Eurographics Conference on Visualization (EuroVis) 2014 H. Carr, P. Rheingans, and H. Schumann (Guest Editors) Volume 33 (2014), Number 3 Stability of Dissipation Elements: A Case Study in Combustion A. Gyulassy 1 , P.T. Bremer 1,2 , R. Grout 3 , H. Kolla 4 , J. Chen 4 , and V. Pascucci 1 1 University of Utah 2 Lawrence Livermore National Laboratory (LLNL) 3 National Renewables National Laboratory (NREL) 4 Sandia National Laboratory (SNL) Abstract Recently, dissipation elements have been gaining popularity as a mechanism for measurement of fundamental properties of turbulent flow, such as turbulence length scales and zonal partitioning. Dissipation elements segment a domain according to the source and destination of streamlines in the gradient flow field of a scalar function f : M → R. They have traditionally been computed by numerically integrating streamlines from the center of each voxel in the positive and negative gradient directions, and grouping those voxels whose streamlines terminate at the same extremal pair. We show that the same structures map well to combinatorial topology concepts developed recently in the visualization community. Namely, dissipation elements correspond to sets of cells of the Morse- Smale complex. The topology-based formulation enables a more exploratory analysis of the nature of dissipation elements, in particular, in understanding their stability with respect to small scale variations. We present two examples from combustion science that raise significant questions about the role of small scale perturbation and indeed the definition of dissipation elements themselves. Categories and Subject Descriptors (according to ACM CCS): I.4.7 [Computer Graphics]: Feature Measurement— Feature representation 1. Introduction Many characteristics of combustion science are described by the dissipation of scalar values in the domain. In their semi- nal work [WP06], Wang and Peters introduced a domain de- composition technique for scalar fields based on the gradient flow behavior. They defined dissipation elements as regions where streamlines following the ascending and descending gradient terminate at the same extremal pair, a local mini- mum and local maximum, and computed them by integrating streamlines starting at the center of every voxel. Their work gained considerable interest in the literature, in particular on the use of dissipation elements to connect scalar fluctuations to the basic processes of turbulence in direct numerical sim- ulations (DNS). However, interpretation of DNS data in this respect is necessarily dependent on the stability of the anal- ysis. For example, Wang and Peters [WP06, WP08] wrote an evolution equation for the length scale distribution function describing the characteristic length of dissipation elements. They were able to incorporate phenomenological models for processes that distort elements (diffusion and strain) as well as processes that split/recombine elements. However, the un- derlying physical processes may be difficult to distinguish from short-wavelength numerical errors in the solution that have a similar, albeit nonphysical, character. Given the value of the understanding that can be derived from robust phe- nomenological insights into such processes, it is crucial that the analysis techniques not give rise to spurious events that confound the effects of genuine physical processes. In this paper, we re-introduce dissipation elements as the collection of d-cells of a d-dimensional Morse-Smale com- plex that are formed by the intersection of the ascending d- manifold of a local minimum and the descending d-manifold of a local maximum. While the segmentation of the domain is the same as proposed by Wang and Peters [WP06], this new formulation has several advantages. The visualization community has considerable experience with exploration of these topological structures; the techniques for comput- ing them, extracting geometric information, and visualizing them in a multi-scale topological hierarchy are well estab- lished. Our software tool first precomputes and stores a dis- crete gradient vector field from which dissipation elements at multiple topological scales can either be extracted inter- c  2014 The Author(s) Computer Graphics Forum c  2014 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. DOI: 10.1111/cgf.12361