Magnetohydrodynamics and Fluid Dynamics: Action Principles and Conservation Laws

  • Gary Webb

Part of the Lecture Notes in Physics book series (LNP, volume 946)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Gary Webb
    Pages 1-7
  3. Gary Webb
    Pages 9-20
  4. Gary Webb
    Pages 21-36
  5. Gary Webb
    Pages 53-67
  6. Gary Webb
    Pages 69-113
  7. Gary Webb
    Pages 115-136
  8. Gary Webb
    Pages 137-166
  9. Gary Webb
    Pages 167-189
  10. Gary Webb
    Pages 191-208
  11. Gary Webb
    Pages 229-253
  12. Gary Webb
    Pages 255-258
  13. Back Matter
    Pages 259-301

About this book


This text focuses on conservation laws in magnetohydrodynamics, gasdynamics and hydrodynamics. A grasp of new conservation laws is essential in fusion and space plasmas, as well as in geophysical fluid dynamics; they can be used to test numerical codes, or to reveal new aspects of the underlying physics, e.g., by identifying the time history of the fluid elements as an important key to understanding fluid vorticity or in investigating the stability of steady flows.

The ten Galilean Lie point symmetries of the fundamental action discussed in this book give rise to the conservation of energy, momentum, angular momentum and center of mass conservation laws via Noether’s first theorem. The advected invariants are related to fluid relabeling symmetries – so-called diffeomorphisms associated with the Lagrangian map – and are obtained by applying the Euler-Poincare approach to Noether’s second theorem.

The book discusses several variants of helicity including kinetic helicity, cross helicity, magnetic helicity, Ertels’ theorem and potential vorticity, the Hollman invariant, and the Godbillon Vey invariant. The book develops the non-canonical Hamiltonian approach to MHD using the non-canonical Poisson bracket, while also refining the multisymplectic approach to ideal MHD and obtaining novel nonlocal conservation laws. It also briefly discusses Anco and Bluman’s direct method for deriving conservation laws. 

A range of examples is used to illustrate topological invariants in MHD and fluid dynamics, including the Hopf invariant, the Calugareanu invariant, the Taylor magnetic helicity reconnection hypothesis for magnetic fields in highly conducting plasmas, and the magnetic helicity of Alfvén simple waves, MHD topological solitons, and the Parker Archimedean spiral magnetic field. The Lagrangian map is used to obtain a class of solutions for incompressible MHD. The Aharonov-Bohm interpretation of magnetic helicity and cross helicity is discussed. In closing, examples of magnetosonic N-waves are used to illustrate the role of the wave number and group velocity concepts for MHD waves. 

This self-contained and pedagogical guide to the fundamentals will benefit postgraduate-level newcomers and seasoned researchers alike.


Advected Invariants in MHD Fluid Relabelling Symmetries Lagrangian maps Lie dragging Fluid helicity and vorticity Topological fluid dynamics

Authors and affiliations

  • Gary Webb
    • 1
  1. 1.CSPARThe University of Alabama in HuntsvilleHuntsvilleUSA

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