Advances in Geosciences www.adv-geosci.net 1680-7340 1680-7359 15 Topics in modern geophysical fluid dynamics 2008 10.5194/adgeo-15-3-2008 http://www.adv-geosci.net/15/3/2008/ http://www.adv-geosci.net/15/3/2008/adgeo-15-3-2008.html http://www.adv-geosci.net/15/3/2008/adgeo-15-3-2008.pdf 3 9 2008-03-12 Towards an analytical understanding of internal wave attractors U. Harlander uwe.harlander@tu-cottbus.de Department of Aerodynamics and Fluid Mechanics, Brandenburg University of Technology (BTU) Cottbus, Siemens-Halske-Ring 14, 03046 Cottbus, Germany Time harmonic inviscid internal wave motions constrained to fully closed domains generically lead to singular velocity fields. In spite of this difficulty, several techniques exist to solve such internal wave boundary value problems. Recently it has been shown that for a domain with the shape of a trapezium, solutions can be written in terms of a double sine Fourier series. However, the solutions were found by a numerical technique and thus not all coefficients of the series are available. Unfortunately, for questions related e.g. to regularization of the inviscid {\em singular} solutions, the knowledge of the asymptotic behavior of the spectrum for large wave numbers is essential. Here we discuss solutions of internal wave boundary value problems for which the spectra are known, at least asymptotically. We further describe shortcomings of the found solutions that need to be overcome in the future. Finally, we sketch applications of the solutions in the context of viscous energy dissipation. Chashechkin, Y D. and Prihodko, Y V.: Regular and singular flow components for stimulated and free oscillations of a sphere in continuously stratified liquid, Doklady Physics, 52, 261–265, 2007. Erdélyi, A., Oberhettinger, W. M F., and Tricomi, F G.: Tables of integral transforms, Vol. 1, McGraw-Hill Book Company Inc., 1954. Harlander, U. and Maas, L. R M.: Two alternatives for solving hyperbolic boundary value problems in geophysical fluid dynamics, J. Fluid. Mech, 588, 331–351, 2007. Maas, L. R M. and Lam, F.-P A.: Geometric focusing of internal waves, J. Fluid Mech., 300, 1–41, 1995. Maas, L. R M., Benielli, D., Sommeria, J., and Lam, F.-P A.: Observation of an internal wave attractor in a confined, stable stratified fluid, Nature, 388, 557–561, 1997. Magaard, L.: Ein Beitrag zur Theorie der internen Wellen als Störungen geostrophischer Strömungen, Deutsche Hydrographische Zeitschrift, 21, 241–278, 1968. Ogilvie, G I.: Wave attractors and the asymptotic dissipation rate of tidal disturbances, J. Fluid Mech., 543, 19–44, 2005. Swart, A., Sleijpen, G. L G., Maas, L. R M., and Brandts, J.: Numerical solution of the two dimensional Poincaré equation, J. Comput. Appl. Math., 200, 317–341, 2007. Vlasenko, V., Stashchuk, N., and Hutter, K.: Baroclinic tides, Cambridge University Press, 2005.