B ‐splines as a tool to solve constraints in a non‐hydrostatic forecast model

The finite‐element method has proved to be a useful tool to discretize the vertical coordinate in hydrostatic forecast models, allowing the definition of model variables at full levels so that no staggering is needed. In the non‐hydrostatic case, a constraint in the vertical operators appears (called C1), which allows the set of semi‐implicit linear equations to be reduced to a single equation in one variable as in the analytic case. Recently, vertical finite elements based on B ‐splines have been used with an iterative method to relax the C1 constraint. In this article, we aim to develop representations of vertical operators fully in terms of B ‐splines, in order to keep the C1 constraint. An invertibility relation between integral and derivative operators is also presented in order to have a bijective relationship between the vertical velocity and the vertical divergence. The final scope of this article is to provide a theoretical framework of development of finite‐element vertical operators to be implemented in the ALADIN–HIRLAM numerical weather prediction system.