Turbulence, the irregular motion of fluids, is a challenging problem in physics. Yet some properties of turbulence appear to be universal, independent of the underlying host fluid supporting the motion. Recent studies have found that turbulence in superfluid helium, a quantum fluid, exhibits two of the most fundamental laws of classical fluid turbulence: the Kolmogorov −5/3 law, and the dissipation anomaly. These laws appear despite the fluid being highly constrained by quantum mechanical effects, and completely lacking kinematic viscosity. Such findings suggest further insight into the universal features of turbulent phenomena can be gained by studying analogies between classical and quantum turbulence.\nAtomic Bose-Einstein condensates (BECs) offer a new platform for the study of quantum turbulence; the geometric control available in BEC experiments offers the possibility of studying quantum turbulence in effectively two-dimensional fluids. As two-dimensional turbulence exhibits dramatically different features from its 3D counterpart, BEC systems allow for further study of the analogies between classical and quantum turbulence. In this thesis we numerically and theoretically study 2D quantum turbulence in BECs within the framework of the Gross-Pitaevskii model. We focus on analogies with classical 2D turbulence, with the aim of identifying common or universal features.\nFirst we investigate coherent vortex structures in negative temperature equilibria via an experimentally accessible flow-field measure. Coherent vortices are shown to produce a clear signal in this measure that is independent of the confinement geometry, and we demonstrate that it can be observed in dynamical simulations.\nSecond, studying a quantum analogue of the two-dimensional cylinder wake, we investigate the phenomenon of Strouhal oscillations. We find that the Strouhal number obeys a universal relation, similar to the classical form, upon introducing a modified superfluid Reynolds number that accounts for the critical velocity for vortex nucleation. Like the classical Reynolds number, the superfluid Reynolds number is found to govern the transition from laminar to turbulent behaviour in the quantum fluid.\nFinally, simulating decaying 2D quantum turbulence for very large vortex numbers, we show that quantum fluids are capable of supporting the direct enstrophy cascade, a fundamental feature of two-dimensional turbulent flows. The quantum fluid manifests key features of the classical cascade, including Batchelor’s −3 law of the inertial range, scaling of the inertial range against the superfluid Reynolds number, and the value of the Kraichnan-Batchelor constant.\nThe findings from this work thus provide some new insight into the universality of fundamental turbulence concepts, and their applicability to quantum fluids.
Turbulence, the irregular motion of fluids, is a challenging problem in physics. Yet some properties of turbulence appear to be universal, independent of the underlying host fluid supporting the motion. Recent studies have found that turbulence in superfluid helium, a quantum fluid, exhibits two of the most fundamental laws of classical fluid turbulence: the Kolmogorov −5/3 law, and the dissipation anomaly. These laws appear despite the fluid being highly constrained by quantum mechanical effects, and completely lacking kinematic viscosity. Such findings suggest further insight into the universal features of turbulent phenomena can be gained by studying analogies between classical and quantum turbulence.\nAtomic Bose-Einstein condensates (BECs) offer a new platform for the study of quantum turbulence; the geometric control available in BEC experiments offers the possibility of studying quantum turbulence in effectively two-dimensional fluids. As two-dimensional turbulence exhibits dramatically different features from its 3D counterpart, BEC systems allow for further study of the analogies between classical and quantum turbulence. In this thesis we numerically and theoretically study 2D quantum turbulence in BECs within the framework of the Gross-Pitaevskii model. We focus on analogies with classical 2D turbulence, with the aim of identifying common or universal features.\nFirst we investigate coherent vortex structures in negative temperature equilibria via an experimentally accessible flow-field measure. Coherent vortices are shown to produce a clear signal in this measure that is independent of the confinement geometry, and we demonstrate that it can be observed in dynamical simulations.\nSecond, studying a quantum analogue of the two-dimensional cylinder wake, we investigate the phenomenon of Strouhal oscillations. We find that the Strouhal number obeys a universal relation, similar to the classical form, upon introducing a modified superfluid Reynolds number that accounts for the critical velocity for vortex nucleation. Like the classical Reynolds number, the superfluid Reynolds number is found to govern the transition from laminar to turbulent behaviour in the quantum fluid.\nFinally, simulating decaying 2D quantum turbulence for very large vortex numbers, we show that quantum fluids are capable of supporting the direct enstrophy cascade, a fundamental feature of two-dimensional turbulent flows. The quantum fluid manifests key features of the classical cascade, including Batchelor’s −3 law of the inertial range, scaling of the inertial range against the superfluid Reynolds number, and the value of the Kraichnan-Batchelor constant.\nThe findings from this work thus provide some new insight into the universality of fundamental turbulence concepts, and their applicability to quantum fluids.
