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

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.