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Article Dans Une Revue Journal of Physics A: Mathematical and Theoretical Année : 2015

Anomalous spectral laws in differential models of turbulence

Résumé

Differential models for hydrodynamic, passive-scalar and wave turbulence given by nonlinear first- and second-order evolution equations for the energy spectrum in the $k$-space were analysed. Both types of models predict formation an anomalous transient power-law spectra. The second-order models were analysed in terms of self-similar solutions of the second kind, and a phenomenological formula for the anomalous spectrum exponent was constructed using numerics for a broad range of parameters covering all known physical examples. The first-order models were examined analytically, including finding an analytical prediction for the anomalous exponent of the transient spectrum and description of formation of the Kolmogorov-type spectrum as a reflection wave from the dissipative scale back into the inertial range. The latter behaviour was linked to pre-shock/shock singularities similar to the ones arising in the Burgers equation. Existence of the transient anomalous scaling and the reflection-wave scenario are argued to be a robust feature common to the finite-capacity turbulence systems. The anomalous exponent is independent of the initial conditions but varies for for different models of the same physical system.
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Dates et versions

hal-01119102 , version 1 (20-02-2015)
hal-01119102 , version 2 (08-05-2015)

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Simon Thalabard, Sergey V. Nazarenko, Sébastien Galtier, Sergey Medvedev. Anomalous spectral laws in differential models of turbulence. Journal of Physics A: Mathematical and Theoretical, 2015, 48 (28), ⟨10.1088/1751-8113/48/28/285501⟩. ⟨hal-01119102v2⟩
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