Readings: Interview - Reinforcing the Foundations of Climate and Weather Prediction

BAMS spoke with Andrew N. Staniforth about his new book, Global Atmospheric and Oceanic Modelling: Fundamental Equations (Cambridge University Press). Before his retirement in 2014, Staniforth led the research and development of dynamical cores for weather and climate prediction at two national centers in Canada and the United Kingdom.

Why write this book?
When I joined Environment Canada 50 years ago, the development of quantitative atmospheric and oceanic models was still in its infancy: available knowledge was highly dispersed; different equation sets of varied complexity were in use; their description was couched in differing notation and vocabulary; and many practical aspects were undocumented. I found this frustrating. What I wished for was a comprehensive textbook that led the reader from first principles to the state of the science for the dynamical cores of such models. For climate and weather forecasting, everything is built upon, or around, their governing equations. 
Since then, many excellent textbooks and journal papers have appeared. Yet I think there is still a need for a textbook that provides a general, unified, comprehensive, and detailed account of the underlying geophysical-fluid-dynamical equations on which global, quantitative, atmospheric, and oceanic models for climate and weather prediction are founded. My book therefore aims to do this.

Who is the book for?
Both novices and experts interested in the foundations of climate and weather prediction. 

The only prerequisite to reading my book is a basic mastery of vector calculus and partial differentiation. This makes it accessible to graduate students and is today’s version of the book I wished for 50 years ago. 
For experts, it provides an accessible source for reference purposes to a broad range of interrelated facets. These facets collectively lead in a unified manner to suitable governing equation sets for present and future atmospheric and oceanic dynamical cores.

How does your book relate to other atmospheric and oceanic textbooks?
It complements other textbooks that, after setting out the governing equations in spherical geometry, do a great job of identifying fundamental geophysical phenomena via judicious further simplification. I view the subject from a somewhat different perspective by instead adopting the principle of minimum simplification; just enough to be practically viable, but no more than absolutely necessary. The resulting governing equations are then more general than, and subsume as special cases, those appearing in other textbooks. 

These equations embody the facts that Earth is not spherical but oblate spheroidal (i.e., slightly squashed), and that gravity physically varies from Pole to Equator—something existing textbooks ignore due to their universal assumption of spherical geopotentials. This restrictive assumption was not important in the past. However, it may very well be so in the future . . . and arguably also in the present. For example, the use of spherical geometry in a forecast model is incompatible with the use of the World Geodetic System’s Reference Ellipsoid and (spheroidal) Geodetic Coordinates to report the position of observational data. This may then lead to systematic errors in climate and weather forecasts.

An important aim of my book is the generalization of dynamical cores from spherical to spheroidal geometry. This includes the development of a physically more realistic representation of gravity in conjunction with a compatible geopotential coordinate system for modeling purposes. I also examine simplified equation sets in spheroidal geometry, such as the spheroidal generalization of the 2D nondivergent barotropic vorticity equation . . . with some surprises.

What are the themes of your book? 
I have endeavored to convey the importance of generality (to better represent physical phenomena); scientific rigor (to avoid introducing spurious errors); dynamical and thermodynamical consistency (to avoid systematic errors); and unification of atmospheric and oceanic modeling (for consistency, including at the atmosphere/ocean/cryosphere interfaces).

How did you organize your book to meet its aims? 
Part I describes how, in a unified manner, to obtain a general set of governing equations for the dynamical core of a quantitative, atmospheric, or oceanic prediction model. Approximations are only made when necessary and justified. 

Part II provides a means to understand the significance, and potential impact, of approximating the general set of governing dynamical-core equations, with emphasis on the importance of maintaining analogues in any approximated set of physical conservation laws embodied in the general unapproximated one. 

Part III develops some steady and unsteady, exact, nonlinear solutions for testing dynamical cores, once constructed. For reference, a comprehensive set of vector identities is assembled in an appendix. 

Chapters can (almost) be read sequentially if one wishes. Since there are many interrelated facets involved in the formulation of governing equations for realistic atmospheric and oceanic forecast models, links are however made to earlier and later chapters to understand how the many pieces of the formulational jigsaw fit together to form a complete picture. Each chapter begins with an abstract and a preamble and ends with concluding remarks. These are designed to motivate the content of a chapter, to describe how it fits into the complete picture, and to show why certain formulation choices are better than others. 

Chapters do not have to be read sequentially. Each chapter aims to be a “scientific adventure,” motivated by a stated goal, with a sign-posted journey to the destination. With a willingness to accept certain affirmations, or to refer to part of another chapter, each chapter can be read almost independently of the others. This helps the reader interested in a particular subject to access the relevant material without needing to first carefully read other chapters. 

Some chapters contain original material developed to answer various questions that arose during writing. This material is aimed at readers interested in exploring the current boundaries of research.

What obstacles did you face while writing it?
My book was half written when COVID-19 first emerged. The pandemic was both a curse (before vaccinations became available) and a blessing in disguise (since it provided the impetus to complete the book). 

Since retirement eight years ago, I have had to implement open-source software for word processing, equation editing, and figure creation on my home computer, as well as (sometimes deficient) operating system updates. During lockdown I encountered serious software and hardware problems and, left to my own devices, had to expend considerable effort to overcome them. Happily, I was then able to get back to writing.

What did you learn in the process? 
Writing a substantial book, with many equations, is highly challenging and time consuming, but intellectually rewarding. As I wrote and tried to explain things, it raised questions in my mind that I needed to investigate. This led to a gradual widening of the scope of my book and to a deepening of my knowledge of the representation of gravity, of thermodynamics, and of exact steady and unsteady nonlinear solutions of the 3D governing equations.

What surprised you? 
How much more difficult it is to write a wide-ranging, interdisciplinary book than a review article on a specific topic.

What are the implications of this work?
My book shows how the governing equations of atmospheric and oceanic dynamical cores could/should be generalized to meet the future needs of quantitative atmospheric and oceanic modeling for weather and climate prediction. The computational overhead of doing so should be relatively small, say 10%–20%. For testing purposes, my book also provides a variety of exact, steady, and unsteady nonlinear solutions to these equations; this includes the spherical geometry of today’s models as a special case. 

I believe that climate and weather forecast models will eventually employ spheroidal coordinates not too dissimilar from those developed in my book. Time will tell, of course, but I see no fundamental reason not to move in this direction now. The instantaneous impact of working in spheroidal instead of spherical geometry can be expected to be small. However, the cumulative impact on predictions of global atmospheric and oceanic circulation may be much greater since the Pole-to-Equator variation of gravity is systematic, and the governing equations are nonlinear, leaving this as an open question. 

The only convincing way to answer this question is to build a model in spheroidal geometry. It can then be used to perform controlled comparative experiments in spheroidal and spherical configurations using the same numerics and physical parameterizations. My book leaves this as a challenge to the younger generation!

What comes next? 
Physical and intellectual health permitting, I wish to investigate some further questions that arose while writing my book as well as doing other things in retirement.
 

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