Shell Models of Magnetohydrodynamic Turbulence

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.

published in Physics Reports

References in corpus (20)

• Anisotropic scaling of magnetohydrodynamic turbulence
• Generation of magnetic field by dynamo action in a turbulent flow of liquid sodium
• A simple mechanism for the reversals of Earth's magnetic field
• Numerical demonstration of fluctuation dynamo at low magnetic Prandtl numbers
• Multi-scale Hall-MHD turbulence in the solar wind
• Turbulence in the sub-Alfvénic solar wind driven by reflection of low-frequency Alfvén waves
• Numerical Measurements of the Spectrum in Magnetohydrodynamic Turbulence
• Problems with kinematic mean field electrodynamics at high magnetic Reynolds numbers
• Numerical simulations of current generation and dynamo excitation in a mechanically-forced, turbulent flow
• Scaling laws in decaying helical hydromagnetic turbulence
• Anisotropic fluxes and nonlocal interactions in MHD turbulence
• Rapidly rotating spherical Couette flow in a dipolar magnetic field: an experimental study of the mean axisymmetric flow
• Influence of electromagnetic boundary conditions onto the onset of dynamo action in laboratory experiments
• Parametric instability of the helical dynamo
• Role of Cross Helicity in Cascade Processes of MHD turbulence
• Cascades and dissipation ratio in rotating MHD turbulence at low magnetic Prandtl number
• Phenomenology of turbulent dynamo growth and saturation
• Influence of inner and outer walls electromagnetic properties on the onset of a stationary dynamo
• Oscillating Ponomarenko dynamo in the highly conducting limit
• A Theory of Inertial Range Similarity in Isotropic Turbulence

Cited by in corpus (8)

• Energy transfers in dynamos with small magnetic Prandtl numbers
• Amplification of large-scale magnetic field in nonhelical magnetohydrodynamics
• A systematic bias in the calculation of spectral density from a 3D spatial grid
• Helicity statistics in homogeneous and isotropic turbulence and turbulence models
• Joint inverse cascade of magnetic energy and magnetic helicity in MHD turbulence
• Transition to turbulent dynamo saturation
• Dynamo transition in a five-mode helical model
• MHD Turbulence in spin-down flows of liquid metals