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Q. J. R. Meteorol. Soc. (2003), 129, pp. 2559–2590
doi: 10.1256/qj.02.151
A mechanism of the Madden–Julian Oscillation based on interactions in the
frequency domain
By T. N. KRISHNAMURTI¤ , D. R. CHAKRABORTY, NIHAT CUBUKCU, LYDIA STEFANOVA and
T. S. V. VIJAYA KUMAR
Department of Meteorology, The Florida State University, USA
(Received 10 July 2002; revised 20 January 2003)
S UMMARY
The surface and boundary-layer uxes of moisture exhibit a large ampli cation as the waves in the
Madden–Julian Oscillation (MJO) time-scales interact with synoptic time-scales of 2 to 7 days. This ampli cation
is clearly seen when the datasets are cast in the frequency domain for computations of the respective uxes.
Those ux relations carry triple-product nonlinearities, and the uxes on the time-scale of the MJO are evaluated
using co-spectra of triadal frequency interactions. The trigonometric selection rules on interactions among these
frequencies are largely satis ed by the time-scales of the MJO and two others that reside in the synoptic timescales. Tropical instabilities provide a rich family of tropical disturbances that appear to be ready and waiting
to interact with the MJO time-scales (since these satisfy the selection rules for non-vanishing interactions).
A consequence of these nonlinear interactions in the frequency domain is a two- to three-fold ampli cation of the
surface uxes. Although this analysis does not address how a small signal in the sea surface temperature on the
time-scale of the MJO arises in a coupled atmosphere–ocean model, we are able to show that its presence enables
a large ampli cation of this time-scale vertically across the planetary boundary layer. Given a low-frequency
ocean with many time-scales, this process ampli es the uxes on the time-scale of the MJO; this ampli cation
eventually feeds back to the ocean via ampli ed surface stresses, and an equilibrium state with a robust MJO in
the coupled system is realized. The datasets for this study were derived from a coupled ocean–atmosphere model
that was able to resolve a robust MJO in its simulations. This study also examines the character of sensible-heat
uxes and momentum within the same framework.
K EYWORDS: Climate modelling
Intraseasonal oscillations
1.
Nonlinear dynamics of waves
I NTRODUCTION
This paper explores the surface and PBL (Table 1 lists acronyms used) uxes of
moisture, sensible heat and momentum in the frequency domain with reference to the
MJO. The pioneering work of Madden and Julian (1971) has seen a major thrust of
research into intraseasonal oscillations in the last 30 years. The MJO is a planetaryscale wave that traverses from west to east in roughly 40 days. It has been identi ed as
a global phenomenon with its largest amplitude in the equatorial tropics. Its signature is
seen in most variables, such as sea-level pressure, zonal wind and divergent circulation.
Since that nding, numerous studies have elucidated the importance of this phenomenon
for its modulation of the monsoon activity and even typhoon behaviour in the western
Paci c Ocean. Several recent studies have even portrayed the links of the MJO with the
onset and decay of the ENSO.
In this study, the formulation for the surface and PBL uxes is based on the
FSU coupled atmosphere–ocean global spectral model (LaRow and Krishnamurti 1998;
Krishnamurti et al. 2000; Cubukcu and Krishnamurti 2002). Triple-product nonlinearities convey interesting scale interactions. In the present study we show that the expressions for the surface uxes in the constant- ux layer theory, and in the PBL ux
theory of numerical models, carry such triple-product nonlinearities. Thus, it is possible
to examine the uxes on a certain time-scale as they arise due to interactions with two
other time-scales. If we designate the Madden–Julian time-scale (20 to 60 days) as a
centre-piece for such computations, we see that a large number of possible interactions with the MJO time-scale arise from pairs of time-scales from within the synoptic
¤
Corresponding author: Department of Meteorology, The Florida State University, Tallahassee, FL 32306-4520,
USA. e-mail: [email protected]
c Royal Meteorological Society, 2003.
°
2559
2560
T. N. KRISHNAMURTI et al.
AGCM
CCM
CGSM
ECMWF
ENSO
FSU
GCM
GSM
MJO
NCAR
NCEP
PBL
SST
WISHE
TABLE 1. L IST OF ACRON YMS
Atmosphere-only General Circulation Model
Climate Community Model
Coupled Global Spectral Model
European Centre for Medium-Range Weather Forecasts
El Niño and Southern Oscillation
Florida State University
General Circulation Model
Global Spectral Model
Madden–Julian Oscillation
National Center for Atmospheric Research
National Centers for Environmental Prediction
Planetary boundary layer
Sea surface temperature
Wind induced surface heat exchange
time-scale, in the range of 2 to 7 days. Tropical instabilities arising from horizontal and
vertical wind shear and convection, provide a rich environment for these tropical higher
frequencies (waves, depressions and storms on time-scales of 2 to 7 days). Given a weak
low-frequency signal in the SST on the time-scale of the MJO, it is possible to perceive
frequency interactions such as to amplify the MJO signal. A demonstration of this is the
goal of this paper. We offer a hypothesis that the surface and PBL uxes invoke nonlinear interactions affecting the growth of the Madden–Julian time-scale. In the frequency
domain, this contributes to the maintenance and ampli cation of the Madden–Julian
time-scale in a coupled atmosphere–ocean system.
The inter-frequency interactions can contribute to either growth or decay of the
Madden–Julian time-scale. This is determined by the sign of the co-spectra evaluated
from the relevant frequencies. For quadratic nonlinearities, a selection rule for nonvanishing exchanges among frequencies requires that the two frequencies r and s, say,
are equal. For triple-product nonlinearities among member frequencies r, s and p, the
trigonometric selection rules require that p D r § s, i.e. the growth of a time-scale
within the Madden–Julian window (with frequency p/ is exactly equal to r C s or
¨r § s. It turns out that p at the time-scale of the MJO generally prefers two highfrequency time-scales within the 2- to 7-day window for the augmentation of moisture
uxes.
Over the ocean the spectra of SSTs exhibit some small power on the MJO timescale (Krishnamurti et al. 1988). Figures 1(a) and (b) show power spectra of the SST
and zonal wind at 850 hPa, respectively, for the Indian, West Paci c and Central Paci c
Oceans. These are based on the coupled-model run covering the period from March
1996 to February 1997 (Cubukcu and Krishnamurti 2002). This shows a plot of power x
frequency against the logarithm of the frequency (along the abscissa). The SST shows a
noticeable signal on the time-scale of the MJO, whereas on the synoptic time-scales the
power is much smaller. We do see a semi-annual signal in the SST from the model output
arising from the north–south seasonal migrations of the warm SST regions near the
equator. The zonal winds in the atmosphere show a larger signal on the synoptic-scale,
as is to be expected, and on the MJO time-scale. The interaction among these time-scales
is the topic of this study. Given a number of synoptic-scale tropical disturbances on the
time-scale of 2 to 7 days (e.g. Yanai and Nitta 1968; Reed and Recker 1971; Reed et al.
1977; Thompson et al. 1979; and several others), a unique opportunity is provided for
interactions between these time-scales and the MJO time-scale in the frequency domain
of the surface and PBLs.
A MECHANISM OF THE MJO
Figure 1.
2561
Power spectra (Power £ frequency against log.frequency //: (a) of sea surface temperatures (SSTs) for
the tropical Indian, West Paci c and Central Paci c Oceans; (b) for the zonal wind at 850 hPa.
In this paper we demonstrate the following aspects of the growth of the
Madden–Julian time-scale over near equatorial latitudes for the oceanic basins, since
the amplitude of the MJO is largest there (Krishnamurti and Gadgil 1985).
²
The mutual interactions between the low-frequency behaviour of the SST and
the higher-frequency synoptic disturbances of the tropical atmosphere over the
atmospheric surface layer (the constant- ux layer).
2562
The mutual interactions of the Madden–Julian time-scale with the synoptic-scale
disturbances in the PBL.
²
The issue of climate model simulation of the MJO has been a topic of great
interest in recent years. Summaries of recent contributions on the topic may be found
in Kemball-Cook et al. (2002) and Maloney (2002). The consensus seems to be that
models with prescribed SSTs are not able to simulate realistic features of the MJO,
whereas the coupled atmosphere–ocean models are somewhat more successful in this
respect. Maloney (2002) explored the effects of WISHE following Emanuel (1987) in
the simulation of the MJO. He noted that a removal of WISHE in the NCAR-CCM3.6
model led to an improved simulation of the MJO, and he concluded that WISHE might
contribute to the growth of modes outside of MJO frequencies. This is just one aspect
of the details of a cumulus parametrization through which MJO sensitivity has been
addressed. A more recent unpublished work from Colorado State University (David
Randal, personal communication) points to the importance of cloud-resolving models
where explicit, rather than parametrized, cumulus convection has been very successful
in the mapping of the MJO. Some of the earlier work of Manabe et al. (1965) where
moist convective adjustment was used has provided some evidence of intraseasonal
oscillations. Further work is clearly warranted to identify the scope of models that
resolve the MJO and its variability.
Simulation of the MJO is regarded as one of the most important components for
the medium- and long-range forecasts of the Asian summer monsoon. In many GCM
studies, it has been observed that the simulation of tropical intraseasonal oscillations
highly depends on the choice of cumulus convection parametrization scheme (Wang
and Schlesinger 1999). In some recent studies, Lee et al. (2001) emphasized the in uence of cloud–radiation interaction and cumulus entrainment constraint in simulating
tropical intraseasonal oscillations using the aqua–planet version of Seoul National University GCM (SNUGCM). Wang and Schlesinger (1999) pointed out the importance
of the boundary-layer relative-humidity thresholds for producing realistic variability of
the MJO in GCMs. In another study, Maloney and Hartman (2001) pointed out that
convective downdraughts are important and control the variability of the tropical MJO.
