B-splines as a Tool to Solve Constraints in Non-Hydrostatic Forecast Model

TL;DR

This paper explores the use of B-splines in finite element methods to handle vertical constraints in non-hydrostatic forecast models, aiming to improve the mathematical framework for vertical operator representation.

Contribution

It develops a theoretical framework for representing vertical operators with B-splines that preserves the C1 constraint in non-hydrostatic models.

Findings

Established invertibility relations between integral and derivative operators.

Developed B-spline based representations of vertical operators.

Provided a theoretical basis for implementation in NWP systems.

Abstract

Finite elements has been proven to be an useful tool to discretize the vertical coordinate in the hydrostatic forecast models allowing to define model variables in full levels so that no staggering is needed. In the non-hydrostatic case a constraint in the vertical operators appears (called C1) that does not allow to reduce the set of semi-implicit linear equations to a single equation in one variable as in the analytic case. Recently vertical finite elements based in B-splines have been used with an iterative method to relax the C1 constraint. In this paper we want to develop properly some representations of vertical operators in terms of B-splines in order to keep the C1-constraint. An invertibility relation between integral and derivative operators between vertical velocity and vertical divergence is also presented. The final scope of this paper is to provide a theoretical framework of development of finite element vertical operators to be implemented in the ALADIN-HIRLAM nwp system