On General Transformations and Variational Principles for the Magnetohydrodynamics of Ideal Fluids. Part 1. Fundamental Principles

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Abstract

A new frozen-in field w (generalizing vorticity) is constructed for ideal magnetohydrodynamic flow. In conjunction with the frozen-in magnetic field h, this is used to obtain a generalized Weber transformation of the MHD equations, expressing the velocity as a bilinear form in generalized Weber variables. This expression is also obtained from Hamilton’s principle of least action, and the canonically conjugate Hamiltonian variables for MHD flow are identified. Two alternative energy-type variational principles for three-dimensional steady MHD flow are established. Both involve a functional R which is the sum of the total energy and another conserved functional, the volume integral of a function ϕ of Lagrangian coordinates. It is shown that the first variation δ1R vanishes if ϕ is suitably chosen (as minus a generalized Bernoulli integral). Expressions for the second variation δ2R are presented.

Original languageEnglish
Pages (from-to)125-139
Number of pages15
JournalJournal of Fluid Mechanics
Volume283
DOIs
Publication statusPublished - 1995

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ideal fluids
variational principles
Magnetohydrodynamics
magnetohydrodynamics
Fluids
steady flow
magnetohydrodynamic flow
vorticity
Hamiltonians
energy
Vorticity
magnetic fields
Magnetic fields

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering

Cite this

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abstract = "A new frozen-in field w (generalizing vorticity) is constructed for ideal magnetohydrodynamic flow. In conjunction with the frozen-in magnetic field h, this is used to obtain a generalized Weber transformation of the MHD equations, expressing the velocity as a bilinear form in generalized Weber variables. This expression is also obtained from Hamilton’s principle of least action, and the canonically conjugate Hamiltonian variables for MHD flow are identified. Two alternative energy-type variational principles for three-dimensional steady MHD flow are established. Both involve a functional R which is the sum of the total energy and another conserved functional, the volume integral of a function ϕ of Lagrangian coordinates. It is shown that the first variation δ1R vanishes if ϕ is suitably chosen (as minus a generalized Bernoulli integral). Expressions for the second variation δ2R are presented.",
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AB - A new frozen-in field w (generalizing vorticity) is constructed for ideal magnetohydrodynamic flow. In conjunction with the frozen-in magnetic field h, this is used to obtain a generalized Weber transformation of the MHD equations, expressing the velocity as a bilinear form in generalized Weber variables. This expression is also obtained from Hamilton’s principle of least action, and the canonically conjugate Hamiltonian variables for MHD flow are identified. Two alternative energy-type variational principles for three-dimensional steady MHD flow are established. Both involve a functional R which is the sum of the total energy and another conserved functional, the volume integral of a function ϕ of Lagrangian coordinates. It is shown that the first variation δ1R vanishes if ϕ is suitably chosen (as minus a generalized Bernoulli integral). Expressions for the second variation δ2R are presented.

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