Turbulent Energy Spectrum via an Interaction Potential
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Turbulent Energy Spectrum via an Interaction Potential Rafail V. Abramov1 Received: 6 February 2020 / Accepted: 1 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract For a large system of identical particles interacting by means of a potential, we find that a strong large scale flow velocity can induce motions in the inertial range via the potential coupling. This forcing lies in special bundles in the Fourier space, which are formed by pairs of particles. These bundles are not present in the Boltzmann, Euler and Navier–Stokes equations, because they are destroyed by the Bogoliubov– Born–Green–Kirkwood–Yvon formalism. However, measurements of the flow can detect certain bulk effects shared across these bundles, such as the power scaling of the kinetic energy. We estimate the scaling effects produced by two types of potentials: the Thomas–Fermi interatomic potential (as well as its variations, such as the Ziegler–Biersack–Littmark potential), and the electrostatic potential. In the near-viscous inertial range, our estimates yield the inverse five-thirds power decay of the kinetic energy for both the Thomas–Fermi and electrostatic potentials. The electrostatic potential is also predicted to produce the inverse cubic power scaling of the kinetic energy at large inertial scales. Standard laboratory experiments confirm the scaling estimates for both the Thomas–Fermi and electrostatic potentials at near-viscous scales. Surprisingly, the observed kinetic energy spectrum in the Earth atmosphere at large scales behaves as if induced by the electrostatic potential. Given that the Earth atmosphere is not electrostatically neutral, we cautiously suggest a hypothesis that the atmospheric kinetic energy spectra in the inertial range are indeed driven by the large scale flow via the electrostatic potential coupling. Keywords Turbulence · Energy spectrum · Interaction potential Mathematics Subject Classification 76F
Communicated by Eliot Fried. This work was supported by the Simons Foundation Grant #636144.
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Rafail V. Abramov [email protected] Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan st., Chicago, IL 60607, USA
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Journal of Nonlinear Science
1 Introduction The phenomenon of turbulence in fluids has first been documented by Leonardo da Vinci, and later by Boussinesq (1877) and Reynolds (1883, 1895). As observed, turbulent motions in fluids appear to be caused by the presence of a strong large scale flow, and manifest in the ranges between the large and viscous scales (the so-called “inertial range”). In 1941, Kolmogorov (1941a, b, c) suggested that the power scaling of the turbulent kinetic energy spectrum could be modeled via an ad hoc dimensional hypothesis. With help of Kolmogorov’s hypothesis, Obukhov (1941, 1949), Chandrasekhar (1949), Corrsin (1951), and others observed that the time-averaged kinetic energy spectrum in many real-world turbulent flows scales in the Fourier space as the inverse five-thirds power of
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