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. 46 B, 2003

Development of High-precision Nonhydrostatic Atmospheric Model(1): Governing Equations

Takehiko SATOMURA and Sayaka AKIBA∗
Graduate School of Science, Kyoto University∗

Present affiliation: Suntory Ltd.

Synopsis
For atmospheric sciences in the 21st century, atmospheric numerical models withhigh precision are indispensable tools to be an exact science. Development of nonhydrostatic models based on fully compressible dynamic equations in particular aregiven high priority because they use the least approximation in dynamics. In thispaper, we examined accuracy of finite difference expressions of equations used in nonhydrostatic fully compressible models. It is found that to make a set of flux form
equations is difficult under the constraint of satisfaction of hydrostatic balance withhigh precision. It is also found the set of equations we propose here are not fully flux
form but have good conservation characteristics.
Keywords : nonhydrostatic model, equations, flux form, conservation
1.

Introduction

Synchronizing with rapid development ofcomputer science and technology, numerical models in meso- to small scale meteorology uses less approximations to the governing equations recently.
Now, most mesoscale models uses compressible
equations as their dynamics, while many cloudmodels uses anelastic equations about fifteen years
ago.
Compressible models have several merits.
One of the merits concerns Poisson solvers for pressure. Over complex terrain, Poisson solvers foranelastic (or Boussinesq) equation systems in terrain following coordinates have been consuming a
big amount of computer resources. However, pressure or its substitute variable is one of prognostic variables in compressible models. Therefore,
the solver of Poisson equation for pressure are nolonger an essential part of the model.

There are demerits of compressible models.
One of the demerits is that the time increments
must be small enough to satisfy the CourantFriedrichs-Lewy condition for sound waves. However, numerical technique such as a time-splitting
method or a full- (or semi-) implicit method canalleviate the problem of the short time increment.
The recent increase of computer power also supports the migration from anelastic to compressiblemodels.
There are several variations of governing dynamic equations among compressible numericalmodels in meteorology. One of the big differences
is whether the model uses fully compressible (without any approximations) equations (e. g., Satomura, 1989; Saito, 1997) or a quasi- compressible(with one or more approximations) equations (e.
g., Klemp and Wilhelmson, 1978; Dudhia, 1993;
Xue et al., 1995). Another difference is whether

the equations are in flux forms or in advectiveforms. In this paper, we focus on the latter difference among fully compressible models and examine the accuracy under the constraint of hydrostatic balance.
2.

We examine the following five forms of twodimensional fully compressible nonhydrostatic dynamical equations. Here, we define as U =ρu, W = ρw, Θ0 = (ρθ)0 .
•Advection form:cvE¶∂p0+ ρ0 g ,
∂z
∂U h
∂p0
GE = −+u.
∂x∂x1ρµThis form was developed by Satoh (2002). In thisequation set, both of the equations in (12) are usedsimultaneously in the model time integration toguarantee the energy conservation.

(1)
(2)
(3)

(4)

•Flux form 1:

(5)

∂U
∂uU
∂wU
∂p0+++=0∂t∂x
∂z∂x
∂W∂uW
∂wW
∂p0+++= −gρ0∂t∂x∂z∂z
1 ∂θ0 U1 ∂θW
∂θ0++
∂t
ρ ∂x
ρ ∂zµ¶
θ ∂U
∂W=+
ρ ∂x
∂z
µ ¶R/cp0∂pcp Rθ p+∂tcvp0
¶µ∂W
ρ ∂θ0
∂U+
−=0×
∂x
∂z
θ ∂t

(13)
(14)

(15)

(16)

This form was used in MRI/JMA-NHM (Saito etal. 2001)•Quasi-flux form 2:

(6)
(7)
(8)

This form was used in WRF model (Klemp et al.,2000).
•Flux form 2:
∂U∂uU
∂wU
∂p0+++=0∂t∂x∂z
∂x
∂W
∂uW
∂wW
∂p0+++= −ρ0 g∂t∂x∂z∂z∂ρ0∂U∂W++=0∂t∂x∂zwhere∂z(12)•Quasi-flux form 1:

This form was used by, for example, Tapp andWhite (1977), Carpenter (1979), Satomura (1989).

