A New Method to Estimate Three-Dimensional Residual-Mean Circulation in the Middle Atmosphere and Its Application to Gravity Wave–Resolving General Circulation Model Data

Kaoru Sato Department of Earth and Planetary Science, University of Tokyo, Tokyo, Japan

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Takenari Kinoshita Department of Earth and Planetary Science, University of Tokyo, Tokyo, Japan

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Kota Okamoto Department of Earth and Planetary Science, University of Tokyo, Tokyo, Japan

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Abstract

A new method is proposed to estimate three-dimensional (3D) material circulation driven by waves based on recently derived formulas by Kinoshita and Sato that are applicable to both Rossby waves and gravity waves. The residual-mean flow is divided into three, that is, balanced flow, unbalanced flow, and Stokes drift. The latter two are wave-induced components estimated from momentum flux divergence and heat flux divergence, respectively. The unbalanced mean flow is equivalent to the zonal-mean flow in the two-dimensional (2D) transformed Eulerian mean (TEM) system. Although these formulas were derived using the “time mean,” the underlying assumption is the separation of spatial or temporal scales between the mean and wave fields. Thus, the formulas can be used for both transient and stationary waves. Considering that the average is inherently needed to remove an oscillatory component of unaveraged quadratic functions, the 3D wave activity flux and wave-induced residual-mean flow are estimated by an extended Hilbert transform. In this case, the scale of mean flow corresponds to the whole scale of the wave packet. Using simulation data from a gravity wave–resolving general circulation model, the 3D structure of the residual-mean circulation in the stratosphere and mesosphere is examined for January and July. The zonal-mean field of the estimated 3D circulation is consistent with the 2D circulation in the TEM system. An important result is that the residual-mean circulation is not zonally uniform in both the stratosphere and mesosphere. This is likely caused by longitudinally dependent wave sources and propagation characteristics. The contribution of planetary waves and gravity waves to these residual-mean flows is discussed.

Current affiliation: National Institute of Information and Communications Technology, Tokyo, Japan.

Corresponding author address: Kaoru Sato, Department of Earth and Planetary Science, University of Tokyo, 7-3-1 Bunkyoku Hongo, Tokyo 113-0033, Japan. E-mail: kaoru@eps.s.u-tokyo.ac.jp

Abstract

A new method is proposed to estimate three-dimensional (3D) material circulation driven by waves based on recently derived formulas by Kinoshita and Sato that are applicable to both Rossby waves and gravity waves. The residual-mean flow is divided into three, that is, balanced flow, unbalanced flow, and Stokes drift. The latter two are wave-induced components estimated from momentum flux divergence and heat flux divergence, respectively. The unbalanced mean flow is equivalent to the zonal-mean flow in the two-dimensional (2D) transformed Eulerian mean (TEM) system. Although these formulas were derived using the “time mean,” the underlying assumption is the separation of spatial or temporal scales between the mean and wave fields. Thus, the formulas can be used for both transient and stationary waves. Considering that the average is inherently needed to remove an oscillatory component of unaveraged quadratic functions, the 3D wave activity flux and wave-induced residual-mean flow are estimated by an extended Hilbert transform. In this case, the scale of mean flow corresponds to the whole scale of the wave packet. Using simulation data from a gravity wave–resolving general circulation model, the 3D structure of the residual-mean circulation in the stratosphere and mesosphere is examined for January and July. The zonal-mean field of the estimated 3D circulation is consistent with the 2D circulation in the TEM system. An important result is that the residual-mean circulation is not zonally uniform in both the stratosphere and mesosphere. This is likely caused by longitudinally dependent wave sources and propagation characteristics. The contribution of planetary waves and gravity waves to these residual-mean flows is discussed.

Current affiliation: National Institute of Information and Communications Technology, Tokyo, Japan.

Corresponding author address: Kaoru Sato, Department of Earth and Planetary Science, University of Tokyo, 7-3-1 Bunkyoku Hongo, Tokyo 113-0033, Japan. E-mail: kaoru@eps.s.u-tokyo.ac.jp

1. Introduction

Material circulation of the middle atmosphere is essentially driven by the momentum deposition of atmospheric waves such as gravity waves and Rossby waves propagating from the troposphere as well as the diabatic heating by radiative processes, while differential latent and sensible heatings are also important for the tropospheric circulation. The circulation in the mesosphere forms one cell with a meridional flow from the high latitudes of the summer hemisphere to the high latitudes of the winter hemisphere around the mesopause. The circulation in the stratosphere is mainly composed of two cells from the tropical region to higher latitudes in the two hemispheres and is called the Brewer–Dobson circulation (hereafter referred to as BDC), named after the two scientists who indicated its existence from ozone and water vapor observations. The breaking and/or dissipation of atmospheric waves do not only cause the momentum deposition but also generate atmospheric turbulence. The geostrophic turbulence associated with Rossby wave breaking is attributable to isentropic irreversible mixing and affects the latitudinal distribution of minor constituents. Thus, it is sometimes considered that the BDC is composed of two elements, that is, the material circulation driven by the waves and radiative forcing and the irreversible mixing by the turbulence.

The transformed Eulerian mean (TEM) formulation was introduced by Andrews and McIntyre (1976) to express the two-dimensional (2D) material circulation as the residual-mean circulation by taking into account the large cancellation between the adiabatic cooling (heating) and the convergence (divergence) of heat flux associated with waves. Dunkerton et al. (1981) showed that the residual-mean circulation approximates well Lagrangian-mean circulation. Through the adiabatic heating/cooling associated with its vertical flow branch, the residual-mean circulation maintains the thermal structure of the middle atmosphere that is far from that expected by radiative balance. The peculiar thermal structure observed in the polar and equatorial regions in the stratosphere and in the polar regions in the mesosphere largely affects the distribution of polar stratospheric clouds in winter and polar mesospheric clouds in summer.

Haynes et al. (1991) proposed the downward control principle using the TEM equations indicating that the zonal-mean streamfunction at a level is determined by vertical integration of the wave forcing above that level in a steady state. As the equation is linear for the wave forcing, this principle is frequently used to diagnose contribution of respective waves to the driving of the BDC and its trend (e.g., Rosenlof 1995; Butchart et al. 2006; Garcia and Randel 2008; Li et al. 2008; Calvo and Garcia 2009; McLandress and Shepherd 2009; Okamoto et al. 2011; Shepherd and McLandress 2011). In particular, the amount of tropical upwelling is used as an index of the troposphere–stratosphere mass exchange associated with the BDC. Butchart et al. (2010) compared 11 chemistry–climate model (CCM) simulations for the twenty-first century in terms of stratospheric climate and circulation. One of the common results from these previous studies using CCMs is that the BDC will have a strengthening trend in response to the climate change of the twenty-first century. According to Butchart et al. (2010), in most models, orographic gravity waves are of similar importance to the resolved waves both in determining the upwelling and its trend. The annual-mean upwelling is attributable to the resolved wave drag by about 67% and to the parameterized orographic gravity wave drag (OGWD) by 30%. The contribution of OGWD to the trend is more important. On average, OGWD explains 59% of the trend in the annual-mean upwelling, although the dependence on the model is large. It is considered that the change of the wave forcing is related to upward movement of the breaking region in the upper flanks of the subtropical jets in association with tropospheric warming induced by increasing greenhouse gases (GHG) (Li et al. 2008; McLandress and Shepherd 2009; Okamoto et al. 2011). It is also worth noting that such a change in BDC can affect the characteristics of the quasi-biennial oscillation (Kawatani and Hamilton 2011).

In addition to the strength, the structural change of the BDC has been investigated, especially in terms of the tropical width in the lower stratosphere. Li et al. (2010) examined trends in the latitudinal width of the upward branch of the BDC in the twenty-first century simulated by a CCM. They showed a narrowing of the upward branch and attributed it to the equatorward shift of Rossby waves’ critical latitudes under the GHG increase. This is in contrast to the widening trend of the latitudinal region, in which the tropical high tropopause is observed over the past few decades, as indicated by Seidel and Randel (2007). Seidel and Randel (2007) showed that the tropical widening is associated with the poleward movement of the subtropical jet.

According to the downward control principle, the vertical flow response of the residual-mean circulation is observed below and around the latitudinal ends of the wave forcing in a steady state (Haynes et al. 1991). The seasonal cycle may not be treated as a steady state and lead to meridional extension of the circulation away from the forcing region (Holton et al. 1995). Okamoto et al. (2011) used CCM data and estimated the residual-mean circulation in December–February directly by its definition and indirectly by using the downward control principle from the Eliassen–Palm flux divergence of resolved waves and parameterized gravity wave drag. The two different estimates for the residual-mean circulation accorded well, suggesting that the steady-state assumption is approximately valid even in the seasonal time scales. Moreover, the principle indicates that meridional flow of the residual-mean circulation should be maintained by nearby wave forcing. Norton (2006) discussed the importance of equatorial Rossby waves generated by tropical heating in the troposphere for the momentum budget to cause the upwelling in the tropical and subtropical regions. Okamoto et al. (2011) indicated by applying a diagnostic method based on the downward control principle to CCM data and reanalysis data that the summer hemispheric part of the winter circulation in the stratosphere is driven by the subgrid-scale gravity waves. The gravity waves are probably convectively generated in the summer subtropical region (Sato et al. 2009b). Seviour et al. (2012) examined upward mass flux at 70 hPa using Interim European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-Interim) data for 1989–2009 and showed that the sum of contributions by resolved waves and parameterized orographic gravity waves is 74%, suggesting shortage of orographic and/or nonorographic gravity wave forcing there. On the other hand, Ueyama and Wallace (2010) used temperature data from satellite observation as an index of the vertical flow of the BDC and examined its relation with eddy heat flux in the high-latitude region. Their results suggest significant correlation between the high-latitude wave forcing and the topical upwelling. They argued that an accumulation of transient responses may explain the broad response in the seasonal time scale. Thus, the role of respective kinds of waves in the formation of BDC is still controversial.

