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Large-Scale Atmosphere-Ocean Dynamics. Vol. I: Analytical Methods And Numerical Models Vol. Ii: Geometric Methods And Models

Posted on: Saturday, 20 March 2004, 06:00 CST

LARGE-SCALE ATMOSPHERE-OCEAN DYNAMICS. VOL. I: ANALYTICAL METHODS AND NUMERICAL MODELS VOL. II: GEOMETRIC METHODS AND MODELS John Norbury and Ian Roulstone, Eds., 2002, Vol. I: 370 pp., Vol. II: 364 pp., Vol. I: $80.00, Vol. II: $80.00, hardbound, Cambridge University Press, Vol. I: ISBN 0-521-80681-X, Vol. II: ISBN 0-521- 80757-3

Much of modern dynamic meteorology is based on studies of balanced models that are constructed to eliminate fast (e.g., inertia gravity) waves or oscillations associated with the full dynamic and thermodynamic equations as used in weather forecasting and climate modeling. Quasigeostrophic (QG) theory, as one example of the balanced models, was developed by Rossby, Ertel, Charney, and Eady in the 1940s (e.g., Charney 1947) and is still the fundamental unifying theory for large-scale extratropical motions. Three decades later, semigeostrophic (SG) theory was developed by Hoskins and Bretherton in the 1970s (e.g., Hoskins 1975) and became a new milestone in dynamic meteorology. The question now-another three decades later-is: what is the next possible landmark in large-scale atmosphere-ocean dynamics?

In an attempt to find the direction toward such a new landmark, the famed Isaac Newton Institute in Cambridge, England, held an intense program on "The Mathematics of Atmosphere and Ocean Dynamics" in 1996. This program, along with a follow-up meeting there in 1997, led to these two volumes, edited by Norbury and Roulstone. In a similar effort, the Chinese Academy of Sciences, the Third World Academy of Sciences, and the World Meteorological Organization have organized international symposia on the physicomathematical problems related to climate modeling and prediction, with the first two held in 2001 and 2002. A summer school on applications of advanced mathematical and computational methods to atmospheric and oceanic problems has also been organized at NCAR.

The first volume contains six chapters, while the second volume contains eight. There appears to be no clear separation of contents between the two volumes, and the publication of two volumes is probably needed because each, with a hard cover, is heavy in weight, as well as mathematically "heavy." There arc 100 pages for the longest chapter (Vol. I, chapter 1) versus only 8 pages for the shortest (II, 3). Less than half of the contributors are meteorologists or oceanographers, while the others are experts in mathematics and other disciplines. Even though there was a five- year intervening period between the end of the original program and the publication of the two volumes, all contributors have done an excellent job in providing up-to-date reviews. In addition to a short preface, both volumes devote 18 pages at the beginning to providing an overall scientific background and introduction for each chapter.

In the first volume, chapter 1 is a pedagogical introduction to the mathematics of atmospheric dynamics intended for mathematicians and physicists who desire a compact introduction to the subject. Chapter (I, 2) derives approximate intermediate models within the framework of Hamilton's variational principle using the Euler- Poincare theorem for ideal continua [see chapter (II, 7)]. The resulting equations possess the Kelvin circulation theorem, conserve potential vorticity (PV) on fluid particles, and conserve volume integrated energy. Therefore, they are substantially more accurate than those from QG or SG for the idealized, moderate Rossby number, mesoscale oceanographic flow problems.

Chapter (I, 3) provides a rigorous account of the asymptotic validity of balanced models based on the governing equations for three-dimensional (3D), rotating, stably stratified fluids under the Boussinesq approximation. However, it is probably too rigorous mathematically for most meteorologists and oceanographers to follow.