However, most AGCMs forced with a slowly varying SST annual cycle are unable to
represent the eastward propagation of convection from the Indian Ocean to the West
Paci c. This indicates that the atmospheric internal dynamics may contain mechanisms
driving the MJO in AGCMs. The implication is that the ocean is necessary for generating the MJO in a model, and this remains an open question. Waliser et al. (1999) have
compared the MJO in coupled and atmosphere-only versions of the same GCM and
showed an improvement in many aspects of simulation of the MJO. Due to the complex
interaction between large-scale dynamics and convection, and between convection and
the ocean surface, representation of the MJO presents a challenge for a coupled model
(Inness and Slingo 2002).
2.
Q UADRAT IC AND TRIPLE - PRODUCT NONLI NEARITIES AND COMPUTATIONS OF CO
SPECTRA
We assume that a given time series of data for two variables u.t/ and v.t/ are
cyclic and discrete in time. Since the FSU coupled model was run for one year, we have
assumed a fundamental periodicity over this time span. Our aim is to examine the role
of the MJO within this period. These series are represented by temporal Fourier series
2563
with discrete frequency (n) and Nyquist frequency (N ) such as:
u.t / D
N »
X
Cnu
nD0
¼
2¼
2¼
u
nt C Sn sin
nt :
cos
N
N
(1)
In particular, Cou D uo (time mean), see Sheng and Hayashi (1990a,b). The sample
frequency co-spectrum Pn .u; v/ is de ned as:
Pn .u; v/ D 12 .Cnu Cnv C Snu Snv /:
(2)
In the wave number domain the co-spectrum would be de ned by:
´
Z 2¼ »³
2¼
2¼
Pn .u; v/ D 1=2¼
Cnu cos
n¸ C Snu sin
n¸
N
N
³ 0
´¼
2¼
n¸
£ Cnv cos n¸ C Snv sin
d¸:
N
However over the frequency domain the co-spectrum would be expressed by:
´
2¼
2¼
u
Pn .u; v/ D
nt C Sn sin
nt
cos
N
N
¡T =2
³
´ ¼ ¿ Z T =2
2¼
2¼
v
v
nt C Sn sin
nt dt
£ Cn cos
dt:
N
N
¡T =2
Z
T =2
»³
Cnu
Thus, in principle, the frequency co-spectrum can be formally replaced by a wave
number co-spectrum, Hayashi (1980).
Given triad interaction in the frequency domain, v.t / in Eq. (2) can be expressed as
a product of two sets b.t/ and c.t/, i.e. v.t / D b.t/c.t /.
Now, Pn .u; v/ takes on the form:
Pn .u; v/ D Pn .u; bc/ D L.n/ D
X
LN .n; r; s/:
(3)
r;s
LN .n; r; s/ in Eq. (3) denotes a contribution to L.n/ due to the speci ed combination
of frequencies r and s, where r and s satisfy either of the selection rules r C s D n or
jr ¡ sj D n. The explicit expression for L.n/ in the frequency domain is given below,
using Fourier transforms of u, b and c in the time domain.
Thus,
´³
´
2¼
2¼
2¼
2¼
u
b
b
L.n/ D
nt C Sn sin
nt
Cr cos
rt C Sr sin
rt
cos
N
N
N
N
¡T =2
³
´ ¼ ¿ Z T =2
2¼
2¼
c
c
st C Ss sin
st dt
£ Cs cos
dt:
(4)
N
N
¡T =2
Z
T =2
»³
Cnu
2564
The above formalism has been extensively discussed by Hayashi (1980) in his
pioneering studies. Thus we may write:
Z T =2 ³ X X X
2¼
2¼
2¼
L.n/ D .1=T /
Cnu Crb Csc cos
nt cos
rt cos
st
N
N
N
¡T =2
n
r
s
2¼
2¼
2¼
nt sin
rt cos
st
N
N
N
2¼
2¼
2¼
nt cos
rt cos
st
C Snu Crb Csc sin
N
N
N
2¼
2¼
2¼
nt sin
rt cos
st
C Snu Srb Csc sin
N
N
N
2¼
2¼
2¼
nt cos
rt sin
st
C Cnu Crb Ssc cos
N
N
N
2¼
2¼
2¼
nt sin
rt sin
st
C Cnu Srb Ssc cos
N
N
N
2¼
2¼
2¼
nr cos
rt sin
st
C Snu Crb Ssc sin
N
N
N ´
2¼
2¼
2¼
nt sin
rt sin
st dt:
C Snu Srb Ssc sin
N
N
N
C Cnu Srb Csc cos
Considering the rst term of this equation
³
´
Z
1 T =2 X X X u b c
2¼
2¼
2¼
Cn Cr Cs cos
nt cos
rt cos
st dt
T ¡T =2
N
N
N
n
r
s
Z T =2 µ X X X
1
2¼
Cnu Crb Csc cos
nt
D
N
2T ¡T =2
n
r
s
»
¼ ¶
2¼
2¼
.r C s/t cos
.r ¡ s/t
£ cos
dt:
N
N
This is an in nite integral and must be made nite for computational purposes. Using
the orthogonal property of Fourier functions, this term takes the following form:
Z
X
1 T =2 X u b c
2¼
2¼
Cn Cr Cs cos2
nt dt C
Cnu Crb Csc cos2
nt dt
T ¡T =2 nDr Cs
N
N
nDr¡s
X
2¼
Cnu Crb Csc cos2
nt dt
C
N
nDs¡r
(only these terms survive under the condition n D r C s, n D r ¡ s, n D s ¡ r).
Applying cos2 A D .1 C cos 2A/=2, this takes the form
2 X 3
6nDrCs 7
6 X 7
7 u b c
16
7C C C :
D 6
7 n r s
46
6nDr¡s
7
X
4
5
nDs¡r
2565
Hence the above equation reduces to:
X 3
X
2
2
C
¡
6 rCsDn 7
6 r CsDn
6
6
X 7
X
7 u b c 16C
16
C
6
7
6
L.n/ D 6
7 Cn Cr Cs C 4 6 r ¡sDn
4 6 r¡sDn
7
6
X
X
4
5
4
C
C
2
r ¡sD¡n
C
X 3
2
3
7
7
7 u b c
7C S S
7 n r s
7
5
r¡sD¡n
C
X 3
6 r CsDn 7
6 rCsDn 7
6
7
6
X
X 7
7 u b c 16
7 u b c
16
¡
C
6
7
6
7S S C :
Sn Cr Ss C 6
C 6
7
7 n r s
r¡sDn
4 6 r ¡sDn
4
6
X 7
X 7
4
5
4
5
C
¡
r¡sD¡n
(5)
r ¡sD¡n
This is used to calculate the net gain or loss of energy (in our problem the uxes) by a
frequency n as it interacts with frequencies r and s.
(a) Surface similarity theory
Our use of surface similarity theory, based on the work of Businger et al. (1971),
can be expressed following Krishnamurti et al. (1998).
The latent-heat ux is given by:
Lh D ½CLH LjV j.Qs ¡ Qa /:
(6)
Ls D ½Cp CS jV j.Ts ¡ Ta /
(7)
Here ½ is the density of air, L is the latent heat of vaporization, jV j is the magnitude of
the surface wind and Qs and Qa denote the speci c humidities over the sea surface and
at the anemometer level. CLH denotes the exchange coef cient for latent heat. This is a
stability-dependent time varying coef cient of the similarity theory.
The expression for sensible-heat ux is given by:
where Cp is the speci c heat of air at constant pressure, CS is the exchange coef cient
for sensible heat, Ta is the temperature of air near the surface (at anemometer level) in
Kelvin and Ts is the surface temperature. The expression for u momentum ux is given
by:
¿ x D ½CM jV j.ua ¡ us /
(8)
where ua and us are zonal winds at anemometer and surface levels, respectively; CM is
the exchange coef cient for momentum. Lh , Ls and ¿ x are the respective uxes of latent
heat, sensible heat and momentum at the top of the constant- ux layer. It is assumed
that CLH D CS D CM , where CLH , CS and CM are the bulk exchange coef cients of
latent heat, sensible heat and momentum respectively; these exchange coef cients are
functions of stability and, as such, are space and time dependent. Daily values of Lh , Ls
and ¿ x on the left-hand side (l.h.s.) of Eqs. (6), (7) and (8), respectively, are obtained as
model output based on surface similarity theory. On the right-hand side (r.h.s.) of these
equations daily values of jV j, jQs ¡ Qa j, (Ts ¡ Ta / and (ua ¡ us / are also obtained
from model output datasets. Using all those values in Eqs. (6), (7), and (8), values of
bulk coef cients are obtained as space- and time-dependent quantities. We used this
methodology for computations of uxes in the frequency domain.
2566
Using Fourier time series for CLH , jV j, and L.Qs ¡ Qa / and taking the co-spectrum
among them on the r.h.s. of expression (6), with the help of formula (5) latent-heat ux
in the frequency domain can be written as:
X 3
2
C
6 r CsDn 7
6
X 7
7
½ 6
C
6
7 [LHC.n/:MVC.r/:DMC.s/]
hLh .n/i D 6
7
2 6 r ¡sDn
X 7
4
5
C
2
r¡sD¡n
¡
X 3
6 rCsDn 7
6
X 7
7
½ 6
C
6
7 [LHC.n/:MVS.r /:DMS.s/]
C 6
7
2 6 r¡sDn
X 7
4
5
C
2
r ¡sD¡n
C
X 3
6 rCsDn 7
6
X 7
7
½ 6
¡
6
7 [LHS.n/:MVC.r /:DMS.s/]
C 6
7
2 6 r¡sDn
X 7
4
5
C
2
r ¡sD¡n
C
X 3
6 rCsDn 7
6
X 7
7
½ 6
C
6
7 [LHS.n/:MVS.r /:DMC.s/];
C 6
7
2 6 r¡sDn
X 7
4
5
¡
r ¡sD¡n
(9)
where .LHC; LHS/, .MVC; MVS/ and .DMC; DMS/ are the cosine and sine coef cients
of CLH , jV j, and L.Qs ¡ Qa / associated with frequencies n, r and s respectively.
