Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 136: 146 – 154, January 2010 Part A Coupling a mass-conserving semi-Lagrangian scheme (SLICE) to a semi-implicit discretization of the shallow-water equations: Minimizing the dependence on a reference atmosphere J. Thuburn, a M. Zerroukat, b† N. Wood b * † and A. Staniforth b† a School of Engineering, Computing and Mathematics, University of Exeter, UK b Met Office, Exeter, UK *Correspondence to: Nigel Wood, Met Office, FitzRoy Road, Exeter EX1 3PB, UK. E-mail: Nigel.Wood@metoffice.gov.uk. † The contributions of M. Zerroukat, N. Wood and A. Staniforth were written in the course of their employment at the Met Office, UK, and are published with the permission of the Controller of HMSO and the Queen’s Printer for Scotland. In a recent paper, a conservative semi-Lagrangian mass transport scheme SLICE has been coupled to a semi-implicit semi-Lagrangian scheme for the shallow-water equations. The algorithm involves the solution at each timestep of a nonlinear Helmholtz problem, which is achieved by iterative solution of a linear ‘inner’ Helmholtz problem; this framework, as well as the linear Helmholtz operator itself, are the same as would be used with a non-conservative interpolating semi- Lagrangian scheme for mass transport. However, in order to do this, a reference value of geopotential was introduced into the discretization. It is shown here that this results in a weak dependence of the results on that reference value. An alternative coupling is therefore proposed that preserves the same solution framework and linear Helmholtz operator but, at convergence of the nonlinear solver, has no dependence on the reference value. However, in order to maintain accuracy at large timesteps, this approach requires a modification to how SLICE performs its remapping. An advantage of removing the dependence on the reference value is that the scheme then gives consistent tracer transport. Copyright c  2010 Royal Meteorological Society and Crown Copyright. Key Words: area remapping; consistent tracer transport; Helmholtz problem; nonlinear transport; reference profile Received 5 March 2009; Revised 21 July 2009; Accepted 14 September 2009; Published online in Wiley InterScience 19 January 2010 Citation: Thuburn J, Zerroukat M, Wood N, Staniforth A. 2010. Coupling a mass-conserving semi-Lagrangian scheme (SLICE) to a semi-implicit discretization of the shallow-water equations: Minimizing the dependence on a reference atmosphere. Q. J. R. Meteorol. Soc. 136: 146 – 154. DOI:10.1002/qj.517 1. Introduction Arguably the primary research focus on semi-implicit semi-Lagrangian (SISL) schemes over the last decade has been the development of locally conservative semi- Lagrangian transport schemes (e.g. Machenhauer et al., 2008, and references therein). With few exceptions, to be discussed below, these schemes have only been applied to the passive transport of scalars. To achieve conservation of total fluid mass, such schemes need to be extended from passive (linear) transport, in which the scalar field does not feed back on the velocity field, to interactive (nonlinear) transport, in which the scalar field, namely the mass, height, or geopotential, interacts with the velocity field. The challenge is to do so whilst ensuring stability of the fast dynamical modes (the gravity waves in a shallow-water context) with the large timesteps that semi-Lagrangian advection permits. One way of achieving this is to couple Copyright c  2010 Royal Meteorological Society and Crown Copyright.