Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. (2012) A three-dimensional monotone and conservative semi-Lagrangian scheme (SLICE-3D) for transport problems M. Zerroukat* and T. Allen Met Office, Exeter, UK *Correspondence to: M. Zerroukat, Met Office, FitzRoy Road, Exeter EX1 3PB, UK. E-mail: mohamed.zerroukat@metoffice.gov.uk The Semi-Lagrangian Inherently Conserving and Efficient (SLICE) scheme is extended to three-dimensional transport on the sphere (SLICE-3D). This paper presents details of the scheme for a longitude–latitude C-type staggering grid. The scheme is applied to a passive transport with an analytically defined wind field that can be considered as an ‘idealized climate circulation’. The performance of SLICE- 3D is assessed against the widely used standard non-conservative Semi-Lagrangian (SL) scheme. The computational performance of the scheme for multiple species transport on parallel machines is also examined. Copyright c  2012 British Crown copyright, the Met Office. Published by John Wiley & Sons Ltd. Key Words: remapping; conservation; advection; sphere; semi-Lagrangian Received 9 December 2010; Revised 12 September 2011; Accepted 21 December 2011; Published online in Wiley Online Library Citation: Zerroukat M, Allen T. 2012. A three-dimensional monotone and conservative semi-Lagrangian scheme (SLICE-3D) for transport problems. Q. J. R. Meteorol. Soc. DOI:10.1002/qj.1902 1. Introduction Semi-Lagrangian (SL) schemes are an integral part of many atmospheric models because of their superior computational efficiency and stability compared to Eulerian schemes. For latitude–longitude grids, this stability property is very important in the polar regions where the grid lines converge. This results in high Courant numbers in the polar regions, which can be problematic for Eulerian schemes which may require very small time steps. The unconditional stability of SL schemes allows the time step to be chosen by accuracy and computational efficiency considerations rather than by a stability constraint. However, a common shortfall of interpolating SL schemes is the lack of mass conservation. Although mass conservation is not critical for short-time simulations, it is very important for long-period simulations such as those of climate studies. Over a long simulation with a non-conservative scheme, the total mass can drift significantly if no correction is applied. Mass conservation has been dealt with either mass-fixing schemes or conservative remappings. Traditionally, mass- fixing schemes (Priestley, 1993; Gravel and Staniforth, 1994; Williamson and Rasch, 1994; Huang, 1997; Bermejo and Conde, 2002; Schraner et al., 2008) have been the main route for restoring diagnostically the global mass conservation constraint. These schemes are very appealing owing to their simplicity and the negligible cost of incorporating them within existing SL models, especially for three-dimensional problems. However, the main criticism of these fixers is the ad hoc nature of where the deficit/surplus is added or subtracted. Recently there have been a number of conservative remapping schemes developed to inherently conserve mass (Ran˘ ci´ c, 1992, 1995; Nair et al., 2002, 2003; Nair and Machenhauer, 2002; Lauritzen et al., 2006, 2010; Lauritzen and Nair, 2008; Zerroukat et al., 2002, 2004, 2005, 2006, 2009). Unlike mass-fixing methods, the conservative schemes, which are basically incremental remappings (Dukowicz and Baumgardner, 2000), are more mathematically based and inherently satisfy local and global mass conservation. To the knowledge of the authors, all the reported conservative remappings are applied to two- dimensional problems. Three-dimensional remappings can be relatively complex and computationally expensive owing to the extra geometric computations needed and this has been the main reason for their non-existence in the literature and their lack of application for any real three-dimensional problems. The complexity of three-dimensional remappings also makes their parallel implementation difficult. The use of a floating (Lagrangian) vertical coordinate can alleviate Copyright c  2012 British Crown copyright, the Met Office. Published by John Wiley & Sons Ltd.