Toward a theory of interstellar turbulence. 1: Weak Alfvenic turbulence

Abstract

We study weak Alfvenic turbulence of an incompressible, magnetized fluid in some detail, with a view to developing a firm theoretical basis for the dynamics of small-scale turbulence in the interstellar medium. We prove that resonant 3-wave interactions are absent. We also show that the Iroshnikov-Kraichnan theory of incompressible, magnetohydrodynamic turbulence -- which is widely accepted -- describes weak 3-wave turbulence; consequently, it is incorrect. Physical arguments, as well as detailed calculations of the coupling coefficients are used to demonstrate that these interactions are empty. We then examine resonant 4-wave interactions, and show that the resonance relations forbid energy transport to small spatial scales along the direction of the mean magnetic field, for both the shear Alfven wave and the pseudo Alfven wave. The three-dimensional inertial-range energy spectrum of 4-wave shear Alfven turbulence guessed from physical arguments reads E(k<SUB>z</SUB>, k<SUB>perpendicular</SUB>) approximately V<SUB>A</SUB>v<SUB>L</SUB>L<SUP>-1/3</SUP>k<SUB>perpendicular</SUB><SUP>-10/3</SUP>, where V<SUB>A</SUB> is the Alfven speed, and v<SUB>L</SUB> is the velocity difference across the outer scale L. Given this spectrum, the velocity difference across lambda<SUB>perpendicular</SUB> approximately k<SUB>perpendicular exp -1</SUB> is v<SUB>lambda (sub perpendicular</SUB>) is approximately v<SUB>L</SUB>(lambda<SUB>perpendicular</SUB>/L)<SUP>2/3</SUP>. We derive a kinetic equation, and prove that this energy spectrum is a stationary solution and that it implies a positive flux of energy in k-space, along directions perpendicular to the mean magnetic field. Using this energy spectrum, we deduce that 4-wave interactions strengthen as the energy cascades to small, perpendicular spatial scales; beyond an upper bound in perpendicular wavenumber, k<SUB>perpendicular</SUB>L is approximately (V<SUB>A</SUB>/v<SUB>L</SUB>)<SUP>3/2</SUP>, weak turbulence theory ceases to be valid. Energy excitation amplitudes must be very small for the 4-wave inertial-range to be substantial. When the excitation is strong, the width of the 4-wave inertial-range shrinks to zero. This seems likely to be the case in the interstellar medium.