地球環境研究，Vol.18（2016）

Comparison of Models Having Positive-only Wave CISK with
the NICAM Outputs about Eastward Propagation ofSuper Clusters in the Equatorial Region―Part 1. Approach from a Full Model―
YOSHIZAKI Masanori＊

キーワード：positive-only wave CISK, super clusters, eastward propagation, equatorial βAbstract

waves and static stability. It is found that splitting disturbances drastically changed into a slowly westward-

As large-scale eastward-propagating（EP）precipita-

propagating one when the static stability becomestion systems（PSs）symmetric about the equator, two

uniform, which originally has a wavy structure in the

disturbances are observed; super clusters（SCs）with atroposphere with large（small）values in the lower

horizontal scale of O（1000 km）and a speed of about 15（upper）layers. It is shown that the variable static sta-

- 25 ms and the Madden Julian Oscillation（MJO）withbility helps to enhance the top-heavy heating mainly

a horizontal scale of O（~5000 km）
and a speed less thandue to the vertical advection of the potential tempera-

10 ms （Wheeler and Kiladis 1999）. When positive-only

ture in the heat equation. From these results, it is antic-

wave conditional instability of the second kind（POWC）

ipated that the splitting disturbances are obtained inis applied, splitting disturbances in the longitudinalthe‘usual’ top-heavy heating by adopting the realistic

direction first appear in the top-heavy heating case, andstatic stability.

-1-1

then become EP SC-like ones（ Yoshizaki, 1 9 9 1 a,
Yoshizaki et al. 2012b）. As long as idealistic models areused in simplified basic fields, however, splitting distur-

1. Introduction

bances are attained only in the ‘unusual’ top-heavy

In the tropics, several organized structures of large-

heating. Then, a present study is investigated how toscale precipitation systems（ PSs ）are frequently

get splitting disturbances in the‘usual’ top-heavy heat-observed with specific horizontal and time scales, suching.

as the Madden-Julian Oscillation（MJO）（e.g., Madden

In the Part 1, the outputs of a dry model with theand Julian 1971, Wheeler and Kiladis 1999）, super clus-

POWC are compared with those of the non-hydrostaticters（SCs）（e.g., Wheeler and Kiladis 1999）, typhoons

icosahedral atmosphere model（NICAM）aqua-planet

and hurricanes. Hayashi and Sumi（1986）found SCs

simulation. Using a full model having the same verticalusing an atmospheric global circulation model in an idea­

grid structure and basic fields as the NICAM, it is con-­listic environment of zonally uniform sea surface tem-firmed that an EP disturbance, whose horizontal struc-perature（SST）（aqua-planet）. Using satellite data,

ture indicates an asymmetric east-west structure

Nakazawa（1988）showed a multi-scale structure（MSS）

similar to the Gill（1980）response pattern, is attainedof PSs, in which eastward-propagating（EP）PSs coexist

and the EP speed is comparable to that of NICAM out-

with westward-propagating（WP）ordinary cloud clus-

puts. Simplification from the full model is pursued byters with a horizontal scale of O（100 km）. In mosteliminating the nonlinear terms and basic zonal windcases（e.g., Nakazawa 1988）, SCs interrelate with intra-

seasonal tropical variability, such as MJOs, with timeand modifying the basic fields of a speed of sound＊	Department of Earth-environment Systems, Rissho University

1Comparison of Models Having Positive-only Wave CISK with the NICAM Outputs about Eastward Propagationof Super Clusters in the Equatorial Region（Yoshizaki）

scales of 30-60 days and a horizontal scale of O（~5,000

insight. However, an essential understanding of funda-km）.A slow eastward speed is unusual when comparedmental physics, especially, EP property, is not attainedwith that of the dry Kelvin wave anticipated from lin-

due to its complexity. Further simplifications are

ear theory（e.g., Takayabu 1994; Wheeler and Kiladis

required using or comparing the NICAM outputs. For

1999）. Takayabu（1994）showed a very shallow equiva-

these purposes, the formulation of the POWC heating islent depth of 15-30 m from cloud disturbances utilizing

introduced because SCs are considered to be driven by

3-hourly geostationary meteorological satellite infrareda large-scale diabatic heating. The fact that the large-

data.

scale diabatic heating takes place in the areas of large-

To explain these characteristic features of SCs,

scale upward motions in the lower troposphere is

although some treated MJOs, many theoretical andcommonly received and, thus, the physics of the POWC

numerical studies have been conducted, such as Kelvin-may be acceptable, although Bretherton（2002）pointed

Rossby coupled-type structure（Hayashi and Sumi 1986）,

out that the wave CISK is somewhat a slippery hypoth-the wave conditional instability of the second kindesis.

