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On the effect of the Coriolis force on the enstrophy cascade

Yuri Cacchiò, Amirali Hannani, Gigliola Staffilani

TL;DR

The paper investigates how the Coriolis force on a $\beta$-plane influences the direct enstrophy cascade in statistically stationary forced-dissipated 2D Navier–Stokes equations. It develops a modified velocity and vorticity Karman–Howarth–Monin framework by extending the velocity field to an enlarged domain and incorporating the Coriolis term, and it proves that, in the small-scale inertial range, the Coriolis contribution vanishes to leading order. Under a weak uniform bound on the vorticity of stationary solutions, the authors establish two precise small-scale asymptotics for the third-order structure functions: $\mathbb{E}\fint_{\mathbb{S}}\fint_{\mathbb{T}\times I} |\delta_{ln}\omega|^2 \delta_{ln}u\cdot n \,dxdn \sim -2\eta l$ and $\mathbb{E}\fint_{\mathbb{S}}\fint_{\mathbb{T}\times I} |\delta_{ln}u|^2 \delta_{ln}u\cdot n \,dxdn \sim \frac{1}{4}\eta l^3$ for $l_\nu \ll l \ll l_I$, validating the small-scale enstrophy cascade despite rotation. This provides the first rigorous treatment of the $eta$-plane NS equations in this turbulence regime and aligns with experimental and numerical observations. The work also clarifies the role of regularity of the invariant measure in controlling the Coriolis-induced contributions to the KHM relations.

Abstract

We study the direct enstrophy cascade at small spatial scales in statistically stationary forced-dissipated 2D Navier-Stokes equations subject to the Coriolis force in the $β$-plane approximation. We provide sufficient conditions inspired by [6,63] to prove that at small scales, in the presence of the Coriolis force, the so-called third-order structure function's asymptotics follows the third-order universal law of 2D turbulence without the Coriolis force. Our result indicates that at small scales, the enstrophy flux from larger to smaller scales is not affected by the Coriolis force, confirming experimental and numerical observations. To the best of our knowledge, this is the first mathematically rigorous study of the above equations.

On the effect of the Coriolis force on the enstrophy cascade

TL;DR

The paper investigates how the Coriolis force on a -plane influences the direct enstrophy cascade in statistically stationary forced-dissipated 2D Navier–Stokes equations. It develops a modified velocity and vorticity Karman–Howarth–Monin framework by extending the velocity field to an enlarged domain and incorporating the Coriolis term, and it proves that, in the small-scale inertial range, the Coriolis contribution vanishes to leading order. Under a weak uniform bound on the vorticity of stationary solutions, the authors establish two precise small-scale asymptotics for the third-order structure functions: and for , validating the small-scale enstrophy cascade despite rotation. This provides the first rigorous treatment of the -plane NS equations in this turbulence regime and aligns with experimental and numerical observations. The work also clarifies the role of regularity of the invariant measure in controlling the Coriolis-induced contributions to the KHM relations.

Abstract

We study the direct enstrophy cascade at small spatial scales in statistically stationary forced-dissipated 2D Navier-Stokes equations subject to the Coriolis force in the -plane approximation. We provide sufficient conditions inspired by [6,63] to prove that at small scales, in the presence of the Coriolis force, the so-called third-order structure function's asymptotics follows the third-order universal law of 2D turbulence without the Coriolis force. Our result indicates that at small scales, the enstrophy flux from larger to smaller scales is not affected by the Coriolis force, confirming experimental and numerical observations. To the best of our knowledge, this is the first mathematically rigorous study of the above equations.

Paper Structure

This paper contains 12 sections, 5 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Turbulence
  3. Result: Turbulence
  4. Statement of the Result
  5. Sketch of the proof and main ideas
  6. Remarks
  7. Preliminaries and Main Results
  8. Modified Velocity KHM relations
  9. Solution extension
  10. Weak Velocity KHM relation
  11. Modified Vorticity KHM relations
  12. Direct cascade

Key Result

Theorem 2.1

Suppose that $L=L(\alpha)<\infty$ and $b=b(\alpha)<\infty$ are a continuous monotone decreasing functions such that $\lim_{\alpha\to 0}L=\lim_{\alpha\to 0}b=\infty$ and suppose that $a=a(\alpha)>-\infty$ is a continuous monotone increasing function such that $\lim_{\alpha\to 0}a=-\infty$. Let $\{u\} In particular, it suffices to choose $l_{\nu}\to0$ satisfying

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.1
  • proof
  • ...and 10 more