## 1. Introduction

In a recent study of Lucarini and Ragone (2011) the energy budgets of a bunch of publicly available climate models and their subsystems (atmosphere, land, ocean) have been investigated with the outcome that almost none of them simulates a closed energy budget. Most atmospheric subsystems feature a positive net energy balance with biases of the order of 1 W m^{−2} indicating a too weak radiative cooling. This spurious heating is partly attributed to the missing reinjection of dissipated kinetic energy as thermal energy. In state-of-the art dry dynamical cores, the latter mechanism is often crudely or not at all included. As shown by Becker (2003), a mechanistic hydrostatic spectral model generates a spurious thermal forcing of about 2 W m^{−2} if the dissipative heating is neglected. Those kinds of errors might drive the model climate towards an improbable dynamical state and thus might contribute to increase the uncertainty in climate sensitivity (Burkhardt and Becker, 2006; Lucarini and Ragone, 2011).

The term ‘dynamical core’ is not clearly defined in the literature. There is an agreement that it contains the resolved fluid flow component (Williamson, 2007). Discrepancies in perception and implementation exist for the diffusion and damping terms in a dynamical core. Some people consider them as purely numerical measures that are needed to suppress nonlinear instability. An alternative and more plausible viewpoint considers them as a physical parametrization of subgrid-scale waves and turbulence. In any case, those added terms serve to prevent the build-up of kinetic energy at the truncation scale.

In this paper we want to be specific and define a *basic* and an *extended* dynamical core. The *basic* dynamical core only considers reversible (frictionless and isentropic) dynamics in which the global integrals of entropy, mass, and total energy are conserved. Mass and entropy follow local conservation laws. Conservation of global total energy reflects local energy conversions between the different sub-energies (kinetic, potential, and internal energies). In addition, the *extended* dynamical core includes the frictional dissipation due to waves and turbulence and allows the global integral of entropy to increase while mass and energy are still conserved. This model configuration can be called adiabatic, because the whole atmosphere is still a closed system. The distinction between a basic and an extended dynamical core is generally impossible for most state-of-the-art atmospheric flow solvers since damping and diffusion are often inherent to the basic numerics, for instance in the form of implicit off-centring in the time integration schemes or diffusive/scale-selective properties of spatial operators. Jablonowski and Williamson (2011) give an overview of diffusion, filters, and fixers in general circulation models (GCMs).

The vertical discretization scheme of Simmons and Burridge (1981) for the primitive equations is an excellent example of an ingredient of the basic dynamical core as defined above. Similar efforts for non-hydrostatic models in terrain-following coordinates have not yet been pursued as rigorously. The intention of the current paper is to fill this gap and present a basic dry dynamical core that fulfils the aforementioned conservation laws by design for spatial discretization, and to a very high degree for time discretization. The spatial discretization uses Poisson brackets that conserve energy by definition by accounting for correct energy conversions between kinetic, internal, and potential energy. Mass and entropy, in the form of the mass-weighted potential temperature,* are conserved because their prognostic equations are employed in flux form. Poisson brackets for non-hydrostatic atmospheric dynamics will be introduced in section 2 and their discretization will be sketched in section 3. Time discretization (section 4) employs a method that considers the integration by parts rule in time.

Once this basic dynamical core has been set up unbounded energy increase is prevented even if the downscale kinetic energy cascade reaches the truncation scale. Only then is the sense to employ a Smagorinsky diffusion scheme in the momentum equation *and* to account for the accompanying dissipative heating according to the energy conservation law evident. This mechanism constitutes a parametrization of unresolved dynamical scales that is necessary to balance the forward cascade of kinetic energy. Only if the Smagorinsky-type diffusion is employed does the dissipative heating indeed lead to entropy increase as required by the second law of thermodynamics. In section 5 it will be shown how this diffusion/dissipation mechanism works, even in a quite complicated grid-staggering environment.

This sketched modelling framework can now serve as a basis for a full moist turbulent GCM as envisaged in Gassmann and Herzog (2008, hereafter GH08).

