A global hexagonal C-grid non-hydrostatic dynamical core (ICON-IAP) designed for energetic consistency

Transcription

1 Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75, January 03 A A global hexagonal C-grid non-hydrostatic dynamical core (ICON-IAP) designed for energetic consistency Almut Gassmann* Leibniz-Institut für Atmosphärenphysik an der Universität Rostock e.v., Kühlungsborn, Germany *Correspondence to: A. Gassmann, Leibniz-Institut für Atmosphärenphysik an der Universität Rostock e.v., Schloss-straße 6, 85 Kühlungsborn, Germany. gassmann@iap-kborn.de This study describes a new global non-hydrostatic dynamical core (ICON-IAP: Icosahedral Nonhydrostatic model at the Institute for Atmospheric Physics) on a hexagonal C-grid which is designed to conserve mass and energy. Energy conservation is achieved by discretizing the antisymmetric Poisson bracket which mimics correct energy conversions between the different kinds of energy (kinetic, potential, internal). Because of the bracket structure this is even possible in a complicated numerical environment with (i) the occurrence of terrain-following coordinates with all the metric terms in it, (ii) the horizontal C-grid staggering on the Voronoi mesh and the complications induced by the need for an acceptable stationary geostrophic mode, and (iii) the necessity for avoiding Hollingsworth instability. The model is equipped with a Smagorinsky-type nonlinear horizontal diffusion. The associated dissipative heating is accounted for by the application of the discrete product rule for derivatives. The time integration scheme is explicit in the horizontal and implicit in the vertical. In order to ensure energy conservation, the Exner pressure has to be offcentred in the vertical velocity equation and extrapolated in the horizontal velocity equation. Test simulations are performed for small-scale and global-scale flows. A test simulation of linear non-hydrostatic flow over a rough mountain range shows the theoretically expected gravity wave propagation. The baroclinic wave test is extended to 40 days in order to check the Lorenz energy cycle. The model exhibits excellent energy conservation properties even in this strongly nonlinear and dissipative case. The Held Suarez test confirms the reliability of the model over even longer timescales. Copyright c 0 Royal Meteorological Society Key Words: global atmospheric modelling; Lorenz energy cycle; hexagonal C-grid Received 09 November 0; Revised March 0; Accepted 6 March 0; Published online in Wiley Online Library June 0 Citation: Gassmann A. 03. A global hexagonal C-grid non-hydrostatic dynamical core (ICON-IAP) designed for energetic consistency. Q. J. R. Meteorol. Soc. 39: DOI:0.00/qj.960. Introduction In a recent study of Lucarini and Ragone (0) 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 W m 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 (003), a mechanistic hydrostatic spectral model generates a spurious thermal forcing of about W m if the dissipative heating is neglected. Those kinds of errors might Copyright c 0 Royal Meteorological Society

2 A hexagonal Non-Hydrostatic C-grid Dynamical Core 53 drive the model climate towards an improbable dynamical state and thus might contribute to increase the uncertainty in climate sensitivity (Burkhardt and Becker, 006; Lucarini and Ragone, 0). The term dynamical core is not clearly defined in the literature. There is an agreement that it contains the resolved fluid flow component (Williamson, 007). 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-theart 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 (0) give an overview of diffusion, filters, and fixers in general circulation models (GCMs). The vertical discretization scheme of Simmons and Burridge (98) 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 massweighted 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 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 This comparatively simple equivalence between the entropy and the potential temperature is valid for a dry model configuration; for a moist atmosphere this equivalence gets lost. 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 Smagorinskytype 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 (008, 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, 007). Icosahedronbased grids configured with grid smoothing, for instance the spring dynamics optimization of Tomita et al. (00), 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 (008) 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., 00), several papers have dealt with the generalization of this reconstruction rule and with the derivation of energyor enstrophy-conserving schemes (Thuburn et al., 009; Ringler et al., 00; 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 Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75 (03)

3 54 A. Gassmann triangles as the dual entities. In contrast, Gassmann (0) 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. (983). 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. (00). The basic and extended dynamical core versions will be investigated in the framework of the baroclinic wave test of Jablonowski and Williamson (006). 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, 994) in which the atmosphere is no longer considered as a closed system, but is open to space and surface.. Continuous dynamics and discretization concept We start out from the Poisson bracket form of the nonhydrostatic compressible dynamics as defined in GH08. In that paper the moist turbulent form of the equations was given. Here our focus lies on the dry dynamics as the fundamental configuration of a full GCM. The general dynamics are described by F t ={F, H}+(F, f r ) + (F, Q) () (equation (45) in GH08). The Poisson bracket constitutes the basic dynamical core and consists of three sub-brackets: ( δf {F, H} = V δv ωa ϱ δh ) dτ () δv ( δf δh V δϱ δv δh δf ) dτ (3) δϱ δv ( δf δh (θ δ θ v ) δh ) δf (θ δ θ δv ) dτ (4) V (equation (46) in GH08) with ϱ the density, θ the potential temperature, θ = ϱθ, v the velocity vector, ω a the absolute vortex vector, dτ the volume element, H the Hamiltonian (energy functional), and F an arbitrary functional of the dynamical variables v, ϱ, and θ. The forcing terms that are added in the extended dynamical core configuration read δf (F, f r ) = δv ϱ ϱv v dτ (5) (F, Q) = V V δf δ θ ε dτ (6) c p π (equations (47) and (48) in GH08). The former (Eq. (5)) is the turbulent friction and the latter (Eq. (6)) the associated frictional heating with ε = ϱv v.. v > 0. Furthermore, π is the Exner pressure and c p the heat capacity at constant pressure. The Hamiltonian of dry atmospheric flow sums the kinetic (K), potential ( ), and internal (I) energy contributions: ( ) H(v, ϱ, θ) = ϱv + ϱgz + ϱc v T dτ, (7) V with g the gravity acceleration, z geometric height, T temperature, and c v the heat capacity at constant volume. Required functional derivatives of H are δh δv = ϱv δh δϱ = v + gz δh δ θ = c pπ. (8) Before sketching the discretization procedure for this dynamics, we have to discuss the choice of the given special form of the bracket (Eqs () (4)). Some important properties are associated with Poisson brackets: antisymmety {A, B} = {B, A}, the Jacobi identity {{A, B}, C} = {A, {B, C}} {B, {A, C}}, and conservation of global invariants (Casimirs). To prove the Jacobi identity, the Poisson bracket must be brought into Lie Poisson form where the bracket expression becomes linear in the dependent variables. The dependent variables of the Hamiltonian are thereby transformed into the translational momenta m = ϱv, density ϱ, and entropy density σ = ϱs, as given in Morrison (998). If the Coriolis force is included, a suitable transformation might alternatively envisage the angular momenta (Roulstone and Brice, 995). Global invariants (Casimirs) for geophysical fluids are discussed by many authors. Here we want to cite Bannon (003), who even extends his investigation to a binary geophysical fluid. The most intriguing Casimir in our case is Ertel s potential vorticity. Vorticity-related Casimirs are most easily accessible via the Poisson bracket form (Eqs () (4)), whereas the Lie Poisson form veils the ubiquitous vorticity nature of the atmospheric flow. The numerical discretization of Poisson brackets will generally lead to a loss of properties like the Jacobi identity or the conservation of Casimirs. One can still define discretizations that conserve additionally one Casimir if the dynamics are brought into a Nambu bracket form (a threefold antisymmetric bracket). Salmon (007) has exploited the potential enstrophy as the Casimir for his discretization of the shallow-water equations. In the case considered in the present paper, however, there are several obstacles that prevent us from exploiting more properties than the antisymmetry of the Poisson bracket (Eqs () (4)):. The discretization of Poisson brackets always presumes a correct behaviour of the linear dynamics. On hexagonal and triangular C-grids, constraints have to Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75 (03)