（CISK）theory（e.g., Takahashi 1987）, the moist Kelvin

Diabatic heating, which is produced by cumulonim-

wave theory with positive-only wave CISK（POWC）

bus clouds, is a key for the development of PSs. It is（e.g., Lau and Peng 1987）, the evaporation-wind feed-

known that precipitation occurs irreversibly and pro-back（e.g., Neelin et al. 1987）, the selective amplification

duces organized PSs. A term of“irreversibly”means

between EP and WP disturbances（e.g., Yoshizaki

that precipitation falls out from the atmosphere due to

1991a）, the wind-induced surface heat exchange（Yanothe gravity and some water substance disappears. Due

et al., 1995）, the frictional-convergence feedback（e.g.,

to irreversible processes, organized PSs generally have

Wang and Li 1994; Wang 2005）, the convectively cou-

inhomogeneous horizontal structures of motions, MSS,

pled equatorial waves（CCEWs）（e.g., Nasuno et al. 2007,

and EP property. On the contrary, non-precipitation

2008）, the extended ones with model-coupled parame-

systems（NPSs）or PSs with real precipitation as well as

terization of the quasi-equilibrium scheme（Mapes 2000;‘negative precipitation’ 1） are homogeneous in the hori-

Majda and Shefter 2001; Majda et al. 2004; Khouider

zontal direction. From Nakajima and Matsuno（1988）

and Majda 2006）, the gross moisture stability（e.g., Ray-and Yoshizaki（2012a）, the NPSs have Benard cell-likemond et al. 2009）and so on.

periodic motions with an aspect ratio of O（1）, but do

Recently, a non-hydrostatic icosahedral atmospheric

not have MSS or EP property.

model（NICAM）simulated EP SCs（Tomita et al. 2005,

A lot of works have mentioned above are re-summa-

Nasuno et al. 2007, 2008）. The NICAM is a global cloud-rized into two groups; NPS and PS groups. The linear

resolving model that is energy-conservative and suit-theories having eikx -type disturbances, where k is a hor-able for long-range simulations（Satoh 2003）. In thisizontal wavenumber and x a horizontal coordinate, aremodel, physics, such as Grabowski’s（1998）microphysi-classified into the NPS group（e.g., Takahashi 1987; Nee-cal cloud parameterization scheme, Mellor-Yamada’s

lin et al. 1987; Yano et al. 1995; Wang and Li 1994;（1974）Level 2 with a moist effect（Smith 1990）, theWang 2005; Mapes 2000; Majda and Shefter 2001;

surface flux（Louis 1979）, and the radiation process

Majda et al. 2004; Khouider and Majda 2006; Raymond（Nakajima et al. 2000）, are included. In this simulation,

et al. 2009）.On the other hand, for example, the POWC

a MSS with a few EP SCs and WP cloud clusters inside

approach can be classified into the PS group（e.g.,

was simulated around the equator. Nasuno et al.（2007,

Hayashi and Sumi 1986; Lau and Peng 1987; Yoshizaki2008）showed that the convectively coupled Kelvin

1991a）. As long as the POWC is used, simulated fea-

waves, which may correspond to SCs, accompany a pairtures look like response patterns of localized heating asof off-equatorial gyres. They also obtained a verticallya limiting case of PSs.

slanted structure of temperature and moisture fields.
The NICAM yielded much useful information and
2

Wheeler and Kiladis（1999）classified large-scale EP
PSs symmetric about the equator into two disturba

地球環境研究，Vol.18（2016）

nces; SCs and MJO. Following Yoshizaki et al.（2012b）,∂U ′ ∂V ′ ∂W ′
+
+
= 0,
∂x ∂y
∂z

where they are separated into SCs as free PSs and
MJO as forced PSs about the longitudinal forcing, the
POWC is applicable to SCs, since the POWC is assumed
to be longitudinally uniform. As well, Yoshizaki（1991a）(5)

by defining the mass - weighted variables as

U ′ ≡ ρ u′, V ′ ≡ ρ v′, W ′ ≡ ρ w′, and B′ ≡ ρ g

θ′
. (6)
θ

pointed out that the EP property of SCs is closely

Here, t is the time; u’, v’, and w’ are deviations of therelated to the splitting disturbances in the top-heavy

eastward, northward, and vertical velocities, respec-

heating. The splitting disturbances first develop in both

tively; x, y, and z are spatial coordinates; p is the preseast and west direction, but soon EP disturbances growsure; θ is the potential temperature; B’ is the mass-

selectively, resulting into the dominance of EP prop-

weighted buoyancy; Q＊ is a diabatic heating; <ρ> is theerty. In this study, thus, the term of‘splitting’ are usedhorizontally averaged density; <Cs> is the horizontally

similarly to‘EP property’. However, idealistic models inaveraged speed of sound waves;2） g is the gravity accele­

the simplified basic fields need the‘unusual’ top-heavy

ration; β is the beta parameter（= 2.3 x 10-11 m-1 s-1）; andheating profiles to get splitting disturbances.

νH is the horizontal viscosity/diffusion. The RayleighThen, the problems are how to get them in thedamping is included to prevent the excessive vertical‘usual’ top-heavy heating. Two approaches are tried inreflection of gravity waves at the upper boundaries.

a series of this study; a full model（Part 1）and a sim-

The coefficient r is set to be 1.1×10-5 s-1 and increases

plest model（Part 2）. In Part 1, the full model has theabove the height of 28 km to its maximum value of 10-4

same structure and number of vertical grid points ands-1. The quantities with a bracket and a prime mean

basic fields as those of the NICAM aqua-planet simula-those averaged in the domain（Lx, Ly）and their devia-tion. The model and basic fields used are explained intions, respectively. The definition of domain will be pre-Section 2. In Section 3, the simulated results of the full

sented in subsection 2b. Two parameters, ε1 and ε2, aremodel are compared with NICAM outputs. In Section 4,

assumed. For the ε1 = 1 case, the nonlinear term3） isthe full model is simplified step-by-step, and the impacts

defined asof the nonlinear terms, basic zonal wind, the speed of

NL 1( A′) = − u ′sound waves, and static stability are studied. Finally,
the discussion and summary are presented in Section 5.

