DSpace at VNU: A Minimal Model of QBO-Like Oscillation in a Stratosp-Troposp Coupled System under a Radiative-Moist Convective Quasi-Equilibrium State

SOLA, 2014, Vol. 10, 112−116, doi:10.2151/sola.2014-023

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A Minimal Model of QBO-Like Oscillation
in a Stratosphere-Troposphere Coupled System
under a Radiative-Moist Convective Quasi-Equilibrium State
Shigeo Yoden1, Hoang-Hai Bui1, 2, and Eriko Nishimoto1
1
Departmemt of Geophysics, Kyoto University, Kyoto, Japan
2
Hanoi University of Science, Vietnam National University, Hanoi, Vietnam

Abstract
We re-examine the internal oscillation dynamically analogous
to the equatorial quasi-biennial oscillation (QBO) that was firstly
obtained by Held et al. (1993; hereafter HHR93) as a radiativeconvective quasi-equilibrium state in a highly-idealized twodimensional regional model with explicit moist convections under
a periodic lateral boundary condition without Coriolis effects. A
QBO-like oscillation with a period of 120.6 days is obtained for
the control experiment with a similar configuration as HHR93.
The QBO-like oscillation is a robust feature, not sensitive to the
choice of model configuration such as domain size and horizontal
resolution, or boundary conditions such as prescribed zonal wind
at the top and sea surface temperature.
The obtained QBO-like oscillations show downward propagation of the zonal mean signals in the stratosphere as revealed
by observations and wave-mean flow interaction theories, while
unlike the observed equatorial QBO, they have a clear signal in
the zonal mean zonal wind and temperature in the troposphere.
The zonal mean precipitation also varies in accordance with the
oscillation, though its day-to-day fluctuation is very large compared to the long-period oscillation.
(Citation: Yoden, S., H.-H. Bui, and E. Nishimoto, 2014: A

minimal model of QBO-Like oscillation in a stratosphere-troposphere coupled system under a radiative-moist convective quasi-equilibrium State. SOLA, 10, 112−116, doi:10.2151/sola.2014023.)

1. Introduction
The quasi-biennial oscillation (QBO) of the equatorial stratosphere is considered as an internal oscillation due to wave-mean
flow interactions under a zonally periodic boundary condition (see
e.g., Baldwin et al. 2001). Classical QBO theories (Lindzen and
Holton 1968; Holton and Lindzen 1972) assumed the separation
of the troposphere where waves are generated, from the stratosphere where interactions take place, by specifying time-constant
wave forcing at the bottom boundary near the tropopause. In a
laboratory analogue of the QBO, a standing internal gravity wave
was forced mechanically at the bottom boundary (Plumb and
McEwan 1978) or at the top (Otobe et al. 1998) of an annulus of
salt-stratified water. The separation is a theoretical idealization
under an assumption of “independent stratospheric variations” in
stratosphere-only models (Yoden et al. 2002).
In the real atmosphere, however, there is no such a clear
boundary separating the stratosphere and the troposphere. Dynamical coupling between them in the extra-tropics has drawn much
attention over recent years (see, e.g., Yoden et al. 2002), whereas
relatively little attention has been paid to the coupling in the tropics, in particular, to the downward influence of the stratosphere
to the troposphere. Only a few observational studies have shown
Corresponding author: Shigeo Yoden, Kyoto University, Department of
Geophysics, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502,
Japan. E-mail: ©2014, the Meteorological
Society of Japan.

some evidence of the downward influence of the stratospheric
QBO. Gray (1984) pointed out an apparent influence of the QBO
on Atlantic tropical cyclone activity for the period of 1950−1982,
although such a statistically significant relationship was not
obtained for a longer dataset including the period of 1983−2008

