QUARTERLY JOURNAL
OF THE
ROYAL
METEOROLOGICAL
Vol. 130
APRIL 2004 Part B
Q. J. R. Meteorol. Soc. (2004), 130, pp. 1171–1194
SOCIETY
No. 599
doi: 10.1256/qj.03.133
Structure, genesis and scale selection of the tropical quasi-biweekly mode
By PIYALI CHATTERJEE1 and B. N. GOSWAMI2∗
of Physics, Indian Institute of Science, Bangalore, India
2 Centre for Atmospheric and Oceanic Sciences, Indian Institute of Science, Bangalore, India
1 Department
(Received 29 July 2003; revised 3 December 2003)
S UMMARY
The quasi-biweekly mode (QBM) and the 30–60 day mode are two major intraseasonal oscillations (ISOs)
in the tropics. The QBM is known to have a major influence in determining the active and break conditions of
the Indian monsoon during the northern summer. A westward-propagating equatorial Rossby wave with quasibiweekly period influences the Australian monsoon during the northern winter. Universality between the summer
and winter QBM is established through analysis of daily circulation and convection data for 10 years. It is shown
that the mean spatial structure of the QBM in circulation and convection resembles that of a gravest meridional
mode equatorial Rossby wave with wavelength of about 6000 km and westward phase speed of approximately
4.5 m s−1 . However, the maximum zonal wind occurs at around 5◦ N (5◦ S) during the northern summer (winter).
The wave structure appears to be translated northward (southward) by about 5◦ during the northern summer
(winter). The relationship between outgoing long-wave radiation and circulation data indicates that the mode is
driven unstable by coupling with moist convection. Similarity in temporal and spatial characteristics of the mode
during the two seasons leads us to propose that the same mechanism governs the genesis and scale selection of
the mode in both the seasons.
An acceptable mechanism for genesis and scale selection of the QBM has been lacking. In the present
study, a mechanism for genesis and scale selection of the observed QBM is proposed. A simple 2 12 -layer model
that includes a steady Ekman boundary layer (BL) formulation incorporating effect of entrainment mixing is
constructed for the convectively coupled equatorial waves. Without influence of the background mean flow,
moist feedback in the presence of frictional BL convergence drives the gravest meridional mode equatorial
Rossby wave unstable with observed wavelength and period but with zonal winds symmetric about the equator.
Potential temperature perturbation associated with the Rossby wave is in phase with relative vorticity perturbation
at low level. The BL drives moisture convergence in phase with the relative vorticity at the top of the BL.
Release of latent heat associated with the BL convergence enhances the potential temperature leading to a positive
feedback. The mean flow over the Indian Ocean and western Pacific at low levels is such that the zero ambient
absolute vorticity or the ‘dynamic equator’ shifts to around 5◦ N (5◦ S) during summer (winter) and results in a
shift of the unstable Rossby waves towards the north (south) by about 5◦ . The resulting structure of the unstable
Rossby mode resembles the observed structure of the biweekly mode. It is shown that neither evaporation–wind
feedback nor vertical shear of the mean flow is crucial for the existence of the mode. However these processes
marginally modify the growth rate and make the structure of the unstable wave more realistic.
K EYWORDS: Boundary layer entrainment Convective coupling Dynamic equator Equatorial Rossby
waves
1.
I NTRODUCTION
The quasi-biweekly mode (also known as 10–20 day mode) is almost ubiquitous
in the tropical western Pacific and Indian Ocean (IO). It can be seen in spectral
analysis of precipitation, convection and many circulation parameters (Murakami and
Frydrych 1974; Murakami 1975; Krishnamurti and Bhalme 1976; Krishnamurti and
∗
Corresponding author: Centre for Atmospheric and Oceanic Sciences, Indian Institute of Science, Bangalore560012, India. e-mail: [email protected]
c Royal Meteorological Society, 2004.
1171
1172
P. CHATTERJEE and B. N. GOSWAMI
Ardunuy 1980; Chen and Chen 1993; Numaguti 1995; Kiladis and Wheeler 1995).
Chen and Chen (1993) carried out a detailed study of the structure and propagation
characteristics of the quasi-biweekly mode (QBM) during the summer monsoon of 1979
using daily FGGE IIIb and outgoing long-wave radiation data. They showed that the
horizontal structure of the QBM consisted of two vortices, one centred at 18◦ N and
the other centred close to but south of the equator. They also showed that the QBM
has a zonal wavelength of approximately 6000 km and westward propagation speed
of about 4–5 m s−1 . The westward-propagating QBM together with the 30–60 day
mode dominate the intraseasonal variability in the tropics. The 30–60 day mode is
manifest in the form of the northward-propagating monsoon intraseasonal oscillation
(ISO) during the boreal summer over the Asian monsoon region (Yasunari 1979, 1981;
Sikka and Gadgil 1980; Krishnamurti and Subramanyam 1982) and that of the eastwardpropagating Madden–Julian Oscillation (MJO) in winter (Madden and Julian 1971,
1994; Hendon and Salby 1994). The QBM is an important component of the monsoon
ISO as its amplitude is comparable to that of the northward-propagating 30–60 day mode
(e.g. Goswami et al. 1998; Goswami and Ajaya Mohan 2001), and its phase relative to
that of the 30–60 day mode determines the rainy (active) and dry (break) spells of the
Indian summer monsoon (Krishnamurti and Ardunuy 1980; Yasunari 1981; Goswami
et al. 2003). The phase relationship between the westward-propagating QBM and the
northward-propagating 30–60 day mode also determine the active and break phases of
the South China Sea monsoon (Chen et al. 2000). It has been recently shown (Sengupta
et al. 2001) that observed biweekly fluctuations in the upper ocean zonal transport
(Schott et al. 1994) in the equatorial IO is driven by the biweekly fluctuations of the
surface stress. The QBM is strong over the western Pacific warm pool and IO region
during the northern summer and is prominent over the central and western Pacific during
the northern winter. Numaguti (1995) analysed Japan Meteorological Agency global
objective analyses of winds at low and upper levels during November and December
1992 and identified a 15–20 day oscillation whose horizontal structure with zonal wave
number 6 and westward phase speed is consistent with an equatorial Rossby wave of
n = 1. The vertical structure of the wave was found to be baroclinic near the equator
but quasi-barotropic in the subtropics. During the intensive observing period of the
Tropical Ocean Global Atmosphere/Coupled Ocean–Atmosphere Response Experiment
from December 1992 to February 1993, four Japanese moored acoustic Doppler current
profilers showed a 14-day peak in power spectra of zonal and meridional currents
at 2◦ S and 2◦ N and meridional current at the equator (Zhu et al. 1998). The quasibiweekly current oscillations are found to have westward phase propagation, eastward
group propagation, and horizontal structure and dispersion relation consistent with an
equatorial mixed Rossby–gravity (MRG) wave. It appears that the biweekly fluctuations
in oceanic currents is forced by the QBM of surface winds similar to the ones found by
Numaguti (1995). Using US National Meteorological Center operational analyses for
December to February for 8 years, Kiladis and Wheeler (1995) identified the existence
of westward-propagating equatorial Rossby waves with zonal wave number 6 and
dominant quasi-biweekly periodicity in the Pacific sector. The mode is found to have
westward phase speed (less than 5 m s−1 west of the dateline) and an eastward group
speed. They also noted that, unlike the MJO or the MRG wave, the vertical structure of
the QBM has a significant barotropic component.
