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Motivated by the spurious variance loss encountered during covariancepropagation in atmospheric and other large-scale data assimilation systems, weconsider the problem for state dynamics governed by the continuity and relatedhyperbolic partial differential equations. This loss of variance is oftenattributed to reduced-rank representations of the covariance matrix, as inensemble methods for example, or else to the use of dissipative numericalmethods. Through a combination of analytical work and numerical experiments, wedemonstrate that significant variance loss, as well as gain, typically occursduring covariance propagation, even at full rank. The cause of this unusualbehavior is a discontinuous change in the continuum covariance dynamics ascorrelation lengths become small, for instance in the vicinity of sharpgradients in the velocity field. This discontinuity in the covariance dynamicsarises from hyperbolicity: the diagonal of the kernel of the covarianceoperator is a characteristic surface for advective dynamics. Our numericalexperiments demonstrate that standard numerical methods for evolving the stateare not adequate for propagating the covariance, because they do not capturethe discontinuity in the continuum covariance dynamics as correlations lengthstend to zero. Our analytical and numerical results demonstrate in the contextof mass conservation that this leads to significant, spurious variance loss inregions of mass convergence and gain in regions of mass divergence. The resultssuggest that developing local covariance propagation methods designedspecifically to capture covariance evolution near the diagonal may prove auseful alternative to current methods of covariance propagation.

Continuum covariance propagation for understanding variance loss in advective systems