(Camargo and Sobel 2010). By analyzing the records of outgoing long-wave radiation and highly reflective cloud index over
decades, Collimore, et al. (1998, 2003) showed a relationship
between the QBO and tropical deep convection through the modulations of tropopause height and cross-tropopause zonal wind
shear.
The use of general circulation models (GCMs) of the atmosphere provides a complementary means to explore possible
stratosphere-troposphere coupling mechanisms relating to the
QBO. Takahashi (1996) first succeeded in GCM simulation of a
QBO-like oscillation for realistic sea surface temperature (SST)
and surface topography, whereas Horinouchi and Yoden (1998)
performed an idealized “aqua-planet” experiment for analyzing
wave-mean flow interactions associated with the QBO. Recently,
four models in the Coupled Model Intercomparison Project Phase
5 (CMIP5) simulated the QBO realistically and projected its
future change (Kawatani and Hamilton 2013). Even though these
high-end numerical models might include the coupling process
associated with the QBO, no attempt has been made to analyze
possible downward influence deep in the troposphere, as far as
we know, perhaps due to too weak signals or limitation in spatial
resolutions of these global models. Any parameterization schemes
on cumulus convections and/or small-scale gravity waves, which
are considered as major sources of model uncertainty, are necessary to simulate the QBO.
Regional cloud-resolving models (CRMs) have been used to
investigate convectively generated stratospheric gravity waves
(Fovell et al. 1992; Alexander et al. 1995) and their possible role
in forcing the QBO in the equatorial stratosphere (Alexander
and Holton 1997). However, downward influence of the QBO
to the troposphere has been beyond the scope of these studies.
Held et al. (1993, hereafter HHR93) introduced another type of
two-dimensional CRM with a periodic lateral boundary condition
and obtained a QBO-like oscillation with a period of about 60

days in a radiative-moist convective quasi-equilibrium state,
though they did not report much about the oscillation without any
description about the dynamical coupling between the stratosphere
and the troposphere.
In this study, we reexamine the HHR93 results by performing
much longer time integrations up to 2 years of a state-of-the-art
regional CRM to describe the oscillation characteristics more
precisely. We also study robustness of the QBO-like oscillation
in a series of experiments by changing some model parameters.
We focus on the phenomenological description in this letter, and
detailed dynamical analyses of the oscillation, including a momentum budget analysis, will be reported in a separated paper.

2. Model and experimental design
The Advanced Research WRF (ARW) version 3 (Skamarock
et al. 2008) is used to conduct a series of two-dimensional
regional simulations of the tropical troposphere and stratosphere.


Yoden et al., QBO-Like Oscillation in a Stratosphere-Troposphere Coupled System
A periodic boundary condition is assumed in the zonal direction
so that the zonally averaged winds are free to evolve. The Coriolis
parameter is set to zero.
The control experiment has a similar configuration as HHR93;
640 km domain width with 5 km horizontal resolution and 130
vertical levels up to 26 km from the surface at the initial state. At
the bottom boundary, SST is uniform and constant at 27°C. At the
top boundary, a traditional Rayleigh damping layer is introduced
for 5 km depth to absorb vertically-propagating gravity waves by
relaxing dependent variables to the reference state given as the
initial condition.

An idealized zonally uniform initial condition is given by the
climatological profiles of temperature and moisture on the equator
(gray solid line in Fig. 1 for temperature) that were created from
the ERA-Interim reanalysis dataset (Dee et al. 2011) and a constant zonal wind of 5 m s−1 in the entire domain. Time integrations
are made for two years with a time increment of Δ t = 10 s.

113

Convective parameterization is turned off in all experiments.
WRF Single-Moment 6-class (WSM6) scheme is used for cloud
microphysics to represent explicit moist convection. As for the
references for this scheme, see Skamarock et al. (2008, Section
8.1.5). For radiation schemes, the Rapid Radiative Transfer Model
(RRTM) (ibid., Section 8.6.1) is used for longwave radiation, and
MM5 (Dudhia) (ibid., Section 8.6.5) for shortwave radiation. We
set the solar declination to the equinox condition and fix the solar
insolation to the daily averaged value (436 W m−2). Planetary
boundary layer scheme is Yonsei University (YSU) PBL (ibid.,
Section 8.5.2) with surface fluxes based on Monin-Obukhov
similarity theory, and the 1.5 order prognostic TKE closure option
(ibid., Section 4.2.4) is used for the eddy viscosities.
Nine experiments (2)−(10) as summarized in Table 1 are
carried out to investigate sensitivity of the QBO-like oscillation
obtained in Control case (1) to model configurations, boundary
conditions, or cloud microphysics.

3. Results

Fig. 1. Vertical profile of temperature at the initial state (gray solid line)
and those of the zonal mean temperature (°C) in quasi-equilibrium states

for ten sensitivity experiments. The statistical equilibrium states are obtained as time averages for complete cycles of the obtained QBO-like
oscillations, for Control case (black line behind green and light blue lines)
and the nine other cases of experiments (color lines as shown inside the
figure).

