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RG theory of spontaneous stochasticity for Sabra model of turbulence

Alexei A. Mailybaev

Abstract

We consider fluctuating Sabra models of turbulence, which exhibit the phenomenon of spontaneous stochasticity: their solutions converge to a stochastic process in the ideal limit, when both viscosity and small-scale noise vanish. In this paper, we develop a renormalization group (RG) approach to explain this phenomenon. Here, RG is understood as an exact relation between the stochastic properties of systems with different dissipative and noise terms, in contrast to the Kadanoff-Wilson coarse-graining procedure, which involves small-scale integration. We argue that the stochastic process in the ideal limit is represented as a fixed point of the RG operator. The existence of such a fixed point confirms not only the convergence in the ideal limit, but also the universality of the spontaneously stochastic process, i.e. its independence from the type of dissipation and noise. The dominant eigenmode of the linearized RG operator determines the leading correction in the convergence process. The RG eigenvalue $ρ\approx 0.84 \exp(2.28i)$ is universal and it turns out to be complex, which explains the rather slow and oscillatory convergence in the ideal limit. These universality predictions are accurately confirmed by numerical data.

RG theory of spontaneous stochasticity for Sabra model of turbulence

Abstract

We consider fluctuating Sabra models of turbulence, which exhibit the phenomenon of spontaneous stochasticity: their solutions converge to a stochastic process in the ideal limit, when both viscosity and small-scale noise vanish. In this paper, we develop a renormalization group (RG) approach to explain this phenomenon. Here, RG is understood as an exact relation between the stochastic properties of systems with different dissipative and noise terms, in contrast to the Kadanoff-Wilson coarse-graining procedure, which involves small-scale integration. We argue that the stochastic process in the ideal limit is represented as a fixed point of the RG operator. The existence of such a fixed point confirms not only the convergence in the ideal limit, but also the universality of the spontaneously stochastic process, i.e. its independence from the type of dissipation and noise. The dominant eigenmode of the linearized RG operator determines the leading correction in the convergence process. The RG eigenvalue is universal and it turns out to be complex, which explains the rather slow and oscillatory convergence in the ideal limit. These universality predictions are accurately confirmed by numerical data.

Paper Structure

This paper contains 24 sections, 1 theorem, 59 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The concept of (Eulerian) spontaneous stochasticity
  3. Fluctuating Navier-Stokes equations
  4. Fluctuating Sabra shell model
  5. Fluctuating LES shell model
  6. Numerical observation of spontaneous stochasticity
  7. Discrete-time fluctuating shell models
  8. Discrete-time algorithm
  9. Properties of discrete-time models
  10. Discrete-time LES shell model
  11. RG relation for DT-LES models
  12. Flow
  13. RG relation between the flows
  14. Spontaneous stochasticity from a fixed-point RG attractor
  15. Ideal limit as a dynamical system
  16. ...and 9 more sections

Key Result

Theorem 1

For $N \ge 2$, the flows are related as where $\mathcal{R}$ is called the RG operator and is defined by the relations

Figures (11)

  • Figure 1: (a) Evolution of the fluctuating Sabra model for $\mathrm{Re} = 10^9$. Turbulent dynamics begins at the blowup time $t_b \approx 0.8$ (vertical line). (b) Turbulent state at the final time $T = 1.5$.
  • Figure 2: (a) Absolute shell velocities $|u_1(t)|$ in fifty realizations of the fluctuating Sabra model for $\mathrm{Re} = 10^9$ with the same initial and boundary conditions. (b) Standard deviations of $|u_1(t)|$ for the Reynolds numbers $\mathrm{Re} = 10^5,\ldots,10^9$. (c,d) PDFs of $\mathrm{Re}\,u_1(T)$ and $\mathrm{Re}\,u_2(T)$ at $T = 1.5$ for the same sequence of Reynolds numbers. The PDFs are calculated using the histogram approach for $4 \times 10^4$ simulations.
  • Figure 3: Universality of spontaneous stochasticity for three different regularizations: the fluctuating Sabra model (\ref{['eq1_5']}), the fluctuating LES model (\ref{['eq1_5SGS']}) and the auxiliary model (\ref{['eqRGV_1']}). Comparison of PDFs for (a) $\mathrm{Re}\,u_1(T)$ and (b) $\mathrm{Re}\,u_2(T)$.
  • Figure 4: Space-time lattice of the discrete-time shell model. Each arrow corresponds to one step of the integration algorithm, in which the time intervals are chosen adaptively at each shell $n$. This algorithm provides a first-order numerical scheme for the original Sabra model at the green nodes.
  • Figure 5: (a) Evolution of the discrete-time Sabra model for $\varepsilon = 0.1$ and $\mathrm{Re} = 10^9$. (b) PDFs of $\mathrm{Re}\,u_1(T)$ for $\mathrm{Re} = 10^7$ and different $\varepsilon = 0.5$, $0.1$ and $0.01$ converging to the PDF of the fluctuating Sabra model.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof