Nonlinear saturation of the dissipative trapped-ion mode by mode coupling

Abstract

The nonlinear saturation of the dissipative trapped-ion mode is analyzed. . I The basic mechanism considered is the process whereby energy in long wavelength th unstable modes is nonlinearly coupled via E x B convection to short wavelength modes stabilized by Landau damping due to both circulating and trapped ions. In the usual limit of the mode frequency small relative to the effective electron collision frequency, a one-dimensional nonlinear partial differential equation for the potential can be derived, as first shown by LaQuey, Mahajan, Tang, and Rutherford. The stability and accessibility of the possible equilibria for this equation are examined in detail, both analytically and numerically. The equilibrium emphasized by LaQuey et al. is shown to be unstable. However, a class of nonlinear saturated states which are stable to linear perturbations is found. Included in the analysis are the effects of both ion collisions and dispersion due to finite ion bananawidth effects. Cross-field transport is estimated and the scaling of the results is considered for tokamak parameters (specifically those for the'• Princeton Large Torus). It is concluded that the anomalous• cross-field transport can be much lower than the estimate of Kadomtsev and Pogutse, for relevant parameters.

Periodic Stationary Patterns Governed by a Convective Cahn--Hilliard Equation

“…The infinite system of equations (3.25) was replaced by a finite system, [7,8,9,10,11,12]. The stability interval is 0.766 < K < 0.838.…”

“…This equation has been derived in a large number of physical contexts; e.g., it describes the instability of oscillations in reaction-diffusion systems [4], the instability of a flame front [5], film flow instability [6,7], and instability of solidification fronts [8,9,10]. Though for the KS equation there exists an interval of locally stable spatially periodic stationary solutions [7,11,12], the most remarkable type of dynamics is spatio-temporal chaos [13,14].…”

Solitary-Wave and Periodic Solutions of the Kuramoto-Velarde Dispersive Equation

“…3, α 0 ≥ 0, α 1 = 0, α 2 = 0, sign α 2 = sign α 3 , is a generalization of the Kuramoto-Velarde (KV) equation corresponding to the case α 1 = 0 and of the dispersive Kuramoto-Sivashinsky (KS) equation which corresponds to the case α 3 = 0. The dispersive (KS) equation is a model equation for long waves on a viscous fluid flowing down an inclined plane [1], as well as for drift waves in plasma [2]. The KV equation is an equation describing slow space-time variations of disturbances at interfaces, diffusion-reaction fronts and plasma instability fronts [3] and [4].…”

Two-dimensional wave dynamics in thin films. I. Stationary solitary pulses

“…With the addition of the dispersive term ‫ץ‬ 3 H / ‫ץ‬x 3 where ␦ is a positive parameter that characterizes the relative importance of dispersion and whose magnitude depends on the particular case considered. The 1D gKS equation has been reported for a wide variety of systems including a falling film with weak surface tension, 7 a film falling down a uniformly heated wall, 8 plasma waves with dispersion due to finite ion banana width, 9 and liquid films sheared by a turbulent gas. 10 Kawahara and Toh 11 constructed numerically stationary solitary pulse solutions of the two-dimensional ͑2D͒…”