
M. Sinhuber, J. Friedrich, R. Grauer, M. Wilczek
Multilevel stochastic refinement for complex time series and fields: A DataDriven Approach
Spatiotemporally extended nonlinear systems often exhibit a remarkable complexity in space and time. In many cases, extensive records of such data sets are difficult to obtain, yet needed for a range of applications. Here, we present a method to generate synthetic time series or fields that reproduce statistical multiscale features of complex systems. The method is based on a hierarchical refinement employing transition probability density functions (PDFs) from one scale to another. We address the case in which such PDFs can be obtained from experimental measurements or simulations and then used to generate arbitrarily large synthetic data sets. The validity of our approach is demonstrated at the example of an experimental dataset of high Reynolds number turbulence.
New Journal of Physics 23 (2021) 63063


J. Friedrich, J. Peinke, A. Pumir, R. Grauer
Explicit construction of joint multipoint statistics in complex systems
Complex systems often involve random fluctuations for which selfsimilar properties in space and time play an important role. Fractional Brownian motions, characterized by a single scaling exponent, the Hurst exponent H, provide a convenient tool to construct synthetic signals that capture the statistical properties of many processes in the physical sciences and beyond. However, in certain strongly interacting systems, e.g., turbulent flows, stock market indices, or cardiac interbeats, multiscale interactions lead to significant deviations from selfsimilarity and may therefore require a more elaborate description. In the context of turbulence, the KolmogorovOboukhov model (K62) describes anomalous scaling, albeit explicit constructions of a turbulent signal by this model are not available yet. Here, we derive an explicit formula for the joint multipoint probability density function of a multifractal field. To this end, we consider a scale mixture of fractional OrnsteinUhlenbeck processes and introduce a fluctuating length scale in the corresponding covariance function. In deriving the complete statistical properties of the field, we are able to systematically model synthetic multifractal phenomena. We conclude by giving a brief outlook on potential applications which range from specific tailoring or stochastic interpolation of wind fields to the modeling of financial data or nonGaussian features in geophysical or geospatial settings.
Journal of Physics: Complexity 2 (2021) 45006


J. Friedrich and R. Grauer
Generalized description of intermittency in turbulence via stochastic methods
We present a generalized picture of intermittency in turbulence that is based on the theory of stochastic processes. To this end, we rely on the experimentally and numerically verified finding by R.~Friedrich and J. Peinke [Phys. Rev. Lett. 78, 863 (1997)] that allows for an interpretation of the turbulent energy cascade as a Markov process of velocity increments in scale. It is explicitly shown that all known phenomenological models of turbulence can be reproduced by the KramersMoyal expansion of the velocity increment probability density function that is associated to a Markov process. We compare the different sets of KramersMoyal coefficients of each phenomenology and deduce that an accurate description of intermittency should take into account an infinite number of
coefficients. This is demonstrated in more detail for the case of Burgers turbulence that exhibits pronounced intermittency effects. Moreover, the influence of nonlocality on KramersMoyal coefficients is investigated by direct numerical simulations of a generalized Burgers equation. Depending
on the balance between nonlinearity and nonlocality, we encounter different intermittency behavior that ranges from selfsimilarity (purely nonlocal case) to intermittent behavior (intermediate case that agrees with Yakhot's mean field theory
[Phys. Rev. E 63 026307 (2001)]) to shocklike behavior (purely nonlinear Burgers case).
Atmosphere 11 (2020) 1003


J. Friedrich, S. Gallon, A. Pumir, R. Grauer
Multipoint fractional Brownian bridges and their applications
We propose and test a method to interpolate sparsely sampled signals by a stochastic process with a broad range of spatial and/or temporal scales. To this end, we extend the notion of a fractional Brownian bridge, defined as fractional Brownian motion with a given scaling (Hurst) exponent H and with prescribed start and end points, to a bridge process with an arbitrary number of prescribed intermediate and nonequidistant points. We demonstrate the validity of our method on a signal from fluid turbulence in a high Reynolds number flow. Furthermore, we discuss possible extensions of the present work to include the nonselfsimilar character of the signal. The derived method could be instrumental within a variety of fields such as astrophysics, particle tracking, specific tailoring of surrogate data, and spatial planning.
Phys. Rev. Lett. 125 (2020) 170602


