Transition to Dissipation in Two- and Three-Dimensional Superflows Cristi´ anHuepe 1 , Caroline Nore 2 , and Marc-Etienne Brachet 3 1 James Franck Institute, University of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637, USA 2 Universit´ e de Paris-Sud, LIMSI, Bˆatiment 508, F-91403 Orsay Cedex, France 3 Laboratoire de Physique Statistique de l’Ecole Normale Sup´ erieure associ´ eau CNRS et aux Universit´ es Paris 6 et 7, 24 rue Lhomond, 75005 Paris, France Abstract. Vortex nucleation in two-dimensional (2D) and three-dimensional (3D) su- perflowspastacylinderisstudied.ThesuperflowisdescribedbyaNonlinearSchr¨odinger like equation. In the 2D case, a continuation method is used to characterize the bifur- cation of stationary states leading to vortex formation. A saddle-node followed by a secondary pitchfork bifurcation that leads to the branch of nucleation solutions (one vortex in an asymmetric field) is found. The dependence of the bifurcation diagram on the ratio of the coherence length to the disc diameter is studied. Using the 2D station- ary vortex nucleation solutions to construct the initial condition, the 3D dynamics of a vortex pinned to the surface of the cylinder is numerically studied. Quasistationary half-ring vortices, pinned at the sides of the cylinder, are generated after a short time. On a longer time scale, a vortex stretching may occur, inducing dissipation and drag. The corresponding 3D critical velocity is found to be well below the 2D one. 1 Introduction Aboveacertaincriticalvelocity,superfluidflowsareknowntoenteradissipative regime. The Nonlinear Schr¨ odinger equation (NLSE) describes the dynamics of superfluid 4 He,attemperatureslowenoughforthenormalfluidtobenegligible. In the homogeneous two-dimensional NLSE flow past a disc, the existence of a transition to dissipation due to the periodic emission of pairs of counterrotat- ing vortices was found in [1]. Although this model system is not quantitatively equivalenttoa 4 Heflow,ithasbeenstudiedindetailforitsuniversalproperties andmathematicalinterest[2,3,4].Acloserconnectiontoexperimentswasgiven by the recent experimental success in producing and manipulating dilute Bose Einstein condensates. The NLSE has been used to accurately describe the dy- namicsofsuchsystems,allowingdirectquantitativecomparisonbetweentheory andexperiment[5].Inparticular,arecentexperimentwhichstudiesthedissipa- tionproducedinaBoseEinsteincondensedgasbymovingabluedetunedlaser beam through it [6] has renewed the interest in dynamics of a NLSE superflow past an obstacle. The present paper reviews some recent work in the NLSE description of a superflow past a cylinder in 2D and 3D [2,3,4,7,8]. The paper is organized as follows. In Sect. 2 we present the hydrodynamic form of the NLSE. In Sect. 3, the numerical tools used in this work are described. The bifurcation diagram C.F. Barenghi, R.J. Donnelly, and W.F. Vinen (Eds.): LNP 571, pp. 297–304, 2001. c  Springer-Verlag Berlin Heidelberg 2001