Free-Surface Variational Principle for an Incompressible Fluid with Odd Viscosity

Abstract

We present variational and Hamiltonian formulations of incompressible fluid dynamics with free surface and nonvanishing odd viscosity. We show that within the variational principle the odd viscosity contribution corresponds to geometric boundary terms. These boundary terms modify Zakharov's Poisson brackets and lead to a new type of boundary dynamics. The modified boundary conditions have a natural geometric interpretation describing an additional pressure at the free surface proportional to the angular velocity of the surface itself. These boundary conditions are believed to be universal since the proposed hydrodynamic action is fully determined by the symmetries of the system.Introduction. Variational principle in hydrodynamics have a long history. We refer to Ref.[1] and references therein for an introduction to the topic. In particular, the Luke's variational principle (LVP) is a variational principle of an inviscid and incompressible fluid with a free surface [2,3]. LVP provides both bulk hydrodynamic equations for an irrotational flow as well as kinematic and dynamic boundary conditions at the free surface boundary [3]. Such principle was later extended to include surface tension and bulk vorticity (for a recent summary see [4]). In this letter, we present a further extension of LVP which accounts for the presence of odd viscosity in isotropic two-dimensional fluids with broken parity.In three dimensions, parity odd terms in the viscosity tensor were known for a long time in the context of plasma in a magnetic field [5] and in hydrodynamic theories of superfluid He-3A [6], where the fluid anisotropy plays a major role. In two dimensions however the odd viscosity is compatible with isotropy of the fluid [7]. The odd viscosity is the parity violating non-dissipative part of the stress-strain rate response of a two-dimensional fluid. The recent interest in odd viscosity is motivated by the seminal paper by Avron, Seiler, and Zograf [8] where it was shown that, in general, quantum Hall states have non-vanishing odd viscosity. The role of odd viscosity (a.k.a. Hall viscosity) in the context of quantum Hall effect has been an active area of research [9-32], but is out of the scope of this work.In the Ref.[7], Avron has initiated the search for odd viscosity effects in classical 2D hydrodynamics. These effects are subtle in the case when the classical twodimensional fluid is incompressible. Recent works have outlined some of observable consequences of the odd viscosity for incompressible flows [33][34][35][36][37][38]. In particular, in Ref.[38] the equations governing the Hamiltonian dynamics of surface waves were derived in the case where bulk vorticity is absent.Let us start by summarizing the main equations of an incompressible fluid dynamics with odd viscosity. In the following we assume that the fluid density is constant and take it as unity. We also neglect all thermal effects. Then, the hydrodynamic equations are the incompressibility condition and the Euler equationHere, v(x, t) is a two-component veloc...

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