Plasma Phys. Control. Fusion 41 (1999) 1497–1515. Printed in the UK PII: S0741-3335(99)07370-4 Hamiltonian magnetic reconnection D Grasso†, F Pegoraro‡, F Porcelli† and F Califano§ † Dipartimento di Energetica, Politecnico di Torino and Istituto Nazionale Fisica della Materia, Italy ‡ Dipartimento di Fisica, University of Pisa and Istituto Nazionale Fisica della Materia, Italy § Dipartimento di Astronomia, Universit` a di Firenze and Istituto Nazionale Fisica della Materia, Italy Received 2 September 1999 Abstract. Magnetic reconnection in two dimensional (2D), collisionless, non-dissipative regimes is investigated analytically and numerically by means of a finite difference code in the nonlinear regime where the island size becomes macroscopic. The cross-shaped structure of the reconnection region, originally reported by Cafaro et al (1998 Phys. Rev. Lett. 80 20) is analysed as a function of the ratio between the ion sound Larmor radius and the inertial skin depth. This cross shape structure is found to survive in the presence of weak dissipation. Further insight on the quasi-explosive behaviour of the magnetic island width as a function of time and on the spatial structure of the perturbed current density is provided. We confirm that the amount of reconnected flux becomes of order unity on the time scale of the inverse linear growth rate. Results in the collisionless limit are interpreted on the basis of the Hamiltonian properties of the adopted collisionless, 2D, fluid model. Thus, collisionless reconnection is a fast, non-steady-state process, fundamentally different from 2D resistive magnetic reconnection, of which the Sweet–Parker model is the classic paradigm. 1. Introduction Magnetic reconnection in collisionless plasmas is a basic nonlinear physics problem. The study of this problem was originally motivated by applications to space plasma processes, such as reconnection events occurring in the Earth magnetotail [1, 2]. Collisionless reconnection is now believed [3] to account for fast relaxations in laboratory fusion plasmas, for instance the so-called sawtooth relaxations in Tokamak experiments [4]. In such devices the plasma temperature is so high that the relaxation time can be shorter [5–7] than the electron–ion collision time. In this regime electron inertia is the main factor responsible for the decoupling of the plasma motion from the magnetic field. The dynamics of a collisionless plasma can be described in terms of a Hamiltonian model. For a fluid plasma model, the Hamiltonian formalism is particularly powerful in that it facilitates the identification of the Casimirs. These are conserved quantities that, in a two-dimensional (2D) geometry, are expressed in terms of integrals of arbitrary functions of special combinations of the magnetic flux, of the velocity stream function and of their Laplacians. The topology of these special fields is frozen in the course of plasma evolution. One question of principle often raised is the very possibility of any significant magnetic reconnection in the absence of dissipation, or equivalently in the presence of topological invariants. Thus, one of the aims of this paper is to show that significant magnetic reconnection is indeed possible in collisionless regimes. At the same time, important differences between Hamiltonian and dissipative reconnection will be elucidated. We shall show that collisionless 0741-3335/99/121497+19$30.00 © 1999 IOP Publishing Ltd 1497