In this talk a joint work with Jerrold Marsden on a theory of the counter-intuitive phenomena of dynamical destabilization under the action of dissipation is presented. While the existence of one class of dissipation-induced instabilities in finite-dimensional mechanical systems was known to Lord Kelvin, until recently it has not been realized that there is another major type of these phenomena hinted by one of theorems due to mechanician Merkin; in fact, these two cases exhaust all the generic possibilities in finite dimensions. We put the main theoretical achievements in a general context of geometric mechanics, thus unifying the current knowledge in this area and the multitude of relevant physical problems scattered over a vast literature.
Next we develop a rigorous notion of dissipation-induced instability in the infinite-dimensional case, which inherent differences from classical finite degree of freedom mechanical systems make uncovering this concept more intricate. In building this concept of dissipation-induced instability we found Arnold's and Yudovich's nonlinear stability methods, for conservative and dissipative systems respectively, along with some new existence theory for solutions to be the essential ingredients. As a paradigm and the first infinite-dimensional example to be carefully analyzed, we use a two-layer quasi-geostrophic beta-plane model, which describes the fundamental baroclinic instability in atmospheric and ocean dynamics.