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 2. Approach from a Simplest Model―
YOSHIZAKI Masanori ＊キーワード：positive-only wave CISK, super clusters, eastward propagation, equatorial βAbstract

nized large-scale precipitation systems（PSs）with specific horizontal and time scales, such as the Madden-

Using a full model with the positive-only wave CISKJulian Oscillation（MJO; e.g., Madden and Julian 1971;

to study the eastward-propagating（EP）property ofWheeler and Kiladis 1999）and super clusters（SCs）（e.g.,

super clusters with a horizontal scale of O（1,000 km）

Nakazawa 1988; Takayabu 1994; Wheeler and Kiladis

in the tropical areas, Yoshizaki（2016）in Part 1 obtained1999）. Among them, SCs have the eastward-propagat-

that the occurrence conditions of splitting disturbancesing（EP）property and an asymmetric east-west struc-on the top-heavy heating are relaxed when realistic

ture with a horizontal scale of O（1,000 km）（e.g.,

static stability is used. In this study, a mode expansion

Nasuno et al. 2007, 2008）. A general detailed introduc-in the vertical direction is applied to a simplest model intion is given in Part 1.

an idealistic troposphere, and linear and nonlinear cases

To explain the EP property of SCs, many theoreticalare studied. In the linear case, the propagation property

and numerical studies have been conducted（e.g.,of growing disturbances is separated into two groups:

Hayashi and Sumi 1986; Takahashi 1987; Lau and Peng（i）a single slowly westward-propagating one and（ii）

1987; Neelin et al. 1987; Yoshizaki 1991a, 1991b; Yano etsplitting ones. However, the（ii）-type disturbances,al. 1995; Wang and Li 1994; Wang 2005; Mapes 2000;

which lead to the EP property, appear only in the

Majda and Shefter 2001; Majda et al. 2004; Khouider‘unusual’ top-heavy heating compared with observa-

and Majda 2006）. Although there are advantages andtions. It is found that the occurrence conditions for the

weaknesses as have described in Part 1, a PS view is（ii）-type disturbances become small in the nonlinear

promoted. Here, the PS view is assumed that the dis-case. The nonlinear term mainly causes the upward

turbance interacts with the diabatic heating due to pre-transport of the potential temperature and helps tocipitation and its physics is inhomogeneous in theenhance the formation of the top-heavy heating. Thus, ithorizontal direction. A positive-only wave CISKis anticipated that the EP property likely appears in the（POWC）
approach（e.g., Hayashi and Sumi 1986; Lau and‘usual’ top-heavy heating when complicated models are

Peng 1987; Yoshizaki 1991a）can be grouped in the PS

used in the realistic basic fields, for example, due to theview.

inclusion of the nonlinear term（Part 2）and the adoptionof variable static stability（Part 1）.

A series of the present study aims at how to get
splitting disturbances in the‘usual’ top-heavy heating,since Yoshizaki（1991）showed that they are obtained

1. Introduction

only in the‘unusual’’top-heavy heating as long as an
idealistic model is used in the simplified basic field. In

Part 1, the EP property of SCs was obtained using a
In the tropics, there are frequently observed orga＊	Department of Earth-environment Systems, Rissho University

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

full model with the POWC, and a simplification was

due to a non-dispersive property is attained for eachmade to understand the EP property. It was found thatmode when the hydrostatic assumption is applied on aa slowly westward-propagating（WP）disturbance newly

non-rotating plane.

emerges from the EP one, when the variable static sta-

In a linear case, the variables of mass-weighted
winds, mass-weighted buoyancy and pressure can bebility become uniform.
In this study, a simplest model is considered in a sim-

decomposed into horizontal and vertical parts, and theplest basic field, where an idealistic troposphere has avertical parts can be separated into two groups:（U’, V’,constant speed of sound waves and constant static sta-p’）and（W’, B’, Q）.

bility. The investigation of two propagating regimes（aThe profiles of Q expanded in the vertical direction

single slowly WP disturbance and splitting ones）is a

are shown in Fig. 1. Compared with Q＊（solid lines inprimary topic. The linear and nonlinear cases are also