Abstract Chapter 5 delves into quantum turbulence in superfluid helium and atomic Bose-Einstein condensates (BECs). The foundation of quantum turbulence research lies in classical fluid dynamics and turbulence, which is why we begin by covering essential concepts like the Reynolds number and Kolmogorov’s -5/3 power law in the energy spectrum. These principles are crucial for understanding quantum turbulence. We then explore the shared aspects of quantum turbulence in superfluid helium and atomic BECs, such as the differences in vortex configurations compared to classical turbulence, energy spectra, cascades, and the distinction between quasiclassical (Kolmogorov) and ultraquantum (Vinen) turbulence, focusing on their decay behaviors. In superfluid helium, we emphasize experimental studies in thermal counterflow and numerical simulations using the Vortex Filament Model (VFM). The advent of visualization experiments in the mid-2000s was transformative, allowing the observation of normal fluid velocity profiles and accelerations via tracer particles, which led to more accurate simulations coupling the VFM with normal fluid dynamics. Another significant topic is the energy spectrum of quantum turbulence, and whether it mirrors classical turbulence remains a key question. In addition to thermal counterflow, localized quantum turbulence generated by oscillating objects has been a major area of study. We present research showing how quantized vortices grow and evolve into turbulence. Superfluid 3He-B also provides a platform for quantum turbulence, with advancements in microfabrication bringing MEMS and NEMS into focus for studying quantum hydrodynamics at micro- and nanoscale, influenced by boundary conditions. In atomic BECs, confined by potentials, turbulence is induced through techniques like shaking or moving potentials. A unique aspect of BECs is two-dimensional quantum turbulence, featuring inverse energy cascades, Onsager vortices, and negative temperature states. Finally, atomic BECs, being compressible, serve as a platform for both vortex and wave turbulence. Multicomponent BECs, such as spinor BECs, introduce spin turbulence, further diversifying the field.
Identified as a unique regime of fluid flow some 500 years ago, turbulence has fascinated and confounded for centuries. Indeed, a fundamental understanding of the turbulence phenomenon remains one of the most important unsolved problems in classical physics. Interestingly, superfluid atomic Bose-Einstein condensates (BECs) can also be agitated into highly excited states of chaotic vortex motion, suggestive of turbulence. However, in contrast to classical fluids, the vortices in these quantum systems exhibit topological stability, the restriction of quantized circulation, and a well defined length scale characterized by the so-called healing length of the superfluid. Such simplifying aspects motivate the study of turbulence in these systems, which may shed new light on the longstanding classical problem. \n\nWhilst a considerable number of studies have been conducted on turbulence in three-dimensional BEC systems, and such studies have uncovered similarities between the classical and quantum counterparts, including the presence of the Kolmogorov k−5/3 law in the kinetic energy spectrum and the associated direct energy cascade, comparatively few studies have been conducted on two-dimensional quantum turbulence. An important question for such systems is whether the inverse energy cascade, a defining feature of classical two-dimensional turbulent fluid flows, may occur in these systems. In addition to a Kolmogorov kinetic energy spectrum, this process is signified by the clustering of same-circulation vortices, which results in the emergence of macroscopic vortex structures from the turbulent flow. \n\nAlthough intermittent few-vortex clustering has been observed in the break-down of superfluid flow around an obstacle, current attempts to observe a sustained inverse cascade have been unsuccessful, largely due to complications which arise due to the compressibility of the dilute Bose gas - a feature not considered in the classical theory.\n\nWe present a study of two-dimensional quantum turbulence in atomic BECs within the framework of damped Gross-Pitaevskii theory. With the motivation of determining the existence of the inverse cascade phenomenon in such systems, we focusing on experimentally realistic forcing procedures, and study the details of kinetic energy spectra, the statistics of vortex clustering, and the transport of kinetic energy through scale space in the resulting turbulent flows.\nWe investigate a range of vortex injection regimes produced through the circular stirring of a harmonically confined system with a repulsive Gaussian obstacle potential, identifying distinct regimes of turbulence. In the incompressible regime relevant to the inverse cascade, we identify numerous problems with the system and forcing procedure. Motivated by these findings, we investigate turbulence in a minimal model of the spindown of a toroidal persistent current. In this system, we identify a regime of forcing and damping in which the inverse cascade is corroborated by three independent measures: I) merging of vortex clusters; II) a Kolmogorov scaling in the incompressible kinetic energy spectrum; and III) Spectral condensation, associated with the emergence of large scale rotating structures from the turbulent flow. We provide clear evidence that the inverse cascade phenomenon, previously observed in a range of classical fluid systems, can also occur in quantum turbulence.
