A Lagrange multiplier approach for the metric terms of semi‐Lagrangian models on the sphere

Abstract The horizontal momentum equation of the shallow‐water equations on the sphere is written in 3‐D vector form using the undetermined Lagrange multiplier method. The direct time discretization of this equation is proposed as an alternative to the usual approach, i.e. time discretization after explicit removal of the constraint by the use of generalized coordinates. The proposed procedure is applicable to any coordinate system and eliminates an instability associated with the metric term in a spherical grid spectral semi‐Lagrangian model. For a polar‐stereographic grid finite‐element model, equivalent results and a gain in efficiency are obtained. The formalism can be extended to multilevel models.