Department of Applied
Mathematics and Theoretical Physics
Instructor: Dr Natalia Berloff
Office: CMS G1.02
Lecture: MR 15
Class web page: www.damtp.cam.ac.uk/user/ngb23/SV/
Vortices are often associated with fluid dynamics: in this course
we shall concentrate on quantized vortices. These are vortices in
nonlinear fields (superfluid flow fields being but one example)
that owe their existence and perseverance to the
topology of the order-parameter field describing a medium with
broken symmetry. The basic Ginzburg-Landau model serves to describe
the vortex core structure in systems as diverse as chemical
patterns, liquid crystals, atomic condensates, superconductors, and
Most of the course will be devoted to superfluid motion that, in its
simplest form, can be formulated as a conservative dynamical system. At the
most basic level, the theory is projected on the classical inviscid
compressible fluid dynamics. But it will be shown that acoustic
dispersion creates an effective dissipation mechanism, replacing
friction, and the latter makes a comeback when more realistic models
The topics covered are:
- Introduction to order parameter space, broken symmetry and
foundations of topological theory of defects.
- Dissipative Ginzburg-Landau equations. Vortices in liquid crystals.
- Vortices in superfluids. Gross-Pitaevskii equation (GPE). Solitary waves
of the GPE: vortex rings and rarefaction pulses and their
stability. Vortex nucleations.
- Superfluid helium. Landau two-fluid model and HVBK model. Motion of
vortex lines. Dissipative and nonlocal GPE.
- Vortices in non-uniform Bose-Einstein condensates.
- Systems with special properties, more intricate topology with
applications to superconductors, systems far from equilibrium,
cosmic strings etc.
Pre-requisites for the course include previous attendance at a first
course in fluid mechanics. Familiarity with solution methods for
partial differential equations will be assumed.
Reading to complement course material
L.M. Pismen ``Vortices in nonlinear fields: from liquid crystals to
superfluids; from non-equilibrium patterns to cosmic strings'', International series of monographs in physics 100, Clarendon Press Oxford, 1999.
R.J. Donnelly: Quantized Vortices in Helium II,
Cambridge University Press, Cambridge, 1991.
P.H. Roberts and N.G. Berloff, ``Nonlinear Schrodinger equation as a model of superfluid helium,'' "Quantized Vortex Dynamics and Superfluid Turbulence" edited by C.F. Barenghi, R.J. Donnelly and W.F. Vinen, Lecture Notes in Physics, volume 571, Springer-Verlag, 2001.
A.L. Fetter and A.A. Svidzinsky ``Vortices in a trapped dilute
Bose-Einstein condensate'', J. Phys.: Condens. Matter 13, R135-R194