NL 2 (θ ′) = − u′

A discussion of the linear steady response problem forthe mesoscale heating is presented in the Appendix.

∂ A′
∂ A′
∂  A′ − v′− W ′   for A' = U ′, V ′, and (7)∂x∂y∂z ρ 
∂θ ′
∂θ ′
∂θ ′
− v′
− w′
.
∂x
∂y
∂z

(8)

For the ε2 =1 case, the basic zonal wind is included.
The diabatic heating is essential to drive the entire
system and the POWC is adopted. Q＊ is defined as wB F

2. Model description and basic fields（z）,where wB is the vertical velocity around the heighta. Model descriptionof 1 km and F（z）is a prescribed function related to the

The equatorial beta plane and hydrostatic approxi-

heating. It is assumed that F（z）

mation are assumed. The governing equations of a full

Cq is a constant. From <wB>

model are given following Saito et al.（2006）asCq

Cq <Q＊（z）>, where0.01 ms-1 from Fig. 2b,

100 is roughly estimated. Here, for the POWC

 ∂U ′∂u∂ U ′ ∂ p′
= − + β y V ′ + ν H ∆ HU ′ + ε1 NL1 (U ′) + ε 2  u+ W ′  − r U ′, (1)
x
∂t
∂x
∂
∂z 


system to be unstable, Cq = 105 is adopted.

∂V ′
∂ p′
∂V ′
=−
− β y U ′ +ν H ∆ HV ′ + ε1 NL1 (V ′) + ε 2 u− r V ′, (2)
∂t
∂y
∂x

tudinal and latitudinal directions are 360 and 60, respec-

0=−

g
∂ p′
p′ + B′,
−
2
∂z
Cs

(3)

∂θ ′
∂θ ′
∂θ
= − w′+ Q * +ν H ∆ H θ ′ + ε1 NL2 (θ ′) + ε 2 u− r θ ′, and (4)
∂t
∂z
∂x

In this study, the numbers of grid points in the longitively, and the horizontal grid sizes are 1 degree（about
111 km around the equator）. The large value of νH isadopted as 4 x 106 m2 s-1 to prevent the instability catastrophe. As the boundary conditions, periodic and wallconditions are assumed in the longitudinal and latitudi3

Comparison of Models Having Positive-only Wave CISK with the NICAM Outputs about Eastward Propagationof Super Clusters in the Equatorial Region（Yoshizaki）

nal directions, respectively. The time step is 60 seconds.

of the NICAM aqua-planet simulation are utilized. Two

In the vertical direction, the number of grid points is

outputs are usable: 3-hourly averaged data with a hori-

54, the grid size is variable from 35.5 m in the lowest

zontal grid size of 7 km and snapshot data with a hori-layer to 4 km, and the top height is 40.3 km. The struc-zontal grid size of 3.5 km at 85 days（marked by anture and number of vertical grid points are the same asarrow in Fig. 1a）.Figure 1a shows a longitude-time sec-

those of NICAM.

tion of the rain mixing ratio at the surface in the

As an initial perturbation, a circular thermal in which

NICAM 7 km outputs（Tomita et al. 2005; Nasuno et al.

excess amplitude of the potential temperature, ∆θ , is 22007）. EP disturbances similar to the observed SCs

K is placed around the equator in the lower tropo-with a speed of 17 ms-1, as well as WP cloud clusters,

sphere（z < 5 km）as

are well reproduced. These features are discussed in∆θ exp { −(x / x a )2 − (y / y a )2 } cos {π (z −z a ) / (2z a ) }, (9)detail in Nasuno et al.（2007, 2008）.A vertical section ofu, w, θ , θ”, and Q＊ at 70 days along the line of 100°

where xa = ya = 4 km, za = 2.5 km, and π is the ratio of–160°longitude is shown in Fig. 1b. θ”is defined as

the circumference to the diameter of a circle（= 3.1416）.

the deviations of θ from the values averaged in thehorizontal domain（100°– 160°longitude and -4°– 4°

b. Basic fields

latitude in this case）. Variables averaged in 1 degree

For the basic fields of our numerical models, outputs(a)are used to demonstrate the general features of SCs.

z = 10.6 km

y (degree)(c)

17 m s-1x (degree)(d) z = 1.4 km

z (km)y (degree)(b)

１

x (degree)x (degree)