Fig. 2a in Part 1）, the maximum points of Q are locatedtreated, and the mechanisms enhancing the top-heavy

lower because of z dependence of <ρ>. For（W’, B’, Q）,

heating by the inclusion of nonlinearity are discussed.

the vertical structure of the n-th baroclinic mode（hn）

To examine the problems noted above, a vertical-mode

can be expressed as

expansion is applied（Section 2）. The propagation pro­perty is examined in the linear case in Section 3, and ad 2 hng d hn  2 2g ++ nπ +
2 hn = 0,
2

d z2
d
z
4 Cs 
Cs

(1)nonlinear extension is discussed in Section 4. In Section

where n is a positive integer（n=1, 2, 3,…）.The analyti-

5, discussion and conclusions are presented.

cal solution of（1）is given as

2. Vertical-mode expansion of the forcing
The governing equations are essentially similar to（1）–（5）in Part 1, although the buoyancy equation（4）’inAppendix is used instead of the θ equation（4）in Part




g
z 
,
hn ( z ) = exp  −2 sin  nπ
2 z

zT 


 2 Cs(2)where zT is the top height. This satisfies the following
relationship as
z


 g
1 T

exp 
2 z  h n ( z ) hn ′ ( z ) dz = δ n n ′ ,
∫
zT 0

 Cs

(3)

1. The details are found in Appendix. Here, only diffe­

where δn n’ is Kronecker’s delta（1 for n = n’ and 0 for n

rent parameters from Part 1 are noted. Q is the forcing≠ n’）. These solutions indicate the orthogonal relation-

defined asg ρθ

Q * . Constant values of <N> andship between different n and n’.
This mode expansion in the vertical direction has<Cs> are specified as 10 s and 300 ms , respectively,

already been used by Fulton and Schubert（1985）,

but the density variation is included as shown in Fig. 2d

Schubert and Masarik（2006）, and Tulich et al.（2007）.

of Part 1. The number of vertical grid points is 32, theCompared with previous studies, the present form ofgrid size is uniform as 500 m, and the top height is 16（2）is a lower-weighted functional one in the verticalkm.

direction. The difference comes from the choice of dif-

-2

-1

-1

In this study, a mode expansion of the forcing in theferent variables, i.e., a mass-weighted z-velocity compo-

vertical direction is tried, similarly to Yoshizaki（1991a）.

nent in the present case W ′(≡ ρ w′) or a z-velocity

There are two advantages compared with the gridcomponent in the previous ones, such as w’. Since themodels in the vertical direction which most researchers

amplitude ofused（e.g., Yamasaki 1969, Hayashi 1970, Lau and Peng

constant in the vertical direction, the amplitude rever-

1987, Chang and Lim 1988）. The first is the linear pro­

sal between the present and previous studies can beperty in the vertical direction. Compared with Chang

anticipated.

and Lim（1988）and so on, the separation between the
propagating and non-propagating features can be easily
done. The second is a simple treatment of horizontal
propagation of each baroclinic modes. A constant speed
181

ρ 2 w in the isothermal atmosphere is

The mode expansion applied to the vertical profile ofQ（Fig. 1a）is approximated as

Q(z) = 0.559 h1(z) + 0.282 h 2(z) −0.053 h 3(z) −0.040 h 4(z) ….

(4)

地球環境研究，Vol.18（2016）

2nd

1st

1st
2nd

z

3rd

1st +2nd +8*3rd +4th

8*3rd
4th
Q

Figure 1

1st +2nd +3rd +4th

Q

4th
Q

Q

⒜ Vertical forcing profiles of（1, 1, 1, 1 ）; first（blue）, second（green）, third（purple）, and fourth（cyan）baroclinic modes, their sum（red）, and Q（black）fromQ＊（solid lines）of Fig. 2 a in Part 1. ⒝（1, 1, 8, 1 ）. The level of wB is shown by
an arrow.

