Odd surface waves in two-dimensional incompressible fluids

Abanov, Alexander G.; Can, Tankut; Ganeshan, Sriram

doi:10.21468/scipostphys.5.1.010

Cited by 61 publications

(74 citation statements)

References 45 publications

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“…There the odd viscosity was introduced phenomenologically, and with no connection to the vortex matter. A comprehensive list of references on recent developments can be found in the paper [29], which studies the effects of the anomalous stress on free surface waves. Despite some similarities, there is an important difference with our study.…”

Section: Edge Dynamics As Action Of the Virasoro-bott Group With Odd mentioning

confidence: 99%

“…Despite some similarities, there is an important difference with our study. In [29] the anomalous stress (38) were phenomenologically imposed on a flow with zero net vorticity. In the vortex matter the anomalous stress and the net vorticity can not be considered separately; Finally we mention the recent work on assemblies of active rotors, see [6] and references therein.…”

Section: Edge Dynamics As Action Of the Virasoro-bott Group With Odd mentioning

confidence: 99%

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Edge Wave and Boundary Layer of Vortex Matter

Bogatskiy

Wiegmann

2019

Phys. Rev. Lett.

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We show that a vortex matter, that is a dense assembly of vortices in an incompressible two-dimensional flow, such as a fast rotating superfluid or turbulent flows with sign-like eddies, exhibits (i) a boundary layer of vorticity (vorticity layer), and (ii) a nonlinear wave localized within the vorticity layer, the edge wave. Both are solely an effect of the topological nature of vortices. Both are lost if the vortex matter is approximated as a continuous vorticity patch. The edge wave is governed by the integrable Benjamin-Davis-Ono equation exhibiting solitons with a quantized total vorticity. Quantized solitons reveal the topological nature of the vortices through their dynamics. The edge wave and the vorticity layer are due to odd viscosity of the vortex matter. We also identify the dynamics with the action of the Virasoro-Bott group of diffeomorphisms of the circle, where odd viscosity parametrizes the central extension. Our edge wave is a hydrodynamic analog of the edge states of the fractional quantum Hall effect.

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Section: Edge Dynamics As Action Of the Virasoro-bott Group With Odd mentioning

confidence: 99%

Section: Edge Dynamics As Action Of the Virasoro-bott Group With Odd mentioning

confidence: 99%

Edge Wave and Boundary Layer of Vortex Matter

Bogatskiy

Wiegmann

2019

Phys. Rev. Lett.

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“…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.…”

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confidence: 99%

“…Therefore, we can start from the Luke's variational principle to produce the bulk hydro equations together with perfect fluid boundary conditions and look for boundary corrections to LVP to obtain the modified boundary conditions on the fluid which are in agreement with (7). In contrast with [38], here we do not use any expansions in ν e and our results do not rely on small surface angle approximations or on assumptions on the structure of the boundary layer.Luke's variational principle. Let us start from the simplest case of the incompressible potential fluid flow, that is, v = ∇θ.…”

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confidence: 99%

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

Abanov

Monteiro

2019

Phys. Rev. Lett.

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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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“…Within the hydrodynamic description of electron transport, non-zero η H influences significantly the structure of the electron flow [45][46][47][48][49][50], which allows one to access η H experimentally [51]. Also, it was argued that the dissipative and Hall viscosity affect the spectrum of edge magnetoplasmons [52][53][54].For noninteracting electrons in the absence of disorder each filled Landau level (LL) gives a contribution to the Hall viscosity equal (2n + 1)/(8πl 2 B ) [25], where n denotes the LL index and l B = c/(eB) stands for the magnetic length. This result is stable against perturbations of the Hamiltonian which preserve translational and rotational invariance [29].…”

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confidence: 99%

Dissipative and Hall Viscosity of a Disordered 2D Electron Gas

et al. 2019

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Hydrodynamic charge transport is at the center of recent research efforts. Of particular interest is the nondissipative Hall viscosity, which conveys topological information in clean gapped systems. The prevalence of disorder in the real world calls for a study of its effect on viscosity. Here we address this question, both analytically and numerically, in the context of a disordered noninteracting 2D electrons. Analytically, we employ the self-consistent Born approximation, explicitly taking into account the modification of the single-particle density of states and the elastic transport time due to the Landau quantization. The reported results interpolate smoothly between the limiting cases of weak (strong) magnetic field and strong (weak) disorder. In the regime of weak magnetic field our results describes the quantum (Shubnikov-de Haas type) oscillations of the dissipative and Hall viscosity. For strong magnetic fields we characterize the effects of the disorder-induced broadening of the Landau levels on the viscosity coefficients. This is supplemented by numerical calculations for a few filled Landau levels. Our results show that the Hall viscosity is surprisingly robust to disorder.Introduction. -Ordinary fluid motion is described by the theory of hydrodynamics, one of whose cornerstones is viscosity, which serves as the source of dissipation. Under certain conditions, charge transport in an electronic system can also be dominated by hydrodynamic viscous flow [1,2]. The discovery of graphene stimulated renewed theoretical [3][4][5][6][7][8][9][10][11] and experimental [12][13][14][15][16][17][18] interest in the hydrodynamic description of charge conduction.In the absence of time-reversal symmetry the viscosity tensor has non-dissipative antisymmetric components. In the presence of a magnetic field B, this nondissipative Hall viscosity (η H ) was studied theoretically in the classical limit of high temperature plasmas [19][20][21][22][23], and for low temperature electron gas [24]. Later, interest in the Hall viscosity was rekindled in quantum systems with a gapped spectrum, due to the connection between η H and geometric response [25][26][27][28][29][30][31], and its expected quantization in the presence of translational and rotational symmetries [29]. It was understood that beyond the Hall conductivity and viscosity there are additional non-dissipative electro-magnetic and geometrical response functions in gapped quantum systems [32][33][34][35][36][37][38][39][40][41][42][43][44]. Within the hydrodynamic description of electron transport, non-zero η H influences significantly the structure of the electron flow [45][46][47][48][49][50], which allows one to access η H experimentally [51]. Also, it was argued that the dissipative and Hall viscosity affect the spectrum of edge magnetoplasmons [52][53][54].For noninteracting electrons in the absence of disorder each filled Landau level (LL) gives a contribution to the Hall viscosity equal (2n + 1)/(8πl 2 B ) [25], where n denotes the LL index and l B = c/(eB) ...

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Odd surface waves in two-dimensional incompressible fluids

Cited by 61 publications

References 45 publications

Edge Wave and Boundary Layer of Vortex Matter

Edge Wave and Boundary Layer of Vortex Matter

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

Dissipative and Hall Viscosity of a Disordered 2D Electron Gas

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