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


  • Almut Gassmann

    Corresponding author
    1. Leibniz-Institut für Atmosphärenphysik an der Universität Rostock e.V., Kühlungsborn, Germany
    • Leibniz-Institut für Atmosphärenphysik an der Universität Rostock e.V., Schloss-straße 6, 18225 Kühlungsborn, Germany.
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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 off-centred 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 time-scales. Copyright © 2012 Royal Meteorological Society

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.

2. Continuous dynamics and discretization concept

We start out from the Poisson bracket form of the non-hydrostatic 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

equation image(1)

(equation (45) in GH08). The Poisson bracket constitutes the basic dynamical core and consists of three sub-brackets:

equation image(2)
equation image(3)
equation image(4)

(equation (46) in GH08) with equation image the density, θ the potential temperature, equation image, v the velocity vector, equation image the absolute vortex vector, dτ the volume element, ℋ the Hamiltonian (energy functional), and ℱ an arbitrary functional of the dynamical variables v, equation image, and equation image. The forcing terms that are added in the extended dynamical core configuration read

equation image(5)
equation image(6)

(equations (47) and (48) in GH08). The former (Eq. (5)) is the turbulent friction and the latter (Eq. (6)) the associated frictional heating with equation image. Furthermore, π is the Exner pressure and cp the heat capacity at constant pressure. The Hamiltonian of dry atmospheric flow sums the kinetic (equation image), potential (Φ), and internal (ℐ) energy contributions:

equation image(7)

with g the gravity acceleration, z geometric height, T temperature, and cv the heat capacity at constant volume. Required functional derivatives of ℋ are

equation image(8)

Before sketching the discretization procedure for this dynamics, we have to discuss the choice of the given special form of the bracket (Eqs (2)–(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 = equation image v, densityequation image, and entropy density equation image, as given in Morrison (1998). If the Coriolis force is included, a suitable transformation might alternatively envisage the angular momenta (Roulstone and Brice, 1995). Global invariants (Casimirs) for geophysical fluids are discussed by many authors. Here we want to cite Bannon (2003), 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 (2)–(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 (2007) 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 (2)–(4)):

  • 1.The discretization of Poisson brackets always presumes a correct behaviour of the linear dynamics. On hexagonal and triangular C-grids, constraints have to 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, 2008; Gassmann, 2011). Sommer and Névir (2009) used a Nambu bracket discretization for shallow-water equations similar to Salmon (2007) on a triangular C-grid which turned out to be problematic in a multilayer shallow-water version of this code (Griewank, 2009). 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, 2011). 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.
  • 2.Non-hydrostatic compressible equations are difficult to cast in a Nambu bracket form. Névir and Sommer (2009) 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. (1983) (see Appendix B of the present paper).

Keeping the form Eqs (2)–(4) is nevertheless advantageous as it clearly highlights the energy conversions that are relevant for understanding the atmospheric energy cycle. Lorenz (1967) highlights the spatially averaged energy conversions as

equation image(9)
equation image(10)

where C is the adiabatic conversion rate, D is the dissipation rate and Qnf is the non-frictional heating rate. The bracket formulation Eqs (2)–(4) matches the adiabatic conversion rate C and we can write

equation image(11)
equation image(12)

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 (2)–(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:

  • 1.divergence, needed in Poisson bracket Eqs (3) and (4);
  • 2.vorticity, needed in Poisson bracket Eq. (3);
  • 3.higher-order advection scheme for tracers, needed in Eq. (4) to specify θ at the faces of a grid box volume;
  • 4.vector reconstruction rule, needed in Poisson bracket Eq. (3);
  • 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

3.1. Grid staggering and nomenclature of terrain-following coordinates

The grid structure is displayed in Figure 1. 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 Ne points inward for the sketched cell; for the neighbouring one it points outward. Ne is needed to specify the direction of the horizontal velocity components. The tangential unit vector Te forms an orthogonal right-hand system with the normal unit vector and the vertical unit vector k. Te 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 dqn (the distance between two centres) and dqt (the distance between two vertices) are available as geometric measures. Areas are composed from ‘plates’ that lie tangential on the sphere. We observe the following area measures:

equation image
Figure 1.

Top and side views of the grid structure. For further explanation see text.

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 equation image. The dqn and dqt 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 1 displays how terrain-following coordinates are incorporated into the grid structure. Geometric heights zk are defined at the main level cell and vertex points. The corresponding half-level values zk+1/2 are the arithmetic means of two adjacent heights. The slope of the local horizontal coordinate surfaces is naturally defined at the edges:

equation image(13)
equation image(14)

To explore the full strength of the vector-invariant formulation in terrain-following coordinates, vectors need to be available in both covariant (with qi basis vectors) and contravariant (with qi 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, 2004). Base vectors of both systems are sketched for the side view in Figure 1. The metric functional determinant equation image is the vertical distance between two height points. The coordinate line increment in the vertical is dqz = 1. 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:

equation image(10)
equation image(16)

The vorticity computation awaits the covariant components:

equation image(17)
equation image(18)

