The global attractor of the Toner-Tu-Swift-Hohenberg equations of active turbulence and its properties

TL;DR

This paper proves the existence of a finite-dimensional global attractor for the Toner-Tu-Swift-Hohenberg equations modeling active turbulence, providing theoretical insights and numerical validation for vortex length scale phenomena in bacterial turbulence.

Contribution

It establishes the existence of a finite-dimensional global attractor for the TTSH equations and links the attractor's properties to the Swift-Hohenberg length scale, supported by numerical simulations.

Findings

Existence of a finite-dimensional global attractor for TTSH equations.

Explicit estimates for the Lyapunov dimension of the attractor.

Numerical simulations confirm theoretical bounds and reveal vortex length scale phenomena.

Abstract

The Toner-Tu-Swift-Hohenberg (TTSH) equations are one of the basic equations that are used to model turbulent behaviour in active matter, specifically the swarming of bacteria in suspension. They combine features of the incompressible Navier-Stokes, the Toner-Tu and Swift-Hohenberg equations, together with the important properties that they are linearly driven, and that the Laplacian diffusion is taken to be negative in combination with hyper-dissipation. We prove that the TTSH equations possess a finite-dimensional compact global attractor on the periodic domain $\mathbb{T}^d$ ($d=2,3$) and we establish explicit estimates for its Lyapunov dimension which agree with the heuristic prediction based on the Swift-Hohenberg length scale. The predominance of this length scale (as a vortex length scale) has been observed in both numerical and experimental studies of bacterial turbulence, so our methods and results provide a rigorous theoretical foundation for this phenomenon. We also carry out pseudospectral direct numerical simulations of these PDEs in dimension $d=2$ through which we obtain Lyapunov spectra for representative parameter values. We show that our numerical results are consistent with the analytically derived rigorous bounds.