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

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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.