Generation of Internal Gravity Waves by an Oscillating Horizontal Elliptical Plate

Abstract

Abstract. Time-harmonic oscillations of a horizontal plate generate internal gravity waves in an unbounded stratified fluid. A plane-wave (Fourier) decomposition is used in which waves with outgoing group velocity are selected. The pressure and the velocity in the far field are estimated in terms of the Fourier transform of the pressure jump across the plate. Explicit solutions are obtained for arbitrary prescribed motions of an elliptical plate. Energy is confined to certain wave beams, bounded by conical characteristic surfaces of the underlying hyperbolic partial differential equation; the solution itself is not axisymmetric. The details of the beam structure are complicated (because each characteristic surface has a vertical axis of symmetry, whereas the elliptical plate does not), but they emerge naturally from the asymptotic method, a method that has wider applicability. Results for analogous piston problems are also obtained; these problems arise when a piece of a rigid horizontal plane is forced to oscillate, thus generating waves above the plane.

Near-field internal wave beams in two dimensions

“…By contrast, for thin forcing, the evolution of the waves is set by the distance normal to the forcing. This can be seen in the inviscid calculations of Oser (1957), Reynolds (1962), Martin & Llewellyn Smith (2011, 2012 b ) and Davis (2012) for a horizontal disc, Hurley (1969) for an inclined plate and Llewellyn Smith & Young (2003) for a vertical plate, or in the viscous calculations of Kistovich & Chashechkin (1999 a , b ) for a two-dimensional inclined plate, Vasil'ev & Chashechkin (2003, 2006 a , b , 2012) for a three-dimensional inclined plate, Tilgner (2000), Bardakov, Vasil'ev & Chashechkin (2007), Davis & Llewellyn Smith (2010), Le Dizès (2015) and Le Dizès & Le Bars (2017) for a horizontal disc, Maurer etal. (2017) and Boury, Peacock & Odier (2019) for a horizontal wave generator and Beckebanze, Raja & Maas (2019) for a vertical wave generator.…”

“…Forcing is assumed to be inviscid, with implications discussed later in § 5.3. Then, using (3.2), (3.3) and the inviscid version of (3.1), either the pressure is prescribed on both sides of the line yielding a velocity discontinuity across it, so that or the velocity is prescribed yielding a pressure discontinuity , so that For a horizontal disc, Gabov & Pletner (1988) considered the former forcing and Martin& Llewellyn Smith (2011, 2012 b ) the latter. We take where is the th derivative of the Dirac delta function, with or .…”