4 A hexagonal Non-Hydrostatic C-grid Dynamical Core 55 be obeyed that guarantee the linear dependency of the three horizontal velocity components and lead to the spurious-mode-free linear wave dispersion relation (Thuburn, 008; Gassmann, 0). Sommer and Névir (009) used a Nambu bracket discretization for shallow-water equations similar to Salmon (007) on a triangular C-grid which turned out to be problematic in a multilayer shallow-water version of this code (Griewank, 009). The results exhibited a characteristic checkerboard pattern in the divergence field, which is a clear indication for the non-fulfilment of the linear dependency among three horizontal vector components (Gassmann, 0). In conclusion, it is required to meet constraints posed by the linear wave dynamics before trying to impose more constraints hidden in any bracket discretization which is guided by the desire to treat the nonlinear equations correctly.. Non-hydrostatic compressible equations are difficult to cast in a Nambu bracket form. Névir and Sommer (009) and also GH08 have followed this path, but they need helicity as a constituting Casimir for one of their sub-brackets. However, helicity is only conserved and thus a true Casimir for incompressible flows. Even if one would follow their line, the numerical implementation turns out to be difficult and numerically expensive because of the additional need to suppress an instability first observed by Hollingsworth et al. (983) (see Appendix B of the present paper). Keeping the form Eqs () (4) is nevertheless advantageous as it clearly highlights the energy conversions that are relevant for understanding the atmospheric energy cycle. Lorenz (967) highlights the spatially averaged energy conversions as (I + )/ t = Q nf + D C (9) K/ t = C D, (0) wherec is the adiabatic conversion rate, D is the dissipation rate and Q nf is the non-frictional heating rate. The bracket formulation Eqs () (4) matches the adiabatic conversion rate C and we can write {I +, H} = [gz (ϱv) + c p π (θϱv)]dτ () V {K, H} = [ϱv gz + ϱvc p θ π]dτ. () V In a similar manner, Eqs (5) and (6) describe the dissipation term D in Lorenz s perception. Even though the validity of the term Poisson bracket as a numerically discretized entity is limited to the single property of the antisymmetic structure in the following, we will retain this phrase throughout the paper. Our task is now to discretize the energy functional H (Eq. (7)) and the Poisson bracket (Eqs () (4)). In both cases, the integrals are discretized as a sum over all grid boxes. The following spatial operators and forcing terms have to be defined on the numerical grid:. divergence, needed in Poisson bracket Eqs (3) and (4);. vorticity, needed in Poisson bracket Eq. (); 3. higher-order advection scheme for tracers, needed in Eq. (4) to specify θ at the faces of a grid box volume; t w e T e u e N e w c r dq t dq n v e dq t k + g k + q z q z q n c g k u q n e k c k g k Figure. Top and side views of the grid structure. For further explanation see text. 4. vector reconstruction rule, needed in Poisson bracket Eq. (); 5. the nonlinear momentum diffusion (Eq. (5)) and the associated frictional heating (Eq. (6)). The first two tasks in particular need careful consideration of those operators in terrain-following coordinates. Note that we do not need to specify a gradient operator. As it is the dual of the divergence, it is generated automatically during the discretization procedure. 3. Spatial discretization 3.. Grid staggering and nomenclature of terrain-following coordinates The grid structure is displayed in Figure. C-grid staggering is used in the horizontal, and L-grid staggering is employed in the vertical. Local coordinate system information at each edge has to be carried explicitly. The normal unit vector N e points inward for the sketched cell; for the neighbouring one it points outward. N e is needed to specify the direction of the horizontal velocity components. The tangential unit vector T e forms an orthogonal right-hand system with the normal unit vector and the vertical unit vector k. T e is necessary to represent the horizontal vorticity components. We introduce abbreviations for edges (e), cells (c), vertices (v) and rhombi (r), which are defined as two triangles that share one edge. Great circle distances dq n (the distance between two centres) and dq t (the distance between two vertices) are available as geometric measures. Areas are composed from plates that lie tangential on the sphere. The areas are not spherical but plates, which compares well to the philosophy usually applied when using grid point models with the geographical coordinate system, where an area is obtained via the discrete approximation a cos(ϕ) λ ϕ. Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75 (03)

5 56 A. Gassmann We observe the following area measures: A e = dq t e dqn e A v = e v dq t e(v) dqn e A c = e c A r = v r dq t e dqn e 4 A v. Note that the edge point bisects the distance between two centres, but the edge point does not lie exactly halfway between two vertices. Therefore we need the additional distance information dq t e(v). The dqn and dq t arcs are perpendicular by definition. It is possible to describe non-uniform grids and even pentagons within that grid structure. Currently, the model is still formulated with shallow-atmosphere approximation, which assumes that distance and horizontal area measures are independent of height. Deep-atmosphere equations would require all given distances and area measures to be dependent on the actual level height. The side view in Figure displays how terrain-following coordinates are incorporated into the grid structure. Geometric heights z k are defined at the main level cell and vertex points. The corresponding half-level values z k+/ are the arithmetic means of two adjacent heights. The slope of the local horizontal coordinate surfaces is naturally defined at the edges: Je,k n = (z c,k z c,k)/dq n e (3) Je,k t = (z v,k z v,k)/dq t e. (4) To explore the full strength of the vector-invariant formulation in terrain-following coordinates, vectors need to be available in both covariant (with q i basis vectors) and contravariant (with q i basis vectors) coordinate systems. Common textbooks on dynamic meteorology generally contain introductory information on the concept of reciprocal coordinate systems (e.g. Zdunkowski and Bott, 004). Base vectors of both systems are sketched for the side view in Figure. The metric functional determinant g is the vertical distance between two height points. The coordinate line increment in the vertical is dq z =. We mention here that we prognose the normal velocity component u and the vertical velocity w that are defined in a locally orthogonal coordinate system with respect to the surface of the sphere. For computation of the divergence we need to provide the contravariant velocity components: q n e,k = u e,k (5) q z c,k+/ = (w c,k+/ u e,k J n e,kc k+/ )/ g c,k+/. (6) The vorticity computation awaits the covariant components: k q n,e,k = u e,k + w e c,k+/ Je,k n (7) q z,c,k+/ = w c,k+/ gc,k+/. (8) In the preceding expressions (Eqs (6) and (7)), horizontal and vertical averaging operations are required. We define all averaging operations to be volume weighted. Thus a vertical averaging from main levels to half levels is just the arithmetic mean, whereas an averaging of a variable ψ, which is defined on cells or edges, from half levels to main levels is a weighted average according to ψ k+/ k = gk+/ ψ k+/ + g k / ψ k / g k. (9) Horizontal averaging from centres to edges ψ c e is again just the arithmetic average. The reverse averaging from edges to cells is often associated with the inner product of two vectors (GH08, equation (7)). It reads exemplarily for the metric term in Eq. (6) u e,k Je,k n c = A c gc,k e c u e,k Je,k n dq n e dqt e ge,k. (0) Finally, we anticipate the vorticity to result in contravariant components ω t and ω z (see next subsection). Those have to be converted into orthogonal components for their use in Eq. (), which is accomplished via η e,k+/ = ω t e,k+/ () for the horizontal vorticity component, and ζ v,k = ωv,k z gv,k + Je,k t (T e i v ) v v ωe,k+/k t (Te i v ) + Je,k t (T e j v ) v v ωe,k+/k t (Te j v ) () for the vertical vorticity component. The special averaging in Eq. () is due to the need for cancellation of some metric terms arising in the covariant velocity computation on the one hand and in the orthogonal vorticity computation on the other. In the preceding expression, i v, j v are unit vectors in east and north directions at the vertex point. Averaging to the vertex point is similar to Eq. (0), e.g. Je,k t (T e j v ) v = Je,k t A (T e j v )dq n dq e t v gv,k e(v) ge,k. (3) e v The projection of two vectors, as for instance T e j v,is performed in Cartesian space with the coordinate system origin in the Earth s centre. 3.. Discrete integral theorems Gauss and Stokes integral theorems are invoked for the definition of divergence and vorticity. By its nature, the Gauss theorem requires contravariant velocity measures. The divergence of a flux F reads ( F c,k = f n e,k ge,k dq t e A γ e(c) c c gc,k +f z c,k+/ e c gc,k+/ A c f z c,k / gc,k / A c ), (4) where γe(c) c =± depends on whether the local coordinate system orientation points outward or inward with respect to cell c. The Stokes integral theorem works with covariant measure numbers and results in a contravariant vorticity component, which has to be reconverted into an orthogonal Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75 (03)

