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

University Of Washington, Seattle WA

Investigators

Abstract

In this project, meteorologists and applied mathematicians are collaborating to study the "weak temperature gradient" (WTG) equations for large-scale tropical atmospheric dynamics. The WTG equations form a "balance model", or set of singular limit equations, asymptotically valid in a parameter regime relevant to the tropical atmosphere. The WTG equations are designed to facilitate study of key aspects of the tropical climate problem, such as the large-scale dynamical roles of moist convection and other diabatic processes. This focus is achieved by eliminating other processes which can be present in solutions to the primitive equations, such as gravity waves and baroclinic instability, in the same way as gravity waves are eliminated in extratropical balance models such as the quasi-geostrophic equations. One goal of this project is to develop new mathematics by studying the WTG equations' properties. Another goal is to solve the equations under geometries, boundary conditions and forcings which represent idealizations of those relevant to the real earth's climate, and then to study the sensitivity of the solutions to key parameters. Achievement of these goals is expected to lead to deeper understanding of both the real climate and more complex mathematical models of it, such as general circulation models (GCMs). The earth's climate system is notoriously complex. Many different physical processes interact in a tangled web of feedbacks to produce the dynamic and variable climate we observe. Unlike a laboratory science, meteorology and oceanography are hindered by the impossibility of controlled experiments which might allow the key mechanisms to be conclusively revealed. We are stuck with the one planet on which we live and cannot change its basic properties to see what happens. Consequently, the science proceeds by observation and by a heavy reliance on numerical simulation with GCMs, which are sophisticated computer programs run on the most powerful computers available. These simulations offer the possibility of a certain kind of controlled climate experiments: we can create virtual earths on the computer, control their properties, and observe their behavior with precision. One obvious limitation of this approach is that the models are imperfect representations of reality. A less obvious, but equally important problem is the models' complexity, which both limits the number and type of simulations which can be done and renders the results nearly as difficult to understand as the real climate system. Because of these problems, there is a need to supplement observational and GCM studies with theoretical studies using models that are simpler than GCMs - if not simple enough to allow solution with pencil and paper, then at least using very simple computer programs that run quickly on a PC. To the extent that these simpler models have key features in common with the real system, their simplicity allows a deeper level of understanding of the basic dynamics of climate and leads to explicit hypotheses that can be tested against observations and GCM simulations. This project harnesses the physical insight of climate scientists and the sophisticated methods of applied mathematicians to develop and study simple models designed specifically to represent the tropical atmospheric component of the earth's climate system.