Anomalous Scaling Exponents in Nonlinear Models of Turbulence

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence. We construct, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. The statistics of the auxiliary linear model are dominated by “statistically preserved structures“ which are associated with exact conservation laws. The latter can be used, for example, to determine the value of the anomalous scaling exponent of the second order structure function. The approach is equally applicable to shell models and to the Navier-Stokes equations.

Figure 1

Log-log plot of the sixth order structure functions of the fields ${\mathit{u}}_{+}$ and ${\mathit{u}}_{-}$ (circles and squares, respectively), for $\mu =1$, as a function of the third order structure functions. The dashed line corresponds to the best fit in the scaling region with slopes 1.77. Lower inset: the sixth order structure function of the two fields as a function of $r$. Upper inset: $z_p\equiv {\zeta }_p/{\zeta }_3-p/3$ computed for the structures functions of ${\mathit{u}}_{+}$ (line) and ${\mathit{u}}_{-}$ (circles).

Figure 2

The sixth order structure function of the field $w_n$ in Eqs. (11) for $\lambda ={10}^{-1}$, ${10}^{-3}$, and ${10}^{-5}$, together with the sixth order structure function for the Sabra model (3) and for the linear model (6), respectively. The structure functions of the field $u_n$ for $\lambda >0$ are not shown since they are indistinguishable from those of the $w_n$. Inset: log-log plot of the fourth-order correlation function $F_{2,2}(k_n,k_7)$ vs $k_n$ calculated for the linear field ($+$) and for the nonlinear field (solid line) at $\lambda =0$.

Figure 3

With the symbols ($+$) the constants $I^{(2)}$ (bottom) and $I^{(4)}$ (top) constructed by projecting the decaying structure function of the linear model on the forced structure function of the nonlinear model. To emphasize the importance of using the correct SPS, we also show the result for $I^{(4)}$ using the dimensional Kolmogorov prediction for $Z^4$ (small dots) and $Z^4=1$ (solid line).