6 A hexagonal Non-Hydrostatic C-grid Dynamical Core 57 form, as described in the previous subsection, to be ready for its use in Eq. (). The vorticities are naturally defined at the edge lines of the grid box volume. For the contravariant vertical vorticity component we find ωv,k z = A v gv,k e v q n,e,k dq n e γ v e(v). (5) Again, we need a measure γe(v) v =±tosignify the positive rotation sense. The contravariant tangential vorticity component is given as ( ωl:={e,k+/} t = q z,c,k+/ dq t e A γ c(l) l e gl c e + k k+/ q n,e,k A e γk(l) l. (6) Also, γ l =± signifies here the positive rotation sense Definition of the discrete Hamiltonian and its functional derivatives The continuous integral of the Hamiltonian (Eq. (7)) is discretized by a sum over grid boxes. Naturally, one would define the horizontal part of the kinetic energy at the cell centre to be built of the face velocity components surrounding that cell. On the hexagonal C-grid, however, there is a strong experimental and theoretical indication that this approach is likely to result in an internal mode instability, first reported by Hollingsworth et al. (983) (abbreviated in the following as Hollingsworth instability). This instability arises if the vector-invariant formulation is not numerically equivalent to the velocity advection form of the horizontal wind equations. We introduce here a more suitable Hamiltonian that is designed such as to erase that instability. The stencil of the horizontal part of the kinetic energy is extended via inclusion of the kinetic energy at the vertices. A similar approach was already proposed in Hollingsworth et al. (983, section 8). Thus let us write the kinetic energy part of the Hamiltonian as H kin = c,k wc,k+/ + α ( ( A c gc,k ϱ c,k A c gc,k e c +α = c,k v c gc,k+/ u e,k A c + w c,k / dq n e e dqt ge,k A c v A v e v gc,k / A c )) u e,k dqn dq e t e(v) ge,k (7) A c gc,k ϱ c,k (Kc,k w + Ku c,k ). (8) Theory leaves the freedom to formulate the kinetic energy with either orthogonal velocity components or with the product of contravariant and covariant components. For the sake of simplicity, in view of the following functional derivatives we choose the former approach. The last line abbreviates the horizontal part of the specific kinetic energy as Kc,k u and the vertical part as Kw c,k. A c v means the kite area which is shared by the cell area and the vertex area. The parameters α and α must obey α + α =. They depend on the actual implementation of the generalized Coriolis term. Appendix B discusses the causes and the cure for Hollingsworth instability in more depth. Next, we have to determine the functional derivatives of the kinetic energy functional with respect to the dynamic quantities of the system (ϱ, u, w). The functional derivative with respect to the density follows immediately from Eq. (8): δh kin δϱ c,k = K w c,k + Ku c,k. (9) To obtain the functional derivative with respect to the vertical velocity component w we rewrite the energy functional as a sum over cells at half levels: H w kin = c,k ( w c,k+/ ϱ c,k + w c,k / = c,k+/ gc,k / gc,k+/ A c A c ) (30) k+/ A c gc,k+/ ϱ w c,k+/ c,k. (3) The resulting averaging rule for ϱ from main levels to half levels indeed turns out to be a simple arithmetic average. The required functional derivative is thus δh kin = ϱ k+/ c,k w c,k+/, (3) δw c,k+/ which gives the vertical mass flux. To discover the functional derivative with respect to the horizontal wind component u, we have to rewrite the horizontal kinetic energy part of H as a sum over the edges at main levels: H u kin = c,k = e,k ( ϱ c,k α +α v c e c u e,k A c v A v e v dq n e dqt e ge,k (33) ) u e,k dqn dq e t e(v) ge,k e ϱ u e,k c,k A e ge,k. (34) The density averaged to the respective edge is now obvious as ϱ e c,k = α ϱ e c,k + α dq t q t v(e) A c v ϱ c,k. (35) e A v e v c v The requested functional derivative, and thus the horizontal mass flux, becomes δh kin = ϱ e c,k u e,k. (36) δu e,k Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75 (03)

7 58 A. Gassmann It should be emphasized here that the special definition of the kinetic energy with α and α gives rise to slightly changed density values at the grid box interfaces compared to the simple arithmetic averages. The full Hamiltonian of the system comprises also the potential and internal energy parts. It reads H = H kin + ϱ c,k (gz c,k + c v T c,k )A c gc,k. (37) c,k The still missing functional derivatives with respect to ϱ and θ are obtained exactly as in the continuous case: δh = Kc,k w δϱ + Ku c,k + gz δh c,k = c p π c,k. (38) c,k δ θ c,k 3.4. Poisson bracket discretization The Poisson bracket (Eqs () (4)) contains three subbrackets that are individually antisymmetric. Therefore we can investigate them separately. Bracket parts (3) and (4) are similar in structure, because they both contain the divergence. For example, we will consider here the bracket part (3) more in depth. When aiming at using the bracket in the prognostic velocity equation, we have to recover the gradient formulation for the second term. Invoking the integration by parts rule for that term and assuming boundary conditions such that V ( )dτ = 0, we find {F, H} ϱ = V ( δf δh δϱ δv + δf δv δh δϱ ) dτ. (39) When replacing the bracket integral (3) by a sum over cells at main levels, the step from (3) to (39) means a rewriting of the second term as a sum over edge points at main levels (where u is defined) and a sum over cell points at half levels (where w is defined). Because δf/δv is required in contravariant components, this step turns out to be tedious and intricate, but essential, as it leads to the consistent metric correction terms for the gradient arising in terrain-following coordinates. Hence let us write Eq. (3) as a sum over grid cells and insert Eqs (5), and (6) with (0) in (4) {F, H} ϱ = c,k + e c δf δϱ c,k ( e c δh ge,k dq t e δu γ e(c) c e,k + δh A c δh c k+ δw c,k+ e,k δu e,kj n A c δh A c + δh c k δw c,k e,k δu e,kj n A c + c,k ( δh δf ge,k dq t e δϱ c,k δu γ e(c) c + e,k e c ( δf A c δf ) dq n e dqt e [ + δf δu e δw c,k+ δf J n e ge δu e gc δw c,k k+ δf Je n ge k gc δu e Je n ge k + δf J n ]) e ge k. (40) gc δu e gc From this expression we get the discretized version of Eq. (39): {F, H} ϱ = c,k ( δf δh ge,k dq t e δϱ c,k δu γ e(c) c e,k e c + δh A c δh c k+ δw c,k+ e,k δu e,kj n A c δh A c + δh c k δw c,k e,k δu e,kj n A c + δf ge,k dq n e δu dqt e e,k e,k δh n + Je,k n δϱ δh z c,k c,k+/ δf gc,k+ δw c,k+ A c δϱ c,k+ k e ( z δh δϱ ) c,k+ Here the vertical gradient of a variable ψ is given as. (4) z ψ c,k+/ = (ψ c,k+ ψ k )/ g c,k+/. (4) The correct sign of the horizontal gradient n ψ e,k = dq n e c e γ c c(e) ψ c,k (43) is automatically generated from γc(e) c. In the second but last line of Eq. (4) the orthogonal horizontal gradient as built from the covariant gradient and the metric correction term becomes unveiled. The first part of Eq. (4) reveals how the contravariant mass flux enters the computation of the divergence. The contravariant vertical mass flux is not simply a multiplication of the density at half levels with the contravariant velocity, ϱ k k+/ q z k+/, but rather (ϱ c,k k+/ w c,k+/ ϱ c,k e u e,k J n e,kc k+/ )/ g c,k+/. (44) The consistency of the metric terms is crucial as spurious gradients or divergences may give rise to noisy results in terrain-following coordinates, which is a matter of continuing concern, especially if the steepness of the coordinate surface slopes increases with finer horizontal mesh sizes. Equipped with knowledge of the discretization of Eq. (3) we can immediately write down the discretized Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75 (03)

