Methods of Analysis

Lesson

Introduction
Trace Moments (TM)
Double Trace Moments (DTM)
- Theoretical estimation
- Empirical estimation: turbulent flow
- Empirical estimation: biology
Probability Distribution Multiple Scaling
Structure Function
Empirical Modal Decomposition (EMD)
Exercises
Conclusion

Empirical estimation: turbulent flow

The characterization of the turbulent flows can be achieved using multifractal formalism, it had been showed in literature that the UM model reproduces well the statistics of the measured field in geophysics. The UM parameters have been estimated for different measured fields in geophysics, such as : velocity, temperature. We will give in this slide the estimation of for the wind velocity field and temperature field. The dimensional analysis performed on the Navier-Stokes equations and the advection-diffusion equation provides the expressions of fluxes which are defined as below:

$\varepsilon_\lambda \approx \Delta v_\lambda^{3}; \quad \quad \phi_\lambda \approx \Delta T_\lambda^{3}; \quad \quad \phi = \varepsilon^{-1/2} \chi^{3/2}$

Where represents the flux for the velocity field, the flux for the temperature field and the temperature variance. The data sets were measured in wind-tunnel, for the velocity: field the sampling frequency is and the time duration is about , for the temperature field: the sampling frequency is and the time duration is .

Double trace moment as a function of the resolution in a log-log plot, showing the scale invariance for the velocity field (a) and temperature field (b).

The estimation of the UM parameters passes through the statistical moment function :

Log K(q, eta) as function of log eta : (a) velocity field, (b) temperature field

The obtained UM parameters are: for the wind velocity and for temperature field .

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