Beyond Linear Fields: the Lie-Taylor Expansion

TL;DR

This paper extends solutions of compressible nonlinear MHD using Lie-Taylor series, demonstrating their physical relevance, including resistivity effects, and proposing broader applications to fluid models.

Contribution

It introduces a Lie-Taylor series expansion for nonlinear MHD fields, enhancing the understanding of the Dolzhansky-Kirchhoff equations and their physical applicability.

Findings

Positive implications for the Dolzhansky-Kirchhoff equations

Resistivity can be incorporated into the D-K model

Lie-Taylor series may provide insights into other fluid models

Abstract

The work extends the linear fields' solution of compressible nonlinear magnetohydrodynamics~(MHD) to the case where the magnetic field depends on superlinear powers of position vector, usually but not always, expressed in Cartesian components. Implications of the resulting Lie-Taylor series expansion for physical applicability of the Dolzhansky-Kirchhoff~(D-K) equations are found to be positive. It is demonstrated how resistivity may be included in the D-K model. Arguments are put forward that the D-K equations may be regarded as illustrating properties of nonlinear MHD in the same sense that the Lorenz equations inform about the onset of convective turbulence. It is suggested that the Lie-Taylor series approach may lead to valuable insights into other fluid models.