Moreover, recent studies show that the BDC is composed of two branches: one is a shallow branch that roughly exhibits hemispheric symmetry in the lowermost stratosphere, and the other is a deep branch observed mainly in the winter hemisphere (e.g., Birner and Bönisch 2011). According to Birner and Bönisch (2011), the transport in the lowermost stratosphere is made by this shallow branch and by isentropic irreversible mixing. The deep branch is slow (time scale of several months to years) and the shallow branch is fast (time scales of days to a few months). The shallow branch is mainly driven by synoptic-scale waves and partly by gravity waves (Plumb 2002; Miyazaki et al. 2010), while the deep branch is mainly by planetary waves (Plumb 2002) and partly by gravity waves (Okamoto et al. 2011).

So far, the BDC has been examined mainly in the 2D meridional cross section. However, there are several studies indicating that the BDC has zonally asymmetric structures. Callaghan and Salby (2002) made a pioneering study to examine three-dimensional (3D) structure of the BDC using an isentropic vertical coordinate. They showed that the cross-isentropic flow is not zonally symmetric around a strongly perturbed polar vortex in NH winter. Hitchman and Rogal (2010) indicated the importance of regional outflow of the tropical convection in Southeast Asia for the formation and maintenance of the column ozone maximum situated to the south of Australia. The outflow reinforces the westerly jet by angular momentum transport and subsequently increases synoptic-scale wave activity embedded in the jet. Sato et al. (2009a) showed by using satellite observations that the stratospheric ozone recovery observed in late spring and summer in the Antarctic strongly depends on the longitude. Lin et al. (2009) used data from the satelliteborne Microwave Sounding Unit (MSU) in 1979–2007 and simulation data from a coupled atmosphere–ocean general circulation model (GCM) to examine a horizontal trend pattern of the temperature in SH winter and spring. They showed that the regional dependence of temperature trend is related to those of both column ozone and eddy heat flux. Randel et al. (2010) indicated the importance of an upward flow on the eastern side of the anticyclonic circulation of the Asian monsoon for the transport of hydrogen cyanide (HCN) into the stratosphere. Convection in the Asian and African monsoon regions is also regarded as a strong source of the gravity waves propagating into the upper stratosphere and mesosphere, which may drive the zonally asymmetric BDC (Sato et al. 2009b). These studies suggest that the BDC likely has significant 3D structure that has not been explored yet.

Recently, Kinoshita and Sato (2013a) derived 3D TEM equations, including 3D residual-mean flow and 3D wave activity flux in the primitive-equation system. The “residual-mean flow” in the equations was obtained as the sum of the 3D time-mean flow and 3D Stokes drift associated with waves. Thus, the derived residual-mean flow is regarded as an approximation of Lagrangian-mean flow. The 3D wave activity flux was obtained so that its divergence corresponds to the wave forcing of the mean flow in the horizontal momentum equations. This formulation by Kinoshita and Sato (2013a) was made without using any dispersion relations and hence is applicable both to Rossby waves and to gravity waves. In the present study, it is shown that this wave activity flux is written as momentum flux using Lagrangian wind perturbations. Moreover, Kinoshita and Sato (2013b) derived another form of 3D wave activity flux describing propagation of the wave packet and discussed its relation to the 3D wave activity flux that appears in the 3D TEM equations obtained by Kinoshita and Sato (2013a). The formulation by Kinoshita and Sato (2013b) uses a unified dispersion relation for Rossby waves and gravity waves that was newly derived.

A problem with the formulas by Kinoshita and Sato (2013a,b) is that the time mean is used instead of the zonal mean. Thus, at a glance, stationary waves cannot be treated. However, the formulation by Kinoshita and Sato (2013a,b) is valid if we can assume that the temporal and/or spatial scales of the mean (more precisely speaking, background) field are much longer than those of the perturbation field. In other words, their formulation can be applied for any wave, including stationary waves, by taking an appropriate mean field. Moreover, taking it into consideration that the average for the flux calculation is inherently needed to remove an oscillatory component of unaveraged quadratic functions on a scale of one-half the wavelength of the wave field, the averaging problem can be overcome by using an extended method of the Hilbert transform, which is introduced in the present paper. The extended Hilbert transform is used to estimate the envelope function of momentum or energy fluxes of the wave field as a substitute of the temporal or spatial “mean.” The present paper describes a new method to examine the 3D material circulation using the formulas by Kinoshita and Sato (2013a) and using the extended Hilbert transform. As an example, 3D structure of the residual-mean circulation in the middle and upper stratosphere and mesosphere is examined, utilizing simulation data from a gravity wave–resolving GCM.

A brief description of the high-resolution GCM data used in the present study is given in section 2. Theoretical consideration of 2D residual-mean circulation and its application to the GCM data are made in section 3. Some results of the 2D analysis in section 3 are not very new but are given because they provide reference materials to lead and validate the theory of 3D residual-mean circulation proposed by the present paper. The 3D theory is given in section 4. The treatment of stationary waves in three dimensions using an extended Hilbert transform is also described. Results of the 3D analysis using the GCM data are shown in section 5. Summary and concluding remarks are made in section 6.

2. Short description of gravity wave–resolving GCM data

Utilized data for the analysis are outputs from the T213L256 GCM developed by Watanabe et al. (2008) (the KANTO model), which covers a height region up to 85 km in the upper mesosphere with horizontal resolution of about 60 km and vertical grid spacings of about 300 m above a height of 10 km. No gravity wave parameterizations were included in this model. Thus, all gravity waves are spontaneously generated. The characteristics of simulated gravity waves depend on artificial diffusion and cumulus parameterizations. The set of tuning parameters of the parameterizations was carefully chosen by conducting several sensitivity tests to obtain gravity wave amplitudes in the lower stratosphere that are comparable to radiosonde observations over the central Pacific in the latitudinal range of 28°N–48°S (Sato et al. 2003). The time integration was made using the Earth Simulator over three model years in which climatology with realistic seasonal variation was specified for the sea surface temperature and stratospheric ozone. Physical quantities were sampled at a short time interval of 1 h. The model succeeded in simulating zonal-mean zonal wind and temperature fields, which are consistent with observations, suggesting that the momentum budget including gravity waves is realistic. Watanabe et al. (2008) illustrated an overview of the model performance including the momentum budget. As described by a series of our previous papers using the model data (e.g., Sato et al. 2009b; Kawatani et al. 2010; Sato et al. 2012), overall characteristics of the simulated gravity waves are realistic. The present study examined contribution of gravity waves, synoptic-scale waves, and stationary and transient planetary waves to the residual-mean circulation. Following our previous studies (Sato et al. 2009b, 2012), small horizontal-scale fluctuations with total wavenumber n ≥ 21 (horizontal wavelengths shorter than 1800 km) are designated as gravity waves (GWs). The components with zonal wavenumbers of s = 1–3 and s = 4–20 are examined as planetary waves (PWs) and synoptic-scale waves (SWs), respectively. Monthly mean PW components and remaining PW components are analyzed as stationary PWs and transient PWs, respectively.

3. Two-dimensional residual-mean circulation

a. Theory on the relation between 2D circulation and E-P flux

The residual-mean circulation for the 2D TEM system is composed of two parts: one is ageostrophic wind and the other is Stokes drift. First, it is shown how each part of the residual-mean circulation is related to respective terms of Eliassen–Palm (E-P) flux divergence.

By including Coriolis effect, the Lagrangian perturbation of zonal wind uL is expressed as
e1
where the square brackets denote the zonal mean; the primes denote the perturbation from the zonal mean; η′ and ζ′ are latitudinal and vertical displacements, respectively; and f is Coriolis parameter.1 This formula expresses that uL is the sum of Eulerian zonal wind perturbation u′ and the latitudinal and vertical advection of angular momentum by the perturbation ([u]yf)η′ + [u]zζ′. Using the relation2
e2
latitudinal and vertical fluxes of zonal momentum are derived as
e3
e4
and
e5
e6
where υ and w are latitudinal and vertical wind components, respectively; φ is geopotential; ρ0 is basic state density; and N is the Brunt–Väisälä frequency. Here, the terms [ηυ′] and [ζw′] are ignored.3 Note that the subscripts 1 and 2 denote momentum fluxes and heat fluxes, respectively: Y1(≡ρ0[uυ′]) and Z1(≡ρ0[uw′]) are momentum fluxes and and are proportional to heat fluxes. It is important that (4) and (6) are equivalent to the formulas of y and z components of E-P flux, namely
e7
Thus, the E-P flux is considered to be the flux of “Lagrangian” zonal momentum. The zonal momentum equation in the 2D TEM system is
e8
where
e9
e10
and [X] is the other nonconservative mechanical forcing and friction (e.g., Andrews et al. 1987).
Comparison between these formulas of residual-mean circulation [(9) and (10)] and the divergence of E-P flux [(7)] indicates that the Stokes drift ([υ]S and [w]S) is related to the divergence of heat fluxes:
e11
Subtraction of (11) from (8) yields the Eulerian zonal-mean zonal momentum equation:
e12
If it can be assumed that the mean flow is steady and [X] is negligible, this equation indicates that the ageostrophic flow ([υ]a and [w]a) is related to the divergence of momentum fluxes:
e13
Moreover, when [u]z is small compared with the other terms, the meridional component of ageostrophic flow is approximately written using momentum fluxes as
e14
where
e15
Validity of the assumption of small [u]z in the middle atmosphere will be discussed in section 5 using the high-resolution GCM data. This equation indicates that if atmospheric waves with negligibly small heat flux are dominant, the residual-mean flow can be estimated only using momentum fluxes. This may be the case for the mesosphere where the gravity wave forcing is dominant. It is worth noting that the estimation of vertical flux of horizontal momentum is possible using mesosphere–stratosphere–troposphere (MST) radar observations, while those of heat flux are generally difficult.
The Stokes drift [υ]S is written using the terms included in the E-P flux similar to [υ]a:
e16
This equation describes the relation between [υ]S and the terms of the E-P flux, although [υ]S is directly calculated by (9). The important part of the argument in this section is that respective wave contributions to [υ]a are estimated by (14) and those to [υ]S are directly calculated by (9).

b. Results of 2D analysis using gravity wave–resolving GCM data

Figure 1 shows meridional cross sections of the E-P flux vector calculated using (7) by arrows and its divergence by colors for all waves (Figs. 1a,e), PWs (Figs. 1b,f), GWs (Figs. 1c,g), and SWs (Figs. 1d,h) in July (top row) and January (bottom row) of the second model year. Note that GWs are resolved waves in the high-resolution GCM used in the present study. The distributions in the respective sections for the 2 months are roughly mirror images of each other.

