Chaos, statistics and inverse cascades in turbulent flows
Describing turbulence has been one of the most important unsolved problems of physics for the last two centuries. Multiple attempts have been made and yet there is no successful theory of turbulence to date. The most common approach to describe turbulence is the statistical one, nevertheless, ideas from other fields such as dynamical system theory, field theories and critical phenomena have re-emerged over the past years. In this work we carry out a series of studies that use ideas from dynamical systems to describe the chaotic properties of hydrodynamic and magnetohydrodynamic turbulence. For this, we use direct numerical simulations (DNS) and a numerical implementation of the EDQNM closure to study the relation between the Lyapunov exponents and different flow characteristics such as Reynolds number and space dimensionality. We perform a thorough numerical analysis of the statistical properties of Lyapunov exponents in homogeneous and isotropic turbulence. We also look at the relation between the Lyapunov exponents and the Reynolds number λ ∼ Re1/2, that was established by David Ruelle in 1979. DNS show that Ruelle’s relation holds, although corrections due to intermittency effects are not observed. We also see that the Lyapunov exponents are a robust measure of the system, which remains stable even for underresolved simulations. We also look at the Lyapunov exponents behaviour for varying spatial dimension. We find that these decrease with increasing dimension. Using the EDQNM closure equations, we find that there is a critical dimension dc ≈ 5.8 above which the fluid is no longer chaotic. Additionally, we study the Lyapunov exponent scaling behaviour while varying the aspect ratio of a rectangular lattice. We see a sudden transition between two and three-dimensional phenomenology, contrary to the smoother transition observed when looking at the two and three-dimensional energy components. Last, we study the decay of turbulent magnetohydrodynamic fluids. Magnetohydrodynamic (MHD) equations are the result of coupling the Maxwell’s equations to the N-S equations. This gives place to a rich phenomenology and gives an appropriate description of many astrophysical systems. In particular, decaying MHD is a candidate to explain the evolution of the large scale cosmic magnetic fields observed in the universe. The transfer from small to large scales of magnetic energy is common in helical flows. However, recent numerical studies showed a strong inverse transfer of magnetic energy also in nonhelical flows. The properties of this inverse transfer are not yet understood and different codes show slightly different properties. We perform a complete analysis, where our results shows a present but not strong inverse energy transfer of nonhelical flows. In addition, we find that this inverse transfer grows with increasing Prandtl number, contrary to what is observed in recent literature.
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