Shell Models of Magnetohydrodynamic Turbulence

TL;DR

This paper reviews shell models of magnetohydrodynamic turbulence, highlighting their development, capabilities, and applications, including dynamo action and wave phenomena, emphasizing their ability to simulate extended inertial ranges efficiently.

Contribution

It provides a comprehensive mathematical framework for MHD shell models and discusses their advantages, limitations, and recent improvements in capturing turbulence dynamics.

Findings

Shell models can simulate turbulence with extended inertial ranges.

They reflect key turbulence features like intermittency and reversals.

Applications include dynamo action and wave phenomena in MHD.

Abstract

Shell models of hydrodynamic turbulence originated in the seventies. Their main aim was to describe the statistics of homogeneous and isotropic turbulence in spectral space, using a simple set of ordinary differential equations. In the eighties, shell models of magnetohydrodynamic (MHD) turbulence emerged based on the same principles as their hydrodynamic counter-part but also incorporating interactions between magnetic and velocity fields. In recent years, significant improvements have been made such as the inclusion of non-local interactions and appropriate definitions for helicities. Though shell models cannot account for the spatial complexity of MHD turbulence, their dynamics are not over simplified and do reflect those of real MHD turbulence including intermittency or chaotic reversals of large-scale modes. Furthermore, these models use realistic values for dimensionless parameters (high kinetic and magnetic Reynolds numbers, low or high magnetic Prandtl number) allowing extended inertial range and accurate dissipation rate. Using modern computers it is difficult to attain an inertial range of three decades with direct numerical simulations, whereas eight are possible using shell models. In this review we set up a general mathematical framework allowing the description of any MHD shell model. The variety of the latter, with their advantages and weaknesses, is introduced. Finally we consider a number of applications, dealing with free-decaying MHD turbulence, dynamo action, Alfven waves and the Hall effect.