In the preceding expressions (Eqs (16) and (17)), 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

equation image(19)

Horizontal averaging from centres to edges equation image 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 (72)). It reads exemplarily for the metric term in Eq. (16)

equation image(20)

Finally, we anticipate the vorticity to result in contravariant components ωt and ωz (see next subsection). Those have to be converted into orthogonal components for their use in Eq. (3), which is accomplished via

equation image(21)

for the horizontal vorticity component, and

equation image(22)

for the vertical vorticity component. The special averaging in Eq. (22) 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, iv,jv are unit vectors in east and north directions at the vertex point. Averaging to the vertex point is similar to Eq. (20), e.g.

equation image(23)

The projection of two vectors, as for instance Te · jv, is performed in Cartesian space with the coordinate system origin in the Earth's centre.

3.2. 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

equation image(24)

where equation image 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 form, as described in the previous subsection, to be ready for its use in Eq. (3). The vorticities are naturally defined at the edge lines of the grid box volume. For the contravariant vertical vorticity component we find

equation image(25)

Again, we need a measure equation image to signify the positive rotation sense. The contravariant tangential vorticity component is given as

equation image(26)

Also, γl = ±1 signifies here the positive rotation sense.

3.3. 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. (1983) (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. (1983, section 8). Thus let us write the kinetic energy part of the Hamiltonian as

equation image(27)
equation image(28)

The last line abbreviates the horizontal part of the specific kinetic energy as equation image and the vertical part as equation image. Acv means the kite area which is shared by the cell area and the vertex area. The parameters α1 and α2 must obey α1 + α2 = 1. 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 (equation image,u,w). The functional derivative with respect to the density follows immediately from Eq. (28):

equation image(29)

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:

equation image(30)
equation image(31)

The resulting averaging rule for equation image from main levels to half levels indeed turns out to be a simple arithmetic average. The required functional derivative is thus

equation image(32)

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:

equation image(33)
equation image(34)

The density averaged to the respective edge is now obvious as

equation image(35)

The requested functional derivative, and thus the horizontal mass flux, becomes

equation image(36)

It should be emphasized here that the special definition of the kinetic energy with α1 and α2 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

equation image(37)

The still missing functional derivatives with respect to equation image and equation image are obtained exactly as in the continuous case:

equation image(28)

3.4. Poisson bracket discretization

The Poisson bracket (Eqs (2)–(4)) contains three sub-brackets 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 equation image, we find

equation image(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 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 (15), and (16) with (20) in (24)

equation image(40)

From this expression we get the discretized version of Eq. (39):

equation image(41)

Here the vertical gradient of a variable ψ is given as

equation image(42)

The correct sign of the horizontal gradient

equation image(43)

is automatically generated from equation image. In the second but last line of Eq. (41) the orthogonal horizontal gradient as built from the covariant gradient and the metric correction term becomes unveiled. The first part of Eq. (41) 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, equation image, but rather

equation image(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 thermodynamic bracket part (4):

equation image
equation image(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 θe,k and θc,k+1/2 such that the resulting transport equation for the potential temperature is of higher-order accuracy. Skamarock and Gassmann (2011) 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 θe,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 (3). 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 equation image is computed as an average over the three neighbouring vorticities at rhombi ζr. 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 (2011) and is again motivated in Appendix A. We suggest here, solely based on experience, the following energy-conserving discretization of the nonlinear Coriolis term:

equation image(46)

where ζa = ζ + f is the absolute vorticity, and the horizontal reconstruction rules for a functional derivative equation image are given for vertices as

equation image(47)

and for rhombi as

equation image(48)

The functional equation image stands for either ℱ or ℋ in Eq. (46). The pre-factor 1/3 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 (2008) 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 ζv-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.

The remaining task is now to discretize the part of Eq. (3) 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

equation image(49)

has to hold in the discrete case in the vertical velocity equation. As the kinetic energy is fixed from Eq. (27), the same inner product rule as occurring in it has to be provided for uzu in the definition of the discrete bracket (Eq. (3)). Therefore, the equation image averaging defined in Eq. (35) has to be inherited:

equation image(50)

3.5. Prognostic model equations and boundary conditions

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 1. As actual model variables we choose ℱ to be the functionals:

equation image(51)
equation image(52)
equation image(53)
equation image(54)
equation image(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 equation image 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

equation image(56)
equation image(57)
equation image(58)
equation image(59)
equation image(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 (41), (45), (46), and (50) of the last section. Formally this reads

equation image(61)

That very expression yields the discretized form of the following continuous equations:

equation image(62)
equation image(63)
equation image(64)
equation image(65)

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 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 (2004) 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 (41) and (45) the following considerations led us to a suitable formulation. We observe from Eq. (17) that the covariant horizontal part is corrected with the covariant vertical part to result in the orthogonal horizontal part, so that we have

equation image(66)

From Eq. (18) 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 equation image for all times, which is approximated by

equation image(67)
equation image(68)

Thus the horizontal velocity equation at the lowermost main level reads

equation image(69)

