Structure functions and scaling exponents

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Structure functions and scaling exponents

To evaluate the moments of vorticity fluctuations it is necessary to perform an average over the statistics of forcing which generates the fluctuations, and over exit-times statistics:

\begin{displaymath} S_p^{\omega}(r) \equiv \langle (\delta_r \omega)^p \rangle ... ...mega^p \rangle_f \langle e^{-p \alpha T_L(r)} \rangle_{T_L(r)} \end{displaymath} (2.29)

Since the velocity field is smooth, two-dimensional and incompressible, particles separate exponentially fast and the statistics of exit-times can be replaced by the statistics of finite time Lyapunov exponents, which measure the growth rate of the logarithm of the distance between two particle during a time $t$. The finite-time Lyapunov exponent and exit-times are thus related in a smooth flow by the condition:

\begin{displaymath} L=r e^{\gamma T_L(r)}. \end{displaymath} (2.30)

We remark that this relation does not hold in the case of non-smooth flows, where the relative dispersion follows a power law [40]. In this case the growth of the distance $\ell $ between two trajectories is dominated by the structure of the flow at the same scale. The scaling law for the intensity of the eddies of size $\ell $ characterized by the Hölder exponent of the rough velocity field, gives origin to a power law for particle separation. Thus the exit-time [画像:$T_L(r)$] value is dominated by the eddy at large scale $L$ and is almost independent from the small initial separation $r$.

Replacing the average over the statistics of exit-times with the statistics of finite-time Lyapunov exponents, and using the asymptotic behavior for the distribution [画像:$P(\gamma,t) \sim t^{1/2} \exp[-S(\gamma) t]$] (see Section (2.4)), one gets the following estimate for moments of vorticity fluctuations

[画像:\begin{displaymath} S_p^{\omega}(r) \sim \langle \Omega^p \rangle \int d \gamma... .../ \gamma} \sim \left({r \over L} \right)^{\zeta^{\omega}_p}\;. \end{displaymath}] (2.31)

The scaling exponents are evaluated from Eq. (2.31) by a steepest descent argument as:

[画像:\begin{displaymath} \zeta^{\omega}_p = \min_{\gamma} \left\{ p,[p \alpha + S(\gamma)]/\gamma \right\}\;. \end{displaymath}] (2.32)

In the range 0ドル< \zeta^{\omega}_2 < 2$ the scaling exponent of the second order structure function for vorticity is related to the spectral slope of vorticity power spectrum

\begin{displaymath} Z(k)=2\pi k \langle \vert\hat{\omega}({\mbox{\boldmath$k$}})\vert^2 \rangle \sim k^{-1-\xi} \end{displaymath} (2.33)

by the relation $\xi = \zeta^{\omega}_2,ドル which gives the explicit dependence on the friction coefficient $\alpha $ of the spectral slope of the energy spectrum:

\begin{displaymath} E(k)=Z(k)/k^2\sim k^{-3-\xi} \end{displaymath} (2.34)

next up previous contents
Next: From active to passive Up: Lagrangian description of the Previous: Fluid trajectories and exit-times Contents

Stefano Musacchio 2004年01月09日