Grid-based calculation of Lagrangian mean

TL;DR

This paper introduces a grid-based iterative method for calculating Lagrangian mean in two-timescale flows, avoiding particle tracking and improving computational efficiency, with applications to shallow-water equations and wave-averaged geostrophic balance.

Contribution

The paper presents a novel grid-based iterative approach to compute Lagrangian mean without particle tracking, reducing computational costs in numerical simulations.

Findings

Method reduces computation, memory, and communication in parallel models.

Accurately calculates Lagrangian mean in various examples.

Validates wave-averaged geostrophic balance in shallow-water equations.

Abstract

Lagrangian averaging has been shown to be more effective than Eulerian mean in separating waves from slow dynamics in two-timescale flows. It also appears in many reduced models that capture the wave feedback on the slow flow. Its calculation, however, requires tracking particles in time, which imposes several difficulties in grid-based numerical simulations or estimation from fixed-point measurements. To circumvent these difficulties, we propose a grid-based iterative method to calculate Lagrangian mean without tracking particles in time. We explain how this method reduces computation, memory footprint and communication between processors in parallelised numerical models. To assess the accuracy of this method several examples are examined and discussed. We also explore an application of this method in the context of shallow-water equations by quantifying the validity of wave-averaged geostrophic balance -- a modified form of geostrophic balance accounting for the effect of strong waves on slow dynamics.