Except in the lower layer, the sum of these four baroclinic modes can approximately represent the forcing

3. Propagation property in a linear caseprofile. At least, several baroclinic modes are needed to

As an example of a linear case, Figure 2 shows theexpress the realistic vertical profiles of the forcing（dia-longitude - time sections of log 10 | w | at z = 3.5 km forbatic heating）. It is noteworthy that the signs of the（1, 0, 0, 0）（, 1, 1, 8, 0）and（1, 0. 8, 0）.In Fig. 2a, a single

first and second baroclinic modes are positive at thenearly stationary growing disturbance is obtained.

level of wB, while those of the third and fourth ones are

Hereafter, this is called a slowly WP disturbancenegative. Hereafter, the dependence of the propagation

because it propagates westward slowly when a longer

property is examined on the basis of the combination ofsimulation is made. Such disturbances are seen for a3

the first four baroclinic modes expressed as（1, 1, 1, 1）,

smaller than 7 in（1, 1, a3, 1）, smaller than 8 in（1, 1, a3,

where each row means h（z）fromn = 1 in order andn0）,and smaller than 2 in（1, 0, a3, 0）（dashed lines in Fig.

the number indicates each amplitude（an）. For example,

6）. Nearly stationary damped disturbances are also

（1, 0, 4, 0）means Q = 0.559 h1（z）– 0.212（= 4 x 0.053）

obtained for（0, 1, 0, 0）（not shown）.On the other hand,

h3（z）. Here, by putting a1 = 1, a2 = 1 or 0, and a4 = 1 or

the preferred disturbances in Figs. 2b and 2c are split-

0, only a3（integer）is changed in the interval of 0 to 8.

ting ones. Such splitting disturbances are usually foundIn this study, the baroclinic mode of a3 is negative atfor cases in which the 3rd baroclinic mode is negatively

the level of wB, and its magnitude is considered tolarge at the level of wB, i.e., a large a3. Thus, the top-

determine the whole propagation property. Figure 1b

heavy heating profiles are likely to produce splittingshows an example of a forcing profile with eight times

disturbances. Consequently, all growing disturbancesof the 3 baroclinic mode（1, 1, 8, 1）. Although this casecalculated in this study can be separated into two

is artificially exaggerated（‘unusual’）, the top-heavygroups:（i）a single slowly WP disturbance and（ii）split-

heating profiles, where the heating is intense aroundting ones, as in Takahashi（1987）and Yoshizaki（1991a）.

the heights of 5 – 10 km, are frequently observed（e.g.,

The question of how such a propagation property isrd

Lin et al. 2004）.

determined then needs to be answered. It is shown thatthe role of each baroclinic mode is determined by the19Comparison of Models Having Positive-only Wave CISK with the NICAM Outputs about Eastward Propagationof Super Clusters in the Equatorial Region（Yoshizaki）(a)(1, 0, 0, 0)(b)(1, 1, 8, 0)(c)(1, 0, 8, 0)

2

0
0

-2

-4

-2

-4

Figure 2-2-4Longitude - time sections of log 10 | w | at z = 3.5 km in the linear case for ⒜（1, 0,
0, 0 ）, ⒝（1, 1, 8, 0 ）, and ⒞（1, 0, 8, 0 ）with contour interval of 2. Areas drawn
by grey indicate those of upward motion. The horizontal sections along the S1,
S2, S3, and S 4 lines are shown in Fig. 3.

(a) (1, 0, 0, 0) 3 day(b) (1, 0, 8, 0) 1 day(c) (1, 0, 8, 0) 2 day(d) (1, 0, 8, 0) 3 dayyFigure 3x orizontal structures of vectors（u, v）, w（color）, and Q（contours）at ⒜ 3 daysH
for（1, 0, 0, 0 ）, and at ⒝ 1 day, ⒞ 2 days, and ⒟ 3 days for（1, 0, 8, 0 ）. Allvariables are normalized in each plane.

