Asymptotic analysis of stability transition in MHD models

Abstract

Summary form only given. Mathematical models of plasma instabilities, such as dense Z-pinches, are often described by systems of nonlinear PDEs with fast varying coefficients, which involve numerous uncertainties in equation parameters, boundary and initial conditions. Instabilities and high dimension of these models may amplify uncertainties and result in unpredictability of simulations, which mirrors unpredictability of real systems. This has two important aspects. One is that underling dynamics may exhibit extreme sensitivity to variation of their parameters, initial and boundary conditions, which has been studied in the context of bifurcation phenomena and deterministic chaos. The second is combinatorial complexity of evaluating all model combinations that arise from possible variations in assumptions, parameters and initial data which prohibits direct evaluation of model uncertainties. Thus, it is important to understand and quantify the limits of predictability of full system simulation in terms of the uncertainties, inherent structure of the model and its components, and the length of the observation interval, and to develop computational approaches minimizing the effect of uncertainties and reducing simulation time while preserving and controlling the accuracy of obtained results. In this paper we outline an asymptotic approach leading to derivation of simplified models of initial complex systems with fast varying coefficients. Each of these simplified models intend to provide to a certain degree inner averaging of individual elaborated simulations of the initial system and present more robust and practically significant results then individual computation events. Stability transition describing by these simplified models could be interpreted as bifurcation phenomena developed due to variation of parameters in systems with constant or slowly varying coefficients which lead to deep classification of complex unstable behavior induced by fast varying parameters.