An analytical connection between temporal and spatio-temporal growth rates in linear stability analysis

TL;DR

This paper derives an exact, easily computable formula linking temporal and spatio-temporal growth rates in linear stability analysis, applicable to various models and useful for stability characterization.

Contribution

It introduces a novel exact formula for complex frequency in spatio-temporal stability, depending only on temporal quantities, with practical applications demonstrated on model equations.

Findings

Quadratic truncation provides accurate estimates of absolute instability transition.

The formula simplifies stability analysis by relying on temporal data.

Error decreases with higher-order truncations.

Abstract

We derive an exact formula for the complex frequency in spatio-temporal stability analysis that is valid for arbitrary complex wave numbers. The usefulness of the formula lies in the fact that it depends only on purely temporal quantities, which are easily calculated. We apply the formula to two model dispersion relations: the linearized complex Ginzburg--Landau equation, and a model of wake instability. In the first case, a quadratic truncation of the exact formula applies; in the second, the same quadratic truncation yields an estimate of the parameter values at which the transition to absolute instability occurs; the error in the estimate decreases upon increasing the order of the truncation. We outline ways in which the formula can be used to characterize stability results obtained from purely numerical calculations, and point to a further application in global stability analyses.