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 split-explicit 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, 1992). 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 sub-process describes the linearized divergences and gradients essential for the linear wave propagation, and another sub-process describes the advection. Regarding the sub-brackets (3) 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 time-splitting ratio in split-explicit schemes is only about 1:3, 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 equation image bracket (4). Instead, the pressure gradient term had to be treated implicitly according to (see equation (93) in GH08)

equation image(70)

We repeat the associated proving steps here for clarity. From Eq. (11) we can obtain the temporal change of internal energy as

equation image(71)

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. (12)) as

equation image(72)

from which the form in Eq. (70) becomes obvious. To simultaneously employ prognostic equations for both pressure variables, equation image (Eq. (64)) and π (Eq. (65)), in the third line of Eq. (71) is equivalent to the usage of a linearized equation of state:

equation image(73)

when determining equation imagen+1 after one has determined πn+1 from

equation image(74)

Linearized versions of the equation of state are frequently used in non-hydrostatic atmospheric models (e.g. Klemp et al., 2007, equation (39); Davies et al., 2005, equation (6.20)).

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 equation image 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 1999, p. 115). Approximating πn−1/2 = (πn + πn−1)/2 and πn = (πn+1/2 + πn−1/2)/2, which gives πn+1/2 = 2πnπn−1/2 = (3πnπn−1)/2, the pressure gradient (70) reads

equation image(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., 2007). Here, however, the weights are not motivated by a traditional linear stability analysis.

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 gh2/2, but rather cvπequation image, where π and equation image 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 2 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.

Figure 2.

Temporal evolution of total energy for a one-dimensional sound wave propagation.

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 equation image. 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 equation image. 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 must be well defined. One cannot assume that the last time step is hydrostatically balanced; therefore the WRF approach as in Klemp et al. (2007) is not adequate. A globally uniform background state (as in COSMO, Doms and Schättler, 2002) 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 NΔt < 2 (Skamarock and Klemp, 1992). Considering the scales (dqn ≤120 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 ei read

equation image(76)
equation image(77)
equation image(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 equation image contributes most to the turbulent frictional heating as the molecular frictional heating εmol is small. Because of the stationary TKE equation we find the total frictional heating to be

equation image(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 equation image is based on a generalized mixing-length approach (Smagorinsky, 1993). The present implementation follows closely the description given in Becker and Burkhardt (2007), 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 Gassmann (2011). 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 u1, u2, and u3. The discrete linear finite difference diffusion operator for u1 reads

equation image(80)

where δii is a discrete one-dimensional Laplacian along a coordinate line. The vector-invariant form is written as

equation image(81)

with the divergence D = 2/3(δ1u1 + δ2u2 + δ3u3) and the rhombus vorticities equation image and equation image. Inserting those into Eq. (81) yields

equation image(82)

which is equivalent to Eq. (80). The form that uses shear and strain deformations reads

equation image(83)

with the strain deformation E = 2/3(δ1u1 − (δ3u3 + δ2u2)/2) and the shear deformations on rhombi equation image and equation image. Inserting E, F2, and F3 in Eq. (83)

equation image(84)

yields again Eq. (80). Hence all three forms (Eqs (80), (81) and (83)) are equivalent because the derivatives in the different directions are commutable. This would not have been achieved if the vorticity in Eq. (81) 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 (2011), 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.

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.

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., 2002).

The horizontal zero trace symmetric stress tensor for the edge e1 reads

equation image(85)

Strain deformation E and shear deformation F are defined as

equation image(86)

As we prognose the normal velocity at each edge, the friction term reads

equation image(87)

where Kh is a nonlinear diffusion coefficient. The divergence has to be taken over the quadrilateral area sketched in Figure 1. Thus, we have

equation image(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

equation image(89)

For the shear deformation at vertices we have to use a similar formula that includes the three adjacent rhombi:

equation image(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:

equation image(91)

Rewriting this sum as a sum over grid cells imitates integration by parts and gives

equation image(92)

which states that the dissipative heating on one grid cell is

equation image(93)

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 Kh with Smagorinsky's ansatz as

equation image(94)

where equation image is a mixing length and Smin is a minimum horizontal wind shear. In our case, Kh 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 F2 + E2 = |S|2 are invariant. Therefore, area-weighted averaging of Kh from edges to cells and rhombi is allowed. The squared mixing length is currently scaled with the edge area, equation image, where csmag is a tunable parameter usually chosen to be 0.125. In the present implementation we set Smin = 1/(270Δt).

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 Kh,ud increasing with height and starting with zero at the lower end of the damping layer.

6. Results

6.1. 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 terrain-following coordinates might be readily tested with the test case proposed by Schär et al. (2002). Therein, a two-dimensional 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 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, 2006) 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 (2011) 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 third-order 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 240 km down to 30 km. The visual convergence was achieved for a mesh size of 60 km.

The Held–Suarez test (Held and Suarez, 1994) inspects an idealized mean climate state. We focus here on the comparison of two 1000-day runs with and without dissipative heating. This elucidates the importance of consistent energetics on time-scales relevant to climate modelling.

