[Paper Review] Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations

Margaret Beck, C. Eugene Wayne|arXiv (Cornell University)|Aug 17, 2011

Fluid Dynamics and Turbulent Flows17 references4 citations

TL;DR

This paper provides a dynamical systems explanation for the metastability of bar states in the 2D Navier-Stokes equations on the torus. By approximating the linearized operator about bar states and showing rapid decay at rate $\mathcal{O}(e^{-\sqrt{\nu}t})$, it explains why these states persist for long times despite slow viscous decay ($\mathcal{O}(e^{-\nu t})$), with numerical evidence confirming the rate is optimal for the full linear operator.

ABSTRACT

Quasi-stationary, or metastable, states play an important role in two-dimensional turbulent fluid flows where they often emerge on time-scales much shorter than the viscous time scale, and then dominate the dynamics for very long time intervals. In this paper we propose a dynamical systems explanation of the metastability of an explicit family of solutions, referred to as bar states, of the two-dimensional incompressible Navier-Stokes equation on the torus. These states are physically relevant because they are associated with certain maximum entropy solutions of the Euler equations, and they have been observed as one type of metastable state in numerical studies of two-dimensional turbulence. For small viscosity (high Reynolds number), these states are quasi-stationary in the sense that they decay on the slow, viscous timescale. Linearization about these states leads to a time-dependent operator. We show that if we approximate this operator by dropping a higher-order, non-local term, it produces a decay rate much faster than the viscous decay rate. We also provide numerical evidence that the same result holds for the full linear operator, and that our theoretical results give the optimal decay rate in this setting.

Motivation & Objective

• To explain the long-lived, quasi-stationary behavior of bar states in 2D incompressible Navier-Stokes flows at high Reynolds numbers.
• To analyze the linearized dynamics about bar states and identify mechanisms for rapid convergence to these states.
• To demonstrate that an approximation of the linearized operator yields decay rates significantly faster than the viscous timescale.
• To provide numerical evidence that the theoretical decay rate is optimal for the full linearized operator.
• To lay the groundwork for extending the linear analysis to the full nonlinear Navier-Stokes equation.

Proposed method

• Approximate the time-dependent linearized operator about bar states by dropping a higher-order, non-local term, resulting in a simplified operator with analyzable spectral properties.
• Use spectral analysis to show that the simplified operator exhibits decay at rate $\mathcal{O}(e^{-\sqrt{\nu}t})$, much faster than the viscous decay rate $\mathcal{O}(e^{-\nu t})$.
• Apply a change of variables $w = \sqrt{1 + \Delta^{-1}} u$ to transform the linear operator into a form with a skew-symmetric component, facilitating spectral analysis.
• Perform numerical computations of eigenvalues for the full linearized operator to compare with the theoretical decay rate from the approximation.
• Use Fourier analysis and the Biot-Savart law to relate vorticity to velocity and define the bar state solutions explicitly in terms of cosine modes.
• Leverage the fact that bar states are maximum entropy solutions of the Euler equation, linking them to physically relevant long-lived states in 2D turbulence.

Experimental results

Research questions

• RQ1 Why do bar states in 2D Navier-Stokes flows persist for long times despite being unstable on the viscous timescale?
• RQ2 Can a dynamical systems approach explain the rapid convergence to bar states observed in numerical simulations?
• RQ3 Is the decay rate of $\mathcal{O}(e^{-\sqrt{\nu}t})$ in the linearized approximation optimal for the full linearized operator?
• RQ4 What is the role of non-self-adjointness in enabling fast decay modes in the linearized dynamics?
• RQ5 Can the analytical framework for bar states be extended to the nonlinear Navier-Stokes equation?

Key findings

• The simplified linearized operator about bar states exhibits decay at rate $\mathcal{O}(e^{-\sqrt{\nu}t})$, which is significantly faster than the viscous decay rate $\mathcal{O}(e^{-\nu t})$.
• Numerical computations of the full linearized operator's eigenvalues confirm that the decay rate $\mathcal{O}(e^{-\sqrt{\nu}t})$ is optimal, not just for the approximation but for the actual system.
• The invariant subspace for the approximate linear operator ensures that solutions converge rapidly to the bar state, providing a mechanism for metastability.
• The mechanism for fast decay differs from that in Burgers' equation, as it arises from small eigenvalues scaling with $\sqrt{\nu}$, not from large coefficients in eigenfunction expansions.
• The linearized operator is non-self-adjoint, which underlies the presence of fast-decaying modes and the separation of time scales.
• The results suggest a pathway to extending the analysis to the full nonlinear Navier-Stokes equation, though the time-dependent nature of the linearized operator complicates standard spectral projection techniques.