Compared to the large volume of literature addressing the structure, genesis and
scale selection of the 30–60 day mode during the last two decades, very few studies
have addressed these issues for the QBM. Some dynamical framework for genesis and
temporal scale selection for the northward-propagating 30–60 day mode of monsoon
THE QUASI-BIWEEKLY MODE
1173
ISO is available (Goswami and Shukla 1984; Keshavamurthy et al. 1986; Nanjundiah
et al. 1992). Using a zonally symmetric general-circulation model, Goswami and Shukla
(1984) showed that the 30–60 day oscillation arises from a feedback between organized
convection and local meridional Hadley circulation. Similarly, a dynamical framework
is available for understanding the genesis and propagation characteristics of the MJO
(Lau and Peng 1987; Emanuel 1987; Neelin et al. 1987; Wang 1988; Wang and Rui
1990, hereafter WR). However, no acceptable dynamical framework for genesis and
scale selection of the westward-propagating QBM is available so far. A qualitative
mechanism in terms of cloud–radiation–atmospheric stability feedback was proposed
by Krishnamurti and Bhalme (1976) for the QBM of the Indian summer monsoon. They
proposed that radiation increases surface temperature and makes the atmosphere unstable, leading to generation of moist convection. Clouds then cut down the solar radiation
and the resulting surface cooling quenches the instability and convection. Once the sky
clears, the radiation initiates the process all over again. However, selection of quasibiweekly period was not established quantitatively. Webster (1983) obtained a QBM in
a zonally symmetric model of the atmosphere intended to study the fluctuations of the
tropical convergence zone. As the zonal scale of the QBM is finite, zonally symmetric
dynamics are not applicable for explaining the genesis or scale selection of this mode.
Goswami and Mathew (1994) used a shallow-water model with moist feedback and a
moisture relaxation time-scale and found that evaporation–wind feedback (EWF) in the
presence of mean westerlies can drive a MRG mode unstable with quasi-biweekly period
and zonal wavelength between 9000 and 12 000 km. This mechanism also may not be
responsible for the observed QBM as the observed structure is similar to an equatorial
Rossby wave rather than a MRG wave and the preferred zonal wavelength is between
6000 and 7000 km.
Dry shallow-water theory with constant mean flow is inadequate to explain the
phase speed, group speed and equivalent depth of the observed tropospheric Rossby
waves (Kiladis and Wheeler 1995). Inclusion of friction, convective coupling and shear
of the mean flow may be important to explain these observed features of the QBM.
Zhang and Webster (1989) showed that the change in structure due to the non-Doppler
effect of the basic state is considerable for equatorial Rossby waves, moderate for MRG
waves and negligible for the other equatorial waves. Wang and Rui (1990) used a twolayer atmosphere with a frictional boundary layer (BL) and concluded that the gravest
meridional mode equatorial Rossby waves remain damped even in the presence of
convective coupling. Could shear of the tropical mean flow drive some equatorial waves
unstable and lead to the observed QBM? Xie and Wang (1996) studied the stability
of low-frequency equatorial waves in a vertically sheared zonal flow in the presence
of convective feedback. They found that the gravest meridional mode Rossby wave
could be driven unstable by vertical shear and, for some reasonable values of heating
parameter and vertical shear, the most unstable mode has a wavelength of 3500 km and
phase speed of 2.8 m s−1 . This may explain some observations of westward-propagating
synoptic waves (Lau and Lau 1990), but does not explain the genesis and scale selection
of the QBM. Using a two-level global primitive-equation instability model, Frederiksen
(2002) showed that the observed three-dimensional basic state could destabilize some
equatorial Rossby waves with the fastest growing mode having a period of 17.9 days and
e-folding time of 12.8 days in the presence of EWFs. Although the study demonstrates
that instability of the observed mean flow could destabilize equatorial Rossby waves, the
growth rate of the fastest growing mode is too slow to be relevant for the observed QBM.
Thus, the origin and scale selection of the QBM has remained an unsolved problem
so far.
1174
P. CHATTERJEE and B. N. GOSWAMI
The QBM associated with the Indian monsoon during the northern summer and the
quasi-biweekly equatorial Rossby mode in the Pacific during the northern winter have
so far been studied in isolation without recognition of any connection between the two.
Using convection and circulation data for 10 years, we highlight the similarity in spatial
and temporal structure of the QBM during the two seasons, and propose that they are
manifestations of a universal tropical QBM modified by different mean flow during the
two seasons. A common mechanism for genesis and scale selection of the QBM is then
provided. It is shown that the QBM is a gravest meridional mode equatorial Rossby
wave driven unstable through a wave–BL–CISK mechanism in the presence of a BL
including the effect of entrainment. The role of the background mean flow is primarily
in translating the horizontal structure of the Rossby wave to the north (south) in summer
(winter) through the movement of the zero ambient absolute vorticity line (the dynamic
equator).
2.
DATA AND METHOD
In order to re-examine the horizontal and vertical structure of the QBM during the
northern summer and winter, we use daily zonal and meridional winds at a number
of vertical levels from the US National Centers for Environmental Protection/National
Center for Atmospheric Research (NCEP/NCAR) reanalysis (Kalnay et al. 1996; Kistler
et al. 2001) for the ten-year period from 1992 to 2001. To investigate the relationship
between circulation and convection on the biweekly time scale, we also use daily
interpolated outgoing long-wave radiation (OLR, Liebmann and Smith 1996) for the
same ten-year period. Daily circulation and convection data are supplemented with
daily satellite estimates of precipitation for six years (1997–2002) from the Global
Precipitation Climatology Project (GPCP, Huffman et al. 2001). Daily anomalies are
constructed as deviations of the daily values from the annual cycle for each year.
For circulation, OLR as well as GPCP rainfall data, the annual cycle is defined as the
sum of the annual mean and the first three harmonics of daily data for each year. Power
spectral density analysis is carried out on unfiltered daily anomalies using the Tukey lag
window (Chatfield 2003) to indicate the amplitude of the QBM. To isolate the QBM, the
anomalies are band-pass filtered using a Lanczos filter (Duchon 1979) to keep periods
between 10 and 20 days. The Lanczos filter has a sharp frequency response within this
band of periods.
3.
U NIVERSALITY IN SPATIAL STRUCTURE OF THE SUMMER AND WINTER QBM S
The existence of a spectral peak with approximate two-week period (the QBM) in
monsoon rainfall as well as in many related parameters was shown in the seminal work
of Krishnamurti and Bhalme (1976). Here, we show that the satellite-derived rainfall
estimates also show this peak. The QBM can be distinguished clearly from synoptic
disturbances (periods less than 10 days) and from lower-frequency oscillations with
period between 30 and 60 days in power spectra of daily GPCP rainfall anomalies over
the northern Bay of Bengal (BoB) (10–20◦N, 80–90◦E) from 1 May to 30 September
between 1997 and 2002 (Fig. 1(a)). The power spectra of daily zonal winds averaged
over the equatorial western Pacific (0–5◦ S, 158–162◦E) and zonal and meridional winds
over the central equatorial IO (2◦ N–2◦ S, 80–85◦ E) are shown in Figs. 1(b), (c) and
(d) respectively, based on daily anomalies from NCEP/NCAR reanalysis for a full
ten years (1992–2001). It may be noted that both zonal winds over the equatorial western
Pacific and IO and meridional winds over the equatorial IO have significant power in
1175
THE QUASI-BIWEEKLY MODE
3
a
b
f x P(f)
1
2
0.5
1
0
1
0
2
10
10
1
2
10
10
d
c
0.6
f x P(f)
1
0.4
0.5
0
0.2
1
2
10
10
Period (days)
0
1
2
10
10
Period (days)
Figure 1. Spectra of (a) precipitation over northern Bay of Bengal, (b) zonal winds over the western equatorial
Pacific, (c) zonal and (d) meridional winds over the equatorial central Indian Ocean. The dashed lines represent
the 95% confidence level based on a theoretical red-noise background. See text for further details.
the 10–20 day range. The equatorial meridional winds carry very little power at the
30–60 day range, while there is significant power in the equatorial zonal winds in this
period range, indicating a certain difference between the structures of the QBM and the
30–60 day mode.
