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

大气和海洋是分层的流体，因此通过内波支持扰动的传播。这些内波可能会非线性变形并因倾覆而破裂，导致周围流体的混合。大气和海洋也显示出可能变得不稳定的强剪切流，产生卷，也可能导致密度的混合和局部均匀化。研究者研究这两个过程中哪一个在给定的流程配置中盛行的问题。在初步工作的基础上，研究人员推测，在浅水状态下存在一个明显的边界，在这个边界以下，动力学不允许剪切不稳定，只留下波浪破碎作为可能的混合机制。在数学方面，他们考虑混合型偏微分方程系统，其中双曲域对应于内波，椭圆域对应于剪切不稳定性。流动的非线性稳定性问题可以用解本身是否能使系统变为椭圆来表述。研究人员已经证明，这种情况不会发生在一个简单的系统中，并将结果扩展到更一般的情况。除了这个稳定性结果之外，他们还提出了一个闭包来量化波浪破裂时发生的混合。了解和量化流体混合是全球天气和气候研究的关键因素。大气和海洋是分层流体：由于温度、盐度和其他影响，其密度（主要）随高度变化。分层的流体允许内部波的传播，这些波最终可能会破坏并混合流体。另一个可能的混合来源是由于剪切不稳定性：在不同速度的流动之间的界面上形成涡流。在这个项目中，调查人员研究这两种影响中哪一种更有可能在给定的环境条件下占上风。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。通过改进对流携带和混合的参数化，提高了大气-海洋耦合模式的预测能力。它还训练本科生和研究生使用应用数学工具来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.