QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Q. J. R. Meteorol. Soc. (2009) Published online in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/qj.519 Canonical Hamiltonian representation of pseudoenergy in shear flows using counter-propagating Rossby waves E. Heifetz*, N. Harnik and T. Tamarin Department of Geophysics, Tel-Aviv University, Israel ABSTRACT: Pseudoenergy serves as a non-canonical Eulerian Hamiltonian of linearized shear flow systems. It is non- canonical in the sense that the canonical Hamilton equations cannot be written when the dynamical variable is taken as the (potential) vorticity. Here we apply the counter-propagating Rossby wave kernel (KRW) perspective to obtain a compact form of the pseudoenergy as a domain integral of the local KRW pseudomomentum carried by the instantaneous KRW phase speed in the mean flow frame of reference. Written this way, with the generalized momenta taken as the KRW pseudomomenta and the generalized coordinates as the instantaneous KRW locations, canonical Hamilton equations can be derived both in their continuous (using functional derivatives) and discrete (using function derivatives) forms. As a simple example of the insight such a formulation can yield, we reexamine the classical stability transition from Rayleigh to Couette flow. In this transition the instability is lost even though the classical necessary conditions of Fjørtoft and Rayleigh are still satisfied. The pseudoenergy-KRW formulation allows to interpret the stabilization both as an inability of the KRWs to phase lock constructively, and in terms of the pseudoenergy becoming negative. These two apparent different rationalizations are shown to be essentially one and the same. Copyright c  2009 Royal Meteorological Society KEY WORDS pseudoenergy; Eulerian canonical Hamiltonian; Rossby wave interaction; shear flow Received 23 May 2008; Revised 24 July 2009; Accepted 2 September 2009 1. Introduction It is common practice in many physical disciplines, to try to describe conserved dynamical systems in a canon- ical Hamiltonian form (e.g. Goldstein, 1969; Peskin and Schroede, 1995). The generalized momenta and coordi- nates of the canonical representation are considered to be the natural physical framework to describe the system. In geophysical fluid dynamics, such a representation is possible in a particle-following Lagrangian framework, where the Hamiltonian is the total integrated energy of all the fluid particles, and the particles’ positions and velocities are the generalized coordinates and momenta. Since a Lagrangian framework is usually too complex for practical use, much effort in developing a Hamilto- nian geophysical fluid dynamics theory has been directed towards its representation in the simpler Eulerian frame- work. However, in the transition to the Eulerian frame- work, the phase space is degenerated due to the inherent ‘particle-relabelling symmetry’ along surfaces of constant (potential) vorticity (hereafter generally PV), which is Lagrangianly conserved (e.g. Shepherd, 1990; Salmon, 1988). Under this symmetry, the Hamiltonian becomes invariant under translation of fluid particles along PV sur- faces. This reduction in phase space prevents a canonical representation of the dynamics in the Eulerian frame- work, so that only a non-canonical representation can be obtained (Shepherd, 1990; Salmon, 1988; Chapter 7 ∗ Correspondence to: E. Heifetz, Department of Geophysics, Tel-Aviv University, Israel, 69978. E-mail: eyalh@cyclone.tau.ac.il of Salmon 1998 gives a comprehensive review of non- canonical Hamiltonian fluid dynamics). In this paper, we will show that, for linear dynamics, we can nonetheless reformulate the Hamilton equations in a canonical form, by using a counter-propagating Rossby wave perspective. Shear flows with basic states that are constant in time and in the zonal direction conserve both pseudoenergy and pseudomomentum, which are exact wave activity invariants of the nonlinear dynamics (they are defined up to the Casimirs of the flow which can be the integrals of any function of the PV). For the linearized dynamics, the pseudoenergy can be shown to become the non- canonical Hamiltonian, whereas the pseudomomentum becomes a conserved Noether current. In the context of linear instability, the conservation of pseudomomentum and pseudoenergy yield the two necessary conditions for modal shear instability – the Rayleigh (1880) and Fjørtoft (1950) conditions, respectively. A mechanistic interpretation of these conditions is obtained in terms of a mutual interaction of two oppositely propagating Rossby waves, which phase lock and reinforce each other in the presence of shear, due to the action-at-a-distance nature of PV anomalies (e.g. Hoskins et al., 1985; Heifetz et al., 2004a,b). This is best illustrated for the simple case where the mean flow PV gradients are concentrated in two localized regions, each supporting a Rossby wave which propagates to the left of the local mean PV gradient. Though the waves are PV localized, each induces a non-local velocity field which affects the other wave by advecting the mean PV gradient. The Rayleigh condition Copyright c  2009 Royal Meteorological Society