We consider an isotropic compressible non-dissipative fluid with broken parity subject to free surface boundary conditions in two spatial dimensions. The hydrodynamic equations describing the bulk dynamics of the fluid and the free surface boundary conditions depend explicitly on the parity breaking non-dissipative odd viscosity term. We construct an effective action which gives both bulk hydrodynamic equations and free surface boundary conditions. The free surface boundary conditions require an additional boundary term in the action which resembles a $1+1D$ chiral boson field coupled to the background geometry. We solve the linearized hydrodynamic equations for the deep water case and derive the dispersion of chiral surface waves. We show that in the long wavelength limit the flow profile exhibits an oscillating vortical boundary layer near the free surface. The layer thickness is controlled by the ratio betwenn odd viscosity (${\nu }_o$) and sound velocity ($c_s$), $\delta \sim {\nu }_o/c_s$. In the incompressible limit, $c_s\to \infty$ the vortical boundary layer becomes singular with the vorticity within the layer diverging as $\omega \sim c_s$. The boundary layer is formed by odd viscosity coupling the divergence of velocity $\nabla \cdot 𝐯$ to vorticity $\nabla \times 𝐯$. It results in non-trivial chiral free surface dynamics even in the absence of external forces. The structure of the odd viscosity induced boundary layer is very different from the conventional free surface boundary layer associated with dissipative shear viscosity.

中文翻译：

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奇偶性破损的二维可压缩流体的流体力学：无耗散情况下的变分原理和自由表面动力学

我们考虑在两个空间维上受自由表面边界条件约束的奇偶性各向同性可压缩非耗散流体。描述流体的整体动力学和自由表面边界条件的流体力学方程式明确取决于奇偶性破坏非耗散奇粘度项。我们构造了一个有效的动作，它给出了整体流体动力学方程和自由表面边界条件。自由表面边界条件在动作中需要一个附加的边界项，类似于$\mathrm{1个}+\mathrm{1个}d$手性玻色子场耦合到背景几何。我们求解深水情况的线性流体动力学方程，并推导出手性表面波的色散。我们表明，在长波长范围内，流量分布在自由表面附近显示出振荡的涡旋边界层。层厚由奇数粘度（${\nu }_{Ø}$）和声速（$C_s$）， $\delta 〜{\nu }_{Ø}/C_s$。在不可压缩的极限$C_s\to \infty$ 涡旋边界层变得奇异，层内的涡旋随着 $\omega 〜C_s$。边界层是由奇数粘度耦合速度的发散形成的$\nabla \cdot 𝐯$ 涡旋 $\nabla \times 𝐯$。即使在没有外力的情况下，它也能产生不平凡的手性自由表面动力学。奇数粘度引起的边界层的结构与与耗散剪切粘度相关的常规自由表面边界层有很大不同。