content equals the heat capacity times the average temperature of the system. Since the heat capacity of water is much higher than that of air, the rate of change of the heat content depends crucially on the volume of water that is included in the system, along with the total volume of the atmosphere: The former can extend from the oceanic mixed layer, of a few tens of meters, to the entire depth of the ocean, of several kilometers. Accordingly, the atmospheric temperature—assumed to be in equilibrium with the appropriate volume of water—will change more or less rapidly for a given change in the radiation balance.
The radiation balance is often expressed, in the context of decade-to-century variability or of paleoclimate studies, as the sum of a feedback term, dependent on global average temperature, and of a prescribed source term. The feedback term, in its simplest form, is linear, i.e., proportional to temperature. In this version of a 0-D model, equilibrium temperature equals the source term divided by the constant of proportionality, or gain. The change in temperature is thus directly proportional to the prescribed change in radiation balance and inversely proportional to the feedback gain. In the context of decade-to-century-scale climate variability, the externally prescribed change in radiation balance is due to an increase in greenhouse-gas concentration, which leads to a more positive balance at the surface, or to an increase in the concentration of aerosols of natural or anthropogenic origin (e.g., volcanoes or industry respectively), which leads to a more negative surface balance; insolation changes are admittedly smaller, on this time scale, and can contribute in either direction.
The characteristic time needed for temperature to reach a new equilibrium when the source term has been changed is inversely proportional to the heat capacity. This time can be as short as a few years or as long as millennia, depending on the amount of water assumed to be in equilibrium with the surface temperature. The importance of these two model constants, gain and heat capacity, in validating such O-D linear models is discussed by Lindzen (1995).
Somewhat more complex than the O-D atmospheric models are the one-dimensional (1-D) models, which fall into two main categories: energy-balance models (EBMs) and radiative-convective models (RCMs). In EBMs, the spatial coordinate treated explicitly is latitude; the rate of change of the heat content of each latitude belt is equal to the incoming radiation minus the outgoing, as in the O-D models discussed first, plus a redistribution term between belts. These models were introduced, independently of each other, by Budyko (1969) and Sellers (1969), to study the expected cooling of climate as a result of increased aerosol loading. Their main feature was the nonlinear dependence of planetary reflectivity, or albedo, on surface temperature, which determined the presence or absence of ice at a given latitude. As a result of this nonlinear ice-albedo feedback, temperature was shown to drop precipitously as the insolation received at the surface decreased below a certain threshold.
Ghil (1976), Held and Suarez (1974), and North (1975), using slightly different formulations of EBMs, showed that this precipitous drop was due to the presence of multiple equilibria and the ensuing hysteresis. The existence of multiple model equilibria and their dependence on insolation changes was confirmed by Wetherald and Manabe (1975) with a somewhat simplified GCM, illustrating for the first time the power of the hierarchic modeling concept. With their increased geographic detail, 1-D models have been used to address such interesting decade-to-century-scale problems as polar amplification of climate perturbations-i.e., the fact that surface air temperature changes are larger at high than at low latitudes. Polar amplification, which has been noted in a number of GCM studies, can be easily understood in the 1-D EBM context (Ghil, 1976).
In RCMs (Ramanathan and Coakley, 1978), the resolved dimension is height, and the main mechanism under study is the interaction of radiation with clouds. A number of papers by Cess (1976), Manabe and Wetherald (1967), Schneider (1972), and others considered the effects of relative humidity, partial cloud cover, cloud height, and other factors on the vertical distribution of heat flux and temperature. RCMs are often used for the off-line testing of radiative and cloudiness formulations, or parameterizations, of sub-grid-scale thermodynamic processes in GCMs. The performance of GCMs is also examined by intercomparison with RCM results, as well as with 0-D and 1-D EBMs.
The second model type, intermediate between the simplest, 0-D models and the most detailed ones, the GCMs, is given by the two-dimensional (2-D) models; these are of three kinds, according to the phenomena they concentrate on and the spatial coordinates they resolve. Zonally symmetric, meridional-plane models essentially combine and extend the main features of 1-D EBMs and RCMs. Considerable attention was given to the development of such models in the 1960s and early 1970s by B. Saltzman and colleagues (e.g., Saltzman and Vernekar, 1971; see also Saltzman's 1978 review). Zonally symmetrized versions of GCMs have also been found useful in the study of intraseasonal and interannual variability (Goswami and Shukla, 1984). Two-dimensional meridional models are widely applied to coupled dynamics-chemistry problems in the stratosphere (Andrews et al., 1987; Kaye, 1987) that arise in connection with ozone depletion in high latitudes.
Another kind of 2-D model extends EBMs to greater geographic detail, resolving both latitude and longitude. In fact, one subset of these, the thermodynamic models of J. Adem and collaborators (e.g., Adem, 1964), preceded the 1-D EBMs introduced by Budyko (1969) and Sellers (1969) that were so thoroughly analyzed in the mid-1970s. The results of the former—like those of the 2-D meridional models in the previous paragraph—attracted less attention