3.3. Wave Forcing and Surface Pressure Change
 We compared the lagged regression of the E-P flux divergence with that of the zonal-mean zonal wind tendency based on PC1 to investigate how the variability of the zonal wind is driven by wave forcing with time. Figure 5 demonstrates that the acceleration due to E-P flux divergence corresponded very well with the tendency of the zonal wind in the stratosphere. An analysis based on zonal wave number decomposition demonstrated that the variability of the E-P flux came primarily from the zonal wave number 1 component (not shown) [KK01]. Although correspondence of the E-P flux divergence with the tendency of the zonal wind was very good in the stratosphere, it was not good in the troposphere, indicating that the E-P flux divergence analysis does not work well in the troposphere. One reason for this is the existence of the surface as a lower boundary in the transformed Eulerian mean (TEM) formulation. The lower boundary condition is very complex in the TEM formulation and acts as a significant source of forcing, making it difficult to analyze the troposphere [HS89]. We overcame this difficulty by analyzing the eddy forcings in the Eulerian mean (EM) formulation, whose lower boundary condition can be treated very simply [HS89].
Figure 5. Same as Figure 2 except for the wave forcing due to E-P divergence (top) and the zonal-mean zonal wind tendency (bottom) due to the PC1. Contour interval is 0.2 ms−1day−1 (0.1 ms−1day−1) for the wave forcing (zonal wind tendency).
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 There are two types of eddy forcings in the EM formulation, mechanical (or momentum) forcing, which primarily forces zonal wind, and thermal forcing, which forces temperature. These forcings are proportional to the meridional divergence of the northward eddy momentum (mechanical forcing) and heat fluxes (thermal forcing) (see Appendix). The northward eddy momentum flux represents northward transport of positive zonal momentum; therefore more zonal momentum leaves the region than enters if its divergence is positive. Meridional divergence of the northward eddy momentum thus decelerates the zonal wind. A similar interpretation can be applied to thermal forcing. However, these two forcings were combined through equations and are subject to variabilities, including both zonal wind and temperature. It is apparent that the characteristics of the EM eddy forcings arise from the characteristics of the eddy fluxes. We therefore examined eddy fluxes first.
 The meridional and vertical components of the E-P flux are proportional to eddy fluxes, so the time evolution of the E-P flux in Figure 4 corresponds well with the evolution of these eddy fluxes. Figure 6 depicts the lagged regression of eddy fluxes due to PC1. There is a local maximum in the eddy momentum flux at the upper troposphere around 50°N. It does not travel, but gradually increases and peaks at lag −5 days, then gradually decreases. This corresponds to the enhanced equatorward propagation of the E-P flux at lag 0 in Figure 4. In contrast, the eddy heat flux has a relatively simple standing structure throughout, and its amplitude increases toward the upper stratosphere. However, the time evolution of the amplitude was very prominent. It was significantly negative at lag −30 days, increased with time, peaked with a significantly positive value at lag +5 days, and then decreased with time. This feature corresponds with the change of the vertical propagation of the E-P flux in Figure 4. Note that the mechanical forcing in the EM formulation is proportional to the meridional divergence in the meridional component of the E-P flux, and the thermal forcing is proportional to the meridional divergence in the vertical component of the E-P flux.
Figure 6. Same as Figure 2 except for the eddy momentum flux (top) and the eddy thermal flux (bottom) due to the PC1. Contour interval is 2 m2s−2 (2 mKs−1) for the eddy momentum (thermal) flux.
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 Figure 7 illustrates the lagged regression of the mechanical and thermal forcings by the PC1. The arrows indicate meridional circulation. There are two forcing centers for mechanical forcing (upper panels), one in the troposphere and the other traveling downward from the stratosphere. The tropospheric forcing appeared around 65°N in the 300 hPa level at lag −15 days and developed in conjunction with forcing from the stratosphere at lag 0. It then became weaker but was still connected with the stratospheric forcing until lag +15 days. This forcing center corresponded to the polar-side gradient of the tropospheric momentum eddy flux in Figure 6a, and is significant compared with that of the subtropical center due to the metric factor. Stratospheric forcing appeared at 80°N in 70 hPa at lag −30 days and propagated downward, reaching 85°N and 300 hPa at lag +15 days. A negative signal above this forcing followed this signal. This can also be traced back to the downward propagating eddy momentum flux from the stratosphere. Wave number decomposition demonstrated that the downward signal came from a wave of the zonal wave number 1 component, and the tropospheric signal came from a combination of wave number 1 to 3 components.
Figure 7. Same as Figure 2 except for mechanical (top) and thermal (bottom) eddy forcings with the Eulerian mean meridional circulation (arrows) due to the PC1. Contour interval is 0.2 ms−1day−1 (0.2 Kday−1) for the mechanical (thermal) eddy forcing. The horizontal (vertical) reference arrow indicates 6 × 10−2 ms−1 (3 × 10−4 ms−1). Small vectors are neglected.
