"On relationships between the fundamental invariants in the hydrodynamics and in the solid state physics ones"
F.V. Dolzhansky, V.M. Ponomarev (Obukhov Institute of Atmospheric Physics, RAS)
Abstract:
V. Arnold revealed that the Kelvin circulation theorem in fluid mechanics is, in a certain sense, analogous to the law of conservation of top angular momentum in mechanics. On the basis of generalized rigid body construction, i.e. the Euler equations of motion of a rigid body with a fixed point in an arbitrary Lie group, and its extensions to the case of body motion in external force fields characterized by scalar or vector potential it is shown that the respective mechanical prototypes of Moffat’s invariant in fluid mechanics (helicity of the velocity field), potential vorticity, and Moffat’s invariant in magnetohydrodynamics (the cross helicity of velocity and magnetic fields) are respectively the kinetic momentum of the rigid body squared, the kinetic momentum projection on the direction of gravity field, and the kinetic momentum projection on the direction of the electric current passing through the electrical neutral medium of low density where an ideal rigid conductor moves. The validity of the Kelvin theorem for the magnetic vector potential and the conservation of the magnetic field helicity (Woltjer invariant) follows from the fact that this field immovable with respect to the fluid. Some examples of fluid motion, equations for which exactly correspond to the equations of rigid body motion are presented.The motion equations of the heavy top in the Coriolis force field could be regarded as a simplest model of the ideal stratified fluid motion. The quasigeostrophic approximation of baroclinic model made according to the procedure that used in geophysical hydrodynamics corresponds to the Euler equations for free gyroscope with dependent variables, which are vertical vorticity and thermal wind components. The vertical stratification may be considered as free parameter. The dissipation and heat conduction, and beta effect, and external driving are taken into account as well.