A simple XY model for cascade transfer

Cascade transfer is the phenomenon that an inviscid conserved quantity, such as energy or enstrophy, is transferred conservatively from large (small) to small (large) scales. As a consequence of this cascade transfer, the distribution of the transferred quantity obeys a universal scaling law independent of the details of large (small) scales. For example, in the energy cascade in fluid turbulence, the energy spectrum follows Kolmogorov's power law [1]. Such behavior is observed even in systems different from ordinary fluids, such as quantum fluid, elastic body, and spin systems. Here, we aim to establish the concept of a universality class for cascade transfer. As a first step toward this end, we propose a simple model representing one universality class [2]. In doing so, we regard cascade transfer as a cooperative phenomenon of unidirectional transport across scales and ask how it emerges from spatially local interactions. The constructed model is a modified XY model with amplitude fluctuations, in which the spin is regarded as the “velocity” of a turbulent field in d dimensions. We show that the model exhibits an inverse energy cascade with the non-Kolmogorov energy spectrum. We also discuss the relation to spin turbulence [3,4] and atmospheric turbulence [5].