A numerical study of vortices and turbulence in quantum fluids

Abstract

Quantum uids possess amazing properties of which two are particularly striking. Firstly they exhibit super uid ow, with the total absence of viscosity. Secondly, there are no excitations when the uid velocity (relative to some obstacle or surface) is slower than a critical value; above this velocity the ow becomes dissipative and macroscopic excitations are created in the form of quantised vortices with xed circulation proportional to Planck’s constant. In this thesis we numerically study the dynamics of quantum uids in the vicinity of obstacles and surfaces, from the production of a single vortex pair to the complex and chaotic motion of turbulent vortex tangles. This approach provides quantitative predictions for atomic Bose-Einstein condensates (BEC) and qualitative insight for super uid helium. We give detailed descriptions of the numerical schemes and present extensive numerical simulation of the Gross-Pitaevskii equation (GPE) and its variants at zero temperature and beyond, in both two and three dimensions. We study the wake that forms behind obstacles in the presence of a super uid ow, modelling atomic BEC experiments with moving laser-induced potentials, and explore the dependence on obstacle shape and size. We nd that suitable obstacles produce classicallike wakes consisting of clusters of vortices of the same polarity. Remarkably, symmetric wakes resemble those observed in classical viscous ow at low Reynolds number, despite the constrained vorticity. The structures are unstable, forming time-dependent asymmetric wakes similar to a classical Bénard–von Kármán vortex street. Motivated by the recentwork of Kwon et al. (Phys. Rev. A 90, 063627 (2014)), we model an atomic BEC experiment in which a trapped, oblate condensate is translated past a stationary, laser-induced obstacle. The critical velocity is exceeded and so vortices nucleate, forming a state of two-dimensional quantum turbulence. We explore the system at both zero-temperature and with thermal dissipation, modelled through a phenomenological term in the GPE. Our simulations provide insight into early stage evolution, not accessible experimentally, and into the decay of vortices by annihilation or passage out of the condensate. We use classical eld methods to simulate homogeneous Bose gases at nite temperature, from strongly non-equilibrium initial distributions to thermalised equilibrium states. We introduce a moving cylindrical potential and study how the thermal component of the gas a ects vortex nucleation. We have found that the critical velocity decreases with increasing temperature and scales with the speed of sound of the condensate. Above the critical velocity, vortices are nucleated as irregular vortex lines, rings, or vortex tangles. Finally we model the surfaces of walls and moving objects (such as wires, grids, propellers or spheres) in the presence of super uid ow, using a real rough boundary obtained via atomic force microscopy. We nd evidence pointing to the formation of a thin ‘super- uid boundary layer’ consisting of vortex loops and rings. As boundary layers usually arise from viscous forces, this is a surprising and intriguing result.