Vladimir Zeitlin Laboratory of Dynamical Meteorology, Ecole Normale Superieure, Paris e-mail: zeitlin@lmd.ens.fr
I will give a brief resume of the theory of nonlinear geostrophic adjustment at small Rossby numbers (Reznik, Zeitlin and BenJelloul, 2001; Zeitlin, Reznik, BenJelloul, 2003) and of its predictions for corrections to the traditional slow dynamics and for inertia-gravity wave emission. I will then present some exact results on the existence and uniqueness of the adjusted (balanced) state at arbitrary Rossby numbers, and will discuss dynamical obstacles for reaching this state and related wave phenomena, following (Zeitlin, Medvedev and Plougonven, 2003; LeSommer, Medvedev, Plougonven and Zeitlin, 2003, Plougonven and Zeitlin, 2005). In particular, symmetric and Rossby-Kelvin instabilities will be considered. As illustrations, I will present results of high-resolution numerical simulations in a simplified model of a straight front (Bouchut, LeSommer, and Zeitlin, 2004). I will then discuss a phenomenon of Lighthill radiation arising at large Rossby numbers and illustrate it quantitatively on a simple example by generalizing the method of Zeitlin (1991) and Plougonven and Zeitlin (2001) to rotating systems. Finally, I will give examples of inertia-gravity wave emission observed in observational data (Plougonven, Teitelbaum and Zeitlin, 2003), and in numerical simulations with the mesoscale models.
References: V. Zeitlin, On the back-reaction of acoustic radiation for distributed vortex structures ,1991, Phys. Fluids A v.3, p. 1677 -1680. G. Reznik, V. Zeitlin and M. Ben Jelloul, Nonlinear theory of the geostrophic adjustment. Part.I : Rotating shallow water model, 2001, J. Fluid. Mech., v.445, p. 93 ? 120. R. Plougonven and V. Zeitlin, Internal gravity wave emission from a pancake vortex: an example of wave-vortex interaction in strongly stratified flows , 2001, Phys. Fluids., v.14, p. 1259 ? 1268. V. Zeitlin, S. Medvedev and R. Plougonven, Frontal adjustment, slow manifold and nonlinear wave phenomena in 1d rotating shallow water. Part 1. Theory.,2002, J. Fluid Mech., v. 481, p. 269-290, . V. Zeitlin, G. Reznik and M. Ben Jelloul, Nonlinear theory of the geostrophic adjustment. Part.II : Two-layer and continuously stratified models , 2003, J. Fluid. Mech.,v.491, p. 207-228. J. Le Sommer, S. Medvedev , R. Plougonven and V. Zeitlin, Singularity formation during relaxation of jets and fronts toward the state of geostrophic equilibrium, 2003, Commun. Nonlin. Sci. & Num. Sim., v. 8, p. 415-442. R. Plougonven, H. Teitelbaum and V. Zeitlin, Inertia-gravity wave generation by the tropospheric mid-latitude jet from FASTEX radiosoundings, 2003, J. Geoph. Res., v. 108, D21, 4686. F. Bouchut, J. LeSommer and V. Zeitlin, Frontal adjustment, slow manifold and nonlinear wave phenomena in 1d rotating shallow water. Part 2. High-resolution numerical simulations, 2004, J. Fluid Mech., v. 514, p. 35-63. R. Plougonven and V. Zeitlin, Lagrangian approach to geostrophic adjustment of frontal anomalies in stratified fluid - 2005, Geoph. Astroph. Fluid Dyn., 99, p. 101 - 135.