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R. Grimshaw

Abstract

The first-order wave equation model of Gill and Schumann, which describes the flow induced on the continental shelf by the longshore wind stress, evaluated at the coast, is extended in three ways. First, we relax the nondivergent approximation and hence are able to estimate the response in the Kelvin wave mode. Second, we include the full wind stress forcing, and allow for an offshore variation in the wind stress. We also include a representation of pressure forcing. Thus we are able to describe the response due to localized forcing systems. Third, we consider the coupling of the deep-ocean region to the shelf region across a continental slope region, modeled as a discontinuity in depth at the shelf break. The results are applied to the response on the shelf due to a localized forcing system, either traveling longshore parallel to the coast, or traveling offshore normal to the coast.

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R. Grimshaw

Abstract

The effects of a variable Coriolis parameter, coastline curvature and variable bottom topography on continental shelf waves are examined for the case when the variations occur on a length scale much greater than the shelf width, which is assumed to be an appropriate length scale for the waves. Explicit formulas are derived for the change in amplitude and phase speed. Particularly simple formulas are derived for Kelvin waves and long, nondivergent shelf waves.

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R. Grimshaw

Abstract

The effect of a mean current on Kelvin waves is considered, including the case when the current is not in geostrophic balance. Because the problem is non-separable, a perturbation scheme is developed for weak shear from which the effect of the mean current on the wave phase speed and decay rate are determined. The perturbation scheme also establishes that in general the Kelvin wave contains all vertical modes. The special case of a two-layer fluid is considered in more detail.

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R. Grimshaw

Abstract

The interaction of a longshore current with a longshore topographic feature is investigated in the barotropic case. It is shown that near resonance, when a long-wave speed is close to zero (in a fixed reference frame), there is enhanced generation of upstream and downstream coastally trapped waves. An evolution equation of the the KdV-type is derived to describe the resonant behavior, and numerical solutions are discussed for a range of parameters describing the forcing terms, the detuning term and dissipation. The analogous situation of resonant generation due to wind stress is developed in an appendix.

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R. Grimshaw

Abstract

For the Helmholtz velocity profile shown in Fig. 1, it is shown that the interface can support an exact steady finite-amplitude wave which radiates internal gravity waves away from the interface.

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R. Grimshaw

Abstract

The interaction of a, wave packet of internal gravity waves with the mean wind is investigated, for the when there is a region of wind shear and also a critical level. The principal equations are the Doppler-shifted dispersion relation, the equation for conservation of wave action, and the mean momentum equation in which the mean wind is accelerated by a “radiation stress” tensor due to the waves. These equations are integrated numerically to study the behavior of a wave packet approaching a critical level, where the horizontal phase speed matches the mean wind. The results demonstrate the exchange of energy from the waves to the mean wind in the vicinity of the critical level, as a function of the initial wave amplitude and the dissipation. For small initial wave amplitudes (so small that changes in the mean wind do not affect the wave packet), the wave packet narrows and grows in magnitude as it propagates toward the critical level, until it reaches a maximum, after which it is strongly dissipated; by contrast, as the initial wave amplitude is increased, the wave packet remains broader, achieves a lower maximum further away from the critical level, and decays less rapidly after the maximum has been reached. The corresponding changes in the mean wind are generally similar to those of the wave packet, with the addition of a small residual value due to dissipation. The interaction between the waves and the mean wind is also studied in the absence of any initial wind shear. The, results demonstrate a transfer of energy to the mean wind and a consequent decay in the amplitude of the wave.

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A. Dorr
and
R. Grimshaw

Abstract

In this paper we consider the effect of the variation of the Coriolis parameter with latitude on barotropic shelf waves, using a β-plane model. Solutions are constructed using the method of inner and outer asymptotic expansions, where the inner expansions hold over the shelf, and the outer expansions hold in the deep ocean. Three cases are identified, depending on the relationship between the shelf wave frequency and the allowed frequencies for deep-ocean Rossby waves. The connection is provided by the matching of the longshore wave-numbers. In the first case, the shelf wave frequency is too large to permit Rossby wave radiation, and the variation of the shelf wave amplitude is governed by conservation of longshore energy flux. In the second case, the shelf wave frequency is sufficiently small to permit Rossby wave radiation at high latitudes, and in the third case there is Rossby wave radiation at all latitudes. In both these cases the longshore shelf wave energy flux decays at a rate determined by the radiated Rossby wave energy flux.

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A. Dorr
and
R. Grimshaw

Abstract

In this paper we consider the scattering of continental shelf waves by a rigid barrier, within the context of the nondivergent, barotropic approximation. An incident wave mode encounters a rigid barrier, normal to the coast, and is scattered into a finite number of reflected wave modes and an infinite set of evanescent wave modes. The evanescent wave modes consist of two branches, a shelf branch and an ocean branch. This latter branch does not seem to have been previously identified in the literature, and is needed here to counter the flow field induced by the incident wave mode at the rigid barrier in the ocean region. The solution is constructed approximately using a Galerkin method at the rigid barrier. It is found that except near the barrier the flow field is dominated by the incident wave and the reflected wave of the same mode number, and has the appearance of an amplitude modulated wave.

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T. Beer
and
R. Grimshaw

Abstract

This note is concerned with equatorward-propagating barotropic non-divergent shelf waves in a channel for which analytic solutions are obtained both for constant Coriolis parameter, and for the Coriolis parameter which varies with latitude. We show that the presence of a topographic barrier, or a turning latitude where the wave frequency equals the maximum allowable frequency for shelf waves, will result in wave reflection, and the subsequent formation of an amplitude-modulated wave.

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R. Grimshaw
and
Yi Zengxin

Abstract

Two models of coastal currents are described that allow fully nonlinear wavelike solutions for the limit of long waves. The first model is an adaptation of a model used by Yi and Warn for finite-amplitude βplane Rossby waves in a channel. It utilizes a particular choice of continental shelf topography to obtain a nonlinear evolution equation for long waves of finite amplitude. The second model describes the waves that form at the vorticity interface between two regions of constant potential vorticity. Again a nonlinear evolution equation is obtained for long waves of finite amplitude. For both model equations, numerical results are presented and compared with the corresponding results for the BDA equation, which is the weakly nonlinear limit for both models.

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