京 都 大 学 防 災 研 究 所 年 報

第 46 号 Ｂ

平成 15 年 4 月

Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 46B, 2003

Accuracy of Terrain Following Coordinates over Isolated Mountain:
Steep Mountain Model Intercomparison Project (St-MIP)

Takehiko SATOMURA*, Toshiki IWASAKI**, Kazuo SAITO***Chiashi MUROI**** and Kazuhisa TSUBOKI****** Graduate School of Science, Kyoto University** Graduate School of Science, Tohoku University*** Meteorological Research Institute

Present affiliation: Numerical Prediction Division, Japan Meteorological Agency**** Numerical Prediction Division, Japan Meteorological Agency
Present affiliation: Meteorological Research Institute*****Hydrospheric Atmospheric Research Center, Nagoya University
Synopsis
Following the recent rapid increase of computational power, resolutions of atmosphericnumerical models increase and steep slopes are resolved in models in mountainous areas.To
examine the accuracy of the z* coordinates, which is one of the most common coordinates
over complex terrain for atmospheric models and to study the ability of representation offlows over steep slope, a Steep Mountain Model Intercomparison Project (St-MIP) wascompleted with three nonhydrostatic atmospheric models. Mountain waves over bell-shaped
mountains with various half-widths and heights ranging from 0.6 to more than 45 degrees of
averaged inclination angles were compared with theoretical calculations and also amongmodels. It is shown that models using the z* coordinates simulates waves unexpectedly wellover very steep mountains provided that the horizontal grid size is small enough to represent
the mountain shape smoothly. However, only a model which uses implicit method both in thehorizontal and vertical direction cannot complete the time integration.
Keywords: steep slope, accuracy, terrain following coordinates, mountain wave, St-MIP
1. Introduction
Recently, resolutions of atmospheric numericalmodels increase significantly with the increase ofcomputer power. The increase of horizontal resolution
permits the model's representation of steep slopes overcomplex terrain.
In atmospheric models, the terrain following
coordinates called z* coordinatesz * = zTz − zszT − z s(1)where zT is the top of the model domain and zs is thesurface terrain height, is commonly employed. It wasshown that the z* coordinates produced spurious vertical velocity just over the steep mountain (Satomura,1989). Although the grid resolution of the model usedby Satomura (1989) was 500 m, it was not smallenough to resolve the steep mountain precisely. Therefore, It is still unclear whether the spurious verticalvelocity found by Satomura (1989) was caused by thetruncation error of z* coordinates over the steep slope
itself or by a multiplier effect with the bad representation of the mountain.
The model intercomparison project named "StMIP" was planned to clarify the model ability in representing flow over steep mountain by comparing simulated results of different models. All models
participated in St-MIP used the z* coordinates.

In this paper, we present main results of themodel intercomparison.
2. Participated models
Three nonhydrostatic compressible models participated in St-MIP: MRI/NPD-NHM, CReSS andTSO. The MRI/NPD-NHM was developed at Meteorological Research Institute and Numerical Prediction
Division of Japan Meteorology Agency (Saito et al.,2001). It uses fully compressible hydrodynamic equations and has options of a horizontally explicit andvertically implicit method (hereafter, HE-VI) and alsoa horizontally and vertically implicit method (HI-VI)
of time integration. The CReSS model was developedby a cooperation of Hydrospheric AtmosphericResearch Center, Nagoya University, and Research
Organization for Information Science \& Technology(Tsuboki and Sakakibara, 2001). The CReSS bases on
the ARPS developed at the CAPS (Xue et al., 1995)
and uses quasi-compressible equations. The TSO
model was developed at Meteorological ResearchInstitute (Satomura, 1989). It uses fully compressibleequations and the horizontally explicit and verticallyimplicit method of time integration. All three models
uses the leap-frog time integration scheme with variations of time splitting method.
3. Simulation conditions
Table 1 shows the simulation conditions of StMIP. There are 7 cases (A1 to A4 and D1 to D3) to besimulated. All cases include one bell-shaped mountainzs =h
1 + ( x − x0 )2 / a 2(2)at the center of the domain, x0, where h is the height ofthe mountain and a is the half width of the mountain.
Here, the Brant-Väisäla frequency, N, was not specified, but constant horizontal velocity, U=10 m s-1, andScorer parameter l=U/N were specified for each case.

