## Abstract

To extend the numerical stability limit over steep slopes, a truly horizontal pressure-gradient discretization based on the ideas formulated by Mahrer in the 1980s has been developed. Conventionally, the pressure gradient is evaluated in the terrain-following coordinate system, which necessitates a metric correction term that is prone to numerical instability if the height difference between adjacent grid points is much larger than the vertical layer spacing. The alternative way pursued here is to reconstruct the pressure gradient at auxiliary points lying at the same height as the target point on which the velocity is defined. This is accomplished via a second-order Taylor-series expansion in this work, using the hydrostatic approximation to transform the second derivatives into first derivatives to facilitate second-order accurate discretization in the presence of strong vertical grid stretching. Moreover, a reformulated lower boundary condition is used that avoids the extrapolation of vertical derivatives evaluated in potentially very thin layers. A sequence of tests at varying degrees of idealization reveals that the truly horizontal pressure-gradient discretization improves numerical stability over steep slopes for a wide range of horizontal mesh sizes, ranging from a few hundreds of meters to tens of kilometers. In addition, tests initialized with an atmosphere at rest reveal that the spurious circulations developing over steep mountains are usually smaller than for the conventional discretization even in configurations for which the latter does not suffer from stability problems.

*Corresponding author address:*Günther Zängl, Deutscher Wetterdienst, Frankfurter Straße 135, D-63067 Offenbach, Germany. E-mail: guenther.zaengl@dwd.de