6.2. 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. (2003) 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. (2003). However, care must be taken if the higher-order horizontal advection scheme developed in Skamarock and Gassmann (2011) is carried over for the transport of the potential temperature in the terrain-following framework. In that paper, higher-order 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

equation image(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

equation image(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 = 0.01/s, U = 10 m/s, Δx = 500 m and Δz = 300 m. Diffusion was switched off beneath 10 000 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. (2002, Figure 13g) 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 13a of Schär et al. (2002).

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.

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. (2003). 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 θe,k and θc,k+1/2 are properly provided and the metric information is supplied there.

6.3. Baroclinic wave test case

The baroclinic wave test case defined in Jablonowski and Williamson (2006) 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 120 km (40 962 cells) and a time step of 120 s. The 26 hybrid pressure levels advocated in Jablonowski and Williamson (2006) are converted into height levels by assuming a suitable vertical temperature profile. If not otherwise stated, the basic dynamical core without diffusion, dissipation, and the sponge layer is run.

Figure 5.

Cross-section of the baroclinic jet defined in Jablonowski and Williamson (2006). Zonal wind contours are thick and temperature contours are thin.

To elucidate that the instability described by Hollingsworth et al. (1983) can occur in our simulations, Figure 6 compares the vertical velocity fields at level 20 (approximately 4700 m height) at day 9 for different model configurations. Those are the energy-conserving TRSK scheme with α1 = 1 (uncorrected for instability), the energy-conserving TRSK scheme with α1 = 3/4 (corrected for instability) and the vertex vorticity scheme (Eq. (46)) without α-correction (α1 = 1). 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 26° 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 α1 = 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 6.

Vertical velocity at level 20 (4700 m height) at day 9 of the developing baroclinic wave test. Top: TRSK energy-conserving scheme with α1 = 1. Middle: TRSK energy-conserving scheme with α1 = 3/4. Bottom: equation (46) with α1 = 1.

Figure 7 displays the relative vorticity fields at day 7 (upper panel) and day 10 (middle panel) for the run with scheme (46). Those fields are plotted from the model output of the vorticities equation image. At day 7, the field is still smooth without remarkable vorticity filaments. By day 10, 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.

Figure 7.

Relative vorticity at level 22 (850 hPa) at days 7 and 10 of the developing baroclinic wave test without diffusion (top and middle panels); bottom: day 10 with diffusion.

Running the test case without diffusion for up to 15 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) − 1 oscillates around zero by ±3 · 10−9. 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 non-hydrostatic 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.

Figure 8.

Temporal evolution of global energy measures of the baroclinic wave test without diffusion and dissipation with scheme (46) (black lines) and the TRSK energy-conserving scheme without α correction (grey lines). Different line styles signify total energy (solid), kinetic energy (dash-dotted), internal energy (long dashed) and potential energy (short dashed).

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 26° 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 second-order 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.

Figure 9.

Day 10 of the developing baroclinic wave at about 850 hPa. Dissipative heating rate (colours) and potential temperature (contours).

In Becker and Burkhardt (2007) it was shown that a complete baroclinic life cycle can only be modelled adequately in the sense of Lorenz (1967) 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 10 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−2 in the last 30 days, which amounts to a spurious cooling of about 0.21 W m−2. It is also interesting to observe the behaviour of the energies after day 22, 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 constant in the late phase. The kinetic energy amounts in both runs are rather similar. This is as expected because the dissipative heating cannot generate kinetic energy by producing available potential energy for conversion into kinetic energy, which would be a perpetuum mobile.

Figure 10.

Temporal evolution of global energy measures of the baroclinic wave test with diffusion and dissipative heating (black lines), and with dissipative heating switched off (grey lines). Line styles are as in Figure 8.

We will now check how much the usually neglected frictional heating affects the energy balance in dependence of different horizontal resolutions. The energy curves of three such energetically inconsistent runs performed with 240, 120 and 60 km grid spacing (10 242, 40 962 and 163 842 cells) are displayed in Figure 11. Even though the chosen experimental set-up is very artificial, the results confirm the common knowledge that coarse-resolution runs tend to be more diffusive than fine-resolution runs. The dissipation rate is therefore larger and the energy budget error made by the neglected heating also becomes larger on coarse meshes. Therefore it is likely that radiation budget biases measured in climate models tend to decrease when the resolution is increased. This effect is indeed found in the ECHAM5 atmosphere model as reported by Roeckner et al.(2006).

Figure 11.

Evolution of the total energy and sub-energies in 40-day runs without frictional heating with 240 km (black), 120 km (red), and 60 km (blue) resolution. Line styles are as in Figure 8.