∂U
∂uU
∂wU
∂Θ0+++ γRπ=0
∂t
∂x∂z
∂x
∂uW
∂wW
∂Θ0
∂W+++ γRπ∂t∂x∂z µ∂z¶π0= −g ρ̄ − ρ0π̄∂U Θ0∂W Θ0∂Θ0++=0∂t∂x∂z∂U∂W∂ρ0++=0∂t∂x∂z∂tg̃ = g −Equations∂u∂u∂u∂π 0+u+w+ cp θ=0∂t
∂x∂z
∂x∂w
∂w
∂w
∂π 0
θ0+u+w+ cp θ= −g∂t∂x∂z∂zΘ∂θ0∂θ0∂θ0+u+w
=0
∂t
∂x∂z
000
∂π
∂ρpigw∂π+u+w−∂t∂x∂zcp θµ¶
Rπ dθ0Rπ ∂u ∂w=−cv θ dt
cv ∂x ∂z∂ρeh
+ ∂U h + ∂W∂z´ ∂t ³ ∂x 0
∂p00
= u ∂p∂x + w ∂z + ρ wg − ρwg ∂p0 + ∂c2s W = Rd (G − W g̃)

(9)
(10)
(11)

∂U∂uU∂wU∂π 0+++ cp ρθ=0
∂t
∂x
∂z∂x
∂W∂uW∂wW∂π 0+++ cp ρθ
∂t∂x
∂z
∂z
θ0= −gρΘ∂π 0∂π∂π 0+u+w∂t µ ∂x∂z¶
Rπ dθ0Rπ ∂u ∂w=−+cv θ dt
cv ∂x∂z00∂θ∂θ∂θ0+u+w=0∂t∂x∂z•Quasi-flux form 3:

(17)

(18)

(19)
(20)

∂U
∂uU∂wU∂p0+++=0∂t∂x∂z∂x∂W∂uW∂wW∂p0+++= −gρ0∂t∂x∂z∂zµ ¶R/cp∂p0
cp R p+∂t
cv p0 p0µ¶∂U θ0∂W θ×+=0∂x∂z∂ρ ∂U∂W++=0
∂t∂x
∂z

(21)
(22)

(23)
(24)

This form was found by authors and firstly used
by Akiba (2002) in her master thesis.
3.

Hydrostatic balance

In this section, accuracy of hydrostatic balance of above equation sets is examined. Hydrostatic balance is one of the most important balance for the atmosphere of the earth. Most of all
atmospheric motions are driven by small amountof imbalance from the hydrostatic balance. If themodel code cannot reproduce the hydrostatic balance precisely, unrealistic flow will appear in themode. Therefore, it is important for nonhydrostatic models to use equations which assure basic
hydrostatic balance.
Let’s start to examine the equation set ”fluxform 1”. This set of equations has π 0 in the righthand-side of eq. (6). This variable is calculated
asπ0=π−πµ¶R /cµ¶R /c
Rd Θ d v
Rd Θ̄ d v=−(25)p0
p0
Normally, p i is nearly equal to π and, therefore,such kind of subtraction possibly causes a serious
cancellation error. This fact indicates that one
must carefully make a discrete form of the equation
to avoid the cancellation error.
The set ”flux form 2” does not have such a
possible cancellation error. This set, however, uses
the energy conservation equation in different forms
eq. (19) two times. As Satoh (2002) also describedin his paper, twice use of same equation in different
forms introduces a kind of inconsistency, while hestated this inconsistency seemed to be small.
The set ”quasi-flux form 1” also has the cancellation error problem. In this set, ρ0 is not aprognostic variable but is calculated as
ρ0 = ρ − ρ̄Similar to the calculation of π 0 in the set ”fluxform 1”, ρ is nearly equal to ρ̄ and special attention should be paid to make a discrete form of theequation set.
The other sets ”advection form”, ”quasi-flux
form 2” and ”quasi-flux form 3” do not suffer fromthe cancellation error due to the subtraction ofhydrostatic variable (π̄ or ρ̄) from its total variable(π or ρ). In this sense, these tree forms are superior
to the ”full flux form 1”, ”full flux form 2” and”quasi-flux form 1”.
4.

