Parameterization and Reduction for Nonlinear Stochastic Systems with Applications to Fluid Dynamics

海洋的动力学表现出几种大规模的持续流，包括中纬度的墨西哥湾流和黑潮作为突出的例子。它们与低纬度和高纬度的其他洋流一起，将大量的热量和动量从热带地区转移到极地地区，影响当地和全球气候。温暖的海水射流还具有生产清洁的无碳海上能源的巨大潜力。因此，了解这些电流的空间和时间变化对我们的社会至关重要。在这个项目中，研究者将分析内在的非线性和外在的随机强迫之间的相互作用，在形成观察到的变化。理清这些相互作用，分析噪声对调节系统的动力学和统计行为的影响，对于许多实际应用来说仍然是巨大的挑战。为了解决这些问题，研究人员将建立一个新的范例的参数化和有效减少随机强迫非线性耗散方程，如那些管理大规模的海洋流动。所提出的方法非常依赖于降维方法最近开发的调查员和他的同事。从该项目中获得的知识有望使人们对大尺度气候模式的基本机制有新的认识。该奖项也将提供机会，研究生在本research.The降维方法，并在这个项目中进一步发展的基础上，一个新的随机参数化技术未解决的小尺度动力学的基本非线性随机偏微分方程。研究人员将推导出明确的公式，近似的小规模的动态方面的大规模的动态和历史的噪声路径，从而导致低维随机方程只涉及大规模的变量。这样的简化方程能够捕捉到原始随机系统的关键动力学特征，并且在理论和数值上都更容易获得。 噪声对经典Rayleigh-Benard对流模式形成和双环流风驱动海洋环流的时变的影响将在拟议的理论框架内进行研究。未解决的小尺度动力学的参数化公式是严格合理的随机不变流形的上下文中。这些公式将在这个项目中扩展到处理参数制度，远离发病的第一个不稳定性使用变分框架。参数化是路径的性质，这是非常适合的情况下，人们不仅有兴趣在统计量，但也随机明智的动力学行为。这些公式涉及到噪声的历史，这将记忆引入到相应的简化方程中。这种记忆效应对于简化方程定性和定量地捕获原始系统的动力学和统计特征起着重要作用，并且已经被说明为负责实现良好的建模性能，即使在已知对其他传统方法具有挑战性的情况下也是如此。这些简化的系统将帮助我们更好地理解噪声对所研究系统的影响，否则计算成本太高而无法获得。通过研究这些减少模型受到各种类型的噪音，所提出的方法也将带来进一步改善潜在的随机models.This奖项反映了NSF的法定使命的可能途径的见解，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The dynamics of the oceans exhibits several large-scale persistent currents, including the Gulf stream and the Kuroshio in the middle latitudes as prominent examples. Together with other currents in low and high latitudes, they transfer substantial amounts of heat and momentum from the tropics to the polar regions, influencing local and global climate. The balmy jet of seawater also carries a great potential for producing clean offshore carbon-free energy. To understand the spatial and time variabilities of such currents is thus of vital importance for our society. In this project, the investigator will analyze the interplay between intrinsic nonlinearity and extrinsic stochastic forcing in shaping the observed variabilities. To disentangle such interactions and to analyze the impact of noise on dynamical and statistical behaviors of the governing systems are still grand challenges for many practical applications. To address these questions, the investigator will establish a new paradigm for the parameterization and the effective reduction of stochastically forced nonlinear dissipative equations, such as those governing large-scale oceanic flows. The proposed approach relies crucially on a dimension reduction methodology developed recently by the investigator and his colleagues. The knowledge gained in this project is expected to bring new understanding of the fundamental mechanisms of large-scale climate patterns. The award will also provide opportunities for the involvement of graduate students in this research.The dimension reduction methodology adopted and further developed in this project is based on a new stochastic parameterization technique for the unresolved small-scale dynamics of the underlying nonlinear stochastic partial differential equations. The investigator will derive explicit formulas that approximate the small-scale dynamics in terms of both the large-scale dynamics and the history of the noise path, leading thus to low-dimensional stochastic equations involving only large-scale variables. Such reduced equations are able to capture key dynamical features of the original stochastic systems and are much more accessible both theoretically and numerically. The impact of noise on both pattern formation in the classical Rayleigh-Benard convection and time-variability of the double-gyre wind-driven ocean circulation will be studied within the proposed theoretic framework. The parameterization formulas of unresolved small-scale dynamics are rigorously justified in the context of stochastic invariant manifolds. These formulas will be extended in this project to handle parameter regimes that are away from the onset of the first instability using a variational framework. The parameterization is pathwise in nature, which is very well suited for cases when one is not only interested in statistical quantities but also trajectory-wise dynamical behaviors. The formulas involve the history of the noise, which introduces memory into the corresponding reduced equations. This memory effect plays a fundamental role for the reduced equations to capture both qualitatively and quantitatively the dynamical and statistical features of the original system, and it has already been illustrated to be responsible for achieving good modeling performance even in situations that are known to be challenging for other traditional methods to operate. These reduced systems will help us understand better the impact of noise on the studied systems, which are otherwise computationally too expensive to obtain. By studying these reduced models subject to various types of noise, the proposed approach will also bring insights into possible ways of further improving the underlying stochastic models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.