  In recent years several local extrema based methodologies have been proposed
to investigate either the nonlinear or the nonstationary time series for
scaling analysis. In the present work we study systematically the distribution
of the local extrema for both synthesized scaling processes and turbulent
velocity data from experiments. The results show that for the fractional
Brownian motion (fBm) without intermittency correction the measured extremal
point density (EPD) agrees well with a theoretical prediction. For a
multifractal random walk (MRW) with the lognormal statistics, the measured EPD
is independent with the intermittency parameter $\mu$, suggesting that the
intermittency correction does not change the distribution of extremal points,
but change the amplitude. By introducing a coarse-grained operator, the
power-law behavior of these scaling processes is then revealed via the measured
EPD for different scales. For fBm the scaling exponent $\xi(H)$ is found to be
$\xi(H)=H$, where $H$ is Hurst number, while for MRW $\xi(H)$ shows a linear
relation with the intermittency parameter $\mu$. Such EPD approach is further
applied to the turbulent velocity data obtained from a wind tunnel flow
experiment with the Taylor scale $\lambda$ based Reynolds number $Re_{\lambda}=
720$, and a turbulent boundary layer with the momentum thickness $\theta$ based
Reynolds number $Re_{\theta}= 810$. A scaling exponent $\xi\simeq 0.37$ is
retrieved for the former case. For the latter one, the measured EPD shows
clearly four regimes, which agree well with the four regimes of the turbulent
boundary layer structures.

Statistical Mechanics;Fluid Dynamics