J. Friedrich and R. Grauer
Markov Property of Velocity Increments in Burgers Turbulence
We investigate the intermittency properties of a turbulent flow without pressure described by the Burgers equation. To this end, we make use of a phe nomenogical description devised by R. Friedrich and J. Peinke [Phys. Rev. Lett. 78, 863 (1997)] that interprets the turbulent energy cascade as a Markov process in scale. The impact of Burgersshocks on the Markov property of the velocity incre ments is discussed and compared to numerical simulations. Furthermore, we give a brief outlook on the use of the Markov property as a possible closure of a hierarchy of multiincrement probability density functions derived directly from the Burgers equation.
in Complexity and Synergetics (Springer), S.C. Müller et al. (eds.) (2018) 39


J. Friedrich, H. Homann, T. Schäfer, R. Grauer
Longitudinal and transverse structure functions in high Reynoldsnumber magnetohydrodynamic turbulence
We investigate the scaling behavior of longitudinal and transverse structure functions in homogeneous and isotropic magnetohydrodynamic (MHD) turbulence by means of an exact hierarchy of structure function equations as well as by direct numerical simulations of two and threedimensional MHD turbulence. In particular, rescaling relations between longitudinal and transverse structure functions are derived and utilized in order to compare different scaling behavior in the inertial range. It is found that there are no substantial differences between longitudinal and transverse structure functions in MHD turbulence. This finding stands in contrast to the case of hydrodynamic turbulence which shows persistent differences even at high Reynolds numbers. We propose a physical picture that is based on an effective reduction of pressure contributions due to local regions of same magnitude and alignment of velocity and magnetic field fluctuations. Finally, our findings underline the importance of the pressure term for the actually observed scaling differences in hydrodynamic turbulence.
New J. Phys. 18 (2016) 125008


J. Köhler, J. Friedrich, A. Ostendorf, and E. L. Gurevich
Characterization of azimuthal and radial velocity fields induced by rotors in flows with a low Reynolds number
We theoretically and experimentally investigate the flow field that emerges from a rodlike microrotor rotating about its center in a nonaxisymmetric manner. A simple theoretical model is proposed that uses a superposition of two rotlets as a fundamental solution to the Stokes equation. The predictions of this model are compared to measurements of the azimuthal and radial microfluidic velocity field components that are induced by a rotor composed of fused microscopic spheres. The rotor is driven magnetically and the fluid flow is measured with the help of a probe particle fixed by an optical tweezer. We find considerable deviations of the mere azimuthal flow pattern induced by a single rotating sphere as it has been reported by Di Leonardo et al. [Phys. Rev. Lett. 96, 134502 (2006)]. Notably, the presence of a radial velocity component that manifests itself by an oscillation of the probe particle with twice the rotor frequency is observed. These findings open up a way to discuss possible radial transport in microfluidic devices.
Physical Review E 93 (2016) 23108


J. Friedrich and R. Friedrich
Generalized vortex model for the inverse cascade of twodimensional turbulence
We generalize Kirchhoff's point vortex model of twodimensional fluid motion to a rotor model which exhibits an inverse cascade by the formation of rotor clusters. A rotor is composed of two vortices with likesigned circulations glued together by an overdamped spring. The model is motivated by a treatment of the vorticity equation representing the vorticity field as a superposition of vortices with elliptic Gaussian shapes of variable widths, augmented by a suitable forcing mechanism. The rotor model opens up the way to discuss the energy transport in the inverse cascade on the basis of dynamical systems theory.
Physical Review E 88 (2013) 53017
