重力惯性表面波
On gravito-inertial surface waves
AI总结：
本文研究了重力惯性表面波的几何描述，通过压力和边界条件的方程分析，揭示了低频波在有固边界情况下能量集中于边界的现象，以及在椭球中转化为球谐函数的特性。
摘要：
在地球环境中，由重力和全球旋转作用形成的波被称为重力惯性波。本文给出了重力惯性表面波的几何描述，考虑了无压缩流体在光滑紧致三维域中的理想模型，受恒定旋转向量作用，并在恒定的布兰特-韦斯托频率下分层。通过压力满足的泊松方程和边界上的凯尔文方程建立谱问题。当波频较小时，泊松方程为椭圆型，可用狄利克雷到诺伊曼算子将凯尔文方程转化为边界上的伪微分方程。发现对于大协向量，波能量集中在边界，并在通用域中表现出表面波吸引子。在椭球中，证明这些波是平方可积的，并在边界上还原为球谐函数。
英文摘要：
In geophysical environments, wave motions that are shaped by the action of gravity and global rotation bear the name of gravito-inertial waves. We present a geometrical description of gravito-inertial surface waves, which are low-frequency waves existing in the presence of a solid boundary. We consider an idealized fluid model for an incompressible fluid enclosed in a smooth compact three-dimensional domain, subject to a constant rotation vector. The fluid is also stratified in density under a constant Brunt-Väisälä frequency. The spectral problem is formulated in terms of the pressure, which satisfies a Poincaré equation within the domain, and a Kelvin equation on the boundary. The Poincaré equation is elliptic when the wave frequency is small enough, such that we can use the Dirichlet-to-Neumann operator to reduce the Kelvin equation to a pseudo-differential equation on the boundary. We find that the wave energy is concentrated on the boundary for large covectors, and can exhibit surface wave attractors for generic domains. In an ellipsoid, we show that these waves are square-integrable and reduce to spherical harmonics on the boundary.
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