Fig. 1.
Fig. 1.

Meridional cross sections of E-P flux vector and its divergence (colors) for (a),(e) all resolved waves (EPFD), (b),(f) planetary waves (PWD), (c),(g) gravity waves (GWD), and (d),(h) synoptic-scale waves (SWD) averaged in (top) July and (bottom) January of the second year. Contours of zonal-mean zonal winds are superimposed on all panels. Contour intervals are every 20 m s−1. Dashed contours show negative values.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1

In the winter mesosphere above about 50 km, the net E-P flux divergence (i.e., contribution by all waves) is negative in most regions of both hemispheres, which is mainly contributed to by GWs, except for the lower part of subtropical westerly region around 60 km. PWs also contribute to the negative E-P flux divergence in most latitudes, except around 60°, depending on the altitude. The positive E-P flux divergence of PWs around 60° is related to the generation of eastward 4-day waves (Watanabe et al. 2009, and references therein). The contribution of SWs is relatively small but negative in the winter hemisphere, except for the region around the jet core. In the summer mesosphere, the net E-P flux divergence is positive and mainly explained by the GW contribution and partly by SW and PW contributions. Such dominant GW contribution to the mesospheric momentum budget is consistent with previous theoretical studies (Lindzen 1981; Matsuno 1982; Holton 1982).

In the lower stratosphere of about 15–20 km, the net E-P flux divergence is negative in most latitude regions. Contribution by PWs is widely distributed and dominant in the winter middle and high latitudes. Contributions of GWs and SWs are also large in midlatitudes of both hemispheres in both months. This significant contribution of GWs in this region was also indicated by Miyazaki et al. (2010) and Okamoto et al. (2011).

In the middle and upper stratosphere of about 25–50 km, the net E-P flux divergence is negative in most latitudes of the winter hemisphere and positive in low latitudes of the summer hemisphere. The negative divergence in the winter hemisphere is mainly due to PWs, as is consistent with previous studies (Plumb 2002, and references therein). The positive divergence in the low latitudes of the summer hemisphere is due to GWs. This divergence forms the summer hemispheric part of the winter circulation (Okamoto et al. 2011).

An interesting point is that the E-P flux divergence associated with GWs is positive in the lower-latitude part of the westerly jet in the winter hemisphere. This feature means that the GWs accelerate the westerly wind in that region. It is also interesting that there is positive E-P flux divergence around 40 km, slightly below the center of the westerly jet in the Southern Hemisphere in July. This positive divergence is due to PWs and partly canceled by SWs. Similar positive divergence below the westerly jet of the winter hemisphere is observed in some other months (not shown), although it is not evident in January (Fig. 1e). The mechanism causing this positive divergence is interesting, but we leave it for future studies.

Because the purpose of the present study is mainly to demonstrate the usefulness of the new method to examine 3D material circulation in the atmosphere, further analysis and discussion is focused on the circulation of the middle and upper stratosphere (i.e., the deep branch of BDC) and mesosphere in July and January.

Figure 2a shows the meridional cross section of [υ]* in July that is directly calculated by (9). Figure 2i is also [υ]*, but estimated from the divergence of the E-P flux using (14) for [υ]a in (9) under the assumption that the mean wind is steady and the vertical shear of the mean wind is negligible. Overall distributions of [υ]* in Figs. 2a and 2i are similar, assuring the validity of the assumption. Slight difference observed particularly in the summer stratosphere is mainly due to vertical advection of the mean wind by the residual-mean flow, as shown in section 5.

Fig. 2.
Fig. 2.

Meridional cross sections of (a) the meridional component of the residual-mean flow ([υ]*), contributions of (b) PWs and (c) GWs that are estimated using the downward control principle (DCP), (d) zonal-mean meridional velocity ([υ]a), its contribution (e) by PW and (f) by GW , (g) meridional component of the Stokes drift ([υ]S), (h) its contribution by PWs, and (i) [υ]* estimated using EPFD due to all waves by DCP that are averaged in July. Contours show zonal-mean zonal winds with an interval of 20 m s−1.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1

The distribution of [υ]a and [υ]S are shown in Figs. 2d and 2g, respectively. In the middle and upper stratosphere (~25–50 km), except around 40 km, southward flow is dominant in [υ]*, which extends from low latitudes of the summer hemisphere to high latitudes of the winter hemisphere (Fig. 2a). This flow is mainly due to [υ]a in low and middle latitudes of both hemispheres and [υ]S in high latitudes of the winter hemisphere. The negative [υ]S in high latitudes of the winter hemisphere is largely canceled by [υ]a, which is consistent with the previous studies (e.g., Dunkerton 1978).

In the mesosphere, [υ]* extends over all latitudes in the winter and summer hemispheres (Fig. 2a). This flow is primarily due to [υ]a and partly due to [υ]S in middle latitudes above about 60 km and all of the latitudes below about 60 km in the winter hemisphere. The transition between the stratospheric circulation and mesospheric one is continuous, which is consistent with the schematic view shown by Dunkerton (1978).

Figures 2b and 2c show contributions by PWs and GWs to [υ]*, respectively. Figures 2e, 2f, and 2h show three dominant components of [υ]*, namely, by PWs, by GWs, and by PWs. It is clear that [υ]a in the winter hemisphere (Fig. 2d) is comparable to by PWs below 60 km (Fig. 2e) and by GWs above (Fig. 2f), while [υ]a in the summer hemisphere is comparable to by GWs (Fig. 2f). The Stokes drift [υ]S (Fig. 2g) is well explained by by PWs (Fig. 2h) in all latitude and height regions.

From these analyses, it is concluded that the strong southward flow [υ]* in the middle and upper stratosphere (i.e., a deep branch of the BDC) and lower mesosphere below about 60 km is roughly divided into three dominant contributions: [υ]a induced by GWs in the summer low latitudes, [υ]a induced by PWs in the winter low and middle altitudes, and [υ]S by PWs in the winter high latitudes that is partly canceled by the PW-induced [υ]a. Moreover, in the middle and upper mesosphere above about 60 km, [υ]* is mainly contributed to by [υ]a due to GWs and partly by [υ]S due to PWs in middle latitudes of the winter hemisphere. It is worth noting here that [υ]* by PWs in the mesosphere has interesting structure around 60°S, that is, positive around 65 km and negative around 50 km. This structure is likely due to 4-day waves generated by in situ baroclinic/barotropic instability (Watanabe et al. 2009). This fact indicates that the baroclinic/barotropic instability in the winter hemisphere contributes at least partly to the residual-mean circulation of the mesosphere.

Figure 3 is as in Fig. 2, but for the vertical component of the residual-mean flow obtained using the continuity equation. Similarity in the distribution in the meridional cross section is also observed between directly calculated [w]* (Fig. 3a) and [w]* estimated from the E-P flux divergence (Fig. 3i).

Fig. 3.
Fig. 3.

As in Fig. 2, but for the vertical component.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1

In the mesosphere, [w]* is primarily downward in the winter hemisphere, while it is generally upward in the summer hemisphere (Fig. 3a). A characteristic upward flow is also observed in 20°–50°S in a height region of 50–70 km. These dominant downward and upward residual-mean flows are mainly contributed to by GWs . Contribution of PWs is relatively weak, but it is upward and downward in higher and lower latitudes than 60°S in the winter hemisphere, respectively. In the equatorial region, a secondary circulation is observed in association with the semiannual oscillation, namely, upward (downward) flow in the easterly (westerly) shear region.

In the middle and upper stratosphere, [w]* is generally downward in the winter hemisphere and upward in the summer hemisphere (Fig. 3a), although [w]* is weak in middle and high latitudes of the summer hemisphere. The upward flow in the summer hemisphere is mainly due to by GWs. The downward flow in latitudes lower than about 50°S of the winter hemisphere is due to by PWs, which is largely canceled by of PWs and by by GWs. The downward flow in latitudes higher than about 50°S is mainly due to by PWs and to by GWs, which is largely canceled by by PWs. It is interesting that the contributions by respective waves to [w]* are different than those to [υ]*, although it is understood from the downward control principle. The meridional cross sections of [υ]* and [w]* and their components in January were roughly mirror images of those in July, although their details are not shown here.

4. Theory of three-dimensional residual-mean circulation

a. Relation between 3D residual-mean flow and 3D wave activity flux

Recently, Kinoshita and Sato (2013a) derived 3D transformed Eulerian mean equations, including 3D Stokes drift and 3D wave activity flux, which are applicable both to Rossby waves and gravity waves. The zonal and meridional momentum equations are written as
e17
e18
where
e19
F1 and F2 are the wave activity fluxes as defined below, overbars and primes denote mean and deviation from the mean, respectively, is the residual-mean flow
e20
and the 3D Stokes drift is written as
e21
e22
e23
Equations (21)(23) are derived without using any dispersion relation. This means that these formulas are applicable both to Rossby waves and to gravity waves.
The wave activity fluxes F1 = (F11, F12, F13) and F2 = (F21, F22, F23) are written as follows:
e24
e25
e26
e27
e28
e29
Note that Y11, Y13, Z11, and Z13 respectively correspond to Y1, Y2, Z1, and Z2 in the 2D TEM equations. It is also worth noting that the residual-mean flow in (17) and (18) was derived as the sum of mean flow and Stokes drift [(21)(23)] in Kinoshita and Sato (2013a), and hence, approximately expresses Lagrangian-mean flow. Formulas of the 3D wave activity flux [(24)(29)] were derived as an additional term having forms of the horizontal momentum equations.
Similar to the 2D theory, it is shown that these fluxes are related to covariance of Lagrangian wind perturbations (uL and υL) and wind fluctuations [υ′ ≡ (u′, υ′, w′)]:
e30
e31
where
e32
e33
This point is not explicitly described in Kinoshita and Sato (2013a).

