Growth of small localised perturbations in surface quasi-geostrophic turbulence

Abstract

The butterfly effect, namely the amplification of an initially localised infinitesimal perturbation, is a fundamental hallmark of chaotic systems. While this phenomenon is well understood in low-dimensional chaotic systems, its implications become more complex in spatially extended systems characterised by multiple temporal and spatial scales, such as geophysical flows. In this letter, we investigate the butterfly effect by examining the evolution of infinitesimal localised perturbations in a minimal model of mesoscale geophysical turbulence. We find that localised perturbations exhibit a highly variable dynamics, including an initial transient regime during which perturbation energy decreases while the error spectrum shifts from the injection scale toward the system’s most unstable configuration. The duration of this transient regime spans a broad range and may persist for several characteristic small-scale times, depending on the perturbation’s size, its initial location and the Reynolds number.