Collaborative Research: Fast Spin Up of Ocean General Circulation Models Using Newton-Krylov Methods

气候系统的数值模式在了解过去气候变率和预测未来气候变化方面发挥着重要作用。在许多研究中，气候模式是由与时间无关或周期性（季节）变化的强迫场驱动的，通常非常希望获得模式的平衡解。现有的方法是基于简单的权宜之计，即整合模型直到瞬变消失，但由于深海需要几千年的时间才能达到平衡，因此这种方法过于昂贵，无法常规使用。本项目的主要目的是发展一种实用而有效的计算周期性强迫海洋环流模式平衡解的方法。一般的方法是将问题表述为一个大型的非线性代数方程组，用一类称为无矩阵牛顿-克雷洛夫的方法来求解，这是非线性方程超线性收敛解的牛顿型方法的组合，以及求解牛顿修正方程的克雷洛夫子空间方法。为了使这种方法适用于具有107个自由度的全局模型，将开发新的无矩阵预处理策略。该方法的“无矩阵”特性使其非常灵活，可用于任何海洋或气候模型。该方法可应用于在任何时期强迫的模式，包括由时间无关强迫驱动的模式，尽管这里主要关注的是季节周期。初步结果表明，该方案可使季节性强迫ogcm的自旋速度比目前的做法提高两个数量级以上。分析了该技术的收敛性，并与传统的“加速”方法比较了其效率。虽然主要目标是标称分辨率为1的海洋气候模式，但该方法也将应用于下一代更高分辨率的模式，包括允许涡流的模式。该技术将用于从海洋再分析产品和末次盛冰期的产品中获得各种气候强迫估计的平衡解。知识价值：深海缓慢的动态调整时间尺度是我们更有效地利用气候模式的主要障碍之一。提出的研究将通过开发有效计算季节强迫ogcm平衡解的实用算法来解决气候模拟中的这一基本问题。这项研究的一个直接结果将是改进对现代海洋环流和末次盛冰期海洋环流的估计。更广泛的影响：通过大大降低获得气候模型平衡解的计算成本，这项研究将使科学家能够解决目前不可行的科学和社会相关问题。这些问题包括系统参数敏感性研究和古气候模拟，这些领域对于表征气候变化模拟中的不确定性尤为重要。所提出的方法的一个关键优点是，它对潜在的海洋或气候模型代码的假设很少，从而确保这项研究的结果可以被尽可能多的研究人员使用。这项工作与正在进行的海洋环流、古海洋学和海洋生物地球化学领域的工作直接相关。更广泛地说，虽然具体目标是解决海洋自旋问题，但用偏微分方程建模的系统的周期解和极限环的计算是非常普遍的，并且所提出的方法可能在其他学科中具有广泛的适用性。这项研究将有助于培养和教育研究生。作为这项研究的一部分开发的数字代码将免费提供给研究界。这项研究的结果将发表在期刊文章中，并在会议上发表。

Numerical models of the climate system play an important role in efforts to understand past climate variability and predict future climate changes. In many studies, climate models are driven by forcing fields that are either time-independent or that vary periodically (seasonally) and it is often highly desirable to obtain equilibrium solutions of the model. Existing methods, based on the simple expedient of integrating the model until the transients have died out, are too expensive to use routinely because the deep ocean takes several thousand years to equilibrate. The principal objective of this project is to develop a practical and efficient method for computing equilibrium solutions of periodically forced ocean general circulation models (OGCMs). The general approach will be to formulate the problem as a large system of nonlinear algebraic equations to be solved with a class of methods known as matrix-free Newton-Krylov, a combination of Newton-type methods for superlinearly convergent solution of nonlinear equations, and Krylov subspace methods for solving the Newton correction equations. To render this approach practical for global models with order (107) degrees of freedom, novel matrix free preconditioning strategies will be developed. The "matrix-free" nature of the proposed approach makes it extremely flexible, allowing its use with any ocean or climate model. The method can be applied to models forced at any period, including those driven by time-independent forcing, although the main focus here is the seasonal cycle. Preliminary results suggest that this scheme can accelerate the spin up of seasonally forced OGCMs by over two orders of magnitude over current practice. The convergence properties of this technique will be analyzed, and its efficiency assessed against traditional "acceleration" methods. While the primary target is ocean climate models with a nominal resolution of one , the method will also be applied to the next generation of higher resolution models, including eddy permitting ones. The technique will be applied to obtain equilibrium solutions for various forcing estimates for both present day climate from ocean reanalysis products, and that of the Last Glacial Maximum. Intellectual merit: The slow dynamical adjustment timescale of the deep ocean is one of the principal obstacles to our ability to make more effective use of climate models. The proposed study will address this fundamental problem in climate simulation by developing practical algorithms for efficiently computing equilibrium solutions of seasonally forced OGCMs. A direct outcome of this research will be improved estimates of the circulation of both the modern ocean, and that of the Last Glacial Maximum. Broader Impacts: By greatly reducing the computational cost of obtaining equilibrium solutions of climate models, this research will allow scientists to address questions of scientific and societal relevance that are currently unfeasible. These questions include systematic parameter sensitivity studies and simulations of paleoclimate, areas that are especially important for characterizing uncertainties in climate change simulations. A key advantage of the proposed approach is that it makes few assumptions about the underlying ocean or climate model code thus ensuring that the results of this research can be used by the widest possible group of researchers. This work is directly relevant to ongoing work in the areas of ocean circulation, paleoceanography, and ocean biogeochemistry. More broadly, while the specific objective is to address the ocean spin up problem, the computation of periodic solutions and limit cycles of systems modeled by partial differential equations is a very general one, and the proposed method is likely to have broad applicability in other disciplines. This research will contribute to the training and education of a graduate student. Numerical code developed as part of this research will be made freely available to the research community. Findings of this study will be published in journal articles and presented at conference meetings.