Predicting Climate Change Via the Fluctuation -Dissipation Theorem: A Practical Computational Strategy for Linear Response on a Chaotic Attractor

行星尺度上的气候动力学对各种全球物理参数变化的响应是当代大气和海洋科学中广泛研究的领域。控制行星气候动力学的物理参数包括太阳辐射、火山活动、温室气体、臭氧、极地冰融化和许多其他参数，这些参数通常通过直接数值模拟来模拟适当的气候模式，该模式通常是一个复杂的非线性偏微分方程或此类方程的系统。涨落耗散定理的主要思想是在行星气候动力学接近统计平衡的默认假设下，对行星气候动力学的统计行为进行建模，然后应用涨落响应公式（涨落响应公式的关键部分）预测气候动力学物理参数微小变化的平均气候响应（气候状态的各种统计量的响应）。与直接的数值模拟不同，这种方法不需要实际模拟各种参数变化的气候发展的每个适当情景（这通常会带来相当复杂的计算问题），而是为气候模式的单个数值模拟提供对大范围小参数变化的各种可能响应。显然，后一种性质将在很大程度上促进气候响应模拟。然而，统计物理学中表述的经典波动耗散定理并不直接适用于气候建模，因为几乎所有非平凡气候模型的统计状态与经典版本的定理在数学上根本不相容。在这里，我们提出了一个新的数学框架来执行气候响应建模，该框架采用了波动耗散定理的适当修正版本，旨在克服这些困难。然而，这一框架需要数学和计算方法的进一步广泛发展，才能实际用于气候响应建模。目前的建议阐述了这一发展的详细战略。

The response of climate dynamics on the planetary scale to changes of various global physical parameters is an area which is being extensively studied in contemporary atmospheric and ocean science. The physical parameters controlling the planetary climate dynamics range from solar radiation, to volcanic activity, greenhouse gas, ozone, polar ice melting, and many others, which are normally modeled via direct numerical simulation for an appropriate climate model, which is typically a complex nonlinear partial differential equation or a system of such equations. In the context of the Fluctuation-Dissipation Theorem, the main idea is to model the statistical behavior of planetary climate dynamics under the tacit assumption that the dynamics is close to its statistical equilibrium, and then to apply the fluctuation-response formula, which is the key part of the Theorem, to predict the average climate response (the response of various statistical quantities of the climate state) to small changes of the physical parameters of the climate dynamics. Unlike a straightforward numerical simulation, this approach does not require one to actually simulate each appropriate scenario of climate development for various changes of parameters (which usually poses a computational problem of substantial complexity), instead providing the whole variety of possible responses to a wide range of small parameter changes for just a single numerical simulation with a climate model. Obviously, this latter property will facilitate climate response modeling to a great extent. However, the classical Fluctuation-Dissipation Theorem as formulated in statistical physics is not directly applicable to climate modeling due to the fundamental mathematical incompatibility of the statistical states of virtually all nontrivial climate models with the classical version of the Theorem. Here we propose a novel mathematical framework to perform climate response modeling with a suitably amended version of the Fluctuation-Dissipation Theorem, which is designed to circumvent these difficulties. However, this framework requires extensive further development of mathematical and computational approaches to become practically usable for climate response modeling. The detailed strategy for this development is set forth in the current proposal.