Some Problems Related to Fluid and Geophysical Fluid Dynamics

研究人员研究了与流体和地球物理流体动力学有关的三类问题。第一类问题涉及不可压缩流体的Navier-Stokes方程与传统湍流理论之间的联系。研究了基于Navier-Stokes方程的三维边界驱动流动的能量耗散率估计和Kolmogorov的湍流能量耗散率的启发式估计之间的联系，以及由Navier-Stokes方程预测的渐近自由度(无论是作为整体吸引子或指数吸引子的分维，还是作为确定系统的模式、节点数或体积的数目)与传统湍流理论中结合启发式Landau-Lifschitz论点的联系。第二类问题是与不可压缩粘性流体有关的边界层行为，其动力学由Navier-Stokes方程控制。分别研究了不透水边界和透水边界情况。通过数值模拟和实验室数据对结果进行了检验。文中还给出了相关的高效数值算法。第三类问题与地球物理流有关，强调基于正压准地转模式的大尺度相干结构的涌现性和持久性。研究了三个原型问题。第一个是地球物理流的选择性衰减问题，它直接从动力学方程中预测Rossby波的出现。第二类是具有小尺度正涡随机强迫的自旋加速问题。三是球面上的正压准地转方程。液体对我们的生活是不可或缺的。它们的范围从我们呼吸的空气和我们饮用的水到覆盖我们地球表面的海洋和大气。研究这些流体的运动对我们来说是至关重要的。在这个阶段，完全的动力学方程的完全解析度超出了当今最强大的计算系统的能力。因此，需要使用近似方法(数值和渐近方法)来提供简化但典型情况下流体行为的定性和定量结果。这里提出的问题是典型的，理解它们将有助于我们理解与流体和地球物理流体动力学有关的基本概念。对这些典型例子的研究将加深我们对更现实和更复杂的问题的理解，这些问题在许多工程应用(如机翼设计或湍流燃烧)以及天气预报、气候学和环境现象中都很重要。

The investigator studies three classes of problems relatedto fluid and geophysical fluid dynamics. The first class ofproblems involves connections between the Navier-Stokes equationsof incompressible fluids and conventional turbulence theory. Theinvestigator studies links between the estimates of energydissipation rate of three-dimensional boundary-driven flows basedon the Navier-Stokes equations and Kolmogorov's heuristicestimate of energy dissipation rate for turbulent flows, andconnections between the asymptotic degrees of freedom predictedby the Navier-Stokes equations (either as the fractal dimensionof the global or exponential attractor, or as the number ofdetermining modes, nodes or volume of the system) and those fromconventional turbulence theory in combination with the heuristicLandau-Lifschitz argument. The second class of problems isboundary layer behavior associated with incompressible viscousfluids whose dynamics are governed by the Navier-Stokesequations. Both the case with non-permeable boundary and the casewith permeable boundary are studied. Results are checked vianumerical simulation and laboratory data. Some relevant efficientnumerical algorithms are developed as well. The third clas ofproblems is related to geophysical flows, emphasizing emergenceand persistence of large scale coherent structures based on thebarotropic quasi-geostrophic model. Three prototype problems arestudied. The first is a selective decay problem for geophysicalflows that predicts the emergence of Rossby waves directly fromthe dynamic equations. The second is a spin-up problem withrandom forcing of small scale positive vortices. The third isthe barotropic quasi-geostrophic equations on the sphere. Fluids are indispensable to our life. They range from theair we breathe and water we drink to oceans and atmosphere thatcover the surface of our the earth. Study of the motion of thesefluids is crucial to us. At this stage full resolution of thecomplete dynamic equations lies beyond the capability of the mostpowerful computing system available today. Hence approximationmethods (both numerical and asymptotic) are needed to providequalitative and quantitative results on the behavior of thefluids for simplified but prototypical situations. The problemsproposed here are prototypical and understanding them willfacilitate our understanding of fundamental concepts related tofluid and geophysical fluid dynamics. Study of these typicalexamples will further our understanding of more realistic andsophisticated problems that are important in many engineeringapplications (such as wing design or turbulent combustion) aswell as in weather prediction, climatology and environmentalphenomena.