Topological Fluid Dynamics

T opological fluid dynamics is a young mathematical discipline that studies topological features of flows with complicated trajectories and their applications to fluid motions, and develops group-theoretic and geometric points of view on various problems of hydrodynamical origin. It is situated at a crossroads of several disciplines, including Lie groups, knot theory, partial differential equations, stability theory, integrable systems, geometric inequalities, and symplectic geometry. Its main ideas can be traced back to the seminal 1966 paper [1] by V. Arnold on the Euler equation for an ideal fluid as the geodesic equation on the group of volume-preserving diffeomorphisms. One of the most intriguing observations of topological fluid dynamics is that one simple construction in Lie groups enables a unified approach to a great variety of different dynamical systems, from the simple (Euler) equation of a rotating top to the (also Euler) hydrodynamics equation, one of the most challenging equations of our time. A curious application of this theory is an explanation of why long-term dynamical weather forecasts are not reliable: Arnold’s explicit estimates related to curvatures of diffeomorphism groups show that the earth weather is essentially unpredictable after two weeks as the error in the initial condition grows by a factor of 105 for this period, that is, one loses 5 digits of accuracy. (Ironically, 15 day(!) weather forecasts for any country in the world are now readily available online at www.accuweather.com.) Another application is related to the Sakharov–Zeldovich problem on whether a neutron star can extinguish by “reshaping” and turning to radiation the excessive magnetic energy. In this introductory article we will touch on these and several other purely mathematical problems motivated by fluid mechanics, referring the interested reader to the book [4] for further details and the extensive bibliography.