A comparison of closures for stochastic advection-diffusion equation

K. D. Jarman, A. M. Tartakovsky

Research output: Contribution to journalArticlepeer-review


Perturbation-based moment equations for advection-diffusion equations with random advection have been shown to produce physically unrealistic multimodality. Despite similarities, macrodispersion theory applied to the same equations produces moments that do not exhibit such multimodal behavior. This is because macrodispersion approximations, whether explicitly or implicitly, involve renormalized perturbations that remove secularity by including select higher-order terms. We consider basic differences between the two approaches, using a low-order macrodispersion approximation to clarify why one produces physically meaningful behavior while the other does not. We demonstrate that using a conventional asymptotic expansion (in the order of velocity fluctuations) leads to equations that cannot produce physically meaningful (macro)dispersion, whether applied to moment equations or macrodispersion theory, and prove that the resulting moment equations are in fact hyperbolic for the special case of pure advection by constant random velocity. We identify higher-order terms that must be added to the conventional expansion to recover second- and fourth-order macrodispersivity approximations. Finally, we propose a closed-form approximation to two-point covariance as a measure of uncertainty, in a manner consistent with the derivation of macrodispersivity. We demonstrate that this and all the macrodispersion-based approximations to moments are more accurate than the alternatives for an example of transport in stratified random media.

Original languageEnglish (US)
Pages (from-to)319-347
Number of pages29
JournalSIAM-ASA Journal on Uncertainty Quantification
Issue number1
StatePublished - 2013
Externally publishedYes


  • Heterogeneous media
  • Macrodispersion
  • Moment equations
  • Random fields
  • Solute transport
  • Stochastic

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Statistics, Probability and Uncertainty
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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