DOI 10.1393/ncc/i2007-10250-x IL NUOVO CIMENTO Vol. 30 C, N. 4 Luglio-Agosto 2007 A note on the full non-linear stability of inviscid, planar flows with constant relative vorticity F. Crisciani( 1 )( ∗ ) and G. Badin( 2 ) ( 1 ) Istituto di Scienze Marine del CNR - Trieste, Italy ( 2 ) Department of Earth and Ocean Sciences, Liverpool University - United Kingdom (ricevuto il 24 Luglio 2007; revisionato il 7 Gennaio 2008; approvato il 18 Gennaio 2008; pub- blicato online il 2 Febbraio 2008) Summary. — The non-linear stability of inviscid, planar flows with constant rela- tive vorticity is proved in the context of the quasi-geostrophic shallow-water theory, for simply connected fluid domains of arbitrary shape. First, the result is obtained relative to the enstrophy and kinetic energy norms and, then, it is extended to a “generalised energy” norm which is expressed through the former. PACS 47.20.Ky – Nonlinearity, bifurcation, and symmetry breaking. PACS 47.15.ki – Inviscid flows with vorticity. 1. – Introduction In this paper we study the full non-linear stability of a special class of two-dimensional, uniformly rotating flows of inviscid, incompressible fluids, governed by the shallow-water equations in the quasi-geostrophic approximation. The main feature of the considered basic states is their constant relative vorticity, a problem formally raised by [1], in a simply connected domain of arbitrary shape. The considered basic flows turn out to be nonlinearly stable with respect to both the enstrophy and energy norms and hence to a generalized norm expressed as a function of the former. The stability of constant vorticity flows has been already considered in the context of Eulerian flows [2]. However, the results reported here represent a generalization in a context of geophysical relevance. 2. – Model equations The governing equations of a uniformly rotating, single-layer incompressible and in- viscid fluid on the f -plane are (2.1) ∂u ∂t + u ∂u ∂x + v ∂u ∂y − fv = −g ∂η ∂x , ( * ) E-mail: fulvio.crisciani@ts.ismar.cnr.it c  Societ` a Italiana di Fisica 337