Here it should be emphasized that these 3D TEM equations are derived only assuming that the temporal or spatial scales of the mean and perturbation fields are separable, although Kinoshita and Sato (2013a) supposed the time mean. More specifically speaking, the mean field is slow (large scale) field and the perturbation field is fast (small scale) field when we consider temporal (spatial) scales. Thus, the 3D momentum equations [(17) and (18)] include time and spatial derivatives for mean-field scales. See Kinoshita and Sato (2013a) for details of the derivation. In summary, the derived 3D formulas hold for any mean if the mean field is distinguished from the perturbation field by their scales. This point is important to estimate the contribution of stationary waves to the residual-mean flow as discussed later.

The residual-mean flow is composed of three terms:
e34
where [] is balanced mean flow (such a flow that satisfies a balance of forces including pressure gradient force), [] is unbalanced mean flow, and [] is Stokes drift.
A simplest balanced mean flow is the geostrophic flow:
e35
e36
For strong flow, we may need to consider the gradient wind balance (Randel 1987).
The unbalanced mean flow is defined as the departure of the mean flow from the balanced mean flow:
e37
The unbalanced mean flow is equivalent to the ageostrophic flow [υ]a in 2D TEM equations.
Similar to the 2D theory discussed in section 3, respective correspondences can be considered between and the divergence of the 3D wave activity flux. First, it is seen from comparison between (21)(23) and (24)(29) that the sum of the mean flow advection by Stokes drift and Coriolis acceleration associated with Stokes drift exactly equals the divergences of heat flux and in the following:
e38
e39
These equations are those to be compared to (11) in the 2D TEM system.
The relation between the unbalanced mean flow and 3D wave activity flux divergence is not as simple as for the 2D theory, but it depends on the definition of the balanced flow. However, by analogy with the 2D theory, the unbalanced mean flow induced by the wave forcing is defined so as to satisfy the following relation when the mean flow is approximately steady and and are negligible:
e40
e41
If we can assume that the terms proportional to , , , , , and fy are small, the 3D unbalanced mean flow is approximately obtained from the wave activity flux:
e42
e43
where
e44
The vertical wind component of the unbalanced mean flow is estimated by the continuity equation:
e45
The validity of the definition of and the assumptions in (42) and (43) can be confirmed by accordance of the zonal mean of estimated 3D flows with the directly calculated zonal-mean 2D flows using the data, as will be shown in section 5. An important point is that the terms on the right-hand sides of (21)(23) and (42)(43) are written with the wave fluxes only. This means that the contributions of respective waves to the 3D residual-mean flows are separately estimated. Hereafter, the wave contribution to the residual-mean flow, that is, the sum of and , is referred to as unbalanced residual-mean flow :
e46

Last but not least, it should be emphasized that there is a role of zonally symmetric fluctuations in the 3D TEM system, although it is treated as the mean field in the 2D TEM system. For example, gravity waves with horizontal-wavenumber vectors pointing meridionally have zonally symmetric but meridionally fluctuating structure. Such gravity waves have significant values of , , and . The divergence of these wave fluxes appears in the mean meridional momentum equation [see (18)] and can cause . Several previous studies (e.g., Lieberman 1999; Miyahara et al. 2000) suggested the importance of such wave activity flux divergence in the meridional momentum equation in the mesosphere and lower thermosphere.

b. Treatment of stationary waves in 3D analysis using an extended Hilbert transform

As already mentioned, when the time mean is used for an average, the 3D residual-mean circulation and wave activity flux cannot be calculated for stationary waves. However, the average is inherently needed for smoothing out an oscillatory component of unaveraged quadratic functions. We propose therefore to use an extended Hilbert transform, which is newly introduced in the present study, for the smoothing.

Hilbert transform is a procedure to obtain an envelope function of a particular wave packet. We extend this procedure to obtain the wave activity flux and Stokes drift whose temporal and/or spatial structure is comparable to the whole scale of the wave packet. In other words, the scale of the mean field is taken as that of the background field, which the “wave packet” interacts with.

The Hilbert transform H[a(t)] of a particular time series a(t) is the time series that is composed of Fourier components of a(t) whose phases are shifted by −π/2 radians, namely,
e47
e48
where ω is the ground-based frequency and ϕω is an arbitrary phase (e.g., Bracewell 1999).
An extended Hilbert transform H[a(x, t)], hereafter referred to as e-HT, of a particular fluctuation field a(x, t) is defined as an arbitrary fluctuation field composed of Fourier components of a(x, t) whose phases are shifted by −π/2 radians:
e49
e50
where k ≡ (k, l, m) is a wavenumber vector; k, l, m are zonal, meridional, and vertical wavenumbers, respectively; and ϕk,ω is an arbitrary phase. An analytic representation of the real function a(x, t) is defined as a complex function A(x, t) {≡a(x, t) + iH[a(x, t)]}. The envelope function Aenv(x, t) of a(x, t) is obtained by using A(x, t):
e51
where A*(x, t) denotes the complex conjugate of A(x, t). This corresponds to an average of a(x, t)2:
e52
where angle brackets mean an average with a scale expressing the overall structure of the wave packet. Similarly, flux quadratics 〈a(x, t)b(x, t)〉 are obtained as
e53
Note again that in this method, the envelope scale is roughly regarded as that of the background field with which the wave packet interacts.

Examples of the estimation of the envelope function using the e-HT are illustrated in Fig. 4. Figure 4a shows a fluctuation field of a particular quantity forming two wave packets. Figures 4b and 4c show the results of the envelope function estimation by applying the e-HT in the x and y directions, respectively. It is clear that the envelope function of the wave packet is successfully obtained with the e-HT in the x direction, while this is not the case for the estimate with the e-HT in the y direction. The failure of the estimate in the y direction is attributable to too few wave crests (less than one) in that direction. In other words, the wave packets cannot be distinguished from the background field in the y direction. Thus, the e-HT should be made in such a direction that the waves can be distinguished from the mean field. For example, quasi-stationary waves are hardly distinguished from the time-mean field, but they can be distinguished from the zonal-mean field. Thus, quasi-stationary waves are extracted as deviation from the zonal mean, and the e-HT should be applied in the zonal direction. In general, the e-HT should be taken in time or spatial direction in which the waves are fluctuating. When waves are fluctuating in more than two directions, the envelope function can be estimated taking the e-HT in only one of the directions, because what we need is to make the phase shift by −π/2 radians.

Fig. 4.
Fig. 4.

An illustration of the estimation method of envelope function using the e-HT. (a) The fluctuation field forming two wave packets. Estimates of Aenv(x, t) using e-HT in the (b) x and (c) y directions. (d) Application of the e-HT to stationary waves. The solid red curve shows a longitudinal (x) profile of a particular quantity a(x, t) that is composed of s = 1, 2, 3 wave components (red dashed curves). H[a(x, t)] in the x direction is shown by the blue curve. Aenv(x, t) is shown by the black curve.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1

Figure 4d illustrates an example of application of the e-HT to stationary waves. The solid red curve shows a longitudinal profile of a particular quantity a(x) that is composed of s = 1, 2, 3 components (red dashed curves). The extended Hilbert transform of a(x) in the x direction is shown by the blue curve. The envelope function Aenv(x) is obtained using (52), as denoted by the thick black curve. It is clear that Aenv(x) describes the longitudinal structure of the planetary wave packet.

It is important that the e-HT can be obtained also for transient waves. Therefore, using the e-HT, it is possible to estimate the wave activity flux and the 3D residual-mean flow using (53) for any wave packet. In the present study, this method using the e-HT is applied to estimate the residual-mean flows associated with GWs and PWs, including both stationary and transient components.

5. Results of the 3D analysis using gravity wave–resolving GCM data

As seen from the results of the 2D analysis in section 3, dominant waves contributing to the residual-mean flow in the middle atmosphere are PWs and GWs. Thus, in this section, we examine contributions of three kinds of wave fields, namely, “all” waves defined as the departure from the zonal mean, PWs having s = 1–3, and GWs having n ≥ 21 using the derived formulas. Moreover, a monthly mean PW field and the deviation from the monthly mean are analyzed as stationary and transient PW components, respectively. These definitions of PWs and GWs are the same as for the 2D analysis. Monthly mean zonal-mean zonal wind and Brunt–Väisälä frequency are used as and N, respectively, for the wave flux calculation. This way to take the mean fields may not be appropriate for the estimation for GWs when PWs having large amplitudes are present. In such a case, should be defined locally. However, in the case of the present study, the difference between the estimates using zonal-mean and locally defined was quite small (not shown). The unbalanced residual-mean flow in the 3D space was obtained using the e-HT (53) by a phase shift in the x direction for the three kinds of wave fields at each time. The Stokes drift was obtained using (21)(23), and the unbalanced mean flow was estimated using (42), (43), and (45). Furthermore, and obtained for respective wave fields were averaged over a month to examine the residual-mean flow for each month.

First, in order to confirm the validity of the 3D analysis method, consistency with the result of the 2D TEM analysis is examined. The strictest comparison may be for the unbalanced mean flow because the formulas were derived under the largest number of assumptions for the mean wind unlike Stokes drift. Figures 5a and 5b show the meridional cross sections of the zonal-mean and [υ]a that are estimated from the momentum fluxes associated with all waves using (43) and (14), respectively. Note that the flow in Fig. 5a includes contribution of stationary waves that is estimated using the e-HT.

Fig. 5.
Fig. 5.

Meridional cross sections of zonal-mean unbalanced mean meridional flow calculated from (a) 3D momentum flux divergence , (b) 2D momentum flux divergence [υ]a, (c) 2D momentum flux divergence plus vertical advection of zonal-mean zonal wind, (d) vertical advection of zonal mean wind, and (e) tendency of zonal-mean zonal wind in July of the second year. Contours show zonal-mean zonal winds with an interval of 20 m s−1.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1

It is important that the zonal-mean (Fig. 5a) and [υ]a (Fig. 5b) agree quite well in terms of the distribution and magnitude, although they are not exactly the same. Moreover, the distributions of the unbalanced mean meridional flow shown in Figs. 5a and 5b accord well with the meridional cross section of [υ]a that was directly estimated (Fig. 2d), indicating that underlying assumptions for the mean wind are approximately valid. The slight difference is observed in the lower part of easterly jet in the summer hemisphere. This mainly comes from the vertical advection of the mean wind (Fig. 5d). The mean zonal wind tendency term is not significant (Fig. 5e). Figure 5c shows the meridional cross section of estimated [υ]a, including the correction term of [see (13)]. The agreement becomes better. These results support that the method to estimate 3D unbalanced mean flow from momentum flux divergence is appropriate. Note that the zonal mean of is exactly and analytically equal to [υ]S, if the wave fluxes for the same wave components are taken into account [see (9) and (22)].

a. The 3D unbalanced residual-mean flow and contribution by PWs and GWs at 10 hPa in July

The maps of the meridional components of , , and at 10 hPa (~32 km) for all waves, PWs, and GWs are shown in Fig. 6 for July of the second model year. Note that the color scale is different for the maps for GW from those for all waves and PWs. The flows shown in Fig. 6 include the contribution of stationary waves estimated using the e-HT. The most important feature is that these flows, and hence, the Brewer–Dobson circulation, are not zonally uniform.

