations induced by winds and atmospheric pressure. The residual spectrum smoothly fills the space between the bands centred on 0, 1, 2, c/d and rises sharply toward `zero' frequency, reflecting a similar pattern in the meteorological spectra. The residual spectrum rises into cusp-like peaks about each of the strong spectral lines, as might be expected from a low-frequency modulation of `tidal carrier frequencies', but detailed analyses fail to confirm this hypothesis. Another feature is a slight 2 c/y `jitter' in the admittances, probably the result of some trilinear interactions. Once the ocean's response to various specified inputs has been determined for a given station, it can serve as a basis for a tide prediction which is perhaps more physical than the harmonic method now in use. The convolution formalism explicitly distinguishes between astronomic inputs and oceanographic response, with Kepler-Newtonian mechanics fully taken into account (in the harmonic method, K.-N. mechanics serves only to identify principal tidal frequencies). Moreover, the response method leads to a systematic expansion for weak nonlinearities. The response method gives better prediction with fewer station constants, but the improvement is small compared to the low frequency residuals. To reduce these we have tried a Wiener-type self prediction with past values of the recorded tide as input function. At Honolulu the residual variance can be reduced by 50% for a prediction time of 40 days. At Newlyn where the effect of local weather (storm tides) is severe, the response method should be generalized to include as additional input functions some pertinent meteorological variables as well as sea level at other tide stations.
Cartwright David Edgar
Munk W. H.
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