Using Fourier time series for CLH , jV j and .Ts ¡ Ta / and taking the co-spectrum
among them in the r.h.s. of expression (8), with the help of formula (5), sensible-heat
ux in the frequency domain can be written as:
X 3
2
C
6 rCsDn 7
6
X 7
7
½Cp 6
C
6
7 [LHC.n/:MVC.r /:DTC.s/]
hLs .n/i D
7
2 6
6 r¡sDn
X 7
4
5
C
2
r ¡sD¡n
¡
X 3
6 r CsDn
6
X
½Cp 6
C
6
C
2 6
6 r ¡sDn
X
4
C
r¡sD¡n
7
7
7
7 [LHC.n/:MVS.r/:DTS.s/]
7
7
5
2
C
2567
X 3
6 r CsDn 7
6
X 7
7
½Cp 6
¡
6
7 [LHS.n/:MVC.r/:DTS.s/]
C
7
2 6
6 r ¡sDn
X 7
4
5
C
2
r¡sD¡n
C
X 3
6 r CsDn 7
6
X 7
7
½Cp 6
C
6
7 [LHS.n/:MVS.r/:DTC.s/];
C
7
2 6
6 r ¡sDn
X 7
4
5
¡
(10)
r¡sD¡n
where .DTC; DTS/ are the cosine and sine coef cients of .Ts ¡ Ta / associated with
frequency s. Using Fourier time series for CLH , jV j and .ua ¡ us / and taking the cospectrum among them on the r.h.s. of expression (10), with the help of formula (5),
u momentum ux in the frequency domain can be written as:
X 3
2
C
6 rCsDn 7
6
X 7
7
½ 6
x
C
6
7 [LHC.n/:MVC.r/:DUC.s/]
h¿ .n/i D 6
7
2 6 r¡sDn
X 7
4
5
C
2
r ¡sD¡n
¡
X 3
6 r CsDn 7
6
X 7
7
½ 6
C
6
7 [LHC.n/:MVS.r/:DUS.s/]
C 6
7
2 6 r ¡sDn
X 7
4
5
C
2
r¡sD¡n
C
X 3
6 r CsDn 7
6
X 7
7
½ 6
¡
6
7 [LHS.n/:MVC.r/:DUS.s/]
C 6
7
2 6 r ¡sDn
X 7
4
5
C
2
½
C
2
r¡sD¡n
C
X 3
6 r CsDn
X
6
6C
6
6 r ¡sDn
6
X
4¡
7
7
7
7 [LHS.n/:MVS.r/:DUC.s/];
7
7
5
(11)
r¡sD¡n
where DUC, DUS are the cosine and sine coef cients of .us ¡ ua / associated with
frequency s:
2568
(b) A simple example of triad interactions
If u, b and c are interacting such that the energy going into frequency n is given
by Eq. (5), under what circumstances is there growth of the wave with that frequency?
The cosine and sine coef cients can be expressed as:
Cnu D ®nu cos ’nu ;
where
Snu D ¡®nu sin ’nu ;
³
®nu
D
2
.Cnu
C
2
Snu /1=2 ;
’nu
D tan
¡1
Su
¡ nu
Cn
(12)
´
:
(13)
If we substitute (12) into the Fourier expansion of u (Eq. (1)), we obtain:
³
´
X
X
2¼
2¼
2¼
u
u
u
u
u
u
u.t/ D
®n cos ’n cos
nt ¡ ®n sin ’n sin
nt D
®n cos
nt C ’n :
N
N
N
n
n
(14)
Thus ’nu can be interpreted as the phase shift of the nth wave in the cosine decomposition
of u. If we substitute expressions (9) into Eq. (5) and focus on a single triad combination
(n; r; s) we get:
1
L.n/ D ®nu ®rb ®sc £
2
.cos ’nu cos ’rb cos ’sc ¡ cos ’nu sin ’rb sin ’sc C sin ’nu cos ’rb sin ’sc
C sin ’nu sin ’rb cos ’sc /;
.cos ’nu
r C s D n;
cos ’rb cos ’sc C cos ’nu
C sin ’nu sin ’rb cos ’sc /; r
.cos ’nu cos ’rb cos ’sc C cos ’nu
¡ sin ’nu sin ’rb cos ’sc /; s
which can be simpli ed to:
sin ’rb sin ’sc ¡ sin ’nu cos ’rb sin ’sc
¡ s D n;
sin ’rb sin ’sc C sin ’nu cos ’rb sin ’sc
¡ r D n;
fcos.’nu ¡ ’rb ¡ ’sc /g;
1 u b c
L.n/ D ®n ®r ®s fcos.’nu ¡ ’rb C ’sc /g;
2
fcos.’nu C ’rb ¡ ’sc /g;
r C s D n;
r ¡ s D n;
s ¡ r D n:
(15)
(16)
Since the ®’s are positive de nite, the maximum possible value of L.n/ is obtained when
the cosine contribution is unity, i.e. when:
’rb C ’sc D ’nu § 2k¼;
’rb
¡’rb
¡
C
’sc
’sc
D
D
’nu
’nu
§ 2k¼;
§ 2k¼;
r C s D n;
r ¡ s D n;
(17)
¡r C s D n;
or, in other words, when ’rb ; ’sc relate to ’nu in the same way that r, s relate to n.
Figure 2(a) is a schematic illustration of the r; s; n frequency space. Here we highlight
those regions of r C s and r ¡ s where the synoptic-scale interacts with the MJO timescale. Outside of the two slanting zones in Fig. 2(a), the MJO does not interact with
the synoptic time-scales. Also shown in the bottom left is a region where r C s lies on
the MJO time-scale. Those are the regions where r and s also individually lie on the
MJO time-scale. Since a robust synoptic time-scale covering the period 2 to 7 days
2569
Figure 2. (a) A schematic diagram illustrating regions on a frequency, r, versus frequency, s, space where
the synoptic time-scales can interact with the Madden–Julian Oscillation (MJO) time-scales. The shaded areas
denote where s ¡ r or r ¡ s can amplify the MJO time-scales via triad interactions. (b) A schematic diagram
illustrating the lines along which three frequencies 9, 48 and 57 interact to amplify the lower frequency (with a
period of 40 days). Since the relationship among frequencies is cyclical, this family of lines reappears at §2k¼ .
(c) An example of three frequencies 9, 48 and 57 showing where the triad interactions can amplify the MJO
time-scale. The phase shifts of the synoptic-scale frequencies, 48 and 57, illustrated by thin dotted and solid lines,
match the phase shift of the MJO, shown by the heavy solid line.
2570
Figure 2.
Continued from previous page.
is known to exist over the tropics, strong interaction within the MJO time-scale is
possible along the shaded region. To illustrate an example of an interaction leading
to growth of the MJO time-scale, consider the contribution of the interaction of waves
n D 9, r D 48 and s D 57 (n D s ¡ r/ to the energy of n D 9. According to Eq. (17),
b C ’ c D ’ u § 2k¼ .
there will be maximum growth of energy at frequency 9 when ¡’48
57
9
The solutions to this equation are displayed as a diagram in Fig. 2(b). An example of a
triad of waves satisfying this equation is shown in Fig. 2(c).
Given the SST, Ts , and the surface speci c humidity Qs , if they carry a small
signal on the MJO time-scale we ask whether that frequency can amplify from triad
interactions at the top of the constant- ux layer. The selection rules tell us that higherfrequency tropical waves can, in principle, provide such a possibility. A question not
addressed in this study is whether the temporal variations of the exchange coef cients
can also contribute to the growth or decay of the MJO via triad interactions.
(c) Disposition of uxes in the PBL
Here, again, our interest is in the speci c algorithm that was used in our coupled
model for the vertical disposition of surface uxes within the PBL. Our formulation of
the PBL follows that of Louis (1979). The vertical ux convergence is expressed by a
vertical diffusion of surface uxes, the general expression is given by:
³
´
@¿
g2 @
2 @¿
½ K
;
D¢¢¢C 2
(18)
@t
@¾
ps @¾
where ps is the surface pressure, ¾ is the vertical sigma coordinate (p=ps /, and ¿ is
a basic dependent variable such as momentum, heat or moisture. The exchange coef cient, K; is determined from a mixing-length theory that involves stability-dependence
via the bulk Richardson number RiB (Louis 1979; Krishnamurti et al. 1998).
In our model the exchange coef cients are expressed by:
@jV j
KH D KQ D ` 2
Fh RiB ;
(19)
@z
2571
and
@jV j
Fm RiB ;
(20)
@z
where Fh (Fm / denotes the non-dimensional heat (momentum) ux, ` is the mixing
length (Blackadar 1962), expressed by ·z=.1 C ·z=¸/ where · is the Von Karman
constant, z is the height of the relevant computational level in the PBL, and ¸is a constant
which denotes an asymptotic mixing length whose values for heat and moisture uxes
are set to 450 m, and to 150 m for momentum.
