Transition from axisymmetric to three-dimensional turbulence

Abstract

In purely axisymmetric turbulence sustained by a linear forcing mechanism, a stable, entirely poloidal flow is observed when the toroidal component of the forcing is below a threshold. We investigate using numerical simulations whether this state persists when toroidal variations of the flow are continuously reintroduced and the forcing is purely poloidal. It is shown how the pressure–strain correlation allows the redistribution of the energy towards the toroidal component. A simple statistical model allows to capture the main physical effects on the level of the global energy balance. This model is then used to investigate the stability of the poloidal state for various toroidal-to-poloidal forcing strengths and different degrees of axisymmetry.

Keywords:

• Axisymmetric turbulence
• non-integer dimension
• transition
• statistical modelling

Acknowledgments

Eric Serre is gratefully acknowledged for providing the numerical code. Z.Q. was supported by a grant from the China Scholarship Council. All simulations were carried out using the facilities of the PMCS2I (École Centrale de Lyon). Laurent Pouilloux and Dan-Gabriel Calugaru are acknowledged for technical assistance.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Another interesting limit is $\alpha \to \mathrm{\infty }$. Indeed, in this limit the system might also tend to an axisymmetric system, since all modes with variations in the θ direction will attenuate through diffusion so rapidly that variations are smoothened out. We have not checked this conjecture and focus here on the range between $\alpha =0$ and $\alpha =1$ where we know the precise behaviour of the limiting cases.

Additional information

Funding

Zecong Qin was supported by a grant from the China Scholarship Council.