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.


b. Results of 2D analysis using gravity wave–resolving GCM data
Figure 1 shows meridional cross sections of the E-P flux vector
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.
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
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,
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).
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
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
4. Theory of three-dimensional residual-mean circulation
a. Relation between 3D residual-mean flow and 3D wave activity flux



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.























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
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.
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.
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
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
Meridional cross sections of zonal-mean unbalanced mean meridional flow calculated from (a) 3D momentum flux divergence
Citation: Journal of the Atmospheric Sciences 70, 12; 10.1175/JAS-D-12-0352.1
It is important that the zonal-mean
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
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
Another interesting feature observed in
In the SH, the three kinds of flows
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
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
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
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
Such nonzonal distribution of
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
Results are shown for 10 hPa in the SH in July of the second year in Fig. 10. The horizontal wind components (
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
Figure 11 shows contributions of PWs and GWs to
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
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,
Last, polar stereographic maps of
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
f. Comparison with 3D time-mean flow
As we mentioned in section 4,
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
On the other hand, the magnitude of
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
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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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 −fη′.
The first equality is derived as
Phase difference between υ′(~Dη′/Dt) and η′ is 90° for the Fourier components having the same wavenumber vector and frequency. Thus, [η′υ′] can be ignored. This is also the case for [ζ′w′]. Moreover, the covariance of the Fourier components with different wavenumber and/or frequency obviously becomes zero.