8 A hexagonal Non-Hydrostatic C-grid Dynamical Core 59 thermodynamic bracket part (4): {F, H} θ = c,k + δh δh c,k+/ δf δw c,k+ δw c,k δ θ c,k ( e c δh δu e,kj n e,k δh θ e,k ge,k dq t e δu γ e(c) c e,k c k+ θ c,k+ A c δh c k e,k δu e,kj n θ c,k A c δf gc,k+ δw c,k+ A δh cθ c,k+ z δ θ c,k+ + δf ge,k dq n e δu dqt e e,k e,k δh θ e,k n + Je,k n δ θ θ c,k+ δh z c,k δ θ c,k+ k e. (45) Here, we have assumed that some averaged values of θ are available at the grid box faces. As yet, they are not specified. The Poisson bracket formalism leaves the freedom of choice for those values. We need values of θ e,k and θ c,k+/ such that the resulting transport equation for the potential temperature is of higher-order accuracy. Skamarock and Gassmann (0) describe how to compute the face values for a resulting second-, third- or fourth-order θ-advection. As can be inferred from the last line of Eq. (45) that the same face values occur in the pressure gradient term. Specifically, we apply the third-order advection scheme in the horizontal and the second-order scheme in the vertical. The horizontal advection algorithm that delivers θ e,k is completed with the terrain-following metric correction applied to the directional Laplace occurring in it. The most intricate modelling step is the discretization of the generalized Coriolis term, resulting from the bracket part (). Without corrupting the antisymmetry of the bracket, it can further be separated into two pieces containing the vertical vorticity ζ and the horizontal vorticity η, respectively. We first consider the part dealing with vertical vorticity. Three issues have to be considered for nonlinear equations on non-equilateral grids: (i) the nonlinear generalized Coriolis term must be energetically neutral; (ii) the tangential wind vector reconstruction must not give rise to a spurious geostrophic mode; and (iii) the discretization should be chosen such that the Hollingsworth instability is unlikely to occur. Requirement (i) is met if we discretize Poisson brackets. The other two conditions are more subtle. Experiments with different versions of the nonlinear Coriolis term suggest that requirement (iii) is the best choice if only the vorticity at the two neighbouring vertices of an edge enters the nonlinear Coriolis term. This vertex vorticity ζ v = r v ζ r/3 iscomputedasanaverage over the three neighbouring vorticities at rhombi ζ r.the perception of natural vorticity on a hexagonal C-grid to be defined as an average of three rhombus vorticities instead of a single triangle vorticity has been explained for the regular hexagonal C-grid in Gassmann (0) and is again motivated in Appendix A. We suggest here, solely based on experience, the following energy-conserving discretization of the nonlinear Coriolis term: {F, H} ζ = hor F 3 v r,k r,k [ ] k ζ a ϱ r,k H hor A r gr,k v r,k + [ ] hor F k ζ a v v,k ϱ v,k H hor A v gv,k, (46) v v,k v,k where ζ a = ζ + f is the absolute vorticity, and the horizontal reconstruction rules for a functional derivative δg/δv hor are given for vertices as δg δv hor v,k = and for rhombi as δg δv hor r,k = δg N e dq n A v gv,k δu dq t e e(v) ge,k (47) e,k A r gr,k v r e v e v δg N e dq n δu dq t e e(v) ge,k. (48) e,k The functional G stands for either F or H in Eq. (46). The pre-factor /3 in the first term of Eq. (46) accounts for the fact that every triangle is covered in parts by three rhombi. Actually, reconstruction coefficients in east and north directions are stored at the mentioned rhombus and vertex locations. This facilitates turning the directions associated with the k operation. In the case of a regular grid and linear equations this procedure delivers the tangential wind reconstruction proposed by Thuburn (008) and thus meets requirement (ii). This condition causes the seemingly counter-intuitive two-sum structure of Eq. (46). Any of the two sums in Eq. (46) meets condition (i) and uses the neighbouring vertex vorticities as required for condition (iii), but only their combination delivers the correct linear and regular-grid limit case for condition (ii). However, for an irregular grid Eq. (46) does not exactly give a stationary geostrophy mode, hence condition (ii) is not exactly fulfilled. Because Eq. (46) uses only vorticities of the immediate neighbouring vertices we call this scheme the vertex vorticity scheme. An alternative approach for the nonlinear Coriolis term has been developed by TRSK. For reconstructing the tangential wind they require a derived linear vorticity equation on triangles to involve a divergence that is established by area-weighted divergences of the three surrounding hexagons. This guarantees that the tendency of the vorticity equation vanishes for a zero divergence. It can also be proven assuming a suitable Helmholtz decomposition that for a zero divergence the geostrophic balance is met and thus the tendency to divergence vanishes (J. Thuburn, personal communication). Condition (ii) is thus accomplished. The nonlinear energy-conserving version of TRSK is not able to employ only neighbouring ζ v -vorticities of the considered edge without sacrificing its energy-conserving nature. Experiments reveal that the TRSK energy-conserving scheme is more susceptible to Hollingsworth instability (requirement (iii)) than the alternatively proposed scheme (Eq. (46)). We will return to this issue in section 6.3. Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75 (03)