sign at the level of wB. Here, we consider（1, 1, 8, 0）in

dently, the 1st baroclinic mode grows with a slowly WP

Fig. 2b. When each baroclinic mode is excited indepen-

property（Fig. 2a）, while the 2nd baroclinic mode is20

地球環境研究，Vol.18（2016）

damped nearly stationary and the 3rd baroclinic mode isFor（1, 0, 8, 0）, with time, EP and WP disturbances

also damped（not shown）. Although the 2 and 3 baro-

indicate different behaviors; that is, the EP one isclinic modes are similarly damped, they play different

always single, while a new WP one repeatedly occurs

roles. When the 1 and 2

baroclinic modes are com-on the eastern side of an old disturbance resulting in abined as（1, 1, 0, 0…）（not shown）, a single slowly WP

wavy train of WP disturbances（Fig. 2c）. Such asym-

property appears as（1, 0, 0, 0）（Fig. 2a）. When the 1stmetric horizontal patterns are frequently found inand 3 baroclinic modes are combined as（1, 0, 8, 0）

POWC simulations（e.g., Yoshizaki 1991a）. This feature（Fig. 2c）, on the other hand, splitting disturbances are

was explained by considering a linear response problem

obtained, as in（1, 1, 8, 0）（Fig. 2b）
. The difference

for a localized heating propagating in the longitudinalbetween the 2 and 3 baroclinic modes is the sign at

direction by Yoshizaki（ 1 9 9 1 b）. The details are

the level of wB. In（1, 0, 8, 0）（Fig. 2c）, the 1 baroclinic

explained in Section 5.

nd

st

nd

rd

rd

nd

rd

st

mode works for the amplification of the disturbanceand the 3rd baroclinic modes work for the propagation.
The oscillatory property in the 3rd baroclinic mode,
which is originally damped with time due to negative

4. Extension to a nonlinear casea. Nonlinear property

buoyancy at the level of wB, is excited by the coupling

In a nonlinear case, only terms related to ε1 in Part 1of the 1 baroclinic mode and changes to the propaga­

are added. However, it is assumed that the vertical pro-ting one in the horizontal direction.

file of the Q is unchanged. Figure 4 shows the longitudest

The results obtained above can be generalized as fol-- time sections of w at z = 3.5 km for（1, 0, 0, 0）and（1,lows, although they are essentially similar to those in

1, 8, 0）.A single slowly WP disturbance is found in Fig.

Yoshizaki（1991a）. When the signs of baroclinic modes

4a, and splitting disturbances are seen in Fig. 4b. Mostat the level of wB are all positive or positive and slightly

cases have propagation properties similar to the linear

negative, single slowly WP growing disturbances arecase, although the obtained amplitudes become compli-

attained. On the other hand, splitting growing distur-cated and sometimes saturated.

bances are obtained when the signs of the baroclinic

In some cases, however, different propagation pro-modes at the level of wB are both positive and moder-perties are found between the linear and nonlinear cases.

ately negative. At least, one baroclinic mode should be

Figure 5 shows the longitude - time sections of log 10 | w |greater than a threshold magnitude at the level of wB

in the linear case and w in the nonlinear case at z = 3.5

for preferred disturbances to be unstable. Generally, thekm for（1, 1, 5, 1）. The linear preferred disturbance

top-heavy forcing profile likely induces the propagationgrouped into a single slowly WP one（Fig. 5a）differs

property. Therefore, the dynamics of the propagationfrom the nonlinear preferred ones, in which splitting

property reported above are different from those of li­-ones are dominant（Fig. 5b）.

near neutral waves controlled by the dispersion relation（e.g., Matsuno 1966）.

The dependence of a3 on the propagation property inlinear and nonlinear cases is summarized in Fig. 6.

Figure 3 shows the horizontal patterns of u, v, w, andHere,（1, 0, a3, 0）（1,,
1, a3, 0）,and（1, 1, a3, 1）are studied.

