Theoretical Study of Inhomogeneity Effects on Three-Wave Parametric Instability: A WKBJ Approach

TL;DR

This paper uses a WKBJ-based approach to analyze how media inhomogeneity influences three-wave parametric instability, deriving a unified amplification formula and highlighting the significance of parameter gradients.

Contribution

It introduces a novel WKBJ-based method to model inhomogeneity effects on three-wave PI, including a unified amplification factor applicable to various PI types.

Findings

Wave number of quasi-mode is a complex root of the dispersion equation.

Derived a unified amplification factor formula for different PI types.

Parameter gradients significantly affect local growth rates when inhomogeneity exceeds 10^{-3}.

Abstract

The mechanisms by which media inhomogeneity affects the three wave parametric instability (PI), including the wave number mismatch and the parameter gradients, are investigated using an approach based on the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) approximation. This approach transforms the coupling wave equations into an amplitude equation and iteratively solves its characteristic polynomials. By analyzing the solutions, we proposed that the wave number of the quasi-mode, a key term in the wave number mismatch of non-resonant type PI, should be a complex root of the quasi-mode's linear dispersion equation. Based on this, we derive a unified amplification factor formula that covers the resonant and non-resonant, the forward-scattered and backward-scattered types of PI. The impact of parameter gradients on the local spatial growth rate becomes significant when the inhomogeneity exceeds 10^{-3}. Considering parameter gradients extends our approach's validity to an inhomogeneity of about 10^{-2}. This approach holds promise for more specific PI modeling in the future.