J. Fluid Mech. (1990), vol. 213, pp. 54S571 Printed in Great Britain 549 Pseudo-advective relaxation to stable states of inviscid two-dimensional fluids By G. F. CARNEVALE’ AND G. K. VALLIS’ Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093, USA Division of Natural Sciences, University of California, Santa Cruz, Santa Cruz, CA 95064, USA (Received 31 March 1989) The continuous transformation of one flow into another of higher or lower energy while preserving the potential vorticity of all particles can be accomplished by advection with an artificial velocity field. Since isolated extremal energy states are stable states, this method can be used to find stable stationary flows on a prescribed isovortical sheet. A series of numerical simulations of this method for two- dimensional fluids that demonstrates its feasibility and utility is presented. Additionally, a corollary to Arnol’d’s nonlinear stability theorems is discussed, which shows that there can be a t most two Arnol’d stable states per isovortical sheet. 1. Introduction I n Vallis, Carnevale & Young (1989, hereafter VCY), we presented a general method for decreasing or increasing the energy of a flow while preserving the circulation on all material curves. Here we consider applications of this method to two-dimensional flows and demonstrate its utility in finding and examining stable states. Kelvin (1887) had investigated the stability of stationary flows by imagining a process analogous to the one we use. By numerical simulation of our method, we can animate the thought experiments proposed so long ago by Kelvin and extend the range of practical application of this approach based on energy extremization. A concept which will be useful in our discussions is the isovortical sheet. Two- dimensional flow is simply the advection of vorticity (potential vorticity in geophysical contexts). Thus it preserves the potential vorticity of all material particles. A useful decomposition of phase space is achieved by grouping all the states which can be obtained fFom each other by a smooth, vorticity-preserving mapping or, in other words, by advection with some divergenceless but otherwise arbitrary flow field for some finite time. A trajectory in phase space must be wholly contained in such an isovortical sheet (Arnol’d 1965~). Furthermore, since energy is conserved in inviscid flow, any trajectory must lie on a constant-energy surface. The isovortical sheet, the constant-energy surface, and their intersection are in general all infinite- dimensional. However, if the intersection is just a point, it then follows that the state represented by that point is a stationary flow. Furthermore, if that point is a maximum or minimum of energy with respect to all isovortical perturbations in a neighbourhood of it, then it represents a stable stationary flow (cf. Arnol’d 1965u, b, 1966). The process which we are considering moves a state point toward ever lower (or higher) energy while remaining within the same isovortical sheet. Thus it can be used