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The ideal Chew–Goldberger–Low (CGL) plasma equations, including the double adiabatic conservation laws for the parallel ( $$p_\parallel$$ ) and perpendicular pressure ( $$p_\perp$$ ), are investigated using a Lagrangian variational principle. An Euler–Poincaré variational principle is developed and the non-canonical Poisson bracket is obtained, in which the non-canonical variables consist of the mass flux $${\boldsymbol {M}}$$ , the density $$\rho$$ , the entropy variable $$\sigma =\rho S$$ and the magnetic induction $${\boldsymbol {B}}$$ . Conservation laws of the CGL plasma equations are derived via Noether's theorem. The Galilean group leads to conservation of energy, momentum, centre of mass and angular momentum. Cross-helicity conservation arises from a fluid relabelling symmetry, and is local or non-local depending on whether the gradient of $$S$$ is perpendicular to $${\boldsymbol {B}}$$ or otherwise. The point Lie symmetries of the CGL system are shown to comprise the Galilean transformations and scalings.
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Going Beyond the MHD Approximation: Physics-based Numerical Solution of the CGL Equations
Abstract We present a new numerical model for solving the Chew–Goldberger–Low system of equations describing a bi-Maxwellian plasma in a magnetic field. Heliospheric and geospace environments are often observed to be in an anisotropic state with distinctly different parallel and perpendicular pressure components. The Chew–Goldberger–Low (CGL) system represents the simplest leading order correction to the common isotropic MHD model that still allows the incorporation of the latter’s most desirable features. However, the CGL system presents several numerical challenges: the system is not in conservation form, the source terms are stiff, and unlike MHD, it is prone to a loss of hyperbolicity if the parallel and perpendicular pressures become too different. The usual cure is to bring the parallel and perpendicular pressures closer to one another, but that has usually been done in an ad hoc manner. We present a physics-informed method of pressure relaxation based on the idea of pitch-angle scattering that keeps the numerical system hyperbolic and naturally leads to zero anisotropy in the limit of very large plasma beta. Numerical codes based on the CGL equations can, therefore, be made to function robustly for any magnetic field strength, including the limit where the magnetic field approaches zero. The capabilities of our new algorithm are demonstrated using several stringent test problems that provide a comparison of the CGL equations in the weakly and strongly collisional limits. This includes a test problem that mimics the interaction of a shock with a magnetospheric environment in 2D.
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- Award ID(s):
- 2009871
- PAR ID:
- 10526985
- Publisher / Repository:
- DOI PREFIX: 10.3847
- Date Published:
- Journal Name:
- The Astrophysical Journal
- Volume:
- 970
- Issue:
- 2
- ISSN:
- 0004-637X
- Format(s):
- Medium: X Size: Article No. 154
- Size(s):
- Article No. 154
- Sponsoring Org:
- National Science Foundation
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