Numerical Studies of Quantum Turbulence

Chapter 3 focuses on the physics of quantized vortices in superfluid helium. We begin by introducing the unique properties of 4He, including its phase diagram, the superfluid transition (lambda transition), and the phenomenon of superfluidity. Following this, we will discuss the two-fluid model, a foundational and widely used phenomenological framework. Next, we delve into thermal counterflow, a crucial area that spurred the initial research on quantum turbulence, superfluid turbulence, and quantized vortices. We will then explore the dynamics of quantized vortices, often described by the Vortex Filament Model (VFM), which plays a central role in understanding quantum turbulence. The equations of motion and basic dynamics of vortices will be presented, and since the vortex motion is both nonlinear and nonlocal, numerical simulations are essential for accurately describing quantum turbulence. We will cover the computational methods used in these simulations, including both “two-way” and “one-way” approaches. In addition to quantum turbulence, we will examine the quantized vortex lattice under rotation, another key phenomenon in superfluidity. Early experiments on vortex lattice formation, the realization of rigid-body rotation, and simple examples of vortex dynamics under rotation will be introduced. Finally, we will discuss recent advancements in visualization experiments in superfluid helium, highlighting the results of visualizing quantized vortices, which have greatly enhanced our understanding of vortex dynamics.

Chapter 1 Energy Spectra of Quantum Turbulence

We study theoretically and numerically the dynamics of quantized vortices in superfluid helium and rotating Bose-Einstein condensates. After reviewing briefly the recent motivation, we discuss these topics with the emphasis on the research done by our group. One of the modern important problems is how superfluid turbulence relates to classical turbulence. First, we show that superfluid turbulence consisting of a vortex tangle has an energy spectrum consistent with the Kolmogorov law. Second, we discuss the vortex states that appear in a rotating channel with counterflow. In the field of atomic-gas Bose-Einstein condensation, the dramatic observations of quantized vortices were made for rotating condensates. By solving numerically the Gross-Pitaevskii equation, we found a whole story of the vortex lattice formation consistent with the observations.

Bose-Einstein condensates of dilute gases are well-suited for investigations of vortex dynamics and turbulence in quantum fluids, yet there has been little experimental research into the approaches that may be most promising for generating states of two-dimensional turbulence in these systems. Here we give an overview of techniques for generating the large and disordered vortex distributions associated with two-dimensional quantum turbulence. We focus on describing methods explored in our Bose-Einstein condensation laboratory, and discuss the suitability of these methods for studying various aspects of two-dimensional quantum turbulence. We also summarize some of the open questions regarding our own understanding of these mechanisms of two-dimensional quantum turbulence generation in condensates. We find that while these disordered distributions of vortices can be generated by a variety of techniques, further investigation is needed to identify methods for obtaining quasi-steady-state quantum turbulence in condensates.

The problem of quantum turbulence in a channel with an inhomogeneous counterflow of superfluid turbulent helium is studied. \ The counterflow velocity $V_{ns}^{x}(y)$ along the channel is supposed to have a parabolic profile in the transverse direction $y$. Such statement corresponds to the recent numerical simulation by Khomenko et al. [Phys. Rev. B \textbf{91}, 180504 (2015)]. The authors reported about a sophisticated behavior of the vortex line density (VLD) $\mathcal{L}(\mathbf{r},t)$, different from $% \mathcal{L}\propto V_{ns}^{x}(y)^{2}$, which follows from the naive, straightforward application of the conventional Vinen theory. It is clear, that Vinen theory should be refined by taking into account transverse effects and the way it ought to be done is the subject of active discussion in the literature. In the work we discuss several possible mechanisms of the transverse flux of VLD $\mathcal{L}(\mathbf{r},t)$ which should be incorporated in the standard Vinen equation to describe adequately the inhomogeneous quantum turbulence (QT). It is shown that the most effective among these mechanisms is the one that is related to the phase slippage phenomenon. The use of this flux in the modernized Vinen equation corrects the situation with an unusual distribution of the vortex line density, and satisfactory describes the behavior $\mathcal{L}(\mathbf{r},t)$ both in stationary and nonstationary situations. The general problem of the phenomenological Vinen theory in the case of nonuniform and nonstationary quantum turbulence is thoroughly discussed.

