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

(A15)

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 ξl(λ,Φ,t), Dl(λ,Φ,t), Tl(λ,Φ,t) and ps(λ,Φ,t) where l=1,2,…,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):

(A16)
(A17)
(A18)
(A19)
(A20)
(A21)
(A22)
(A23)
(A24)
(A25)

This scheme has a specific problem. Let us assume a reference atmosphere at rest with T=Tref(p) and horizontally varying surface pressure ps = psg due to orography Фs. 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

(A26)

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 ps instead of ln ps as a prognostic variable.

Let us define

(A27)

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

(A28)
(A29)

The mass of the atmosphere is defined by the mean surface pressure pref which, in the present model, is constant by definition. For zero orography we have pref=p00=1013 mb. When orography is included, we can implicitly define a reference surface pressure distribution psg(λ,Ф) corresponding to T=Tref(p) by the root of the rhs of (A29):

(A30)

In KMCM, Eq. (A30) is solved for psg , and pref  is defined as the horizontal average of psg. Obviously, both (A28) and (A29) vanish for the reference state. This suggests to use the 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:

(A31)
(A32)
(A33)
(A34)

The reference temperature Tref(p) used in KMCM is defined as

(A35)

This profile can be adjusted with respect to 6 parameters, fixing Tref at the pressure levels Pbo, Pto and Ptr .These levels are assumed to correspond to bottom, tropopause, and some middle atmospheric pressure level. Then, ξ must satisfy

(A36)

The conditions (A36) allow to eliminate the coefficients ξ0,ξ1,ξ2 and ώ. The procedure is described in Becker et al. (1997, appendix) for the vertical profile of the equilibrium temperature. The default parameter setup for Tref  is Pbo 1013 mb, Ptr 110 mb, Pto 0.1 mb, Tbo 280 K, Ttr 210 K and Tto 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 → 0. Hence, the geostrophic error above the tropopause can hardly be eliminated. Therefore KMCM utilizes the more realistic profile.

Owing to (A35), Eq. (A29) becomes

(A37)

and the reference surface pressure psg(λ,Φ) is found by numerically determining the root of the rhs of (A37).

Figure 2: (a) Default reference temperature profile of KMCM. (b) Reference temperature used in ECHAM. The latter is calculated according to Tc := T00(p/p00)β with T00=288 K and β=1/5.256.

In section 5.2 (in Becker habil), KMCM is used with a T42/L24 resolution. For ps = P00, 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

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