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 The features of thermal forcing are particularly interesting. A strong standing meridional dipole forcing pattern with a negative high-latitude center from the surface to the upper stratosphere appeared at lag −30 days. The stratospheric forcing decreased in amplitude as time passed and almost vanished at −15 days, after which its amplitude increased with changing polarity, and a deep standing meridional dipole structure with positive high-latitude forcing was created again around the peak period of lag +5 days. It gradually decreased its amplitude, but retained its structure at lag +15 days. Although stratospheric forcing affects the signal as an oscillator, tropospheric forcing exhibits some equatorward propagation with time. The meridional dipole structure is a direct result of the meridional gradient of the standing structure of the eddy heat flux in Figure 6b. Wave number decomposition analysis revealed that this came primarily from a wave of the zonal wave number 1 component in the stratosphere and a combination of components 1 to 3 in the troposphere. Thermal forcing was small in the troposphere, but became substantial toward the upper stratosphere. In contrast, the magnitude of the mechanical forcing in the stratosphere was as great as that in the troposphere.
 The meridional circulation (vector in Figure 7) indicates that the mean vertical velocity was upward (downward) if the thermal forcing was positive (negative), and the meridional velocity was equatorward (poleward) if the mechanical forcing was positive (negative). In fact, the lagged correlation patterns of the meridional and vertical velocities were very similar to those of mechanical and thermal forcings (Figure 8) because advection and eddy forcing should be nearly balanced in a slow variability system (PJO). Data of the vertical velocity was available only up to 100 hPa in the reanalysis data. Also, the correlation of meridional wind near the surface was substantial (0.5) at lag 0 to +15 days, which indicates the importance of meridional circulation for coupling of the stratosphere to the surface [Kodera and Kuroda, 2003]. This result suggests that the meridional circulation was primarily driven by eddy forcings.
Figure 8. Same as Figure 2 except for the lagged correlation of the meridional (top) and vertical (bottom) velocities due to the PC1. Contour interval is 0.1 but is shown only for the 95% significance region (greater than 0.25), except for the 0 contour.
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 We examined this in further detail by using an EM model (see Appendix) to analyze the eddy-forced meridional circulation and surface pressure change. Equations were applied for the anomalous fields defined as departures from climatology. The stability of the mean field was estimated from the winter mean (December to March) temperature. Figure 9 illustrates the meridional circulation with the stream function (upper panels) and the SLP tendency (lower panels) by using eddy forcings calculated from the regression by PC1. EM wind (vectors) is shown only up to 100 hPa for comparison with the observation (Figure 7). The present results are identical to those obtained by regression of the circulations evaluated at each moment by the model, due to the linearity of the equation. It is clear that the overall features of the observed meridional circulation are reproduced well from this calculation, particularly for the upper troposphere to lower stratosphere and high-latitude areas. The area of increasing surface pressure corresponded well with the downward flow, and the area of decreasing surface pressure, with the upward flow. Note that the surface pressure change can be reproduced by setting an adequate lower boundary condition of DΦ/Dt = 0 [HS89]. The time evolution of the SLP tendency from lag −30 to 0 days corresponded well with the formation of AO and was maximized at lag 0 with a value of −0.7 hPa/day at the polar cap. After that, it gradually decreased and almost vanished at lag +30 days, in correspondence with the weakened meridional circulation.
Figure 9. Calculated meridional circulation (top) and the surface-pressure tendency (bottom) from the observed eddy forcings in Figure 7. Arrows indicate velocities on the meridional plane. The horizontal (vertical) reference arrow indicates 6 × 10−2 ms−1 (3 × 10−4 ms−1). The contour interval is 5 × 108 kgs−1 for the mass stream function, and the unit of vertical axis is 0.01 hPaday−1 for the surface-pressure tendency. Small vectors are neglected.
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 This analysis demonstrates the SLP tendency due to eddy forcings alone and should not be directly compared with the observed tendency. In fact, the observed SLP tendency is composed mainly of eddy forcing and frictional forcing and they are almost balanced, as will be described later. However, frictional forcing follows a given forcing, and thus the calculated SLP tendency should be a good indicator for the actual SLP anomaly, as can be seen by comparing the lower panels of Figures 4 and 9.
 We analyzed the roles of eddy forcings in the meridional circulation and the SLP by applying mechanical and thermal forcing separately. Deep dipole stratosphere-troposphere meridional circulation due to the thermal forcing (Figure 10b) was prominent throughout. Anomalous meridional circulation at high-latitude changed its polarity from clockwise at lag −30 days to counterclockwise at lag 0 to +15 days, corresponding to the change of polarity of the eddy thermal forcing. The polar SLP tendency changed from positive at lag −30 days to negative at lag 0 to +15 days, corresponding with the change of polarity of the meridional circulation. Comparing these results with that of the total forcing indicates that the deep meridional circulation seen in Figure 9 (and Figure 7) is due to thermal forcing, but the effect of the polar cap SLP is only about 40% at the stage when the AO-like signal appears (called as the annular stage).
