Book Description

Excerpt from Statistical Mechanics of Stationary Homogeneous Hydromagnetic Turbulence The statistical mechanics of stationary, homogeneous, hydromagnetic turbulence is treated by a method which does not require the assumption of small departure from a state of detailed balance. The equations of motion under cyclic boundary conditions are expressed in terms of the external driving forces and linearly independent dynamic variables formed from the spatial Fourier coefficients of velocity and magnetic fields. The nonlinear interaction is exhibited as the sum of individually conservative elementary interactions among triads of modes. A functional equation is constructed giving necessary and sufficient conditions for a distribution of time functions to satisfy the equations of motion, it involves only the second-, third-, and fourth-order moments of the distribution functional. By using the fact that each mode interacts with many others, a procedure is developed in which each elementary interaction is introduced as a perturbation on the total motion. The moments appearing in the functional equation are expressed by means of this procedure in terms of the diagonal elements of the covariance matrices of the dynamic variables and external forces and the distribution-averaged infinitesimal impulse response matrix of the system. Closed equations for these quantities are obtained so that the power spectra in frequency and wave number of velocity and magnetic fields, and the energy exchange among the modes, may be determined from the driving force power-spectrum. A variational method is given for defining and constructing the most nearly normal distribution functional with the previously derived second-, third-, and fourth-order moments, thereby determining the higher moments. The consistency with the present theory is confirmed for the cascade hypothesis fundamental to the Kolmogorov similarity theory of the inertial range. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.