9 60 A. Gassmann The remaining task is now to discretize the part of Eq. () dealing with horizontal vorticity. As in the horizontal plane, we have to be aware of an analogue of Hollingsworth instability in the vertical plane. To avoid it, the cancellation u z u zu = 0 (49) has to hold in the discrete case in the vertical velocity equation. As the kinetic energy is fixed from Eq. (7), the same inner product rule as occurring in it has to be provided for u z u in the definition of the discrete bracket (Eq. ()). Therefore, the e averaging defined in Eq. (35) has to be inherited: {F, H} η = δf δw e,k+ η T e δh ϱ e,k+ δw δf δu e,k = e,k α +α e c e e,k+ e e,k+ A e ge,k η ϱ η δh ϱ e,k+ δu A c v v c A v e v k e + δf k+ δu e,k+ k e + δh k+ δu e,k+ N e N e A e ge,k+ e δh k + δf δw e,k δw c,k+ c,k+ k+ e,k+ η δh ϱ e,k+ δu k+ e,k+ dq n e dqt e ge,k+ dq t e(v) dqn e ge,k Prognostic model equations and boundary conditions (50) In principle, the Poisson bracket predicts the evolution of every arbitrary functional F. Specifically, we are interested in the behaviour of the prognostic values at desired grid points marked with an index. As actual model variables we choose F to be the functionals: F u := u e,k = e,k u e,k F w := w c,k+ = F ϱ := ϱ c,k = c,k F θ := θ c,k = c,k F π := π c,k = c,k c,k+ ϱ c,k θ c,k π c,k δ e,k e,k A e ge,k A e ge,k (5) w c,k+ δ c,k c,k δ c,k+ c,k+ A c gc,k+ (5) A c gc,k+ A c gc,k A c gc,k (53) δ c,k c,k A c gc,k A c gc,k (54) δ c,k c,k A c gc,k A c gc,k. (55) Here, two thermodynamic variables instead of one are given. The reason is related to the time discretization scheme and will be explained in section 4. There it will be highlighted that the two thermodynamic variables π and θ are in fact connected via the equation of state (see Eq. (73)). The functional derivatives needed for the evaluation of the bracket parts are now δf u δu = δf w δw = δf ϱ δϱ = δf θ δ θ = δf π δ θ = { 0 if e, k e, k A e ge,k if e, k = e, k { 0 ifc, k + c, k + A c gc,k+ if c, k + = c, k + { 0 if c, k c, k A c gc,k if c, k = c, k { 0 if c, k c, k A c gc,k if c, k = c, k { 0 if c, k c, k Rπ c v θ (56) (57) (58) (59) A c gc,k if c, k = c, k. (60) As we saw, a lengthy treatise was necessary before arriving at the point were we can see the actual prognostic equations of the basic model configuration. Those equations occur naturally if Eqs (56) (60) are inserted into the Poisson sub-brackets (4), (45), (46), and (50) of the last section. Formally this reads F t ={F, H} ζ +{F, H} η +{F, H} ρ +{F, H} θ. (6) That very expression yields the discretized form of the following continuous equations: v t = ω a ϱ ϱv c pθ π (K + gz) (6) ϱ = (ϱv) (63) t θ = (θϱv) (64) t π = Rπ (θϱv). (65) t c v θ The main advantage of using the bracket form is that the antisymmetric nature of the bracket is automatically carried over to the discretized equations, and energy conservation as being generated from spatial numerics is guaranteed. This is even possible in a complicated numerical environment with (i) the occurrence of terrain-following coordinates with all the metric terms in it, (ii) horizontal C-grid staggering on the Voronoi mesh and the complications induced by the need for an acceptable stationary geostrophic mode, and (iii) the necessity for avoiding the Hollingsworth-instability. The main strength of the approach is certainly point (i). There is now no longer a doubt concerning how to discretize metric terms in the equations. In the past, derivation of the equations was split into two steps. A first step was done to simplify the equations still in the continuous frame. A hydrostatic background pressure was removed to reduce the absolute values of the terms involved in computation of the horizontal pressure gradient term. At the same time, an explicit buoyancy term appeared. Buoyancy could thus only be described properly with respect to this ab initio chosen hydrostatic state. In a second step, the spatial discretization was independently done term by term on the Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75 (03)

10 A hexagonal Non-Hydrostatic C-grid Dynamical Core 6 already reformulated equations. It was ignored that there are cancellations in the first step (in the continuous frame) that cannot be expected to occur in the discrete frame. Thus the resulting discretized equations were no longer consistent with the original equations. In contrast to that approach, we start out by discretizing the Poisson bracket immediately, and are therefore sure that the original dynamical content of the equations survives all successive steps. In the context of terrain-following coordinates, we also have to inspect the lower boundary condition. It was observed in Gassmann (004) that the lower boundary condition for the metric correction pressure gradient term was crucial for the stability of a simulation of a resting, stably stratified atmosphere over a mountain. With the present knowledge of the structure of the gradient terms in Eqs (4) and (45) the following considerations led us to a suitable formulation. We observe from Eq. (7) that the covariant horizontal part is corrected with the covariant vertical part to result in the orthogonal horizontal part, so that we have t u e,k = t q n e,k Jn e,k [ g t w] c,k+ + [ g t w] c,k g c,k e. (66) From Eq. (8) we know that the covariant vertical velocity equation is directly proportional to the orthogonal vertical velocity equation. The orthogonal vertical velocity equation at the surface level is nothing other than prescribing a vanishing q z for all times, which is approximated by w c,k= t w c,k= = Je nu c ek=, (67) = J n e tu e c k=. (68) Thus the horizontal velocity equation at the lowermost main level reads u = c p θ n π e,k= n (K + gz) e,k= t e,k= + J n e,k= [ gc,k=+ N e ω a ϱ ϱv e,k= (c p θ z π + z (K + gz)) g c,k=+ c,k= ]e gc,k= t w g c,k=. (69) c,k= We have to evaluate this equation using Eq. (68) in the last term, which calls for a horizontal implicit solver for u at the lowest model level. Experience suggests a fast convergence with only a few iteration steps. The upper boundary condition is also specified to be a rigid lid. As the uppermost level is flat, there are no obstacles that prevent a direct implementation of this condition. 4. Time integration scheme Many non-hydrostatic models are formulated with splitexplicit time integration, which collects fast wave terms and slow advective terms to be stepped forward with different time-step sizes and different time integrators (for the main ideas refer to Skamarock and Klemp, 99). Inspection of the Poisson bracket reveals, however, that all arising terms except the generalized Coriolis term belong to the fast wave part because it is not allowed to tear the bracket ingredients of one sub-bracket apart. Otherwise the spatial antisymmetry of the bracket would be destroyed. Traditionally, however, the bracket parts (3) and (4) dealing with flux divergences were separated in the time-splitting procedure so that one subprocess describes the linearized divergences and gradients essential for the linear wave propagation, and another subprocess describes the advection. Regarding the sub-brackets () and (3), some terms in the generalized Coriolis term and the kinetic energy term must cancel out in order to avoid Hollingsworth instability. As the kinetic energy term is the counterpart of mass flux divergence, it also belongs to the fast dynamics. We must therefore conclude that the whole nonlinear equation set has to be integrated with the same time step. If one chooses an explicit approach in the horizontal and an implicit method in the vertical, the time step is given by the CFL condition of horizontal acoustic wave propagation. This seems certainly a high price to pay in terms of efficiency. In practice, however, the timesplitting ratio in split-explicit schemes is only about :3, thus determined by the Mach number, so that the gain from the splitting procedure is not large. GH08 was already concerned with a time integration scheme that obeys the integration by parts rule in time in the same manner as the integration by parts rule in space is the background of Poisson bracket parts (3) and (4). In mathematical literature, such types of time integrators are called symplectic. It was found in GH08 that the implicit midpoint rule guarantees energy conservation for all the hydrodynamic terms, but not for the thermodynamics, which is hidden in the {F, H} θ bracket (4). Instead, the pressure gradient term had to be treated implicitly according to (see equation (93) in GH08) ( cv c p θ π c p θ π n+ + R ) π n. (70) c p c p We repeat the associated proving steps here for clarity. From Eq. () we can obtain the temporal change of internal energy as I t ={I, H} θ = (c v π n+ (θϱv) + Rπ n (θϱv))dτ V = (c v π θ n+ n+ θ n + Rπ n c v θ n π n+ π n V t Rπ n )dτ t c v π n+ θ n+ c v π n θ n = dτ. (7) t V In the step from the third line to the fourth the previously mentioned integration by parts rule in time becomes visible. Performing the spatial integration by parts in the second line we obtain the corresponding sub-part of the change in kinetic energy (Eq. ()) as K ={K, H} t θ = ϱvc p θ ( c v π n+ + R π n )dτ (7) V c p c p Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75 (03)