6.4. Held–Suarez test

The Held–Suarez test (Held and Suarez, 1994) was suggested as a standard test case to assess the quality of dynamical cores of global models. Of special interest are the mean eddy heat and momentum transport by extratropical baroclinic disturbances. How do those properties differ in our model if we add the dissipative heating back to the internal energy? We start the model again from the same initial conditions as before and run the model for 1250 days with Held–Suarez forcing. In the run with frictional heating we also convert the kinetic energy removed by Rayleigh friction into heat, even though this friction is a spurious physical process in the strict sense, as it is not given as the divergence of a stress tensor. Figure 12 shows the evolution of the total energies for the first 500 days. The total energy contained in the atmosphere has reached an equilibrated state after about 250 days in both runs. Hence meaningful statistics can only be computed for the last 1000 days. Figure 12 reveals also that the amount of energy contained in the atmosphere differs slightly between the two realizations. The run with dissipative heating is slightly more energetic and the longer-term total energy variations are slightly stronger. This behaviour remains during the entire length of the model run (not shown).

Figure 12.

Evolution of the total energies for the first 500 days of the Held–Suarez test. Black line: with dissipative heating. Grey line: without dissipative heating.

Figure 13 displays the zonally averaged eddy transports of momentum and heat, and the temperature of the run with dissipative heating with contours. Those mean properties are comparable to the results of other models (e.g. Lin, 2004; Wan et al., 2008). The absolute values of the eddy fluxes are slightly smaller than given in the literature, but they depend on the horizontal and vertical resolution and the accuracy of the numerical methods of the respective model. Here we want to turn the attention to the difference between the runs without and with dissipative heating, which are displayed using colours in Figure 13. The run without dissipative heating has notably smaller absolute values of eddy momentum transport. There is no region where the eddy momentum transport is increased in its absolute value. The additional eddy activity in the run with dissipative heating is remarkable at the poleward side. Consistently, the eddy heat transport is also shifted poleward for the run with dissipative heating. The largest differences occur at the poleward flank in the upper troposphere and at the equatorward flank in the lower troposphere. The mean temperature shows a slightly colder troposphere for the run without dissipative heating. The warming and cooling pattern are overlaid because the whole eddy activity and thus also regions of rising and sinking air are shifted between the two runs. All the discussed properties imply that more energy is processed and more dynamic activity is present in the run with dissipative heating.

Figure 13.

Zonal and temporal means of the Held–Suarez test. Contours show the run with dissipative heating. Colours show differences: from run without dissipative heating to run with dissipative heating. Left: eddy momentum flux. Middle: eddy heat flux. Right: temperature.

7. Discussion and summary

The intention of this paper is to demonstrate that the non-hydrostatic hexagonal C-grid dynamical core can be formulated and implemented in full energetic consistency not only for the resolved dynamics part, but also for the diffusion/dissipation part.

When studying the climate of the Earth, a consistent energetic cycle is of importance, as the primary question is concerned with the energy balance and tiny changes of this balance over longer time periods. In climate change simulations, such changes involve many feedbacks and forcings. However, one should not forget that for these kinds of numerical experiments it is desirable that unavoidable spurious energy sources and sinks should be smaller than the interesting changes in radiative forcing. The proposed model is intended to contribute to the goal of minimizing possible energy budget errors and retaining physical credibility at the same time.

The presented model uses discretization of the Poisson bracket, which mimics correct energy conversions and thus total energy conservation by construction. We have shown that the corresponding discretization works well together with terrain-following coordinates and that it passes the crucial Schär test case as required. The full strength of the Poisson bracket approach could also be carried over to the specific grid structure of a quasi-uniform hexagonal C-grid. The complication that arises in the form of Hollingsworth instability has been understood and erased. Higher-order advection of tracers or tracer-like quantities is easily possible within the given approach. A beneficial influence on the phase error of the baroclinic wave could be demonstrated in Skamarock and Gassmann (2011) if a higher-order transport scheme for potential temperature was applied. In the present paper it could also be shown that diffusion and dissipation can be formulated in a consistent manner on the hexagonal C-grid. This was elucidated with the extended baroclinic wave test to mimic the Lorenz energy cycle.

In addition to spatial discretization, also temporal discretization is formulated under the consideration of energy conservation while keeping all the linear stability properties. If one chooses an explicit time-stepping approach, the disadvantage of the presented approach is revealed as the need for integrating the whole equation set with the time-step restriction for horizontally propagating acoustic waves. As an aside to state-of-the-art explicit time integrators, we have demonstrated that an energetically correct off-centred implicit weight for the pressure gradient term corresponds to a particular temporal extrapolation of the Exner pressure in the comparable forward–backward time integration scheme.

The new non-hydrostatic model at hand is especially suitable for the study of atmospheric processes that are not mainly driven by large-scale quasi-geostrophic motions but are dominated by the excitation, propagation and breaking of gravity waves. Hence it will be suitable for the study of mesoscale phenomena such as fronts, convective events and gravity-wave breaking.

In this paper the description has been restricted to dry atmosphere dynamics. Inclusion of moisture requires again a careful consideration of conservation laws. GH08 have already provided the pathway to energetic consistency for that case. Towards a full GCM it would also be desirable to check all required parametrizations for their inner consistency.