To establish the robustness of the characteristic features of the QBM and the
similarity in structure and propagation characteristics of the mode during northern
summer and winter, the mean horizontal and vertical structure and westward phase
propagation are investigated using NCEP/NCAR reanalysed winds and OLR for
10 years (1992–2001). A phase composite technique is used to bring out the mean characteristics during summer (1 June–30 September) and winter (1 December–28 February)
seasons separately. The composites can be constructed with respect to a reference time
series. The amplitude time series of the dominant mode of variations of the filtered
anomalies may represent such a reference time series. Therefore, a combined empirical
orthogonal function (EOF) of 10–20 day filtered zonal and meridional winds at 850 hPa
and OLR anomalies for ten summer seasons (June 1–September 30) is carried out in the
region between 30◦ N–25◦ S and 40–120◦E. The first two EOFs (not shown) essentially
represent the horizontal structure and westward propagation of the QBM with their
principal components (PCs) strongly correlating at a lag of 4 days. An index of the
QBM is defined as {PC1(t) + PC2(t − 4)}/2 and shown in Fig. 2, normalized by its
own standard deviation. The index is used as the reference time series to define phases
of evolution of the QBM and for constructing composite structures corresponding to
different phases. In order to construct the composites based on relatively stronger quasibiweekly oscillations, all the peaks of the index exceeding +1 normalized unit (one
standard deviation) are identified and assigned a phase 8, with phase 1 and phase 15
corresponding to 7 days before and 7 days after the peak. Composites of 10–20 day
filtered wind anomalies at 850 hPa and OLR corresponding to all 15 phases are constructed, based on all the selected events (about 40 in number) over the 10-year period.
1176
P. CHATTERJEE and B. N. GOSWAMI
Figure 2. Quasi-biweekly mode index derived from the first two principal components of the combined empirical
orthogonal function of 10–20 day filtered zonal and meridional winds at 850 hPa and outgoing long-wave radiation
for 10 years, normalized by its own standard deviation.
Having identified the dates for the 15 phases of the mode from the index, composites of
10–20 day filtered winds at a number of vertical levels are also obtained to construct the
mean vertical structure of the mode.
Two contrasting phases separated by seven days (phase 8 and phase 1) are shown
in Figs. 3(a) and (b), while the relative vorticity averaged around the equator for all
phases as a function of longitude is shown in Fig. 3(c). The double vortex structure,
with centres around 18◦ N and 3◦ S, seen in our composite bears close resemblance to
that obtained by Chen and Chen (1993) who analysed the spatial structure of the QBM
in detail during 1979. It is interesting to note the complete reversal of the anomalous
wind and OLR patterns in 7–8 days. The size of the vortices indicates a half wavelength
of approximately 3000 km. The westward phase speed from Fig. 3(c) is approximately
4.5 m s−1 and is consistent with approximate speeds noted in earlier studies, such as
Krishnamurti and Ardunuy (1980, their Fig. 4(a)) and Chen and Chen (1993, their Fig. 3
(left panel)). Figure 3(c) also indicates an eastward group speed of about 1.7 m s−1 for
the mode. The double vortex horizontal structure of the QBM is reminiscent of that of
the n = 1 equatorial Rossby wave, with the exception that the largest zonal winds are
not centred on the equator but around 5◦ N. This may be interpreted as a translation of
the equatorial Rossby mode to the north by about 5◦ . It is interesting to note that this
location coincides roughly with the mean position of the zero absolute vorticity line at
low levels over the IO during the northern summer (see Fig. 2(d) of Tomas and Webster
1997). Significant wind and OLR anomalies associated with the composited structure
is indicative of considerable coherence in structure between events and establishes the
robustness of the QBM.
The horizontal structure of the mode during the boreal winter was also examined
in a similar manner by conducting an EOF analysis of 10–20 day filtered zonal and
meridional winds at 850 hPa and OLR for the winter season (1 November–31 March)
for 10 years over the western Pacific (20◦ N–30◦ S, 120◦ E–160◦W), and by constructing a
QBM index based on the first two PCs. This index was used as a reference time series for
identifying the 15 phases of the mode in a manner similar to that used in the summer case
and composite anomalies for all the phases were constructed. The composite structure
of wind and OLR anomalies for two phases (phase 8 and phase 1) separated by 7 days
are shown in Fig. 4. It is also characterized by a double vortex structure, with the
difference that the stronger vortex is now located in the southern hemisphere and the two
vortices are now centred around 5◦ S. The mode has a westward phase speed of about
THE QUASI-BIWEEKLY MODE
1177
Figure 3. Northern summer composites of filtered wind and outgoing long-wave radiation (OLR) anomalies:
(a) for phase 8 with positive (negative) OLR anomalies shaded (dash contoured with same interval as shading)
and (b) for phase 1, with shading/contouring convention reversed. Wind speeds are in m s−1 (scale shown at top),
OLR in W m−2 . (c) shows relative vorticity of composited winds averaged between 5◦ N and 5◦ S as a function of
the 15 phases, with vorticity in units of 10−6 s−1 .
5.0 m s−1 and an eastward group speed of about 2.5 m s−1 . The wavelength of this
wave is estimated to be between 6000 km and 7000 km. The horizontal structure and
propagation characteristics of the mode are consistent with those found by Numaguti
(1995) and Kiladis and Wheeler (1995) of equatorial Rossby waves with quasi-biweekly
period. Thus, the winter structure could also be interpreted as an equatorial Rossby wave
translated to the south by about 5◦ . It may be noted that the two vortices are tilted in the
horizontal plane. The southern vortex is tilted from south-east to north-west while the
northern one is tilted from north-east to south-west. Such a tilt is consistent with an
unstable westward-propagating Rossby mode (Xie and Wang 1996). We note (Figs. 3(a)
and (b)) that the southern vortex of the mode during the northern summer is also tilted
from south-east to north-west. However, the tilt of the northern vortex during summer is
non-uniform in the east and the west. This may be due to the effect of vertical shear of
mean winds in this region.
1178
P. CHATTERJEE and B. N. GOSWAMI
Figure 4.
As Fig. 3, but for the northern winter.
The composite vertical structure corresponding to phase 8 of the mode during the
northern summer is shown in Fig. 5. Vertical sections are shown along the lines indicated
in Fig. 3(a). Similar vertical sections for the mode during the northern winter were
also examined and found to be similar. The vertical structure is very similar to the one
shown by Chen and Chen (1993) and consistent with that found by Numaguti (1995) and
Kiladis and Wheeler (1995) during the northern winter. The same phase often extends
from the surface to about 200 hPa and changes sign around 150 hPa. However, the largest
amplitude is mostly located in the lower atmosphere. Thus, the vertical structure has a
significant baroclinic as well as a barotropic component. The meridional circulation
shown in Fig. 5(a) also indicates a significant baroclinic component.
Similarity in preferred temporal scale (quasi-biweekly period), zonal scale (wave
number 6), horizontal structure (n = 1 equatorial Rossby wave), westward phase propagation and eastward group propagation of both summer and winter QBM indicates
that a universal mechanism may be responsible for both the phenomena. Close association between OLR and circulation indicates strong convective coupling for the mode.