The horizontal grid increment, dx, was determined to be smaller than 1/5 of the half width of themountain, a, to resolve the mountain accurately. One
exception of this rule was case D3 where dx was a/4
owing to save computational time. The vertical gridincrement, dz, was determined to be smaller than dx
and also to capture the vertical structure of flow overthe mountain. The horizontal grid number is morethan 2000 to prevent cyclic lateral boundaries modifying the simulated results. The height of integration
domain is higher than 15 km except cases A3 and A4
where no noticeable vertically propagating wave
won’t appear. If one’s model requires a kind of sponge
layer to diffuse reflected gravity waves from the upper
free-slip rigid wall boundary, it should be placed at
higher than 10 km. The lower boundary was specified
to be free-slip and rigid surface.
The last column is the non dimensional mountainheight scaled by the Scorer parameter, hl, which indicates the applicability of linear theory. When hl <<1(series A), the mountain is so small compared with thevertical stratification that the linear theory has a goodaccuracy. When hl ≈1 (series D), the mountain is highenough to take into account the nonlinear effects (e.g.
Smith, 1977).
The second column from the last shows theparameter al indicating what type of disturbances
would be observed in each case (e.g. Smith, 1977).
When al >> 1 (case A1), linear theory predicts thatnearly hydrostatically balanced mountain waves willappear, while nonhydrostatic mountain waves should
be observed when al ≈1 (case A2). When al << 1, thelinear theory predicts that wave structures diminish
and simple flow which climbs over and goes down themountain will be simulated.
The column θ is the average slope angle of themountain calculated from the height and the half
width of the mountain. Case A1 is very gentle slopeand cases A4 and D2 are 45°.
The final integration time was specified to theeach case: 300 min, 100 min, 20 min and 10 min for
A1, A2, A3 and A4, respectively, and 100 min forseries D. These values were determined to assure

Table 1. Simulation conditions
a (m)

h (m)

l (m-1)dx, dz (m)θ (deg)

a*l

h*l

A1

5000

100

2x10 -3

1000, 250

0.57

100.2

A2500

100

2x10 -3

100, 100

5.7

10.2

A3100

100

2x10 -3

20, 20

26.5

0.2

0.2

A450

100

2x10 -3

5, 5

45

0.1

0.2

D1500

500

1x10 -350, 50

26.5

0.5

0.5

D2250

500

1x10 -350, 50

45

0.25

0.5

gravity waves forced by the mountain of propagating
faraway from the mountain at the end of simulations.

(a)4. Results
4.1 Structures
Figure 1 shows the vertical velocity around themountain in case A1. For MRI/NPD-NHM, resultswith the option HI-VI are shown when the results withHE-VI option are the same as those of HI-VI. Resultsof MRI/NPD-NHM and TSO are the almost identicalto the results of the linear theory. The amplitude ofwaves in CReSS, however, seems to be weaker thanother models and linear theory.
Figures 2 and 3 show the vertical velocity incases A2 and A3, respectively. Model results seem tobe identical to the results of the linear theory. In thesecases, results of CReSS are similar to those of theother models.
Figure 4 shows the vertical velocity in case A4.
In this case, MRI/NPD-NHM with the option HI-VI
cannot perform the integration to the end of specified
time owing to the unknown reason. The same modelwith the option HE-VI can, however, complete thetime integration. This figure, therefore, shows theresults of MRI/NPD-NHM with HE-VI option.
Because the mountain is narrow comparing to its
height and also the scale of excited disturbances arecomparable to the mountain, it was difficult to compare directly patterns of simulated vertical velocity
with those of linear theory. It is possible to say, however, that the patterns of vertical velocity of models
are similar to the linear theory.
Figures 5 and 6 show the vertical velocity incases D1 and D2, respectively. Even in these highly
nonlinear cases, patterns of vertical velocity simulatedby the models are similar to the linear theory, although
theoretically calculated amplitude of vertical velocity
in case D2 is noticeably smaller than that of simulatedresults. This discrepancy is perhaps caused by thenonlinear effects of the mountain. The model MRI/NPD-NHM with the option HI-VI again failed tocomplete the time integration in case D2, while thatwith HE-VI could complete.