Turbulence appears in most natural and man-made flows. However, the analysis of turbulence is particularly difficult. Links between microscopic fluid dynamics and statistical signatures of turbulence appear unobtainable from the postulates of fluid dynamics making turbulence one of the most important unsolved theoretical problems in physics. Two-dimensional quantum turbulence (2DQT), an emerging field of study, involves turbulence in two-dimensional (2D) flows in superfluids, such as Bose-Einstein condensates (BECs). In 2D superfluids, a turbulent state can be characterized by a disordered distribution of numerous vortex cores. The question of how to effectively and efficiently generate turbulent states in superfluids is a fundamental question in the field of quantum turbulence. Therefore, experimental studies of vortex nucleation and the onset of turbulence in a superfluid are important for achieving a deeper understanding of the overall problem of turbulence. My PhD dissertation involves the study of vortex nucleation and the onset of turbulence in quasi-2D BECs. First, I discuss experimental apparatus advancements that now enable BECs to be created in a hybrid optical-magnetic trap, an atom trapping configuration conducive to 2DQT experiments. Next, I discuss the design and construction of a quantum vortex microscope and initial vortex detection tests. Finally, I present the first experiments aimed at studying 2DQT carried out in the updated apparatus. Thermal counterflow in superfluid helium, in which the normal and superfluid components flow in opposite directions, is known to create turbulence in the superfluid. However, this phenomenon has not been simulated or studied in dilute-gas BECs as a possible vortex nucleation method. In this dissertation, I present preliminary data from the first experiments aimed at understanding thermal counterflow turbulence in dilute-gas BECs.
Turbulence in classical fluids has been the subject of scientific study for centuries, yet there is still no complete general theory of classical turbulence connecting microscopic physics to macroscopic fluid flows, and this remains one of the open problems in physics. In contrast, the phenomenon of quantum turbulence in superfluids has well-defined theoretical descriptions, based on first principles and microscopic physics, and represents a realm of physics that can connect the classical and quantum worlds. Studies of quantum turbulence may thus be viewed as a path for progress on the long-standing problem of turbulence.A dilute-gas Bose-Einstein condensate (BEC) is, in most cases, a superfluid that supports quantized vortices, the primary structural elements of quantum turbulence. BECs are particularly convenient systems for the study of vortices, as standard techniques allow the microscopic structure and dynamics of the vortices to be investigated. Vortices in BECs can be created and manipulated using a variety of techniques, hence BECs are potentially powerful systems for the microscopic study of quantum turbulence.This dissertation focuses on quantized vortices in BECs, specifically experimental and numerical studies of their formation, dynamics, and decay, in an effort to understand the microscopic nature of vortices as elements of quantum turbulence. Four main experiments were performed, and are described in the main chapters of this dissertation, after introductions to vortices, experimental methods, and turbulence are presented. These experiments were aimed at understanding various aspects of how vortices are created and behave in a superfluid system. They involved vortex dipole nucleation in the breakdown of superfluidity, persistent current generation from a turbulent state in the presence of energy dissipation, decay of angular momentum of a BEC due to trapping potential impurities, and exploration of the spontaneous formation of vortices during the BEC phase transition. These experiments represent progress towards enhanced understanding of the formation, dynamics, and decay of vortices in BECs and thus may be foundational to more general studies of quantum turbulence in superfluids.
We argue that turbulence in superfluids is governed by two dimensionless parameters. One of them is the intrinsic parameter q which characterizes the friction forces acting on a vortex moving with respect to the heat bath, with q −1 playing the same role as the Reynolds number Re= UR/ν in classical hydrodynamics. It marks the transition between the “laminar” and turbulent regimes of vortex dynamics. The developed turbulence described by Kolmogorov cascade occurs when Re≫1 in classical hydrodynamics, and q ≪1 in superfluid hydrodynamics. Another parameter of superfluid turbulence is the superfluid Reynolds number Re s = UR/κ , which contains the circulation quantum κ characterizing quantized vorticity in superfluids. This parameter may regulate the crossover or transition between two classes of superfluid turbulence: (i) the classical regime of Kolmogorov cascade where vortices are locally polarized and the quantization of vorticity is not important; (ii) the quantum Vinen turbulence whose properties are determined by the quantization of vorticity. A phase diagram of the dynamical vortex states is suggested.
We review turbulence in ultracold quantum gases, using the scalar contact-interaction Bose-Einstein condensate as the reference system for quantized circulation, compressibility, vortices, sound, and cascades. We focus on the quantitative diagnostics that connect helium and classical phenomenology to microscopic wave-function dynamics: incompressible and compressible kinetic-energy spectra, wave-occupation spectra, spectral fluxes, vortex-resolved correlations, and velocity statistics. These diagnostics distinguish equilibrium vortex organization, decaying turbulent relaxation, forced cascade dynamics, and weak-wave turbulence, and show why power laws alone are insufficient evidence for a cascade. We survey experiments on two-dimensional Onsager clustering, three-dimensional vortex-line turbulence, box-trap wave cascades, engineered dissipation, and turbulent equations of state. We close by briefly placing the contact-interaction scalar superfluid system in a broader landscape of nonlocal, multicomponent, fermionic, and driven-dissipative quantum fluids, where turbulence concepts can be tested for universality.