Most of this work was accomplished within the ICON project, which is funded by the Max Planck Institute for Meteorology (MPI-M), Hamburg, and Deutscher Wetterdienst (DWD), Offenbach. For programming support I am indebted to Leonidas Linardakis (grid generation) and Rainer Johanni (parallelization). I thank Bill Skamarock for fruitful discussions on Hollingsworth instability. I am especially indebted to John Thuburn and another reviewer for helpful comments and discussions.

Appendix A

Perception of the dual grid

In TRSK it is assumed that the vorticity on a hexagonal C-grid is defined on triangles. Here we want to motivate an alternative viewpoint which is most convincing when considering equilateral grids, but does not cover the irregular grid case. Performing a comparison of the hexagonal C-grid with the well-known quadrilateral C-grid will lead us to the perception that vorticity should be considered as an average over three rhombus vorticities that share one triangle.

The semi-discretized linear shallow-water equations on a quadrilateral C-grid read

equation image(A1)
equation image(A2)
equation image(A3)

Here, overbars signify arithmetic average operators along the indicated coordinate lines; δx and δy abbreviate centred finite differences. If we derive the divergence equation we obtain

equation image(A4)

The vorticity is naturally placed at the corner of the quadrilateral grid box. We observe that it occurs as averaged from corners to centres in the previous equation. If we derive the vorticity equation, it is convenient to average it to the centre of the main grid box too:

equation image(A5)

because then, using also the continuity equation (A3), we can directly derive the characteristic equation without further manipulation:

equation image(A6)

Let us now perform the same steps for the equilateral hexagonal C-grid. For that, consider the drawing in Figure 3 and the nomenclature as introduced in section 5. The semi-discretized linear shallow water equations now read

equation image(A7)
equation image(A8)
equation image(A9)
equation image(A10)

Here we have to note that a special averaging operator equation image, e.g. equation image, is necessary to reconstruct the tangential wind. This operator has been found by Thuburn (2008) and guarantees that the overspecified gradient components of a scalar ψ (with ψ located at the centres of hexagons) are linear dependent in the sense of

equation image(A11)

Applying this equation image-averaging in the previous momentum equations means that if an initial wind field obeys equation image it will do so for all times. This averaging guarantees also that the geostrophic mode is stationary. Let us now derive the divergence equation as we have done in the quadrilateral C-grid case. One obtains

equation image(A12)


equation image(A13)


equation image(A14)

The latter vorticities are to be found at edges of the hexagons. Their circulation refers to rhombi and they read

equation image(A15)
equation image(A16)
equation image(A17)

The last expression in Eq. (A13) tells us that ζv is averaged from corners to centres to yield ζa in the same way as ζ was averaged from corners to centres in the quadrilateral C-grid case to yield equation image. However, ζv in the hexagonal C-grid case is itself an average of three rhombus vorticities, as visible from Eq. (A14). Hence a vorticity defined as a circulation around a single triangle is of no relevance as it is not the counterpart of a corner vorticity in the quadrilateral C-grid. Rather, the three rhombi as in Eq. (A14) have to be considered as one entity, and they should be considered as the counterpart of a corner vorticity in the quadrilateral C-grid. Proceeding now with the derivation of the vorticity equation as in the quadrilateral C-grid case, one obtains

equation image(A18)

and the characteristic equation becomes, by using again the continuity equation (A10),

equation image(A19)

In this case we obtain three physically meaningful solutions: one stationary Rossby mode and two inertio-gravity modes.

From the comparison of Eqs (A19) and (A6) we find that a double xy-averaging on the quadrilateral grid is equivalent to doubled tilde-averagings in all three directions on the hexagonal C-grid.

The perception of the averaged rhombi as the dual grid (where the vorticity is defined) was already motivated in Gassmann (2011) for the example of the discrete vector-invariant Laplacian of the wind field. In the present paper we have made use of this recognition when deriving the diffusion term in section 5. In Gassmann (2011) it has been shown that the deeper reason for the perception of the averaged rhombi is the desire to keep the components of the vector Laplacian linear dependent so that Eqs (A7)–(A9) stay linear dependent also if diffusion terms are added.

A last heuristic point concerns the nonlinear shallow-water equations. The energy-conserving velocity equations of Sadourny (1975) read

equation image(A20)
equation image(A21)

with kinetic energy equation image. The Coriolis term is here omitted for brevity. If we ignore the fluid depth h in our considerations, this scheme yields exactly the discrete form of the equivalence

equation image(A22)

which is known not to trigger Hollingsworth instability. The energy-conserving scheme of Sadourny (1975) employs only the immediate neighbouring vorticities of an edge to compute the generalized Coriolis term. Intuitively and heuristically it is thus not astonishing that the vertex vorticity scheme (Eq, (46)) that uses also only immediate neighbouring ζv-values does not show Hollingsworth instability. A proof of the discrete form of Eq. (A22) in the case of the hexagonal C-grid would be desirable but has not yet been found by the author.

