Internal-gravity wave propagation in a range-dependent waveguide with forcing at the bottom

Internal-Gravity wave propagation in a range-dependent waveguide with forcing at the bottom No´e Lahaye and Stefan G. Llewellyn Smith Department of Mechanical and Aerospace Engineering (MAE), University of California San Diego nlahaye@eng.ucsd. edu Abstract We compute the propagation of waves forced at the bottom of an inhomogeneous stratified moving fluid, using vertical normal modes and horizontal rays. The problem is separated into a range-dependent nonlinear eigenproblem that defines the vertical structure of the solution and a set of coupled inhomogeneous ordinary differential equations with non- constant coefficients in the horizontal. Under the adiabatic approximation, i.e. neglecting the energy transfer between different modes due to variation of the medium properties in the horizontal, the latter reduces to a set of uncoupled ODEs. Results of the implemen- tation of the method are given in an idealized case. Introduction A common property of ocean, atmospheric and other geophysical systems, is that both their vertical extent and the typical vertical lengthscale of variation of the medium’s properties are small compared to the horizontal ones. If the fluid is vertically bounded (e.g. by physical boundaries) or if there is a mechanism that results in a vertical trapping of the propagating waves, the physical system forms a waveguide, in which constructive and destructive interferences of vertically bouncing waves result in a discrete set of vertical modes that propagate horizontally (i.e. along the waveguide). This is the approach we take here, which couples range-dependent normal modes and horizontal rays. While this method has been widely used in ocean acoustics (e.g. Burridge and Weinberg, 1977; Desaubies and Dysthe, 1995; Jensen et al., 2011) and in atmospheric infrasound propagation (Brekhovskikh and Godin, 1999; LePichon et al., 2009), it has received rela- tively little attention in the context of internal wave propagation, especially with a moving background state. Notable exceptions are the work of Keller and Mow (1969), who used a very similar approach to study the internal-wave dynamics in an ocean with a sloping bottom, and more recent papers by Godin (2002, 2012), who gave a generalized formula- tion for acoustic-gravity waves in range-dependent waveguide with a background velocity field. Most studies (if not all) address the free propagation of waves (with eventually attenuation) or the response to a point source in the waveguide. In many geophysical situations, such as mountain-generated waves in the atmosphere or tidal conversion by a ridge in the ocean, waves are forced at the boundary (usually the bottom) over a finite region in space. In this paper, we derive a similar set of equations in a moving fluid under the Boussinesq approximation, describing the steady-state response to a forcing mechanism at the bottom of the domain. The latter is explicitly included in the formulation and has a finite size in the horizontal. After deriving the linearized equation and expressing the solution in terms of range-dependent vertical modes and a set of coupled-mode equations (§ 2), we give some results of the numerical implementation of the method, in a case where a stratified fluid VIII th Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016