Rotation of trajectories in lagrangian stochastic models of turbulent dispersion

We present a new measure for the rotation of Lagrangian trajectories in tur bulence that simplifies and generalises that suggested by Wilson and Flesch ( Boundary-Layer Meteorol. 84, 411-426). The new measure is the cross prod uct of the velocity and acceleration and is directly related to the area, r ather than the angle, swept out by the velocity vector. It makes it possibl e to derive a simple but exact kinematic expression for the mean rotation < d s > of the velocity vector and to partition this expression into terms < dS > that are closed in terms of Eulerian velocity moments up to second or der and unclosed terms. The unclosed terms < ds'> arise from the interactio n of the fluctuating part of the velocity and the rate of change of the flu ctuating velocity. We examine the mean rotation of a class of Lagrangian stochastic models tha t are quadratic in velocity for Gaussian inhomogeneous turbulence. For some of these models, including that of Thomson (J. Fluid Mech. 180, 113-153), the unclosed part of the mean rotation < ds'> vanishes identically, while f or other models it is non-zero. Thus the mean rotation criterion clearly se parates the class of models into two sets, but still does not provide a cri terion for choosing a single model. We also show that models for which < ds'> = 0 are independent of whether th e model is derived on the assumption that total Lagrangian velocity is Mark ovian or whether the fluctuating part is Markovian.