We also examined the standard deviation of winds and OLR for the QBM and found
THE QUASI-BIWEEKLY MODE
Figure 5.
contoured)
meridional
a factor of
1179
(a) Latitude–pressure section of zonal wind composites (negative values shaded and positive values
along 85◦ E for phase 8 of the northern summer quasi-biweekly mode, together with vectors of
wind and vertical pressure velocity (with a negative sign). Vertical pressure velocity is scaled by
100. (b) and (c) show longitude–pressure sections of zonal wind composites along 10◦ N and 7◦ S,
respectively. The longitude and latitudes of the sections are marked on Fig. 3(a).
(not shown) that it has large amplitude only over warm waters between the date line and
the central IO. This fact also supports the QBM being a convectively coupled oscillation.
Can moist feedback drive the n = 1 equatorial Rossby wave unstable with maximum
growth corresponding to a wavelength of about 6000 km and westward phase speed
of about 4 m s−1 ? If so, what leads to the translation of the mode to the north by about
5◦ ? In the next section, the stability of moist equatorial Rossby waves in the presence
of a steady BL and modification by the mean summer and winter monsoon flow are
studied.
4.
S TABILITY OF CONVECTIVELY COUPLED EQUATORIAL ROSSBY WAVES
To study the stability of the equatorial waves under convective coupling, we
construct a 2 12 -layer model. The vertical structure of the model is shown in Fig. 6.
1180
P. CHATTERJEE and B. N. GOSWAMI
Two layers of free atmosphere provide the minimum vertical resolution required to
represent the barotropic and baroclinic components of the QBM. The BL serves two
important purposes. It provides a better estimate of moisture convergence through a
better representation of mean humidity profile. Secondly, the phase relationship between
the potential temperature and the frictional convergence in the BL may be different
from that between potential temperature and wave convergence (wave–CISK), and has
the potential to lead to an instability. The BL also facilitates coupling between the
barotropic and baroclinic components. Our basic model without the mean flow is similar
to Model A of WR, where a ‘rigid lid’ upper boundary condition (vertical pressure
velocity, ω = ωu = 0 at pressure p = pu = 125 hPa) is used, except that we do not make
the long-wave approximation. The formulation of the non-adiabatic heating, Q2 , in our
model is identical to that used by WR. With the inclusion of mean background flow,
our model is similar to that of Xie and Wang (1996) but with a BL. However, the
BL formulation of our model is significantly different from that of WR. Therefore,
the formulation of the BL will be described in some detail in section 4(a). Since
the parametrization of non-adiabatic heating is similar to WR, it will be described
briefly in section 4(b). The vertical discretization is such that the momentum equations
are written at levels 1 and 3 while the thermodynamic energy equation is written at
level 2 (see Fig. 6). For convenience of analysis and physical understanding, we define
a baroclinic and a barotropic component by χc = (χ3 − χ1 )/2 and χt = (χ3 + χ1 )/2,
respectively, where χ represents zonal, u, and meridional, v, wind speeds, geopotential
height anomaly, φ, and the mean background zonal flow, U , at the respective levels.
The non-dimensionalized equations for the barotropic and baroclinic components
may be written as
∂ut
∂uc
∂U c
∂ut
∂U t
+ Ut
+ Uc
− vt y −
+vc
∂t
∂x
∂x
∂y
∂y
∂uc ∂vc
∂φt
+
(1a)
+ξ
+ 2κpωe + rut = −
∂x
∂y
∂x
∂vt
∂vt
∂vc
∂φt
+ Ut
+ Uc
+ yut + rut = −
∂t
∂x
∂x
∂y
∂U t
∂uc
∂ut
∂U c
∂uc
+ Ut
+ Uc
− vc y −
+vt
∂t
∂x
∂x
∂y
∂y
γ ∂uc ∂vc
∂φc
+
+
+ δpωe + ruc =
2 ∂x
∂y
∂x
∂vc
∂vc
∂vt
∂φc
+ Ut
+ Uc
+ yuc + rvc = −
∂t
∂x
∂x
∂y
∂
+N
∂t
∂ut ∂vt
+
= −ωe
∂x
∂y
∂φc
φc + U 2
− yvt U c + (1 − B)ωe
∂x
∂uc ∂vc
+(1 − I )
+
+ (ut + uc ) = 0.
∂x
∂y
(1b)
(1c)
(1d)
(1e)
(1f)
THE QUASI-BIWEEKLY MODE
Zu = 0
1181
p = 125 hPa
u
u ,v ,I
1
1
1
p1= 325 hPa
Z2, Q2
p2= 525 hPa
u ,v ,I
3
3
3
Ze
p3= 725 hPa
pe= 925 hPa
Mixed Layer
p = 1013 hPa
s
Figure 6. The vertical structure of the model.
In the above equations, γ = (U u − 2U 2 + U e )/2, κ = (U u + 2U 2 − 3U e )/8p,
δ = (U u − U e )/4p and ξ = 2δp. Here ωe represents the vertical pressure velocity
at the top of the BL. The Rayleigh friction coefficient r and the Newtonian cooling
coefficient N are assumed to be equal in this study. The parameters I and B in Eq. (1f)
come from the non-adiabatic heating rate Q2 (described below) and represent change in
static stability due to wave and BL convergence respectively. Formulation of the BL and
the non-adiabatic heating are briefly described below.
(a) Boundary layer in the model
For our BL, we use the steady mixed-layer formulation of Stevens et al. (2002),
using generalized Ekman balance to find a relation between bulk winds (which are also
a proxy for the surface winds) and winds and pressure gradients at the top of the BL.
Bulk winds are defined as
h
(uB , vB ) =
{up (p), vp (p)} dz,
0
where h is the BL height. The mixed-layer winds obey a balance between the Coriolis
force, the pressure force and the Ekman term. The Ekman term has two components,
the entrainment or exchange of momentum between BL and upper layers, and the
Rayleigh friction due to surface roughness. Stevens et al. (2002) show that incorporation
of entrainment helps better simulation of divergence of the vector winds in the equatorial
region. It also provides a conceptual basis for the use of anisotropic Rayleigh friction in
zonal and meridional directions required to explain the momentum balance for tropical
surface winds.
1182
P. CHATTERJEE and B. N. GOSWAMI
The steady Ekman balance may be written as
∂φe We
Wd
+
(ve − vB ) −
vB
∂y
h
h
∂φe We
Wd
+
(ue − uB ) −
uB ,
−βyvB = −
∂x
h
h
βyuB = −
where h is the BL height, We is the entrainment velocity, Wd is the drag coefficient and
ue , ve and φe are free atmospheric winds and geopotential at the top of the BL. The bulk
winds in the BL in dimensional form may be written as:
#i
u e #i #e
βy
∂φe
∂φe
uB = 2
+ 2
(2a)
−
#i + (βy)2
#i2 + (βy)2 ∂y
#i + (βy)2 ∂x
∂φe
∂φe
ve #i #e
βy
#i
vB = 2
+
− 2
.
(2b)
#i + (βy)2
#i2 + (βy)2 ∂x
#i + (βy)2 ∂y
These equations are identical to the linear bulk model equations of Stevens et al. (2002,
their Eqs. 16(a) and (b)). The parameters #i = (We + Wd )/ h and #e = We / h have values
shown in Table 1. Stevens et al. (2002) showed that for certain values of parameters #i
and #e , the BL model simulates the observed mean surface winds over the Pacific quite
well. Continuity of the vertical velocity at the interface between the BL and the free
atmosphere gives
ωe = (ps − pe )∇ · uB
We neglect (βy)2 in comparison with #i2 as (βy)2 #i2 within 20◦ of the equator.