Fig. 6.
Fig. 6.

Horizontal maps of (a)–(c) wave-induced 3D residual-mean flow, (d)–(f) unbalanced mean flow, and (g)–(i) Stokes drift due to (left) all waves, (middle) PWs, and (right) GWs at 10 hPa in July of the second year (colors). Contours show monthly mean geopotential heights with an interval of 0.5 × 103 m. Note that color scales are different between the panels of all waves and PWs and those for GWs.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1

In the NH, negative for all waves (Fig. 6a) is dominant in the Indian and African monsoon regions and mainly contributed to by due to GWs (Fig. 6f). It is also interesting that weak positive is observed in SH low latitudes for GWs. This is due to the eastward GW force, as already indicated in section 3b. It is worth noting here that weak by GWs is consistent with theoretical characteristics of linear inertia–gravity waves: the momentum fluxes (Z11) are larger than heat fluxes (Z13) (e.g., Sato et al. 1997).

Another interesting feature observed in by GWs is significantly negative and positive values to the west and east of the southern Andes, respectively. This is explained by the fact that zonal wind variances associated with topographically forced GWs are confined over the mountains, although a part of GW energy propagates leeward by the mean wind perpendicular to the wavenumber vector (e.g., Preusse et al. 2002; Sato et al. 2012). This feature is also indicated by Kinoshita and Sato (2013a).

In the SH, the three kinds of flows , , and by all waves are dominated by PWs. Unbalanced mean flow υa is mainly observed in the Western Hemisphere. It is negative in latitudes lower than about 40°S and positive in higher latitudes. On the other hand, negative is longitudinally widely distributed. The positive is canceled by the negative in the higher latitudes of the Western Hemisphere. However, because the dominant regions of positive and negative are slightly different, weak equatorward is observed in a part of the Western Hemisphere even in the winter hemisphere. The residual-mean flow by waves in the Eastern Hemisphere is negative and mainly explained by .

b. Contribution of stationary and transient waves in three dimensions in July

Further examination was made by dividing PWs into stationary and transient wave components. The results of the 3D analysis for 10 hPa are shown in Fig. 7. It is clear that the dominant in the SH western longitudes seen in Fig. 6e is mainly explained by stationary PWs. The Stokes drift by stationary PW is also dominant in the Western Hemisphere. Contours in Fig. 7 show time-mean geopotential height. The meandering contours indicate that the amplitude of stationary PWs is certainly large in the Western Hemisphere. On the other hand, the distribution of and by transient PWs is more zonally uniform compared with that by stationary PWs.

Fig. 7.
Fig. 7.

As in Fig. 6, but for (a),(c),(f) stationary and (b),(d),(f) transient PWs.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1

c. The 3D unbalanced residual-mean flow at 10 hPa in January

Figure 8 shows by all waves, by PWs, and by GWs in January of the second model year at 10 hPa. The unbalanced meridional mean flow is again not zonally uniform. Strong northward flow in NH is observed over the Pacific and in North America in low and middle latitudes and in Eurasia in middle and high latitudes. This flow is mainly due to PWs, although the distributions of by all waves and by PWs do not accord as well as for those in July. The GW contribution to in NH is small but has negative values in the western Pacific and eastern North America to the western Atlantic in middle latitudes. These regions correspond to storm tracks. Thus, GWs contributing to the negative likely originate from jet–front systems. In the SH, there is systematic northward flow by GWs in low latitudes, although total is weak. Roughly speaking, the longitudinal distribution is uniform, but values are slightly enhanced to the east of continents, that is, Africa, Australia, and South America. This suggests a possible role of GWs that are generated by convection associated with the monsoon in these regions.

Fig. 8.
Fig. 8.

As in Fig. 6, but for (a) wave-induced 3D residual-mean flow and contributions by (b) PWs and (c) GWs at 10 hPa in January of the second year.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1

d. The 3D unbalanced residual-mean flow at 0.05 hPa in July and January

Results of the 3D analysis in the mesosphere are shown in Fig. 9 for by all waves, by PWs, and by GWs in July and January of the second model year at 0.05 hPa (~70 km). Note that color scales are the same for all maps for all waves, PWs, and GWs, unlike the maps at 10 hPa. In July (January), is generally southward (northward) in the entire region at this level. It is clear that GW contribution is dominant, although PW contribution is also large in low latitudes of the winter hemisphere in both months.

Fig. 9.
Fig. 9.

As in Fig. 6, but for (a),(d) wave-induced 3D residual-mean flow and contributions by (b),(e) PWs and (c),(f) GWs at 0.05 hPa in (top) July and (bottom) January of the second year. Color scales are different than those of Fig. 6.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1

The distribution of is not zonally uniform. In the summer hemisphere, due to GWs is large in several subtropical regions, that is, 80°–160°E and 80°–60°W in the NH in July and 20°–80°E, 140°E–150°W, and 60°–10°W in the SH in January. On the other hand, the mean wind is primarily zonally uniform as seen from zonally elongated geopotential contours in Fig. 9. This feature is common for the stratosphere and lower mesosphere (not shown). This is because planetary waves originating from the troposphere hardly propagate into the easterly wind in the middle atmosphere in summer (the Charney–Drazin theorem; Andrews et al. 1987). Thus, it is likely that the longitudinal variation of is reflected by GW sources. In fact, these strong regions are located to the east of monsoon region. The longitudinal difference between the dominant regions of by GWs and the monsoon regions at 0.05 hPa is larger than that at 10 hPa, suggesting eastward propagation of the GW packets from the source. In the winter hemisphere, the distribution of is complicated. Vertical filtering of GW in the background field that is modified by larger-scale waves and nonzonal GW source distribution are likely mechanisms (Smith 2003; Sato et al. 2009b).

Such nonzonal distribution of by GW drag may be a source of PWs, which have large amplitudes in the upper mesosphere and lower thermosphere (Smith 2003). It is worth noting here again that by PWs is at least partly related to baroclinic/barotropic instability in which unstable fields are formed by GW forcing (Watanabe et al. 2009, and references therein). The results of the 3D analysis in the present study suggest the possibility that the phase structure of PWs generated by the instability is determined by the longitudinal distribution of GW drag. This is an interesting issue for future studies.

Although the results were mainly shown for July and January of the second year so far, we also examined the data in July of the first and third years. The results are generally consistent with those of the second year. Notable differences between the three years are dominant longitudes of stationary PW and the strength of the flows in the monsoon regions induced by GW. However, these are probably explained by interannual variability of the atmospheric circulation.

e. The 3D residual-mean flow in the polar stereographic map

In this subsection, we show the polar stereographic maps of () induced by waves, where is estimated using (42), (43), and (45) and is calculated using (21)(23).

Results are shown for 10 hPa in the SH in July of the second year in Fig. 10. The horizontal wind components ( and ) in Fig. 10a tend to be westward in low latitudes and eastward in high latitudes. However, they are not zonally uniform. The westward flow in low latitudes and eastward flow in high latitudes are confluent to the south of Australia and merged into the eastward flow around the polar night jet. The westward flow in low latitudes is mainly due to unbalanced mean flow, and eastward flow is a mixture of unbalanced mean flow and Stokes drift (Figs. 10c,e).

Fig. 10.
Fig. 10.

Polar stereographic projection maps of (a),(b) the residual-mean flow induced by waves, (c),(d) unbalanced mean flow, and (e),(f) Stokes drift at 10 hPa in the SH in July of the second year; (top) arrows show horizontal component vectors and (bottom) colors show vertical components. Contours show monthly mean geopotential heights with an interval of 0.5 × 103 m.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1

The vertical wind component is relatively complicated, as shown in Fig. 10b. Strong downward flow is observed in longitudes clockwise from 60°W to 60°E along the polar night jet. In the remaining longitude region, including the date line, strong downward flow extends toward lower latitudes (~30°S), although downward flow is also observed inside the polar vortex. These downward flows are mainly due to unbalanced mean flow . The Stokes drift is strong except for the longitude region clockwise from 60° to 120°E, and it is downward (upward) inside (outside) the polar vortex. The downward inside the polar vortex is largely canceled by upward .

Figure 11 shows contributions of PWs and GWs to . Note that color scale and unit vectors are different for GWs from those in Fig. 10, while those for PWs are the same. Compared with Fig. 10, it is seen that overall structure is mainly determined by the residual-mean flow induced by PW forcing. However, it is worth noting that characteristic downward and upward flows are observed around the southern Andean region, which is likely associated with topographically forced GW.

Fig. 11.
Fig. 11.

Polar stereographic projection maps of the residual-mean flow induced by (a),(b) PWs and (c),(d) GWs in the SH; (top) arrows show horizontal component vectors and (bottom) colors show vertical components. Contours show monthly mean geopotential heights with an interval of 0.5 × 103 m. Note that unit vectors for (a) and (c) are different.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1

Next, polar stereographic maps of 3D residual-mean flow at 10 hPa in the NH in July of the second year and those in the SH and NH in January of the second year are shown in Fig. 12. In both summer hemispheres, horizontal components of are large only in subtropical regions (Figs. 12a,c). The upward flow is strong in the Asian and African monsoon region in the NH in July, while longitudinal variation of is not large in the SH in January (Figs. 12b,d). The strong upward flow in the NH monsoon region is consistent with the previous study by Randel et al. (2010), who examined the upward transport of minor constituents using a satellite observation and CCM simulation.

Fig. 12.
Fig. 12.

