A Macroscopic Plasma Lagrangian and its Application to Wave Interactions and Resonances by Yueng-Kay Martin Peng June 1974 SUIPR Report No. 575 National Aeronautics and Space Administration Grant NGL 05-020-176 National Science Foundation Grant GK-32788X (NASA-CR-138649) A MACROSCOPIC PLASMA N74-27231 LAGRANGIAN AND ITS APPLICATION TO WAVE INTERACTIOUS AND RESONANCES (Stanford Univ.) 262 p HC $16.25 CSCL 201 Unclas G3/25 41289 INSTITUTE FOR PLASMA RESEARCH STANFORD UNIVERSITY, STANFORD, CALIFORNIA NIZ~a A MACROSCOPIC PLASMA LAGRANGIAN AND ITS APPLICATION TO WAVE INTERACTIONS AND RESONANCES by Yueng-Kay Martin Peng National Aeronautics and Space Administration Grant NGL 05-020-176 National Science Foundation Grant GK-32788X SUIPR Report No. 575 June 1974 Institute for Plasma Research Stanford University Stanford, California I A MACROSCOPIC PLASMA LAGRANGIAN AND ITS APPLICATION TO WAVE INTERACTIONS AND RESONANCES by Yueng-Kay Martin Peng Institute for Plasma Research Stanford University Stanford, California 94305 ABSTRACT This thesis is concerned with derivation of a macroscopic plasma Lagrangian, and its application to the description of nonlinear three- wave interaction in a homogeneous plasma and linear resonance oscilla- tions in a inhomogeneous plasma. One approach to obtain -the--Lagrangian is via the inverse problem of the calculus of variations for arbitrary first and second order quasilinear partial differential systems. Necessary and sufficient conditions for the given equations to be Euler-Lagrange equations of a Lagrangian are obtained. These conditions are then used to determine the transformations that convert some classes of non-Euler-Lagrange equations to Euler-Lagrange equation form. The Lagrangians for a linear resistive transmission line and a linear warm collisional plasma are derived as examples. Using energy considerations, the correct macroscopic plasma Lagrangian is shown to differ from the velocity-integrated Low Lagrangian, by a macroscopic potential energy that equals twice the particle thermal kinetic energy plus the energy lost by heat conduction. The generalized variables are the macroscopic plasma cell position (Eulerian coordinates) defined in Lagrangian coordinates, and the vector and scalar potentials APReCPr.A. AOp iii" ' TAhi NO)T FED defined in Eulerian coordinates. With the continuity and heat flow equations treated as constraints, the Euler-Lagrange equations are shown to be the force law and Maxwell's equations. The effects of viscosity, heat conduction, and elastic collisions are included in this variational principle. The corresponding macroscopic Hamiltonian, and the micro- scopic Hamiltonian corresponding to the Low Lagrangian, are also derived. Under the assumptions of scalar pressure and adiabatic processes, the macroscopic Lagrangian is approximated by expansions in weak pertur- bations of the generalized variables. The averaged Lagrangian method is then used to derive nonlinear three-wave coupling coefficients in a warm homogeneous two-component plasma. The effects of wave damping are included phenomenologically in the coupled mode equations. The general results are then specialized to make detailed quantitative comparisons between theory and available experimental results on parame- trically excited ion-acoustic waves. The approximate quadratic Lagrangian is also used to estimate the electrostatic (Tonks-Dattner) resonance properties of an inhomogeneous plasma. The Rayleigh-Ritz procedure is applied directly to the Lagrangian corresponding to a system of Euler-Lagrange equations. Use of an appro- priate set of trial functions then leads to frequency and eigenfunction estimates in excellent agreement with the existing theoretical and experi- mental results for a low pressure positive column. Since this method mainly involves evaluating finite integrals, and solving algebraic eigen- value equations, it is found to be more efficient than numerically solving differential equations, and more accurate than inner-outer expansions. iv TABLE OF CONTENTS Page ABSTRACT. .......................... iii LIST OF TABLES. ...... ...... viii LIST OF ILLUSTRATIONS . ... ... ............ ix LIST OF SYMBOLS . ....... .* xvi ACKNOWLEDGMENTS . * * * * xl 1. INTRODUCTION . ........................ 1 2. INVERSE PROBLEM OF THE CALCULUS OF VARIATIONS . ........ 2.1 Introduction . .............. .. 5 2.2 First Order Differential Equations .... ....... 7 2.2.1 Necessary Conditions . ... ...... .. 8 2.2.2 Sufficient Conditions ...... 9 2.2.3 Linear Differential Equations . .. ...... 12 2.3 Second Order Differential Equations. .......... 13 2.3.1 Necessary Conditions ... ..... ..... 14 2.3.2 Sufficient Conditions .............. 15 2.3.3 Nonlinear Differential Equations of the Second Rank .. ..... o o...... ....... 