The development of ionizational (electrothermal) instability under moving two-temperature inert gas plasma in across magnetic field is investigated. The gasdynamic flow
parameters, the local and average values of temperature and concentration, local conductivity for electron plasma component, basic integral characteristics of magnetohydrodynamic (MHD) channels were defined. The electron energy balance is brought such the Joule heating
power is equal the rate of energy transfer to the heavy plasma component at the elastic collisions and the rate of energy transfer for ionization. This energy balance is conserved in a stable plasma and it is not conserved in unstable plasma. The defect of electron energy is
about 1/3 or 1/2 from Joule heating energy, meaning large part of energy is lost in fluctuations.