Figure 1 shows vertical profiles of the zonal mean temperature
in quasi-equilibrium state for Control case (black line behind
green line) and the nine other cases of the experiments, together
with the initial state (gray solid line). In all the experiments except
for Warm rain case with Kessler scheme (Skamarock et al. 2008,
Section 8.1.1), QBO-like oscillations are obtained as described
below (Figs. 2 and 3). The zonal mean temperature for the nine
cases (1)−(9) shows similar lapse rate (7.7 K km−1) as the observed
climatology (i.e., the initial state), and has lower values about
5−10 K than the climatology through the troposphere; SST_30
case (orange line) is about 5 K lower, SST_25 (dark blue line) and
Fine (gray dashed line) cases are the lowest over 10 K, and the
others are in between. The tropopause for the nine cases is located
at 11−13 km, several km below the climatology. In Hitop cases
(red line and black dashed line), temperature in the stratosphere is
much lower than the climatology because of the lack of shortwave
heating due to ozone.
Note that the Warm rain experiment has a very different
vertical profile of the zonal mean temperature (red dashed line). In
this quasi-equilibrium state without QBO-like oscillation, moist
convections are not very active and much smaller lapse rate of
3.2 K km−1 is maintained up to the elevated tropopause at the
height of 24 km just below the Rayleigh damping layer. The
changed lapse rate would be a consequence of the very different
spatial distributions of clouds and moisture that give different

diabatic heating by the atmospheric radiation and cloud microphysics.

Table 1. List of the ten experiments performed in this study. The first two columns are case number and name, and
the third one gives brief description of the difference from Control case (1) given as; time step, Δ t = 10 s; horizontal resolution, Δ x = 5 km and grid number, Nx = 128 for the domain width of 640 km; vertical resolution, ∆z~200
m and grid number, Nz = 130 for the domain height Ztop = 26 km at the initial state; prescribed zonal wind at the top
boundary, Utop = 5 m s−1; SST = 27°C; and WRF Single-Moment 6-class scheme for cloud microphysics. The fourth
and fifth columns are the mean and the standard deviation of the oscillation period in the unit of days, both of which
are calculated with the periods estimated from the zonal mean zonal winds for all the levels below the Rayleigh
damping layer. Four cycles of the oscillation are used for the estimation except for Fine case (3), in which only one
cycle is used for the estimation (denoted by *).
Case number

Case name

Description

Mean period
[days]

Standard Deviation
[days]

(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)

(9)
(10)

Control
Control_0
Fine
Coarse
Wide
Hitop
Hitop_0
SST_25
SST_30
Warm rain

See the caption
Utop = 0 m s−1
Δ t = 5 s, Δ x = 2 km, Nx = 320
Δ x = 10 km, Nx = 64
Nx = 256 (double domain)
Ztop = 40 km, Nz = 200
Ztop = 40 km, Utop = 0 m s−1
SST = 25°C
SST = 30°C
Kessler microphysics

120.6
135.0
124.6*
121.3
112.3

134.8
132.8
111.9
133.2
−

3.0
1.7
3.1*
0.5
1.4
0.9
0.4
0.6
0.7
−


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Fig. 2. Time-height sections of the zonal mean zonal wind (m s−1) for four cases of (a) Control, (b) Control_0, (c) Hitop, and (d) Hitop_0. Negative values
are shaded. A pair of vertical lines in each plot indicates the interval of full four cycles of oscillation.

Fig. 3. Statistical values of the zonal mean zonal wind variations (m s−1); the time mean (thick line), the time mean plus/minus one standard deviation (gray
box), and maximum/minimum values (thin whiskers), for (a) Control, (b) Control_0, (c) Coarse, (d) Wide, (e) Hitop, (f) Hitop_0, (g) SST_25, and (h)
SST_30. All values are calculated from the four oscillation cycles starting from day 100.

Figure 2 shows QBO-like oscillations of the zonal mean zonal

wind for (a) Control, (b) Control_0, (c) Hitop, and (d) Hitop_0
cases. All of the cases show clear oscillations both in the stratosphere and the troposphere with a kink around the tropopause. If
we look at zero-wind lines in the plot of Hitop_0 case as an example, we can easily identify the downward propagation of the oscillation from the bottom of the Rayleigh damping layer (~30 km) to
the tropopause (~13 km) at a mean speed of roughly 170 m day−1
(the descending time is about 100 days). The downward propagation is slower in Control_0 case with 120 m day−1 (about 65 days
for the height range from 20 km to 12 km). Downward propagation of the oscillation continues to the surface at a mean speed of
about 260 m day−1 (about 50 days from 13 km to 0 km) in Hitop_0
case.
The QBO-like oscillation of the zonal mean zonal wind is
symmetric with respect to the zero wind in the cases with Utop =
0 m s−1 (Figs. 2b, d), whereas positive wind phase is longer and its
maximum wind speed is larger in the cases with Utop = 5 m s−1 (Figs.

