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Energy fluxes in quasi-equilibrium flows
- Alexandros Alexakis, Marc-Etienne Brachet
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- Journal:
- Journal of Fluid Mechanics / Volume 884 / 10 February 2020
- Published online by Cambridge University Press:
- 17 December 2019, A33
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We examine the relation between the absolute equilibrium state of the spectrally truncated Euler equations (TEE) predicted by Kraichnan (J. Fluid Mech., vol. 59 (4), 1973, pp. 745–752) to the forced and dissipated flows of the spectrally truncated Navier–Stokes (TNS) equations. In both of these idealized systems, a finite number of Fourier modes is kept contained inside a sphere of radius $k_{max}$, but, while the first conserves energy, in the second, energy is injected by a body-force $\boldsymbol{f}$ and dissipated by the viscosity $\unicode[STIX]{x1D708}$. For the TNS system, stochastically forced with energy injection rate ${\mathcal{I}}_{{\mathcal{E}}}$, we show, using an asymptotic expansion of the Fokker-Planck equation, that in the limit of small $k_{max}\unicode[STIX]{x1D702}$ (where $\unicode[STIX]{x1D702}=(\unicode[STIX]{x1D708}^{3}/{\mathcal{I}}_{{\mathcal{E}}})^{1/4}$, the Kolmogorov length scale) the flow approaches the absolute equilibrium solution of Kraichnan with an effective ‘temperature’ such that there is a balance between the energy injection and the energy dissipation rate. We further investigate the TNS system using direct numerical simulations in periodic cubic boxes of size $2\unicode[STIX]{x03C0}/k_{0}$. The simulations verify the predictions of the model for small values of $k_{max}\unicode[STIX]{x1D702}$. For intermediate values of $k_{max}\unicode[STIX]{x1D702}$, a transition from the quasi-equilibrium ‘thermal’ state to Kolmogorov turbulence is observed. In particular, we demonstrate that, at steady state, the TNS system reproduces the Kolmogorov energy spectrum if $k_{max}\unicode[STIX]{x1D702}\gg 1$. As $k_{max}\unicode[STIX]{x1D702}$ becomes smaller, then a bottleneck effect appears, taking the form of the equipartition spectrum $E(k)\propto k^{2}$ at small scales. As $k_{max}\unicode[STIX]{x1D702}$ is decreased even further, so that $k_{max}\unicode[STIX]{x1D702}\ll (k_{0}/k_{max})^{11/4}$, the equipartition spectrum occupies all scales approaching the asymptotic equilibrium solutions found before. If the forcing is applied at small scales and the dissipation acts only at large scales, then the equipartition spectrum appears at all scales for all values of $\unicode[STIX]{x1D708}$. In both cases, a finite forward or inverse flux is present even for the cases where the flow is close to the equilibrium state solutions. However, unlike the classical turbulence, where an energy cascade develops with a mean energy flux that is large compared to its fluctuations, the quasi-equilibrium state has a mean flux of energy that is subdominant to the large flux fluctuations observed.
On the thermal equilibrium state of large-scale flows
- Alexandros Alexakis, Marc-Etienne Brachet
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- Journal:
- Journal of Fluid Mechanics / Volume 872 / 10 August 2019
- Published online by Cambridge University Press:
- 13 June 2019, pp. 594-625
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In a forced three-dimensional turbulent flow the scales larger than the forcing scale have been conjectured to reach a thermal equilibrium state forming a $k^{2}$ energy spectrum, where $k$ is the wavenumber. In this work we examine the properties of these large scales in turbulent flows with the use of numerical simulations. We show that the choice of forcing can strongly affect the behaviour of the large scales. A spectrally dense forcing (a forcing that acts on all modes inside a finite-width spherical shell) with long correlation times may lead to strong deviations from the $k^{2}$ energy spectrum, while a spectrally sparse forcing (a forcing that acts only on a few modes) with short correlated time scale can reproduce the thermal spectrum. The origin of these deviations is analysed and the involved mechanisms is unravelled by examining: (i) the number of triadic interactions taking place, (ii) the spectrum of the nonlinear term, (iii) the amplitude of interactions and the fluxes due to different scales and (iv) the transfer function between different shells of wavenumbers. It is shown that the spectrally dense forcing allows for numerous triadic interactions that couple one large-scale mode with two forced modes and this leads to an excess of energy input at the large scales. This excess of energy is then moved back to the small scales by self-interactions of the large-scale modes and by interactions with the turbulent small scales. The overall picture that arises from the present analysis is that the large scales in a turbulent flow resemble a reservoir that is in (non-local) contact with a second out-of-equilibrium reservoir consisting of the smaller (forced, turbulent and dissipative) scales. If the injection of energy at the large scales from the forced modes is relative weak (as is the case for the spectrally sparse forcing) then the large-scale spectrum remains close to a thermal equilibrium and the role of long-range interactions is to set the global energy (temperature) of the equilibrium state. If, on the other hand, the long-range interactions are dominant (as is the case for the spectrally dense forcing), the large-scale self-interactions cannot respond fast enough to bring the system into equilibrium. Then the large scales deviate from the equilibrium state with energy spectrum that may display exponents different from the $k^{2}$ spectrum.
Lack of universality in MHD turbulence, and the possible emergence of a new paradigm?
- Annick Pouquet, Marc-Etienne Brachet, Ed Lee, Pablo Mininni, Duane Rosenberg, Vadim Uritsky
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- Journal:
- Proceedings of the International Astronomical Union / Volume 6 / Issue S271 / June 2010
- Published online by Cambridge University Press:
- 12 August 2011, pp. 304-316
- Print publication:
- June 2010
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We review some of the recent results obtained in MHD turbulence, as encountered in many astrophysical objects. We focus attention on the lack of universality in such flows, including in the simplest case (no externally imposed magnetic field, no forcing, unit magnetic Prandtl number). Several parameters can foster such a breakdown of classical Kolmogorov scaling, such as the presence of velocity-magnetic field correlations, or of magnetic helicity and the role of the interplay between nonlinear eddies and Alfvén waves. A link with avalanche processes is also discussed. These findings have led to the conjecture of the emergence of a new paradigm for MHD turbulence, as a possibly unsettled competition between several dynamical phenomena.