Noise-expansion cascade: an origin of randomness of turbulence

Abstract

Randomness is one of the most important characteristics of turbulence, but its origin remains an open question. By means of a ``thought experiment'' via several clean numerical experiments based on the Navier-Stokes equations for two-dimensional turbulent Kolmogorov flow, we reveal a new phenomenon, which we call the ``noise-expansion cascade'' whereby all micro-level noises/disturbances at different orders of magnitudes in the initial condition of Navier-Stokes equations enlarge consistently, say, one by one like an inverse cascade, to macro-level. More importantly, each noise/disturbance input may greatly change the macro-level characteristics and statistics of the resulting turbulence, clearly indicating that micro-level noise/disturbance might have great influence on macro-level characteristics and statistics of turbulence. Besides, the noise-expansion cascade closely connects randomness of micro-level noise/disturbance and macro-level disorder of turbulence, thus revealing an origin of randomness of turbulence. This also highly suggests that unavoidable thermal fluctuations must be considered when simulating turbulence, even if such fluctuations are several orders of magnitudes smaller than other external environmental disturbances. Hopefully, the ``noise-expansion cascade'' as a fundamental property of the NS equations could greatly deepen our understandings about turbulence, and besides is helpful for attacking the fourth millennium problem posed by Clay Mathematics Institute in 2000.