11 6 A. Gassmann from which the form in Eq. (70) becomes obvious. To simultaneously employ prognostic equations for both pressure variables, θ (Eq. (64)) and π (Eq. (65)), in the third line of Eq. (7) is equivalent to the usage of a linearized equation of state: π = (ϱθ) R cv ( R p 00 d ln π = R c v d ln(ϱθ) ) R cv π n+ π n π n = R c v (ϱθ) n+ (ϱθ) n (ϱθ) n, (73) when determining θ n+ after one has determined π n+ from π n+ = π n t Rπ n (θϱv). (74) c v θ n Linearized versions of the equation of state are frequently used in non-hydrostatic atmospheric models (e.g. Klemp et al., 007, equation (39); Davies et al., 005, equation (6.0)). One could argue that the described mechanism does not hold when considering time reversal, because the implicit weights in Eq. (70) are not time-centred. However, the reverted usage of the implicit weights is still possible in that case because Eqs (64) and (65) have the same physical meaning: they are pressure equations. Entropy is not created (or even destroyed) within the considered part of the time integration scheme because the spatial integral of ϱθ remains constant during the procedure. Formal time reversal is thus covered by the approach. We aim at a horizontally explicit scheme with the intention of avoiding a three-dimensional implicit solver, which would be either too expensive or require too many development resources that are currently not available. We reformulate the horizontal part of the equations in such a way as to arrive at a forward backward formulation, which is a slight deviation from the pure symplectic scheme. Formally we would thus require the gradient (Eq. (70)) to lag half a time level behind and the divergence to be half a time level ahead (Durran 999, p. 5). Approximating π n / = (π n + π n )/andπ n = (π n+/ + π n / )/, which gives π n+/ = π n π n / = (3π n π n )/, the pressure gradient (70) reads ( cv c p θ n π n+ + R ) π n c p c p = c p θ n ( cv c p (3π n π n ) + R = c p θ n (( cv c p + ) π n + ) (π n + π n ) c p ( R c p ) π n ). (75) This kind of extrapolation, but with empirically chosen weights, is usually explained by the need for stabilizing a split-explicit time integration scheme (Klemp et al., 007). Here, however, the weights are not motivated by a traditional linear stability analysis. Applying a flux limiter in the advection scheme for θ prevents even spurious local entropy destruction or generation. Figure. Temporal evolution of total energy for a one-dimensional sound wave propagation. The deeper reason for the pretended damping which is visible in the implicit weights in Eq. (70) and the extrapolation weights in Eq. (75) is that the internal energy is not a quadratic nonlinear product like the comparable potential energy in the shallow-water system gh /, but rather c v π θ, whereπ and θ have different exponents of p. Therefore, the implicit or extrapolation weights should not be interpreted as an artificial add-on for the sake of linear numerical stability but as a representative of an inherent physical necessity. To elucidate the difference between the forward backward scheme with and without extrapolation, we examine an arbitrary one-dimensional sound wave propagation in a background wind. In this experiment, all other terms are temporally discretized as usual and the advection terms are treated with a Runge Kutta second-order scheme. Figure depicts the evolution of the total energy during this experiment. It is obvious that the non-extrapolated Exner pressure term leads to a permanent increase of total energy. The initial increase of total energy in the experiment with extrapolation is due to the imbalance in the first time step because a previous time step is not available for extrapolation. Repeating the same experiment with the shallow-water equations (not shown here) reveals that energy is only conserved without an extrapolation in the geopotential gradient term, as expected, because the potential energy in the shallow-water equations is a pure quadratic quantity. Summarizing the horizontal part of the time integration scheme, we mention that the forward backward scheme with the mentioned extrapolation in the pressure gradient term is employed. The divergence term is backward as usual. The remaining horizontal terms are treated within a Runge Kutta second-order scheme. Regarding the vertically implicit part of the time integration scheme, the solver is formulated for the vertical mass flux. This gives a 5-band diagonal matrix to solve, which accounts at once for the acoustic waves and vertical advection of w and ϱ. Vertical advection of horizontal velocity and potential temperature is discretized with the implicit midpoint rule. Unlike many vertical solvers do, we do not explicitly define a buoyancy term gθ / θ. This step would assume a linearization around a hydrostatic background state. This linearization would contradict the inherent nonlinearity of the bracket concept. Furthermore, this hydrostatic pressure Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75 (03)

12 A hexagonal Non-Hydrostatic C-grid Dynamical Core 63 must be well defined. One cannot assume that the last time step is hydrostatically balanced; therefore the WRF approach as in Klemp et al. (007) is not adequate. A globally uniform background state (as in COSMO, Doms and Schättler, 00) is not satisfactory, either. The drawback of avoiding the explicit buoyancy term is that buoyancy oscillations cannot be handled implicitly, which restricts the allowable time step to the condition N t < (Skamarock and Klemp, 99). Considering the scales (dq n 0 km) at which we want to use the model and the maximum values of N 0.03/s that may occur in the model domain, most of the applications would certainly fall into the category where the explicit time step is dictated by acoustic waves rather than buoyancy oscillations. 5. Discretization of turbulent friction and dissipative heating To allow the kinetic energy to cascade downscale over the resolvable limit of a model demands a subgrid-scale turbulence closure scheme, which is a physical rather than a numerical concept. This parametrization comprises momentum diffusion and dissipative heating in order to have a branch that converts kinetic energy of turbulence into internal energy. The involved energy conversions can be motivated by considering Reynolds-averaged equations in which the involved sub-energy equations for resolved kinetic energy, turbulent kinetic energy (TKE), and the internal energy e i read ϱ dˆv / = (...) + ϱv dt v : ˆv (76) 0 = (...) v p ϱv v : ˆv ε mol (77) ϱ dê i dt = (...) + v p + ε mol. (78) The second line represents a stationary equation for turbulent kinetic energy. The third equation reveals that the subgrid-scale work of the turbulent velocity fluctuations against the pressure gradient force v pcontributes most to the turbulent frictional heating as the molecular frictional heating ε mol is small. Because of the stationary TKE equation we find the total frictional heating to be ε := v p + ε mol = ϱv v : ˆv. (79) In the context of the present paper we restrict ourselves to horizontal momentum diffusion and omit vertical diffusion. The numerical treatment of the diffusion tensor ϱv v is based on a generalized mixing-length approach (Smagorinsky, 993). The present implementation follows closely the description given in Becker and Burkhardt (007), but applies it to the non-equilateral hexagonal C-grid. In order to illustrate the peculiarity of the diffusion mechanism on the hexagonal C-grid we restrict the considerations to the case without terrain-following coordinates, which would complicate the issue further at this place but is actually implemented into ICON-IAP. We also omit all vertical-level indices k. Before we proceed, some important findings regarding the perception of the diffusion operator on the regular hexagonal C-grid have to be emphasized here and are repeated from Figure 3. Grid structure of the regular hexagonal C-grid. The dashed lines and indices will become relevant in the explanations given in Appendix B. Gassmann (0). Suppose that we have a regular hexagonal grid structure as displayed in Figure 3. The directions of the coordinate lines are associated with the arrows. We name the normal vector components u, u,andu 3. The discrete linear finite difference diffusion operator for u reads ˆ u = /3(δ + δ + δ 33 )u, (80) where δ ii is a discrete one-dimensional Laplacian along a coordinate line. The vector-invariant form is written as ˆ u = (δ ζ 3 δ 3 ζ )/ 3 + δ D, (8) with the divergence D = /3(δ u + δ u + δ 3 u 3 ) and the rhombus vorticities ζ = / 3(δ 3 u δ u 3 ) and ζ 3 = / 3(δ u δ u ). Inserting those into Eq. (8) yields ˆ u = /3(δ u δ u δ 33 u + δ 3 u 3 ) + /3(δ u + δ u + δ 3 u 3 ), (8) which is equivalent to Eq. (80). The form that uses shear and strain deformations reads ˆ u = (δ F 3 δ 3 F )/ 3 + δ E, (83) with the strain deformation E = /3(δ u (δ 3 u 3 + δ u )/) and the shear deformations on rhombi F = / 3(δ 3 u + δ u 3 /) and F 3 = / 3(δ u + δ u /). Inserting E, F,andF 3 in Eq. (83) ˆ u = /3(δ u + δ u / + δ 33 u + δ 3 u 3 /) + /3(δ u δ 3 u 3 / δ u /) (84) yields again Eq. (80). Hence all three forms (Eqs (80), (8) and (83)) are equivalent because the derivatives in the different directions are commutable. This would not have been achieved if the vorticity in Eq. (8) or the shear deformation in Eq. (83) were given using only velocity components of single triangles instead of rhombi and the St Andrew s cross differences had been replaced by a difference between vertex points. As discussed in Gassmann (0), the rhombi as dual grid entities owe their peculiarity to the fact that the Laplacian of a Helmholtz decomposition of a horizontal vector has to account for the linear dependency of its components on a trivariate C-staggered mesh. Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75 (03)