Following Louis (1979), the analytical formulae for the surface similarity theory,
Fh and Fm are given as:
1
Fh D Fm D
; RiB ¸ 0;
(21)
.1 C 5RiB /2
for the stable case, and
KM D ` 2
1 C 1:286jRiB j1=2 ¡ 8RiB
; RiB < 0;
(22)
1 C 1:286jRiB j1=2
1 C 1:746jRiB j1=2 ¡ 8RiB
Fm D
; RiB < 0;
(23)
1 C 1:746jRiB j1=2
for the unstable case. Here, the bulk Richardson number over an atmospheric layer is
given by:
¿
2

g @µ
@jV j

:
RiB D
(24)
@z
µ @z
Fh D
The heat ux is given by:
@µ
@jV j
@µ
· 2 z2
@jV j
RiB
@µ
Fh RiB
D ½Cp l 2
D ½Cp
@z
@z
@z
.1 C ·z=¸/2 @z .1 C 5RiB /2 @z
FH D ½Cp KH
FH D ½Cp
2 /
· 2 z2
@jV j .RiB C 1:286RiB jRiB j1=2 ¡ 8RiB
@µ
2
1=2
@z
.1 C ·z=¸/ @z
1 C 1:286jRiB j
(25)
for the unstable case.
Using Fourier time series for the term involving RiB , @µ=@z and @jV j=@z, and
by taking co-spectra among them in the r.h.s. of expression (25) with the help of
formula (5), the heat ux in the frequency domain is given by:
X 3
2
C
6 rCsDn 7
6
X 7
2
2
6C
7
½Cp
· z
6
7 [RIC.n/:PTC.r/:VTC.s/]
hFH .n/i D
6
7
2 .1 C ·z=¸/2 6 r¡sDn
7
X
4
5
C
2
r ¡sD¡n
¡
X 3
6 r CsDn 7
6
X 7
6
7
½Cp
C
6
7 [RIC.n/:PTS.r/:VTS.s/]
C
6
7
2 .1 C ·z=¸/2 6 r ¡sDn
7
X
4
5
C
· 2 z2
r¡sD¡n
2572
2
C
X 3
6 r CsDn
6
X
6
½Cp
¡
6
C
2 .1 C ·z=¸/2 6
6 r ¡sDn
X
4
C
· 2 z2
2
7
7
7
7 [RIS.n/:PTC.r/:VTS.s/]
7
7
5
r¡sD¡n
C
X 3
6 r CsDn
6
X
6C
½Cp
6
C
2 .1 C ·z=¸/2 6
6 r ¡sDn
X
4
¡
· 2 z2
7
7
7
7 [RIS.n/:PTS.r/:VTC.s/]:
7
7
5
(26)
r¡sD¡n
Values of RIC.n/ and RIS.n/ will be different for stable and unstable cases. Here
.RIC; RIS/, .PTC; PTS/ and .VTC; VTS/ are the cosine and sine coef cients of the
terms involving RiB , @µ =@z and @jV j=@z associated with frequencies n, r and s;
respectively.
The moisture ux is given by:
@q
@jV j
@q
· 2 z2
@jV j
RiB
@q
FQ D ½KQ
Fh RiB
;
D ½l 2
D½
2
2
@z
@z
@z
.1 C ·z=¸/ @z .1 C 5RiB / @z
2 /
· 2 z2
@jV j .RiB C 1:286RiBjRiB j1=2 ¡ 8RiB
@q
FQ D ½
;
(27)
2
1=2
.1 C ·z=¸/ @z
@z
1 C 1:286jRiB j
for the unstable case. Using Fourier time series for the terms involving RiB , @q=@z and
@jV j=@z, and taking the co-spectrum among them on the r.h.s. of expression (27) with
the help of formula (5), the expression of moisture ux in the frequency domain, FQ .n/
is the same as FH .n/ except that PTC and PTS are to be replaced by PQC and PQS, the
Fourier temporal coef cients of @q=@z.
The momentum ux is given by:
@v
@jV j
@v
· 2 z2
@jV j
RiB
@v
FM D ½KM
Fm RiB
;
D pl 2
D½
2
2
@z
@z
@z
.1 C ·z=¸/ @z .1 C 5RiB / @z
2 / @v
· 2 z2
@jV j .RiB C 1:746RiBjRiB j1=2 ¡ 8RiB
FM D ½
;
(28)
.1 C ·z=¸/2 @z
@z
1 C 1:746jRiB j1=2
for the unstable case. Using Fourier time series for the terms involving RiB , @v=@z and
@jV j=@z, and taking co-spectra among them on the r.h.s. of expression (28) with the help
of formula (5), the expression for FM .n/ in the frequency domain is the same as that for
FH .n/ except that PTC and PTS are replaced by P V C and P V S, the Fourier temporal
coef cients of @v=@z.
The frequency interactions in the PBL are determined entirely by the variation of
the exchange coef cient K . Our premise is that these uxes in the frequency domain
build up within the PBL, thus enhancing the amplitude of the Madden–Julian time-scale
via the triad interactions. Thus we expect that a small oceanic signal in the SST on the
Madden–Julian time-scale is enhanced at the top of the atmospheric constant- ux layer,
and a further enhancement occurs across the PBL. These are illustrated in the results of
computations presented in the next section.
3.
2573
FSU COUPLED OCEAN – ATMOSPHERE MODEL AND THE MJO
(a)
Description of the coupled model
(i) Ocean model. The ocean model employed here is a modi ed version of the Max
Planck Institute ocean model (Sterl 1991; Latif et al. 1994). The model has 17 irregularly
spaced vertical layers. The upper part of the ocean is of greater interest since turbulent
activity due to external forcing is effective there; therefore, the upper 300 m have ner
vertical resolution with the thickness of layers from the top being 20, 20, 20, 20, 20, 30,
30, 40, 50, 50, 100, 200, 500, 1000, 1500 and 2000 m, respectively. There is variable
resolution in the meridional direction where: within §10± of the equator the resolution
is constant at 0.5± ; between §10± and §20± it increases to 1± ; and outside of §50±
it is set to 5± . Constant resolution of 5± is used in the zonal direction. The equations
of motion are solved using nite-differencing techniques on a horizontally staggered
E-grid scheme (Arakawa and Lamb 1977).
The time integration of the full system with a time step of 120 minutes is carried out
by the method of fractional steps. This method allows for each individual equation to be
separated into components which are integrated individually and then combined to give
the solution of the full system. For turbulent diffusion, a constant mixing coef cient of
1000 m2 s¡1 is used in order to calculate the horizontal mixing processes, whereas in
the vertical a formula dependent on the Richardson number is used. Surface forcing is
represented by the exchange of heat and momentum at the air–sea interface.
(ii) Atmospheric model. The numerical weather prediction model used in this study is
the FSUGSM described in Krishnamurti et al. (1998). The horizontal and vertical resolution of the model are exible; for this study we use a horizontal resolution truncated at
wave number 42 (T42) which gives an approximate grid mesh of 2.8± latitude/longitude
and a vertical resolution of roughly 0.5 km described by 14 sigma-levels between
roughly 50 and 1000 hPa. A semi-implicit time integration scheme is used with a time
step of 20 minutes to represent the time derivatives in the model equations. The highfrequency gravity-wave oscillations are suppressed by semi-implicit time differencing
wherever they appear in the model equations. The initial data sources for the FSUGSM
comprise the global analysis of ECMWF and the SSTs from NCEP. The prognostic
model variables are: vorticity, divergence, dew-point depression, surface pressure, and a
variable which combines the geopotential height and logarithm of the surface pressure.
The model physics includes:
²
²
²
²
²
²
²
modi ed Kuo scheme for cumulus parametrization (Krishnamurti et al. 1983);
shallow convective adjustment (Tiedke 1984);
dry convective adjustment;
large-scale condensation;
surface-layer parametrization by similarity theory (Businger et al. 1971);
PBL parametrization includes a diffusive formulation based on mixing length
theory dependent on Richardson number (Manobianco 1989);
parametrization of short-wave radiation by Lacis and Hansen (1974), and longwave radiation by Harshvardhan and Corsetti (1984).
The salient elements of this coupled model are outlined in Fig. 3.
(b) Datasets
The datasets used in the present study are global daily values for an entire year
(March 1996 to February 1997, inclusive) extracted from the coupled-model output
2574
Figure 3.
A schematic diagram describing the Florida State University (FSU) Coupled Ocean–Atmosphere
model.
(Cubukcu and Krishnamurti 2002). The variables relevant to the study of uxes include
winds, temperature, bulk coef cients and humidity in the atmosphere, and the ocean
temperatures of the coupled model. Here area averaging is carried out in areas: between
latitudes §11± and longitudes 130± and 150± E for the warm pool; between longitudes
180± and 160± W for the central tropical Paci c; and between 60± and 110± E for the
Indian Ocean. The same algorithms that were used within the coupled-model forecasts
are used for the computation of uxes in the frequency domain. It should be noted that
the total uxes of the coupled model’s output are not used directly, since the computation
of uxes in the frequency domain calls for the explicit computation of the co-spectra of
the triad interactions.
At each transform grid point of the spectral model, where the physical processes
are evaluated, we construct a time series for these triple-product variables. These are
360-day time series, one entry per day for each variable, denoting the value at the top of
the constant- ux layer for the period from March 1996 to February 1997. This is based
entirely on the model output datasets from the coupled model. This database is used to
construct the frequency co-spectra for the triads described earlier.
(c) The MJO in the FSUCGSM
The coupled model has previously been applied to numerous climatological problems (e.g. LaRow and Krishnamurti 1998; Krishnamurti et al. 2000; Cubukcu and
Krishnamurti 2002). The results have demonstrated that the FSUCGSM is an excellent
2575
(a)
0
-1 2
12
4
4
4
-4
12
-4
-12
4
10
-4
2
-1
12
12
12
-12
12
4
4
20 -4
12
-1
2
12
12
30
-44
12
12
4
12
4 -4 -12
40
412
-4
12
12
-4
4
-12-4
-4
124
4
50
12
12
-4
12
-4
4
60
12
4 1
2
4
70
12
12
4
12
-4
12
4
-4
4
80
12
-12
4
12
-4
60E
4
120E
4 -4
12
90
0
12
180
120W
60W
0
(b)
Figure 4. (a) A coupled-model result based on the eastward passage of the Madden–Julian Oscillation.
