## Abstract

A semi-Lagrangian treatment of the 1D passive tracer equation using prescribed piecewise-continuous interpolating functions is considered. Whether the process of localizing the upstream position affects the adjoint counterpart of the associated tangent linear model when the Courant number varies spatially is evaluated. An analysis is done and the accuracy of the numerical adjoint is found to be reduced when the upstream positions are irregularly distributed around the grid point at which the adjoint is evaluated. This irregular distribution follows an acceleration or a deceleration pattern and is a consequence of a discontinuity in the integer value of the Courant number. The cubic Lagrange interpolation reduces this lack of accuracy when compared to the linear interpolation. Numerical 1D adjoint experiments confirm the analysis. The adjoint of the 2D passive tracer equation on the sphere is studied in the context of a solid body rotation. Numerous irregular distributions are induced by a sequence of discontinuities in the integer value of the longitudinal Courant number near the poles. Numerical 2D adjoint experiments reveal that the cubic spline interpolation could also be affected, but to a lesser extent than the cubic Lagrange interpolation. Those errors are visible even if the Lipschitz criterion giving bounds to the intensity in the variation of the Courant number is respected.

*Corresponding author address:* Recherche en PrÃ©vision NumÃ©rique, 2121 Route Trans-Canadienne, Dorval, QuÃ©bec, H9P 1J3, Canada.

Email: monique.tanguay@ec.gc.ca