# Vertical discretization

The model equations are prepared for numerical computation using an energy and angular momentum conserving finite-difference scheme introduced by Simmons and Burridge (1981) and modified by a reference state reformulation (Simmons and Chen, 1991). First of all, the intermediate hybrid levels

must be fixed. The corresponding intermediate pressure levels

are known from (Al). Full pressure levels and centered pressure differences are defined as

and

In KMCM, *lev* is arbitrary, and the distribution of levels can be adjusted by a few parameters. The discretization method yields partial differential equations for the hybrid level representations of the prognostic variables *Î¾ _{l}(Î»,Î¦,t), D_{l}(Î»,Î¦,t), T_{l}(Î»,Î¦*

*,t)*and

*p*

_{s}(*Î»,Î¦*

*,t)*where

*l*=1,2,â€¦,

*lev*. The corresponding tendencies are computed by evaluating the right hand sides of (A4)-(A8) at full model levels. The hybrid level representations of diffusion, dissipation, and gravity wave tendencies at full hybrid levels is also straightforward.

A finite-difference representation of the dynamical core in an angular-momentum and energy conserving way is as follows (Simmons and Burridge, 1981):

This scheme has a specific problem. Let us assume a reference atmosphere at rest with *T=T _{ref}(p)* and horizontally varying surface pressure

*p*due to orography

_{s}= p_{sg}*Ð¤*. In this trivial example all tendencies must vanish. And this is, of course, the case for the continuous equations. However, in finite-difference form, the pressure gradient term and the geopotential gradient give rise to physically inconsistent tendencies, known as the geostrophic wind error. This error is defined as

_{s}Owing to (A16)-(A20) the residuum (A26) is generally different from zero and produces significant errors above the tropopause, as already mentioned by Simmons and Burridge (1981). This problem can be avoided by expanding (A16)-(A19) with respect to a reference state. One way to perform such a modification has been proposed by Simmons and Chen (1991) and is incorporated for instance in the climate model ECHAM (DKRZ, 1992). A similar method is applied in KMCM. Differences to previous methods arise from the choice of the reference temperature and from using p_{s} instead of ln p_{s} as a prognostic variable.

Let us define

and rewrite the continuous formulations of the pressure gradient term and the gradient of the geopotential

The mass of the atmosphere is defined by the mean surface pressure p_{ref} which, in the present model, is constant by definition. For zero orography we have *p _{ref}=p_{00}=*1013 mb. When orography is included, we can implicitly define a reference surface pressure distribution

*p*

_{sg}(*Î»,Ð¤*

*)*corresponding to

*T=T*by the root of the rhs of (A29):

_{ref}(p)In KMCM, Eq. (A30) is solved for *p _{sg}* , and

*p*is defined as the horizontal average of

_{ref}*p*. Obviously, both (A28) and (A29) vanish for the reference state. This suggests to use the

_{sg}*rhs*of (A28) as a basis for finite differencing since the geostrophic wind error is zero by definition. Accordingly, (A16)-(A19) are reformulated in the following way:

The reference temperature *T _{ref}(p)* used in KMCM is defined as

This profile can be adjusted with respect to 6 parameters, fixing *T _{ref}* at the pressure levels

*P*and

_{bo}, P_{to}*P*.These levels are assumed to correspond to bottom, tropopause, and some middle atmospheric pressure level. Then, Î¾ must satisfy

_{tr}The conditions (A36) allow to eliminate the coefficients *Î¾ _{0,}Î¾_{1,}Î¾_{2}* and ÏŽ. The procedure is described in Becker et al. (1997, appendix) for the vertical profile of the equilibrium temperature. The default parameter setup for

*T*is

_{ref}*P*1013 mb,

_{bo}*P*110 mb,

_{tr}*P*0.1 mb,

_{to}*T*280 K,

_{bo}*T*210 K and

_{tr}*T*220K. The corresponding temperature profile is shown in Fig. 2a. It may be compared to the reference profile used in ECHAM (DKRZ, 1992, Eq. (2.4.2.14)). Throughout the troposphere both reference states are quite similar. Above the tropopause, (A35) yields reasonable temperatures. The ECHAM profile (Fig. 2b) has no tropopause and tends to zero for p â†’ 0. Hence, the geostrophic error above the tropopause can hardly be eliminated. Therefore KMCM utilizes the more realistic profile.

_{to}Owing to (A35), Eq. (A29) becomes

In section 5.2 (in Becker habil), KMCM is used with a T42/L24 resolution. For p_{s} = P_{00}, pressure levels in mb are from ground to top:

987, 903, 789, 684, 589, 503, 426, 357, 296, 242, 196, 155, 121, 92.3, 68.5, 49.3, 33.9, 22.1, 13.5, 7.43, 3.71, 1.86, 0.93, 0.31

In this model version, gravity wave effects are switched off. For the model runs presented in chapter 6 (in Becker habil), the vertical resolution consists of 60 hybrid layers. The corresponding pressure levels in mb are:

987, 915, 825, 743, 667, 598, 535, 478, 426, 379, 336, 297, 262, 231, 203, 177, 155, 135, 117, 93.7, 68.4, 49.9, 36.5, 26.6, 19.4, 14.2, 10.4, 7.56, 5.52, 4.03, 2.94, 2.15, 1.57, 1.14, 0.835, 0.609, 0.445, 0.325, 0 .237, 0.173, 0.126, 0.0922, 0.0673, 0.0491, 0.0359, 0.0262, 0.0191, 0.0140, 0.0102, 0.00744, 0.00543, 0.00396, 0.00289, 0.00211, 0.00154, 0.00113, 0.00082, 0.00060, 0.00047, 0.00022