Q at 1.75 km at 3 days for（1, 0, 0, 0）and at 1, 2 and 3

In the linear case, faster propagation is generallydays for（1, 0, 8, 0）. For（1, 0, 0, 0）（Fig. 3a）, a quasi-

obtained in larger a3 cases of splitting disturbances.

stationary disturbance moving slightly westward has a

However, the occurrence conditions for splitting distur-compact circular shape. For（1, 0, 8, 0）（Figs. 3b, 3c andbances are‘unusual’. The term of‘unusual’ means that

3d）, on the other hand, a disturbance expands outward

the heating profile is too‘top-heavy’ compared with thein a radial direction first and soon the trapped structure

observations or the NICAM outputs. In the nonlinear

in the latitudinal direction appears due to the equatorial

cases, on the contrary, the threshold magnitudes frombeta. When this behavior is viewed in the longitude -the slowly WP disturbance to splitting ones become

time section, splitting disturbances are attained.

smaller than those in the linear case. This suggests that21Comparison of Models Having Positive-only Wave CISK with the NICAM Outputs about Eastward Propagationof Super Clusters in the Equatorial Region（Yoshizaki）

the nonlinear effect not only produces the amplitude

averaged in the domain of 10°longitude and -4°– 4°lat-

saturation of preferred disturbances but also weaken

itude. Except for the time change of θ’, two terms of

the occurrence condition for slitting disturbances.
Because the realistic atmosphere is nonlinear, splitting
disturbances are more likely than they are in the linear
case.

||Q＊ || （heating term）and

− w′

dθ
（linear term）aredz

dominant in the linear case（Fig. 7a）,while three termsof the heating, linear terms, and − u ′b. Role of the nonlinear terms to enhance the top-

∂θ ′
∂θ ′
∂θ ′
− v′
− w′
∂x
∂y
∂z

（nonlinear term）dominate in the nonlinear case（Fig.

heavy heating

7b）. Since the vertical profiles of the heating and linear

It is necessary to explain why nonlinearity decreases

terms are similar in both cases, only a nonlinear term isthe threshold values. Figure 7 shows the vertical pro-focused on. Compared with these two terms, the nonli­

files of heat budgets for（1, 1, 5, 1）at 6 days, denoted asnear term has a higher vertical structure; i.e., negative|| ||. The areas of intense upward motion are chosen andvalues in the lower troposphere（< 9 km）and positive(a) (1, 0, 0, 0)(b) (1, 1, 8, 0)
0.02
0.04
- 0.020.02- 0.04

0.04
0.04

0.1
- 0.040.12

Figure 4- 0.02L ongitude - time sections of w at z = 3.5 km in the nonlinear case for ⒜（1, 0, 0,0 ）and ⒝（1, 1, 8, 0 ）with contour interval of 0.02 ms-1. Areas drawn by grey
indicate those of upward motion.
(a) (1, 1, 5, 1)(b) (1, 1, 5, 1)
0.06

6

0.04
0.02

4

- 0.022
0-2-4

Figure 522L ongitude - time sections of ⒜ log 10 | w | with contour interval of 1 in the linear
case and ⒝ w in the nonlinear case at z = 3.5 km with contour interval of 0.02
. Areas drawn by grey indicate those of upward motion.
ms- 1 for（1, 1, 5, 1 ）地球環境研究，Vol.18（2016）

values in the upper troposphere. A similar vertical pro-idealistic troposphere with constant static stability andfile is obtained for a quasi-nonlinear case,constant speed of sound waves should be discussed

only − w′

1）

in which∂θ ′is included in the linear case among∂zfrom a different standpoint from Part 1.
In this study, the grouping of growing disturbances

three terms in the nonlinear term（Fig. 7c）. Since theabout propagation was tried. In the linear case, twoquasi-nonlinear term works to advect θ’upward, thegroups of growing disturbances are obtained as

nonlinear case has the tendency of the top-heavy heat-

Yoshizaki（1991a）（: i）a single slowly WP one and（ii）

ing more than it does in the linear case. As the top-

splitting ones, depending on the vertical profiles of theheavy heating likely induces splitting disturbances, it isheating. When the signs of baroclinic modes at the levelconcluded that nonlinearity relaxes the occurrence con-of wB are all positive or positive and slightly negative, adition for splitting disturbances.

