© 2014 Elsevier Ltd. There have been noticeable advancements in developing parametric covariance models for spatial and spatio-temporal data with various applications to environmental problems. However, literature on covariance models for processes defined on the surface of a sphere with great circle distance as a distance metric is still sparse, due to its mathematical difficulties. It is known that the popular Matérn covariance function, with smoothness parameter greater than 0.5, is not valid for processes on the surface of a sphere with great circle distance. We introduce an approach to produce Matérn-like covariance functions for smooth processes on the surface of a sphere that are valid with great circle distance. The resulting model is isotropic and positive definite on the surface of a sphere with great circle distance, with a natural extension for nonstationarity case. We present extensive numerical comparisons of our model, with a Matérn covariance model using great circle distance as well as chordal distance. We apply our new covariance model class to sea level pressure data, known to be smooth compared to other climate variables, from the CMIP5 climate model outputs.