Superfluids are extraordinary fluids characterised by the complete absence of viscosity, with low-temperature helium (below 2.1 K) and atomic Bose–Einstein condensates as the most prominent examples. They are macroscopic manifestations of quantum mechanics and are routinely studied in the lab today. A defining feature of superfluids is their concentration of vorticity into extremely thin filaments—topological defects known as quantum vortices—where circulation is quantised. Despite their inviscid and quantum nature, superfluids share striking similarities with high-Reynolds-number classical flows and can be regarded as the skeleton of turbulent flows.

One of the most fundamental equations describing the dynamics of superfluids is the Gross-Pitaevskii model. Beneath its apparent mathematical simplicity lies a remarkably rich myriad of physical phenomena, including non-linear wave dynamics, quantum vortex nucleation, vortex dynamics and reconnection, and turbulence. Although this model is formally derived for weakly interacting Bose gases, its hydrodynamical description is rich enough to give an excellent qualitative description of superfluid helium.

In this lecture, I will first give an introduction to the Gross-Pitaevskii model and present its most fundamental solutions: density waves, quantum vortices and the connection to hydrodynamics. Then, I will follow a journey across scales, presenting theoretical and numerical results that range from the process of vortex reconnection to a fine comparison between classical and quantum turbulence, where the dynamics of intricate vortex tangles lead to very complex statistics. Throughout the lecture, I will highlight the differences and similarities between classical and quantum fluids, uncovering the universal phenomena they share. In the final part of the lecture, I will briefly illustrate how the Gross–Pitaevskii model can be extended to incorporate the dynamics of moving and reacting objects, and how it may be generalised to offer a more accurate description of superfluid helium.