On mass - conservation in high - order high - resolution rigorous 1 remapping schemes on the sphere

6 It is the purpose of this short article to analyze mass conservation in high-order rigorous 7 remapping schemes, which contrary to flux-based methods, relies on elaborate integral con8 straints over overlap areas and reconstruction functions. For applications on the sphere these 9 integral constraints may be violated primarily due to inexact or ill-conditioned integration 10 and we propose a generic, local and multi-tracer efficient method that guarantees that the 11 integral constraints are satisfied in discrete space irrespective of the accuracy of the numeri12 cal integration method and slight inaccuracies in the computation of overlap areas. We refer 13 to this method as enforcement of consistency as it is based on integral constraints valid in 14 continuous space. The consistency enforcement method is illustrated in idealized transport 15 tests with CSLAM in HOMME (Conservative Semi-LAgrangian Multi tracer scheme in the 16 High Order Method Modeling Environment) where the analytic integrals, that were found to 17 be ill-conditioned at certain resolutions and flow conditions, have been replaced with robust 18 quadrature. This violates mass-conservation, however, with the consistency enforcement 19 method mass-conservation is inherent even with low-order quadrature and renders rigor20 ous remap schemes such as CSLAM (that was previously limited to gnomonic cubed-sphere 21 grids) mass-conservative on any spherical grid. 22