A new mathematical framework can shed light on small-scale turbulent flow

Unlocking the mysteries of synchronization in turbulent dynamics.

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The importance of weather forecasting is well known. It relies on predicting air movement in the atmosphere, characterized by turbulent flows resulting in chaotic eddies of air. 

Scientists use a method relying on Data Assimilation (DA) to predict atmospheric turbulence. This approach integrates various information sources to infer details about small-scale turbulent eddies from larger ones. 

A crucial parameter called the critical length scale is identified within DA, representing the point where all relevant small-scale eddy information can be extrapolated from larger ones. Despite consensus on a common value for this critical scale, its origin, and relationship with Reynold’s number, indicating turbulence level, still need to be clarified. 

A team of researchers, led by Associate Professor Masanobu Inubushi from the Tokyo University of Science, Japan, has recently proposed a mathematical framework to address this issue. Researchers can shed light on small-scale turbulent flow through this framework to better understand the turbulence phenomena. 

They treated the process of DA as a stability problem. Researchers have theoretically explained the critical scale in atmospheric turbulence by framing the phenomenon as the ‘synchronization of a small vortex by a large vortex.’ This explanation, rooted in the ‘stability problem of synchronized manifolds,’ marks the first successful theoretical understanding of the critical scale in predicting turbulent flows. 

In a cross-disciplinary approach, the research team combined chaos theory and synchronization theory to explore the critical length scale in atmospheric turbulence prediction. Focusing on the Data Assimilation (DA) manifold, they conducted a stability analysis, revealing that the critical length scale is crucial for successful DA. 

The study highlighted the importance of transverse Lyapunov exponents (TLEs) in determining the DA process’s success or failure. Notably, the researchers concluded that the critical length scale increases with the Reynolds number, providing clarity on its dependence. 

Associate Professor Masanobu Inubushi from the Tokyo University of Science, Japan, said“This new theoretical framework has the potential to significantly advance turbulence research in critical problems such as unpredictability, energy cascade, and singularity, addressing a field that physicist Richard P. Feynman once described as ‘one of the remaining difficulties in classical physics.” 

Journal Reference:

  1. Masanobu Inubushi, Yoshitaka Saiki, Miki U. Kobayashi, and Susumu Goto. Characterizing Small-Scale Dynamics of Navier-Stokes Turbulence with Transverse Lyapunov Exponents: A Data Assimilation Approach. Physical Review Letters. DOI: 10.1103/PhysRevLett.131.254001