Three‐Wave Interaction in Cold Semi‐Infinite Plasmas

Abstract

AhstreetWe have considered the theory for the resonant nonlinear interaction between three waves in B cold semi-infinite magnetized plasma end ciilculnted the coupling coefficients.The linear theory of surface waves has been developed in detail (see e.g. [ 1-31) and the nonlinear theory of such modes has recently received increased attention [e.g, 4-61. I n the present paper we introduce further ideas in this field.W e shall consider a semi-infinite electron-ion plasma with essentially constant density iV, (= N , = N i ) for x > 0. The region .z < 0 is assumed to be filled with a material with frequency-independent dielectric constant E,, which for siniplicity we decide to be the sattie as for vacuum. We describe our plasma by means of the cold fluid equations (except for our extension in (14)), and assume that there are no particle dyifts in the rinperturbed state. A constant magnetic field Bo, which is supposed to be present, is by definition in the xy-plane, where 0 is the angle between B, and the positive x-axis.The linear theory for such a semi-infinite plasma is well known [e.g. 2,3]. The nonlinear theory, for the particular direction Bo = BOO, has been discussed in [5]. We shall here extend the non-linear theory by considering the oblique case [2], where B, is n o t normal to the plasma boundary.We denote the current density by J and start with the Maxwell equationRestricting ourselves to longitudinal waves, we introduce coniplex potentials @, such that the electric field can be written as E = -V@ + C.C. Froin (1) we then obtain ~vhere we have integrated over the region --oo < x < m, 0 < y < I , 0 < z < 1.Separating the current density J = J I + JnI into a linear, J,, and a non-linear, J n l , part, we obtain from (2) after partial integration drC'@* -(-iiwe(~>)) * I @ + C.C. = -dl.(F@*) -J,l $-C.C.