Mass conservation

Another point to be considered is the aspectof the energy conservation. Generally, finite difference schemes which are not full flux forms are
difficult to conserve the total energy. To examine
how much energy will be lost or added by time integration, a heat island experiment was conducted.
In this paper, results using the ”quasi-flux form 2”and ”quasi-flux form 3” were shown. The experiment conditions were:• the centered both in time and space finite difference schemes were used for the Arakawa A
grid system.
• the grid increment was 200 m both in the horizontal and vertical directions,• the domain size was 80 km in the horizontal
and 10 km in the vertical direction,• both the horizontal and the vertical boundaries were rigid wall,• the horizontal and the top boundaries werefree slip, and the bottom boundary was theviscous boundary,• the right half of the bottom surface was 5 K
higher than the left half,• one-equation turbulent closure scheme wasused.
Figure 1 shows the time change of total mass
for the ”quasi-flux form 2”. The mass monotonically increased with time. The increase rate ofmass was1601 dM≈
M dt
8 × 104 45 × 3600
= 4.6 × 10−9 sec−1 .

(26)This mass increase rate indicates that the average pressure will increase about 1 hPa after 2

Fig. 3 Potential temperature after 6 hours integration for the ”quasi-flux form 3”. Contour interval is 0.2 K.

Fig. 1 Change of total mass with time for the”quasi-flux form 2”. The unit of the ordinate is kg.

Fig. 4 Same as Fig. 3 except horizontal velocity.
Contour interval is 0.5 m s−1 .

not only conserved the total mass, but also captured the characteristics of the heat island circulation well.
5.
Fig. 2 Same as Fig. 1 except for the ”quasi-flux
form 3”.

days integration. This is a rather large change ofmass for climate models.
Figure 2 shows the time change of total mass
for the ”quasi-flux form 3”. In this case, the mass
initially decreased only about 0.9 kg and then kept
almost the constant value.
Figures 3 and 4 show the potential temperature and the horizontal velocity after 6 hours integration using the ”quasi-flux form 3”. It is shownthat a mixed layer was formed over the heated surface up to the 1.2 km and convective motions dominated in the mixed layer. The flow toward theheated surface existed below 400 500 m height,
and the reverse flow was observed above. Therefore, the simulation using the ”quasi-flux form 3”

Conclusion

Six forms of two-dimensional nonhydrostatic
fully compressible hydrodynamic equations werecompared from the view point of the hydrostaticbalance. It was found that tree forms named ”fullflux form 1” used in WRF model (Klemp et al.,2000), ”full flux form 2” (Satoh, 2002) and ”quasiflux form 1” used in MRI/JMA-NHM (Saito etal., 2001) possibly suffered from the cancellation
error by subtracting a value nearly equal to theother value. Other three forms ”advection form”(e. g., Tapp and White, 1977; Carpenter, 1979;
Satomura, 1989), ”quasi-flux form 2” and ”quasiflux form 3” found by authors were free from suchthe cancellation error.
Mass conservation was also examined for the”quasi-flux form 2” and ”quasi-flux form 3”. By
simulating a heat island circulation, it is foundthat the total mass of ”quasi-flux form 2” monotonically increased while the total mass of ”quasi-flux form 3” was almost constant during the simulation.
As a conclusion, the ”quasi-flux form 3” isbest form from the view points of numerical calculation error and mass conservation among thesix forms examined in this paper. Probably the”quasi-flux form 3” can be used in climate simulation models in the future because of its less numerical error and good mass conservation characteristics.
Acknowledgements
This study was supported by the Category 7of MEXT RR2002 Project for Sustainable Coexistence of Human, Nature and the Earth. Figures
are drawn by the GFD-DENNOU Library.
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高精度非静力学大気モデルの開発（１）方程式系

里村雄彦・秋庭清香 ∗京都大学大学院理学研究科
現所属: サントリー（株）∗要旨

精密科学を指向する 21 世紀の気象学にとって，高精度の大気モデルは必要不可欠の道具である。特に，対象現象のスケールに制限のない完全圧縮流体力学方程式に基づく非静力学モデルの構築への要請は高く，既に幾つかの試みが発表されている。本研究では，完全圧縮非静水圧流体力学方程式系のいくつかの表現形式の差分化を行ったときの精度について検討した。その結果，静水圧平衡を精度良く満たしつつ完全なフラックス形式の差分方程式をつくることは，状態方程式の非線型性によって困難であること，完全なフラックス形式ではなくても保存性の優れたものがあることが見いだされた。
キーワード : 非静力学，方程式系，フラックス形式，保存性