On the two-dimensional extension of one-dimensional algebraically growing waves at neutral stability

TL;DR

This paper extends the study of algebraically growing waves at neutral stability from one to two spatial dimensions, revealing how dimensionality affects long-term behavior and decay rates of wave solutions.

Contribution

It introduces a two-dimensional extension of previously studied 1D wave operators and analyzes how solutions evolve over time in higher dimensions.

Findings

Long-time behavior shows a decay factor of t^{-1/2} in 2D.

Regions that grew in 1D now are neutral in 2D.

Solutions depend on a similarity variable contracting space and time.

Abstract

This work considers two linear operators which yield wave modes that are classified as neutrally stable, yet have responses that grow or decay in time. Previously, King et al. (Phys. Rev. Fluids, 1, 2016, 073604:1-19) and Huber et al. (IMA J. Appl. Math., 85, 2020, 309-340) examined the one-dimensional (1D) wave propagation governed by these operators. Here, we extend the linear operators to two spatial dimensions (2D) and examine the resulting solutions. We find that the increase of dimension leads to long-time behaviour where the magnitude is reduced by a factor of $t^{\frac{-1}{2}}$ from the 1D solutions. Thus, regions of the solution which grew algebraically as $t^{\frac{1}{2}}$ in 1D now are algebraically neutral in 2D, whereas regions which decay (algebraically or exponentially) in 1D now decay more quickly in 2D. Additionally, we find that these two linear operators admit long-time solutions that are functions of the same similarity variable that contracts space and time.