Figure 1 （a）Longitude- time section of the rain mixing ratio at the surface from0 to 5 × 10- 4 kg kg- 1 obtained by NICAM 7 km output（Tomita et al
2005; Nasuno et al. 2007）. The eastward propagation of precipitation systems is roughly 17 ms-1.（b）Vertical structure of vectors（u, w）, w（color）,
θ（dashed lines; contour interval is 5 K）, θ”（thick dashed lines; contour interval is 1 K with purple（> 0 ）and cyan（< 0 ）colors）, and Q＊（solid lines; contour interval is 2 x 10- 4 K s- 1 ）along the bold line in（a）at
70 days. θ”is defined as deviations of θ from the values averaged in
the domain of 100°– 160°longitude and - 4 °– 4 °latitude.（c）Horizontal
structure of vectors（u, v）, w（color）, θ（dashed lines; contour interval
is 1 K）, and Q＊（solid lines; contour interval is 5 x10- 4 K s- 1 ）at z = 10.6
km at 70 days.（d）Same as Fig. 1 c except at z = 1.4 km.
4地球環境研究，Vol.18（2016）The main diabatic heating occurs at an interval of 115°

of <Q＊ >, <w>, < θ >, < u >, and <ρ>. The vertical– 145°longitude, and SCs are seen in this range. Warmprofiles denoted by solid lines are selected by averaging（cold）θ”is located on the eastern（western）side ofthe domain of 40°– 50°longitude（Lx = 10°）and -4°– 4°

the heating. Figures 1c and 1d show the horizontal pat-latitude（Ly = 8°）from snapshot data of 3.5 km at 85＊

terns of u, v, w, θ , and Q at the heights of 10.6 km

days. In this area, the most intensive precipitation isand 1.4 km, respectively. Convergence at z = 1.4 km

related to the EP disturbance. Compared with those in

and divergence at 10.6 km are found around the areas

the 7-km resolution in the same domain at 85 days, Q＊

of upward motions. It is noteworthy that the off-equato-

and w are stronger in intensity by more than 2. How-

rial vortices found on both sides of upward motions,ever, similar shape and intensity for Q＊ are obtained atwhich are an equatorial Rossby wave-like response pat-70 days, when the isolated heating is remarkable.

tern, are seen, especially remarkably on the northernhemisphere at z = 10.6 km. These features can be

3. Results of a full model and its comparison

obtained at any time; however, they are more remark-with NICAM outputs

able when the diabatic heating is localized like around70 days. In this way, the large-scale horizontal struc-

Hereafter, the outputs of a full model are presented.

tures obtained by the NICAM generally look like a Gill

Figure 3a is a longitude - time section of W’ along the（1980）response pattern.

equator at z = 3.7 km. At the initial time, simulated dis-

The domains（Lx, Ly）are specified for the calculation

turbances split in both eastward and westward direc-of the basic fields. Figure 2 shows the vertical profiles

tions, but the WP disturbance soon disappears, while

(a)

(b)

<w>

<Q*>
(c)

(d)

<ρ>

<θ>
<ρ>
<θ>
<θ>

<u>

<θ>
Figure 2

Mean vertical profiles of（a）Q ,（b）w ,（c）θ , and（d）u and ρ. Solidlines denote variables averaged in the domain of 40°– 50°longitude and- 4 °– 4 °latitude at 85 days obtained by NICAM 3.5 km output. Dottedlines in（a）and（b）mean variables averaged in the same domain and days
as the solid lines, except NICAM 7 km output. Dashed lines in（a）
（b）
,
and
（d）
denote variables averaged in the domain of 115 °– 145 °longitude and- 4 °– 4 °latitude at 70 days obtained by NICAM 7 km output.
＊

5Comparison of Models Having Positive-only Wave CISK with the NICAM Outputs about Eastward Propagationof Super Clusters in the Equatorial Region（Yoshizaki）

(a)

(b)

0.001

0.002
0.001
wmax

Figure 3 （a）Longitude- time section of W at the equator at z = 3.7 km in the fullmodel with the same grid location as the NICAM and the basic fields
obtained by the NICAM aqua-planet simulation. Areas of upward anddownward motions are drawn by solid and dashed lines with contour
interval of 10- 3 ms-1, respectively. Areas drawn by grey indicate those of
upward motion.（b）Temporal variation of the maximum W at z = 3.7 km.

the EP one becomes dominant as a single disturbance.

is dominant.

It is noteworthy that the EP speed is slow at first

The vertical section of U, W’, θ , θ”
, and Q＊ is（about 9.6 ms-1）, then becomes fast, and is finally uni-

shown in Fig. 4a. The upright feature of convection

form（about 21 ms ）after about 15 days. The round-the-reaching a height of 17 – 18 km is seen, and conver-world time is approximately 21 days by using thisgence（divergence）in the lower（upper）troposphere is

speed. Figure 3b is a temporal variation of the maxi-

found. Different heights of the intense updraft are

mum values of W’ at z = 3.7 km. Maximum values grownoticed on the eastern and western sides（about 18 km

with time and become saturated and then uniform after

and about 15 km, respectively）. Slightly leftward tilting

about 20 days. The final propagation speed of a pre-of θ”is also found with a height below 15 km. The

ferred disturbance is synchronously related to its maxi-

horizontal sections of U’, V’, W’, θ , and Q＊ at themum amplitude.

heights of 10.4 km and 1.4 km are shown in Figs. 4b-1

It is noteworthy that the horizontal structure, con-and 4c, respectively. A baroclinic structure is obvious.

sisting of a single small area of intense upward motion

The Kelvin wave-like response pattern is noticed on theand widespread areas of weak downward ones, iseastern side of the heating, while the equatorial Rossby

established after about 4 days. Such a feature is fre-wave-like off-equatorial gyres are remarkable on thequently found in the POWC-type cases（e.g., Lau andwestern side. These features are similar to the Gill

Peng 1987）and is also similar to the stage around 70（1980）response pattern, as anticipated from the PSdays in Fig. 1a, in which one intense precipitation area

view.