Shown are the 200 hPa zonal wind anomalies (m s¡1 / on a Hovmüller diagram for a 90-day period during the
summer of 1996 in the equatorial belt 5± S to 5± N. (b) Power spectra of the 200 hPa zonal wind anomalies for the
tropical Indian, West Paci c and Central Paci c Oceans.
2576
tool for climatological studies. Cubukcu and Krishnamurti (2002) showed the model’s
performance in simulating the tropical intra-annual oscillations such as the MJO to be
quite successful.
The entire context of this paper is based on the same numerical simulation used by
Cubukcu and Krishnamurti (2002). A simple example of MJO signal as simulated by
the FSUCGSM is shown in Fig. 4(a), which displays the time–longitude cross-section
of tropical 200 hPa zonal wind anomalies for three consecutive months towards the end
of the time integration (October–December 1996). The anomalies are relative to a oneyear mean of the model output; positive values denote westerly anomalies. The zonal
wind at 200 hPa is often used in various climate diagnostics bulletins to illustrate the
eastward passage of the MJO—a phenomenon that was captured quite nicely in our oneyear coupled-model run. The model output, as illustrated, shows a robust MJO, with the
largest westerly wind anomalies of the order of 20 m s¡1 . The power spectra of 200 hPa
winds shown in Fig. 4(b) highlights the peak signals in the time range of 25–60 days
that represent the MJO. From Figs. 1(a) and (b), and 4(b), a strong signal in the power
spectra of the SSTs, and the zonal winds at 850 hPa and 200 hPa levels, can be seen for
several frequencies within the 20- to 60-day time-scales. It does not show a strong band
covering the entire MJO time-scale as noted by Salby and Hendon (1994) and Hendon
and Salby (1994). However, several spectral peaks covering the entire range from 25 to
60 days can be seen on these gures. In this study we have limited our examination to
30–50 days, primarily based on the SST spectra and our previous studies.
4.
C OMPUTATION OF FLUXES IN THE FREQUENCY DOMAIN
The results presented in this paper are based on the following computations of the
uxes. The letters are used to label corresponding panels of plots displayed in Figs. 5
to 11.
(a) Total uxes on the time-scale of the MJO for the sensible heat, latent heat and
momentum across the constant- ux layer.
(b) Total uxes in the constant- ux layer on the time-scale of the MJO arising from
interaction of the MJO with the synoptic time-scale of from 2 to 7 days.
(c) Fluxes contributed by salient (strongest contributing) triad interactions in the
surface layer.
(d) Salient triad interaction frequencies contributing to (a) in the constant- ux layer.
(e) Total uxes on the time-scale of the MJO for the sensible heat, latent heat and
momentum in the PBL at the 850 hPa level.
(f) Total uxes in the PBL on the time-scale of the MJO arising from interaction of
the MJO time-scale with synoptic time-scale of 2 to 7 days.
(g) Fluxes contributed by the salient triad interactions in the PBL.
(h) Salient triad interaction frequencies contributing to (e) in the PBL.
In this section uxes of latent and sensible heat are expressed in W m¡2 , and momentum
uxes in N m¡2 . We consider three domains for illustrating these uxes in the frequency
domain:
²
²
²
Indian Monsoon: 11.2± S to 11.2± N, 61.88±E to 109.69±E;
Central Paci c Ocean: 11.2± S to 11.2± N, 160.31± W to 180± W;
Western Paci c Ocean: 11.2± S to 11.2± N, 151.88± E to 132.19±E.
We did not include a domain over the Atlantic since the MJO signal was weak in that
sector.
2577
a
10.
9.0
91894
88
7.2
3.65.4
83 86
1.880
1.7
8
0
1.88
3.683
3.
d
7,57,50
9,49,40
6,57,51
9,57,48
6,57,51
8,43,35
8,57,49
6,57,51
9,49,40
8,57,49
8,49,41
7,57,50
6,57,51
8,57,49
6,57,51
7,57,50
6,57,51
7,57,50
6,57,48
9,57,48
7,57,50
8,49,41
8,49,41
9,49,40
9,49,40
7,57,50
6,57,51
9,49,40
9,49,40
9,49,40
7,57,50
7,57,50
7,57,50
9,57,48
6,57,51
e
f
15.4
19
1157..9.7
4
11.2N
8.4N
5.6N
1.195
0.862
0.528
2.8N
0
2.8S
5.6S
8.4S
11.2S
60
70
190.4.96
12.41
4
17.7
10.9
5.03
19.9
22.2
1 .1
g
h
95
1
9
.5 2
1.81.5
6229
80
9.46
79.91.40.9
86 4
6.51
5.03
3.55
.9
19
22.2
15.417.7
8
4
15.
2 .9
.
3
1 10
8.7
6.4
8
3
52
c
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
b
8.727
5.25
6.9892
3.5
1.23
788
7.9
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
90
100
110 60
8,57,49
8,57,49
8,49,41
9,49,40
7,57,50
8,57,49
8,57,49
8,49,41
8,49,41
7,57,50
8,49,41
8,49,41
8,49,41
8,49,41
8,57,50
9,49,40
6,57,51
6,57,51
6,57,51
7,57,50
8,49,41
7,57,50
6,57,51
9,57,48
9,57,48
8,57,49
8,57,49
9,57,48
9,57,48
9,49,40
8,57,49
8,57,49
8,57,49
9,57,48
9,57,48
70
80
90
100
110
Figure 5. Latent-heat uxes (W m¡2 / over the Indian Ocean region. (a) Total latent-heat uxes on the time-scale
of the Madden–Julian Oscillation (MJO) across the constant- ux layer. (b) Total uxes of latent heat across the
constant- ux layer on the time-scale of the MJO arising from interaction of the MJO with the synoptic time-scale
of 2 to 7 days. (c) Fluxes of latent heat contributed by salient (strongest contributing) triad interactions in the
surface layer. (d) Salient triad interaction frequencies contributing to latent-heat uxes on the time-scale of the
MJO across the constant- ux layer. (e) Total latent-heat uxes on the time-scale of the MJO in the planetary
boundary layer (PBL) at 850 hPa. (f) Total latent-heat uxes in the PBL on the time-scale of the MJO arising from
interaction of the MJO time-scale with the synoptic time-scale of 2 to 7 days. (g) Latent-heat uxes contributed by
the salient triad interactions in the PBL. (h) Salient triad interaction frequencies contributing to latent-heat uxes
on the time-scale of the MJO in the PBL at 850 hPa.
2578
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
4.60
6
3.1
1.72
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
3
0.5
6
0.3
0.18
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
.2.9
14
12
.7
11
1 0.95.2
8.0
a
b
5
2 .2
1.24
4.60
1.72
16
3.4.
60
4 8
6.0
7.
8.942
3 .2 7
1.24
25
2.3.
27
4.2
19
32
5.3
6.
4.29
6.04
c
d
0.70
0.3
6
0.53
0.18
6
0.30.70
3
0.5
0.87
1
1.2.0
25
1.39
1.57
6,57,51
9,49,40
9,57,48
9,57,48
7,57,50
6,57,51
6,57,51
9,57,48
8,57,49
8,57,49
9,57,48
9,49,40
9,57,48
8,57,49
8,49,41
8,49,41
6,49,43
9,57,48
8,49,41
8,49,41
7,57,50
7,57,50
7,57,50
7,57,50
7,57,50
7,57,50
7,57,50
7,57,50
0.70
e
9.2
5.5
6.7 8.0
6.7
8.0
9.2
10.5
11.7
12.9
15.14.2
4
6.7
1 1.7
5
11.2N
0.7
8.4N
5
0. 7
5.6N
2.8N
0.60
0
2.8S
5.6S
8.4S
11.2S
130
135
f
9.58
9
8.7
3 1
7.28.0
4
6.4
6
5.6
4 .8.089
4
31
4.3.
09
4.88
5.66
6.44
7.23
8.01
8.79
7.23
6.44
10.5
g
h
0.90
1.05
0 .45
0.0.7
605
0.90
11.0.159
1.34
1.49
1.64
140
Figure 6.
145
150
6,49,43
6,57,51
6,57,51
6,57,51
6,57,51
6,57,51
6,57,51
6,57,51
6,57,51
6,57,51
9,57,48
9,49,40
6,57,51
6,57,51
8,57,49
8,57,49
9,57,48
7,57,50
8,49,41
8,49,41
7,57,50
7,57,50
7,57,50
7,57,50
9,57,48
7,57,50
7,57,50
7,57,50
130
135
140
145
150
Same as Fig. 5 but for the West Paci c region.
(a) Latent-heat uxes
The results for latent-heat uxes in the frequency domain are illustrated in Figs. 5,
6 and 7. These cover the results from our computations for the Indian, West Paci c and
Central Paci c Oceans. Each illustration comprises eight panels (a) to (h) corresponding
to the ux computations listed at the beginning of this section.
Overall the results of these latent-heat uxes over the Indian Ocean show that the
total surface uxes (Fig. 5(a)) on the time-scale of the MJO are largest in the trade wind
belt of the southern hemisphere and the south-west monsoon ow over the Arabian
Sea. These maximum total uxes are of the order of 10 W m¡2 . The uxes over the
equatorial belt are smaller. In general, the uxes over land areas between 90± and 110± E
2579
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
a
2.13
3.29
0
5.4
7.5
9.6
31
11.2N
8.4N
5.6N
2.8N
0
2.8S
91
5.6S 0.23
8.4S
11.2S
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
b
15.
18.07
1396
.85
11
.74
9.6
3
7.5
1
5.40
3.29
9.6
3
4.71
6 .01
6.
0.4545
1
1.
