The irregular, swirling motion of fluids we call turbulence can be found everywhere, from stirring in a teacup to currents in the planetary atmosphere. This phenomenon is governed by the Navier–Stokes equations—a set of mathematical equations that describe how fluids move. Despite being known for nearly two centuries, these equations still pose major challenges when it comes to making predictions. Turbulent flows are inherently chaotic, and tiny uncertainties can grow quickly over time. In real-world situations, scientists can only observe part of a turbulent flow, usually its largest and slowest moving features. Thus, a long-standing question in fluid physics has been whether these partial observations are enough to reconstruct the full motion of the fluid.
Researchers studying three-dimensional turbulence, such as the kind found in smoke or stirred water, or air flow around a moving car, have made substantial progress on this topic over the past few decades. They showed that if one continuously observes the flow down to a sufficiently fine scale, it is possible to mathematically recover the smaller unobserved motions. In these systems, however, the required level of detail is very high; observations must extend down to extremely small scales where energy from turbulence is lost as heat. Whether the same idea applies to two-dimensional turbulence, which behaves very differently, has remained largely unclear and the comparative studies on two-dimensional and three-dimensional studies remain unexplored so far.
Against this backdrop, Associate Professor Masanobu Inubushi from the Department of Applied Mathematics, Tokyo University of Science, Japan, and Professor Colm-Cille Patrick Caulfield from the Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK, conducted a study to shed light on this problem. This research was conducted during Dr. Inubushi’s overseas research at the University of Cambridge. Their work, made available online on January 22, 2026 and published in Volume 1,027 of the Journal of Fluid Mechanics on January 25, 2026, focuses on a well-established mathematical model of two-dimensional turbulence, along with a comparative study on three-dimensional flows, and involves numerical simulations to test how much observational detail is needed to reconstruct the full flow. Notably, their study has been selected as the cover article of the journal , highlighting its impact.
It is important to understand that two-dimensional turbulence is not just a simplified version of the three-dimensional case. Instead of energy cascading mainly toward smaller and smaller swirls, it can also move in the opposite direction, from small scales to large ones. This difference underlies many large-scale features of weather and ocean circulation that are simply not seen in three-dimensional systems.
To tackle the problem, the researchers used a technique called data assimilation, which dynamically combines observational data with mathematical models. Simply put, they assumed that the large-scale motion of the fluid was known from observations, while the smaller-scale motion is initially unknown. They then tested whether the small scales can be recovered over time by letting the equations evolve. To measure whether this reconstruction succeeds in a robust way, they relied on tools from chaos theory known as Lyapunov exponents, which quantify how fast errors grow or shrink in a dynamical system.
Their results revealed a clear and surprising difference between two- and three-dimensional turbulence. In the two-dimensional case, the team found that it is enough to observe the flow only down to the scale at which energy is injected into the system. Unlike three-dimensional systems, observations do not need to reach down to the tiniest scales of discernible motion. As Dr. Inubushi explains, “The present study initiates a new direction of research into two-dimensional turbulence by introducing a novel approach based on synchronization. Through the use of data assimilation and Lyapunov analysis, we demonstrated that the ‘essential resolution’ of observations for flow field reconstruction in forced two-dimensional turbulence is surprisingly lower than the equivalent essential resolution in forced three-dimensional turbulence.”
In essence, in two-dimensional turbulence, the large-scale structures contain enough information to determine the smaller ones. The researchers attribute this to the way information moves across scales in two dimensions, where interactions between large and small motions are stronger and more direct than in three dimensions.
Although this study is theoretical, its implications do extend beyond mathematics. Two-dimensional turbulence is a key element in simplified models of the atmosphere and oceans. Understanding how much information is needed to accurately reconstruct flows in such systems can help guide future approaches to modelling and prediction. “Predicting fluid motion in the atmosphere and oceans is important for everyday applications such as weather forecasting,” notes Dr. Inubushi.
By providing fresh insights into the Navier–Stokes equations, this work provides a stronger foundation for future advances in climate modelling, data-driven forecasting, and a broader understanding of fluid motion. The results may inform future weather forecasting approaches. In particular, the study shows, in a highly idealized setting, that large-scale observations can be sufficient to infer smaller-scale flow structures, which is a key issue for prediction in the presence of the so-called butterfly effect.
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