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  • Author or Editor: GEORGE W. PLATZMAN x
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George W. Platzman

Abstract

A method is designed to calculate normal modes of natural basins. The purpose is to determine the period and configuration of free oscillations in a way that provides for the full two-dimensionality of the problem, and thus avoids limitations such as are inherent in the traditional channel approximation. The method—called resonance iteration—is amenable to detailed numerical analysis.

As a test, the lowest positive and negative modes in a rotating square basin of uniform depth are calculated for a range of rotation speeds, with results that agree well with existing evaluations of this case. Similar agreement was found in an application to Lake Erie, a basin that conforms to the channel approximation. The method was then applied to two basins where it was expected to give results differing from those obtained by earlier methods: Lake Superior and the Gulf of Mexico. The fundamental gravitational mode of Lake Superior was found to have a period of 7.84 hr, which is 9% greater than the value known from the channel approximation, and is in virtually exact agreement with a recent spectral analysis. Phases of this mode also agree with observation.

The Gulf of Mexico is of particular interest because of the still-unresolved role of normal modes in the tidal regime of that basin. With the basin completely closed, the method of resonance iteration gave a period of 7.48 hr for the slowest gravitational mode and produced a single, positive amphidromic system that imparts to this mode approximately the character of a longitudinal oscillation on the nearly west-east axis of Mexico Basin. With the Gulf open through the Yucatan Channel and the Straits of Florida, the structure of this mode, is not altered qualitatively and the period is lowered to 6.68 hr. The most significant effect of these “ports” is that they elicit an additional gravitational oscillation—the so-called Helmholtz mode—which has a period much longer than that of the slowest seiche-type oscillation, and nodal points only at the ports. This co-oscillating mode is found to have a period of 21.2 hr. The proximity of its period to that of the diurnal tide points to a revival of the traditional conception that the tidal regime in the Gulf of Mexico is affected appreciably by resonance.

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George W. Platzman

Abstract

In preceding parts of this study a set of normal modes was constructed as a basis for synthesizing diurnal and semidiurnal solutions of Laplace's tidal equations. The present part describes a procedure by which such solutions can be computed as eigenfunction expansions. Since the calculated normal modes are nondissipative, it is necessary to incorporate dissipation into the synthesis procedure. This is done by a variational treatment of the tidal equations.

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George W. Platzman

Abstract

Diurnal and semidiurnal tides of second and third degree are synthesized from 60 normal modes with period in the range 8 to 96 h. Diurnal tides, especially those of second degree, can be represented by remarkably few modes. The principal lunar diurnal constituent, for example, consists almost entirely of a single forced mode excited mainly in the Pacific and Indian Oceans. Semidiurnals are spectrally more heterogeneous, and more resonant, than diurnals, but some specific features can be attributed to individual modes. Several of the most energetic modes in the principal lunar semidiurnal constituent are prominent in the Atlantic Ocean. Together with the fact that diurnally excited modes are relatively weak in that region, this presumably accounts for the observed tendency for the total tide to be predominantly semidiurnal in the Atlantic but mixed diurnal and semidiurnal in many parts of the Pacific and Indian Oceans.

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George W. Platzman

Abstract

The linearized primitive equations for a barotropic world ocean are discretized by means of first-order, piecewise-linear finite elements. Surface elevation and Stokes/Helmholtz velocity potentials are adopted as dependent variables. On any segment of the ocean boundary, specification of elevation and simple radiation are allowed as alternative conditions. The discretized mass and momentum equations are designed to make the finite-element solution satisfy global energy balance exactly. They also permit an arbitrary choice of axes at each node of the grid and thereby avoid the “pole” problem. The model is tested by applying it to a rotating rectangular basin and to a domain consisting of the Atlantic and Indian Oceans.

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George W. Platzman

Abstract

A finite-element model recently designed for calculation of oceanic normal modes is amended here with a prescription for the proper treatment of multiple connectivity, the main effects of which are likely to be located in the Southern Ocean. For this purpose line bases, as well as point bases, are needed in the finite-element representations of volume-flux and energy-flux streamfunctions. The dynamical conditions that should be met by the circulation of velocity and energy on each boundary are found to arise as natural boundary conditions for the finite-element equations.

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George W. Platzman

Abstract

Normal modes are calculated for a homogeneous ocean occupying a connected domain consisting of the North Atlantic, South Atlantic, and Indian Oceans. Coastal configuration and bathymetry are resolved on a grid of 675 six-degree Mercator squares. The calculation is based upon the Lanczos process and is more efficient than resonance iteration. Twenty-six gravity modes were found with periods greater than 8 h, the slowest being a fundamental mode of about 67 h. The North Atlantic co-oscillates with the South Atlantic at a period of about 42 h, and has strong resonances at 23, 21, 14.4, 12.8, 8.6 and 8.3 h. Eleven topographically-induced modes of rotational type were found with periods less than 100 h; the fastest of these is a 44 h mode in the Weddell Sea. In the 6° model the fastest rotational mode of the North Atlantic is a 55 h topographic wave most prominent near the Grand Banks of Newfoundland.

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George W. Platzman
,
Gary A. Curtis
,
Kirk S. Hansen
, and
Richard D. Slater

Abstract

We have calculated normal modes with period between 8 and 80 h for a domain consisting of the Arctic, Atlantic, Indian and Pacific oceans. In this period range the numerical model has 56 modes, of which 13 are topographic vorticity waves all slower than 30 h. The trapping sites for these modes are the Siberian Shelf, the Icelandic Plateau, the Grand Banks, the Falkland Plateau and Patagonian Shelf, the Kerguelen Plateau, the New Zealand and Fiji Plateaus, and the Hawaiian Ridge. Predominantly planetary vorticity waves do not appear in the model at periods less than 80 h. The 41 modes found between 30 and 8 h include basin modes in the North Atlantic, Indian and equatorial Pacific; quarter-wave resonances in the Arabian Sea, Bay of Bengal and Gulf of Guinea; and Kelvin waves on the Antarctic coast, the Pacific North American coast and the New Zealand coast. Several vorticity and gravity modes exhibit an eastward circumglobal flow of energy that is confined to equatorial latitudes except where deflected southward by the continental land masses of the Southern Hemisphere. Tide-gage data are cited as consistent with 25.8 and 14.1 h basin modes in the North Atlantic and with the three Kelvin waves noted above, whose periods are 28.7, 15.5 and 10.8 h, respectively.

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