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Field theory of the inverse cascade in two-dimensional turbulence

Jackson R. Mayo*

  • Department of Physics, Princeton University, Princeton, New Jersey 08544-0708, USA

  • *Present address: Combustion Research Facility, Sandia National Laboratories, Livermore, CA 94551–0969, USA. Electronic address: jmayo@sandia.gov
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Phys. Rev. E 72, 056316 – Published 14 November, 2005

DOI: https://doi.org/10.1103/PhysRevE.72.056316

Abstract

A two-dimensional fluid, stirred at high wave numbers and damped by both viscosity and linear friction, is modeled by a statistical field theory. The fluid’s long-distance behavior is studied using renormalization-group (RG) methods, as begun by Forster, Nelson, and Stephen [Phys. Rev. A 16, 732 (1977)]. With friction, which dissipates energy at low wave numbers, one expects a stationary inverse energy cascade for strong enough stirring. While such developed turbulence is beyond the quantitative reach of perturbation theory, a combination of exact and perturbative results suggests a coherent picture of the inverse cascade. The zero-friction fluctuation-dissipation theorem (FDT) is derived from a generalized time-reversal symmetry and implies zero anomalous dimension for the velocity even when friction is present. Thus the Kolmogorov scaling of the inverse cascade cannot be explained by any RG fixed point. The β function for the dimensionless coupling ĝ is computed through two loops; the ĝ3 term is positive, as already known, but the ĝ5 term is negative. An ideal cascade requires a linear β function for large ĝ, consistent with a Padé approximant to the Borel transform. The conjecture that the Kolmogorov spectrum arises from an RG flow through large ĝ is compatible with other results, but the accurate k−5∕3 scaling is not explained and the Kolmogorov constant is not estimated. The lack of scale invariance should produce intermittency in high-order structure functions, as observed in some but not all numerical simulations of the inverse cascade. When analogous RG methods are applied to the one-dimensional Burgers equation using an FDT-preserving dimensional continuation, equipartition is obtained instead of a cascade—in agreement with simulations.

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Phys. Rev. E 72, 056316– Published 14 November, 2005
  • Received 25 October 2004

© 2005 American Physical Society

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