Impact of physically relevant and numerically induced diapycnal mixing and meso-scale dissipation on meridional mass and tracer transports in the Southern Ocean
Final Report Abstract
It is well known that in numerical models the advective transport relative to fixed or moving grids needs to be discretised with sufficient accuracy to minimise the spurious decay of tracer variance (spurious mixing). In this project a general analysis of discrete variance decay (DVD) caused by advective and diffusive fluxes has been established. Lacking a general closed derivation for the local DVD rate, two non-invasive methods to estimate local DVD during model runtime are discussed. Whereas the first was presented recently by Burchard and Rennau (2008), the second is a newly proposed alternative. This alternative analysis method is argued to have a more consistent foundation. In particular, it recovers a physically sound definition of discrete variance in a Finite-Volume cell. The diagnosed DVD can be separated into physical and numerical (spurious) contributions, with the latter originating from discretisation errors. Based on the DVD analysis, a 3D dissipation analysis is developed to quantify the physically and numerically induced loss of kinetic energy. This dissipation analysis provides a missing piece of information to assess the discrete energy conservation of an ocean model. Analyses are performed and evaluated for three test cases, with complexities ranging from idealised 1D advection to a realistic ocean modelling application to the Western Baltic Sea. In all test cases the proposed alternative DVD analysis method is demonstrated to provide a reliable diagnostic tool for the local quantification of physically and numerically induced dissipation and mixing. These new numerical analysis methods have then been applied for studying the effect of various advection schemes on eddy-induced mixing and restratification in a re-entrant Eady channel. The spurious dissipation and mixing of these advection schemes were quantified in idealised experiments of lateral shear and baroclinic instabilities for configurations with large and small Rossby numbers. In addition, a two-dimensional barotropic shear instability test case is used to examine numerical dissipation of momentum advection in isolation, without any baroclinic effects. Effects of advection schemes on the evolution of background potential energy and the dynamics of the restratification process are analysed. The advection schemes for momentum and tracers are considered using several different methods including a recently developed local dissipation analysis. As highly accurate but computationally demanding schemes we apply WENO and MP5, and as more efficient lower-order Total Variation Diminishing (TVD) schemes we use among others the SPL-max-1/3 and a Third-Order-Upwind scheme. The analysis shows that the MP5 and SPL-max-1/3 schemes provide the most accurate results. Following our comprehensive analysis of computational costs, the MP5 scheme is approximately 2.3 times more expensive in our implementation. In contrast to the configuration with a small Rossby number, in which significant differences between schemes are apparent, the different advection schemes behave similarly for a larger Rossby number. Regions of high numerical dissipation are shown to be associated with low grid Reynolds numbers. The major outcome of the present study is that generally positive global numerical dissipation and positive background potential energy evolution delay the restratification process. Using the WOCE climatology by Gouretski and Koltermann (2004), maps of meso-scale eddy kinetic energy (EKE), diffusivities for mixing along isopycnals (isopycnal diffusivity) and for the advective effect of meso-scale eddies (skew diffusivity) are created using properties of the fastest growing unstable baroclinic waves and a simple ad hoc scaling of the amplitudes from linear stability theory. Amplitudes of EKE compare well with near-surface observational estimates based on satellite data and results of an eddy-permitting model, but show a low bias in regions where eddies are not generated locally but propagate into. Largest diffusivities are found in the deep Antarctic Circumpolar Current, and in the shallow western boundary and low latitude west-ward currents. It was also shown that the vertical structure of the diffusivities can be explained to a large extent by the effect of the planetary vorticity gradient which leads to a decrease of skew diffusivities at the surface (bottom) and to a downward (upward) shift of the steering level, and thus the maximum of isopycnal diffusivities, for eastward (westward) flow.
Publications
- (2012). Quantification of numerically induced mixing and dissipation in discretisations of shallow water equations. Int. J. Geomath., 3, 51-65
Burchard, H.
- (2013). A global map of meso-scale eddy diffusivities based on linear stability analysis. Ocean Modell., 72, 198-209
Vollmer, L. and C. Eden
- (2013). Quantification of numerical and physical mixing in coastal ocean model applications. In: Ansorge, R., H. Bijl, A. Meister, and T. Sonar (eds.): Oberwolfach workshop: Recent developments in the numerics of non-linear hyperbolic conservations laws and their use in science and engineering. Springer, Berlin, Heidelberg, pp. 89-103
Burchard, H., and U. Gräwe
- (2014). Quantification of spurious dissipation and mixing Discrete variance decay in a Finite-Volume framework. Ocean Modell., 81, 49-64
Klingbeil, K., M. Mohammadi-Aragh, U. Gräwe, and H. Burchard
(See online at https://doi.org/10.1016/j.ocemod.2014.06.001) - (2015). Advantages of vertically adaptive coordinates in numerical models of stratified shelf seas. Ocean Modell., 92, 56-68
Gräwe, U., P. Holtermann, K. Klingbeil, and H. Burchard
(See online at https://doi.org/10.1016/j.ocemod.2015.05.008) - (2015). The impact of advection schemes on restratifiction due to lateral shear and baroclinic instabilities. Ocean Modell., 94, 112-127
Mohammadi-Aragh, M., K. Klingbeil, N. Brüggemann, C. Eden, and H. Burchard
(See online at https://doi.org/10.1016/j.ocemod.2015.07.021)