6

地球環境研究，Vol.18（2016）

(b)
(a)

(b)

-１

-１

(c)

3

2

1

(c)

y (degree)

-2

１ 2

０

x (degree)

Figure 4 （a）
 Vertical structure of vectors（U, W’）, W’（color）, θ（dashed lines;
contour interval is 5 K）, θ”（thick dashed lines; contour interval is
1 K with purple（> 0 ）and cyan（< 0 ）colors）, and Q＊（solid lines;
contour interval is 10- 4 K s- 1 ）along the equator at 21 days. θ”isdefined as deviations of θ from the values averaged in the domainof -140°– -80 °longitude and - 4 °– 4 °latitude.（b）Horizontal structure of vectors（U, V’）, W’（color）, θ（dashed lines; contour interval
is 1 K）, and Q＊（solid lines; contour interval is 10- 4 K s- 1 ）at z = 10.6
km at 21 days.（c）Same as Fig. 1 c except at z = 1.4 km.
Here, the propagation speed and structures are com-

tor in Figs. 1c and 1d. This difference may come frompared with those of the NICAM aqua-planet simulation.

whether the Hadley circulation exist（Fig. 3）or not

The propagation speed in this case is 21 ms and is con-（Figs. 4 and 5）. In spite of this difference, the windsidered to be a similar magnitude to that in Fig. 1afields are similar to large-scale response patterns as（Tomita et al. 2005; Nasuno et al. 2007）, because bothreported above. The question of why similar wind fields

are in the range of 15 – 25 ms of observed SCs in

are induced by different extensions of the heating needs

Wheeler and Kiladis（1999）. For the vertical structure,

to be asked. Linear steady response problems of large-

the upright features of upward motions and θ” arescale and mesoscale heating are studied in Appendix.

remarkable in Fig. 4a compared with Fig. 1b and theFrom this study, large-scale response patterns of winds

results of Nasuno et al.（2008）. These are due to a com-are obtained similarly even when the mesoscale heating

pact distribution of the diabatic heating in the longitudi-

is specified to be aligned in the longitudinal or latitudi-

nal direction.

nal direction. Therefore, it is concluded that the differ--1-1

For the horizontal structure, a similar pattern about

ence of horizontal extension of the heating is not

the wind fields, especially in the upper troposphereserious as far as detailed differences are not concerned.

（Fig. 4b）
, is found to that of Fig. 1c: the dominant Kel-A full model reproduces the EP property and asym-

vin wave-like zonal wind on the eastern side of themetric east-west structure. However, it does not simu-

heating and the equatorial Rossby wave-like off-equato-

late observed features, such as a MSS and verticallyrial vortical circulation on the western side. However, aslanted structures of PSs（e.g., Nakazawa 1988）. Theselarge latitudinal extension of Q and W’ is noticed in

may stem from the large horizontal diffusion / viscosity

Figs. 4b and 4c, while time-dependent scattered meso-

and a simultaneous adjustment time of the POWC

scale（∼100 km）heating is seen aligned along the equa-

adopted in this study. Due to large horizontal diffusion /＊7

Comparison of Models Having Positive-only Wave CISK with the NICAM Outputs about Eastward Propagationof Super Clusters in the Equatorial Region（Yoshizaki）

viscosity, the large-scale preferred disturbance isa. Elimination of the nonlinear terms and basic

uniquely selected, resulting in neither smaller-scale ones

zonal wind

nor a MSS. In addition, the simultaneous adjustment

Here, simplification starts by changing the values oftime of the POWC might produce vertically upright

ε1 and ε2. The W-field and the maximum values in the

structures.

nonlinear cases with half the basic zonal wind and without it are shown in Figs. 5a and 5c, respectively. Com-

4. S
 implification by changing the model andbasic fields

pared with Fig. 3a, the basic zonal wind still works forthe development of the EP disturbance. In a linear case,a preferred disturbance with a maximum growth rate

The full model used in Section 3 had the same struc-increases exponentially with time, and, thus, logarithmic

ture and number of vertical grid points as those offorms of w are utilized for plotting. The log 10 | w | fields

NICAM, and the basic fields of u, θ , and ρ obtained byin the linear cases with and without the basic zonal

the NICAM are also used. Owing to these treatments,

wind are shown in Figs. 5b and 5d, respectively. Even

the EP property was obtained. However, the full modelin these cases, the basic zonal wind plays a supporting

still has various factors, such as the nonlinear terms,

role on the EP property. It is concluded that both thebasic zonal wind and so on. To that end, the full model

nonlinear terms and basic zonal wind play a consider-is simplified step-by-step, eliminating the nonlinear

able role on the maintenance of EP property.

terms and basic zonal wind and modifying the basic
fields of a speed of sound waves and static stability.
<N> is the horizontally averaged static stability defined
as

N =

g ∂ θ
. The simplification process is shownθ ∂zb. Modification of the basic fields of a speed ofsound waves and static stability
The basic fields of the speed of sound waves andstatic stability are further modified. In Fig. 6a, the basic
field of <Cs>, where the original one varies with an

in Table 1.

amplitude of about 40 ms-1 around 300 ms-1 in the troposphere（not shown）, changes into a uniform one（300Table 1

8

Simplification processes. The model becomes simpler from left toright. The parameters of ε 1 and ε 2 are specified 1 or 0.5 or 0.
The symbols of V, U, and S denotes“variable”,“uniform”, and“stretched”, respectively. A, B, C, and D of static stability means
Lines A, B, C, and D in Fig.7, respectively. The boundaries fromgrey areas to white ones denote changes of model or basic fields.