As in Figs. 10a and 10b, but for (a),(b) NH in July of the second year, (c),(d) SH in January of the second year, and (e),(f) NH in January of the second year. Note that color scales and unit vectors are different between the summer and winter hemispheres.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1

On the other hand, in the NH in January has an interesting structure. The zonal component tends to be westward in low latitudes and eastward in high latitudes, as in the SH in July (Fig. 12e). Strong downward flow is observed in two longitudinal regions clockwise from 30° to 120°W and from 170° to 50°E around the polar night jet. The strong downward flows around the jet extend to lower latitudes having a spiral-like form. It is also interesting that upward flow is observed even in the NH in this month (i.e., winter).

Last, polar stereographic maps of at 0.05 hPa in the mesosphere are shown (Fig. 13). In the winter hemisphere (Figs. 13a,b,g,h), the polar vortex is larger than that at 10 hPa. The horizontal component is generally eastward and poleward around the westerly jet. The downward flow area is spread toward 120°W in the SH in July, while it is confined to high latitudes in the Western Hemisphere in the NH in January. Moreover, a strong downward flow is observed around the southern Andes and Antarctic Peninsula in Fig. 13b in the SH in July, which is likely due to gravity waves forced topographically in that region. It is important that the distributions of observed in Figs. 13b and 13h have some similarity to those at 10 hPa in Figs. 10b and 12f, respectively. This fact can be explained by the downward control principle indicating that the vertical flow in the stratosphere is largely affected by wave forcing in the mesosphere and above.

Fig. 13.
Fig. 13.

As in Figs. 10a and 10b, but for 0.05 hPa in respective months and respective hemispheres in the mesosphere.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1

In the summer hemisphere shown in Figs. 13c–f, the distribution of is relatively zonally uniform compared with that in the winter hemisphere. The zonal component is large and generally eastward in the subtropical region. There is slight hemispheric difference in the distribution of upward motion: the upward motion is stronger and distributed in higher latitudes in the NH than that in the SH. The distribution can be affected by the location and strength of the (simulated) easterly jet, which modifies lateral propagation of GW (Sato et al. 2009b). Further detailed discussion is beyond the scope of the present paper, however.

f. Comparison with 3D time-mean flow

As we mentioned in section 4, is the sum of and (). To see the relative strength of and , () is calculated. Results are shown in Fig. 14. Note that the color scales for and for are different for 10 hPa and are the same for 0.05 hPa.

Fig. 14.
Fig. 14.

Horizontal maps of monthly mean meridional wind at (top) 10 and (bottom) 0.05 hPa in (left) July and (right) January. Note that color scales for 10 hPa are different than those for Fig. 10.

Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1

At 10 hPa, the magnitude of is much larger than that of in the winter hemisphere. This is the case for the summer hemisphere, although is not visible with this color scale. This result indicates that is dominant in by its magnitude. However, it should be emphasized that does not contribute to the zonal-mean mass transport in the meridional cross section.

On the other hand, the magnitude of is comparable to that of at 0.05 hPa. However, the distribution of is largely different from in the winter hemisphere. The planetary-scale pattern with positive and negative values is observed in , reflecting the existence of large-amplitude PWs. In the summer hemisphere, is similar to both in the magnitude and distribution. This means that in the summer hemisphere is small, and is primarily determined by , which is mainly due to GWs.

6. Summary and concluding remarks

A new method to estimate 3D material circulation driven by waves in the atmosphere was proposed based on recently derived formulas by Kinoshita and Sato (2013a). The formulas are applicable to both Rossby waves and gravity waves. Although this theory considered the time mean for the averaging of flux calculation, the underlying assumption for the formulation is that the temporal and/or spatial scales of the mean (more precisely speaking, background) field are much longer than those of the perturbation field. Thus, the formulas can be applied also to stationary waves. The 3D residual-mean flow is divided into three components, that is, balanced mean flow, unbalanced mean flow, and Stokes drift. The last two components are induced by wave forcing, and the sum of their zonal mean is equivalent to the 2D residual-mean flow in the TEM system. It was shown that the unbalanced mean flow is estimated by the momentum flux divergence, while the Stokes drift is directly calculated by the divergence of heat flux and . Moreover, by taking into account that the averaging is inherently needed to remove an oscillatory component of unaveraged quadratic function on a scale of one-half the wavelength of the wave field, we proposed the utilization of an extended Hilbert transform. This extended Hilbert transform was newly introduced in the present paper. Here the whole scale of the wave packet corresponds to the scale of the “mean” field with which the wave packet interacts.

By applying this method to the outputs from simulation by a gravity wave–resolving general circulation model, the 3D structure of the residual-mean circulation in the middle and upper stratosphere and mesosphere was examined for January and July. Characteristics of the residual-mean flow in January and July were roughly a mirror image of each other. An important result was that the residual-mean circulation is not zonally uniform in any altitude region.

In the middle and upper stratosphere, the zonal-mean meridional component of the residual-mean circulation was from the subtropical region of the summer hemisphere to the high latitudes of the winter hemisphere. This meridional flow was divided into three parts according to the dominant terms of the wave activity flux convergence: poleward Stokes drift by PWs in the winter hemisphere high latitudes that was largely canceled by equatorward unbalanced mean flow due to PWs; poleward unbalanced mean flow by PWs in middle and low latitudes of the winter hemisphere; and equatorward unbalanced mean flow by GWs in the summer hemisphere. In the winter hemisphere high latitudes, the poleward Stokes drift and equatorward unbalanced mean flow were large in different longitude regions. Thus, even in the winter hemisphere, there were some longitude regions where equatorward flow was dominant. In the summer hemisphere, the unbalanced mean flow was strong in and slightly to the east of the monsoon region. This is likely because the monsoon convection is a dominant source of GWs propagating eastward.

In the mesosphere, GWs were the most important wave to drive the residual-mean circulation. In addition, the contribution of PW generated by baroclinic/barotropic instability was not negligible in the winter hemisphere. The distribution of the residual-mean flow is not zonally uniform, which is likely because of nonzonal GW source distribution in both winter and summer hemispheres, the filtering of GWs in the polar night jet largely disturbed by PWs in the winter stratosphere, and the characteristics of the instability in the winter mesosphere. There was some similarity in the structure of the vertical component of the residual-mean flow between the stratosphere and mesosphere. This resemblance was roughly understood by the downward control principle.

The atmosphere is coupled vertically by various kinds of waves with various scales originating from nonzonal sources and propagating three dimensionally. It is considered that the 3D analysis proposed by the present study must improve our understanding of the vertical coupling processes, including wave–wave interaction as well as wave–mean flow interaction. Moreover, it is interesting to examine barotropic/baroclinic instability in terms of the three-dimensional structure.

Application of the extended Hilbert transform to obtain eddy and flux quadratics as shown by the present study is available also for the other equation systems such as the quasigeostrophic system. The use of this method is effective to estimate fluxes and variances associated with wave packets, because the phase interferences among multiple sinusoidal waves are properly included. This point is an advantage compared with the other formulas of 3D wave activity flux proposed by previous studies, which inherently assume a monochromatic wave.

Acknowledgments

This study is supported by Grant-in-Aid for Scientific Research (A) 25247075 of the Ministry of Education, Culture, Sports and Technology (MEXT), Japan. The authors thank Theodore G. Shepherd and Alan Plumb for constructive comments. Thanks are also due to Yuki Yasuda for his providing schematic figure to illustrate the extended Hilbert transform and useful discussion on the theoretical aspect of the present study. The model simulation was conducted using the Earth Simulator by the KANTO project members. A part of the simulation was made as a contribution to the Innovative Program of Climate Change Projection for the 21st Century supported by MEXT.

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  • Andrews, D. G., J. R. Holton, and C. B. Leovy, 1987: Middle Atmosphere Dynamics. Academic Press, 489 pp.

  • Birner, T., and H. Bönisch, 2011: Residual circulation trajectories and transit times into the extratropical lowermost stratosphere. Atmos. Chem. Phys., 11, 817827, doi:10.5194/acp-11-817-2011.

    • Search Google Scholar
    • Export Citation
  • Bracewell, R., 1999: The Fourier Transform and Its Applications. McGraw-Hill, 640 pp.

  • Butchart, N., and Coauthors, 2006: Simulations of anthropogenic change in the strength of the Brewer–Dobson circulation. Climate Dyn., 27, 727741, doi:10.1007/s00382-006-0162-4.

    • Search Google Scholar
    • Export Citation
  • Butchart, N., and Coauthors, 2010: Chemistry–climate model simulations of twenty-first century stratospheric century stratospheric. J. Climate, 23, 53495374.

    • Search Google Scholar
    • Export Citation
  • Callaghan, P. F., and M. L. Salby, 2002: Three-dimensionality and forcing of the Brewer–Dobson circulation. J. Atmos. Sci., 59, 976991.

    • Search Google Scholar
    • Export Citation
  • Calvo, N., and R. R. Garcia, 2009: Wave forcing of the tropical upwelling in the lower stratosphere under increasing concentrations of greenhouse gases. J. Atmos. Sci., 66, 31843196.

    • Search Google Scholar
    • Export Citation
  • Dunkerton, T. J., 1978: On the mean meridional mass motions of the stratosphere and mesosphere. J. Atmos. Sci., 35, 23252333.

  • Dunkerton, T. J., C.-P. F. Hsu, and M. E. McIntyre, 1981: Some Eulerian and Lagrangian diagnostics for a model stratospheric warming. J. Atmos. Sci., 38, 819843.

    • Search Google Scholar
    • Export Citation
  • Garcia, R. R., and W. J. Randel, 2008: Acceleration of the Brewer–Dobson circulation due to increases in greenhouse gases. J. Atmos. Sci., 65, 27312739.

    • Search Google Scholar
    • Export Citation
  • Haynes, P. H., M. E. McIntyre, T. G. Shepherd, C. J. Marks, and K. P. Shin, 1991: On the “downward control” of extratropical diabatic circulations by eddy-induced mean zonal forces. J. Atmos. Sci., 48, 651678.

    • Search Google Scholar
    • Export Citation
  • Hitchman, M. H., and M. J. Rogal, 2010: Influence of tropical convection on the southern hemisphere ozone maximum during the winter to spring transition. J. Geophys. Res., 115, D14118, doi:10.1029/2009JD012883.

