Hamiltonian structure of the Sabra shell model of turbulence : Exact calculation of an anomalous scaling exponentV

{ We show that the Sabra shell model of turbulence, which was introduced recently, displays a Hamiltonian structure for given values of the parameters. The requirement of scale independence of the ux of this Hamiltonian allows us to compute exactly a one-parameter family of anomalous scaling exponents associated with 4th-order correlation functions. The eld of turbulence and turbulent statistics cannot pride itself on a large number of exact results. One of the best known exact results is the \fourth-fth law" of Kolmogorov, which pertains to the third-order moment of the longitudinal velocity diierences, xing its scaling exponent 3 to unity 1]. This follows from the conservation of energy, a quadratic invariant, by the Euler part of the Navier-Stokes equations. This famous result reappeared also in simpliied toy models of turbulence, like shell models, since the conservation of a quadratic invariant was built into their deenition 2]. Nevertheless, the high degree of simpliication involved in the shells model did not lead so far to additional exact results. In this letter we report a discovery of a Hamiltonian structure of the Sabra shell model of turbulence that was introduced recently 3]. The Hamiltonian structure exists for given values of the parameters of the model, and it does not coincide with the usual quadratic invariants; in fact it is cubic in the velocities. There is a Hamiltonian density H n which is local in shell-space, and its existence implies a ux of a local conserved density. The consequence of the constancy of this ux in shell-space xes the value of a fourth-order scaling exponent which is \anomalous" in the sense that it must involve the existence of a renormalization scale. This result is both exact and nontrivial, constituting a kind of \boundary condition" on theories of anomalous scaling in such models. The Sabra model of turbulence, like all shell models of turbulence, describes truncated uid mechanics in wave number space in which we keep N + 1 \shells". Denoting the n-th wave c EDP Sciences