Clebsch confinement and instantons in turbulence

The turbulence in incompressible fluid is represented as a field theory in 3 dimensions. There is no time involved, so this is intended to describe stationary limit of the Hopf functional. The basic fields are Clebsch variables defined modulo gauge transformations (symplectomorphisms). Explicit formulas for gauge invariant Clebsch measure in space of generalized Beltrami flow compatible with steady energy flow are presented. We introduce a concept of Clebsch confinement related to unbroken gauge invariance and study Clebsch instantons: singular vorticity sheets with nontrivial helicity. This is realization of the “instantons and intermittency” program we started back in the 1990s.1 These singular solutions are involved in enhancing infinitesimal random forces at remote boundary leading to critical phenomena. In the Euler equation vorticity is concentrated along the random self-avoiding surface, with tangent components proportional to the delta function of normal distance. Viscosity in Navier–Stokes equation smears this delta function to the Gaussian with width [Formula: see text] at [Formula: see text] with fixed energy flow. These instantons dominate the enstrophy in dissipation as well as the PDF for velocity circulation [Formula: see text] around fixed loop [Formula: see text] in space. At large loops, the resulting symmetric exponential distribution perfectly fits the numerical simulations2 including pre-exponential factor [Formula: see text]. At small loops, we advocate relation of resulting random self-avoiding surface theory with multi-fractal scaling laws observed in numerical simulations. These laws are explained as a result of fluctuating internal metric (Liouville field). The curve of anomalous dimensions [Formula: see text] can be fitted at small [Formula: see text] to the parabola, coming from the Liouville theory with two parameters [Formula: see text], [Formula: see text]. At large [Formula: see text] the ratios of the subsequent moments in our theory grow linearly with the size of the loop, which corresponds to finite value of [Formula: see text] in agreement with DNS.

A variational formulation for steady surface water waves on a Beltrami flow

This paper considers steady surface waves ‘riding’ a Beltrami flow (a three-dimensional flow with parallel velocity and vorticity fields). It is demonstrated that the hydrodynamic problem can be formulated as two equations for two scalar functions of the horizontal spatial coordinates, namely the elevation η of the free surface and the potential Φ defining the gradient part (in the sense of the Hodge–Weyl decomposition) of the horizontal component of the tangential fluid velocity there. These equations are written in terms of a non-local operator H ( η ) mapping Φ to the normal fluid velocity at the free surface, and are shown to arise from a variational principle. In the irrotational limit, the equations reduce to the Zakharov–Craig–Sulem formulation of the classical three-dimensional steady water-wave problem, while H ( η ) reduces to the familiar Dirichlet–Neumann operator.