Forced oceanic waves

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Abstract
This paper concerns the linear response of the ocean to forcing at a specified frequency and wave number in the absence of mean currents. It discusses the details of the forcing function, the general properties of the equations of motion, and possible simplifications of these equations. Two representations for the oceanic response to forcing are described in detail. One solution is in terms of the normal modes of the ocean. The vertical structure of these modes corresponds to that of the barotropic and baroclinic modes; their latitudinal structure corresponds to that of inertia‐gravity and Rossby waves. These waves are eigenfunctions of Laplace's tidal equations (LTE) with the frequency as eigenvalue. The description in terms of vertically standing modes is particularly useful if the forcing is nonlocal, because only these modes can propagate into undisturbed regions. The principal result is that it is extremely difficult for baroclinic (but not barotropic) disturbances to propagate horizontally away from a forced region. Instabilities of the Gulf Stream excite disturbances that are confined to the immediate neighborhood of the current; disturbances due to instabilities of equatorial currents do not propagate far latitudinally. A second representation of the oceanic response to forcing is in terms of vertically propagating, or vertically trapped, latitudinal modes. These modes are eigenfunctions of LTE with the equivalent depth h (not the frequency) as eigenvalue. Both positive and negative eigenvalues h are necessary for completeness. The modes with h > 0 consist of an infinite set of inertia‐gravity waves and a finite set of Rossby waves which either propagate vertically or form vertically standing modes. The latitudinally gravest modes are equatorially trapped and have been observed in the Atlantic and Pacific oceans. The modes with h < 0 are necessary to describe the oceanic response to nonresonant forcing. In the vertical this response attenuates with increasing distance from the forcing region. Because of the shallowness of the ocean the large eastward traveling atmospheric cyclones in mid‐latitudes and high latitudes force a response down to the ocean floor. Interaction with the bottom topography will result in smaller‐scale disturbances and will affect the frequency spectrum of the response when bottom‐trapped waves are excited.

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