Internal waves of permanent form in fluids of great depth

Abstract

This paper presents a general theoretical treatment of a new class of long stationary waves with finite amplitude. As the property in common amongst physical systems capable of manifesting these waves, the density of the (incompressible) fluid varies only within a layer whose thickness h is much smaller than the total depth, and it is h rather than the total depth that must be considered as the fundamental scale against which wave amplitude and length are to be measured. Internal-wave motions supported by the oceanic thermocline appear to be the most promising field of practical application for the theory, although applications to atmospheric studies are also a possibility.The waves in question differ in important respects from long waves of more familiar kinds, and in § 1 their character is discussed on the basis of a comparison with solitary-wave and cnoidal-wave theories on customary lines, such as apply to internal waves in fluids of limited depth. A summary of some simple experiments is included at the end of § 1. Then, in § 2, the comparatively easy example of two-fluid systems is examined, again to illustrate principles and to prepare the way for the main analysis in § 3. This proceeds to a second stage of approximation in powers of wave amplitude, and its leading result is an equation (3·51) determining, for arbitrary specifications of the density distribution, the form of the streamlines in the layer of heterogeneous fluid. Periodic solutions of this equation are obtained, and their properties are discussed with regard to the interpretation of internal bores and wave-resistance phenomena. Solutions representing solitary waves are then obtained in the form \[ f(x) = a\lambda^2/(x^2+\lambda^2), \] where xis the horizontal co-ordinate and where aΛ = O(h2). (The latter relation between wave amplitude and length scale contrasts with the customary one, aΛ2 = O(h3)). The main analysis is developed with particular reference to systems in which the heterogeneous layer lies on a rigid horizontal bottom below an infinite expanse of homogeneous fluid; but in § 4 ways are given to apply the results to various other systems, including ones in which the heterogeneous layer is uppermost and is bounded by a free surface. Finally, in §5, three specific examples are treated: the density variation with depth is taken, respectively, to have a discontinuous, an exponential and a ‘tanh’ profile.

Computation of Time-Periodic Solutions of the Benjamin–Ono Equation

“…We did not take advantage of this structure when we developed our numerical method; hence, our approach can also be used for nonintegrable problems. We also note that bifurcation within the family of multiphase solutions has not previously been discussed in the literature, nor has the remarkable dynamics of the Fourier coefficients of these solutions beyond the original derivation by Benjamin (1967) of the form of the traveling waves for this equation.…”

Computing the Hilbert Transform on the Real Line

“…We also implemented the method on the Benjamin-Ono (BO) equation (28) u, + uux + ^{Uxx} = 0, -oo < x < oo, which, together with a suitable initial condition u(x, 0), describes waves in stratified fluids [3,17]. This equation is similar to the well-known Kortewegde Vries equation (KdV), which replaces the term ß^{uxx} with uXXx ■ Like the KdV equation the BO equation admits soliton solutions.…”

“…For this class of functions the integral (1) exists for almost all y, and it defines a function in the same class [7, p. 311]. As for applications, the Hubert transform plays an important role in optics [4], signal processing [11], and waves in stratified fluids [3,17].…”

Slow solitary waves in multi-layered magnetic structures

“…10-19. In particular, it has been shown 10,11,13 that, in the longwavelength approximation, the propagation of nonlinear slow sausage waves in a magnetic slab is described by the Benjamin-Ono ͑BO͒ equation, previously derived for waves in fluids with the infinite depth. 20,21 Although the representation of magnetic filaments by magnetic slabs enabled us to understand many important properties of magnetic flux tube oscillations, this approximation is not particularly realistic. Real magnetic flux tubes are inhomogeneous and, in particular, characterized by a continuous dependence of equilibrium quantities on the radial distance.…”