2021/09/06 by Shay Gilpin, Gilpin, Shay, Tomoko Matsuo +3 · 1 citation
Earth and Planetary Sciences · Engineering · Environmental Science · Mathematics · #35L65 (Primary) #47B38 #65M22 #65M32 #65M75 #65M80 (Secondary) #86A10 #Analysis of PDEs (math.AP) #Climate variability and models #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Meteorological Phenomena and Simulations #math.AP #msc:35L65 #msc:47B38 #msc:65M22 #msc:65M32 #msc:65M75 #msc:65M80 #msc:86A10
paper · pdf · doi:10.48550/arxiv.2109.02530
arxiv created 2021/09/06 · openalex publication_date 2021/09/06 · arxiv updated 2021/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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.