Physicists at HSE University Reveal How Vortices Behave in Two-Dimensional Turbulence
Researchers from the Landau Institute for Theoretical Physics of the Russian Academy of Sciences and the HSE University's Faculty of Physics have discovered how external forces affect the behaviour of turbulent flows. The scientists showed that even a small external torque can stabilise the system and extend the lifetime of large vortices. These findings may improve the accuracy of models of atmospheric and oceanic circulation. The paper has been published in Physics of Fluids.
Turbulence is the seemingly random motion of water or air that arises when streams of fluid begin to move at very high speeds. In a turbulent state, flows continuously mix and fragment, with vortices breaking down into smaller ones, sometimes dissipating completely or, conversely, merging into larger structures.
Although scientists understand the conditions under which turbulent flows arise, describing their behaviour and evolution requires complex mathematical models with many parameters. To better grasp the dynamics of real systems—such as the Earth's atmosphere and oceans—researchers often simplify the problem by modelling turbulence on a two-dimensional plane rather than in three-dimensional space.
The laws governing turbulence in three-dimensional and two-dimensional systems differ fundamentally. In three-dimensional turbulence, energy moves through a direct cascade: large flows break down into smaller ones, whose energy is eventually dissipated as heat. In contrast, turbulence in flat systems behaves differently—a two-dimensional configuration drives energy through an inverse cascade, where small vortices tend to merge into larger ones.
Vladimir Parfenyev, Research Fellow at the Landau Institute for Theoretical Physics of the Russian Academy of Sciences and at the HSE Laboratory for Condensed Matter Physics, and Alisa Shikanian, master's student at the HSE Faculty of Physics, modelled the behaviour of turbulent vortices on a plane, studying the dynamics of a fluid confined within a square cell.
They investigated a scenario in which a constant external torque was applied to a two-dimensional system, as if the flows were being gently twisted from the outside. The results showed that even a very small external torque can extend the lifetime of large vortices and stabilise the system’s behaviour.
Mathematical modelling made it possible to determine how the thickness of the boundary layer near the walls—where energy is dissipated—depends on the parameters of the system under study. In the simulation, the fluid velocity near the boundaries behaved similarly to that observed in laboratory experiments with soap films conducted by other research groups: it increased with distance from the walls according to the same type of pattern. At the same time, large vortices formed within the system, and the inverse energy cascade continued until it reached the full size of the system.
Vladimir Parfenyev
'The most important result for us is the clear understanding that friction of the fluid against the boundaries alone is not enough to stop the inverse energy cascade in a two-dimensional system. Fluid vortices will always tend to merge into larger structures. One way or another, energy in the system accumulates on a large scale—this is how order emerges from chaos,’ says Vladimir Parfenyev.
The scientists’ findings deepen our understanding of the processes that govern the formation of large-scale structures in two-dimensional turbulence and provide a solid foundation for future research in this field. Geoscientists can use these results to improve predictive models of oceanic and atmospheric currents.
The research was conducted with support from the Russian Science Foundation, Project 23-72-30006.
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