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Numerical simulation of tides in the World Ocean: 3. A solution to the spectral problem

Numerisches Modell der Gezeiten im Weltozean: 3. Lösung der Spektralaufgabe

Simulation numérique de marées dans l'Océan mondial: 3. Une solution au problème du spectre

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Summary

The spectral problem is formulated as an initial boundary-value problem for free oscillations excited by an arbitrary initial disturbance. Numerical solution to this problem shows that the spectrum of eigenoscillations of the World Ocean in a range of periods from 8 hours and above contains more than 30 energetically significant modes. In particular, in semi-diurnal and diurnal spectral bands, there are modes with periods 11.96; 12.50; 12.68; 12.82 and 22.50; 23.87; 25.90 h quite close to the periods of main harmonics of the tide-generating potential. This fact, together with qualitative similarity of spatial pattern of these modes and observed tides, can be a direct proof of the resonance origin of semi-diurnal and diurnal tides in the World Ocean.

Zusammenfassung

Die Spektralaufgabe wird als Anfangs-Randwertaufgabe für Eigenschwingungen, die durch eine beliebige Anfangsstörung angeregt sind, betrachtet. In der numerischen Lösung sind über 30 energetisch signifikante Eigenschwingungen mit einer Periode von 8 Stunden und mehr enthalten. Insbesondere treten in den zwölfstündigen und vierundzwanzigstündigen Spektralbereichen Perioden von 11,96; 12,50; 12,68; 12,82 und 22,50; 23,87; 25,90 h auf, die in der Nähe der halb- und eintägigen Gezeitenkräfte liegen. Dieser Umstand und auch die qualitative Ähnlichkeit dieser Eigenschwingungen mit den beobachteten Gezeiten können als direkter Beweis dafür dienen, daß die zwölfstündigen und vierundzwanzigstündigen Gezeiten im Weltozean durch Resonanz entstehen.

Résumé

Le problème du spectre est formulé en tant que problème initial de valeur aux limites pour des oscillations libres excitées par une perturbation initiale arbitraire. La solution numérique à ce problème montre que le spectre d'oscillations propres de l'Océan mondial dans une gamme de périodes de 8 heures et au-dessus contient plus de 30 modes ayant une signification énergétique. En particulier, dans les bandes du spectre diurnes et semi-diurnes, il y a des modes avec des périodes de 11,96; 12,50; 12,68, 12,82 et 22,50; 23,87; 25,90 h tout à fait proches des périodes d'harmoniques majeures du potentiel générateur de marée. Ce fait, ainsi que la similitude qualitative du type spatial de ces modes et des marées observées, peut être une preuve directe du fait que les marées diurnes et semi-diurnes dans l'Océan mondial naissent par résonance.

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Abbreviations

f :

Coriolis parameter

g :

gravity acceleration

H :

ocean depth

k :

unit vector directed vertically upwards

L1 :

tidal Laplace's operator

L2 :

operator characterizing effects of loading and self-attraction of ocean tides

ℒ:

L1+L2

n :

No. of spherical harmonic in expansion

n :

unit vector directed along outer normal to coastal line

t :

time

u :

depth-averaged vector of tidal current velocity

w :

vector with components (u, ζ)

w * :

vector, complex-conjugated withw

γn :

Love's reduction factor ofn-th order

σ:

frequency of eigenoscillation

ϱ0 :

mean water density

ϱ⊕ :

mean density of Earth's matter

ζ:

deviation of ocean free surface from its undisturbed state

ɛ:

parameter varying within (0,1)

γ=iσ:

eigennumber of Laplace's operator ℒ

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Author notes

  1. Vadim Yu. Gotlib &  Boris A. Kagan

    Present address: Leningrad Branch, P. P. Shirshov Institute of Oceanology, Academy of Sciences USSR, 30 Pervaya Liniya, 199053, Leningrad, USSR

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    1. Vadim Yu. Gotlib
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    2. Boris A. Kagan
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    Gotlib, V.Y., Kagan, B.A. Numerical simulation of tides in the World Ocean: 3. A solution to the spectral problem. Deutsche Hydrographische Zeitschrift 35, 45–58 (1982). https://doi.org/10.1007/BF02226268

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