TEMPORAL ACCUMULATION OF FIRST-ORDER LINEARIZATION ERROR FOR SEMI-LAGRANGIAN PASSIVE ADVECTION

The tangent linear model (TLM) is obtained by linearizing the governin g equations around a space- and time-dependent basic state referred to as the trajectory. The TLM describes to first-order the evolution of perturbations in a nonlinear model and it is now used widely in many a pplications including four-dimensional data assimilation. This paper i s concerned with the difficulties that arise when developing the tange nt linear model for a semi-Lagrangian integration scheme. By permittin g larger time steps than those of Eulerian advection schemes, the semi -Lagrangian treatment of advection improves the model efficiency. Howe ver, a potential difficulty in linearizing the interpolation algorithm s commonly used in semi-Lagrangian advection schemes has been describe d by Polavarapu et al., who showed that for infinitesimal perturbation s, the tangent linear approximation of an interpolation scheme is corr ect if and only if the first derivative of the interpolator is continu ous at every grid point. Here, this study is extended by considering t he impact of temporally accumulating first-order linearization errors on the limit of validity of the tangent linear approximation due to th e use of small but finite perturbations. The results of this paper are based on the examination of the passive advection problem, In particu lar, the impact of using incorrect interpolation schemes is studied as a function of scale and Courant number. For a constant zonal wind lea ding to an integral value of the Courant number, the first-order linea rization errors are seen to amplify linearly in time and to resemble t he second-order derivative of the advected field for linear interpolat ion and the fourth-order derivative for cubic Lagrange interpolation. Solid-body rotation experiments on the sphere show that in situations where linear interpolation results in accurate integrations, the limit of validity of the TLM is nevertheless reduced. First-order cubic Lag range linearization errors are smaller and affect small scales. For th is to happen requires a wind configuration leading to a persistent int egral value of the Courant number. Regions where sharp gradients of th e advected tracer field are present are the most sensitive to this err or which is nevertheless observed to be small. Finally, passive tracer s experiments driven by winds obtained from a shallow-water model inte gration confirm that higher-order interpolation schemes (whether corre ct or not) give similar negligible linearization errors since the prob ability of having the upstream point being located exactly on a grid p oint is vanishingly small.