Information barriers for noisy Lagrangian tracers in filtering random incompressible flows

An important practical problem is the recovery of a turbulent velocity field from Lagrangian tracers that move with the fluid flow. Here, the filtering skill of L moving Lagrangian tracers in recovering random incompressible flow fields defined through a finite number of random Fourier modes is studied with full mathematical rigour. Despite the inherent nonlinearity in measuring noisy Lagrangian tracers, it is shown below that there are exact closed analytic formulas for the optimal filter for the velocity field involving Riccati equations with random coefficients for the covariance matrix. This mathematical structure allows a detailed asymptotic analysis of filter performance, both as time goes to infinity and as the number of noisy Lagrangian tracers, L, increases. In particular, the asymptotic gain of information from L-tracers grows only like ln L in a precise fashion; i.e., an exponential increase in the number of tracers is needed to reduce the uncertainty by a fixed amount; in other words, there is a practical information barrier. The proofs proceed through a rigourous mean field approximation of the random Ricatti equation. Also, as an intermediate step, geometric ergodicity with respect to the uniform measure on the period domain is proved for any fixed number L of noisy Lagrangian tracers. All of the above claims are confirmed by detailed numerical experiments presented here.