地球環境研究，Vol.18（2016）(a) ε1 = 1, ε2 = 0.5(b) ε1 = 0, ε2 = 1
-4

-3

-1

-2

-3
-4
-2

(d) ε1 = 0, ε2 = 0(c) ε1 = 1, ε2 = 0

-4

-3

-4

-3

Figure 5 （a）
Longitude - time section of W and the maximum values at theequator at z = 3.7 km for the nonlinear case with a half of basic
wind.（c）Same as（a）except no basic wind.（b）Longitude - time
section of log 10 | w | at the equator at z = 3.7 km for the linear
case with the basic wind.（d）Same as（b）except no basic wind.
Areas drawn by grey indicate those of upward motion.

(a)

(b)
-3

-4
-3

-3

-4
-4

-3

-4

(c)

-4

-3

(d)
14
-3

10

-2

-4
-3

16

-4

86
4 2

-3

Figure 6 （a）
 Longitude - time section of log 10 | w | at the equator at z = 3.7 km
with a uniform Cs for（a）original <N> 2 case（Line A in Fig. 7 ）,（b）approximated <N> 2 one（Line B in Fig. 7 ）,（c）smaller <N> 2 one（Line C in Fig. 7 ）, and（d）uniform <N> 2 one（Line D in Fig. 7 ）,respectively. Areas drawn by grey indicate those of upward motion.

ms-1）. In this case, its impact is small. On the otherthe wavy structure than that of Line B. Line D is a uni-

hand, the change of <N> has a large impact on the EPform <N>2 case. Figures 6b, 6c, and 6d correspond toproperty. Figures 7a and 7b show the vertical profiles

the cases of Lines B, C, and D, respectively. In Figs. 6bof <N> and θ , respectively. Line A in Fig. 7 refers toand 6c, the EP property remains, although it weakens.

the original <N> case, and a wavy structure between 0

However, a single WP disturbance appears in Fig. 6d

and 16 km（called“troposphere”）are obvious: <N> has

when the uniform <N>2 is specified. It is noteworthy

large values in the lower layer（called “lower tropo-that the growing disturbance appears only in Fig. 6d,

sphere”）and small values in the upper layer（called

while all EP and WP disturbances are seen in Figs. 5“upper troposphere”）. Line B is the approximation ofand 6a, 6b, and 6c, although they decay with time.

the original <N> , and Line C is a smaller amplitude of

Therefore, a different propagation regime appears

2

2

2

2

2

9

Comparison of Models Having Positive-only Wave CISK with the NICAM Outputs about Eastward Propagationof Super Clusters in the Equatorial Region（Yoshizaki）(a)

Z (km)(b)

Z (km)

D

B

C

A

C

B
A

D

<N>2
Figure 7<θ>Vertical profiles of（a）<N> （unit: 10 s ）and（b）< θ >（unit:
K）. Lines A, B, C, and D denote original, approximated, smaller<N>12, and uniform cases, respectively. <N>12 means a deviation of static stability from the constant value.
2

when the wavy structure of <N>2 disappears. The reason for the disappearance of the EP property in the uniform <N>2 case will be discussed in the next-4π (z − z M )
∂ B′
2
2
2
= −W ′ N 0 − W ′ N 1 + ... ⇒ − W0 cos 
N 0
z
2
∂t
M


+

subsection.
c. Reason for the disappearance of the EP prop--2π (z − z M ) 
W0 α   3 π (z − z M )  + ...
 + sin  sin 2   2 zM 
 2 zM  (13)Here, only the second term on right-hand side is dis-2

erty in a uniform <N> casecussed. It is noteworthy that –W’<N>12 has positive val-

Modifying（4）, the buoyancy equation can be derived.

ues in the upper troposphere and negative ones in theHere, only the first term of the right hand side arelower one. This term works as the heating（cooling）in

written as

the upper（lower）
troposphere in the buoyancy equation.

∂ B′
2= −W ′ N + ...
∂t

(10)

That is, the vertical advection of <N>12 has the same

From Fig. 7（Line A）, <N>2 in the troposphere can beapproximated as the sum of a constant part <N>02 andvariable part <N>12 asN2π ( z − z M ) 22
2≡ N 0 + N 1 , where N 1 ≈ − α sin . (11)
 zM<N>12 is simply assumed to be a sinusoidal function of z,
where α is a positive constant and zM is roughly 8 km
（Fig. 8）. The vertical velocity is roughly representedasπ (z − z M )W ' ≈ W0 cos  G ( x, y ) 2 zM (12)where W0 is a positive constant and G（x, y）means thehorizontal function. Then, the buoyancy equation（10）can be rewritten, excluding the horizontal part, as
10

Figure 8

Schematic profiles of W’, <N>12, and–W’<N>12.

