Multiscaling in the randomly forced and conventional Navier–Stokes equations

We present an overview of some results we have obtained recently (A. Sain, Manu and R. Pandit, Phys. Rev. Lett. 81 (1998) 4377) from a pseudospectral study of the randomly forced Navier–Stokes equation (RFNSE) stirred by a stochastic force with zero mean and a variance ∼k4−d−y, with k the wavevector and the dimension d=3. These include the multiscaling of velocity structure functions for y⩾4 and a demonstration that the multiscaling exponent ratios ζp/ζ2 for y=4 are in agreement with those obtained for the Navier–Stokes equation forced at large spatial scales (3dNSE). We also study a coarse-graining procedure for the 3dNSE and examine why it does not lead to the RFNSE.