The term quantum turbulence denotes the turbulent motion of quantum fluids, systems such as superfluid helium and atomic Bose–Einstein condensates (BEC). Quantum fluids differ from ordinary fluids in three respects: (1) they exhibit two-fluid behavior at nonzero temperatures, (2) they can flow freely, without the dissipative effect of viscous forces, and (3) their local rotation is constrained to discrete vortex lines of known size and strength. These quantised vortices can be seen in the image on the right which displays the density of a rotating BEC; dark areas denote low density and the small black 'dimples' indicate the presence of a vortex. This is in contrast to the the eddies in ordinary (classical) fluids, which are continuous and can have arbitrary size, shape, and strength. Recent experiments and numerical simulations have highlighted quantitative connections (as well as fundamental differences) between turbulence in quantum fluids and turbulence in ordinary fluids (classical turbulence). I primarily perform high resolution numerical simulations using either the vortex filament method or seeking solutions to the Gross-Pitaevskii equation.

### Research Highlights

**Kelvin Wave Cascade** Whilst there are similarties between quantum and classical turbulence, at small scales the similarities cease. In classical fluid turbulence, energy is dissipated through viscosity. However, in zero-temperature superfluid turbulence there is no viscosity, so how is energy dissipated? The Kelvin-wave cascade has been proposed to explain this surprising effect.

A Kelvin wave is a rotating sinusoidal or helical displace- ment of the core of vortex filaments away from its unperturbed position, they can be triggered in a number of waves, but vortex reconnection is probably the dominant mechanism.The Kelvin-wave cascade is the process in which the nonlinear interaction of Kelvin waves creates waves of shorter and shorter wavelength.

At high temperatur mutual friction quickly damps out the shorter Kelvin waves, but at low temperatures the cascade proceeds unhindered, until the wavenumber is large enough that sound is efficiently radiated away (phonon emission) by rapidly rotating vortices. Rival theories of the cascade were proposed. In 2014 myself and Jason Laurie provided numerical evidence that weakly nonlinear Kelvin wave interactions are governed by the nonlocal wave turbulence theory proposed by L'vov and Nazarenko.

**Coherent structures** Using a numerical model of quantum turbulence, we showed that the total vortex line density can be decomposed into two parts: one formed by metastable bundles of coherent vortices, and one in which the vortices are randomly oriented. We showed that the former is responsible for the observed Kolmogorov energy spectrum, and the latter for the spectrum of the vortex line density fluctuations. Theses results help to explain puzzling experimental measurements of the vortex line density in a superfluid wind tunnel made by Philippe Roche and collaborators.

**Acceleration statistics**

We numerically determined the one-point superfluid acceleration statistics in counterflow turbulence and demonstrated how the mean velocity and acceleration scale with counterflow velocity and temperature.

Counterflow turbulence is the original and perhaps easiest way to observe quantum turbulence in the laboratory. A prototypical experiment consists of a channel which is closed at one end and open to the helium bath at the other end. At the closed end, a resistor inputs a steady flux of heat into the channel. The heat is carried away from the resistor towards the bath by the normal fluid component, whereas the superfluid component flows towards the resistor to maintain zero total mass flux. If the relative velocity of superfluid and normal fluid is higher than a small critical value, the laminar counterflow of the two fluids breaks down and a tangle of vortex lines appears, thus limiting the heat conducting properties of helium-4.

In this study we also showed that the probability density function of one-point acceleration statistics should follow a power-law distribution, with a −5/3 exponent. Previous experimental and numerical studies have shown that the one-point velocity statistics also follow a power-law distribution, due the singular nature of the velocity field induced by a quantised vortex. Our numerical results support these arguments; more important we see excellent agreement between out numerics and experimental results from the Prague group.

**Visualizing Pure Quantum Turbulence in Superfluid ^{3}He**

Superfluid

^{3}He-B in the zero-temperature limit offers a unique means of studying quantum turbulence through the Andreev reflection of quasiparticle excitations by the vortex flow fields. We validated the experimental visualization of turbulence in

^{3}He-B by showing the relation between the vortex-line density and the Andreev reflectance of the vortex tangle in the first simulations of the Andreev reflectance by a realistic 3D vortex tangle. A previous study argued that fluctuations of the Andreev-reflected signal can be interpreted as fluctuations of the vortex line density in. Our combined numerical-experimental results showed that the fluctuations of the vortex line density and of the Andreev reflection are indeed correlated.