Intrinsic relationship between synchronisation thresholds and Lyapunov vectors: evidence from large eddy simulations and shell models

An important parameter characterising the synchronisation of turbulent flows is the threshold coupling wavenumber. This study investigates the relationship between the threshold coupling wavenumber and the leading Lyapunov vector using large eddy simulations and the SABRA model. Various subgrid-scale stress models, Reynolds numbers and different coupling methods are examined. A new scaling relation is identified for the leading Lyapunov exponents in large eddy simulations, showing that they approximate those of filtered direct numerical simulations. This interpretation provides a physical basis for results related to the Lyapunov exponents of large eddy simulations, including those related to synchronisation. Synchronisation experiments show that the peak wavenumber of the energy spectrum of the leading Lyapunov vector coincides with the threshold coupling wavenumber, in large eddy simulations of box turbulence with standard Smagorinsky or dynamic mixed models as well as in the SABRA model, replicating results from direct numerical simulations of box turbulence. Although the dynamic Smagorinsky model exhibits different behaviour, the totality of the results suggests that the relationship is an intrinsic property of a certain class of chaotic systems. We also confirm that conditional Lyapunov exponents characterise the synchronisation process in indirectly coupled systems as they do in directly coupled ones, with their values insensitive to the nature of the master flow. These findings advance the understanding of the role of the Lyapunov vector in the synchronisation of turbulence.