Data Assimilation (DA) is an important mathematical method for predicting turbulent flows for weather forecasting. However, the origins of the critical length scale, a crucial parameter in this method, and its dependence on the Reynolds number are not well understood. Now, researchers have developed a novel theoretical framework that treats DA as a stability problem to explain this parameter. This framework can contribute significantly to turbulence research and inspire novel data-driven methods to predict turbulence.
Weather forecasting is important for various sectors, including agriculture, military operations, and aviation, as well as for predicting natural disasters like tornados and cyclones. It relies on predicting the movement of air in the atmosphere, which is characterized by turbulent flows resulting in chaotic eddies of air. However, accurately predicting this turbulence has remained significantly challenging owing to the lack of data on small-scale turbulent flows, which leads to the introduction of small initial errors. These errors can, in turn, lead to drastic changes in the flow states later, a phenomenon known as the chaotic butterfly effect.
To address the challenge of limited data on small-scale turbulent flows, a data-driven method known as Data Assimilation (DA) has been employed for forecasting. By integrating various sources of information, this approach enables the inference of details about small-scale turbulent eddies from their larger counterparts. Notably, within the framework of DA methods, a crucial parameter known as the critical length scale has been identified. This critical length scale represents the point below which all relevant information about small-scale eddies can be extrapolated from the larger ones. Reynold’s number, an indicator of the turbulence level in fluid flow, plays a pivotal role in this context, with higher values suggesting increased turbulence. However, despite the consensus generated by numerous studies regarding a common value for the critical scale, an explanation of its origin and its relationship with Reynold’s number remains elusive. More