Systematic Mathematical Strategies for Multi-Scale Stochastic Modeling and Uncertainty in Atmosphere/Ocean Science

当代科学面临的重大挑战之一是建立大气及其耦合气候系统的综合预测模型。这是当今科学中最困难的多尺度问题之一，因为在许多时空尺度上存在着令人难以置信的强相互作用的各向异性非线性过程；当代综合计算机模式（GCM）目前无法在适合季节预测和气候变化预估的时间尺度上充分解决或参数化许多这些相互作用。因此，大气/海洋动力学中的多尺度问题为开发新的系统多尺度策略提供了重要的原型，这些策略在从纳米技术到大分子动力学到蛋白质折叠等其他科学学科中都有价值。这些努力的社会影响也很大；据最近估计，如果提前一个月预测厄尔尼诺现象，就能在全球范围内节省1000亿美元。推动气候研究的基本问题是预测1至14天的天气，预测季节至年度时间尺度上的气候变化，最后，预测十年和百年时间尺度上的气候变化，以及量化与这些预测相关的不确定性。最近一个引人注目的观测发现是热带变化对所有这些问题的深远影响。热带的主要影响发生在云团、超星团和行星尺度动力学的相互作用和组织中，这是一个本质上完全非线性的多尺度过程。对于气候变化而言，水蒸气是最重要的温室气体，而云中的微物理过程是辐射反馈的关键机制。事实上，平均云量仅改变4%就能抵消二氧化碳对气候变化的影响。目前的证据表明，一些全球行星遥相关模式，如太平洋-北美涛动，通常概括了热带对中纬度大气的天气和气候影响。由于不可能运行气候变化的大气/海洋耦合综合数值模式，因此涉及这些基本大尺度模式的简化模式至关重要。Majda建议通过现代应用数学工具继续研究中期气候预报的一些最重要的问题和绊脚石，围绕以下新策略：1)热带地区云、对流和行星波的多尺度相互作用；2)大气/海洋未解决特征的随机模拟，以及通过信息论量化复杂系统的不确定性和预测能力。这将导致对大气和海洋未解决尺度的确定性和随机参数化的新策略，这些过程的潜在重要的低阶动态随机模型，以及对天气和气候变化预测的不确定性进行更严格的量化。这类研究通常对科学和工程的其他学科有额外的潜在好处；应用偏微分方程和数值分析的新问题也有望出现。

One of the grand challenges of contemporary science is a comprehensive predictive model for the atmosphere and coupled climate system. This is one of the most difficult multi-scale problems in science today because there is an incredible range of strongly interacting anisotropic nonlinear processes over many spatio-temporal scales; contemporary comprehensive computer models, GCM's, are currently incapable of adequately resolving or parametrizing many of these interactions on time scales appropriate for seasonal prediction as well as climate change projections. Thus, the multi-scale problems in atmosphere/ocean dynamics serve as important prototypes for developing new systematic multi-scale strategies which are valuable in other scientific disciplines ranging from nanotechnology to macro-molecular dynamics to protein folding, etc. The societal impacts for these efforts are also large; it has been estimated recently that a one-month increase in lead time for El Nino prediction would save $100 billion worldwide. Basic questions which drive climate research are the prediction of the weather from 1 to 14 days, the prediction of climate variations on seasonal to yearly time scales, and finally, climate-change projections on decadal and centennial time scales as well as quantifying the uncertainty associated with these predictions. One of the striking recent observational discoveries is the profound impact of tropical variations on all of these problems. The primary influence of the tropics occurs through the interaction and organization of clouds into clusters, super-clusters, and planetary-scale dynamics, an inherently fully nonlinear multi-scale process. For climate change, water vapor is the most important greenhouse gas and the microphysical processes in clouds are a key mechanism for radiative feedback. In fact, only a 4% change in average cloudiness would overwhelm the effects of CO2 in climate change. Current evidence suggests that a few global planetary teleconnection patterns, such as the Pacific North America Oscillation, often summarize the weather and climate impact of the tropics for the mid-latitude atmosphere. Since it will be impossible to run resolved coupled atmosphere/ocean comprehensive numerical models for climate change, reduced models involving these basic large scale patterns are of central importance. Majda proposes to continue work on some of the most important issues and stumbling blocks for medium-range climate forecasting through the tools of modern applied mathematics, centering around novel strategies for: 1) multi-scale interaction of clouds, convection, and planetary waves in the tropics; 2) stochastic modeling of unresolved features for both the atmosphere/ocean and for quantifying uncertainty and predictive capability in complex systems through information theory. This will lead to new strategies for deterministic and stochastic parametrization of unresolved scales for the atmosphere and ocean, potentially significant low-order dynamic stochastic models for these processes, and more rigorous quantification of uncertainty in weather and climate-change predictions. Such research often has additional potential benefit for other disciplines in science and engineering; also novel issues for applied PDE's and numerical analysis are expected to arise.