Description
Reduced models are important for describing the multiscale nonlinear dynamics typical of fluid and plasma turbulence. In this spirit, Hybrid Lattices (HLs) are proposed in order to provide a novel framework to investigate turbulent cascades in Fourier space, bridging regular grids and models based on logarithmic discretizations. The approach combines a regular spectral region, capable of resolving large-scale coherent structures, with a self-similar construction that span many decades in scale in the spirit of shell models.
Shell models reproduce turbulent cascades across wide temporal and spatial ranges, capturing features such as power-law spectra and intermittency. These models drastically decimate the Fourier space through a one-dimensional sequence of logarithmically spaced modes, simplifying the complexity of triadic couplings. Recently, logarithmic lattices and nested polyhedra models have extended these ideas to more than one dimension. However, the logarithmic spacing factor, typically chosen as the golden ratio, causes the Cartesian components of wave-numbers to be non-integer, and prevents transfer between a regular central region and the logarithmic part.
To address these limitations, first the concept of recurrent lattices where introduced, where the outer regions are expanded through recurrence relations, generating an asymptotically logarithmic lattice where Cartesian components of wave-numbers remain integer-valued. They preserve triadic interactions linking regular and recurrent regions, enabling cascade across them. This approach was first demonstrated for shell models.
The Hybrid lattices are then formed by using a regular central part of the grid in Fourier space that is expanded using the recurrence relation. An efficient algorithm have recently been developed for computing convolutions on such lattices. Using this algorithm different turbulent systems are formulated and then investigated. First, considering the one-dimensional Burgers equation. Secondly, in the two-dimensional incompressible Navier–Stokes turbulence, we examine both forward and dual cascade regimes.
Finally, preliminary results on the applicability of HLs to anisotropic plasma turbulence are presented, by considering the Hasegawa–Wakatani model, whose turbulent dynamics generate zonal flow structures. In this setting, the lattice is constructed by employing logarithmic discretization along the poloidal direction, while maintaining full spectral resolution in the radial direction to accurately capturing zonal flow and density profiles.