Dynamical structure factors in models of turbulence

We investigate the dynamical scaling behavior of the time-dependent structure functions, $S_q(r,\tau )=〈\left[u\right(r,\tau \left)-u\right(0,0)\mathrm{}{]}^q〉,$ in the one-dimensional, stochastic Burgers equation as a function of the exponent β that characterizes the scale of noise correlations. We present and analyze the exact equations satisfied by $S_2(r,\tau )$ and a related correlation function to argue that (a) $\partial S_2(r,\tau )/\partial \tau$ exhibits a discontinuity at $\tau =0\mathrm{}$ with an effective dynamical exponent given by $1+\beta /3\mathrm{}$ and (b) the dynamical scaling exponent $z$ is unity for intermediate times (a result equivalent to Taylor’s hypothesis). Various numerical checks of these results are presented. Finally, the corresponding exact equations for the structure functions in the case of the Navier-Stokes equation are presented, and by analogy with the one-dimensional Burgers equation it is shown how Taylor’s hypothesis can arise in homogeneous turbulence.