Adjoint for advection schemes on the sphere in ICON model

Among the most advanced and sophisticated methods for state analysis of an atmospheric system is the four dimensional variational data assimilation. The numerically challenging task of this approach is the development and application of the adjoint model components. For tracer transport in fluid dynamics accuracy of numerical advection schemes is vital. It is even more important for applications in space-time variational data assimilation with adjoint model version. We propose novel straightforward and efficient approach - artificial source term method - for adjoint advection solver development. It has several benefits compared to traditional adjoint model building technique. One of the attractive features of the new approach is that it reuses existing advection solver code, thus resulting into significant reduction of time needed for adjoint solver development. The stability, accuracy and convergence of the adjoint schemes are investigated. The method is implemented and evaluated for the linear advection equation on the sphere in the Icosahedral Nonhydrostatic Model (ICON). Adjoint solvers developed by the conventional and the methods are compared against each other on a collection of standard advection test cases and variational data assimilation test cases developed here. The advantages of the artificial source term method is especially obvious in case of monotonic advection equation solvers, granting, e.g. absence of oscillations and nonphysical negative concentrations.