    • Search Google Scholar
    • Export Citation
  • Holton, J. R., 1982: The role of gravity wave induced drag and diffusion in the momentum budget of the mesosphere. J. Atmos. Sci., 39, 791799.

    • Search Google Scholar
    • Export Citation
  • Holton, J. R., P. H. Haynes, M. E. McIntyre, A. R. Douglass, R. B. Rood, and L. Pfister, 1995: Stratosphere-troposphere exchange. Rev. Geophys., 33, 403439.

    • Search Google Scholar
    • Export Citation
  • Kawatani, Y., and K. Hamilton, 2011: The quasi-biennial oscillation in a double CO2 climate. J. Atmos. Sci., 68, 265283.

  • Kawatani, Y., K. Sato, T. J. Dunkerton, S. Watanabe, S. Miyahara, and M. Takahashi, 2010: The roles of equatorial trapped waves and internal inertia–gravity waves in driving the quasi-biennial oscillation. Part I: Zonal mean wave forcing. J. Atmos. Sci., 67, 963980.

    • Search Google Scholar
    • Export Citation
  • Kinoshita, T., and K. Sato, 2013a: A formulation of three-dimensional residual-mean flow applicable both to inertia–gravity waves and to Rossby waves. J. Atmos. Sci., 70, 15771602.

    • Search Google Scholar
    • Export Citation
  • Kinoshita, T., and K. Sato, 2013b: A formulation of unified three-dimensional wave activity flux of inertia–gravity waves and Rossby waves. J. Atmos. Sci., 70, 16031615.

    • Search Google Scholar
    • Export Citation
  • Li, F., J. Austin, and J. Wilson, 2008: The strength of the Brewer–Dobson circulation in a changing climate: A coupled chemistry-climate model simulation. J. Climate, 21, 4057.

    • Search Google Scholar
    • Export Citation
  • Li, F., R. S. Stolarski, S. Pawson, P. A. Newman, and D. Waugh, 2010: Narrowing of the upwelling branch of the Brewer-Dobson circulation and Hadley cell in chemistry-climate model simulations of the 21st century. Geophys. Res. Lett., 37, L13702, doi:10.1029/2010GL043718.

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    • Export Citation
  • Lieberman, R. S., 1999: The gradient wind in the mesosphere and lower thermosphere. Earth Planets Space, 51, 751761.

  • Lin, P., Q. Fu, S. Solomon, and J. M. Wallace, 2009: Temperature trend patterns in southern hemisphere high latitudes: Novel indicators of stratospheric change. J. Climate, 22, 63256341.

    • Search Google Scholar
    • Export Citation
  • Lindzen, R. S., 1981: Turbulence and stress owing to gravity wave and tidal breakdown. J. Geophys. Res., 86, 97079714.

  • Matsuno, T., 1982: A quasi one-dimensional model of the middle atmosphere circulation interacting with internal gravity waves. J. Meteor. Soc. Japan, 60, 215226.

    • Search Google Scholar
    • Export Citation
  • McLandress, C., and T. G. Shepherd, 2009: Simulated anthropogenic changes in the Brewer–Dobson circulation, including its extension to high latitudes. J. Climate, 22, 15161540.

    • Search Google Scholar
    • Export Citation
  • Miyahara, S., D. Yamamoto, and Y. Miyoshi, 2000: On the geostrophic balance of mean zonal winds in the mesosphere and lower thermosphere. J. Meteor. Soc. Japan, 78, 683688.

    • Search Google Scholar
    • Export Citation
  • Miyazaki, K., S. Watanabe, Y. Kawatani, Y. Tomikawa, M. Takahashi, and K. Sato, 2010: Transport and mixing in the extratropical tropopause region in a high vertical resolution GCM. Part I: Potential vorticity and heat budget analysis. J. Atmos. Sci., 67, 12931314.

    • Search Google Scholar
    • Export Citation
  • Norton, W. A., 2006: Tropical wave driving of the annual cycle in tropical tropopause temperatures. Part II: Model results. J. Atmos. Sci., 63, 14201431.

    • Search Google Scholar
    • Export Citation
  • Okamoto, K., K. Sato, and H. Akiyoshi, 2011: A study on the formation and trend of the Brewer-Dobson circulation. J. Geophys. Res., 116, D10117, doi:10.1029/2010JD014953.

    • Search Google Scholar
    • Export Citation
  • Plumb, R. A., 2002: Stratospheric transport. J. Meteor. Soc. Japan, 80, 793809.

  • Preusse, P., A. Dörnbrack, S. D. Eckermann, M. Riese, B. Schaeler, J. T. Bacmeister, D. Broutman, and K. U. Grossmann, 2002: Space-based measurements of stratospheric mountain waves by CRISTA: 1. Sensitivity, analysis method, and a case study. J. Geophys. Res., 107, 8178, doi:10.1029/2001JD000699.

    • Search Google Scholar
    • Export Citation
  • Randel, W. J., 1987: The evaluation of winds from geopotential height data in the stratosphere. J. Atmos. Sci., 44, 30973120.

  • Randel, W. J., M. Park, L. Emmons, D. Kinnison, P. Bernath, K. A. Walker, C. Boone, and H. Pumphrey, 2010: Asian monsoon transport of pollution to the stratosphere. Science, 328, 611613.

    • Search Google Scholar
    • Export Citation
  • Rosenlof, K. H., 1995: Seasonal cycle of the residual-mean meridional circulation in the stratosphere. J. Geophys. Res., 100, 51735191.

    • Search Google Scholar
    • Export Citation
  • Sato, K., D. O’Sullivan, and T. J. Dunkerton, 1997: Low-frequency inertia-gravity waves in the stratosphere revealed by three-week continuous observation with the mu radar. Geophys. Res. Lett., 24, 17391742.

    • Search Google Scholar
    • Export Citation
  • Sato, K., M. Yamamori, S. Ogino, N. Takahashi, Y. Tomikawa, and T. Yamaouchi, 2003: A meridional scan of the stratospheric gravity wave field over the ocean in 2001 (MeSSO2001). J. Geophys. Res., 108, 4491, doi:10.1029/2002JD003219.

    • Search Google Scholar
    • Export Citation
  • Sato, K., Y. Tomikawa, G. Hashida, T. Yamanouchi, H. Nakajima, and T. Sugita, 2009a: Longitudinal dependence of ozone recovery in the Antarctic polar vortex revealed by balloon and satellite observations. J. Atmos. Sci., 66, 18071820.

    • Search Google Scholar
    • Export Citation
  • Sato, K., S. Watanabe, Y. Kawatani, Y. Tomikawa, K. Miyazaki, and M. Takahashi, 2009b: On the origins of mesospheric gravity waves. Geophys. Res. Lett., 36, L19801, doi:10.1029/2009GL039908.

    • Search Google Scholar
    • Export Citation
  • Sato, K., S. Tateno, S. Watanabe, and Y. Kawatani, 2012: Gravity wave characteristics in the Southern Hemisphere revealed by a high-resolution middle-atmosphere general circulation model. J. Atmos. Sci.,69, 1378–1396.

  • Seidel, D. J., and W. J. Randel, 2007: Recent widening of the tropical belt: Evidence from tropopause observations. J. Geophys. Res.,112, D20113, doi:10.1029/2007JD008861.

  • Seviour, W. J. M., N. Butchart, and S. C. Hardiman, 2012: The Brewer-Dobson circulation inferred from ERA-Interim. Quart. J. Roy. Meteor. Soc., 138, 878888.

    • Search Google Scholar
    • Export Citation
  • Shepherd, T. G., and C. McLandress, 2011: Strengthening of the Brewer–Dobson circulation in response to climate change: Critical-layer control of subtropical wave breaking. J. Atmos. Sci., 68, 784797.

    • Search Google Scholar
    • Export Citation
  • Smith, A. K., 2003: The origin of stationary planetary waves in the upper mesosphere. J. Atmos. Sci., 60, 30333041.

  • Ueyama, R., and J. M. Wallace, 2010: To what extent does high-latitude wave forcing drive tropical upwelling in the Brewer-Dobson circulation? J. Atmos. Sci., 67, 12321246.

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  • Watanabe, S., Y. Kawatani, Y. Tomikawa, K. Miyazaki, M. Takahashi, and K. Sato, 2008: General aspects of a T213L256 middle atmosphere general circulation model. J. Geophys. Res., 113, D12110, doi:10.1029/2008JD010026.

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  • Watanabe, S., Y. Tomikawa, K. Sato, Y. Kawatani, K. Miyazaki, and M. Takahashi, 2009: Simulation of the eastward 4-day wave in the Antarctic winter mesosphere using a gravity wave resolving general circulation model. J. Geophys. Res., 114, D16111, doi:10.1029/2008JD011636.

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1

This formula for uL is different from the Lagrangian perturbation u1 in Andrews and McIntyre (1976) and Andrews et al. (1987) by an additional term −′.

2

The first equality is derived as , where W, V, H, and Z are “analytic expressions” of w′, υ′, η′, and ζ′, respectively, and is the intrinsic frequency. The analytic expressions are described in detail in section 4. The second equality is derived from the thermodynamic equation where D/Dt is Lagrangian time derivative. Thus, ζ′ ≈ −φz/N2.

3

Phase difference between υ′(~′/Dt) and η′ is 90° for the Fourier components having the same wavenumber vector and frequency. Thus, [ηυ′] can be ignored. This is also the case for [ζw′]. Moreover, the covariance of the Fourier components with different wavenumber and/or frequency obviously becomes zero.

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  • Andrews, D. G., and M. E. McIntyre, 1976: Planetary waves in horizontal and vertical shear: The generalized Eliassen–Palm relation and the mean zonal acceleration. J. Atmos. Sci., 33, 20312048.

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    • Export Citation
  • Andrews, D. G., J. R. Holton, and C. B. Leovy, 1987: Middle Atmosphere Dynamics. Academic Press, 489 pp.

  • Birner, T., and H. Bönisch, 2011: Residual circulation trajectories and transit times into the extratropical lowermost stratosphere. Atmos. Chem. Phys., 11, 817827, doi:10.5194/acp-11-817-2011.

