Internal gravity waves and boundary layers from an oscillating circular disc

The generation of internal gravity waves by oscillating bodies, a classical topic from the late 1960s and early 1970s, has been revived about a decade ago in connection with the generation of the internal tide by the oscillation of the barotropic tide over deep-ocean topography, an important topic for ocean mixing and the energy balance of the Earth-Moon system (Garrett 2003; Kunze & Llewellyn Smith 2004; Garrett & Kunze 2007). Most investigations of the problem are inviscid, based on the free-slip boundary condition at the body or topography and the inviscid internal wave equation within the fluid. Simple geometries were studied first, namely circular or elliptical cylinders (Appleby & Crighton 1986; Hurley 1997) in two dimensions and a sphere (Hendershott 1969; Appleby & Crighton 1987; Voisin 1991) or a spheroid (Sarma & Krishna 1972; Lai & Lee 1981) in three dimensions, appropriate for solution in separable coordinates. For more involved geometries or real ocean topographies, a general-purpose method is required that may be implemented numerically. The boundary integral method was proposed in Gorodtsov & Teodorovich (1982) in a steady formulation and applied to the cylinder (Sturova 2001), and in Gabov & Shevtsov (1983, 1984) in an unsteady formulation and applied to various bodies including a plate either horizontal (Gabov 1985), inclined (Gabov & Pletner 1985) or vertical (Gabov & Krutitskii 1987) in two dimensions and a horizontal circular disk (Gabov & Pletner 1988) in three dimensions. However, it is only two decades later, after independent introduction in an oceanographic context for topographies of increasing complexity (Llewellyn Smith & Young 2003; Petrelis, Llewellyn Smith & Young 2006; Balmforth & Peacock 2009; Echeverri & Peacock 2010), that the method finally gained visibility. Such inviscid approaches provide the radiated energy but not the wave profiles. For the latter, a posteriori addition of the viscosity is necessary, implemented in Ivanov (1989), Makarov, Neklyudov & Chashechkin (1990), Hurley & Keady (1997), Voisin (2003) and Echeverri, Yokossi, Balmforth & Peacock (2011) and compared with experiment for the cylinder (Sutherland, Dalziel, Hughes & Linden 1999; Sutherland & Linden 2002; Zhang, King & Swinney 2007) and the sphere (Flynn, Onu & Sutherland 2003; Voisin, Ermanyuk & Flor 2011; Ermanyuk, Flor & Voisin 2011). The addition is not fully consistent, though, in that the effect of viscosity is taken into account on the propagation of the waves (in the wave equation) but not on their generation (in the boundary condition). This approximation rests on the large value of the Stokes number S = ωa^2/ν, with a the size of the body or topography, ω the frequency of oscillation and N the buoyancy frequency. Physically, it implies that only the waves are retained while the other two components of the motion are neglected: the Stokes boundary layer at the body or topography, and an internal boundary layer within the fluid (Kistovich & Chashechkin 2007). In order to obtain all three components, explicit consideration of the no-slip condition is required. For specific geometries and on the approximation that the no-slip condition holds not only at the body or topography but also at its continuation through the fluid, the full viscous problem has been solved for an inclined plate (Kistovich & Chashechkin 1999; Il'inykh, Kistovich & Chashechkin 1999) in two dimensions and a vertical cylinder (Il'inykh, Smirnov & Chashechkin 1999; Kistovich & Chashechkin 2001), an inclined plate (Vasil'ev & Chashechkin 2003) and a horizontal circular disc performing torsional (Il'inykh & Chashechkin 2004) or translational (Bardakov, Vasil'ev & Chashechkin 2007) oscillations in three dimensions. In each case the results were compared with experiment. The approach is summarized in Chashechkin, Kistovich & Smirnov (2001) and Chashechkin, Baidulov & Chashechkin (2006). For translational oscillations of the disc, the introduction of techniques inherited from rotating flows (Vedensky & Ungarish 1994; Tanzosh & Stone 1995), Stokes flows (Tanzosh & Stone 1996; Zhang & Stone 1998) and acoustic waves (Davis & Nagem 2003, 2004) allows the approximation to be relaxed and the condition of continuity of the stresses to be imposed through the fluid in the plane of the disc (Davis & Llewellyn Smith 2010). The present communication considers the connections between these approaches for the only configuration to which all approaches have been applied: the vertical oscillations of a horizontal circular disc. With the disc as a limit of an oblate spheroid, the inviscid solution of Sarma & Krishna (1972) in separable coordinates is shown to coincide with an original solution by the boundary integral method, of double-layer type, and the alternative solution in Gabov & Pletner (1998) to be in error. Addition of viscosity according to Voisin (2003) yields wave profiles compared to the experiments of Bardakov, Vasil'ev & Chashehchkin (2007). Owing to the geometry, the Stokes boundary layer is absent and the internal boundary layer inherits its properties. The role of this layer is assessed using Davis & Llewellyn Smith (2010). In the limit of large S the flow is seen to reduce to the superposition of waves forced by the free-slip condition at the disk, and a boundary layer of thickness 1/S ensuring adaptation from the actual no-slip condition at the disk to the effective free-slip condition for the waves.