## MEMBERS

### Program Members

From a mathematical point of view, fluid motions are composed of vortices and waves. Typical examples of vortical structure include the patterns of two long lines of clouds behind an aircraft wing and a column of air bubbles formed above the drain hole in a Japanese bathtub. If a flow did not have vorticity, then it would be virtually identical to the motion of a rigid body in that its motion would be instantaneously determined once its boundary is given. However, it is vorticity that enables vigorous motion of a fluid with complicated spatial patterns. Many flows are in a turbulent state. Turbulence is a dynamical system with infinite degrees of freedom where various vortices of different sizes form a hierarchical structure, nonlinearly interacting with one another.

My current research interest lies in mathematical analyses of vortex motion. In particular, I have a few results, in the theory of three-dimensional vortex motion, that precede other groups. In 1994, I became the first recipient of the Ryuumon Award, which is given to a young researcher from the Japan Society of Fluid Mechanics, for our work on three-dimensional motion of a vortex filament. Moreover, we recently succeeded in deriving formulas for the traveling velocity of a vortex ring, which agree well with experiments. These formulas covers not only high Reynolds number but also low Reynolds number regions. In addition, we discovered a new unstable mechanism of a vortex ring, and my coauthor received the Ryuumon Award for this work (2006). Our group is currently developing a new theory on the continuous spectra of a vortex and on nonlinear interactions between a vortex and waves.

Vortices are the source for instability of flows. They also play a significant role in the attenuation of kinetic energy. Vortices are responsible for enhancing or suppressing mixing of different substances, increasing or decreasing drag and lift forces acting on moving bodies, and generating aeolian tone. Over a diversity of field as represented by micro-machines, flows in and around living organisms, automobiles, aircrafts and chemical plants, the manufacturing industry demands the most advanced fluid dynamics. Fluid dynamics is recently expanding its frontier to environmental issues, such as wind power generation. As fluid dynamics continues to encompass a broad range of fields, including engineering, earth science, physics, and mathematics, I am working in a trans-disciplinary field. Albeit advances in flow measurement techniques and development of computers by which complicated flow structure is captured, it is human beings who use these advances to extract meanings from the data. Now that the quality of data has been improved, we are requested to develop mathematical treatment of vortex motion and wave dynamics as the Math-for-Industry, which enables deep mathematical structures to be extracted

Euler first developed an approach to analyze fluid motion using partial differential equations in the 18th century, but an entire century passed before Helmholtz published his seminal paper (1858) opening up the research field of vortex motion. Helmholtz demonstrated that “in the absence of viscosity, vortex lines are frozen into the fluid”. This implies that link and knot types of vortex lines do not change with time. One of the thrusts in fluid dynamics in the latter half of the 20th century was to dig up topological meaning of Helmholtz’s laws and to find its applications as initiated by Arnol’d (1966). However, this treatment is as yet limited to two-dimensional flows.

Conventional fluid dynamics was built upon the framework of the Eulerian description, which sets aside a particle picture and starts from a differential relationship between quantities of the spatial-temporal field. This may be regarded as the origin of nonlinearity in flow equations, producing discrepancies from the particle treatment. On the other hand, the Lagrangian approach takes the displacement field of fluid particles as the basic variables, and is capable of treating vortex motion with rigorously maintaining topological invariants. The latter approach provides a common ground to treat motions of molecules, a solid body, fluid, elastic body and even plasma. The Lagrangian description has a high degree of extensibility. A macroscopic system is complex. My group’s research goal is to establish a mathematical framework in which interactions between waves and average flows can be calculated.

As regards education, I exploit the style of freely and flexibly incorporating mathematical methods based on a knowledge of physics. A substantial effort is made on training graduate students. In addition, students join my group from abroad. Beginning in 2006, the Graduate School of Mathematics introduced a long-term internship, and two graduate students from my group took part in this program, both of whom received an excellent evaluation. One of them was offered an intern at a manufacturing firm, and this student was so successful that the company asked him to join the firm before receiving a doctor’s degree.

My group also actively conducts international interchanges of researchers. I won Visiting Professorship of the Japan Society for the Promotion of Science (JSPS) (Long Term) and visited University of Cambridge for ten months in 1996 to conduct collaboration with Prof. Moffatt on motion of a vortex ring. After my return to Japan, prominent, world-renowned researchers have frequently visited my group to exchange information on the research frontiers. Since 2001, my group has hosted five researchers under the Invitation Fellowships Program (Short Term) of the JSPS. These activities have led us to grow an international network of world-class researchers on Topological Fluid Mechanics, including Moffatt, the pioneer, Ricca (University of Milan), Holm (Imperial College London) and Khesin (University of Toronto). I intend on leveraging this network to train young researchers.