Beyond linear fields: the Lie–Taylor expansion

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 expansi...

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Bibliographic Details
Published inProceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Vol. 473; no. 2197; p. 20160525
Main Author Arter, Wayne
Format Journal Article
LanguageEnglish
Published England The Royal Society Publishing 01.01.2017
EditionRoyal Society (Great Britain)
Subjects
Analytic Solution
Catastrophe
Compressibility
Computational fluid dynamics
Fluid flow
Ideal Mhd
Lorenz equations
Magnetohydrodynamic turbulence
Magnetohydrodynamics
Mathematical models
Nonlinear
Resistivity
Series expansion
Taylor series
ideal MHD
nonlinear
catastrophe
analytic solution
resistivity
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Summary: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.
Bibliography:ObjectType-Article-1
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ISSN:1364-5021
1471-2946
DOI:10.1098/rspa.2016.0525