FOUR-DIMENSIONAL INTEGRAL EQUATIONS FOR THE MHD DIFFRACTION WAVES IN PLASMA

Introduction Let consider low frequency oscillations those can be excited and propagate in plasma. Therefore we shall limit ourselves sufficiently slow development of macroscopic processes. Such a supposition is necessary for the possible application in hydrodynamic description that together with electromagnetic one for the field is expressed by equations of magnetic hydrodynamics [1] relatively to medium velocity u r ( , ) t , intensity of a magnetic field b r ( , ) t and a density of media ρ( , ) r t . In linear magnetic hydrodynamics a wave packet contains seven types of characteristics: two rapid magnetoacoustic waves, two slow magnetoacoustic waves, two Alfven waves and entropy wave. The state vector Ψ describing the considered wave packet at each spatial point M x y z ( , , ) and at any time moment t is determined for initial and boundary conditions of the above waves. Under definite physical conditions, for example, at the meeting of the solar wind with the Earth magnetic field, or at sudden inclusion of an electromagnetic field, or during collisions of two gas masses and so on arise strong discontinuities under which not only derivatives of the MHD-values are discontinuous along spatialtemporary coordinates, but these values themselves are also discontinuous ones. Jumps of MHD-values on a surface of discontinuities are determined according to integral laws of conservation or integral balance equations. As to differential equations of magnetohydrodynamics the solutions of those are inaccessible at differential of discontinuous values on the surface of discontinuity they can be represented in the integral form [2] completely equivalent to differential equations (induction and Navier-Stores equations) and also initial and boundary conditions above mentioned. The questions of evolution in magnetohydrodynamics Here it should be noted two factors connecting to the problem of evolution in magnetic hydrodynamics occur. The first factor may already refereed to the classical one and it has been considered sufficiently well in literature [3] and we named it conditionally evolution of discontinuities in space. As it seemed setting of boundary conditions for discontinuities is not sufficient to determine discontinuity moving of the MHD-medium by the only one method. One needs to take into account an increase of entropy and also wave stability in reference to splitting it into several discontinuous or automodel waves. Such waves in magnetic hydrodynamics are called evolutionary ones. For them infinitely small disturbances of MHD values evolves with time remaining small. The nonevolutionary wave is instantly splitted (in case of an ideal medium). The problem of evolution of initial disturbance has here a unique solution if a number of expanding waves (a number of unknown disturbances) is equal to a number of independent boundary conditions. In this case the initial discontinuity is evolutionary one. Otherwise the problem has either innumerable quantity of solutions or the solution of this problem is inaccessible generally, i.e. discontinuity is nonevolutionary and splitted. The evolution conditions of shock wave easily to find, analyzing the linear boundary conditions, written down in laboratory system of coordinates. In brief these evolutionary conditions can be formulated as follows. Relatively the Alfven disturbances exist two domains of