TOWARD A THEORY OF INTERSTELLAR TURBULENCE .1. WEAK ALFVENIC TURBULENCE

We study weak Alfvenic turbulence of an incompressible, magnetized flu id 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 t hat the Iroshnikov-Kraichnan theory of incompressible, magnetohydrodyn amic turbulence-which is widely accepted-describes weak 3-wave turbule nce; consequently, it is incorrect. Physical arguments, as well as det ailed calculations of the coupling coefficients are used to demonstrat e that these interactions are empty. We then examine resonant 4-wave i nteractions, and show that the resonance relations forbid energy trans port 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 Alfve n turbulence guessed from physical arguments reads E(k(z), k(p)erpendi cular to) similar to V-A v(L) L(-1/3)k(perpendicular to)(-10/3), where V-A is the Alfven speed, and v(L) is the velocity difference across t he outer scale L. Given this spectrum, the velocity difference across lambda(perpendicular to) similar to k(perpendicular to)(-1) is v(lambd a perpendicular to) similar to v(L)(lambda(perpendicular to)/L)(2/3). We derive a L kinetic equation, and prove that this energy spectrum is a stationary solution and that it implies a positive flux of energy i n k-space, along directions perpendicular to the mean magnetic field. Using this energy spectrum, we deduce that 4-wave interactions strengt hen as the energy cascades to small, perpendicular spatial scales; bey ond an upper bound in perpendicular wavenumber, k(perpendicular to) L similar to (V-A/v(L))(3/2), weak turbulence theory ceases to be valid. Energy excitation amplitudes must be very small for the 4-wave inerti al-range to be substantial. When the excitation is strong, the width o f the 4-wave inertial-range shrinks to zero. This seems likely to be t he case in the interstellar medium. The physics of strong turbulence i s explored in Paper II.