Asymptotic Models for Internal Waves

The mathematical theory of waves on the interface between two layers of immiscible fluid of different densities has attracted interest because it is the simplest idealization for internal wave propagation and because of the challenging modeling, mathematical and numerical issues that arise in the analysis of this system. The recent survey article of Helfrich and Melville [20] provides a rather extensive bibliography and a good overview of the properties of steady internal solitary waves in such systems as well as for more general density stratifications. The compendium [22] of field observations comprised of synthetic aperture radar (SAR) images of large-amplitude internal waves in different oceans together with associated physical data shows just how varied can be the propagation of internal waves. This variety is reflected in the mathematical models for such phenomena. Because of the range of scaling regimes that come to the fore in real environments, the literature on internal wave models is markedly richer in terms of different types of model equations than is the case for surface wave propagation (see, e.g. [8,10] and the references therein).

The idealized system that will be the focus of the discussion here, when it is at rest, consists of a homogeneous fluid of depth d 1 and density ρ 1 lying over another homogeneous fluid of depth d 2 and density ρ 2 > ρ 1 . The bottom on which both fluids rest is presumed to be horizontal and featureless while the top of fluid 1 is restricted by the rigid lid assumption, which is to say, the top is viewed as an impenetrable, bounding surface. This is a standard assumption, and is reckoned to be a good one when the pycnocline is far from the top, which is when d 1 is large relative to the wavelength of a disturbance. In the present work, two general classes of waves will be countenanced. Both of these require that the deviation of the interface be a graph over the flat bottom, so overturning waves are not within the purview of our theory (see Figure 1 for a definition sketch). The first, which is referred to as the one-dimensional case, are long-crested waves that propagate principally along one axis, say along the x-direction in a standard xyz Cartesian frame in which z is directed opposite to the direction in which gravity acts. Such motions are taken to be sensibly independent of the y-coordinate and can be successfully modeled in the first instance by the two-dimensional Euler system involving only the independent variables x, z and of course time t. Because the interface is a graph over the bottom, these asymptotic models then depend only upon x ∈ R and t, and hence the appellation ‘one-dimensional’. Among one-dimensional models, the simplest are those in which one further assumes that the waves travel only in one direction, say in the direction of increasing values of x. Models which we will call ’two-dimensional’ are not restricted by the long-crested presumption, and are consequently more general than the one-dimensional models. They are derived from the full three-dimensional Euler system and their dependent variables depend upon the spatial variable X = (x, y) ∈ R 2 and time t.

One-dimensional, unidirectional, weakly nonlinear models such as the Korteweg-de Vries (KdV) equation , the Intermediate Long Wave (ILW) equation [23,25] or the Benjamin-Ono equation [5] have been extensively used and compared with laboratory experiments [24,31,36]. While much of our qualitative appreciation of the interaction between the competing effects of nonlinearity and dispersion in surface and internal wave propagation has been informed by these sorts of equations, they are of somewhat limited validity (c.f. [1]). Weakly nonlinear models in two-dimensions have been derived by Camassa and Choi [14]. Nguyen and Dias [29] have derived and studied a Boussinesq-type system in a weakly nonlinear regime. Fully nonlinear models were obtained in the one-dimensional case by Matsuno [28], and in the two-dimensional case by Camassa and Choi [15]. We mention also the interesting paper by Camassa et al. [12] where the aforementioned models are compared, in the one-dimensional case, with experimental observations and numerical integrations of the full Euler system. In [14,15,28] the analysis commences with the full Euler system formulation and the asymptotic models are obtained by formally expanding the unknowns with respect to a small parameter. It is not easy using this approach to provide a rigorous justification of the asymptotic expansion, except perhaps within the setting of analytic functions. A different approach has been carried out by Craig, Guyenne and Kalisch [17] in the one-dimensional case. These authors use the Hamiltonian formulation of the Euler equations (due originally to Zakharov [37] for surface waves and to Benjamin and Bridges [6] for internal waves) and expand the Hamiltonian with respect to the relevant small parameters. This method provides a hierarchy of Hamilt

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