THE TOP LYAPUNOV EXPONENT FOR A STOCHASTIC FLOW MODELING THE UPPER OCEAN TURBULENCE ∗ LEONID I. PITERBARG † SIAM J. APPL. MATH. c  2001 Society for Industrial and Applied Mathematics Vol. 62, No. 3, pp. 777–800 Abstract. A stochastic model is proposed for multiparticle Lagrangian motion in the upper ocean. The model is based on hydrodynamics equations with random forcing, includes a few well interpreted and well estimated parameters, and implies a common description of the one-particle motion via a Langevin equation for the particle velocity. The dependence of the top Lyapunov exponent on the model parameters is studied as a part of a Lagrangian predictability problem. In particular, it is found that the Coriolis effect can radically improve the prediction of a Lagrangian particle position based on observations of other particles. Key words. Lyapunov exponent, stochastic flow, Lagrangian motion, predictability, oceano- graphic applications AMS subject classifications. 76F25, 76F55, 86A05, 62M20 PII. S0036139999366401 Introduction. This paper originated from the following Lagrangian prediction problem: to evaluate the position of a current-following particle in an ill-known flow, given its initial position and observations of several other particles released at approx- imately the same time. The first Lagrangian particle (a synonym for current-following particle) is called the predictand and the observed particles are called predictors. This problem is motivated by some oceanographic applications such as search-and-rescue operations, monitoring the spread of pollutants, and forecasting fish larvae (see [14]). In the first application, the predictand is a lost object and the predictors are drifters— special floating devices widely used in ocean explorations [5, 6]. Real oceanic currents typically have strong fluctuating (turbulent) components. This makes it impossible to compute exactly the particle trajectories using dynamical equations. An adequate approach is to assume that the particles exist in a stochastic flow. It is clear that the prediction error at any time crucially depends on whether at least one predictor is close enough to the predictand. For incompressible flows, particles tend to get away from one another (see [23, 24, 15]); therefore one can expect that the limit of predictability is related to the existence time of a cluster. By cluster I mean a group of particles initially located very close to one another, and its existence time is defined as the time interval between the release and the moment when the minimal distance between the particles becomes greater than the correlation radius of the velocity fluctuations. In turn, the cluster existence time depends on how fast the distance between two par- ticles grows, i.e., on the top Lyapunov exponent (TLE), called simply the Lyapunov exponent. Thus, as a first approximation, the predictability limit is the reciprocal of the TLE. Explicit formulas for the TLE in stochastic flows are known only in a limited number of situations, such as Brownian isotropic flow (see [11] and [2]). Un- fortunately, Brownian flow is not an appropriate model for real oceanic flows. One of the most common features of oceanic Lagrangian motion is a significant correlation ∗ Received by the editors December 14, 1999; accepted for publication (in revised form) April 9, 2001; published electronically December 28, 2001. This work was supported by ONR grant N00014- 99-0042. http://www.siam.org/journals/siap/62-3/36640.html † Center of Applied Mathematical Sciences, University of Southern California, Los Angeles, CA 90089-1113 (piter@math.usc.edu). 777