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Statistical Theory of Magnetohydrodynamic Turbulence: Recent Results Mahendra K

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Statistical Theory of Magnetohydrodynamic Turbulence: Recent Results Mahendra K Physics Reports 401 (2004) 229–380 www.elsevier.com/locate/physrep Statistical theory of magnetohydrodynamic turbulence: recent results Mahendra K. Verma Department of Physics, Indian Institute of Technology, Kanpur 208016, India Accepted 4 July 2004 editor: I. Procaccia Abstract In this review article we will describe recent developments in statistical theory of magnetohydrodynamic (MHD) turbulence. Kraichnan and Iroshnikov first proposed a phenomenology of MHD turbulence where Alfvén time-scale dominates the dynamics, and the energy spectrum E(k) is proportional to k−3/2. In the last decade, many numerical 5 simulations show that spectral index is closer to 3 , which is Kolmogorov’s index for fluid turbulence. We review recent theoretical results based on anisotropy and Renormalization Groups which support Kolmogorov’s scaling for MHD turbulence. Energy transfer among Fourier modes, energy flux, and shell-to-shell energy transfers are important quantities in MHD turbulence. We report recent numerical and field-theoretic results in this area. Role of these quantities in magnetic field amplification (dynamo) are also discussed. There are new insights into the role of magnetic helicity in turbulence evolution. Recent interestingresults in intermittency, large-eddysimulations, and shell models of magnetohydrodynamics are also covered. © 2004 Published by Elsevier B.V. PACS: 47.27.Gs; 52.35.Ra; 11.10.Gh; 47.65.+a Keywords: MHD turbulence; Turbulent energy cascade rates; Field theory E-mail address: [email protected] (M.K. Verma). URL: http://home.iitk.ac.in/∼mkv. 0370-1573/$ - see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.physrep.2004.07.007 230 M.K. Verma / Physics Reports 401 (2004) 229–380 Contents 1. Introduction ........................................................................................232 2. MHD: Definitions and governing equations ..............................................................236 2.1. MHD approximations and equations..............................................................236 2.2. Energy equations and conserved quantities ........................................................239 2.3. Linearized MHD equations and their solutions; MHD waves .........................................242 2.4. Necessity for statistical theory of turbulence .......................................................244 2.5. Turbulence equations in spectral space ............................................................245 2.6. Energy equations ..............................................................................249 3. Mode-to-mode energy transfers and fluxes in MHD turbulence .............................................250 3.1. “Mode-to-mode” energy transfer in fluid turbulence .................................................250 3.1.1. Definition of mode-to-mode transfer in a triad ..............................................251 3.1.2. Solutions of equations of mode-to-mode transfer ...........................................252 3.2. Shell-to-shell energy transfer in fluid turbulence using mode-to-mode formalism ........................253 3.3. Energy cascade rates in fluid turbulence using mode-to-mode formalism ...............................254 3.4. Mode-to-mode energy transfer in MHD turbulence .................................................255 3.4.1. Velocity mode to velocity mode energy transfer.............................................256 3.4.2. Magnetic mode to magnetic mode energy transfers ..........................................257 3.4.3. Energy transfer between a velocity mode to a magnetic mode .................................257 3.5. Shell-to-shell energy transfer rates in MHD turbulence ..............................................260 3.6. Energy cascade rates in MHD turbulence ..........................................................260 3.7. Digression to infinite box .......................................................................261 4. MHD turbulence phenomenological models .............................................................263 4.1. Kolmogorov’s 1941 theory for fluid turbulence .....................................................263 4.2. MHD turbulence models for energy spectra and fluxes ..............................................265 − / 4.2.1. Kraichnan, Iroshnikov, and Dobrowolny et al.’s (KID) phenomenology—E(k) ∝ k 3 2 ..........265 − / 4.2.2. Marsch, Matthaeus and Zhou’s Kolmogorov-like phenomenology—E(k) ∝ k 5 3 ...............266 4.2.3. Grappin et al.—Alfvénic turbulence ......................................................267 −5/3 4.2.4. Goldreich and Sridhar—E(k ⊥ ) ∝ k ⊥ .................................................268 − / 4.2.5. Verma—effective mean magnetic field and E(k) ∝ k 5 3 ....................................270 −2 4.2.6. Galtier et al.