single slowly WP growing disturbance is attained. On
the other hand, splitting growing disturbances are
obtained when the signs of baroclinic modes at the level

5. Discussion and conclusions

of wB are both positive and moderately negative. At

In Part 1, the EP property of SCs in the tropicalleast, one baroclinic mode should be greater than aareas was simulated using a full model with the POWC.

threshold magnitude at the level of wB for preferred

However, this is not enough to get a deep understand-disturbances to be unstable. The（ii）-type disturbancesing of the EP property. Here, a simplest model in an

likely appear in the top-heavy heating, although their

(a)

c

30
0

(m s-1)(b)c(1,0,a3,0)
2

4

6

8

a3

6

8

a3

6

8

a3

-30
30
0

(m s-1)(1,1,a3,0)
2

4

-30

(c)

c

30
0

(m s-1)Figure 6(1,1,a3,1)
2

4

-30

 ependence of the propagation property on the amplitude of the third baroclinicD
mode（a 3 ）in the nonlinear（solid）and linear（dashed）cases for ⒜（1, 0, a3, 0 ）, ⒝, and ⒞（1, 1, a3, 1 ）. Dotted lines in ⒞ denote a quasi-nonlinear case.
（1, 1, a3, 0 ）23Comparison of Models Having Positive-only Wave CISK with the NICAM Outputs about Eastward Propagationof Super Clusters in the Equatorial Region（Yoshizaki）

Figure 7

Vertical profiles of heat budgets in ⒜ linear, ⒝ nonlinear, and ⒞ quasi-nonlinear
cases for（1, 1, 5, 1 ）at 6 days, averaged in the domain of 10 degrees in each
upward motion; ⒜ -10°‒ 0 °longitude, ⒝ 90°‒ 100°longitude, and ⒞ 45°‒ 55°longitude. All variables are normalized in each plane using the maximum values of ||Q＊ ||.

occurrence conditions are ‘unusual’. In the nonlinear

becomes complicated and the basic fields used becomecase, meanwhile, these conditions become small.

realistic. That is, the inclusion of the nonlinear term

Because the nonlinear term mainly makes the upward（Part 2）and the adoption of variable static stability

transport of the potential temperature, it is concluded（Part 1）help to induce the top-heavy heating and thethat nonlinearity induces the top-heavy heating more（ii）-type splitting disturbances are likely realized in thethan in the linear case.

‘usual’ top-heavy heating.

A series of the present works（Parts 1 and 2）areThen, why is only EP property chosen from（ii）-type

summarized as follows: A simplest model in an idealis-

splitting disturbances? One candidate explanation fortic basic field generally has （a i）-type WP disturbance inthis question is an asymmetric selection between thea‘usual’ top-heavy heating（Part 2）. On the other hand,

EP and WP disturbances in the top-heavy heating due

a full model in realistic basic fields（with the‘usual’ top-

to the equatorial beta（Yoshizaki 1991a, 1991b）. In theheavy heating）simulates the EP disturbances, which

linear theories（Yoshizaki 1991b）, the growing EP andoriginate from（ii）-type splitting ones（Part 1）. From

WP disturbances in the POWC cases correspond to athese results, it is anticipated that the vertical profiles

response problem of the EP and WP localized heatingof the top-heavy heating are enhanced as the modelin the Rayleigh damping / Newtonian cooling. When the24

地球環境研究，Vol.18（2016）

horizontal distributions of the heating are approximatedfrom the POWC outputs, it is found in the response
problem that the EP prescribed heating enhances anU '   u~ (t , x, y )   ~
 V '  =  v (t , x, y )  f ( z ),  ~
 p′   φ (t , x, y ) ~ (t , x, y ) W '   w
  ~
 B '  =  b (t , x, y )  h ( z ) Q   q~ (t , x, y )   upward motion only on the heating area, while the WP

and considering the combination of（3）’and（5）’, a fol-prescribed heating generates upward motions on thelowing equation about the vertical part is derived:

