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M. Sinhuber, J. Friedrich, R. Grauer, M. Wilczek

Multi-level stochastic refinement for complex time series and fields: A Data-Driven Approach

Spatio-temporally 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 multi-scale 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 https://iopscience.iop.org/article/10.1088/1367-2630/abe60e https://arxiv.org/abs/2102.08016

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 self-similar 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 self-similarity and may therefore require a more elaborate description. In the context of turbulence, the Kolmogorov-Oboukhov 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 Ornstein-Uhlenbeck 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 non-Gaussian features in geophysical or geospatial settings.

Journal of Physics: Complexity 2 (2021) 45006 https://iopscience.iop.org/article/10.1088/2632-072X/ac2cda  https://doi.org/10.1088/2632-072X/ac2cda https://arxiv.org/abs/2105.03223

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 Kramers-Moyal expansion of the velocity increment probability density function that is associated to a Markov process. We compare the different sets of Kramers-Moyal 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 Kramers-Moyal 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 self-similarity (purely nonlocal case) to intermittent behavior (intermediate case that agrees with Yakhot's mean field theory [Phys. Rev. E 63 026307 (2001)]) to shock-like behavior (purely nonlinear Burgers case).

Atmosphere 11 (2020) 1003 https://www.mdpi.com/2073-4433/11/9/1003/pdf https://doi.org/10.3390/atmos11091003 https://arxiv.org/abs/1610.04432

J. Friedrich, S. Gallon, A. Pumir, R. Grauer

Multi-point 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 non-equidistant 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 non-self-similar 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 https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.125.170602 https://doi.org/10.1103/PhysRevLett.125.170602 https://arxiv.org/abs/2005.07591

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 Burgers-shocks 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 multi-increment probability density functions derived directly from the Burgers equation.

in Complexity and Synergetics (Springer), S.C. Müller et al. (eds.) (2018) 39 http://www.tp1.rub.de/~grauer/publications/friedrich-grauer-2018.pdf https://doi.org/10.1007/978-3-319-64334-2_4

J. Friedrich, H. Homann, T. Schäfer, R. Grauer

Longitudinal and transverse structure functions in high Reynolds-number magneto-hydrodynamic turbulence

We investigate the scaling behavior of longitudinal and transverse structure functions in homogeneous and isotropic magneto-hydrodynamic (MHD) turbulence by means of an exact hierarchy of structure function equations as well as by direct numerical simulations of two- and three-dimensional 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 http://iopscience.iop.org/article/10.1088/1367-2630/18/12/125008/meta http://dx.doi.org/10.1088/1367-2630/18/12/125008 https://arxiv.org/abs/1609.05790

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 https://www.tp1.ruhr-uni-bochum.de/addon/tp1/publications/PhysRevE.93.023108.pdf https://doi.org/10.1103/PhysRevE.93.023108 https://arxiv.org/abs/1508.05215

J. Friedrich and R. Friedrich

Generalized vortex model for the inverse cascade of two-dimensional turbulence

We generalize Kirchhoff's point vortex model of two-dimensional 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 like-signed 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 https://www.tp1.ruhr-uni-bochum.de/addon/tp1/publications/PhysRevE.88.053017.pdf https://doi.org/10.1103/PhysRevE.88.053017 https://arxiv.org/abs/1111.5808