A variational approach to parametric instabilities in inhomogeneous plasmas I: Two model problems

A variational formalism is introduced in the theory of three-wave parametric instabilities in inhomogeneous plasmas. This minimum pump strength principle (MPSP) is then applied to two model problems, the first being the Rosenbluth model equations [Phys. Rev. Lett. 29, 565 (1972)]. By choosing appropriate trial functions, the MPSP is used to solve for the complex eigenfrequency of the most unstable mode. The wave vector mismatch is assumed to be of the form κ(x)=κ(n)(0)xn/n exclamation point, where n is any positive integer. The results are compared to numerical solutions of the same eigenvalue problem. The second problem is the Liu, Rosenbluth, and White Raman sidescattering model [Phys. Fluids 17, 1211 (1974)], which is treated for any positive-integer power law density profile. The choice of trial functions, the role of symmetry, and various useful approximations are discussed