Quantum turbulence: Theoretical and numerical problems

The quantum turbulence is a collection of the interacting quantum vortex loops and filaments. The main goal of this work is to systematize the information on the current state of the research on the reconnecting quantized vortex loops in the superfluid helium: the geometry, the dynamics, the properties of the vortex loops, the energy spectrum before and after the reconnections. The paper discusses the possible role of the reconnections in the formation of the turbulent spectrum. In addition, in this paper the main methods and approaches to the study of the reconnecting vortex loops and quantum turbulence are discussed.

The term quantum turbulence denotes the turbulent motion of quantum fluids, systems such as superfluid helium and atomic Bose-Einstein condensates, which are characterized by quantized vorticity, superfluidity, and, at finite temperatures, two-fluid behavior. This article introduces their basic properties, describes types and regimes of turbulence that have been observed, and highlights similarities and differences between quantum turbulence and classical turbulence in ordinary fluids. Our aim is also to link together the articles of this special issue and to provide a perspective of the future development of a subject that contains aspects of fluid mechanics, atomic physics, condensed matter, and low-temperature physics.

Recent experiments have shown that micron-size particles can be trapped in quantized vortices in superfluid helium. We present results of a dynamically self-consistent calculation of the nonlinear interaction of a sphere and a vortex. We find that, depending on the regime, the sphere can be trapped in the vortex or break free from it. We also find that, if trapped, the sphere can travel along the vortex. We analyze the role played by the various forces acting on the sphere during the interaction.

When two vortices cross, each of them breaks into two parts and exchanges part of itself for part of the other. This process, called vortex reconnection, occurs in classical and superfluids, and in magnetized plasmas and superconductors. We present the first experimental observations of reconnection between quantized vortices in superfluid helium. We do so by imaging micrometer-sized solid hydrogen particles trapped on quantized vortex cores and by inferring the occurrence of reconnection from the motions of groups of recoiling particles. We show that the distance separating particles on the just-reconnected vortex lines grows as a power law in time. The average value of the scaling exponent is approximately 1/2, consistent with the self-similar evolution of the vortices.

A dramatic effect of quantized vortices in superfluid helium on the rate of coalescence of suspended impurities has been predicted; such a catalytic process should result in the formation of fiber-like structures having primarily nanothickness. That should be valid for any impurity and it may be used as a basis for a universal method of producing nanowires and nanotubes. Experiments on the imbedding of molecular hydrogen into liquid helium have supported these conclusions. They showed that: (i) in normal liquid He the coalescence led to the formation of spherical microparticles carried by turbulent motion of the liquid; (ii) in the superfluid only very long filaments were observed, which behaved as quantized vortices should do. These filaments are fiber-like hydrogen crystals and survive the transition of the liquid helium to the normal state. The promises for using this phenomenon in basic and applied sciences are outlined.

Superfluid turbulence has been one of the most important problems of superfluid hydrodynamics, in which quantized vortices play a significant role. The recent research on superfluid turbulence enters a new stage rather different from the old ones chiefly devoted to thermal counterflow. After describing the current motivation on these topics, we discuss our research which studies the energy spectrum both by the vortex filament model and the Gross-Pitaevskii model. Both energy spectra are consistent with the Kolmogorov law, which shows a close similarity between superfluid (quantum) and classical turbulence.

The generation of particles to observe quantized vortex dynamics in superfluid helium

Transition from laminar to turbulent states of classical viscous fluids is complex and incompletely understood. Transition to quantum turbulence (QT), by which we mean the turbulent motion of quantum fluids such as helium II, whose physical properties depend on quantum physics in some crucial respects, is naturally more complex. This increased complexity arises from superfluidity, quantization of circulation, and, at finite temperatures below the critical, the two-fluid behavior. Transition to QT could involve, as an initial step, the transition of the classical component, or the intrinsic or extrinsic nucleation of quantized vortices in the superfluid component, or a simultaneous occurrence of both scenarios—and the subsequent interconnected evolution. In spite of the multiplicity of scenarios, aspects of transition to QT can be understood at a phenomenological level on the basis of some general principles, and compared meaningfully with transition in classical flows.

We establish a statistical relationship between the inverse energy cascade and the spatial correlations of clustered vortices in two-dimensional quantum turbulence. The Kolmogorov spectrum k^{-5/3} on inertial scales r corresponds to a pair correlation function between the vortices with different signs that decays as a power law with the pair distance given as r^{-4/3}. To test these scaling relations, we propose a forced and dissipative point vortex model that captures the turbulent dynamics of quantized vortices by the emergent clustering of same-sign vortices. The inverse energy cascade developing in a statistically neutral system originates from this vortex clustering that evolves with time.