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

Abstract

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.