(b)

(c)

(d)

4.2 Momentum flux
One of the quantitative tests of representation ofgravity waves is comparison of simulated momentumflux
M ≡ ∫ ρu ′w ′dx ,(3)where dashed variables are deviation from the horizontal average, with the true values. To examine theperformance of the models strictly, theoretical
momentum flux including nonlinear effects (for exam-Fig. 1. Vertical velocity in case A1 after 5 hours integration. (a) Linear theory, (b) MRI/NPDNHM, (c) TSO, and (d) CReSS. Solid lines
are positive values, and dashed lines in (a)and shaded areas in (b)-(d) are negative values. Contour interval is 0.05 m s-1

(a)

(a)

(b)

(b)

(c)

(d)

Fig. 2. Same as Fig. 1 except case A2
and after 100 min integration.
Contour interval is 0.1 m s-1

(c)

(d)

Fig. 3. Same as Fig. 1 except case A3
and 20 min integration. Contourinterval is 0.25 m s-1

(a)

(a)

(b)

(b)

(c)

(c)

(d)

(d)

Fig. 4. Same as Fig. 1 except case A4 andafter 10 min integration. Contourinterval is 1 m s-1

Fig. 5. Same as Fig. 1 except case D1 andafter 100 min integration. Contour interval is 0.2 m s-1

(a)

(a)

(b)

(b)

(c)

(c)

(d)

Fig. 6. Same as Fig. 1 except case D2 andafter 100 min integration. Contour interval is 0.5 m s-1

Fig. 7. Vertical profile of momentum fluxby (a) MRI/NPD-NHM, (b) TSO,and (c) CReSS normalized by theflux obtained from the linear theory. Plotted numbers in (b) indicate integration time.ple, calculated from Long’s equation (Long, 1953))
should be used. In this paper, as the first step toward
the strict evaluation, we compare the simulated
momentum flux with that of linear theory.
Figure 7 shows momentum flux in case A1. BothMRI/NPD-NHM and TSO reproduced the momentum
flux similar to the linear theory: normalized valuenearly equals to 1 (i.e., equals to the linear theoreticalvalue), and the flux is almost constant with height.
However, the momentum flux of CReSS decreases
with height rapidly. The vertical group velocity ofmountain waves (gravity waves) was smaller in
CReSS than the other models and also the linear theory. Therefore, mountain waves did not reach to thehigh altitude and the momentum flux associated withthe mountain waves was much smaller than the othermodels and the linear theory. The reason why the vertical group velocity was small was not clear yet.
Figure 8 shows the vertical profile of the momentum flux in case A2. Again, models except CReSS
reproduced the characteristics of momentum flux profile of mountain waves.
Figures 9 and 10 shows the momentum flux incases A3 and A4, respectively. In these cases, simulated momentum flux was clearly larger than the linear
momentum flux just above the mountain. Because thenonlinear analytical solutions are not yet obtained, wecannot deduce a conclusion that models using z* coordinates produced incorrect momentum flux from thesefigures.
Figures 11 and 12 show the momentum flux incases D1 and D2, respectively. In contrast to the small
mountain cases A3 and A4, momentum fluxes in these
nonlinear cases kept characteristics of those in linear
theory. The value was, however, rather larger than thatpredicted by the linear theory.