2a, c). Dependence of the asymmetric nature of the oscillation
on the top boundary condition Utop is clear throughout the stratosphere and the troposphere.
The oscillation period is not very different for all the cases as
listed in Table 1 (the fourth column); from 111.9 days for SST_25
to 135.0 days for Control_0. The estimation of the mean period
of the oscillation is robust with small standard deviation, 3.1 days
at most, throughout the stratosphere and troposphere (the last
column). Note that the oscillation period becomes longer as SST
increases from 25 to 30°C, as opposed to the expectation of shorter period due to more convections.
Figure 3 shows the time mean and variations of the zonal
mean zonal wind for eight cases. The time mean (thick line) is
almost 0 m s−1 in Control_0 (b) and Hitop_0 (f) cases with Utop =
0 m s−1 due to the symmetric nature of the oscillation as described
above. In the cases with Utop = 5 m s−1 except for Hitop case (e),
on the other hand, the time mean is greater than Utop in the stratosphere and the upper troposphere. The variable range also shows

Yoden et al., QBO-Like Oscillation in a Stratosphere-Troposphere Coupled System
an almost symmetric profile with respect to the zero wind in the
two cases with Utop = 0 m s−1 (Figs. 3b, f), whereas it is asymmetric in the cases with Utop = 5 m s−1, largely due to the non-zero
values of the mean zonal wind and skewness of the variations.
These asymmetric features of the oscillation are attributable to
the artificial top boundary condition for the zonal wind and the
asymmetry becomes smaller if the top boundary is moved upward
(e).
The amplitude of the oscillation has two peaks as shown in
Fig. 3; in Hitop cases it has the maximum in the stratosphere at
~24 km and the second maximum at ~11 km, a few km below the
tropopause with a local minimum at ~13 km. In the other cases
with Ztop = 26 km, the stratospheric maximum is comparable to or
smaller than the tropospheric maximum due to the influence of the
top boundary damping.
Some other aspects of the QBO-like oscillation in the
stratosphere-troposphere coupled system are shown in Fig. 4 for
Hitop_0 case. Time variation of the zonal mean zonal wind in the
mid-stratosphere is characterized by rapid transition to the opposite sign and very gradual approach to one of the extreme values
alternatively, in similar way as the observed QBO. On the other
hand, the time variation in the troposphere shows more gradual
increase and decrease.
Figure 4b shows a time-height section of the zonal mean
temperature anomaly from the time mean. In the stratosphere, the
descent of warm anomalies is clear around the timing of the rapid
transition of the mean zonal wind, suggesting the importance of
vertical turbulent mixing associated with the large vertical shear in
the transition phase. Tropospheric temperature also shows periodic
variations associated with the QBO-like oscillation, though there
is little phase lag through the troposphere in contrast to the mean

zonal wind oscillation. The zonal mean daily precipitation (Fig.
4c) also shows the time variation associated with the QBO-like
oscillation in the low-pass filtered component (thick blue line),
though high frequency components are dominant and produce
quite irregular variations.

Fig. 4. Several aspects of the QBO-like oscillation in Hitop_0 case; (a)
time variations of the zonal mean zonal wind (m s−1) at the heights given
by gray thin lines, (b) time-height section of the anomaly of the zonal
mean temperature (K) from the time mean, and (c) time variation of the
zonal mean daily precipitation (mm) and its 21-day running mean (thick
blue line).

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4. Discussion
We demonstrated the QBO-like oscillations of the zonal mean
zonal wind have a clear signal even in the troposphere, in which
organized convective momentum transport (Lane and Moncrieff
2010) might be important because of tilted convective structures
by vertical shear of the mean zonal wind (not shown). We also
showed the zonal mean temperature and precipitation vary
periodically in accordance with the mean zonal wind oscillations.
Further investigation on the interrelation between the mean zonal
wind oscillations and moist convections in the troposphere is
under way.
The present experimental framework is highly idealized and
simplified if compared to the real atmosphere. This is a twodimensional model on a non-rotating plane without Coriolis
effects, instead of the three-dimensional atmosphere on the rotating spherical earth. Clear features of the QBO-like oscillations
in the troposphere obtained in this model may be weakened or