地球環境研究，Vol.18（2016）

role as the top-heavy heating case. The conclusionlarge（small）values in the lower（upper）troposphere.

obtained here, in which the reduction of <N> variation

The vertical advection of variable static stability has

is connected to the weakening of the EP property, isthe same role to enhance the top-heavy heating.

2

consistent with the result that splitting disturbances

In Part 2, using a simplest model, which is a different

likely occur in the top-heavy heating, which will be dis-standpoint from this study, two propagation regimes

cussed in Part 2.

and their formation mechanism is discussed. In the simplest model, a slowly WP disturbance is anticipated inthe‘usual’ observed heating profiles, differently from

5. Discussion and summary

the preferential EP disturbances in the full model. The

The EP property of SCs in the equatorial areas isinteresting and important problems. Thorough explana-connection of the results in Parts 1 and 2 is one of theconcerns.

tions will lead to an understanding of the fundamental
physics of tropical meteorology. To explain the EP
property of SCs as simply as possible, the POWC was
adopted.

Acknowledgements
The author is deeply indebted to Drs. T. Nasuno,

A full model, which has the same structure and num-

Japan Agency for Marine-Earth Science and Technolo­gy

ber of vertical grid points as those of the NICAM as（JAMSTEC）
and H. Tomita, RIKEN, and Prof. M. Satoh,well as basic fields obtained by the NICAM outputs,

University of Tokyo / JAMSTEC, for the use of thewas first applied. A finite-amplitude EP disturbance is

NICAM outputs and many discussions. Thanks are

attained with a speed of about 21 ms , comparable toextended to Drs. T. Nozawa, Okayama University and

the NICAM outputs. Its horizontal structure indicates

T. Horinouchi, Hokkaido University for giving many

the asymmetric structure in the longitudinal direction,comments for improving the manuscript.

-1

similarly to the Gill response pattern, i.e., the Kelvinwave-like zonal patterns on the eastern side of the disturbance and the equatorial Rossby wave-like off-equa-Appendix

torial vortical patterns on the western side. When Figs.

A linear response problem about the localized steady

1c/1d and 4b/4c are compared, however, a large differ-

heating is studied assuming an equatorial beta planeence in the horizontal distributions of the heating isand hydrostatic approximation. The forcing Q is pre-noticed, i.e., the mesoscale heating aligned along thescribed as functional forms. Considering a vertical mode

equator obtained by the NICAM and the large-scale

and assuming an infinite <Cs>, constant < θ > except

heating obtained by the full model. The linear

for static stability, and constant <ρ> in（1）-（5）, non-

responses of winds for the various horizontal exten-

dimensional governing equations are given as

sions of the heating are compared. It is shown that similar horizontal patterns of winds are obtained in spite of
the various extensions of the heating and, thus, this difference does not seriously affect the conclusion.
Next, simplification of the full model is pursued toachieve a deep understanding of the EP property. The
following steps are conducted; eliminating the nonlinear
terms and basic zonal wind and modifying the basic∂ u* 1∂ p*= y*v* −− λ* u* ,∂ t* 2
∂ x*

(A1)

∂ v*
∂ p*
1
= − y* u * −− λ*v* ,
∂ t*
∂ y*
2

(A2)

∂ p*
∂u ∂v
= − * − * − Q* − λ* p* ,∂ t*
∂ x* ∂ y*(A3)

w* = λ* p* + Q* .

(A4)fields of a speed of sound waves and static stability. It is

Here, non-dimensional variables are denoted by asteri­shown that the EP disturbance changes into a WP one

sks. Non-dimensional units of horizontal distance（NH/2

when the static stability becomes uniform, which origi-

1/2βπ）1/2 and time（π/2β NH）
roughly correspond tonally has a wavy structure in the troposphere with

1,050 km and 5.7 hr in dimensional forms, respectively,
11

Comparison of Models Having Positive-only Wave CISK with the NICAM Outputs about Eastward Propagationof Super Clusters in the Equatorial Region（Yoshizaki）

when N = 10-2 s-2, H = 16 km, and β = 2.3x10-11 m-1 s-1 arethis simulation.

used. Gill（1980, 1982）obtained large-scale response pat-

Figure A1 shows snapshops of horizontal winds andterns assuming a long-wave approximation, where thep＊ , which corresponds to - θ’in the lower troposphere,

left-hand side of（A2）is neglected. In this case, a time-at the lower layer at t＊ = 2 and t＊ = 7. The mesoscale

integration is adopted as an initial value problem with-heating with a square size of 0.1 is specified as（a）aout such assumption. A symmetric heating profile about

piece on the equator,（b）three pieces aligned along thethe equator（y＊ = 0）in the latitudinal direction is speci-

equator with an interval of 0.4, and（c）three pieces

fied. The closed domain is used between the range ofaligned along the latitude with an interval of 0.4. On the（-25, 25）in the longitudinal direction and（0, 5）in theother hand, the large-scale heating with a Gaussian