2
.885
0.66099
20.0
6
00
6,57,51
6,57,51
6,57,51
6,57,51
6,57,51
9,57,48
9,57,48
9,57,48
9,49,40
7,49,42
6,57,51
6,57,51
6,57,51
6,49,43
6,57,51
6,57,51
6,57,51
6,57,51
6,57,51
6,57,51
6,57,51
6,57,51
6,57,51
6,57,51
8,49,41
7,49,42
6,57,51
6,57,51
7,57,50
7,57,50
7,57,50
9,57,48
9,57,48
6,57,51
7,57,50
7,49,42
7,57,50
7,57,50
7,57,50
7,57,50
7,57,50
7,57,50
7,38,31
7,57,50
7,57,50
7,57,50
7,57,50
7,57,50
9,57,48
f
218
0.0.5
17.0
9.09.7
99.7.0
15.5
14.0
6.1
12.5
10.9
14.0
15.5
1.
35
6 .1
h
8,57,49
9,43,34
6,57,51
6,57,51
6,57,51
6,57,51
8,57,49
7,57,50
7,57,50
6,57,51
6,57,51
6,57,51
6,57,51
9,57,48
9,49,40
6,49,43
6,57,51
6,57,51
6,57,51
6,57,51
9,57,48
6,49,43
6,49,43
6,49,43
8,57,49
8,57,49
8,57,49
0.73
9,57,48
7,57,50
7,57,50
8,57,49
9,57,48
9,57,48
8,57,49
7,57,50
7,57,50
7,57,50
7,49,40
7,49,42
7,49,42
7,49,42
7,57,50
7,57,50
7,57,50
7,49,40
7,49,42
7,49,42
7,49,42
14
1. 5
3
1.
8,57,49
1.56.35 14
1 1.
0.93
1.14
Figure 7.
8.3
7.5
6.8
7.5 6.8
8.3
9.0
9.7
10.4
g
190
.3 0
01 7
d
160
1.3
1.1
006
0.606.8
98
952
0.4
545
91
0.23
20.0
11.2N
8.4N
3
5.6N
0. 9
2.8N
0
2.8S 1.5
6
5.6S
8.4S
11.2S
180
185
3.42
c
e
11.17
9.88
8.5
90
7.3
6.0
1
4.71
3.42
2.13
195
200
180
185
190
195
200
Same as Fig. 5 but for the Central Paci c region.
are small. Around 30 to 50% of these total uxes (on the MJO time-scale) come from
the triad interaction of the MJO time-scale with two other frequencies on synoptic timescales, Fig. 5(b). The single salient time-scale that contributes the most to the latent-heat
uxes on the MJO time-scale is found to be in the synoptic time-scales of 2 to 7 days.
This seems to be true over the entire Indian Ocean domain. These uxes and the salient
triads are shown in Figs. 5(c) and (d).
The total uxes of latent heat within the PBL at 850 hPa on the time-scale of the
MJO are shown in Fig. 5(e). These total uxes are almost twice those at the surface. A
surprising aspect is the lack of continuity of the 850 hPa uxes with respect to those
at the surface. Over the active monsoon region east of 80± E the uxes at 850 hPa are
quite large, and the pattern of uxes bears little resemblance to those at the surface
2580
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
a
b
3.221
2.171
1.121
6.86
3.50
1
1.12
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
c
d
0.4
0.235685
5
8,18,10
9,13,4
8,13,5
9,10,1
9,10,1
9,10,1
9,10,1
8,11,3
8,11,3
8,13,5
6,8,2
6,9,3
6,7,1
7,9,2
7,9,2
9,14,5
6,9,3
6,7,1
9,10,1
9,10,1
9,11,2
9,10,1
6,9,3
9,14,5
6,9,3
9,11,2
9,11,2
7,9,2
9,11,2
7,10,3
6,8,2
8,8,2
8,11,3
6,11,5
6,9,3
e
f
14 .64
9.81
4.99
4.894
2.480
2.480
4.99
4.894
11.2N
8.4N 0.410.8270
47
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
60
70
Figure 8.
g
80
h
90
100
110 60
9,13,4
9,13,4
8,22,14
8,9,1
6,7,1
8,16,8
8,16,8
6,7,1
9,11,2
9,10,1
6,9,3
6,8,2
9,10,1
7,9,2
7,8,1
8,13,5
7,9,2
9,22,13
9,10,1
9,10,1
6,13,7
9,11,2
8,11,3
9,11,2
9,10,1
7,18,11
7,9,2
9,11,2
9,11,2
8,9,1
7,13,6
6,9,3
7,8,1
6,11,5
7,10,3
70
80
90
100
110
Same as Fig. 5 but for sensible-heat uxes (W m¡2 / over the Indian Ocean region.
level, especially over land areas. Steady input of moisture laterally into the monsoon
region from the Oceans can contribute to the excessive uxes at the 850 hPa level
over the surface level. Around 50 to 75% of these total uxes (on the time-scale of
the MJO) arise from the interaction of the MJO time-scale with high-frequency motions
(i.e. on the 2- to 7-day time-scale), Fig. 5(f). Nearly 10% of these total uxes can be
attributed to a contribution from the single local salient triad, Fig. 5(g). These salient
triads are displayed in Fig. 5(h). A typical salient triad of frequencies here is 9, 57,
48, roughly corresponding to 40, 6 and 7 days, respectively. It is interesting to note
again that the interactions between members of the MJO time-scale and the synoptic
time-scale (2 to 7 days) carry these triads that contribute the largest to the uxes on
the time-scale of the MJO. This is true over the entire Indian Ocean domain. The data
2581
a
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
b
0.5 97
0.966
0.597
1.106
15
1.6
0 .9 66 5
1.77
1.775
1.615
c
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
d
0.0595
0.0595
6,20,14
7,16,9
6,21,15
6,21,15
7,24,17
6,11,5
9,10,1
8,10,2
9,10,1
8,11,3
9,10,1
9,16,7
6,7,1
9,10,1
9,10,1
9,10,1
7,16,9
8,13,5
8,9,1
8,10,2
9,11,2
7,9,2
8,21,13
8,10,2
9,10,1
7,9,2
7,20,13
7,9,2
e
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
f
1.70
2.71
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
130
1.70
3.11
4.52
2.71
4.80
4.80
4 .5
2
g
h
0 .1 28
0.128
0.242.355
0
135
Figure 9.
140
145
150
9,10,1
6,8,2
6,8,2
9,13,4
6,7,1
9,10,1
9,10,1
9,10,1
9,10,1
9,12,3
7,10,3
9,18,9
9,10,1
9,10,1
9,10,1
9,10,1
7,10,3
6,7,1
6,7,1
6,7,1
7,15,8
7,15,8
7,9,2
7,9,2
7,9,2
7,9,2
7,9,2
7,9,2
130
135
140
145
150
Same as Fig. 5 but for sensible-heat uxes (W m¡2 / over the West Paci c region.
considered for the present study cover the period from March 1996 to February 1997,
and hence the underlying thermodynamical effects of El Niño may have come into play
in possibly shifting the latent-heat ux maxima from the West Paci c to the Indian
Ocean. This maximum of latent-heat ux reaches 26.5 W m¡2 at (11.2± N, 75.94± E).
These results convey some of the most important results of this study. A good proportion
of the total latent-heat ux (on the time-scale of the MJO) arises from the interaction
of the MJO time-scale with the disturbances at the synoptic time-scale (2 to 7 days).
A small signal in the SST on the time-scale of the MJO is successively ampli ed
via these triad interactions, rst over the constant- ux layer and next within the PBL.
2582
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
a
b
4.575
3.115
1.656
9.88
6.68
3.49
3.49
1.656
3.115
6.68
c
d
0.2305
0.1
190
0.2305
0.1190
e
5.82
10.67
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S 0.241
8.4S
11.2S
180
185
Figure 10.
6,11,5
6,11,5
6,11,5
6,11,5
9,10,1
6,13,7
6,21,15
6,16,10
6,9,3
6,9,3
7,13,6
8,10,2
8,10,2
6,12,6
9,10,11
6,7,1
6,7,1
6,7,1
6,8,2
6,22,16
6,22,16
9,10,1
9,10,1
6,7,1
6,7,1
6,13,7
6,29,23
7,13,6
7,11,4
9,10,1
7,8,1
9,10,1
9,10,1
9,10,1
7,9,2
7,8,1
7,8,1
9,11,2
6,7,1
7,14,7
7,15,8
7,15,8
7,14,7
9,24,15
8,15,7
8,15,7
7,15,8
8,15,7
9,11,2
f
1520
.51.36
10.67
5.82
9.85
6.72
3.59
3.59
6.72
15.51
g
h
0.45 6
0.241
190
195
200
6,10,4
6,10,4
6,10,4
6,10,4
6,10,4
6,7,1
6,9,3
6,9,3
6,8,2
6,19,13
6,19,13
6,10,4
6,7,1
6,9,3
6,9,3
6,9,3
6,19,13
9,10,1
9,10,1
6,29,23
6,9,3
6,7,1
9,15,6
9,10,1
9,10,1
9,10,1
7,10,3
7,10,3
9,10,1
8,9,1
9,10,1
9,10,1
9,10,1
9,10,1
7,29,22
7,14,7
7,9,2
7,9,2
7,29,22
7,29,22
7,29,22
7,9,2
7,14,7
7,14,7
7,9,2
7,9,2
7,9,2
7,9,2
9,11,2
180
185
190
195
200
Same as Fig. 5 but for sensible-heat uxes (W m¡2 / over the Central Paci c region.
Tropical disturbances arising from the tropical instabilities abound on the synoptic timescale, and the presence of a non-zero SST uctuation on the MJO time-scale facilitates
the rapid ampli cation of the MJO via these triad interactions.