13 64 A. Gassmann To be able to employ Smagorinsky diffusion we have to use the form (83) of the diffusion operator, but generalize it for the irregular grid. It should be pointed out here that the equivalence of the different forms of the diffusion operator, as discussed above, gets lost in the irregular grid case. The irregularity of the grid on the sphere is not strong besides the pentagon points if the grid is optimized in some way, for instance with the spring dynamics approach (Tomita et al., 00). The horizontal zero trace symmetric stress tensor for the edge e reads S e = (EN e N e + FN e T e + FT e N e ET e T e ). (85) Strain deformation E and shear deformation F are defined as E = u q n v q t F = v q n + u q t. (86) As we prognose the normal velocity at each edge, the friction term reads u e t =... ϱ h ( ϱk h EN e ϱk h FT e ), (87) where K h is a nonlinear diffusion coefficient. The divergence has to be taken over the quadrilateral area sketched in Figure. Thus, we have u t =... ( ( ϱkh E) ϱ q n + ( ϱk ) hf) q t. (88) It is obvious that strain deformation is required at cell centre points and shear deformation at vertex points. The reconstruction of the gradients in Eq. (86) is obtained by a piecewise constant reconstruction. One yields for the strain deformation at cell centres u q n v q t c(e) = A c e c(e) u e γ c e(c) dqt e ((N e N e ) (N e T e ) ). (89) For the shear deformation at vertices we have to use a similar formula that includes the three adjacent rhombi: v q n + u r(e) q t = u e γe(v) v A dqn e r v r e v ((N e T e ) (N e N e ) ). (90) The three rhombi covering one vertex triangle contribute equally to the shear deformation. To find the amount of kinetic energy lost by friction we multiply the prognostic equation for u by the mass flux and sum over all edges: ( u e ϱ e e A e e ϱ e e + dq t e dq n e v e c e 3 ( ϱ c K h,c E c(e) )γ c c(e) ) ( ϱ r K h,r F r(e) )γv(e) v. (9) r v Rewriting this sum as a sum over grid cells imitates integration by parts and gives ( A c ( ϱ c K h,c ) E c(e) u e γe(c) c A dqt e c c e c ) + A v c ϱ r K h,r F r(e) u e γe(v) v A c 3 dqn e, (9) v c r v v r which states that the dissipative heating on one grid cell is ε = ϱ c K h,c E c(e) u e γe(c) c A dqt e c e c A v c ϱ r K h,r F r(e) u e γe(v) v A c 3 dqn e. (93) v c r v From this formula it does not seem immediately clear that ε is always positive, and the entropy only increases by this friction. However, this can be verified for an equilateral grid. The proof only holds if the diffusion coefficient is first interpolated to hexagons and rhombi. This is in opposition to the intuition that suggests positioning the diffusion coefficient at vertices instead of at rhombi and multiplying by the averaged shear deformations afterwards. We conclude this section by defining the diffusion coefficient K h with Smagorinsky s ansatz as K h = l h v r e v e v max( S, S min ), (94) where lh is a mixing length and S min is a minimum horizontal wind shear. In our case, K h is naturally given at the edges as this is the point where coordinate system invariance is given for that coefficient. F and E depend on the local edge coordinate system, but F + E = S are invariant. Therefore, area-weighted averaging of K h from edges to cells and rhombi is allowed. The squared mixing length is currently scaled with the edge area, lh = c smaga e,wherec smag is a tunable parameter usually chosen to be 0.5. In the present implementation we set S min = /(70 t). This momentum diffusion strategy may also be used to supply the model with an upper damping layer to prevent wave reflection at the rigid model top. For that purpose one might choose another coefficient K h,ud increasing with height and starting with zero at the lower end of the damping layer. 6. Results 6.. Overview of the experiments In order to demonstrate the properties of the dynamical core, three essential tests are presented here that check for different aspects of the discretization. Non-hydrostatic scales and the formulation of terrainfollowing coordinates might be readily tested with the test case proposed by Schär et al. (00). Therein, a twodimensional non-rotating flow over a hilly mountain is simulated of which it is known that the orographically forced smallest-scale structures decay with height and the coarser ones form a non-hydrostatic gravity wave. This test case focuses on linear wave processes and can thus be run Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75 (03)