smeared by the influences of such complicated processes in the
real atmosphere. However, we think the use of a hierarchy of
numerical models, including this type of idealized simple one, is
useful to deepen our dynamical understanding of the equatorial
QBO and to reduce the gap between an idealized theory and the
complex real atmosphere (Hoskins 1983; Held 2005). We can
regard the present model as a minimal model, or a maximally
simplified model to study the dynamics of the stratospheretroposphere coupling process associated with the equatorial QBO.
Takahashi (1993) made a unique experiment with a twodimensional model along the equator derived from a GCM without the rotation of the earth, and obtained a QBO-like oscillation
with a period less than 100 days, with clear signal even in the
troposphere and associated change in the direction of precipitation
movement (his Figs. 1b and 3). The model resolution was very
coarse with the truncation zonal wavenumber of 10, and the
convective parameterization of Kuo scheme was retained. Some
aqua-planet experiments with three-dimensional GCMs show
QBO-like oscillations with a hint of associated variations in the
troposphere (Horinouchi and Yoden 1998). However, most of the
analyses in these studies were focused on the stratospheric part of
the oscillation. It would be timely to reinvestigate the dynamics of
the QBO-like oscillations obtained in these hierarchies of idealized two- and three-dimensional global models from a viewpoint
of stratosphere-troposphere dynamical coupling in the tropics.
The experimental framework introduced by HHR93 was
quite unique in the sense that a self-sustained radiative-moist
convective quasi-equilibrium state was obtained in a CRM. In this
minimal model, Rayleigh damping layer placed at the top boundary plays an important role to sustain a quasi-equilibrium state by
absorbing vertically-propagating gravity waves and preventing
their artificial reflection (Klemp et al. 2008). Only a very weak
cooling trend is discernible in the stratosphere for the two year
integrations as shown in Fig. 4b. The altitude of the top boundary
and the prescribed reference value in the Rayleigh damping layer

are also important to determine the oscillation amplitude and the
downward propagation speed of the oscillation signals as shown
in Figs. 2 and 3. High frequency gravity waves with a large
vertical group velocity are artificially attenuated in this damping
layer to influence the zonal-mean momentum budget in the layer
and below. As shown in Fig. 1, the thermal structure near the top
boundary is also influenced by the relaxation process of artificial
cooling in Warm rain case and heating in the other cases.
Yoden and Holton (1988) studied symmetric or asymmetric
features of a QBO-like oscillation in a simple stratosphere-only
model, and showed that time variation of the zonal mean zonal
wind is symmetric if the wave forcing is symmetric (or, a standing
wave) at the bottom boundary, whereas it is not if the wave forcing
is not symmetric. In this study with the minimal model of QBOlike oscillation, we demonstrated some examples of asymmetric
time variations due to asymmetric (i.e., nonzero) zonal wind
forcing given by the top boundary condition in Control and Hitop
cases (Figs. 2a, c). On the other hand, in Control_0 and Hitop_0


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cases with the symmetric forcing, the time variation of the zonal
mean zonal wind is almost symmetric (Figs. 2b, d), which is implicitly indicative of symmetric nature of wave momentum fluxes
generated by moist convections in a statistical sense.

5. Conclusions
We reexamined the QBO-like oscillation reported by Held
et al. (1993), and robustly reproduced such oscillations in the

nine cases (Table 1) except for Warm rain case with Kessler cloud
microphysics. The oscillation period is from 111.9 to 135.0 days,
which is not that sensitive to the choice of experimental parameters. The QBO-like oscillations show downward propagation of
zonal mean signals in the stratosphere, similar as the observations
(Fig. 2). Even in the troposphere, the zonal mean temperature and
precipitation are also modulated in association with the QBO-like
oscillation of the zonal mean zonal wind (Fig. 4).
The present model can be regarded as the minimal model that
can produce a QBO-like oscillation in the stratosphere-troposphere
coupled system under a radiative-moist convective quasi-equilibrium state, and the model would be useful for better understanding
the QBO dynamics. Detailed dynamical analyses including momentum budget in the oscillation will be reported in a separated
paper.

Acknowledgements
We thank Dale Durran for his helpful discussion and comments on our numerical experiments. This work was supported
by JSPS KAKENHI (S) Grant Number 24224011. The stay of
HHB as a research fellow was supported by Kyoto University’s
Global COE Program ‘‘Sustainability/Survivability Science for a
Resilient Society Adaptable to Extreme Weather Conditions’’ for
FY2009-13.

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Manuscript received 25 March 2014, accepted 19 June 2014
SOLA: https://www. jstage. jst. go. jp/browse/sola/