Momentum forcing by the waves resolved in the
reanalyses
The time–height cross sections of the forcing by equatorial
waves, averaged over 5∘ N–5∘ S regions, are
shown in Fig. , where model-level data
from ERA-I have been used for recent years (2003–2010).
For all figures in this paper except Fig. ,
the ticks on the horizontal axis correspond to 1 January
of the given years.
The eastward forcing by the Kelvin waves appears in the QBO phase of
strong westerly shear. The MRG waves induce westward forcing in
both phases of the westerly and easterly shear, with comparable
magnitudes between the phases and KC15.
The MRG wave forcing is primarily by the westward propagating mode
not only in the easterly shear but also in the westerly shear
(not shown), which may suggest the possibility of stratospheric
generation of the wave above the easterly jet
seeand KC15.
For the Kelvin and MRG waves, the altitude
and magnitude of the maximum forcing in each QBO cycle vary
significantly. The IG waves provide eastward and westward
forcing in the westerly and easterly shear phases, respectively.
The Rossby wave forcing is strong in the upper stratosphere.
Unlike the other waves, the Rossby wave forcing is not aligned
with the strong-shear phases of the QBO
at altitudes below 30 km. Rather, it has
significant magnitudes in the northern winters and summers and
is weakened in the following seasons. In addition, this forcing
does not appear in the strong easterlies of the QBO, as the
Rossby waves do not propagate easily with the easterly
background wind. These features in the vertical structure of
the equatorial wave forcing are generally similar between the
reanalysis data sets (not shown). Here, we select
three levels, 50, 30, and 10 hPa,
to assess the wave forcing in the reanalyses
in detail. Note that the level of 10 hPa is close to
the upper limit of the sonde sounding assimilated to the
reanalyses.
Time–height cross sections of the zonal momentum forcing by
the Kelvin, MRG, IG, and Rossby waves (from top to bottom) averaged over
5∘ N–5∘ S,
obtained using the model-level data of ERA-I over the period 2003–2010 (shading).
The MRG wave forcing is multiplied by 3.
The zonal mean wind over 5∘ N–5∘ S is superimposed
at intervals of 10 ms-1 (contour).
The thin solid, dashed, and thick solid lines indicate westerly, easterly, and zero wind, respectively.
Zonal momentum forcing by the Kelvin, MRG, IG, and Rossby waves
averaged over 5∘ N–5∘ S at 30 hPa for the period 1979–2010, as well as
the net forcing by all resolved waves (from top to bottom)
obtained using the p-level data of ERA-I (blue), MERRA (red), CFSR (green), and JRA-55 (orange).
The phase of the maximum easterly and westerly in each QBO cycle at 30 hPa
is indicated by the dashed and solid vertical lines, respectively.
The difference between upper and lower bounds of the wave forcing
calculated from each data set is also indicated (gray shading).
Figure shows the zonal forcing given by the
Kelvin, MRG, IG, and Rossby waves at 30 hPa in
1979–2010, as obtained using the p-level data of the four
reanalyses, as well as the net forcing due to all resolved
waves. The spread between the four reanalyses (i.e., the
difference between upper and lower bounds of the wave forcing
estimated from each data set) is also indicated (gray shading).
The phases of the maximum easterly (westerly) in each QBO cycle
at 30 hPa are indicated by the dashed (solid) vertical
lines in Fig. . The temporal evolution of the
equatorial wave forcing is, at the first order, consistent
between the data sets. The peak magnitude of the Kelvin wave
forcing in the E–W phase shows similar cycle-to-cycle variations
in all reanalyses. For instance, the Kelvin wave forcing in the
four reanalyses is strong in 2010
(7.1–8.7 ms-1month-1) and weak in 1992
(2.8–4.7 ms-1month-1;
here, the month in the unit of forcing refers to 30 days
regardless of the month).
Prior to around 1993,
the MRG wave forcing in the reanalyses seems relatively sporadic
and weak compared to afterward, although the forcing in 1980 and
1985 has exceptionally large peaks in MERRA. The magnitude of
the MRG wave forcing reaches ∼2 ms-1month-1. The IG wave forcing varies
between -3 and 4 ms-1month-1, following the
QBO phase. The Rossby wave forcing magnitude is less than or
similar to ∼2 ms-1month-1 in most years,
except in 1980, 1988, and 2008 for CFSR and ERA-I
(3–3.5 ms-1month-1). The net wave forcing
has large positive peaks in the E–W phases
(3.4–11 ms-1month-1), due mainly to the
Kelvin waves, and is negative during the W–E phases
(1.5–5.2 ms-1month-1) by the IG, MRG, and
Rossby waves (Fig. ). The peak forcing ranges
during the E–W and W–E phases are summarized for each wave in
Table 2.
