Collaborative Research: The Weak Temperature Gradient Equations for Tropical Atmosphere Dynamics

在这个项目中，气象学家和应用数学家正在合作研究大尺度热带大气动力学的“弱温度梯度”（WTG）方程。WTG方程形成一个“平衡模型”，或一组奇异极限方程，在与热带大气有关的参数体系中渐近有效。wt方程的设计是为了促进热带气候问题的关键方面的研究，如湿对流和其他绝热过程的大尺度动力作用。这个重点是通过消除其他过程来实现的，这些过程可以出现在原始方程的解中，如重力波和斜压不稳定性，就像在温带平衡模型中消除重力波一样，如准地转方程。本研究的目标之一是通过研究WTG方程的性质来发展新的数学。另一个目标是求解几何、边界条件和强迫下的方程，这些方程代表了与真实地球气候相关的理想化方程，然后研究解对关键参数的敏感性。这些目标的实现预计将导致对真实气候和更复杂的气候数学模型，如大气环流模式（GCMs）有更深入的了解。众所周知，地球的气候系统非常复杂。许多不同的物理过程在错综复杂的反馈网络中相互作用，产生了我们所观察到的动态多变的气候。与实验室科学不同，气象学和海洋学由于不可能进行有控制的实验而受到阻碍，而有控制的实验可能使关键的机制得到决定性的揭示。我们被困在我们生活的这个星球上，无法改变它的基本属性来观察会发生什么。因此，科学通过观察和严重依赖gcm的数值模拟来进行，gcm是在最强大的计算机上运行的复杂计算机程序。这些模拟提供了某种可控气候实验的可能性：我们可以在计算机上创建虚拟地球，控制它们的特性，并精确地观察它们的行为。这种方法的一个明显的局限性是，模型是对现实的不完美表示。不太明显但同样重要的问题是模型的复杂性，这既限制了可以进行的模拟的数量和类型，又使模拟结果几乎和真实的气候系统一样难以理解。由于这些问题，有必要用比GCM更简单的模型来补充观测和GCM研究的理论研究——如果不够简单到可以用铅笔和纸来解决，那么至少可以使用非常简单的计算机程序来在PC上快速运行。在某种程度上，这些更简单的模型与真实系统具有共同的关键特征，它们的简单性使我们能够更深入地了解气候的基本动力学，并导致可以根据观测和gcm模拟进行检验的明确假设。该项目利用气候科学家的物理洞察力和应用数学家的复杂方法来开发和研究专门为代表地球气候系统的热带大气成分而设计的简单模型。

In this project, meteorologists and applied mathematicians arecollaborating to study the "weak temperature gradient" (WTG) equations forlarge-scale tropical atmospheric dynamics. The WTG equations form a"balance model", or set of singular limit equations, asymptotically validin a parameter regime relevant to the tropical atmosphere. The WTGequations are designed to facilitate study of key aspects of the tropicalclimate problem, such as the large-scale dynamical roles of moistconvection and other diabatic processes. This focus is achieved byeliminating other processes which can be present in solutions to theprimitive equations, such as gravity waves and baroclinic instability, inthe same way as gravity waves are eliminated in extratropical balancemodels such as the quasi-geostrophic equations. One goal of this projectis to develop new mathematics by studying the WTG equations' properties.Another goal is to solve the equations under geometries, boundaryconditions and forcings which represent idealizations of those relevant tothe real earth's climate, and then to study the sensitivity of thesolutions to key parameters. Achievement of these goals is expected tolead to deeper understanding of both the real climate and more complexmathematical models of it, such as general circulation models (GCMs).The earth's climate system is notoriously complex. Many differentphysical processes interact in a tangled web of feedbacks to produce thedynamic and variable climate we observe. Unlike a laboratory science,meteorology and oceanography are hindered by the impossibility ofcontrolled experiments which might allow the key mechanisms to beconclusively revealed. We are stuck with the one planet on which we liveand cannot change its basic properties to see what happens. Consequently,the science proceeds by observation and by a heavy reliance on numericalsimulation with GCMs, which are sophisticated computer programs run on themost powerful computers available. These simulations offer thepossibility of a certain kind of controlled climate experiments: we cancreate virtual earths on the computer, control their properties, andobserve their behavior with precision. One obvious limitation of thisapproach is that the models are imperfect representations of reality. Aless obvious, but equally important problem is the models' complexity,which both limits the number and type of simulations which can be done andrenders the results nearly as difficult to understand as the real climatesystem. Because of these problems, there is a need to supplementobservational and GCM studies with theoretical studies using models thatare simpler than GCMs - if not simple enough to allow solution with penciland paper, then at least using very simple computer programs that runquickly on a PC. To the extent that these simpler models have keyfeatures in common with the real system, their simplicity allows a deeperlevel of understanding of the basic dynamics of climate and leads toexplicit hypotheses that can be tested against observations and GCMsimulations. This project harnesses the physical insight of climatescientists and the sophisticated methods of applied mathematicians todevelop and study simple models designed specifically to represent thetropical atmospheric component of the earth's climate system.