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Sufficient Conditions for Dual Cascade Flux Laws in the Stochastic 2d Navier–Stokes Equations

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Abstract

We provide sufficient conditions for mathematically rigorous proofs of the third order universal laws capturing the energy flux to large scales and enstrophy flux to small scales for statistically stationary, forced-dissipated 2d Navier–Stokes equations in the large-box limit. These laws should be regarded as 2d turbulence analogues of the 4/5 law in 3d turbulence, predicting a constant flux of energy and enstrophy (respectively) through the two inertial ranges in the dual cascade of 2d turbulence. Conditions implying only one of the two cascades are also obtained, as well as compactness criteria which show that the provided sufficient conditions are not far from being necessary. The specific goal of the work is to provide the weakest characterizations of the “0-th laws” of 2d turbulence in order to make mathematically rigorous predictions consistent with experimental evidence.

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Notes

  1. A solution is spatially homogeneous if \(u(\cdot ,\cdot )\) has the same law as \(u(\cdot ,\cdot + y)\) for all \(y \in {\mathbb {R}}^2\). See e.g. [2, 6, 38] for more discussions statistical symmetries.

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Jacob Bedrossian

  2. Department of Mathematics, Imperial College London, London, SW7 2AZ, UK

    Michele Coti Zelati

  3. Division of Applied Mathematics, Brown University, Providence, RI, 02906, USA

    Sam Punshon-Smith

  4. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Franziska Weber

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  1. Jacob Bedrossian
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  2. Michele Coti Zelati
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Correspondence to Franziska Weber.

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Funding

J.B. was supported by NSF CAREER grant DMS-1552826 and NSF RNMS 1107444 (Ki-Net). M.C.Z. was supported by the Royal Society through a University Research Fellowship (URF\({\backslash }\)R1\({\backslash } \)191492). S.P-S. was supported by NSF DMS-1803481. F.W. was supported in part by NSF DMS-1912854.

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Appendix A. Isotropic Sixth Order Tensors

Appendix A. Isotropic Sixth Order Tensors

We need the following lemma in Section 5 in order to provide high order expansions in the energy balance:

Lemma A.1

(Expression for an isotropic sixth order tensor) We have

Proof

The left hand side is an isotropic sixth order tensor. From [43], we know that it is a linear combination of 15 fundamental isotropic tensors of the form \( \delta _{i,j}\delta _{k,m}\delta _{p,q}\) and all permutations of i, j, k, m, p, q in this expression. Since i, j, k, m, p, q are interchangeable in \(\fint _{\mathbb {S}\,}n^i n^j n^k n^m n^p n^q \,{\mathrm {d}}n\), they must all occur with the same factor, and therefore

for some constant \(\kappa \in {\mathbb {R}}\). It remains to compute \(\kappa \). We have for \(i=j=k=m=p=q\)

In this case, none of the 15 terms vanishes and therefore \(15\kappa = \frac{5}{16}\) and hence \(\kappa =\frac{1}{48}\). \(\quad \square \)

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Bedrossian, J., Coti Zelati, M., Punshon-Smith, S. et al. Sufficient Conditions for Dual Cascade Flux Laws in the Stochastic 2d Navier–Stokes Equations. Arch Rational Mech Anal 237, 103–145 (2020). https://doi.org/10.1007/s00205-020-01503-9

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