  The integration of physical relationships into stochastic models is of major
interest e.g. in data assimilation. Here, a multivariate Gaussian random field
formulation is introduced, which represents the differential relations of the
two-dimensional wind field and related variables such as streamfunction,
velocity potential, vorticity and divergence. The covariance model is based on
a flexible bivariate Mat\'ern covariance function for streamfunction and
velocity potential. It allows for different variances in the potentials,
non-zero correlations between them, anisotropy and a flexible smoothness
parameter. The joint covariance function of the related variables is derived
analytically. Further, it is shown that a consistent model with non-zero
correlations between the potentials and positive definite covariance function
is possible. The statistical model is fitted to forecasts of the horizontal
wind fields of a mesoscale numerical weather prediction system. Parameter
uncertainty is assessed by a parametric bootstrap method. The estimates reveal
only physically negligible correlations between the potentials. In contrast to
the numerical estimator, the statistical estimator of the ratio between the
variances of the rotational and divergent wind components is unbiased.