Appendix B

Hollingsworth instability


The history of NWP models follows a pathway from filtered quasi-geostrophic equations via quasi-hydrostatic primitive equations to unfiltered non-hydrostatic compressible equations. During the first stage of this development –the quasi-geostrophic age –it was observed that spurious small-scale noise could amplify in grid point models such that the forecast crashed. This was called nonlinear instability. It was suggested to add numerical diffusion to prevent such behaviour, but Arakawa (1966) observed that conserving total energy, circulation and potential enstrophy for the nonlinear vorticity advection also prevents the build-up of this instability. The Arakawa Jacobian was a great invention, but it was very hard to understand at the time because the underlying theoretical foundation that simplifies the derivation was not yet at hand. Salmon and Talley (1989) and Salmon (2005) showed that the Arakawa Jacobian is nothing other than a discretized Nambu bracket.

Because of the success of the Arakawa Jacobian in preventing nonlinear instability and being so deeply theoretically founded, people aimed at inheriting this concept to the quasi-hydrostatic primitive equations. Doing so, they observed again an instability in the computations as reported by Hollingsworth et al. (1983), hereafter H83, and also by Arakawa and Lamb (1981). They both had to alter the kinetic energy definition in their models (H83; Takano and Wurtele, 1982, cited from Arakawa, 2000). Then, the equivalence of the vector-invariant form, which is a requirement for the formulation of energy/enstrophy conservation, and the advection form of the nonlinear momentum advection term is fulfilled numerically at least in the linear sense. In other words, kζ × vh + ∇Kh = vh · ∇vh does indeed hold, which was not the case before. H83 analysed the instability and found that it occurs for internal but not for external modes.

Practical experience with the primitive equations revealed that for grid point models the energy-conserving scheme of Sadourny (1975) was sufficient, and the Arakawa Jacobian properties no longer seemed so important. Furthermore, in the 1970s and 1980s, much modelling effort was put into the development of spectral models where this kind of problem does not appear. Meanwhile, however, new kinds of computational meshes are of interest for global modelling because they are better suited to massively parallel architectures than the spectral models. The Poisson bracket discretization given in the present paper shares the vector-invariant formulation for the horizontal advection operator with the grid point models in which Hollingworth instability was found 30 years ago.

The aim of the current Appendix is twofold. First, we want to shed light on the mechanism of the instability and, secondly, we want to show how we can avoid instability on the hexagonal C-grid staggered mesh by redefining the kinetic energy.

Alternative explanation of Hollingsworth instability

Simulations with hydrostatic or non-hydrostatic and ICON-IAP and MPAS (Skamarock et al., 2011) models that use the TRSK energy-conserving scheme for the nonlinear Coriolis term reveal small-scale noise in the velocity fields that occur in the baroclinically unstable zonal jet (Figure 5) even without a triggered baroclinic wave (Figure 6). It is observed that the noise only appears equatorward of the jet maximum and is furthermore modified by the underlying grid structure of the icosahedron as the main building block for geodesic grids. Therefore one can claim that this instability occurs already if the task is simply to keep a geostrophically and hydrostatically balanced flow stable. If we have found a mechanism that explains the instability, that theory must consider nearly balanced flows and explain the different behaviour to the north and to the south of the zonal jet.

Taking the viewpoint of a historian one might ask: What was the main difference between the quasi-geostrophic model (in which such an instability did not occur) and the primitive equation model (in which such an instability did occur)? In the former model generation a divergence equation,

equation image(B1)

did not occur. Filtering out gravity waves and omitting comparatively small terms, only its stationary version survived as the so-called balance equation:

equation image(B2)

which in its linearized form yields simply the geostrophic balance:

equation image(B3)

One should note that in a quasi-geostrophic model there was not any need to derive the balance equation (B2) or (B3) from the underlying prognostic equations for u and v that later belonged to the dynamical core of the primitive equation model. The balance equation was chosen and fixed by the modelling strategy. Therefore it was not an issue whether

equation image(B4)

was numerically true. It was simply assumed and set that this equivalence did hold.

Inspired by the experiment visualized in Figure 6, we consider a zonal flow, omit all zonal derivatives, and consider the quasi-geostrophic equations. We assume that the balance equation contains, now intentionally, an error term equation image which originates from the numerical non-cancellation:

equation image(B5)

Then, the quasi-geostrophic equations (vorticity equation, balance equation, and temperature equation) read

equation image(B6)
equation image(B7)
equation image(B8)

Quasi-geostrophic interpretation usually considers the geopotential tendency equation:

equation image(B9)

and the omega equation:

equation image(B10)

Both of these essential equations of dynamic meteorology have now an error term on the right-hand side. There will be a perturbation in geopotential field which corresponds to the actual state of equation image. This perturbation is inherited to the wind field in every time step so that equation image will change over time. For the vertical velocity field we find a remarkable dependence of the spoiling term on the stability parameter σ: the less stable the stratification, the more pronounced is the effect of the error term on the ω-field. This explains the differences in the behaviour poleward and equatorward of the jet in Figure 5. To the north in the Northern Hemisphere, the temperature field reveals almost an isothermal atmosphere, which is thus very stably stratified. To the south, however, we have a less stable stratification in which the temperature decreases considerably with height. The vertical velocity field will thus be more disturbed at this side of the jet. In correspondence, H83 observed already the dependence of the growth rate of the instability on the inverse of the internal mode phase speed, which is faster the larger the Brunt–Vaisala frequency. Furthermore, vertical velocity will not suffer from the spurious term if the horizontal wind is independent of height and thus equation image. The perturbation occurs only in a baroclinic atmosphere, where significant vertical wind shear is present. Therefore, according to Figure 5, the perturbation also vanishes further equatorward where the vertical wind shear becomes insignificant.