latitudinal one. The grid sizes Δ x＊ and Δ y＊ are 0.01,

shape is specified in（d）.At t＊ = 2, the initial impulse is

and the time step Δ t＊ is 0.01. The time-integration isseen, especially in（a）, around the radius of 2 from theperformed until t＊ = 10, when the reflection of gravity（0, 0）point. The inner areas correspond to influence

waves from the boundaries does not significantly affectregions. Hereafter, our focus is limited to t＊ = 7, when(a) t* = 2(b) t * = 2t *= 7t* = 7(c) t* = 2(d) t* = 2

t* = 7t* = 7

Figure A 1

12H orizontal distributions of horizontal winds and p＊’at the lower layer at t＊ = 2 and t＊ = 7
for（a）one-piece mesoscale heating case with a square size of 0.1 on the equator,（b）threepiece mesoscale heating case with an interval of 0.4 along the equator,（c）three-piecemesoscale heating case with an interval of 0.4 along the latitude, and（d）large-scale heating case with a Gaussian shape. In（d）, the forcing is denoted by solid lines with an intervalof 0.2. The heating is assumed to be symmetric about the equator（y＊ = 0 ）and drawn bybold solid lines. The dashed lines of negative p＊’are drawn with an interval of 0.2. The
scales in each figure are normalized arbitrary. Winds are plotted with an interval of 0.1.

地球環境研究，Vol.18（2016）

nearly steady fields are attained. When four figures are

Nakajima, K. and T. Matsuno, 1988: Numerical experiments

compared, the details, such as vortical motions near theconcerning the origin of cloud clusters in the tropicalheating, are different. However, when large-scale features are compared, it is indicated that the Kelvinwave-like zonal winds are dominant on the eastern sideof the heating, while an equatorial Rossby wave-likeatmosphere. J. Meteor. Soc. Japan, 66, 309-329.
Nakajima, T., M. Tsukamoto, Y. Tsushima, A. Numaguti, andT. Kimura, 2000: Modeling of the radiative process in an
atmospheric general circulation model. Appl. Opt., 39,4869-4878.

vortical circulation off the equator is predominant on

Nakazawa, T., 1988: Tropical super clusters within intrasea-

the western side. In other words, asymmetric horizon-

sonal variations over the western Pacific. J. Meteor. Soc.

tal structures due to the equatorial beta are seen evenwhen the size of the heating is small. It is concluded
that the difference of the heating extension is not serious.

Japan, 66, 823-839.
Nasuno, T., H. Tomita, S. Iga, H. Miura and M. Satoh, 2007:
Multiscale organization of convection simulated with
explicit cloud processes on an aquaplanet. J. Atmos. Sci.,
64, 1902-1921.
Nasuno, T., H. Tomita, S. Iga, H. Miura and M. Satoh, 2008:

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14

Footnotes:
1）‘Negative precipitation’ is hard to imagine because real
precipitation is irreversible. Therefore, PSs with ‘negative precipitation’ should be considered to be fictitious
ones.
2）The values of static stability and speed of sound waves
are independently treated, although they should be interrelated.
3）In this study, the term“nonlinearity”is used when（7）and（8）are included, although the POWC itself has nonlinearity in the horizontal direction.

地球環境研究，Vol.18（2016）

赤道域のスーパークラスタの東進に関する NICAM の計算結果と positive-onlywave CISK を持つモデルの比較． Part 1. フルモデルからのアプローチ
吉

𥔎

正

憲＊

立正大学 ・ 地球環境科学部

＊

要

旨：
赤道対称モードの大規模東進じょう乱として、1000km オーダーの水平スケールをもち15 – 25 ms-1の速度のスーパークラスタ（SC）と5000km オーダーの水平スケールを持ち10 ms-1以下の速度のマッデンジュリアン振動が観測される（Wheeler and Kiladis 1999）。positive-only wave CISK を使うと、top-heavy な熱源分布のとき、東西に分裂するじょう乱が発生しいずれ東進じょう乱が卓越する（Yoshizaki, 1991a, Yoshizaki et al 2012b）。しかし、単純なモデルで単純な環境場を与えると、伝搬性のじょう乱は‘非常に大きな’top-heavy な熱源分布の場合にしか発現しない。ここでは、‘もっともらしい大きさ’の top-heavy な熱源分布で伝搬性のじょう乱が得られることを目指した。
NICAM と同じ鉛直グリッドと環境場を持つフルモデルを用いることによって、Gill（1980）パターンと同じような
東西に非対称な構造の東進擾乱を再現し、また NICAM 出力と同じような東進速度をもつことを示した。さらに非線型項や基本場の東西風を消去したり音波や大気安定度の大きさを変えたりする実験を行い、東進する擾乱がゆっくりと西進するものに急変するのは、対流圏下層では大きく対流圏上層では小さい大気安定度を鉛直一様にするときに起こることがわかった。その理由は、大気安定度の鉛直方向の依存性が top-heavy な熱源分布を強化するためである。
これから、単純な環境場から大気安定度を実況に近いものに変えることにより、‘通常’の top-heavy な熱源分布でも伝搬性のじょう乱が得られると期待される。

キーワード： positive-only wave CISK、スーパークラスター、東進特性、赤道ベータ

15