## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 5850-5862 |

Number of pages | 13 |

Journal | Journal of Climate |

Volume | 24 |

Issue number | 22 |

DOIs | |

State | Published - Nov 2011 |

Externally published | Yes |

### Bibliographical note

KAUST Repository Item: Exported on 2020-10-01Acknowledged KAUST grant number(s): KUS-C1-016-04

Acknowledgements: 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).

This publication acknowledges KAUST support, but has no KAUST affiliated authors.

## Keywords

- Climate models
- Energy budget/balance
- Statistical techniques
- Temperature
- Time series

## ASJC Scopus subject areas

- Atmospheric Science