Comment on Geometrical Control of Active Turbulence in Curved Topographies

TL;DR

This paper critiques a recent study on active turbulence on a torus, clarifying the limitations of the numerical methods used and providing counterexamples to the original conclusions.

Contribution

It identifies issues with the vorticity-stream function approach on non-simply connected surfaces and offers clarifications and counterexamples to the previous findings.

Findings

Vorticity-stream function approach is not suitable for torus geometries.

Counterexamples demonstrate limitations of the original method.

Clarification of the mathematical and numerical issues involved.

Abstract

In the recent letter Pearce et. al. Phys. Rev. Lett. 122, 168002 (2019) the authors investigate the turbulent dynamics of a two-dimensional active nematic liquid crystal which is constrained to the surface of a torus. The underlying model combines an incompressible surface Navier-Stokes equation with friction and active forcing with a surface Landau-de Gennes model for nematic liquid crystals. To numerically solve the surface Navier-Stokes equation a vorticity-stream function approach is considered. This approach is not appropriate on surfaces that are not simply connected - such as the considered torus - due to non-trivial harmonic vector fields. We will explain the underlying situation and provide details and examples to rebut the argumentation in Pearce et. al. Phys. Rev. Lett. 122, 168002 (2019).