    • Search Google Scholar
    • Export Citation
  • Bracewell, R., 1999: The Fourier Transform and Its Applications. McGraw-Hill, 640 pp.

  • Butchart, N., and Coauthors, 2006: Simulations of anthropogenic change in the strength of the Brewer–Dobson circulation. Climate Dyn., 27, 727741, doi:10.1007/s00382-006-0162-4.

    • Search Google Scholar
    • Export Citation
  • Butchart, N., and Coauthors, 2010: Chemistry–climate model simulations of twenty-first century stratospheric century stratospheric. J. Climate, 23, 53495374.

    • Search Google Scholar
    • Export Citation
  • Callaghan, P. F., and M. L. Salby, 2002: Three-dimensionality and forcing of the Brewer–Dobson circulation. J. Atmos. Sci., 59, 976991.

    • Search Google Scholar
    • Export Citation
  • Calvo, N., and R. R. Garcia, 2009: Wave forcing of the tropical upwelling in the lower stratosphere under increasing concentrations of greenhouse gases. J. Atmos. Sci., 66, 31843196.

    • Search Google Scholar
    • Export Citation
  • Dunkerton, T. J., 1978: On the mean meridional mass motions of the stratosphere and mesosphere. J. Atmos. Sci., 35, 23252333.

  • Dunkerton, T. J., C.-P. F. Hsu, and M. E. McIntyre, 1981: Some Eulerian and Lagrangian diagnostics for a model stratospheric warming. J. Atmos. Sci., 38, 819843.

    • Search Google Scholar
    • Export Citation
  • Garcia, R. R., and W. J. Randel, 2008: Acceleration of the Brewer–Dobson circulation due to increases in greenhouse gases. J. Atmos. Sci., 65, 27312739.

    • Search Google Scholar
    • Export Citation
  • Haynes, P. H., M. E. McIntyre, T. G. Shepherd, C. J. Marks, and K. P. Shin, 1991: On the “downward control” of extratropical diabatic circulations by eddy-induced mean zonal forces. J. Atmos. Sci., 48, 651678.

    • Search Google Scholar
    • Export Citation
  • Hitchman, M. H., and M. J. Rogal, 2010: Influence of tropical convection on the southern hemisphere ozone maximum during the winter to spring transition. J. Geophys. Res., 115, D14118, doi:10.1029/2009JD012883.

    • Search Google Scholar
    • Export Citation
  • Holton, J. R., 1982: The role of gravity wave induced drag and diffusion in the momentum budget of the mesosphere. J. Atmos. Sci., 39, 791799.

    • Search Google Scholar
    • Export Citation
  • Holton, J. R., P. H. Haynes, M. E. McIntyre, A. R. Douglass, R. B. Rood, and L. Pfister, 1995: Stratosphere-troposphere exchange. Rev. Geophys., 33, 403439.

    • Search Google Scholar
    • Export Citation
  • Kawatani, Y., and K. Hamilton, 2011: The quasi-biennial oscillation in a double CO2 climate. J. Atmos. Sci., 68, 265283.

  • Kawatani, Y., K. Sato, T. J. Dunkerton, S. Watanabe, S. Miyahara, and M. Takahashi, 2010: The roles of equatorial trapped waves and internal inertia–gravity waves in driving the quasi-biennial oscillation. Part I: Zonal mean wave forcing. J. Atmos. Sci., 67, 963980.

    • Search Google Scholar
    • Export Citation
  • Kinoshita, T., and K. Sato, 2013a: A formulation of three-dimensional residual-mean flow applicable both to inertia–gravity waves and to Rossby waves. J. Atmos. Sci., 70, 15771602.

    • Search Google Scholar
    • Export Citation
  • Kinoshita, T., and K. Sato, 2013b: A formulation of unified three-dimensional wave activity flux of inertia–gravity waves and Rossby waves. J. Atmos. Sci., 70, 16031615.

    • Search Google Scholar
    • Export Citation
  • Li, F., J. Austin, and J. Wilson, 2008: The strength of the Brewer–Dobson circulation in a changing climate: A coupled chemistry-climate model simulation. J. Climate, 21, 4057.

    • Search Google Scholar
    • Export Citation
  • Li, F., R. S. Stolarski, S. Pawson, P. A. Newman, and D. Waugh, 2010: Narrowing of the upwelling branch of the Brewer-Dobson circulation and Hadley cell in chemistry-climate model simulations of the 21st century. Geophys. Res. Lett., 37, L13702, doi:10.1029/2010GL043718.

    • Search Google Scholar
    • Export Citation
  • Lieberman, R. S., 1999: The gradient wind in the mesosphere and lower thermosphere. Earth Planets Space, 51, 751761.

  • Lin, P., Q. Fu, S. Solomon, and J. M. Wallace, 2009: Temperature trend patterns in southern hemisphere high latitudes: Novel indicators of stratospheric change. J. Climate, 22, 63256341.

    • Search Google Scholar
    • Export Citation
  • Lindzen, R. S., 1981: Turbulence and stress owing to gravity wave and tidal breakdown. J. Geophys. Res., 86, 97079714.

  • Matsuno, T., 1982: A quasi one-dimensional model of the middle atmosphere circulation interacting with internal gravity waves. J. Meteor. Soc. Japan, 60, 215226.

    • Search Google Scholar
    • Export Citation
  • McLandress, C., and T. G. Shepherd, 2009: Simulated anthropogenic changes in the Brewer–Dobson circulation, including its extension to high latitudes. J. Climate, 22, 15161540.

    • Search Google Scholar
    • Export Citation
  • Miyahara, S., D. Yamamoto, and Y. Miyoshi, 2000: On the geostrophic balance of mean zonal winds in the mesosphere and lower thermosphere. J. Meteor. Soc. Japan, 78, 683688.

    • Search Google Scholar
    • Export Citation
  • Miyazaki, K., S. Watanabe, Y. Kawatani, Y. Tomikawa, M. Takahashi, and K. Sato, 2010: Transport and mixing in the extratropical tropopause region in a high vertical resolution GCM. Part I: Potential vorticity and heat budget analysis. J. Atmos. Sci., 67, 12931314.

    • Search Google Scholar
    • Export Citation
  • Norton, W. A., 2006: Tropical wave driving of the annual cycle in tropical tropopause temperatures. Part II: Model results. J. Atmos. Sci., 63, 14201431.

    • Search Google Scholar
    • Export Citation
  • Okamoto, K., K. Sato, and H. Akiyoshi, 2011: A study on the formation and trend of the Brewer-Dobson circulation. J. Geophys. Res., 116, D10117, doi:10.1029/2010JD014953.

    • Search Google Scholar
    • Export Citation
  • Plumb, R. A., 2002: Stratospheric transport. J. Meteor. Soc. Japan, 80, 793809.

  • Preusse, P., A. Dörnbrack, S. D. Eckermann, M. Riese, B. Schaeler, J. T. Bacmeister, D. Broutman, and K. U. Grossmann, 2002: Space-based measurements of stratospheric mountain waves by CRISTA: 1. Sensitivity, analysis method, and a case study. J. Geophys. Res., 107, 8178, doi:10.1029/2001JD000699.

    • Search Google Scholar
    • Export Citation
  • Randel, W. J., 1987: The evaluation of winds from geopotential height data in the stratosphere. J. Atmos. Sci., 44, 30973120.

  • Randel, W. J., M. Park, L. Emmons, D. Kinnison, P. Bernath, K. A. Walker, C. Boone, and H. Pumphrey, 2010: Asian monsoon transport of pollution to the stratosphere. Science, 328, 611613.

    • Search Google Scholar
    • Export Citation
  • Rosenlof, K. H., 1995: Seasonal cycle of the residual-mean meridional circulation in the stratosphere. J. Geophys. Res., 100, 51735191.

    • Search Google Scholar
    • Export Citation
  • Sato, K., D. O’Sullivan, and T. J. Dunkerton, 1997: Low-frequency inertia-gravity waves in the stratosphere revealed by three-week continuous observation with the mu radar. Geophys. Res. Lett., 24, 17391742.

    • Search Google Scholar
    • Export Citation
  • Sato, K., M. Yamamori, S. Ogino, N. Takahashi, Y. Tomikawa, and T. Yamaouchi, 2003: A meridional scan of the stratospheric gravity wave field over the ocean in 2001 (MeSSO2001). J. Geophys. Res., 108, 4491, doi:10.1029/2002JD003219.

    • Search Google Scholar
    • Export Citation
  • Sato, K., Y. Tomikawa, G. Hashida, T. Yamanouchi, H. Nakajima, and T. Sugita, 2009a: Longitudinal dependence of ozone recovery in the Antarctic polar vortex revealed by balloon and satellite observations. J. Atmos. Sci., 66, 18071820.

    • Search Google Scholar
    • Export Citation
  • Sato, K., S. Watanabe, Y. Kawatani, Y. Tomikawa, K. Miyazaki, and M. Takahashi, 2009b: On the origins of mesospheric gravity waves. Geophys. Res. Lett., 36, L19801, doi:10.1029/2009GL039908.

    • Search Google Scholar
    • Export Citation
  • Sato, K., S. Tateno, S. Watanabe, and Y. Kawatani, 2012: Gravity wave characteristics in the Southern Hemisphere revealed by a high-resolution middle-atmosphere general circulation model. J. Atmos. Sci.,69, 1378–1396.

  • Seidel, D. J., and W. J. Randel, 2007: Recent widening of the tropical belt: Evidence from tropopause observations. J. Geophys. Res.,112, D20113, doi:10.1029/2007JD008861.

  • Seviour, W. J. M., N. Butchart, and S. C. Hardiman, 2012: The Brewer-Dobson circulation inferred from ERA-Interim. Quart. J. Roy. Meteor. Soc., 138, 878888.

    • Search Google Scholar
    • Export Citation
  • Shepherd, T. G., and C. McLandress, 2011: Strengthening of the Brewer–Dobson circulation in response to climate change: Critical-layer control of subtropical wave breaking. J. Atmos. Sci., 68, 784797.

    • Search Google Scholar
    • Export Citation
  • Smith, A. K., 2003: The origin of stationary planetary waves in the upper mesosphere. J. Atmos. Sci., 60, 30333041.