14 A hexagonal Non-Hydrostatic C-grid Dynamical Core 65 without diffusion. The two-dimensionality in an x-z plane is achieved in our model by running it with two lines of hexagons that form a narrow double periodic channel. The baroclinic wave test (Jablonowski and Williamson, 006) stresses the numerical issues in several ways. During the early phase of the test, the maintenance of the geostrophic balance can be verified. Hollingsworth instability could be a the cause for the destruction of this balance. In the later phase, when the baroclinic wave has drawn available energy from the mean flow, we can check for correct energy conversions and observe the energy cycle. Available potential energy is converted into kinetic energy, which is in turn converted into unavailable potential energy via dissipative heating. Total energy has to be conserved during the integration. In Skamarock and Gassmann (0) the baroclinic wave test has already been presented with the focus on the role of the higher-order transport scheme for scalars on phase errors of this wave development. It was found that the thirdorder transport scheme for the potential temperature could significantly reduce the longitudinal phase delay of the wave compared to the second-order transport scheme. For that study the model was run on horizontal resolutions from 40 km down to 30 km. The visual convergence was achieved for a mesh size of 60 km. The Held Suarez test (Held and Suarez, 994) inspects an idealized mean climate state. We focus here on the comparison of two 000-day runs with and without dissipative heating. This elucidates the importance of consistent energetics on time-scales relevant to climate modelling. 6.. Schär test case The flow over an orographically modulated mountain is a crucial test, as it checks the consistency of the metric terms. As pointed out in Klemp et al. (003) the test will fail if the order of approximation in the different metric terms does not match. As we derived our equations using the Poisson bracket discretization, which generates consistent metric terms, we do not expect to find the spurious solutions discussed in Klemp et al. (003). However, care must be taken if the higher-order horizontal advection scheme developed in Skamarock and Gassmann (0) is carried over for the transport of the potential temperature in the terrain-following framework. In that paper, higherorder horizontal transport is achieved via the inclusion of a directional Laplacian of the tracer in the upstream located cell. For the third-order scheme, the edge value of the tracer is obtained by θ e,k = θ e e,k x 6 x θ upstream, (95) in the case of flat coordinate lines. Note that the x-direction is perpendicular to the cell edge. The terrain-following coordinates have to be included here for the Laplacian so that we obtain θ e,k = θ e e,k ( x x 6 θ θ ) z x z upstream. (96) Figure 4 compares two simulations using either Eq. (96) or Eq. (95), respectively. The model was run for 6 h in a configuration with N = 0.0/s, U = 0 m/s, x = 500 m Figure 4. Vertical velocity pattern of an idealized flow over a hilly mountain. Run (a) uses (96), and run (b) uses (95) inside the horizontal advection scheme for potential temperature. and z = 300 m. Diffusion was switched off beneath m height and a sponge layer was employed above. From the similarity of the winds in Figure 4(a) with the analytic reference solution given in Schär et al. (00, Figure 3g) one can conclude that the new model is able to reproduce the correct wave pattern and does not exhibit any problems with terrain-following coordinates. The use of the incorrect Laplacian (Eq. (95)) (Figure 4b) displays the same error pattern as visible in Figure 3a of Schär et al. (00). Using Eq. (96) it is not necessary that the contravariant vertical velocity has a higher-order metric correction term as suggested in Klemp et al. (003). It retains its second-order metric correction term as given in Eq. (44). The results support the philosophy that the transport of the tracer-like quantity θ is independent of the discretization of the Poisson bracket as long as the interface values θ e,k and θ c,k+/ are properly provided and the metric information is supplied there Baroclinic wave test case The baroclinic wave test case defined in Jablonowski and Williamson (006) is essential for testing the new numerics in a quasi-geostrophic regime. Figure 5 displays a meridional cross-section of the baroclinic jet. The set-up of the test case is slightly altered by introducing the initial perturbation in the wind field in the Southern Hemisphere, too. This allows us to observe the life cycle of baroclinic waves with attention to the different positions of the pentagons on both hemispheres. The following experiments are all run with an approximate grid spacing of 0 km (40 96 cells) and a time step of 0 s. The 6 hybrid pressure levels advocated in Jablonowski and Williamson (006) are converted into height levels by assuming a suitable vertical temperature Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75 (03)

15 66 A. Gassmann Figure 5. Cross-section of the baroclinic jet defined in Jablonowski and Williamson (006). Zonal wind contours are thick and temperature contours are thin. profile. If not otherwise stated, the basic dynamical core without diffusion, dissipation, and the sponge layer is run. To elucidate that the instability described by Hollingsworth et al. (983) can occur in our simulations, Figure 6 compares the vertical velocity fields at level 0 (approximately 4700 m height) at day 9 for different model configurations. Those are the energy-conserving TRSK scheme with α = (uncorrected for instability), the energy-conserving TRSK scheme with α = 3/4 (corrected for instability) and the vertex vorticity scheme (Eq. (46)) without α-correction (α = ). The instability is clearly visible in the uncorrected TRSK scheme, especially on the equatorward flank of the jet. The noisy pattern occurs independently of the triggered wave. It repeats itself around the globe downstream and poleward of the pentagons. They can be retraced to trigger the noise initially. The pentagons are located at about 6 north and south and they are misaligned by half an icosahedron edge length between the hemispheres. Therefore the streaky pattern on both hemispheres are not mirror-inverted. The maximal absolute values of w in the uncorrected case dominate over the physically meaningful signal of the correct baroclinic wave development. As shown in Appendix B, the chosen value of α = 3/4 minimizes the non-cancellation problem for the TRSK scheme. Surprisingly, the scheme vertex vorticity scheme (Eq. (46)) is not prone to instability. Figure 7 displays the relative vorticity fields at day 7 (upper panel) and day 0 (middle panel) for the run with scheme (46). Those fields are plotted from the model output of the vorticities ζ v = r v ζ r/3. At day 7, the field is still smooth without remarkable vorticity filaments. By day 0, the cyclone development had reached a mature stage with very thin vorticity filaments that had started to collapse. Running a model without horizontal diffusion becomes questionable beyond this point. Running the test case without diffusion for up to 5 days reveals conservation of total energy up to a high degree for runs without the mentioned instability. This is visible from the black solid line in Figure 8. Until day 9 the total energy error E(t)/E(t = 0) oscillates around zero by ± However this value cannot be judged correctly without considering how much energy is involved in energy conversion. Most of the atmospheric energy is in fact unavailable energy. Variations in total energy should therefore be small compared to changes in sub-energies. Figure 8 also displays the evolution of the kinetic, internal and potential energies with respect to their initial values. The evolution of these sub-energies in the two runs without Hollingsworth instability is almost indistinguishable and is given here for the case of the vertex vorticity scheme (Eq. (46)). The rapid growth of the baroclinic wave only starts after day 6. Then, the kinetic energy increases at the expense of internal and potential energies. The latter two sub-energies are not completely coupled in the nonhydrostatic frame as they would be in the hydrostatic frame where they form the total potential energy. A stronger decoupling is suspicious because the test case does not deal with non-hydrostatic scales of motion. In that context, let us consider the evolution of the energy curves of the run without correction of Hollingsworth instability (grey lines in Figure 8). The increase in internal energy and the steeper decrease of potential energy indicate that the achieved state of motion lacks physical credibility. Hence the energetics of that case confirm besides the strange vertical velocity pattern that the simulation features unrealistic dynamics. The test case is now extended in time to 40 days and the model is supplied with the horizontal diffusion scheme, but without upper damping. The lower panel of Figure 7 elucidates the action of diffusion on the vorticity field. The diffusion scheme has now smoothed out small-scale structures and eroded extreme values of the relative vorticity compared to the run without diffusion. Figure 9 shows the instantaneous dissipative heating rates induced by diffusion. These heating rates are very small values, of the order of maximal several hundredth kelvins per day. Their pattern reveals frontal structures as the most deformative flow zones. Smagorinsky diffusion is thus very selective with respect to where and how much it damps. However and this is a peculiarity of the icosahedral grid structure the diffusion also acts appreciably at the pentagon points. This is visible as small green dots at about 6 north and south in Figure 9. The reason is that the accuracy of many of the numerical operators is no longer of almost second order in space. In fact, solely the gradient operator remains of secondorder accuracy near the pentagons. All other operators, like divergence, rotation, and vector reconstructions, have considerable numerical errors. Those are smoothed out, again with a diffusion of reduced accuracy. Longer runs reveal, however, that the pentagon points are no obstacle for the simulations and stronger dynamics will drive the described effect almost imperceptibly. In Becker and Burkhardt (007) it was shown that a complete baroclinic life cycle can only be modelled adequately in the sense of Lorenz (967) if the momentum diffusion is accompanied by coexisting dissipative heating. This heating increases the unavailable part of the energy, while total energy is conserved. Figure 0 compares two model runs in which the diffused energy is either fed back as dissipative heating (black lines) or the dissipative heating is switched off (grey lines), respectively. The latter choice is usually employed in GCMs for climate or numerical weather prediction (NWP) applications. The total energy decrease in the run without heating reveals a drop of the same order of magnitude as the generated kinetic energy. The total amount of energy decreases by about 0.54 MJ m in the last 30 days, which amounts to a spurious cooling of about 0. W m. It is also interesting to observe the behaviour of the energies after day, when the climax of the growth phase had passed. The ongoing energy loss in the run without dissipative heating is indeed mainly reducing the internal energy part. This is in contrast to the run with dissipative heating, where the internal energy remains Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 39: 5 75 (03)