Turbulent cascade arrests and the formation of intermediate-scale condensates

Energy cascades lie at the heart of the dynamics of turbulent flows. In a recent study of turbulence in fluids with odd viscosity X. M. de Wit et al. [Nature (London) 627, 515 (2024)0028-083610.1038/s41586-024-07074-z], the two dimensionalization of the flow at small scales leads to the arrest of the energy cascade and selection of an intermediate scale, between the forcing and the viscous scales. To demonstrate the generality of the phenomenon and its existence for a wide class of turbulent systems, we study a shell model that is carefully constructed to have three-dimensional turbulent dynamics at small wave numbers and two-dimensional turbulent dynamics at large wave numbers. The large scale separation that we can achieve in our shell model allows us to examine clearly the interplay between these dynamics, which leads to an arrest of the energy cascade at a transitional wave number and an associated accumulation of energy at the same scale. Such pile-up of energy around the transitional wave number is reminiscent of the formation of condensates in two-dimensional turbulence, but, in contrast, it occurs at intermediate wave numbers instead of the smallest wave number.