The same as in Fig. , except using the model-level
data (black) along with the p-level data (blue) for ERA-I.
Although the evolution of the wave forcing is generally
consistent between the reanalyses, some robust differences in
forcing magnitude are shown in Fig. . The
positive peaks of the IG wave forcing are always larger in CFSR
than in the other data sets, and the Rossby wave forcing tends to
be larger in CFSR and ERA-I than in MERRA and JRA-55. There are
differences between the reanalyses of up to about
2 ms-1month-1 for the Kelvin, IG, and Rossby
waves, and about 1 ms-1month-1 for the MRG
waves (Fig. ). The difference in the net wave
forcing is up to about 4 ms-1month-1. There
are many potential causes for this spread of forcing magnitudes
between the reanalyses. For instance, each reanalysis used
a different assimilation method, assimilated different
observational data, and essentially used a different forecast
model (e.g., in terms of model dynamics and resolutions). In
addition, the species and numbers of assimilated observational
data for a single reanalysis are dependent on time, particularly
the satellite data. This makes the further investigation of
temporal variations in wave forcing complicated. Therefore, in
this study, we focus on assessing the range of wave forcing
revealed by the reanalyses and do not speculate on the causes
of the spread, or temporal variations, in the reanalyses.
The same as in (a) Fig. and
(b) Fig. , except for the vertical
advection of zonal wind.
Figure shows the wave forcing at 30 hPa
calculated using the model-level data of ERA-I (ERA-I_ml)
along with that using the p-level data of ERA-I. The plot
exhibits robust differences in Kelvin and IG wave forcing
between the two data sets. The peaks of the Kelvin wave forcing
in the E–W phase from ERA-I_ml range from 6.7 to
13 ms-1month-1, which are
2–4 ms-1month-1 larger than those from ERA-I.
The IG wave forcing from ERA-I_ml has positive and negative
peaks that are 0.8–2.7 ms-1month-1 larger
than those from ERA-I. The differences in the MRG and Rossby
wave forcing depend on the year and are typically less than
∼1 ms-1month-1. The net wave forcing in
the E–W (W–E) phase is 4–9 ms-1month-1
(1–4 ms-1month-1) larger in the model-level
result than in the p-level output.
The differences in forcing magnitude between the two ERA-I
data sets are mainly a result of the vertical interpolation
process. When perturbations in the model-level data are
interpolated to the p levels, those parts of waves with short
vertical wavelengths are inevitably damped. For example, when
a p level is centered between two model levels, waves with
a vertical wavelength of 2Δv are totally filtered
out by the interpolation, where Δv is the
vertical spacing between the two model levels. The filtering
rate of waves with larger vertical wavelengths depends on the
interpolation method. Waves with a wavelength of
4Δv will be filtered at a rate of 50 % in
terms of their variance under linear interpolation, although
this will decrease if a higher-order method is used. Given that
Δv in the lower stratosphere is approximately
1.4 km in ERA-I, waves with vertical wavelengths shorter
than about 5.6 km might be significantly damped in the
ERA-I p-level data. These wavelengths are close to the lower
bound of the dominantly observed Kelvin waves (6–10 km)
and MRG waves (4–8 km) . It is
important that radiative damping, which induces the wave forcing
in the atmosphere, is more prevalent in short vertical-scale
waves e.g., than in longer
waves that may be contained in both data sets. This results in
substantial differences between the two data sets, as shown in
Fig. . The same may also be true for the other
reanalyses. Unfortunately, not all the reanalyses provide
model-level data sets. However, the vertical resolution of the
native models in the lower stratosphere is comparable across all
reanalyses (Table 1). Thus, the magnitude of the wave forcing
obtained from the p-level data sets of reanalyses other than
ERA-I (Fig. ) should also be considered as
underestimated, potentially by amounts comparable to those in
ERA-I.
Phase-maximum magnitudes of the Kelvin, MRG, IG, and Rossby wave
forcing, net-resolved wave forcing, X‾, and X‾∗
[ms-1month-1] at 30 hPa in the E–W and W–E phases
for the period 1979–2010, obtained using the p-level data sets and the
ERA-I model-level data set. Details of X‾ and
X‾∗ can be found from the text along with Eqs. ()
and (). Positive forcing is denoted by bold font.