Using the assumption that the free atmospheric pressure gradient and divergence at the
top of the BL (φe and ∇ · ue ) could be approximated by those in the lower layer in our
model (φ3 and ∇ · u3 ), we have the following expression for non-dimensional ωe :
F TC2o k 2 /L2 + l 2 /L2 + ikβ/#i L
ωe =
(3)
(φt + φc ) + '∇ · uc ,
2
#i + F #e /2
where F = (ps − pe )/p and ' = #e F /{2(#i + F #e /2)}. The meridional mode number, l, in Eq (3) is a parameter in the model. Eq. (3) can now be used to substitute for φt
in Eqs. (1c)–(1f) to convert them into an eigensystem.
(b) Formulation of non-adiabatic heating
The anomalous non-adiabatic heating, Q2 , in the model is proportional to the
latent heat released due to anomalous precipitation. The parametrization of Q2 in our
study is identical to that used by Wang (1988) and WR. Assuming that the basic state
evaporation exactly balances the basic state precipitation and that the rate of local
anomalous moisture change is small, the anomalous precipitation P is the sum of
anomalous convergence of moisture in the vertical column of unit cross-section and
the local anomalous evaporation rate E and may be expressed as
ps
ps
dp
∂q(p)
dp
+ E = −b
ω(p)
+ bE
∇ · {q(p)v}
P =b −
g
∂p
g
pu
pu
b
= − {ω2 (q 3 − q 1 ) + ωe (q e − q 3 )} + bE,
g
1183
THE QUASI-BIWEEKLY MODE
TABLE 1.
B
CD
Co
Cp
I
L
Lw
l
N
qe
q1
q3
S2
T
α
β
p
#i
#e
µ
C ONSTANTS , SCALES AND STANDARD VALUES OF NON - DIMENSIONAL PARAMETERS
Boundary layer (BL) forcing
Drag coefficient in the BL
Velocity scale
Specific heat of dry air
at constant pressure
Change in stability due to
internal wave convergence
Length scale
Latent heat of
condensation of water
Meridional mode number
Non-dimensional µ
Mean moisture in BL
Mean moisture in layer 1
Mean moisture in layer 3
Static stability
Time scale
Non-dimensional
static stability
Gradient of the
Coriolis parameter
Pressure difference
between two layers
Newtonian cooling coefficient
= (2qe − q1 − q3 )/α
=
√
(0.5S2 p2 )
2.8
1.5 ×10−3
49 m s−1
1004 J kg−1 K−1
= (q3 − q1 )/α
√
= (Co /β)
0.6
1432 km
= (Co β)−1/2
2.54 × 106 J kg−1
1.7
0.0760
0.017 kg kg−1
5.5 × 10−4 kg kg−1
0.0063 kg kg−1
3 × 10−6 m2 s−2 Pa−2
0.344 days
= (cp p2 p)(Rgas bLw )−1 S2
0.0096
√
= µ/ (βCo )
2.3 × 10−11 m−1 s−1
= (p3 − p1 )
400 hPa
2.7 × 10−2 /350 s−1
0.8 × 10−2 /350 s−1
2.5 × 10−6 s−1
where b is an efficiency factor taken to be 90%. q(p) is the background mean specific
humidity considered a function of height alone. In a vertically continuous model, the
heating Q(p) is such that
ps
dp
Q(p)
= Lw P .
g
pu
On vertical discretization we then have Q2 = bgLw P /p. The anomalous evaporation
is proportional to the anomalous surface winds and the vertical gradient of mean
moisture at the surface. In the potential temperature equation, the dissipation term due
to Newtonian cooling is proportional to the potential temperature θ = −(φ3 − φ1 )/p.
With this background, the expression for Q2 may be written as
Q2 = −
2Cp p2
bLw
[ω2 (q 3 − q 1 ) + ωe (q e − q 3 )] + µ
φc + ∗ u B
p
Rgas p
where ∗ = 2(q s − q a )Lw CD Cp p2 (U e /|U e |)/(bqcritHRgas ), with q a = 0.0185 kg kg−1 ,
q s = 0.020 kg kg−1 , qcrit = 2Cp Co 2 p2 /(bRgas Lw p), and H = p/(gρair ). The ∗ uB
term appears from the linearization of the evaporation, E, and represents the EWF.
Values of climatological mean humidities q e , q 3 , q 1 (at these levels) are shown in
Table 1. ω2 in the expression for Q2 can be expressed in terms of ωe and baroclinic divergence ωc (= ∇ · uc ) using the continuity equation. The parameters I, B (see
Table 1) appear in the model equations via the convergence terms in the Q2 equation.
1184
P. CHATTERJEE and B. N. GOSWAMI
Assuming wave solutions of the form exp{i(kx − σ t)}, with zonal wave number,
k, and angular frequency, σ , and then finite central differencing in the y-direction, we
can convert the system of equations to an eigensystem having 5M eigenvalues and
eigenvectors, where M is the number of grid points. Our grid is defined to be within
60◦ N to 60◦ S with a spacing of 2◦ . The meridional boundary conditions in the absence
of shear are φc = φt = 0 at y = ±Y . The eigenvalue σ could be complex, σ = σr + i<,
with < representing the growth rate. As the seat of the QBM is the western Pacific and
the eastern IO, the background parameters (such as mean humidity, static stability and
mean winds) used in the model are representative of this region. The non-dimensional
parameters used in the model are listed in Table 1. The control case was repeated with
1◦ grid spacing and no significant change in the character (frequency, growth rate and
horizontal structure) of the unstable mode was found between the two resolutions.
5.
R ESULTS
(a) The control case: with convective feedback but no mean flow, no EWF
To begin, we investigate the stability of the equatorial waves in the absence of
mean shear, advection and EWF. All the terms with U in Eqs. (1a) to (1f) are dropped
and in Eq. 1(f) is set to zero. Solution of the eigenvalue problem shows that a
n = 1 Rossby wave is unstable with a distinct maximum in its growth rate as shown
in Fig. 7(b). The maximum growth occurs for k = 1.33 or a wavelength of 6750 km
with a corresponding time period of 16 days. The amplitude-doubling time for the
unstable Rossby wave is about 3.6 days. The frequency and wavelength of the most
unstable wave corresponds to a westward phase speed of 4.8 m s−1 . The horizontal
structure of the mode is shown in Fig. 8(a). The two vortices are symmetric about the
equator. The selection of zonal wavelength and period (and hence the phase speed) are
in good agreement with those of the observed QBM during northern summer or winter.
The latitudinal tilt of the two vortices is also similar to that of the observed QBM during
winter. In the absence of any mean flow, the unstable Rossby mode, however, does not
show the northward (southward) shift during the northern summer (winter).
To gain insight regarding sources of energy for the instability, we turn to the energy
equations. The equation for total energy has the following form after integrating over
the entire latitudinal domain and one wavelength in the zonal direction:
∂
+ 2r εtot = (B − 1)φc ωe − (1 − I )φc ωc − uc · ∇φc − ut · ∇φt ∂t
+Y λ/2
with ≡
dy dx (4)
−Y
−λ/2
and εtot = Kc + APEc + Kt with Kc , Kt representing baroclinic and barotropic components of eddy kinetic energy and APEc being the eddy available potential energy.
