  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.
