线性和非线性区域中惯性波吸引子的直接数值模拟
Direct numerical simulations of an inertial wave attractor in linear and nonlinear regimes
摘要：
在均匀旋转的流体中，惯性波沿着与旋转轴倾斜一定角度的射线传播，该角度取决于波的频率。在封闭区域中，来自边界的多次反射可能导致惯性波集中在称为波吸引子的特定结构上。这种结构以前已经从理论的角度，在实验室实验中，在线性数值计算和一些最近的数值模拟研究。本文对惯性波吸引子进行了二维直接数值模拟。在线性区域中，我们首先恢复Ogilvie（2005）的线性计算和渐近理论的结果，Ogilvie（2005）考虑了一个典型的问题，该问题涉及线性内波聚焦成以稳定状态下的波吸引子为中心的窄光束。梁的速度分布及其与Ekman数的比例关系，以及耗散率的渐近值，被发现是符合线性理论。我们还发现，作为梁建立周围的波吸引子，所施加的力输入的功率达到其极限值更迅速地比耗散率，饱和时，只有梁已达到其最终厚度。在非线性区域，光束受到强烈的影响，变得不稳定的亚谐波不稳定性。这种不稳定性将能量转移到具有较短波长和较低频率的次级波。一个狭窄的惯性波束的不稳定性的发病调查通过一个单独的线性分析和结果，如发病的不稳定性，被发现是一致的波吸引子的全球模拟。在最近的实验室内重力波的实验中，也看到了这种二次波的激发理论上描述的这项工作。
英文摘要：
In a uniformly rotating fluid, inertial waves propagate along rays that are inclined to the rotation axis by an angle that depends on the wave frequency. In closed domains, multiple reflections from the boundaries may cause inertial waves to focus on to particular structures known as wave attractors. Such structures have previously been studied from a theoretical point of view, in laboratory experiments, in linear numerical calculations and in some recent numerical simulations. In the present paper, two-dimensional direct numerical simulations of an inertial wave attractor are presented. In the linear regime, we first recover the results of the linear calculations and asymptotic theory of Ogilvie (2005) who considered a prototypical problem involving the focusing of linear internal waves into a narrow beam centred on a wave attractor in a steady state. The velocity profile of the beam and its scalings with the Ekman number, as well as the asymptotic value of the dissipation rate, are found to be in agreement with the linear theory. We also find that, as the beam builds up around the wave attractor, the power input by the applied force reaches its limiting value more rapidly than the dissipation rate, which saturates only when the beam has reached its final thickness. In the nonlinear regime, the beam is strongly affected and becomes unstable to a subharmonic instability. This instability transfers energy to secondary waves possessing shorter wavelengths and lower frequencies. The onset of the instability of a narrow inertial wave beam is investigated by means of a separate linear analysis and the results, such as the onset of the instability, are found to be consistent with the global simulations of the wave attractor. The excitation of such secondary waves described theoretically in this work has also been seen in recent laboratory experiments on internal gravity waves.
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