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The integration of physical relationships into stochastic models is of majorinterest e.g. in data assimilation. Here, a multivariate Gaussian random fieldformulation is introduced, which represents the differential relations of thetwo-dimensional wind field and related variables such as streamfunction,velocity potential, vorticity and divergence. The covariance model is based ona flexible bivariate Mat\'ern covariance function for streamfunction andvelocity potential. It allows for different variances in the potentials,non-zero correlations between them, anisotropy and a flexible smoothnessparameter. The joint covariance function of the related variables is derivedanalytically. Further, it is shown that a consistent model with non-zerocorrelations between the potentials and positive definite covariance functionis possible. The statistical model is fitted to forecasts of the horizontalwind fields of a mesoscale numerical weather prediction system. Parameteruncertainty is assessed by a parametric bootstrap method. The estimates revealonly physically negligible correlations between the potentials. In contrast tothe numerical estimator, the statistical estimator of the ratio between thevariances of the rotational and divergent wind components is unbiased.

A Matern based multivariate Gaussian random process for a consistent model of the horizontal wind components and related variables