Collaborative Research: Stability, Wave Breaking and Mixing in Stratified Flows

MilewskiDMS-0604635 TabakDMS-0604520 大气和海洋是分层的流体，因此支持扰动通过内波传播。这些内波可能会发生非线性变形并因翻转而破裂，从而导致周围流体的混合。 大气和海洋也显示出强烈的剪切流，这可能变得不稳定，产生滚动，也可能导致混合和局部均匀化的密度。 研究人员研究的问题，这两个过程中占主导地位，在一个给定的流动配置。 根据初步工作，研究者推测，在浅水区有一个尖锐的边界，在该边界以下，动力学不允许剪切不稳定性，只留下波浪破碎作为可能的混合机制。 在数学方面，他们认为系统的偏微分方程的混合型，其中双曲域对应的内波和椭圆域剪切不稳定性。 流的非线性稳定性问题可以用解本身是否能使系统成为椭圆形来表述。 研究人员已经证明，这不可能发生在一个简单的系统，并在这里扩展到更一般的情况下的结果。 除了这个稳定性的结果，他们还提出了一个封闭的方法来量化波浪破碎时发生的混合。 了解和量化流体混合是全球天气和气候研究的关键因素。 大气和海洋是分层的流体：由于温度、盐度和其他影响，流体的密度（主要）随高度而变化。 分层的流体允许内波的传播，这些波最终可能会破碎并混合流体。另一个可能的混合源是由于剪切不稳定性：在不同速度的流体之间的界面处形成涡流。 在这个项目中，研究人员研究这两种效应中哪一种在给定的环境条件下更有可能占上风。 这样的研究具有深远的意义：大气和海洋混合层控制着两者之间的耦合，因此对气候的演变起着关键的控制作用。 这项工作通过改进大气-海洋耦合模式中流体卷吸和混合的参数化，提高了模式的预报能力。 它还培训本科生和研究生使用应用数学工具促进自然科学的发展。

MilewskiDMS-0604635TabakDMS-0604520 The atmosphere and ocean are stratified fluids and as suchsupport the propagation of disturbances through internal waves. These internal waves may deform nonlinearly and break byoverturning, leading to the mixing of the ambient fluid. Boththe atmosphere and ocean also display strong shear flows that maybecome unstable, producing rolls that can also lead to mixing andlocal homogenization of the density. The investigators study theissue of which of these two processes prevails in a given flowconfiguration. Based on preliminary work, the investigatorsconjecture that in the shallow water regime there is a sharpboundary below which the dynamics disallow shear instabilities,leaving only wave breaking as the possible mixing mechanism. Inmathematical terms, they consider systems of partial differentialequations of mixed type, where the hyperbolic domain correspondsto the internal waves and the elliptic domain to shearinstability. The question of nonlinear stability of the flow canthen be formulated in terms of whether the solutions themselvescan make the system become elliptic. The investigators haveproved that this cannot happen for a simple system and hereextend the result to much more general scenarios. In addition tothis stability result, they propose a closure that quantifies themixing taking place when waves break. Understanding and quantifying fluid mixing is a keyingredient in global weather and climate studies. The atmosphereand ocean are stratified fluids: fluids whose density varies(primarily) with height due to temperature, salinity and othereffects. Stratified fluids allow for the propagation of internalwaves, and these waves may eventually break and mix the fluid. Another possible source of mixing is due to shear instabilities:the formation of eddies at the interface between flows ofdifferent speeds. In this project the investigators study whichof these two effects is more likely to prevail given the ambientconditions. Such a study has far-reaching implications: theatmospheric and ocean mixing layers control the coupling betweenthe two, and hence exert a critical control on the evolution ofthe climate. The work advances the predictive capabilities ofcoupled atmosphere-ocean models, by improving theirparameterization of fluid entrainment and mixing. It also trainsundergraduate and graduate students in the use of appliedmathematical tools for the advancement of the natural sciences.