The results of these same computations for the Central and the West Paci c Ocean
are presented in Figs. 6 and 7. These domains are both essentially oceanic except over
Australia. Overall we nd quite similar results over the three ocean basins. The southern
hemisphere trades carry the largest moisture and latent-heat uxes on the MJO timescale over the western Paci c Ocean. For moisture uxes on the MJO time-scale the
largest contribution comes from triad interaction with the synoptic time-scale over all
tropical ocean basins. The salient triads such as 7, 57, 50 (which translate roughly to
2583
a
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
0.027
b
268
-0.0
-0.0060
5
-0.0060
0.0
27 5
c
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
8.9
-3
8.9510
d
-3
510
9,10,1
8,9,1
8,9,1
8,9,1
8,9,1
8,10,2
8,10,2
8,13,5
8,9,1
8,9,1
8,10,2
6,8,2
8,10,2
8,13,5
8,9,1
9,10,1
9,10,1
9,11,2
8,9,1
8,9,1
7,9,2
9,25,16
8,9,1
8,9,1
8,9,1
8,9,1
8,9,1
8,10,2
8,9,1
6,7,1
9,12,13
6,9,3
7,10,13
9,12,13
9,12,13
e
0.00
5
6
0.179
65
06
0.30
f
0.0 373
0.0
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
0.0065
0.052
11.2N
8.4N
5.6N
2.8N
0
2.8S
5.6S
8.4S
11.2S
60
g
h
0.0621
0.0 414
0.0207
70
Figure 11.
80
90
100
110 60
8,10,2
8,9,1
8,9,1
8,9,1
9,10,1
8,9,1
8,9,1
8,9,1
8,9,1
8,9,1
8,9,1
8,9,1
8,9,1
8,13,5
9,11,2
7,9,2
8,9,1
8,9,1
6,16,10
6,9,3
7,9,2
7,13,6
6,13,7
6,16,10
9,16,7
7,9,2
7,13,6
7,13,6
7,13,6
9,16,7
9,12,3
7,13,6
7,13,6
7,13,6
7,13,6
70
80
90
100
110
Same as Fig. 5 but for momentum uxes (N m¡2 / over the Indian Ocean region.
52, 6 and 5 days) contribute about 25% of the total uxes of moisture on the MJO timescale over a speci c region. We also note that, overall, the near equatorial belt is not
a large contributor to uxes on the MJO time-scale. These salient triads are also quite
active over the date line, where they again seem to contribute nearly 15% of the total
uxes (on the MJO time-scale) in the PBL (at 850 hPa level). The signi cant nonlinear
inter-time-scale triad interactions in latent-heat ux involve a pair of frequencies around
40 and 57, corresponding to high-frequency oscillations on the MJO time-scale at
the two layers and over the three regions considered, h6; 57; 51i, h9; 49; 40i and
h9; 49; 40i exhibit the most prominent-scale interactions having values of 1.25, 0.82
and 1.78 W m¡2 over the Central Paci c, West Paci c and Indian Oceans, respectively.
This ampli cation of the MJO over the western Paci c is consistent with the robust
2584
MJO seen over this region on the familiar x ¡ t Hovmüller diagrams of the zonal winds
(Fig. 4).
(b) Sensible-heat uxes
Figures 8, 9 and 10 illustrate the results for sensible-heat ux over the Indian,
Central and West Paci c Oceans, respectively, each comprising eight panels. Panel (a)
shows the total sensible-heat ux in the constant- ux layer on the time-scale of 20 to
60 days. The largest values of these uxes are around 3 to 10 W m¡2 . These total
surface uxes are largest in the trade wind belt of the southern hemisphere, the southwest monsoon ows of the Arabian Sea and the monsoon trough over India. The near
equatorial belt has a minimum of sensible-heat ux on the MJO time-scale. Panels (b)
show the contributions to the uxes on the MJO time-scale arising from all possible
interactions with high-frequency motion (i.e. 1- to 10-day time-scale). Roughly 30 to
50% of the total sensible-heat ux on the MJO time-scale appears to come from the
interaction with these high-frequency motions. The pattern of the total uxes and the
MJO time-scale uxes resulting from high-frequency interaction are similar, and the
maxima of the uxes generally occur over regions of strong surface winds. The ux
contribution by the single largest triad and the speci c salient triads are illustrated in
panels (c) and (d). A typical triad is 8, 18 and 10, here the three frequencies 8, 18 and
10 interact, where 8 D 18 ¡ 10 satis es the selection rule. The time-scales of these are
roughly 45, 20 and 36 days. This is clearly not one of the triads where the MJO timescale interacts with the synoptic time-scale. If we look across the Indian Ocean, we nd
that the salient triads contributing to an enhancement of the uxes on the MJO timescale are not in the high frequencies. This is not surprising since the thermal structure
on the scale of tropical disturbances is quite at and the Bowen ratio quite small over
the tropics. In this sense the decomposition of uxes on the time-scale of the MJO are
quite different for latent- and sensible-heat uxes. Panels (e) illustrate the total ux of
sensible heat within the PBL at the 850 hPa level. The PBL uxes are about 25 to 50%
larger than the surface uxes, due to lateral convergence of uxes on the MJO timescale within the PBL. The contribution to the 850 hPa total uxes that arises from triad
interaction with the synoptic-scale are shown in panels (f). Again, we note that roughly
50% of the total uxes on the time-scale of the MJO are contributed by the interaction
with the high-frequency motions on the time-scale of 2 to 7 days, even though these
are not the salient triads. Panels (g) and (h) show the sensible-heat uxes, within the
PBL at 850 hPa, arising from single salient triads and the speci c triads respectively.
Here again we note that the strongest sensible-heat uxes are not contributed from
the interaction of the MJO time-scale with the synoptic time-scale. Frequencies much
lower than the synoptic time-scales provide the salient triads for interaction with the
MJO time-scales. In general the sensible-heat uxes in the frequency domain call for
interactions of the MJO time-scale with another member of the MJO time-scale and a
much lower frequency. These uxes are small, and do not convey anything important
in respect of the growth of the MJO uxes across the PBL. We have also examined the
results of the sensible-heat ux computations over the western and eastern Paci c. In all
cases the maximum uxes occur over the region of strongest surface and PBL winds.
The results for the three ocean basins were quite similar in terms of the composition of
salient triads.
The uxes of sensible heat are found to be at a maximum over the Central Paci c,
where they are of the order of 16 and 25 W m¡2 at (11.2± N, 160.31± W) in the surface
layer, and between the surface layer and the free atmosphere, respectively. Similarly, the
values of the uxes of sensible heat over West Paci c are of the order of 2 and 6 W m¡2
2585
at (11.2± N, 151.88± E) in the above mentioned two layers. The maximum values of
the sensible-heat ux in the respective layers over Indian Ocean are of the order of
15 and 23 W m¡2 . Thus we notice systematic strengthening of uxes of sensible heat
on the MJO time-scale from the surface layer to the free atmosphere through the PBL.
At almost all grid points this trend has been maintained.
(c) Momentum uxes
The results for momentum uxes on the time-scale of the MJO for the Indian Ocean
are presented in Figs. 11(a) to (h). The largest upward ux of momentum is over the
southern trade winds and in the south-westerly monsoon ow. Between the surface level
and the PBL these uxes are approximately doubled. Overall, the patterns of uxes at the
surface and the PBL (850 hPa level) are quite similar. The most interesting aspects of the
momentum uxes are the distribution and composition of the salient triads at the surface
and the PBL. A number of these triads show interactions between two members of the
MJO family and a lower frequency (i.e. lower than the MJO time-scale). For example,
8, 9, 1 denotes an interaction among 44- and 45-day oscillations and the annual cycle.
Overall, this entire pattern of salient triads is very different from that for the moisture
uxes. The contribution of the high frequencies interacting with the MJO (Fig. 11(b))
is larger. What this means is that higher frequencies, i.e. the synoptic time-scales, do
contribute signi cantly to the overall total even though they are not a part of the salient
triads. This is also true for the sensible-heat uxes. In that sense the moisture uxes
contributed by the high frequencies at the surface and in the PBL are quite unique, in
that they readily interact with the MJO to amplify the latter. The results for the Central
and West Paci c Oceans are quite similar to those over the Indian Ocean and are not
displayed here.
(d ) Total MJO time-scale uxes versus total uxes including all time-scales
The uxes discussed so far in this paper are those on the time-scale of the MJO. It is
important to know how those contributions on the time-scale of the MJO compare with
the total annual-mean surface uxes (from March 1996 to February 1997 inclusive),
where all time-scales are included. The total surface ux of latent heat for the three
oceanic basins, Indian, West Paci c and Central Paci c are shown in Figs. 12(a), (b)
and (c), respectively. Overall, the largest oceanic uxes are of the order of 140 W m¡2 ,
and occur near the borders of these domains where the surface winds are strongest.
The total latent-heat uxes on the time-scale of the MJO are only around 10 W m¡2 .
The total uxes shown here include contributions for all time-scales; a substantial
portion are contributed on the synoptic time-scales. These total uxes invariably increase
between the surface and 850 hPa. These are illustrated in Figs. 12(d), (e) and (f) for the
three ocean basins. The increase of uxes is largest over the land areas of the monsoon
region over the Indian Ocean, where there appears to be a doubling of uxes between
the surface and 850 hPa. Figures 12(g), (h) and (i) show the increase of uxes between
the surface and 850 hPa for the contributions of the three oceanic areas on the MJO
time-scale. It is clear that a large proportion (approximately 60%) of the increase of
total uxes between the surface level and the PBL arises from the contribution on the
MJO time-scale. This increase in the MJO time-scale is largely attributed to the triad
interactions with the synoptic time-scales. It is this increase of the original signal on the
MJO time-scale, starting from the SSTs over the ocean, which appears quite striking in
these datasets.