Designing a new dynamical core for the application on the spherical Earth requires the definition of a computational grid. Regarding a latitude–longitude grid the polar singularities and the associated small grid distances prevent us from using a grid-point model, especially if we want to avoid not physically motivated filter techniques for internal gravity waves near the poles. In recent years, a multitude of other grids have been suggested, among them the cubed sphere grid and icosahedron based hexagonal and triangular grids (cf. Williamson, 2007). Icosahedron-based grids configured with grid smoothing, for instance the ‘spring dynamics’ optimization of Tomita *et al.* (2002), yield a very homogeneous coverage of the the Earth with grid cells. Therefore we chose a grid configuration based on the icosahedron for our model.

Among the different choices for grid staggering the C-grid sticks out because of favourable numerical properties when discretizing, for example, shallow-water equations as the prototype problem for fluid flow. In a C-grid the mass point is defined in the centre of a grid box and the normal velocity components are located at the grid box interfaces. Therefore the discrete-wave dispersion for gravity waves does not reveal stationary waves on the shortest resolvable scale. Geopotential gradients imply flow divergence/convergence in the flow field (or vice versa) in a very local way. The associated conversions between potential and kinetic energy conform to a local integration by parts rule. The C-grid gives the opportunity of a natural definition of vorticity. Hence the vorticity dynamics can easily be uncovered on this type of grid staggering.

In recent years, the historical obstacles regarding applicability of the hexagonal C-grid were successively erased. Due to the appearance of three and not two horizontal velocity components, an additional wave mode shows up in the discrete-wave dispersion equation on an equilateral mesh. If its frequency does not vanish a spurious Rossby mode appears that spoils the vorticity dynamics severely. Thuburn (2008) showed how to understand this spurious geostrophic mode and how its frequency can be forced to vanish by a special reconstruction rule for tangential wind. For the non-uniform hexagonal C-grid, which we recognize as a Voronoi diagram (Ringler *et al.*, 2010), several papers have dealt with the generalization of this reconstruction rule and with the derivation of energy- or enstrophy-conserving schemes (Thuburn *et al.*, 2009; Ringler *et* *al.*, 2010; hereafter TRSK). However, there are still issues that are not yet fully understood. Among them is the perception of the dual grid (where the vorticity is defined) as the counterpart of the hexagonal primal grid (where the divergence is defined). TRSK assume and set triangles as the dual entities. In contrast, Gassmann (2011) found from the Laplacian of the Helmholtz decomposition of a horizontal vector that the vorticity is to be located at a ‘trinity’ of rhombi (see also Appendix A of the present paper). Another problem not yet tackled in the literature dealing with the hexagonal C-grid is the prevention of a nonlinear instability, first observed for a quadrilateral C-grid by Hollingsworth *et al.* (1983). The vector-invariant form of the velocity advection term can give rise to an inadequate balance equation, which in turn spoils the vertical wind field so that the model might crash if no hindering measures are taken. Appendix B of the present article will be dedicated to this issue.

A quality check of the new ICON-IAP model will be presented in section 6. Here we focus on three widely accepted test cases that reveal the quality of the fluid solver. Non-hydrostatic scales and terrain-following coordinates are well captured, as will be demonstrated with the linear flow over a mountain with modulated small-scale topography –a test case suggested by Schär *et al.* (2002). The basic and extended dynamical core versions will be investigated in the framework of the baroclinic wave test of Jablonowski and Williamson (2006). This test case is usually only run until the baroclinic wave has reached its mature state and dissipation has not yet started. Within this moderately nonlinear development the basic dynamical core shows excellent energy conservation. Extending this test case further in time and switching on the diffusion/dissipation terms demonstrates energy conservation even in that strongly nonlinear case. This test case configuration is a valuable intermediate evaluation step with a special focus on consistent energetics before performing the Held–Suarez test (Held and Suarez, 1994) in which the atmosphere is no longer considered as a closed system, but is open to space and surface.