With that limitation, a broad range of models are treated, chosen both for their scientific significance and broader interest and to illustrate key concepts and mathematical tools. Though no implementation details are considered, the treatment is moderately technical and assumes familiarity with simple vector and multivariable calculus, differential equations, and fluid mechanics. (Introduction to Climate Modelling is used as a text for a one semester graduate course, aimed at students who may have only a basic knowledge of climate science but who have a background in physics, or perhaps in engineering or applied mathematics.)
A rapid introduction covers the basics of the climate system, the purpose and limitations of modelling and something of its history, some examples of current climate models, and the hierarchy of models (where "more complex" is by no means "better").
A basic zero-dimensional radiative balance model produces a simple analytic solution, which then illustrates the use of finite difference methods in finding "the numerical solution of an ordinary differential equation of first order". Context for this model is given in an overview of the climate sensitivity and the major feedbacks to greenhouse forcing.
Turning to energy and matter transport, Stocker looks at equations for diffusion, advection, and advection-diffusion; numerical solutions to a simplified advection equation are then derived. This introduces numerical stability and the Courant–Friedrichs–Lewy criterion, as well as different discretization schemes — "Euler forward in time, centered in space" and so forth — and touches on the problems with non-physical "numerical diffusion" produced by approximations.
Turning to energy transport on a larger scale, Stocker presents some simple meridional energy balance models. Atmospheric heat transport can be partitioned into a mean meridional flow and fluxes due to stationary and transient eddies; the ocean heat transport can be partitioned into ocean gyres, meridional overturning circulation, Ekman circulation, and eddy diffusivity. In both cases, "sub-scale transports need to be parametrised due to the limitations imposed by the grid resolution".
With the large scale ocean circulation, Stocker considers the equations of motion and continuity, the special case of shallow water equations, and the Stommel model for flows driven by the wind. Some of the mathematical methods introduced include the use of different reference frames, initial and boundary value conditions, iterative methods, and grids; spectral models are briefly touched on.
A simplified approach to the general circulation of the atmosphere shows how meridional flows can involve thermally direct (Hadley) and indirect (Ferrel) cells. The Lorenz-Saltzman model is presented as an example of a chaotic system that can generate spontaneous abrupt changes or self-sustained oscillations.
Coupled atmosphere-ocean models require modelling of heat, water, and momentum boundary fluxes, with salinity as an additional complication; older models needed flux corrections to prevent drift.
A final chapter looks at the possibilities of multiple equilibria — with "tipping points" between them — taking as an example the abrupt changes found in polar ice cores and their possible explanation by a bipolar seesaw in the Atlantic circulation, and applying coupled models to the latter's future under greenhouse warming. The possibility of multiple equilibria in a simple atmospheric energy balance model ("Snowball Earth" scenarios) is also touched on.
Introduction to Climate Modelling could be used to lure students from other disciplines to research on circulation models, but is perhaps most valuable for offering a plunge — more than a shallow dip — into the topic for the much broader range of scientists who will use climate models in their work and could benefit from an understanding of their internal complexities. It also makes a nice approach for anyone else with the necessary background in physics and mathematics who wants a better understanding of an area of science which has taken on a role in important public policy decisions.