(b) Scale selection
To investigate the cause of the scale selection we examine the energy balance in
Eq (4). In our model, the scale selection is brought about by an interplay between the
BL convergence term (B − 1)φc ωe , the terms uc · ∇φc , ut · ∇φt representing work
done against pressure forces and the wave convergence term (1 − I )φc ωc . The phase
relationship between φc , ωc and ωe are shown in Fig. 8(b). As φc and ωc are in phase
for this wave, the wave convergence term cannot give rise to growth in the present case,
1185
THE QUASI-BIWEEKLY MODE
0
7
(a)
(b)
6
103 *
10 Vr
Ŧ0.5
Ŧ1
Ŧ1.5
Ŧ2
5
4
0
1
2
3
4
3
3
10
5
4
Figure 7.
5
10
10
O (km)
k
(a) The dispersion curve for the Rossby wave driven unstable by convective feedback. (b) The growth
curve, <, versus wavelength, λ, for the same mode.
2
(a)
1.5
1
Y
0.5
0
Ŧ0.5
Ŧ1
Ŧ1.5
Ŧ2
Ŧ3
Ŧ2
Ŧ1
0
1
2
X
4.5
(b)
3
(c)
2
1.5
1
Ŧ1.5
Ŧ4.5
Ŧ1
0
Ŧ1
0
1
Zonal phase (S)
1
1.5
2
2.5
k
Figure 8. (a) The non-dimensional horizontal structure of the maximally growing mode. Latitudes (Y) and
longitudes (X) are also in non-dimensional units. Winds are denoted by arrows and contours of φc are shown at
intervals of 0.04 units. (b) The zonal phases of φc (solid), −ωc (dashed), ωe (dashed + dotted) at y = 0.4572
(6◦ N). (c) The different terms in the spatially averaged baroclinic energy εtot equation, (B − 1)φc ωe (thin
solid), −(1 − I )φc ωc (thick dashed + dotted), −uc · ∇φc (dashed), −ut · ∇φt and their sum (thick solid)
as a function of wave number k. (See text for details.)
so long as I < 1. The wave convergence does reduce the static stability but does not
produce overturning and instability. On the other hand, the positive phase relationship
between φc and ωe indicates that BL convergence of moisture can lead to an instability
as B > 1. The energy budget is demonstrated in Fig. 8(c). The two conversion terms are
also sources of perturbation energy. The scale selection arises as a result of difference
1186
P. CHATTERJEE and B. N. GOSWAMI
in scale dependence of the source and sink terms. Thus, the steady ageostrophic BL
convergence in our model plays a crucial role in making the Rossby mode unstable as
well as in the scale selection. Insight regarding the role of the BL convergence in driving
the mode unstable may be obtained by reducing Eqs. (1a) to (1f) to a set of two coupled
equations in vc and ωe . The two equations may be written as
1
2
2 ηωe
1
C
dω
B
−
1
ik
d2 vc
ik
I
e
− 1 − k 2 − η2 vc =
−
+ I2
(5a)
3
C C
dη
dη2
I2
I4
1
2I2 η dωe
d 2 ωe
−
1
dη2
C2 + I 2 η2 dη
1
2B
2I
2
(γ C2 − k 2 B)
2
1
C
2ik
I
η
k
η
ik
I
2
− −k −
+
+ I2
−
ωe
1
ψCγ
ψ
ψC3 γ
C
C3 + I2 η2 C
1
1
3
1
I4
CI4
I2 η2 dvc ik I 2 η2
+
(ηvc ).
=
1+
1+
ψγ
dη
ψγ
C2
C2
(5b)
= 1 − B, Here B
I = 1 − I , C = iσ + #, η = y/I1/4 and γ = 1 + (k 2 I/C2 ).
From the above equations it is evident that when B = 1, the forcing to the vc
equation vanishes. Then the equation is identical to the parabolic cylinder function
equation with eigenvalues given by
1
ik C2
2
2
I
− 1 − k = 2n + 1, n = 0, 1, 2, . . . .
C I2
It represents moist neutral equatorial waves, including the Rossby waves, whose phase
speeds are slowed down by the convective feedback. The equations also show that the
structure of the convectively coupled unstable mode would be modified from that of a
pure Rossby wave through coupling. Equation (5b) shows that the baroclinic component
of the Rossby wave drives the BL convergence ωe .
The instability mechanism is essentially a wave–BL–CISK and is illustrated
schematically in Fig. 9. The internal wave-convergence does play a role in lowering the
lapse rate and slowing down the waves (via the term I φc ωc ), but is not sufficient to make
the moist parcel unstable. CISK is realized in the model by release of latent heat due to
condensation of the moisture primarily from BL convergence. The equatorial Rossby
waves and their structure affects the BL convergence, ωe , in such a way that an increase
in potential temperature, −φc , due to wave convergence gives rise to an increase in ωe
and vice versa. The structure of the Rossby wave is such that φc is out of phase with
the vorticity, ζc . The cyclonic vorticity at the top of the BL drives Ekman convergence
(ωe < 0). The in-phase relationship between ωe and φc leads to increase in perturbation energy (Eq. (5)). Equivalently, the Ekman convergence (ωe < 0) gives rise to latent
heat release (Q2 > 0) in the region of positive potential temperature perturbation, −φc ,
leading to enhancement of the original temperature perturbation.
How do the wave number, frequency and growth rate of the maximally growing
Rossby mode depend on the convective heating parameters I and B? Sensitivity tests of
the unstable Rossby mode to a wide range of plausible convective coupling strengths
(increase in either B or I) show that increase in convective coupling is associated
with increase in growth rate, decrease in frequency (increase in period) and increase
1187
THE QUASI-BIWEEKLY MODE
]c (=’ u v) <0
Eq
]c (=’ u v) >0
y
P3=700mb
y
Eq
Z <0
e
Ze>0
Pe=925mb
T(=ŦI )
c
]c
(ŦZ )
e
Q
2
Figure 9. Schematic illustrating Wave–Boundary Layer–CISK. uc and vc are represented by arrows and the
boundary-layer convergence, ωe , by contours (solid for positive and dashed for negative). ζc represents the curl of
the baroclinic velocity component and θ represents the potential temperature perturbation −φc . Note that ζc and
Ekman convergence, −ωe , are in phase.
in wavelength of the maximally unstable mode. However, the quasi-biweekly period
(between 13 and 19 days) and a wavelength between 5500 and 7500 km is found to be
a robust feature over a wide range of reasonable convective coupling parameters. It is
plausible that the observed quasi-periodicity of the mode is a result of temporal and
spatial variation of the convective coupling strength. A compilation of results from the
sensitivity tests show that the wave number corresponding to maximally growing mode
1
is a monotonic decreasing function of (B − 1)(1 − I )− 4 , whereas the maximum growth
rate < is a monotonic increasing function of the same parameter (Fig. 10). On the
other hand, the angular frequency corresponding to the maximally growing mode is
a monotonically increasing function of (B − 1)(1 − I )−1 .
(c) Role of background mean flow, no EWF
As discussed in several previous studies (e.g. Zhang and Webster 1989), the
background mean flow can introduce a significant non-Doppler effect on the structure
of the equatorial Rossby waves. One simple non-Doppler effect we envisage is what
we term the ‘dynamic equator’ effect. As the meridional gradient of ambient potential
vorticity (PV) is responsible for the restoring force of the Rossby waves, a background
mean flow modifies the ambient PV (through modification of the absolute vorticity
η = f + ζ ), and influences the structure of the Rossby wave by affecting meridional
displacements through PV conservation. In the absence of any mean background flow,
η = 0 around the equator and the equator defines an axis around which the meridional
structure of the Rossby waves could be either symmetric or antisymmetric. If the
background mean flow is such that the η = 0 line is away from the equator, we claim
that the Rossby waves will ‘see’ this axis as the ‘dynamic equator’ and the meridional
structure of the Rossby waves may now be defined around this axis. Compared to a pure
Rossby wave structure, such a structure in the presence of a background mean flow may
1188
P. CHATTERJEE and B. N. GOSWAMI
2.6
Ŧ0.08
0.2
(a)
(c)
(b)
0.18
Ŧ0.1
2.2
0.16
Ŧ0.12
2
0.14
Ŧ0.14
1.8
max
Vr
*
k
max
max
2.4
0.12
Ŧ0.16
1.6
0.1
Ŧ0.18
1.4
0.08
1.2
0.06
Ŧ0.2
1
2
3
Ŧ1/4
(BŦ1)(1ŦI)
Figure 10.
