Revisiting barotropic instability from the perspective of wave evolution theory

Abstract. The instability of Rossby waves has been a long-standing topic in dynamical meteorology. The classic theoretical analysis had provided in–depth physical understanding of the problem. However, developing a systematic and quantitative comprehension of wave energy and amplitude evolution remains challenging. With an eye to such issues, this investigation provides a novel and practicable algorithm to solve the wave action conservation equation. Theoretical analysis establishes that wave packet energy evolves through two competing factors: direct proportionality to intrinsic frequency and inverse proportionality to group velocity magnitude. Energy density attains extremal values at turning points where group velocity magnitudes become extremized. To ensure a ray can be reflected by a turning point, zonal phase speed must be smaller than an upper limit determined by dispersion relation at the turning point. Crucially, a specific zonal phase speed range emerges below this maximum threshold where concurrent transient growth of both wave energy and amplitude occurs when a ray is moving toward the turning point, with the upper limit corresponding to optimal wave development conditions. Numerical experiments on a prototype westerly jet reveal distinct instability mechanisms: substantial transient growth–capable of triggering nonmodal instability–arises when rays moves toward turning points, while exponential amplification characteristic of modal instability develops at inflection points with positive energy growth rate. The derived zonal phase speed thresholds and transient growth metrics form a diagnostic framework applicable to observed atmospheric flows, enabling quantitative evaluation of both modal and nonmodal instability potentials. By unifying wave evolution dynamics with classical instability criteria, this work provides an operational bridge between theoretical predictions and real–world flow diagnostics.