E–W
W–E
p-level
model-level
p-level
model-level
Kelvin
2.8–8.7
6.7–13
MRG
0.6–2.1
0.6–1.8
0.2–1.8
0.6–2.6
IG
0.9–3.9
2.5–4.3
0.6–3.0
1.9–5.4
Rossby
0.7–2.7
0.6–2.9
0.7–3.5
0.9–3.8
Net-resolved
3.4–11
8.0–19
1.5–5.2
3.3–7.5
X‾
5.8–17
3.1–11
6.6–21
11–18
X‾∗
5.8–14
11–21
Estimated momentum forcing by the waves unresolved
in the reanalyses
As mentioned in Sect. 2, the term X‾ in
Eq. () represents the zonal forcing by unresolved
mesoscale gravity waves and turbulent diffusion, and is also
influenced by the resolved-scale processes that are erroneously
represented in the reanalyses. If one assumes that the
resolved-scale processes are well represented in the reanalyses,
the forcing by unresolved processes can be approximated as
X‾. In this section, we calculate the vertical
advection of zonal wind (the second term on the right-hand side
of Eq. (), ADVz hereafter) and estimate the range of
X‾ in the reanalyses. A discussion of the above
assumption is included in the next section.
Figure a shows ADVz, obtained using the p-level
data of the four reanalyses. The peak magnitude of ADVz in the
W–E phase is around 10 ms-1month-1, and that
in the E–W phase is typically 1–4 ms-1month-1
(excluding the anomalously large peaks in 1983 and 1986–1987 in
CFSR). Note that ADVz in the W–E phase is much larger than the
net-resolved wave forcing in the same phase
(1.5–5.2 ms-1month-1; Table 2), and the two
terms have opposite signs. There exist some robust ADVz
features in the W–E phase: ADVz is very similar in ERA-I and
JRA-55, and ADVz in MERRA is about half of that in ERA-I or
JRA-55 in many years. As a result, the spread between the
reanalyses is quite large (∼10 ms-1month-1) in this phase
(Fig. a).
The large spread in the W–E phase between the different
reanalyses suggests that the ADVz values obtained from the
reanalyses are highly uncertain. Moreover, it is speculated
that this spread may result in a large spread in X‾,
as will be seen later. Therefore, the difference in ADVz
between the reanalyses is further investigated by comparing
w‾∗ and the vertical shear of zonal wind
(u‾z). Figure a shows the climatologies
of w‾∗ obtained from each data set. The
profiles of w‾∗ from ERA-I and JRA-55 are in
good agreement. However, below 30 hPa,
w‾∗ in MERRA is much smaller than in the other
data sets, and that in CFSR is much larger than in the others
above 10 hPa. The profiles of w‾∗ in
ERA-I show only slight differences between the p- and
model-level data. In previous studies by and
, the annual-mean ascent rate was inferred
from the observed H2O to be about
0.26–0.35 mms-1 near 30 hPa. In
Fig. a, w‾∗ at 30 hPa in
ERA-I, CFSR, and JRA-55 is within this range of values. The
smaller value of w‾∗ in MERRA causes ADVz to
be underestimated (see Fig. a) and contributes
to the large spread of ADVz.
(a) Mean residual vertical velocity and
(b) standard deviation
of the monthly and zonal mean wind shear for the period 1979–2010 averaged over
5∘ N–5∘ S, obtained using the p-level data of ERA-I (blue), MERRA (red), CFSR (green), and JRA-55 (orange)
as well as the model-level data of ERA-I (black).
Figure b shows the standard deviation of u‾z obtained
from each reanalysis data set.
These values are governed by the magnitude of u‾z
that alternates between positive and negative with
the QBO phase. Note that the difference in monthly and zonal
mean wind between the reanalyses is small (not shown).
Therefore, u‾z is mainly dependent on the intervals
between the p levels. The standard deviation of u‾z in ERA-I,
CFSR, and JRA-55 is similar, as they have the same p levels.
MERRA has one more p level, at 40 hPa, and thus the
magnitude of u‾z near 40 hPa in MERRA is
larger than in the others. In all of the reanalyses, the
limited sampling across vertical levels causes the magnitude of
u‾z obtained from the p-level data sets to be
underestimated compared to u‾z from the model-level
data (Fig. b). This implies that, as for the wave
forcing, the ADVz values from the p-level data sets should also
be considered as underestimations. The ADVz obtained from
ERA-I_ml is presented in Fig. b. It can be seen
that ADVz in the W–E phase from ERA-I_ml is consistently
2–4 ms-1month-1 greater than that from the
p-level data.
Although this magnitude of difference between the p- and
model-level data seems small in Fig. b, it can
have a significant effect in the estimation of X‾
which has typical values of ∼10 ms-1month-1
as will be shown later.
The Coriolis force and meridional advection
terms in Eq. () are generally small near the
equatorial lower stratosphere (not shown).
The same as in Fig. , except for the terms
(a) X‾, (b) X‾∗, and (c) as in Fig. for X‾∗
(see the text for a definition of these terms).
