Slow modes in passive scalar turbulent advection

This communication describes the computation of slow modes in a generic shell model of passive turbulent advection. It is argued that the propagator for the correlation functions possesses a ladder of slow modes associated with each zero mode. These slow modes decay algebraically fast, as opposed to exponential. Using the explicit form of the propagator of the 2-point structure function, we show that the slow modes structure for the generic case is analogous to the case of a Gaussian correlated advecting field, for which the differential structure of the operator allows a direct computation of these modes. Numerical computations of the slow modes are performed and compare very well to our results.

Figure 1

(Color online) Numerical computation of Eq. (10) for the model described by Eqs. (5, 6). The parameters are $ϵ=0.5$, $\lambda =2$, $k_0=0.25$, $\nu =\kappa ={10}^{-9}$. The number of shells used was 30, with Gaussian $\delta$-correlated forcing on shells 2 and 3 for the velocity field. The initial condition for the scalar was initiated at shells 19 and 20 with random phases and constant amplitude. The average is taken over 13 000 realizations. The time units are natural, with a span corresponding to the eddy turn over the forcing scale $(\approx 125)$. The first data set is simply ${\sum }_n⟨{\mid {\theta }_n\left(t\right)\mid }^2⟩$. The other data sets correspond to the successive slow modes, starting from the zero mode, i.e., ${\sum }_n⟨{\mid {\theta }_n\left(t\right)\mid }^2⟩F_n$, where $F_n$ is the forced second order structure function, as obtained by direct numerical computation. The vertical scale is arbitrary and the curves have been scaled so as to cross at the same point.

Figure 2

(Color online) Numerical evaluation of the slow modes for the Kraichnan model on shells, Eqs. (20, 21, 22, 23, 24, 25). Here $j=1,\dots ,10$. The parameter $\xi =4∕3$ and $n*=10$. The eddy turnover time at that shell is approximately ${10}^{-2}$. The horizontal scale is counted in the units of the turnover time of shell 0.