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An introductory survey of the global energy balance climate models is presented with an emphasis on analytical results. A sequence of increasingly complicated models involving ice cap and radiative feedback processes are solved, and the solutions and parameter sensitivities are studied. The model parameterizations are examined critically in light of many current uncertainties. A simple seasonal model is used to study the effects of changes in orbital elements on the temperature field. A linear stability theorem and a complete nonlinear stability analysis for the models are developed. Analytical solutions are also obtained for the linearized models driven by stochastic forcing elements. In this context the relation between natural fluctuation statistics and climate sensitivity is stressed.

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There are several dozen papers that study the effects of cloud horizontal inhomogeneity on the retrievals of cloud optical thickness, but only a few of them deal with cloud droplet sizes. This paper is one of the first comprehensive attempts to fill this gap: It takes a close theoretical look at the radiative effects of cloud 3‐D structure in retrievals of droplet effective radii. Under some general assumptions, it was found that ignoring subpixel (unresolved) variability produces a negative bias in the retrieved effective radius, while ignoring cloud inhomogeneity at scales larger than a pixel scale (resolved variability), on the contrary, leads to overestimation of the domain average droplet size. The theoretical results are illustrated with examples from Large Eddy Simulations (LES) of cumulus (Cu) and stratocumulus (Sc) cloud fields. The analysis of cloud drop size distributions retrieved from both LES fields confirms that ignoring shadowing in 1‐D retrievals results in substantial overestimation of effective radii which is more pronounced for broken Cu than for Sc clouds. Collocated measurements of broken Cu clouds by Moderate Resolution Imaging Spectrometer (MODIS) and Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) are used to check simulations and theory with observations. The analysis of ASTER and MODIS data and associated derived products recommends against blindly using retrieved effective radii for broken cloud fields, especially if one wants to relate aerosol amounts to cloud droplet sizes.

Geophysical data rarely show any smoothness at any scale, and this often makes comparison with theoretical model output difficult. However, highly fluctuating signals and fractal structures are typical of open dissipative systems with nonlinear dynamics, the focus of most geophysical research. High levels of variability are excited over a large range of scales by the combined actions of external forcing and internal instability. At very small scales we expect geophysical fields to be smooth, but these are rarely resolved with available instrumentation or simulation tools; nondifferentiable and even discontinuous models are therefore in order. We need methods of statistically analyzing geophysical data, whether measured in situ, remotely sensed or even generated by a computer model, that are adapted to these characteristics. An important preliminary task is to define statistically stationary features in generally nonstationary signals. We first discuss a simple criterion for stationarity in finite data streams that exhibit power law energy spectra and then, guided by developments in turbulence studies, we advocate the use of two ways of analyzing the scale dependence of statistical information: singular measures and qth order structure functions. In nonstationary situations, the approach based on singular measures seeks power law behavior in integrals over all possible scales of a nonnegative stationary field derived from the data, leading to a characterization of the intermittency in this (gradient‐related) field. In contrast, the approach based on structure functions uses the signal itself, seeking power laws for the statistical moments of absolute increments over arbitrarily large scales, leading to a characterization of the prevailing nonstationarity in both quantitative and qualitative terms. We explain graphically, step by step, both multifractal statistics which are largely complementary to each other. The geometrical manifestations of nonstationarity and intermittency, “roughness” and “sparseness”, respectively, are illustrated and the associated analytical (differentiability and continuity) properties are discussed. As an example, the two techniques are applied to a series of recent measurements of liquid water distributions inside marine stratocumulus decks; these are found to be multifractal over scales ranging from ≈60 m to ≈60 km. Finally, we define the “mean multifractal plane” and show it to be a simple yet comprehensive tool with many applications including data intercomparison, (dynamical or stochastic) model and retrieval validations.