An improved regularization for time‐staggered discretization and its link to the semi‐implicit method

A regularized time-staggered discretization of the shallow-water equations has recently been proposed. Here, a new form of the regularization operator is presented. This form addresses a weakness in the original formulation so that now the discretization preserves the analytic forced response. Since the reformulation takes account of the forcing terms in the equations, if the unregularized equations are in balance, in the sense that r.(Du/Dt) vanishes, then the regularized equations maintain this balance. of the regularization is governed by two parameters: , which is a smoothing length scale; and , which is a further smoothing parameter. Among other results, linear analysis of the continuous equations showed that the forced response of the regularized equations is close to that of the unregularized equations provided is chosen so that 1 and is chosen to be much smaller than the Rossby radius of deformation, i.e., 2 L 2 . The regularized equations were then discretized using an explicit, time-staggered leapfrog scheme. Again, linear analysis of the discrete equation set showed that the regularized, time-staggered leapfrog discretization only yields a similar result to the analytic forced response when (/L R) 2 1 and 2 1. (In contrast the semi-implicit discretization yields the exact analytic forced response.) Additionally, the discussion of the forced response was in terms of the regularized depth field. Further details of the scheme and its advantages can be found in F05. Here a further development of the regularization procedure is proposed which addresses the shortcom- ings of the forced response of the regularized equations (both the continuous and the explicit, time-staggered discrete equations). The resulting forced response, now in terms of the unregularized depth field, main- tains exactly the analytic response. The new scheme has the advantage over the original one, that the regularization only impacts the unbalanced components of the flow. The structure of the paper is similar to that of F05. The new regularization procedure for both the continuous equations and the discrete ones is set out in section 2. Those equations are linearized in section 3 enabling the eect