# Local and non-local energy spectra of superfluid $^3$He turbulence

## TL;DR

This study investigates the energy spectra of superfluid helium-3 turbulence, revealing temperature-dependent spectral behaviors and enhanced intermittency compared to classical turbulence, using combined numerical simulations and analytic models.

## Contribution

It introduces a non-local energy transfer analysis for superfluid helium-3 turbulence, extending previous models and confirming new spectral regimes across different temperatures.

## Key findings

- Confirmed subcritical energy spectrum with power-law exponents between 5/3 and 3 at low temperatures.
- Discovered scale-invariant spectrum with exponents up to approximately 9 at higher temperatures.
- Observed significantly increased intermittency near critical temperatures, surpassing classical turbulence levels.

## Abstract

Below the phase transition temperature $Tc \simeq 10^{-3}$K He-3B has a mixture of normal and superfluid components. Turbulence in this material is carried predominantly by the superfluid component. We explore the statistical properties of this quantum turbulence, stressing the differences from the better known classical counterpart. To this aim we study the time-honored Hall-Vinen-Bekarevich-Khalatnikov coarse-grained equations of superfluid turbulence. We combine pseudo-spectral direct numerical simulations with analytic considerations based on an integral closure for the energy flux. We avoid the assumption of locality of the energy transfer which was used previously in both analytic and numerical studies of the superfluid He-3B turbulence. For T<0.37 Tc, with relatively weak mutual friction, we confirm the previously found "subcritical" energy spectrum E(k), given by a superposition of two power laws that can be approximated as $E(k)~ k^{-x}$ with an apparent scaling exponent 5/3 <x(k)< 3. For T>0.37 Tc and with strong mutual friction, we observed numerically and confirmed analytically the scale-invariant spectrum $E(k)~ k^{-x}$ with a (k-independent) exponent x > 3 that gradually increases with the temperature and reaches a value $x\simeq 9$ for $T\approx 0.72 Tc$. In the near-critical regimes we discover a strong enhancement of intermittency which exceeds by an order of magnitude the corresponding level in classical hydrodynamic turbulence.