Discrete variance decay analysis of spurious mixing

Abstract

This work examines the use of discrete variance decay of tracers to estimate locally in space and time the numerical mixing caused by different processes during a tracer transport step. Expressions for local discrete variance decay (DVD) rates are directly derived from discrete tracer equations without any assumptions on discrete fluxes of the second moment. They relate the DVD rates to the fluxes of the first moment through the faces of scalar control cell. Mixing associated with advective and diffusive fluxes is thus estimated. The new framework avoids the need for second-moment flux definition when solved directly on finite-volume cell faces but still invokes certain second-moment fluxes when the face DVD rates are partitioned to cells sharing the face. These implied discrete fluxes depend on the partitioning and are non-unique. For third- or higher-order advection schemes, the DVD rates are contaminated by dispersive errors intrinsic to the approach, introducing uncertainty to the locality of any estimates produced by it. Additional temporal averaging or coarse-graining is thus necessary. Through the application of this technique, numerical mixing is found to be correlated with the distribution of eddy kinetic energy. Numerical mixing induced by vertical advection is found to be relatively small and correlated with the distribution of buoyancy fluxes. The explored high-order schemes are found to demonstrate levels of spurious mixing which may locally exceed physical mixing.