heating area as well as on its eastern side. In the
POWC instability case, the EP and WP disturbances

d2 h
g dh
+
+ c h = 0,22
dz
Cs d z(6)′

have different destinies after the response patterns

where c is a constant. Here, f（z）is proportional to

become large. The EP disturbance grows without hindrance keeping in an isolated manner, while the WP

dh. The solution of（6）’is given to satisfy the boundarydz

disturbance makes a wavy train of disturbances on its

conditions; h（0）= h（zT）= 0, where 0 and zT are theeastern side, reducing its growth more significantly

positions at the lower and upper boundaries, respec-

than the EP one. Due to this east-west asymmetric fea-

tively:ture（for example, Fig. 2c）, the EP disturbance isselected as a preferred disturbance.
Finally, the necessary conditions for the realization of
the EP property as like observed SCs are summarized;
1）full models with the driving heating such as the
POWC, 2）the realistic basic fields, 3）the equatorial




g
z 
,
z 2 sin  nπhn ( z ) = exp −
2 
C
2
z
s
T 




(7)′

where n is a positive integer. Using（7）’,（6）’can berearranged as

d 2 hng d hn  2 2g ++n π +
hn = 0.
2
2
2
dz
4 Cs 
Cs d z (8)′beta, and 4）
large-scale disturbances. In this manner, theEP property of PSs observed in the tropics can be sim-

References

ply understood as a response of the localized heating

Chang, C.-P., and H. Lim, 1988: Kelvin wave-CISK: A possible
mechanism for the 30-50 day oscillations. J. Atmos. Sci.,

moving in the longitudinal direction.

45, 1709-1720.
Fulton, S. R. and W. H. Schubert, 1985: Vertical normal

Appendix:transformations: Theory and application. Mon. Wea. Rev.,
113, 647-658.

Using the same definitions and assuming ε1 = ε2 = 0

Gill, A. E., 1980: Some simple solutions for heat-induced trop-from（1）（5）
in Part 1, the governing equations are writ-

ical circulation. Quart. J. Roy. Meteor. Soc., 106, 447-462.
Gill, A. E., 1982: Atmosphere-Ocean Dynamics. Academic

ten as

∂U ′
∂ p′
=−
+ β y V ′ +ν H ∆ H U ′ − r U ′,
∂t
∂x

(1)′

∂V ′
∂ p′
=−
− β y U ′ + ν H ∆ HV ′ − r V ′,
∂t
∂y

(2)′

0=−

g
∂ p′
p′ + B′,
−
2
∂z
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(3)′

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The cases with other nonlinear terms sometimes lead tomode and cloud decomposition of large-scale convectively

computational instability or show complicated patterns.

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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 2. 単純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：
第１部（Yoshizaki、2016）では、positive-only wave CISK を持つフルモデルを用いて、熱帯域の水平スケール約

1000km のスーパークラスタの東進特性を調べ、大気安定度の鉛直分布を実況に近くすると、もっともらしい top-heavy
な熱源分布で、じょう乱は東西に分かれ東進じょう乱が選択された。ここでは、理想化された対流圏の場でもっとも単純なモデルに鉛直方向のモード展開を適用し、線型および非線型の場合について top-heavy な熱源分布とじょう乱の伝搬特性を調べた。線型の場合、成長する擾乱の伝搬特性は、（i）一つの擾乱がゆっくりと西進するタイプと（ii）
東西に二つに分かれて伝搬するタイプの二つに分かれた。しかしながら、東進する擾乱に相当する（ii）のタイプは、第１部でみたように‘非常に大きい’top-heavy な熱源分布のときだけあらわれた。それに対して、非線型にすると、（ii）のタイプの発生条件が緩くなった。非線型にすると、擾乱による温位の上層への輸送が起こり、top-heavy な熱源分布になるためである。実況に近い大気安定度や非線型の導入により top-heavy な熱源分布の条件が緩和したことから、複雑なモデルと実況に近い環境場を用いれば、東進特性は通常の top-heavy な熱源分布でも発現すると期待される。

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

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