Chapter (I, 4) reviews various new mathematical developments in atmosphere and ocean dynamics (also covered in other chapters) at an appropriate level for seientists with graduate training in dynamic meteorology. In particular, it is thought-provoking to learn of actual applications of these new theories to numerical model design and data assimilation and predictability studies. Chapter (I, 5) presents some mathematical ideas on rearrangements of Lagrangian fluid volumes. In fact, the SG equations can be interpreted as a sequence of minimum energy states over a set of rearrangements! The application of these ideas to the decomposition of weather forecast errors is very refreshing. Chapter (I, 6) provides a new interpretation of theories of atmosphere-ocean dynamics [e.g., from (I, 4) and (I, 5)] based on statistical methods. For instance, SG flow can be characterized as the most likely evolution of minimum energy states under the large-scale constraints of the system. Furthermore, motivated by the success of statistical physics in providing a model for equilibrium thermodynamics, the maximum entropy principle is applied to define balance for the 3D Boussinesq equations.

In the second volume, chapter 1 gives a step-by-step account of the basics of Hamilton's principle as used in the derivation of balanced models, which was pioneered by Salmon in the 1980s (Salmon 1988). This approach ensures that the resulting model can retain approximations to the conservation laws of the original primitive equations. It also introduces the powerful energy-Casimir method for the derivation of stability criteria. Many examples in this chapter are helpful for the understanding of these new methods.

Chapter (II, 2) uses a toy model of four coupled nonlinear ordinary differential equations, which govern the high- and low- frequency oscillations of an elastic pendulum, to illustrate many important mathematical ideas. Chapter (II, 3) demonstrates that solutions to the energy transfer equation of an ensemble of interacting internal gravity waves include the stationary Rayleigh- Jeans solution with energy equipartition and the Kolmogorov solution with a constant energy flux through the wave spectrum. Chapter (II, 4) introduces an innovative integral transform, which is a generalization of the Hilbert transform, to study shear flow, including the dynamics of the continuous spectrum.

Chapter (II, 5) reviews many aspects of transformation theory, such as the Legendre duality (applicable to vortex dynamics and numerical analysis), lift transformation (including an example inspired by f-plane SG theory), and canonical transformation. Chapter (II, 6) presents Legendre-transformable SG theories as an extension of the original SG theory (Hoskins 1975). For instance, the new formulation avoids the deficiency of the original SG theory on the hemisphere or with curved fronts. It also enables idealized simulations of the entire life cycle of fully nonlinear baroclinic waves without using nonphysical diffusion coefficients mandated by the consideration of computational stability.

Chapter (II, 7) presents a number of models of geophysical fluid dynamics at various levels of approximation in Euler-Poincare form based on reduction of variational principles. New modifications of the Euler-Boussinesq equations and primitive equations are also proposed that adaptively filter high-frequency waves and, hence, make them good candidates for long-term numerical integration. Chapter (II, 8) presents elementary tutorial material to review the background of SG theory (e.g., its Hamiltonian structure, Lengendre duality, contact structure, and the convexity of certain potential functions) and its possible higher-accuracy analogues based on complex-valued canonical coordinates.

Overall, these two volumes bring together fresh ideas from geometry, analytical methods, and dynamic system theory that have the potential to further develop large-scale atmosphere-ocean dynamics theory and to benefit data assimilation and numerical prediction of weather and climate. However, despite the efforts of some contributors, most chapters are probably still too difficult for researchers to follow without advanced dynamics and mathematics backgrounds. The introductory material at the beginning of the book is helpful for readers to gain some overall understanding of the subject, but a chapter should also be added to provide an elementary discussion of all relevant mathematical concepts and terms. A table of contents should also be added for each chapter. These two volumes are worth purchasing for fluid dynamicists, mathematicians, and physicists who are interested in geophysical fluid dynamics. Some of the chapters [e.g., (1, 4)] may also be used as teaching material for advanced graduate dynamics courses.

REFERENCES

Charney, J. G., 1947: The dynamics of long waves in a baroclinic westerly current. J. Meteor., 4, 135-162.

Hoskins, B. J., 1975: The geostrophic momentum approximation and the semi-geostrophic equations. J. Atmos. Sci., 32, 233-242.

Salmon, R., 1988: Semigeostrophic theory as a Diracbracket projection. J. Fluid Mech., 196, 345-358.

-XUBIN ZENG

Xubin Zeng is an associate professor in the Department of Atmospheric Sciences, University of Arizona.

Copyright American Meteorological Society Feb 2004

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