Magnetohydrodynamics And Fluid Dynamics Action Principles And Conservation Laws

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.

The book describes Fluid Dynamics, Magnetohydrodynamics, and Classical Thermodynamics as branches of Lagrange’s Analytical Mechanics. The approach presented is markedly different from the treatment given to them in traditional text books. A Hamilton-Type Variational Principle as the proper mathematical technique for the theoretical description of the dynamic state of any fluid is formulated. The scheme is completed proposing a new group of variations regarding the evolution parameter.

This book revises the evolution of ideas in various branches of magnetohydrodynamics (astrophysics, earth and solar dynamos, pinch, MHD turbulence and liquid metals) and reviews current trends and challenges. Uniquely, it contains the review articles on the development of the subject by pioneers in the field as well as leading experts, not just in one, but in various branches of magnetohydrodynamics, such as liquid metals, astrophysics, dynamo and pinch.

This book is an introductory text on magnetohydrodynamics (MHD) - the study of the interaction of magnetic fields and conducting fluids.

This book is about Lie group analysis of differential equations for physical and engineering problems. The topics include: -- Approximate symmetry in nonlinear physical problems -- Complex methods for Lie symmetry analysis -- Lie group classification, Symmetry analysis, and conservation laws -- Conservative difference schemes -- Hamiltonian structure and conservation laws of three-dimensional linear elasticity -- Involutive systems of partial differential equations This collection of works is written in memory of Professor Nail H. Ibragimov (1939–2018). It could be used as a reference book in differential equations in mathematics, mechanical, and electrical engineering.

This book is primarily intended to enable postgraduate research students to enhance their understanding and expertise in Fluid Mechanics and Magnetohydrodynamics (MHD), subjects no longer treated in isolation. The exercises throughout the book often serve to provide additional and quite significant knowledge or to develop selected mathematical skills, and may also fill in certain details or enhance readers’ understanding of essential concepts. A previous background or some preliminary reading in either of the two core subjects would be advantageous, and prior knowledge of multivariate calculus and differential equations is expected.

This book is an introduction to terrestial magnetohydrodynamics. It is a compendium of introductory lectures by experts in the field, focussing on applications in industry and the laboratory. A concise overview of the subject with references to further study.