Construction of a multi-dimensional Lagrangian hydrodynamics model of a plasma using the Lie group theory of point transformations

摘要

Conservative macroscopic governing equations for a plasma in the magnetohydrodynamic (MHD) approximation were derived from the symmetry properties of the microscopic action integral. Invariance of the action integral under translation and rotation symmetries was verified. Noether's Theorem was used to express the invariance equations as two sets of microscopic equations that represent the Euler-Lagrange (E-L) equations for the plasma and an equation in divergence form, called a conservation law, that is the first integral of the Euler-Lagrange equations. The specific forms of the conservation laws were determined from the invariance properties admitted by the action integral. Invariance under translations gave the conservation law for the translational momentum balance while invariance under rotations gave the conservation law for the angular momentum balance. The ensemble average of the microscopic equations was taken to give the kinetic representation of the equations. The momentum integrals in the kinetic equation were evaluated to give the fluid representation of the system. The fluid representation was then expressed in the MHD limit to give the one-fluid representation for the plasma. The total derivatives in the conservation laws were evaluated for the kinetic and fluid representations to verify that the expressions are first integrals of the respective E-L equations. The symmetry properties of the conservation laws in auxiliary form were determined to test the system, of equations for mapping properties that may allow the nonlinear conservation laws to be expressed as nonlinear or linear expressions. The results showed that no nonlinear to linear mapping was possible for the governing equations with charge distributions. The quasi-neutral governing equations admitted a scaling group that allows mapping from the source nonlinear equations to nonlinear target equations that contain one less independent variable.;The translation conservation laws were used in a two-dimensional computer simulation of the confined eddy problem to demonstrate an application of the equations. Comparison of the results produced by the code using the conservation law governing equations with previous work is limited to qualitative comparison since differences in the numerical input and graphics software make direct quantitative comparison impossible. Qualitative comparison with previous work show the results to be consistent.