Collaborative Research: Mathematical Studies of Certain Geophysical Models

计算机对大范围或全球范围的现象的预测，例如天气或气候预测，需要在预测的准确性和可用的计算资源之间进行折衷。因此，得出可靠和值得信赖的全球气候模型是一项宏大的科学挑战。利用某些地球物理平衡，例如地球自转平衡(由于地球自转)或静力平衡(由于海洋和大气的浅薄)，地球物理学家推导出合理但不那么复杂的平衡模型。因此，对于相关的空间和时间尺度，严格证明这些模型的有效性是至关重要的。拟议项目的重点是非线性海洋动力学模型和湍流亚格子模型解的分析、统计和数值性质。这个项目的第一个方面是：展示这些简化的地球物理模型中的一些模型的存在、唯一性和对初始数据的连续依赖。特别是两层纬向急流模型、行星地转“温跃层”模型、具有退化变化海底地形的湖泊方程和二维原始方程。这是验证这些模型的推导的第一步，也是最重要的一步。为了证明流体动力学模型的长期行为，人们必须比较它们吸引不变集的统计性质，而不是比较个别解。要做到这一点，有必要把重点放在包含一些耗散机制的模型上。这个项目解决了与这些模型的渐近推导以及数值耗散对其解的影响有关的问题，其中包括边界层分析。这个项目的第二个方面是：在两层地转纬向喷流模型的背景下，推导出新的大涡模拟模型，即所谓的阿尔法模型。阿尔法模型被断言能在大范围的大尺度上重现正确的能谱。建议使用严格的分析工具对这一说法进行调查。文中还建议对新建立的两层地转纬向急流α模式进行计算试验，以验证上述论断。此外，还建议探索将阿尔法模型方法作为子网格模型来实现。气候预测中的重大挑战是，控制海洋和大气动力学的数学方程太难分析研究，而且计算成本仍然高得令人望而却步。事实上，根据物理依据和收集的实验数据，众所周知，大气和海洋的湍流涉及广泛的空间和时间尺度。这反过来又使最强大和最先进的计算机无法访问它们。然而，由于地球的自转和其他地球物理情况，如海洋和大气的浅--在某种意义上，它们比它们的深度宽得多--地球物理学家利用某些地球物理平衡来推导简化的平衡模型。这个项目的第一个主题是：建立这些非线性约化模型解的存在性和正则性。这是证明这些模型的推导及其与有关长度和时间尺度的物理观测的一致性的关键一步。此外，在全球气候预测中，人们对气候的长期统计特征感兴趣。该项目的第二个主题是制定一种系统的方法，在海洋和大气动力学的背景下推导和研究新的平均模式，这些模式能够可靠地再现正确的长期统计数据。

Computer predictions of phenomena on large or global scales, for exampleweather or climate forecasts, need to compromise between accuracy of the predictions and available computing resources. It is therefore a grandscientific challenge to derive global climate models which are reliable and trustworthy. Exploiting certain geophysical balances, such as geostrophic balance (due to earth rotation) or hydrostatic balance (due to the shallowness of the ocean and atmosphere) geophysicists derive reasonable, yet less complex, balanced models. It is therefore essential to justify rigorously the validity of these models, for the relevant spatialand time scales. The focus of the proposed project is on the analytical,statistical and numerical properties of solutions to nonlinear ocean dynamics models and turbulent sub-grid models. The first aspect of this project is to: show existence, uniqueness and continuous dependence on initial data, to some of these reduced geophysical models. In particular, a two-layer zonal jet model, a planetary geostrophic ``thermocline'' model, the lake equations with degenerate varying bottom topography and the two-dimensional primitive equations. This is the first and the most essential step in validating the derivation of these models. In order to justify the long-time behavior of fluid dynamical models, one has to compare the statistical properties of their attracting invariant sets, rather than compare individualsolutions. To do so, it is necessary to focus on models which include some mechanism of dissipation. This project addresses questions related to the asymptotic derivation of these models and the effect of numerical dissipation on their solutions, which include boundary layer analysis. The second aspect of thisproject is to: derive new large-eddy simulation models, the so-called alpha-models, in the context of the two-layers geostrophic zonal jet models. The alpha-models are asserted to reproduce the right energy spectrum for a wide range of large scales. It is proposed to investigate this claim using rigorous analytical tools. It is also proposed to perform computational tests on the newly derived two-layers geostrophic zonal jet alpha-model to verify the above assertion. Furthermore, it is proposed to explore the implementation of the alpha-models approach as sub-grid models. The grand challenge in climate prediction is that the mathematical equations governing the ocean and atmosphere dynamics, are too difficult to study analytically, and still prohibitively expensive computationally. Indeed, it is well established, based on physical grounds and collected experimental data, that atmospheric and oceanic turbulent flows involve a broad spectrum of spatial and time scales. This in turn makes them inaccessible to the most powerful and state-of-the-art computers. However, due to the rotation of the earth and other geophysical situations, such as the shallowness of the oceans and the atmosphere - in the sense that they are much wider than they are deep - geophysicists take advantage of certain geophysical balances to derive simplifiedbalanced models. The first theme of this project is to: establish existence and regularity of solutions to some of these nonlinear reduced models. This isa crucial step in justifying the derivation of these models and theirconsistency with the physical observations for the relevant length and time scales. Furthermore, in global climate prediction one is interested in the long-time statistical features of the climate. The second theme of this project is to develop a systematic approach for deriving and studying new averaged models, in the context of ocean and atmosphere dynamics, which are reliable in reproducing the correct long-term statistics.