  Motivated by the spurious variance loss encountered during covariance
propagation in atmospheric and other large-scale data assimilation systems, we
consider the problem for state dynamics governed by the continuity and related
hyperbolic partial differential equations. This loss of variance is often
attributed to reduced-rank representations of the covariance matrix, as in
ensemble methods for example, or else to the use of dissipative numerical
methods. Through a combination of analytical work and numerical experiments, we
demonstrate that significant variance loss, as well as gain, typically occurs
during covariance propagation, even at full rank. The cause of this unusual
behavior is a discontinuous change in the continuum covariance dynamics as
correlation lengths become small, for instance in the vicinity of sharp
gradients in the velocity field. This discontinuity in the covariance dynamics
arises from hyperbolicity: the diagonal of the kernel of the covariance
operator is a characteristic surface for advective dynamics. Our numerical
experiments demonstrate that standard numerical methods for evolving the state
are not adequate for propagating the covariance, because they do not capture
the discontinuity in the continuum covariance dynamics as correlations lengths
tend to zero. Our analytical and numerical results demonstrate in the context
of mass conservation that this leads to significant, spurious variance loss in
regions of mass convergence and gain in regions of mass divergence. The results
suggest that developing local covariance propagation methods designed
specifically to capture covariance evolution near the diagonal may prove a
useful alternative to current methods of covariance propagation.
