Emergent isotropy of a wave-turbulent cascade in the Gross-Pitaevskii model

TL;DR

This study investigates how turbulence in a Bose-Einstein condensate modeled by the Gross-Pitaevskii equation naturally evolves towards isotropy at small scales despite anisotropic forcing, revealing self-similar cascade behavior.

Contribution

It demonstrates the emergence of isotropy in wave turbulence within the Gross-Pitaevskii model under anisotropic forcing, highlighting the robustness of isotropic steady states.

Findings

Anisotropy decreases at high momenta in steady state

Self-similar cascade front propagates in momentum space

Isotropy is robust against drive amplitude variations

Abstract

The restoration of symmetries is one of the most fascinating properties of turbulence. We report a study of the emergence of isotropy in the Gross-Pitaevskii model with anisotropic forcing. Inspired by recent experiments, we study the dynamics of a Bose-Einstein condensate in a cylindrical box driven along the symmetry axis of the trap by a spatially uniform force. We introduce a measure of anisotropy $A(k,t)$ defined on the momentum distributions $n(\boldsymbol{k},t)$, and study the evolution of $A(k,t)$ and $n(\boldsymbol{k},t)$ as turbulence proceeds. As the system reaches a steady state, the anisotropy, large at low momenta because of the large-scale forcing, is greatly reduced at high momenta. While $n(\boldsymbol{k},t)$ exhibits a self-similar cascade front propagation, $A(k,t)$ decreases without such self-similar dynamics. Finally, our numerical calculations show that the isotropy of the steady state is robust with respect to the amplitude of the drive.