What happens in a primitive equation model is the adjustment towards the already ill-posed quasi-geostrophic state. This process is accomplished via gravity waves. These appear then as the small-scale noise in Figure 6.

Summarizing, we find that the origin of the instability is indeed due to the non-cancellation problem. The spurious fields in the simulation reveal qualitatively such expected behaviour. A similar problem would have been present already in quasi-geostrophic models if the balance equation had been derived directly from the velocity equations. However, as the balance equation was properly set by hand, the problem did not occur and only came to the fore during the age of primitive equation modelling.

Minimization of the non-cancellation for the hexagonal C-grid

In the following, we consider an equilateral hexagonal C-grid as the most simplified version of the distorted mesh that we find on the sphere. We inspect the TRSK energy-conserving scheme, because it exhibits noise. We observe that the nonlinear Coriolis term possesses a stencil that is much larger than the stencil that covers the gradient of the kinetic energy.

To avoid Hollingsworth instability we have to show that the following cancellation does hold or is approximately met via a minimization constraint:

equation image(B11)

Here, capital letters signify Cartesian velocity vector components, in contrast to small letters, which symbolize the trivariate velocity vector components. The first term belongs to the generalized Coriolis term and the second term is the gradient of the kinetic energy.

Consider now the grid structure of the hexagonal C-grid (Figure 3). The cancellation (Eq. (B11)) has to hold in the prognostic equation for u1,i,j. Contributions to U occur only along the two dashed lines to the left and right of point u1,i,j. We do not have the Cartesian wind components directly available. Therefore we have to use expressions of the locally available normal wind components as functions of the Cartesian U and V components:

equation image(B12)
equation image(B13)
equation image(B14)

The vorticities, which are defined at the rhombi, now read

equation image(B15)
equation image(B16)
equation image(B17)

Figure 14 displays the frequencies of occurrence of these rhombus vorticities in the generalized Coriolis term for the TRSK energy-conserving scheme.

Figure 14.

Frequencies of occurrence of rhombus vorticities in the generalized Coriolis term with the TRSK energy-conserving scheme.

The kinetic energy is usually defined at grid cells. Its U2/2-part reads

equation image(B18)

To enlarge the stencil of the kinetic energy it is necessary to blend the kinetic energy at cell centres with the kinetic energy at vertices:

equation image(99)

As we do not yet know how this is done best in order to minimize the non-cancellation problem, we define

equation image(B20)

with a parameter α1 which has to be determined.

Linearizing the gradient of the kinetic energy and considering the line i + 1/2, we can determine the contributions to every addend in Eq. (B11). The related coefficients are given in Table 1. In the table, the division by Δxi is omitted because this is a constant factor in the current investigation and plays no further role.

Table 1. Coefficients to Eq. (B11) along the i + 1/2 line.
position on i + 1/2 linekinetic energy term ckCoriolis term cc
j + 1 + 1/41/12 − α1/12−1/24
j + 1 − 1/41/6 + α1/12−2/24
j + 1/41/12 + α1/6−7/24
j − 1/4−1/12 − α1/67/24
j − 1 + 1/4−1/6 − α1/122/24
j − 1 − 1/4−1/12 + α1/121/24

It is immediately evident that there is no single α1 that brings all contributions to zero. Therefore we have to invoke the least squares method to minimize the overall effect of the non-cancellations. The functional to minimize reads

equation image(B21)

Setting ∂I/∂α1 = 0 one obtains α1 = 3/4.

Even though the whole derivation so far was performed for an equilateral mesh, we do not have another choice than to use it for the distorted grid on the sphere in the same way. This seems reasonable, as the found α1 values are only a best estimate and do not guarantee the perfect cancellation. In order to erase potentially appearing instabilities due to grid distortion, we still have the possibility to tune the diffusion parameters.

The same analysis performed for the nonlinear Coriolis term proposed in Eq. (46) yields α1 = 6/7. Interestingly, the experiment shown in Figure 6 does not even reveal an instability without any correction for that scheme. The reason for this behaviour is still unclear. Experiments with the TRSK energy-conserving scheme and slightly different α1-coefficients from the suggested value α1 = 3/4 were also stable (W. C. Skamarock, personal communication). It is speculated that the clue to the whole issue might again be hidden in the overspecification of the velocity components in a trivariate coordinate system. This overspecification problem is not taken into account by the minimization algorithm suggested here.

  • *

    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 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 a2 cos(φλΔφ.

  • 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.

  • §

    Applying a flux limiter in the advection scheme for θ prevents even spurious local entropy destruction or generation.