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With the advent of wide-spread global and continental-scale spatiotemporaldatasets, increased attention has been given to covariance functions on spheresover time. This paper provides results for stationary covariance functions ofrandom fields defined over $d$-dimensional spheres cross time. Specifically, weprovide a bridge between the characterization in \cite{berg-porcu} forcovariance functions on spheres cross time and Gneiting's lemma\citep{gneiting2002} that deals with planar surfaces.  We then prove that there is a valid class of covariance functions similar inform to the Gneiting class of space-time covariance functions\citep{gneiting2002} that replaces the squared Euclidean distance with thegreat circle distance. Notably, the provided class is shown to be positivedefinite on every $d$-dimensional sphere cross time, while the Gneiting classis positive definite over $\R^d \times \R$ for fixed $d$ only.  In this context, we illustrate the value of our adapted Gneiting class bycomparing examples from this class to currently established nonseparablecovariance classes using out-of-sample predictive criteria. These comparisonsare carried out on two climate reanalysis datasets from the National Centersfor Environmental Prediction and National Center for Atmospheric Research. Forthese datasets, we show that examples from our covariance class have betterpredictive performance than competing models.

Towards a Complete Picture of Stationary Covariance Functions on Spheres Cross Time