The adiabatic kernel of the nonhydrostatic (NH) HIRLAM with the semi-implicit semi-Lagrangian (SISL) integration scheme is presented in this paper. Our investigation continues the work, initiated in the Parts  I - III (Rõõm 2001, Männik and Rõõm 2001, Rõõm and Männik 2002), where the fundamentals of NH atmospheric dynamics in pressure-related coordinates were presented, and, on that basis, the NH explicit-Eulerian and semi-implicit (SI) Eulerian versions of HIRLAM were developed.
SISL has become the popular integration scheme in all advanced weather forecast systems in last two decades. The main advantage of SISL (in comparison with competing schemes like the SI Eulerian scheme or time-split-explicit scheme) is the significantly enhanced overall computational efficiency, which is achieved through substantial gain in numerical stability at the increased time step.
The SISL-ideology to integrate the hydrostatic (HS) primitive equations numerically was first proposed by Robert (1981, 1982), who proceeded from an earlier positive experience with the SI Eulerian scheme Three time level SI Eulerian scheme was proposed by Robert (1969); the first baroclinic multi-level SI Eulerian scheme for HS primitive equations was described in (Robert, Henderson, Thurnbull 1972). A baroclinic, multi-level, HS primitive-equation, three-time-level SISL model was first presented by Robert, Yee and Richie (1985). An alternative approach with two-time-level scheme was developed by Temperton and Staniforth (1987). In operational forecasts, SISL was implemented in the middle of the nineties of the last century. At ECMWF the two-time-level SISL was operationally launched in 1995 (Ritchie et al 1995). For HIRLAM, the two-time-level SISL scheme was introduced by McDonald and Haugen (1992), and further developed by McDonald (1995). Finally, McDonald (1998, 1999) carried out a further extensive investigation to improve the departure point evaluation. Developed by him non-iterative departure point calculation algorithm is currently in use at the operational HIRLAM.
The first NH, fully compressible (i.e. making use of complete, non-simplified set of dynamic equations) SISL was proposed already in 1990 (Tanguay, Robert and Laprise, 1990), but for operational forecasting it has gained momentum during the last years in connection with model transition into NH-resolution domain.
Adiabatic dynamics, applied in current NH SISL scheme, is the White model (White 1989), which represents a simplified version of complete NH pressure-coordinate equations. Roughly speaking, the White model is the simplest generalization of the hydrostatic, primitive-equation, pressure-coordinate dynamics which incorporates the vertical momentum equation and takes vertical acceleration into consideration. This closeness to HS model makes implementation of NH dynamics into existing HS environment of HIRLAM rather straightforward. The White model derivation from general elastic pressure-coordinate equations with description of main qualities is presented in detail in (Rõõm 2001). As comparison with the exact analytical solutions (Rõõm and Männik 1989), and with the 'full' elastic model (French NH Aladin) on the non-linear test flows have demonstrated (Männik 2003), there is no large difference between 'exact dynamics' and the White model results. The White model has been already applied with success in heretofore developed three-time-level, explicit-Eulerian (Männik and Rõõm 2001), and SI Eulerian (Rõõm, Männik 2002, Männik, Rõõm, Luhamaa 2003) schemes. In those models, an additional approximation of the surface pressure adjustment was introduced, which gave reason to call that approach 'anelastic pressure coordinate model', as the acoustic travelling waves were ompletely eliminated from dynamics By the way, using of terminology 'anelastic' served us a disservice, as it was often confused with anelasticity interpretations in shallow convection (constant reference density r = const) or deep convection (fixed reference density r = r0(z)) models. Actually, with the term 'anelastic' we tried just to underline that model lacks acoustic waves - exactly like the HS primitive-equation model does, likewise being anelastic with respect to the internal (vertically propagating) sound waves. In the current NH SISL model, we will restore the non-adjusted pressure treatment of the original White model, which, however, could be still called 'semi-anelastic' because it lacks internal acoustic mode due to non-divergence of three-dimensional (3D) velocity in pressure-coordinates.
The main reason for discarding with surface pressure adjustment was that the implicit treatment of linear development in SISL does not require such an approximation anymore. Adjustment is actually essential in the explicit-Eulerian scheme where it yields significant growth of computational efficiency, expressed in the increase of achievable time step, while in the implicit schemes, the time-step rise is achieved by other, independent means (just by implicit treatment of linear forces). More considerable reason, however, was the experimentally established fact that dynamics with the adjustment approximation may lead to a discontinuity of nonhydrostatic geopotential field at surface, when the time step exceeds the criticalCritical time step in the sense of the Courant-Fiedrichs-Lewy stability criterion. Finally, the non-adjusted, non-simplified model is simpler to deal with in the formal plane. Thus, the non-adjusted surface pressure evolution is restored in this paper.
Another major novelty consists in modifications of geopotential and surface pressure treatment prior to discretization. The neutral reference states are subtracted from geopotential and surface pressure in the very beginning, in the continuous equations already, confining the treatment to evolution of geopotential and pressure fluctuations. Ideologically this approach is similar to 'Eulerian advection of orography' method by Richie and Tanguay (1996). However, in the current treatment, the modification is applied before any discretization, which refers to generality of such an approach. The aim of the modification is elimination of large, dynamically passive fields, otherwise just being a source of additional noise in the numerical scheme.
The last model-specific modification, yet not the least one, is the application of height-dependent reference temperature T0(p) together with the accompanying height-dependent reference-state Brunt-Väisälä frequency N(p), both giving increased stability, as the non-linear residuals are minimized in the vertical development equations.
The NH model altogether aims to be an organic and straightforward extension of the HS SISL core to NH resolutions. Thus, except the necessary modifications of dynamic equations, almost all the numerical scheme is maintained from the HS parent. This includes the use of the two-time-level time-stepping with the complete maintenance of the departure point calculation procedures (McDonald 1998, 1999) and interpolation routines. And, of course, the diabatic counterpart, consisting the so-called 'physics', which is not concern of adiabatic core development, is maintained untouched, and is overtaken from HS model without any change and modification.

Footnotes:

Three time level SI Eulerian scheme was proposed by Robert (1969); the first baroclinic multi-level SI Eulerian scheme for HS primitive equations was described in (Robert, Henderson, Thurnbull 1972). .
By the way, using of terminology 'anelastic' served us a disservice, as it was often confused with anelasticity interpretations in shallow convection (constant reference density r = const) or deep convection (fixed reference density r = r0(z)) models. Actually, with the term 'anelastic' we tried just to underline that model lacks acoustic waves - exactly like the HS primitive-equation model does, likewise being anelastic with respect to the internal (vertically propagating) sound waves. .
Critical time step in the sense of the Courant-Fiedrichs-Lewy stability criterion..

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