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Kolmogorov's dissipation number and determining wavenumber for dyadic models

Mimi Dai, Margaret Hoeller, Qirui Peng, Xiangxiong Zhang

Abstract

We study some dyadic models for incompressible magnetohydrodynamics and Navier-Stokes equation. The existence of fixed point and stability of the fixed point are established. The scaling law of Kolmogorov's dissipation wavenumber arises from heuristic analysis. In addition, a time-dependent determining wavenumber is shown to exist; moreover, the time average of the determining wavenumber is proved to be bounded above by Kolmogorov's dissipation wavenumber. Additionally, based on the knowledge of the fixed point and stability of the fixed point, numerical simulations are performed to illustrate the energy spectrum in the inertial range below Kolmogorov's dissipation wavenumber.

Kolmogorov's dissipation number and determining wavenumber for dyadic models

Abstract

We study some dyadic models for incompressible magnetohydrodynamics and Navier-Stokes equation. The existence of fixed point and stability of the fixed point are established. The scaling law of Kolmogorov's dissipation wavenumber arises from heuristic analysis. In addition, a time-dependent determining wavenumber is shown to exist; moreover, the time average of the determining wavenumber is proved to be bounded above by Kolmogorov's dissipation wavenumber. Additionally, based on the knowledge of the fixed point and stability of the fixed point, numerical simulations are performed to illustrate the energy spectrum in the inertial range below Kolmogorov's dissipation wavenumber.

Paper Structure

This paper contains 13 sections, 15 theorems, 133 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fixed point
  4. Existence of fixed point and properties of fixed point
  5. Stability of fixed point
  6. Analogue of Kolmogorov's turbulence theory
  7. Determining wavenumber
  8. Estimates of the determining wavenumber
  9. Dyadic NSE
  10. Model with both forward and backward energy cascades
  11. Numerical simulations
  12. Model with forward energy cascade
  13. Model with both forward and backward energy cascades

Key Result

Theorem 3.1

There exists a fixed point $(a^*,b^*)\in \ell^2\times \ell^2$ to (sys-2). Equivalently, there exists a solution $\{(A_j,B_j)\}$ to (stat-2-gen) with $(A,B)\in H^{-\frac{1}{3}\theta}\times H^{-\frac{1}{3}\theta}$.

Figures (3)

  • Figure 1: Forward cascade with $\theta=1$.
  • Figure 2: Forward cascade with $\theta=2$.
  • Figure 3: Forward and backward cascades with $\theta=1$.The slope of the best line fit for the kinetic spectrum is around $-\frac{8.14}{3}$.

Theorems & Definitions (19)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 3.1
  • Lemma 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 5.1
  • ...and 9 more