Figure a shows the value of X‾ at
30 hPa obtained from the p-level data sets of the
reanalyses. The positive peaks of X‾ in the E–W
phase range from 5.8 to 17 ms-1month-1, and
the negative peaks in the W–E phases vary from 6.6 to
21 ms-1month-1. X‾ in the E–W
phase is about 50 % larger than the net resolved wave
forcing (3.4–11 ms-1month-1), and that in the
W–E phase is much larger than the net resolved wave forcing
(1.5–5.2 ms-1month-1). The spread in
X‾ between the reanalyses is up to
10 ms-1month-1, except in 1983 and 1986–1987,
when the ADVz in CFSR has abnormally large peaks
(Fig. a). The large spread in X‾
could be expected because of the large spread in ADVz
(Fig. a). From Fig. a, we can see
that a large portion of the spread in ADVz is due to the
underestimated vertical velocity in MERRA. Additionally, the
zonal wind shear is underestimated in all of the p-level
data sets. Therefore, we attempt to partly correct the estimates
of X‾ via an additional calculation
(X‾∗). In this calculation, ERA-I_ml is
considered as reference data for all the terms in
Eq. (), except for the wave forcing term.
X‾∗ is estimated as
X‾∗=u‾t-v‾∗f-(acosϕ)-1u‾cosϕϕ+w‾∗u‾zr-ρ0acosϕ-1∇⋅F,
where a superscript r denotes terms calculated using the
reference data,
and the E–P flux divergence term is calculated using the
respective reanalyses.
X‾∗ is plotted in
Fig. b. The negative peaks of X‾∗ in
the W–E phase are larger than those of X‾ by
5–12 ms-1month-1, particularly for MERRA.
The changes in positive peaks do not appear to be large. The
spread in X‾∗ is up to ∼4 ms-1month-1, which results from the spread
in resolved wave forcing (see Eq. ). Finally,
X‾∗ in ERA-I_ml is shown in Fig. c.
The positive peaks of X‾∗ in the E–W phase in
ERA-I_ml are 3.1–11 ms-1month-1, and the
negative peaks in the W–E phase are
11–18 ms-1month-1.
These values of X‾∗ are
comparable with those estimated by .
The positive peaks are
smaller than those of the Kelvin wave forcing, suggesting that
the peak magnitudes of the net mesoscale gravity wave forcing in the
E–W phase at 30 hPa might be smaller than those of the
Kelvin wave forcing. In contrast, the large negative values of
X‾∗ suggest that gravity waves are the
dominant contributors to QBO in the W–E phase, assuming that the
turbulent diffusion is not of comparable magnitude. These
results are consistent with those from previous studies using
mechanistic, general circulation, or mesoscale models
e.g.,.
The wave forcing estimates at 50 and 10 hPa are also
presented in Tables 3 and 4, respectively. From Tables 2–4,
it is shown that the Kelvin wave forcing in the E–W phase tends
to increase with height from 2.7–9.2 ms-1month-1
at 50 hPa to 2.2–15 ms-1month-1 at
10 hPa, and the IG wave forcing from 0.5–2.5 to
0.5–6.2 ms-1month-1. The Rossby wave forcing
exhibits an abrupt change between 30 and 10 hPa, and it
reaches 14 ms-1month-1 at 10 hPa in
the W–E phase (see also Fig. ).
X‾∗ depends significantly on the height, so that
it is twice as large at 10 hPa as at 50 hPa in
both phases. This may
reflect an increase in mesoscale gravity wave forcing at
10 hPa in both phases of the QBO. However, it should be
noted that the spread in resolved wave forcing, ADVz, and
X‾∗ at 10 hPa across all reanalyses is
2–3 times larger than that at 30 hPa (not shown),
implying less reliability of the forcing estimates at this
altitude. This result might be due to fewer constraints acting
on the wind and temperature fields near 10 hPa in the
reanalyses, owing to the vertical coverage of radiosonde
observations.
We additionally calculated the wave forcing estimates averaged
over 10∘ N–10∘ S at 30 hPa
(Figs. S1–S3 in the Supplement). The results are generally
similar with those for 5∘ N–5∘ S
(Figs. 2, 3, 6), except that the Kelvin (MRG) wave forcing
is about 31 % (10–70 %) smaller when averaged over
10∘ N–10∘ S.
The same as in Table 2, except at 50 hPa.
E–W
W–E
p-level
model-level
p-level
model-level
Kelvin
2.7–6.8
4.6–9.2
MRG
0.6–1.6
0.6–1.7
0.6–2.3
0.8–2.2
IG
0.5–2.3
1.3–2.5
0.4–2.4
1.4–3.7
Rossby
1.1–5.0
1.3–3.6
0.7–4.0
1.2–3.1
Net-resolved
2.8–8.8
5.4–11
0.9–6.4
2.7–6.2
X‾
3.7–10
2.2–4.3
0.5–17
6.9–13
X‾∗
3.5–8.7
7.7–16