—weak turbulence and E(k ⊥ ) ∝ k ⊥ .........................................270 4.3. Absolute equilibrium states .....................................................................270 4.4. Spectrum of magnetic helicity and cross helicity ...................................................273 4.5. Dynamic alignment ............................................................................273 4.6. Selective decay ...............................................................................274 4.7. “Phase” sensitivity of global quantities............................................................275 5. Solar wind: a testbed for MHD turbulence ...............................................................275 6. Numerical investigation of MHD turbulence .............................................................278 6.1. Numerical solution of MHD equations usingpseudo-spectral method ..................................279 3 5 6.2. Numerical results on energy spectra ( 2 or 3 ) ......................................................281 6.3. Numerical results on anisotropic energy spectra ....................................................283 6.4. Numerical results on energy fluxes ...............................................................284 6.5. Shell-to-shell energy transfer-rates in MHD turbulence ..............................................288 7. Renormalization group analysis of MHD turbulence ......................................................291 M.K. Verma / Physics Reports 401 (2004) 229–380 231 7.1. Renormalization groups in turbulence ............................................................291 7.1.1. Yakhot–Orszag(YO) perturbative approach ................................................292 7.1.2. Self-consistent approach of McComb and Zhou ............................................292 7.1.3. Callan–Symanzik equation for turbulence .................................................292 7.2. Physical meaningof renormalization in turbulence ..................................................293 7.3. “Mean magnetic field” renormalization in MHD turbulence ..........................................293 7.4. Renormalization of viscosity and resistivity usingself-consistent procedure .............................300 7.4.1. Nonhelical nonAlfvénic MHD (HM = HK = Hc = 0) .......................................300 7.4.2. Nonhelical Alfvénic MHD (HM = HK; c → 1) ...........................................308 7.4.3. Helical nonAlfvénic MHD (HM = 0; HK = 0; c = 0) ......................................312 7.5. RG calculations of MHD turbulence usingYO’s perturbative scheme ..................................312 7.5.1. Fournier, Sulem, and Pouquet ...........................................................313 7.5.2. Camargo and Tasso ....................................................................313 7.5.3. Liangand Diamond ....................................................................313 7.5.4. Berera and Hochberg ...................................................................313 7.5.5. Longcope and Sudan ...................................................................313 7.6. Callan–Symanzik equation for MHD turbulence ....................................................314 7.7. Other analytic techniques in MHD turbulence ......................................................314 8. Field-theoretic calculation of energy fluxes and shell-to-shell energy transfer ..................................314 8.1. Field-theoretic calculation of energy fluxes ........................................................315 8.1.1. Nonhelical nonAlfvénic MHD (HM = HK = Hc = 0) .......................................315 8.1.2. Nonhelical Alfvénic MHD (HM = HK = 0, c → 1) ........................................320 8.1.3. Helical nonAlfvénic MHD (HM = 0,HK = 0,Hc = 0) ......................................322 8.2. Field-theoretic calculation of shell-to-shell energy transfer ...........................................327 8.2.1. Nonhelical contributions ................................................................328 8.2.2. Helical contributions ...................................................................330 8.3. EDQNM calculation of MHD turbulence ..........................................................331 8.3.1. Pouquet et al. on nonhelical flows (H M = H K = 0) .........................................333 8.3.2. Pouquet et al. on helical flows ...........................................................333 8.3.3. Grappin et al. on Alfvénic MHD flows ....................................................334 8.3.4. EDQNM for 2D MHD flows ............................................................334 9. Field theory of anisotropic MHD turbulence .............................................................334
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