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Register a new userThe Godbillon-Vey Invariant as Topological Vorticity Compression and Obstruction to Steady Flow in Ideal Fluids
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Thomas Machon
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If the vorticity field of an ideal fluid is tangent to a foliation, additional conservation laws arise. For a class of zero-helicity vorticity fields the Godbillon-Vey (GV) invariant of foliations is defined and is shown to be an invariant purely of the vorticity, becoming a higher-order helicity-type invariant of the flow. GV non-zero gives both a global topological obstruction to steady flow and, in a particular form, a local obstruction. GV is interpreted as helical compression and stretching of vortex lines. Examples are given where the value of GV is determined by a set of distinguished closed vortex lines.
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We show the Godbillon-Vey invariant arises as a `restricted Casimir invariant for three-dimensional ideal fluids associated to a foliation. We compare to a finite-dimensional system, the rattleback, where analogous phenomena occur.
Recently F. Huang [Commun. Theor. Phys. V.42 (2004) 903] and X. Tang and P.K. Shukla [Commun. Theor. Phys. V.49 (2008) 229] investigated symmetry properties of the barotropic potential vorticity equation without forcing and dissipation on the beta-plane. This equation is governed by two dimensionless parameters, $F$ and $beta$, representing the ratio of the characteristic length scale to the Rossby radius of deformation and the variation of earth angular rotation, respectively. In the present paper it is shown that in the case $F e 0$ there exists a well-defined point transformation to set $beta = 0$. The classification of one- and two-dimensional Lie subalgebras of the Lie symmetry algebra of the potential vorticity equation is given for the parameter combination $F e 0$ and $beta = 0$. Based upon this classification, distinct classes of group-invariant solutions is obtained and extended to the case $beta e 0$.
In a particle physics dynamics, we assume a uniform distribution as the physical measure and a measure-theoretic definition of entropy on the velocity configuration space. This distribution is labeled as the physical solution in the remainder of the article. The dynamics is governed by an assumption of a Lagrangian formulation, with the velocity time derivatives as the momenta conjugate to the velocity configurations. From these definitions and assumptions, we show mathematically that a maximum entropy production principle selects the physical measure from among alternate solutions of the Navier-Stokes and Euler equations, but its transformation to an Eulerian frame is not established here, a topic that will be considered separately.
We prove a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary-foliation; this is a secondary invariant for longitudinal Dirac operators on type-III foliations. Moreover, employing the Godbillon-Vey index as a pivotal example, we explain a new approach to higher index theory on geometric structures with boundary. This is heavily based on the interplay between the absolute and relative pairings of K-theory and cyclic cohomology for an exact sequence of Banach algebras which in the present context takes the form $0to Jto Ato Bto 0$, with J dense and holomorphically closed in the C^*-algebra of the foliation and B depending only on boundary data. Of particular importance is the definition of a relative cyclic cocycle $(tau_{GV}^r,sigma_{GV})$ for the pair $Ato B$; $tau_{GV}^r$ is a cyclic cochain on A defined through a regularization, `a la Melrose, of the usual Godbillon-Vey cyclic cocycle $tau_{GV}$; $sigma_{GV}$ is a cyclic cocycle on B, obtained through a suspension procedure involving $tau_{GV}$ and a specific 1-cyclic cocycle (Roes 1-cocycle). We call $sigma_{GV}$ the eta cocycle associated to $tau_{GV}$. The Atiyah-Patodi-Singer formula is obtained by defining a relative index class $Ind (D,D^partial)in K_* (A,B)$ and establishing the equality <Ind (D),[tau_{GV}]>=<Ind (D,D^partial), [tau^r_{GV}, sigma_{GV}]>$. The Godbillon-Vey eta invariant $eta_{GV}$ is obtained through the eta cocycle $sigma_{GV}$.
We derive general conditions of slip of a fluid on the boundary. Under these conditions the velocity of the fluid on the immovable boundary is a function of the normal and tangential components of the force acting on the surface of the fluid. A problem on stationary flow of an electrorheological fluid in which the terms of slip are specified on one part of the boundary and surface forces are given on the other is formulated and studied. Existence of a solution of this problem is proved by using the methods of penalty functions, monotonicity and compactness. It is shown that the method of penalty functions and the Galerkin approximations can be used for the approximate solution of the problem under consideration.
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