This paper presents derivations of some analytical forms for spatial correlations of evolving random fields governed by a white-noise-driven damped diffusion equation that is the analog of autoregressive order 1 in time and autoregressive order 2 in space. The study considers the two-dimensional plane and the surface of a sphere, both of which have been studied before, but here time is introduced to the problem. Such models have a finite characteristic length (roughly the separation at which the autocorrelation falls to 1/e) and a relaxation time scale. In particular, the characteristic length of a particular temporal Fourier component of the field increases to a finite value as the frequency of the particular component decreases. Some near-analytical formulas are provided for the results. A potential application is to the correlation structure of surface temperature fields and to the estimation of large area averages, depending on how the original datastream is filtered into a distribution of Fourier frequencies (e.g., moving average, low pass, or narrow band). The form of the governing equation is just that of the simple energy balance climate models, which have a long history in climate studies. The physical motivation provided by the derivation from a climate model provides some heuristic appeal to the approach and suggests extensions of the work to nonuniform cases.
We acknowledge partial support from both the Harold J. Haynes Endowment at Texas A&M University and NSF Grants CMG ATM-0620624 and DMS-1007504. This publication is based in part on work supported by Award KUS-C1-016-04 made by King Abdullah University of Science and Technology (KAUST).