James F. McKenzie

Local and nonlocal advected invariants and helicities in magnetohydrodynamics and gas dynamics I: Lie dragging approach

In this paper advected invariants and conservation laws in ideal magnetohydrodynamics (MHD) and gas dynamics are obtained using Lie dragging techniques. There are different classes of invariants that are advected or Lie dragged with the flow. Simple examples are the advection of the entropy S (a 0-form), and the conservation of magnetic flux (an invariant 2-form advected with the flow). The magnetic flux conservation law is equivalent to Faradays equation. The gauge condition for the magnetic helicity to be advected with the flow is determined. Different variants of the helicity in ideal fluid dynamics and MHD including: fluid helicity, cross helicity and magnetic helicity are investigated. The fluid helicity conservation law and the cross-helicity conservation law in MHD are derived for the case of a barotropic gas. If the magnetic field lies in the constant entropy surface, then the gas pressure can depend on both the entropy and the density. In these cases the conservation laws are local conservation laws. For non-barotropic gases, we obtain nonlocal conservation laws for fluid helicity and cross helicity by using Clebsch variables. These nonlocal conservation laws are the main new results of the paper. Ertels theorem and potential vorticity, the Hollman invariant, and the Godbillon?Vey invariant for special flows for which the magnetic helicity is zero are also discussed.

Local and nonlocal advected invariants and helicities in magnetohydrodynamics and gas dynamics: II. Noether's theorems and Casimirs

Conservation laws in ideal gas dynamics and magnetohydrodynamics (MHD) associated with fluid relabelling symmetries are derived using Noethers first and second theorems. Lie dragged invariants are discussed in terms of the MHD Casimirs. A nonlocal conservation law for fluid helicity applicable for a non-barotropic fluid involving Clebsch variables is derived using Noethers theorem, in conjunction with a fluid relabelling symmetry and a gauge transformation. A nonlocal cross helicity conservation law involving Clebsch potentials, and the MHD energy conservation law are derived by the same method. An Euler Poincare variational approach is also used to derive conservation laws associated with fluid relabelling symmetries using Noethers second theorem.

Multi-symplectic magnetohydrodynamics

A multi-symplectic formulation of ideal magnetohydrodynamics (MHD) is developed based on a Clebsch variable variational principle in which the Lagrangian consists of the kinetic minus the potential energy of the MHD fluid modified by constraints using Lagrange multipliers, that ensure mass conservation, entropy advection with the flow, the Lin constraint and Faradays equation (i.e the magnetic flux is Lie dragged with the flow). The analysis is also carried out using the magnetic vector potential

Alfvén wave mixing and non-JWKB waves in stellar winds

tilde{bf A}

Klein-Gordon equations for toroidal hydromagnetic waves in an axi-symmetric field

where

Multi-symplectic magnetohydrodynamics: II, addendum and erratum

alpha=tilde{bf A}{bfcdot}d{bf x}

Hydromagnetic waves in a compressed-dipole field via field-aligned Klein-Gordon equations

is Lie dragged with the flow, where

On the unstable mode merging of gravity-inertial waves with Rossby waves

{bf B}=nablatimestilde{bf A}

Hall Magnetohydrodynamics with Electron inertia

. The symplecticity conservation laws are shown to give rise to the Eulerian momentum and energy conservation laws in MHD. Noethers theorem for the multi-symplectic MHD system is derived, including the case of non-Cartesian space coordinates, where the metric plays a role in the equations.

Integrable, Nonlinear Travelling Waves and Whistler Oscillitons in Two-Fluid, Electron-Proton Plasmas

Alfv?n wave mixing equations used in locally incompressible turbulence transport equations in the solar wind contain as a special case, non-Jeffreys?Wentzel?Kramers?Brouillon (non-JWKB) wave equations used in models of Alfv?n wave driven winds. We discuss the canonical wave energy equation; the physical wave energy equation, and the JWKB limit of the wave interaction equations. Lagrangian and Hamiltonian variational principles for the waves are developed. Noether?s theorem is used to derive the canonical wave energy equation which is associated with the linearity symmetry of the equations. A further conservation law associated with time translation invariance of the action, applicable for steady background wind flows is also derived. In the latter case, the conserved density is the Hamiltonian density for the waves, which is distinct from the canonical wave energy density. The canonical wave energy conservation law is a special case of a wider class of conservation laws associated with Green?s theorem for the wave mixing system and the adjoint wave mixing system, which are related to Noether?s second theorem. In the sub-Alfv?nic flow, inside the Alfv?n point of the wind, the backward and forward waves have positive canonical energy densities, but in the super-Alfv?nic flow outside the Alfv?n critical point, the backward Alfv?n waves are negative canonical energy waves, and the forward Alfv?n waves are positive canonical energy waves. Reflection and transmission coefficients for the backward and forward waves in both the sub-Alfv?nic and super-Alfv?nic regions of the flow are discussed.