On Batchelor passive advection by a finite-time correlated random velocity field

The Batchelor passive advection is an advection by a smooth velocity field. If the velocity field is a delta-correlated in time random Gaussian process, then the problem is reduced to quantum mechanics of fluctuating velocity gradient u'(t). For the finite-time correlated velocity field, such a reduction does not exist. To illustrate this point, the second moment of a passively advected magnetic field is considered, and the stochastic calculus is used to find finite-time corrections to its growth rate. The growth rate depends on large scale properties of the velocity field. Moreover, the problem is not universal with respect to the short-time regularization: different regularizations give different answers for the growth rate.