4
1
2
3
Ŧ1/4
(BŦ1)(1ŦI)
4
0
2.5
5
7.5
10
Ŧ1
(BŦ1)(1ŦI)
Dependence of (a) scale selection and (b) the growth rate <max on (B − 1)(1 − I )− 4 . (c) shows
sensitivity of the frequency, σrmax , to (B − 1)(1 − I )−1 . See text for details.
1
appear to be shifted to the north or south. The curl of the mean winds during the Indian
summer monsoon causes the zero background absolute vorticity line to shift to about
5◦ N at low levels. We hypothesize that the observed shift of the horizontal structure of
the observed QBM is due to the ‘dynamic equator’ effect on the unstable Rossby mode.
To test this hypothesis, we study the stability of the equatorial waves in the presence
of mean background flow in the tropics characteristic of the northern summer over the
Asian monsoon region. Zonal mean zonal winds are required at the two free atmospheric
layers in our model centred on 725 and 325 hPa. Meridional profiles of mean zonal
winds at the lower and upper layers are obtained from climatological mean zonal winds
during July from NCEP/NCAR reanalysis averaged between 40◦ E and 120◦ E at 700 hPa
and 300 hPa, respectively. The meridional profile of July mean zonal wind at 850 hPa
is also derived, as a sensitivity experiment is planned to be carried out with this profile.
As our objective is to see the role of the mean wind shear over the tropical region
on the equatorially trapped waves, the following idealization is made on the observed
wind profiles. Keeping the magnitude and horizontal shear between 25◦ N and 25◦ S, the
observed profiles are tapered to have no horizontal shear north and south of 30◦ latitude.
The idealized mean zonal winds used in the model are shown in Fig. 11.
Keeping the strength of convective feedback the same as in the control case
(i.e. values of I and B the same, as in Table 1) and without including the EWF ( = 0),
the eigenvalue problem (Eqs. (1a) to (1e)) was again solved by including the 700 and
300 hPa mean zonal winds at levels 3 and 1 respectively. The n = 1 Rossby mode is
again found to be unstable with maximum growth rate corresponding to a period of
19 days and wavelength of 7100 km (westward phase speed = 4.3 m s−1 ) and doubling
time of 4 days. The inclusion of the background mean flow with shear has actually
resulted in a slight decrease in the growth rate of the unstable mode compared to the
control case. The horizontal structure of the unstable mode is shown in Fig. 12. It may
be noted that the two vortices are no longer centred on the equator but shifted to the north
by about 3◦ , with the northern vortex being stronger than the southern one. While the
1189
THE QUASI-BIWEEKLY MODE
10
7
4
1
Ŧ2
Ŧ5
Ŧ8
Ŧ11
Ŧ3
Ŧ2
Ŧ1
0
1
2
3
Figure 11. Model mean winds (m s−1 ) over the tropical Indian Ocean averaged between 40◦ E and 110◦ E: U at
850 hPa (solid), U 3 at 700 hPa (dashed), U 1 at 300 hPa (dashed + dotted). The x-axis is non-dimensional latitude
(1 unit = 12.8◦ ).
2
Y
1
0
Ŧ1
Ŧ2
Ŧ3
Ŧ2
Ŧ1
0
1
2
X
Figure 12. The horizontal structure of the maximally growing mode with observed mean flow at the low level
(700 hPa) and upper level (300 hPa) during the northern summer over the Indian monsoon region. The winds are
denoted by arrows and φc contours are shown at intervals of 0.04 units. The axes are as Fig 8(a).
inclusion of the mean flow at 700 and 300 hPa results in only minor changes in the scale
selection of the unstable Rossby mode, it leads to significant north–south asymmetry and
a northward shift of about 3◦ in the horizontal structure of the mode. It may be noted that
the zero absolute vorticity corresponding to the mean flow at 700 hPa shown in Fig. 11 is
indeed around 3◦ N. Thus, the northward shift of the unstable Rossby mode is consistent
with our proposed ‘dynamic equator’ effect. However, we note that the shift of the model
unstable mode is still smaller than that of the observed QBM (Fig. 3). The observed shift
of the zero absolute vorticity line is rather small at 700 hPa but is largest at lower levels
(see Fig. 2 of Tomas and Webster 1997). The location of the ‘dynamic equator’ is at
1190
P. CHATTERJEE and B. N. GOSWAMI
2
(a)
10 V
r
Ŧ0.3
Ŧ0.8
1.5
Ŧ1.3
Ŧ1.8
0.5
1.5
2.5
1
K
5
(b)
4.5
0.5
Y
102 *
(d)
4
0
3.5
3
4
8
6
12
Ŧ0.5
O (10 km)
(c)
2
Ŧ1
Ŧ1
Ŧ4
Ŧ1
0
1
Zonal phase (S)
Ŧ1.5
Ŧ3
Ŧ2.5
Ŧ2
Ŧ1.5
Ŧ1
Ŧ0.5
0
0.5
1
1.5
X
Figure 13. Dispersion relation, growth rate and non-dimensional horizontal structure of the unstable Rossby
wave in the sensitivity experiment. The background mean flow at 700 hPa and 300 hPa are used for advection
terms while 850 hPa mean zonal winds are used to define the ambient η (see text). (a) The dispersion curve for
the Rossby wave. (b) Growth rate < versus wavelength, λ, for the same mode. (c) The zonal phases of φc (solid),
−ωc (dashed), ωe (dashed + dotted) at y = 0.4572 (6◦ N). (d) The horizontal structure of the maximally growing
mode. The winds are denoted by arrows and φc by contours at intervals of 0.04 units. Axes are as Fig. 8(a).
around 5◦ N with the zonal mean profile at 850 hPa shown in Fig. 11. The larger shift
of the ‘dynamic equator’ at the lower levels appears to be instrumental in the observed
shift of the structure of the QBM. However, the low vertical resolution of our model
prevents us from including the effect of the lower-level mean flow in the model.
To demonstrate that the 850 hPa mean winds may have a stronger ‘dynamic equator’
effect on the unstable Rossby mode, we carried out a sensitivity experiment. This experiment is similar to the previous experiment with mean flow but the ‘dynamic equator’
terms (y − ∂U t /∂y) in Eqs. (1a) and (1c) (indicated by braces) are calculated using
850 hPa mean zonal winds rather than those at 700 hPa. The eigenvalue problem was
solved and the n = 1 Rossby mode was found to be unstable with maximum growth
corresponding to a period of 17 days, wavelength of 6020 km (westward phase speed
of 4.1 m s−1 ) and doubling time of 4.6 days. The dispersion relation for the mode,
and dependence of growth rate on wave number are shown in Figs. 13(a) and 13(b)
respectively, while the horizontal structure of the most unstable mode is shown in
Fig. 13(c). While the temporal and spatial scale selection is not significantly affected
and the Rossby wave with quasi-biweekly period and approximately 6000 km wavelength still remains the dominant unstable mode, its horizontal structure is significantly
modified. The north–south asymmetry of the two vortices and the northward shift to
about 6◦ N are now very similar to the observed structure of the QBM (Fig. 3).
