Zonal flows driven by libration in rotating spherical shells: the case of periodic characteristic paths

TL;DR

This paper explores the nonlinear dynamics of zonal flows driven by libration in spherical shells, revealing scaling laws and boundary effects through asymptotic theory and numerical simulations at very low Ekman numbers.

Contribution

It extends previous linear studies by developing a nonlinear framework with asymptotic analysis and numerical validation for libration-driven zonal flows in spherical shells.

Findings

Nonlinear interactions are localized near boundary reflection regions.

The width of the interaction region scales as E^{1/3}.

The mean zonal flow amplitude scales as E^{1/6} generally, and as E^{-1/2} near the rotation axis.

Abstract

This work investigates the weakly nonlinear dynamics of internal shear layers and the mean zonal flow induced by the longitudinal libration of an inner core within a spherical shell. Building on the work of He et al. (J. Fluid Mech., vol. 939, 2022, A3), which focused on linear dynamics, we adopt a similar setup to explore the nonlinear regime using both asymptotic theory and numerical computations, with Ekman numbers as low as $E=10^{-10}$. A specific forcing frequency of $\widehat{\omega}=\sqrt{2}\widehat{\Omega}$, where $\widehat{\Omega}$ denotes the rotation rate, is introduced to generate a closed rectangular path of characteristics for the inertial wave beam generated at the critical latitude. Our approach extends previous results by Le Diz\`es (J. Fluid Mech., vol. 899, 2020, A21) and reveals that nonlinear interactions are predominantly localized around regions where the wave beam reflects on the boundary. We derive specific scaling laws governing the nonlinear interactions: the width of the interaction region scales as $E^{1/3}$, and the amplitude of the resulting mean zonal flow scales as $E^{1/6}$ in general. However, near the rotation axis, where the singularity of the self-similar solution becomes more pronounced, the amplitude exhibits a scaling of $E^{-1/2}$. In addition, our study also examines the nonlinear interactions of beams which are governed by different scaling laws. Through comparison with numerical results, we validate the theoretical predictions of the asymptotic framework, observing good agreement as the Ekman number decreases.