Characterization of Steady Solutions to the 2D Euler Equation

Abstract

bstract Steady fluid flows have very special topology. In this paper we describe necessary and sufficient conditions on the vorticity function of a 2D ideal flow on a surface with or without boundary, for which there exists a steady flow among isovorticed fields. For this we introduce the notion of an antiderivative (or circulation function) on a measured graph, the Reeb graph associated to the vorticity function on the surface, while the criterion is related to the total negativity of this antiderivative. It turns out that given topology of the vorticity function, the set of coadjoint orbits of the symplectomorphism group admitting steady flows with this topology forms a convex polytope. As a byproduct of the proposed construction, we also describe a complete list of Casimirs for the 2D Euler hydrodynamics: we define generalized enstrophies which, along with circulations, form a complete set of invariants for coadjoint orbits of area-preserving diffeomorphisms on a surface.

A characterization of 3D steady Euler flows using commuting zero-flux homologies

“…Etnyre and Ghrist, developing an idea suggested by Sullivan, showed the equivalence between Reeb flows of a contact form and non-vanishing Beltrami fields with constant proportionality factor [5,6]; the case of general Beltrami fields corresponds to volume-preserving geodesible flows (or Reeb flows of stable Hamiltonian structures), as noticed by Rechtman [12,13]. More recently, Cieliebak and Volkov [4] constructed steady Euler flows that are not geodesible, and Izosimov and Khesin [8] characterized the vorticity functions of 2-dimensional steady Euler flows using Reeb graphs. Nevertheless, we are still far from having a deep understanding of the space of stationary solutions to the Euler equations.…”

Classification of Casimirs in 2D Hydrodynamics

“…The correspondence between abstract measured Reeb graphs and those corresponding to functions on surfaces with boundary now should include compatibility conditions for the surface and the Reeb graph beyond equality of total volumes ş Γ dµ " ş M ω and of the corresponding homology, as it should account for certain discrete information discussed above. Note that the would-be compatibility condition should be consistent with that for the case of Morse functions constant on boundary components and considered in [5] as a limiting case. (Formally speaking, for simple Morse functions constants on the boundary have to be ruled out and can be considered only in the limit, since the restriction of functions to the boundary is to be Morse.…”

“…Finally, note that all objects in the present paper are infinitely smooth (see the case of finite smoothness in [6]). To the best of our knowledge, a complete description of Casimirs in 2D fluid dynamics has not previously appeared in the literature in a self-contained form, while various partial results could be found in [3,9,10,11,5]. In the last section we present a few examples, show how the main notions can be extended to the case of surfaces with boundary, emphasize the main difficulties and formulate open questions in the latter setting.…”

On zonal steady solutions to the 2D Euler equations on the rotating unit sphere

“…The study on the local structure of steady solutions has also attracted recent attention, see [10,11,21,28]. The domain in which the motion takes place actually plays an important role in the structure of the stationary solutions and its properties.…”