Towards High-Quality Visualization of Superfluid Vortices

Abstract

Superfluidity is a special state of matter exhibiting macroscopic quantum phenomena and acting like a fluid with zero viscosity. In such a state, superfluid vortices exist as phase singularities of the model equation with unique distributions. This paper presents novel techniques to aid the visual understanding of superfluid vortices based on the state-of-the-art non-linear Klein-Gordon equation, which evolves a complex scalar field, giving rise to special vortex lattice/ring structures with dynamic vortex formation, reconnection, and Kelvin waves, etc. By formulating a numerical model with theoretical physicists in superfluid research, we obtain high-quality superfluid flow data sets without noise-like waves, suitable for vortex visualization. By further exploring superfluid vortex properties, we develop a new vortex identification and visualization method: a novel mechanism with velocity circulation to overcome phase singularity and an orthogonal-plane strategy to avoid ambiguity. Hence, our visualizations can help reveal various superfluid vortex structures and enable domain experts for related visual analysis, such as the steady vortex lattice/ring structures, dynamic vortex string interactions with reconnections and energy radiations, where the famous Kelvin waves and decaying vortex tangle were clearly observed. These visualizations have assisted physicists to verify the superfluid model, and further explore its dynamic behavior more intuitively.

On the Rotating Nonlinear Klein--Gordon Equation: NonRelativistic Limit and Numerical Methods

“…Equation (1.1) has been used to describe the creation and dynamics of the quantized vortices of a galaxy in 2D or 3D models as a "cosmic rotating superfluid". The vortices generation that obeys the relativistic Feynman relation, quantum turbulence in the form of a vortex tangle that follows mechanism for matter creation during the big bang era, and the formation and transmission of Kelvin waves correspond well to observations [28,31,39,53,54]. For generalisations of (1.1) to curved spacetime Klein-Gordon models, e.g.…”

“…Polar Coordinates. It has been popular to simulate the quantized vortices dynamics on a disk-shaped domain in polar coordinates [2,6,12,53,28,59,34], where the angular momentum operator reads simply as derivative with respect to the angle, i.e. L z = −i∂ θ .…”

“…Such superfluids [37] are often described by a complex-valued scalar field as Ginzburg and Landau proposed an order parameter for the phase coherence [26]. The nonlinear Klein-Gordon equation has then become a popular superfluid model, which is able to cover the entire velocity range, especially the relativistic region [28,55,31,39,53,54,15,48,23,49,21,52]. As a model for describing a rotating galaxy in Minkowski metric, the nonlinear Klein-Gordon equation in a rotating frame [28,31,39,53,54] is used.…”

“…The nonlinear Klein-Gordon equation has then become a popular superfluid model, which is able to cover the entire velocity range, especially the relativistic region [28,55,31,39,53,54,15,48,23,49,21,52]. As a model for describing a rotating galaxy in Minkowski metric, the nonlinear Klein-Gordon equation in a rotating frame [28,31,39,53,54] is used. Also, the nonlinear Klein-Gordon equation appears as a relativistic generalization of the Gross-Pitaevskii equation, a nonlinear Schrödinger equation that serves as a basic model for Bose-Einstein Condensates [2], also in the rotating case, see e.g.…”

“…In order to efficiently study the dynamics, we propose numerical methods for solving (1.3) in polar or Cartesian coordinates. The polar coordinates are popular for studying vortices dynamics in Schrödinger equation or Ginzburg-Landau equation in the literature [2,6,34,12,53,28,59]. We shall present a semi-implicit Fourier spectral method in the polar coordinates for the RKG, which simultaneously diagonalizes both rotating terms.…”