In this book, we report on research in methods of computational magneto- hydrodynamics supported by the United States Department of Energy under Contract EY-76-C-02-3077 with New York University. The work has re- sulted in a computer code for mathematical analysis of the equilibrium and stability of a plasma in three dimensions with toroidal geometry but no sym- metry. The code is listed in the final chapter. Versions of it have been used for the design of experiments at the Los Alamos Scientific Laboratory and the Max Planck Institute for Plasma Physics in Garching. We are grateful to Daniel Barnes, Jeremiah Brackbill, Harold Grad, William Grossmann, Abraham Kadish, Peter Lax, Guthrie Miller, Arnulf Schliiter, and Harold Weitzner for many useful discussions of the theory. We are especially indebted to Franz Herrnegger for theoretical and pedagogical comments. Constance Engle has provided outstanding assistance with the typescript. We take pleasure in acknowledging the help of the staff of the Courant Mathematics and Com- puting Laboratory at New York University. In particular we should like to express our thanks to Max Goldstein, Kevin McAuliffe, Terry Moore, Toshi Nagano and Tsun Tam. Frances Bauer New York Octavio Betancourt September 1978 Paul Garabedian v Contents Chapter 1. Introduction 1 1. 1 Formulation of the Problem 1 1. 2 Discussion of Results 2 Chapter 2. The Variational Principle 4 4 2. 1 The Magnetostatic Equations 6 2. 2 Flux Constraints in the Plasma . 7 2. 3 The Ergodic Constraint .

In this book, we describe in detail a numerical method to study the equilibrium and stability of a plasma confined by a strong magnetic field in toroidal geometry without two-dimensional symmetry. The principal appli cation is to stellarators, which are currently of interest in thermonuclear fusion research. Our mathematical model is based on the partial differential equations of ideal magnetohydrodynamics. The main contribution is a computer code named BETA that is listed in the final chapter. This work is the natural continuation of an investigation that was presented in an early volume of the Springer Series in Computational Physics (cf. 3]). It has been supported over a period of years by the U.S. Department of Energy under Contract DE-AC02-76ER03077 with New York University. We would like to express our gratitude to Dr. Franz Herrnegger for the assistance he has given us with the preparation of the manuscript. We are especially indebted to Connie Engle for the high quality of the final typescript. New York F. BAUER October 1983 O. BETANCOURT P. GARABEDIAN Contents 1. Introduction 1 2. Synopsis of the Method 3 1. Variational principle 3 2. Coordinate system 6 3. Finite Difference Scheme 8 1. Difference equations ....................... " 8 2. Island structure ............................. 10 3. Accelerated iteration procedure .............. . . .. 12 Nonlinear Stability 15 4. 1. Second minimization . . . . . . . . . . . . . . . . .. . . 15 . . . . . 2. Test functions and convergence studies . . . . . . . .. . . 17 . 3. Comparison with exact solutions ................. 19 5. The Mercier Criterion 22 1. Local mode analysis . . . . . . . . . . . . . . . . .. . . 22 . . . . . 2. Computational method . . . . . . . . . . . . . . . .. . . 23 . . . ."

Lawrence Gellert has long been a mysterious figure in American folk and blues studies, gaining prominence in the left-wing folk revival of the 1930s for his fieldwork in the U.S. South. A "lean, straggly-haired New Yorker," as Time magazine called him, Gellert was an independent music collector, without formal training, credentials, or affiliation. At a time of institutionalized suppression, he worked to introduce white audiences to a tradition of black musical protest that had been denied and overlooked by prior white collectors. By the folk and blues revival of the 1960s, however, when his work would again seem apt in the context of the civil rights movement, Gellert and his collection of Negro Songs of Protest were a conspicuous absence. A few leading figures in the revival defamed Gellert as a fraud, dismissing his archive of black vernacular protest as a fabrication -- an example of left-wing propaganda and white interference. A Sound History is the story of an individual life, an excavation of African American musical resistance and dominant white historiography, and a cultural history of radical possibility and reversal in the defining middle decades of the U.S. twentieth century.