To investigate the role of the northern winter mean zonal winds on the unstable
Rossby mode, mean zonal wind profiles were constructed at 850, 700 and 300 hPa by
averaging between 100 and 180◦ E and tapering them beyond 25◦ N and 25◦ S to have
no horizontal shear north and south of 30 degrees latitude. The eigenvalue problem was
solved with the mean flow characteristic of the northern winter, and the n = 1 Rossby
THE QUASI-BIWEEKLY MODE
1191
mode was found to be unstable with maximum growth rate corresponding to quasibiweekly period and ≈6000 km wavelength but the horizontal structure shifted to the
south of the equator by about 3◦ (not shown). Thus, for both the summer and winter
QBMs, the basic scale selection mechanism is intrinsic to the convective coupling but
the observed modification of the structure of the QBM from a ‘pure’ Rossby mode is
due to the influence of the background mean flow.
(d) Effect of evaporation–wind feedback
Having shown that the mode is driven unstable even in the absence of EWF, the
effect of EWF on the unstable Rossby mode is also investigated. First, the role of EWF is
studied in the absence of background mean flow. The eigenvalue problem corresponding
to Eqs. (1a) to (1e) is again solved after including . The inclusion of EWF modifies the
thermodynamic equation as in Eq. (1f). It is found that the n = 1 Rossby mode is still
unstable with maximum growth rate corresponding to approximately 15 day period and
6000 km wavelength. However, both positive or negative (corresponding to either
uniform easterly or westerly background mean flow) increases the growth rate of the
mode. This conclusion is found to remain valid even when the EWF is introduced
in the presence of the mean background flow. Thus, the EWF does not introduce any
qualitative change in either the scale selection or the structure of the unstable Rossby
mode.
6.
S UMMARY AND C ONCLUSIONS
The mean structure of the of the QBM (or the 10–20 day mode) is re-examined
using daily circulation and convection data for ten years (1992–2001). It is shown that
the mode is convectively coupled and that the horizontal structure corresponds to an
equatorial Rossby wave with approximate wavelength of 6000 km and westward phase
speed of 4–5 m s−1 translated to the north of the equator by about 5◦ during the northern
summer and to the south by about 5◦ in the northern winter. The vertical structure is
found to have a barotropic component in addition to a significant baroclinic component.
No convincing mechanism for genesis and scale selection for the QBM has so far been
available. Based on the observational evidence, a hypothesis is advanced that the same
genesis and scale selection mechanism governs both summer and winter QBMs. It is also
proposed that the QBM is a gravest meridional mode equatorial Rossby wave driven
unstable by moist convective feedback and translated to the north (south) in northern
summer (winter) by the ‘dynamic equator’.
The hypothesis is tested with a model having two layers of free atmosphere and
a steady Ekman BL. It is shown that the n = 1 Rossby wave is driven unstable by
convective feedback with maximum growth rate corresponding to a period of 16 days
and wavelength of 6750 km (westward phase speed of 4.8 m s−1 ). When the dynamic
equator is coincident with the geographic equator, as in the transition seasons, the
horizontal structure of the unstable Rossby wave is centred on the geographic equator.
It is shown that the dynamic equator translates the unstable Rossby wave to the north
of the equator by about 5◦ during the northern summer and to the south by a similar
amount in the northern winter.
It is shown that neither baroclinic instability nor EWF is required for the basic
genesis or scale selection of the unstable Rossby mode. Inclusion of EWF increases the
growth rate of the maximally growing mode without significantly influencing the quasibiweekly scale selection. The vertical shear of the mean winds modifies the horizontal
structure to agree better with observed structure of the QBM.
1192
P. CHATTERJEE and B. N. GOSWAMI
Due to the shift of the Rossby wave structure to the north or south of the equator,
the equatorial structure of the low-level winds associated with the observed QBM bears
some resemblance to that of the MRG wave. Could a convectively coupled MRG
wave explain the scale section for the observed QBM? In addition to the unstable
Rossby mode, our model does yield an unstable MRG wave with maximum growth rate
corresponding to a period of 4–5 days and wavelength of about 5000 km with growth
rate of about one third that of a Rossby wave. Since the unstable MRG wave depends
on different physical processes, the convective feedback, the vertical shear and EWF are
not capable of modifying the dispersion relation of the westward-propagating branch of
the MRG so that its frequency falls in the 10–20 day range. Hence we rule out the MRG
mode as a possible mechanism for the 10–20 day monsoon intraseasonal mode.
To a large extent, our model is similar to that of WR. While they concluded that
the gravest meridional mode equatorial Rossby waves were damped in their model, the
n = 1 Rossby mode is unstable in our model. Qualitatively different results obtained
by the two rather similar models warrants a clarification. This difference may be
attributed to two differences in the two model formulations. Unlike in WR, the longwave approximation is not made in the present study. Further, the BL formulation in our
model is significantly different. We do not believe that the long-wave approximation has
much to do with the difference in the results between the two models. The difference
arises primarily from the difference in the BL formulation. The main difference in our
BL formulation is the inclusion of entrainment mixing, which is required for better
divergence simulation of tropical surface winds (Stevens et al. 2002). It may be recalled
that the wave energy generation is related to the phase relationship between frictional
vertical velocity ωe and φc . While the phase relation between ωe and φc for Rossby
waves was destructive in the WR formulation, our BL allows a constructive phase
relationship between ωe and φc for Rossby waves and leads to generation of wave
energy.
Is the equatorial Rossby wave description of the QBM consistent with its strong
association in modulating Indian monsoon rainfall? Arrival of the northern cyclonic
(anticyclonic) vortex of the QBM over the northern BoB centred on 18◦ N enhances
(weakens) meridional shear of zonal winds and enhances (weakens) synoptic activity
and increases (decreases) rainfall on a large scale (Chen and Chen 1993). A gravest
meridional mode free equatorial Rossby wave would have largest convergence/divergence centred on 10–13◦N and could not influence the mean flow and cyclogenesis in
the northern BoB. However, the modified Rossby wave does have cyclonic vorticity
centred on 18◦ N and hence could influence the mean monsoon circulation and synoptic
activity. Thus, the translation of the Rossby wave by the dynamic equator allows the
equatorial wave structure of the QBM to interaction with the Indian monsoon.
The surface wind stress associated with the QBM has a vortex centred on the
equator. Such a wind stress forcing has a significant projection into oceanic MRG waves,
resulting in forcing of MRG waves in the ocean. An ocean model forced with such
winds shows that the observed quasi-biweekly oscillation of several parameters of the
equatorial IO (Schott et al. 1994) can be explained in terms of oceanic MRG waves
(Sengupta et al. 2001).
The present investigation into the mechanism of scale selection of the QBM has
provided useful insights which were lacking in previous studies. We draw attention to
the fact that the spatio-temporal characteristics of the summer and winter QBMs are
very similar, modified only by different mean flows in the two seasons. We also provide
a mechanism for genesis and scale selection of the QBM and show that a universal
mechanism governs genesis and scale selection of the summer and winter QBMs.
THE QUASI-BIWEEKLY MODE
1193
ACKNOWLEDGEMENTS
This work is supported partially by a grant from the Department of Ocean Development, New Delhi through the Indian National Centre for Ocean Information Services. Constructive comments by two anonymous reviewers on an earlier version of the
manuscript has led to improvement in clarity of presentation. We thank R. Vinay for
preparing some of the figures.
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