Noisy Lagrangian Tracers for Filtering Random Rotating Compressible Flows

Abstract

The recovery of a random turbulent velocity field using Lagrangian tracers that move with the fluid flow is a practically important problem. This paper studies the filtering skill of $L$-noisy Lagrangian tracers in recovering random rotating compressible flows that are a linear combination of random incompressible geostrophically balanced (GB) flow and random rotating compressible gravity waves. The idealized random fields are defined through forced damped random amplitudes of Fourier eigenmodes of the rotating shallow-water equations with the rotation rate measured by the Rossby number $\varepsilon$. In many realistic geophysical flows, there is fast rotation so $\varepsilon$ satisfies $\varepsilon \ll 1$ and the random rotating shallow-water equations become a slow–fast system where often the primary practical objective is the recovery of the GB component from the Lagrangian tracer observations. Unfortunately, the $L$-noisy Lagrangian tracer observations are highly nonlinear and mix the slow GB modes and the fast gravity modes. Despite this inherent nonlinearity, it is shown here that there are closed analytical formulas for the optimal filter for recovering these random rotating compressible flows for any $\varepsilon$ involving Ricatti equations with random coefficients. The performance of the optimal filter is compared and contrasted through mathematical theorems and concise numerical experiments with the performance of the optimal filter for the incompressible GB random flow with $L$-noisy Lagrangian tracers involving only the GB part of the flow. In addition, a sub-optimal filter is defined for recovering the GB flow alone through observing the $L$-noisy random compressible Lagrangian trajectories, so the effect of the gravity wave dynamics is unresolved but effects the tracer observations. Rigorous theorems proved below through suitable stochastic fast-wave averaging techniques and explicit formulas rigorously demonstrate that all these filters have comparable skill in recovering the slow GB flow in the limit $\varepsilon \rightarrow 0$ for any bounded time interval. Concise numerical experiments confirm the mathematical theory and elucidate various new features of filter performance as the Rossby number $\varepsilon$, the number of tracers $L$ and the tracer noise variance change.