(a)

(b)

(c)

5. Conclusion
In order to understand the behavior of truncation
error in the z* terrain following coordinates, results ofmountain wave simulation under controlled conditions by 3 different models were compared in theframe work of St-MIP. Contrary to the previous presumption, it was not able to notice a clear error in thevertical velocity patterns even in the steepest casewhere the slope angle was 45°. Both the verticalvelocity pattern and the momentum flux over steep
and high mountain was smooth and reasonable values.
We would conclude, therefore, that we canemploy the z* coordinates in high resolution models
which may contain steep slopes about 45°.
Acknowledgements
This study was supported by the Grand-in-Aid for
Scientific Research B-1-12440124 and the Category 7

Fig. 8. Same as Fig. 7 except for case A2.

(a)(a)(b)(b)(c)(c)Fig. 9. Same as Fig. 7 except for case A3.

Fig. 10. Same as Fig. 7 except for case A4.
The ordinate of (b) covers wider
range to indicate the profile of themomentum flux.

(a)(a)(b)(b)(c)(c)Fig. 11. Same as Fig. 7 except for case D1.

Fig. 12. Same as Fig. 7 except for case D2.

of MEXT RR2002 Project for Sustainable Coexistence of Human, Nature and the Earth. Figures are
drawn by the GFD-DENNOU Library.

References
Long, R. R., 1953: Some aspects of the flow of stratified fluids. I, A theoretical investigation. Tellus,
5, 42-58.
Saito, K., T. Kato, H. Eito and C. Muroi, 2001: Documentation of the Meteorological Research Institute\/Numerical Prediction Division unifiednonhydrostatic model, Tech. Rep. Meteor. Res.
Inst., 42, 133 pp.
Satomura, T., 1989: Compressible flow simulations on
numerically generated grids, J. Meteor. Soc.
Japan, 67, 473--482.
Smith, R. B., 1977: The influence of mountains on theatmosphere, Adv. Geophys., 21, 87-230.
Tsuboki, K., and A. Sakakibara, 2001: CReSS Users'sGuide, HyARC, Nagoya University, 210 pp.
Xue, M., K. K. Droegemeier, V. Wong, A. Shapiro andK. Brewster, 1995: Advanced Regional prediction System ARPS version 4.0 User's Guide,
Center for Analysis and Prediction Storms, 380pp.

急斜面モデル比較プロジェクト (St-MIP) による地形準拠座標系の精度検証

里村雄彦 *・岩崎俊樹 **・斉藤和雄 ***・室井ちあし **** ・坪木和久 ******

京都大学大学院理学研究科*****

東北大学大学院理学研究科

気象研究所，現所属：気象庁予報部数値予報課****

気象庁予報部数値予報課，現所属：気象研究所*****

名古屋大学地球水循環研究センター

要旨

最近の急速な計算機パワーの増加にともなって大気数値モデルの分解能が上がり，山岳地域においては高分解能化により急な斜面がモデル内で表現されるようになった。気象学のモデルで一般的な地形準拠座標系である z*
座標系の急斜面上での誤差を調べ，高分解能モデルの再現性を検証するため，圧縮性方程式を用いた 3 種の非静力学大気モデルを比較する研究「急斜面モデル比較プロジェクト（St-MIP）」を行った。その結果，z* 座標系は，孤立山岳を十分解像するほど細かな水平格子間隔を用いる限り，先行研究でして懸念されているほど打ち切り誤差は大きくなく，45 度を超えるような急斜面を持つ孤立山岳上の流れも精度良く表現していることが分かった。また，水平方向と鉛直方向をともに陰解法で解くモデルでは，急斜面での計算ができないことも分かった。
キーワード：非急斜面，精度，地形準拠座標系，山岳波，St-MIP