2586
11.2N
8.4N
INDIAN OCEAN
140
5.6N
0
5.6S
CENTRAL PACIFIC
14 0
120
100
80
14 0
120
100
2.8N
2.8S
WESTERN PACIFIC
1 20
80
100
120
120
140
8.4S
11.2S
a
5.6N
2.8N
0
2.8S
c
10
9 10.04
7.5
21
8.82
11.2N
8.4N
b
1.27
.0
4
10 .04
10.04
27 .5
9
10.0
10.04
4
1.27
5.6S
8.4S
11.2S
d
e
11.2N
8.4N
f
26
9.
26
9.
2
14.2
4.3 1
5.6N
0
2.8S
1
4.31
9.26
2.8N
4 .3
1
9.2 6
9.2 6
7
9 .1
4. 3
1
5.6S
8.4S
11.2S
60
4 .3
70
80
90
100
110
130
g
135
140
145
1
150
h
180
185
190
195
200
i
Figure 12. (a) Total ux of latent heat (W m¡2 ) over the Indian Ocean; (b) and (c) as (a) but for the West
Paci c and Central Paci c Ocean basins, respectively. (d), (e), (f) As (a), (b) and (c) but differences in total uxes
(W m¡2 ) between the 850 hPa and surface levels. (g), (h), (i) As (a), (b) and (c) but ux contributions on the
time-scale of the Madden–Julian Oscillation.
5.
C ONCLUDING REMARKS
Tropical instabilities have been enumerated by many scientists: Yanai and Nitta
(1968) on barotropic shear ow instabilities; Lindzen (1974) on Wave CISK¤ ; Rennick
(1977) on combined barotropic–baroclinic instabilities plus CISK; Moorthi and Arakawa
(1985) on vertical shear and convection; and several others. The focal point of these
studies is the mechanisms of growth of tropical wave disturbances whose time-scale is
of the order of 2 to 7 days (Reed et al. 1977). These disturbances can be found over
¤
Conditional Instability of the Second Kind.
2587
most of the tropical oceanic basins. They are, so to speak, waiting to interact with the
MJO time-scale of 20 to 60 days. A number of these tropical wave disturbances satisfy
the trigonometric selection rule, Fig. 2(a):
MJO frequency n D r § s
(the tropical wave frequencies).
This relationship appears to be more readily met by the pre-existing tropical wave
disturbance and a weak MJO signal over the tropical oceanic SSTs, or in the surfacelayer ows. From Eq. (9) we can see that the intraseasonal frequency, n; is associated
with exchange coef cient CLH , which is a function of stability and therefore a function
of Richardson number (RiB /. RiB depends on surface potential temperature and wind
stress for computation of latent-heat uxes in the surface layer. SST affects these two
parameters very close to the sea surface, therefore, the intraseasonal variation of CLH is
involved with SST. The humidity uxes are affected much more by the SST oscillations
(Krishnamurti et al. 1988). The term .Qs ¡ Qa / in Eq. (6) does have variations at
intraseasonal time-scales, but synoptic time-scales involved with it are found to be
important in nonlinear interactions between time-scales as far as their contributions to
the MJO time-scale are concerned. The signal in intraseasonal SST in the surface layer
associated with stability-dependent CLH gets ampli ed in the PBL, though CLH in the
PBL is independent of SST.
It is this feature that makes it possible for the surface uxes of moisture to amplify
over the constant- ux layer, and it is this same feature that allows for the further
ampli cation of the MJO over the PBL. Our parametrization of cumulus convection
carries, at most, quadratic nonlinearities, which can convey in-scale (MJO to MJO)
information from the cloud base to the cloud top. Thus, an MJO signal of the SST can
be passed on through the deep convective cloud base to the tropical upper troposphere.
It is our premise that a similar extension of this analysis over the deep convective layers
would reveal the passage of this signal from the cloud base to most of the tropics via
divergent circulations. Over tropical regions of convection the mid-troposphere vertical
motion shows a signal on the time-scale of the MJO; the same feature was also noted
over subtropical regions of descent. The present work is, of course, based entirely on
model output from a coupled model that happened to simulate a somewhat realistic
MJO signal. The model output was better suited for demonstrating the boundary-layer
ampli cation of the MJO signal. This is an internally consistent dataset where the
large-scale variables, especially in the PBL, were part of known model equations.
Those equations were used, with the same model algorithms, to address the triad
interactions in the frequency domain. What is presented here does not preclude other
existing theories on the MJO, most of which are based on climate diagnostics and reanalysis datasets.
According to Fjortoft (1960), cascading of energy can take place either way,
i.e. from larger-scales to smaller-scales (temporal and spatial) or vice versa. In the
present study interactions of oscillations slower than the MJO, with the MJO and higher
frequencies, are not considered. Enhanced uxes of latent heat do feed convection on
the MJO-scale. Propagation of clouds on this scale was noted by Yasunari (1980).
Steady import of moisture laterally into the monsoon region from the oceans as well as
from the southern trades carrying the largest moisture into Paci c are important features
in the surface layer. Tropical disturbances arising from the tropical instabilities abound
on the synoptic time-scale, and the presence of a small SST signal on the MJO timescale facilitates the rapid ampli cation of the MJO through nonlinear triad interactions.
We have also looked at the issues of sensible-heat and momentum transports in
the frequency domain. While these results are most revealing and complementary to
2588
the above ndings, we feel that the ampli cation of moisture uxes is the central issue
for this MJO theory. Further work is clearly needed to understand the scale interaction
of sensible-heat and momentum uxes. The role of the ocean has not been adequately
addressed here. A starting point in our analysis was the given fact that a small signal on
the time-scale of the MJO exists in the SST over the upper ocean. If a MJO frequency
in SST were to be absent (identically zero) then the triad interactions with the tropical
waves would cease to exist. What about the need for an initial MJO signal in the oceanic
SST that undergoes ampli cation over the atmosphere? How does that oceanic signal
come about? The answer to this question requires further work. Oceanic behaviour is
inherently of lower frequency than the atmosphere. If we accept that, and allow for
some non-vanishing signal (however small) in the SST elds on the time-scale of the
MJO to be present, given a plethora of tropical disturbances on the synoptic timescale of 2 to 7 days in the lower troposphere we can provide a simple explanation.
These disturbances arise from prevalent tropical instabilities. These time-scales provide
rich possibilities for triad interactions with a MJO time-scale because they are able to
satisfy the selection rules for triad interactions. This permits the MJO time-scale to
amplify in the PBL. The resulting winds acquire a stronger signal on the MJO timescale; these in turn convey the MJO time-scale to the ocean via surface stresses to
amplify the SST signal. This positive feedback can establish an equilibrium state in the
ocean and in the atmosphere, with both exhibiting the MJO signal. Thus we perceive the
presence of the synoptic-scale disturbances in the tropics, on time-scales of 2 to 7 days,
as an essential element for the excitation of the MJO.
We have not considered the changes in the atmospheric basic states as such, as
described in Molinari et al. (1997), Hartmann and Maloney (2001) and Maloney and
Hartmann (2001). In the frequency domain even the so-called basic-state’s variation
can be part of the MJO. That could have been revealed from an analysis in the ‘wave
number–frequency domain’ where wave number zero (evidently the basic state) would
have exhibited variations. That is beyond the scope of this study as we have concentrated
here only on the ‘frequency–frequency’ interactions.
It may be premature to consider the scenario portrayed here as a theory for the
MJO. Further work is clearly needed to isolate the effects of the respective capabilities
of coupled versus uncoupled models in the simulation of the MJO. Given a picture, such
as Fig. 4, where for many months an intraseasonal wave traverses the globe, it seems
that the oceanic contributions may have to be important. An AGCM, with prescribed
SST, can carry a MJO signal primarily by the aforementioned scenario for a long period
of time, simply from the inertia of the initial state of the surface layer and PBL where
the proposed mechanism can operate for a few months. Evidently, lacking an oceanic
support on the MJO time-scale, this signal could weaken in the AGCMs. Atmospheric
internal dynamics may generate the MJO, but the role of ocean may be important for
its enhancement. Computations of uxes over land in the frequency domain may give
more clues about the physical and dynamical mechanisms of the origin of the MJO.
This problem will be addressed in our future work.
Computation of energy uxes in the frequency domain is a powerful tool for
isolating the role of an important time-scale such as the MJO and the ENSO. In a
number of recent studies, Sheng and Hayashi (1990a,b) explored global energetics in
the frequency domain. They focused on the maintenance of the eddy kinetic energy
on the time-scale of the MJO. In these studies several frequencies covering ENSO,
annual, MJO, and storm-scales (1–10 days) were examined using 6 years of daily global
datasets. A major nding of this study was that the high-frequency (2- to 7-day) timescales were a major source of energy for the MJO. That transfer mainly occurs from
2589
three-component triad interactions, i.e. kinetic to kinetic energy exchanges invoking
nonlinear dynamics. A smaller gain of energy for the MJO occurs from the in-scale
vertical overturnings of potential to kinetic energy. The roles of the annual cycle and
larger time-scales in the maintenance of the MJO were smaller. The main message
from these studies is that tropical high-frequency disturbances play a major role in the
maintenance of the MJO. The present study of the PBL uxes in the frequency domain
complements these results.
A CKNOWLEDGEMENTS
The research work reported here was supported by NOAA grants number
NA96GPO400, NA06GPO512, NA16GP1365, and FSU Research Foundation Center
of Excellence Award. The visit of Dr D. R. Chakraborty to FSU was sponsored by the
Indian Institute of Tropical Meteorology, Pune, India. We acknowledge the data from
the European Centre for Medium-Range Weather Forecasts especially through the help
of Dr. Tony Hollingsworth.
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