Generation of internal waves from rest: extended use of complex coordinates, for a sphere but not a disk

Abstract

The anisotropy created by stratification and or rotation places restrictions, severe if viscosity is present, on the construction of analytic solutions to wave generation and scattering problems. Consequently, much literature is devoted to frequency space and so careful consideration of oscillatory motion generated from rest is advisable. Moreover, the use of complex coordinates in the inviscid case has been incompletely presented. The detailed inversion of the Fourier time transform for a breathing or heaving sphere demonstrates an expanded, perhaps more crucial, role for the complex coordinates and shows that the known phase changes in the energy propagation regions are present throughout the St Andrew's cross that circumscribes the sphere. However, the different solution structure for the heaving disk requires and allows a more direct calculation. Though the inclusion of rotation does not affect the dynamics, it enables the significance of their relative magnitude to be identified and reference to rotation only results achieved.

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.…”

“…When the forcing is a body of simple shape, oscillating in an inviscid fluid, a combination of coordinate stretching and analytic continuation allows the calculation of the waves at an arbitrary distance from the body. This method, introduced by Bryan (1889) for inertial waves and Hurley (1972) for internal waves, has been applied to circular and elliptic cylinders by Hurley (1972Hurley ( , 1997 and Appleby & Crighton (1986), and to spheres and spheroids by Hendershott (1969), Krishna & Sarma (1969), Sarma & Krishna (1972), Lai & Lee (1981), Appleby & Crighton (1987), Voisin (1991), Rieutord et al (2001) and Davis (2012). The waves manifest themselves as a set of critical rays, with singular amplitude at the rays and phase jumps across them.…”

Boundary integrals for oscillating bodies in stratified fluids

“…Appleby & Crighton proceeded differently, for both the circular cylinder (1986) and the sphere (1987), decomposing the wave field into zones delimited by the critical rays, and investigating the properties of the stretched coordinates in each zone, concluding for the sphere that ‘this illustrates the difficulty of working in more comprehensible coordinates’. Davis (2012) combined the two approaches together for the sphere, expressing, in each zone, the stretched coordinates in terms of real algebraic functions of the Cartesian coordinates.…”

“…We then write where the determination of the square roots is set by the replacement (4.36), yielding the combinations consistent with the determinations in § 4 of Davis (2012). The pressure is obtained as and the velocity as where is a unit vector along the -axis.…”

“…1973). The first theoretical studies were modelled after these experiments, and considered cylinders either circular (Hurley 1972; Appleby & Crighton 1986) or elliptic (Hurley 1997; Hurley & Hood 2001), and spheres (Hendershott 1969; Appleby & Crighton 1987; Voisin 1991; Rieutord, Georgeot & Valdetarro 2001; Davis 2012) or spheroids (Krishna & Sarma 1969; Sarma & Krishna 1972; Lai & Lee 1981). The free oscillations of a sphere, displaced from its neutral buoyancy level then released, were also considered (Larsen 1969).…”