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Alfven discontinuity Alfven wave amplitude angles assume axis boundary conditions called characteristic collisions components conservation law consider const coordinate corresponding curl deformation velocities tensor depend determined differential direction discontinuity surface dissipation electric conductivity electric field electrohydrodynamical electron and ion energy entropy equal equation of motion equilibrium evolutionarity conditions ferrofluid flow fluid flux density force frequency front function gradient group polar hydrodynamics ideal magnetohydrodynamics increases inequality instability integral integral curve ionizing shock wave kinetic linear magnetic field lines magnetic surface magnetohydrodynamics Maxwell equation medium MHD quantities MHD waves momentum netic nonvanishing Ohm's law particles perturbations phase velocity pinch piston plane plasma pressure region Reynolds number right-hand side rotation self-similar waves shear shock wave slow magnetosound wave solution stability stationary switch-on temperature thermal conductivity toroidal transverse turbulence unperturbed vanishes vector viscosity viscosity coefficients wave moves zero
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