The conditional Lyapunov exponents and synchronisation of rotating turbulent flows

The synchronisation between rotating turbulent flows in periodic boxes is investigated numerically. The flows are coupled via a master–slave coupling, taking the Fourier modes with wavenumber below a given value km as the master modes. It is found that synchronisation happens when km exceeds a threshold value kc, and kc depends strongly on the forcing scheme. In rotating Kolmogorov flows, kcη does not change with rotation in the range of rotation rates considered, η being the Kolmogorov length scale. Even though the energy spectrum has a steeper slope, the value of kcη is the same as that found in isotropic turbulence. In flows driven by a forcing term maintaining constant energy injection rate, synchronisation becomes easier when rotation is stronger. Here, kcη decreases with rotation, and it is reduced significantly for strong rotations when the slope of the energy spectrum approaches −3. It is shown that the conditional Lyapunov exponent for a given km is reduced by rotation in the flows driven by the second type of forcing, but it increases mildly with rotation for the Kolmogorov flows. The local conditional Lyapunov exponents fluctuate more strongly as rotation is increased, although synchronisation occurs as long as the average conditional Lyapunov exponents are negative. We also look for the relationship between kc and the energy spectra of the Lyapunov vectors. We find that the spectra always seem to peak at approximately kc, and synchronisation fails when the energy spectra of the conditional Lyapunov vectors have a local maximum in the slaved modes.