18 2.4 Differential Transformations . o .......... 21 2.4.1 Differential Transformation of Linear Differential Equations. ........... 22 2.4.2 Resistive Transmission Line . ........... o 23 2.4.3 Warm Collisional Plasma .............. 25 2.5 Discussion . .......................28 3. LAGRANGIAN AND HAMILTONIAN DENSITIES FOR PLASMAS. ... ...... 31 3.1 Introduction . ..................... 31 3.2 Lagrangian Density . ...... .. ............ .. 36 3.2.1 Macroscopic Approximation to the Low Lagrangian. .37 3.2.2 Macroscopic Potential, U . ............ 42 3.2.3 Application of Hamilton's Principle ..... 46 v Page 3.3 Hamiltonian Density. 56 3.3.1 Lagrangian Coordinates, , . 57 3.3.2 Modified Legendre Transformation . 61 3.3.3 Hamilton Equations. ..... 63 3.3.4 Poisson Brackets. .. 65 3.3.5 Energy Conservation Equation. 67 3.3.6 Microscopic Hamiltonian Density . 69 3.3.7 Entropy for Plasmas . 73 3.4 Discussion of the Macroscopic Potential, . 74 3,4.1 Relation of U to Viscosity and Heat Conduction . 75 3.4.2 Loss of Particle Discreteness in Applying Macroscopic Approximation . ..... 76 3.4.3 Definition of Plasma Cell Boundary. .. 78 3.4.4 Relations Among the Variational Principles of Various Models. ................ 79 3.5 Discussion .................... 81 4. THREE-WAVE INTERACTIONS: THEORY . .............. 84 4.1 Introduction ................... 84 4.2 Perturbation Approximation to L . ........... 86 4.2.1 Definition of Perturbations . 87 4.2.2 Expansions of , , n ',P . ............... 89 4.2.3 Expansion of the Lagrangian . 93 4.3 Nonlinear Wave Coupling Coefficients . 96 4.3.1 Averaged Lagrangian . .............. 96 4.3.2 Linear Waves . ................. 99 4.3.3 Wave Coupling Coefficients . 100 4.4 Parametric Wave Amplification . ............ 104 4.4.1 Parametric Amplification Without Wave Damping .. ................ .. 105 4.4.2 Wave Damping. .j..... ....... .. 112 4.4.3 Effects of Other Nonlinear Processes and Plasma Inhomogeneity ............... 120 4./ Discussion . 125 vi Page THREE-WAVE INTERACTION: APPLICATIONS..... ..... ...126 5.1 Introduction• ............ .... .126 5.2 Nonlinear Interaction of Ion-Acoustic Waves . ... .133 5.2.1 The Experiment and Interpretation . ..... .. 133 5.2.2 Ion-Acoustic Waves (I) and Coupling Coefficient: 0 = 0 *....... ... .. 139 5.2.3 Dependence of ZO on e (0) .............141 5.3 Excitation of an Ion-Acoustic Wave by Two Electron Plasma Waves ... ....... ..... .144 5.3.1 Electron Plasma Waves (P) and Coupling Coefficients: 8 = 0 ....... ....144 5.3.2 Evolution of Fi and es ....... .146 5.4 Excitation of an Ion-Acoustic Wave by Two Whistlers: Collinear Propagation . ..... .. ........... 148 5.4.1 Experimental Observations. ..... ..... 149 5.4.2 Whistler Waves (R) and the Coupling Coefficient: 0 =,0 . ...... ...... 192 5.4.3 Evolution of i and es . ........ .. 154 Excitation of an Ion-Acoustic Wave by TwoWhistlers: Oblique Propagation .. ................. 197 5.5.1 Obliquely Propagating Ion-Acoustic Waves (I"). K" = [K" 0, K" . 158 5.5.2 Obliquely Propagating Whistlers (R ), K = Kx O, Kj . .160 5..3 Synchronism Conditions and Coupling Coefficient ..... .............. 161 ,.,.4 Evolution of 8. and s . .164 5.6 Discussion........ .167 6. CONCLUSIONS. .. .............. 169 REFERENCES ........ ................ .174 APPENDIX A SOLUTION OF ULTRAHYPERBOLIC EQUATIONS . ... .. .181 APPENDIX B VARIATIONAL CALCULATIONS FOR RESONANCE OSCILLATIONS OF INHOMOGENEOUS PLASMAS by Y-K. M. Peng and F.W. Crawford vii LIST OF TABLES Number Page 5.1 Some derivations of wave coupling coefficients using (a) the averaged Lagrangian method and (b) the itera- tive method, with (c) related experimental work. The abbreviations used below represent, C: cold plasma, M: macroscopic warm plasma, and m: microscopic warm plasma. With respect to the static magnetic field, we have, II: parallel propagation, -1-: quasiparallel propagation, 1: perpendicular propagation, and X: oblique propagation. With respect to the directions of the phase velocities, we have, = 0: collinear waves, and ® 0: noncollinear waves . ... 127 viii LIST OF ILLUSTRATIONS Figure Page 2.1 Resistive transmission line. .............. 24 3.1 In phase space, a particle trajectory (-) can be specified by its Lagrangian (initial) coordinates, ( O0,0,0), or its Eulerian (present) coordinates, (x,v,t). The polarization vector, , used by Sturrock (1958), connects the real particle trajec- tory (---) to a specified trajectory (--) ........ 32 3.2 Particles of a species that occupy a spatial volume V at time t are assumed to occupy a volume V0 at t = 0. This approximation is acceptable when t is sufficiently close to t = 0 ... 39 3.3 The size of AV is assumed to be much larger than the particle mean free path so that the same particles of