 Figure 10a clearly indicates that the meridional circulation due to mechanical forcing was restricted primarily to inside the troposphere, centered around 55 to 60°N at 600 hPa. Its counterclockwise circulation gradually enlarged from lag −15 days, peaked around lag 0, and then gradually decreased. The tendency of the polar SLP also changed, corresponding with the change of meridional circulation. We found that approximately 60% of the polar SLP change resulted from eddy mechanical forcing at the annular stage.
 We also applied wave forcing separately in the stratosphere and troposphere (Figure 11). We separated the stratosphere and troposphere at 270 hPa for all latitudes, following Sigmond et al.  for simplicity. The results indicated that about 40% of the effect of polar-SLP tendency came from the stratosphere, and that it contributes to about 60% of the total thermal forcing and 30% of the total mechanical forcing (not shown). Note that the general features of meridional circulation and the SLP tendency due to stratospheric (tropospheric) forcing are very similar to those of thermal (mechanical) forcing, corresponding with the dominance of each forcing in each domain (Figure 7).
 Zonal wave number decomposition revealed that the greatest contribution to the meridional circulation and the effect of the SLP came from eddy forcing of the wave of the zonal wave number 1 component at the annular stage (Figure 12). The general features due to the zonal wave number 1 component were very similar to those from total eddy thermal forcing (Figure 10b). This corresponded with the large forcing in the stratosphere and revealed deep meridional circulation. It is notable that nearly 65% of the total decreased polar cap SLP tendency around the annular stage came from a zonal wave number 1 component. The residual portion is well explained by tropospheric forcing of zonal wave number 2 and 3 components, while the contribution from wave numbers equal to or greater than 4 was very small. The contribution of the zonal wave number 2 component dominated the meridional circulation in the troposphere, particularly in the lower troposphere.
Figure 12. Same as Figure 9 except for wave forcing from all waves (ALL), zonal wave number 1 (WN1), 2 (WN2), 3 (WN3), and 4 and higher components (WN4+) at lag 0 day.
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 No external forcings, except from eddies, were included in the previous analysis. Therefore the reaction due to wave forcings must be stationary, and the zonal wind and temperature must increase linearly with time (A9). In reality, this cannot occur, and an increase of the zonal wind and temperature should be balanced by frictional and radiative forcings. We calculated the residual of the primitive equation from six-hourly reanalysis data sets to estimate these forcings. The calculated meridional circulation and SLP tendency from non-eddy forcing at lag days −30, 0, and +30 are indicated in Figure 13. It can be seen that the frictional forcing at the surface was sufficiently large to create considerable meridional circulation, particularly in the subtropics. It is also notable that the SLP tendency was almost the same as that due to eddy forcing (Figure 9), except with the opposite signal. Assimilation data was created through a violation of the balance of equations by introducing the observational data. Therefore we cannot attempt to evaluate non-eddy forcing precisely by this method. However, we believe that its overall qualitative character is correct.
 We included the Rayleigh friction and Newtonian cooling terms in the equation and integrated the model with prescribed eddy forcings related with the PJO to examine the consistent time evolution of the zonal wind and meridional circulation for comparison with the observation. The Rayleigh friction coefficient (λ) and Newtonian cooling rate (α) are expressed by simple functional forms of height from surface (z) and are expressed as follows:
Constant values are R0 = 1/(0.5 days), R1 = 1/(30 days), and N1 = 1/(10 days); z is expressed in meters. The value of R0 and the functional dependence on height are roughly estimated from the frictional forcing calculated from the reanalysis data, and the small value of R1 is introduced for stability of the integration. The Newtonian cooling rate is determined so that the timescale of the radiative damping becomes 20 days in the troposphere and four days in the upper stratosphere. Time integration was then performed by the Matsuno (Euler backward) scheme, with a time step of 0.1 days and the initial condition set at day −35 (Figure 14). It was evident that the overall poleward and downward movement of the zonal wind and formation of the AO-like SLP structure at day 0 are reproduced well by the integration, though the wind in the upper stratosphere differs substantially from the observation. The meridional circulation is stronger than in the eddy-only case (Figure 9) due to the impact of frictional forcing on the zonal winds and becomes comparable to the observation (Figure 7). Thus the decreasing SLP of the polar cap due to eddy forcing around day 0 is balanced with the increasing SLP due to the meridional circulation arising from frictional forcing at the surface. Thus the role of frictional forcing through zonal winds, as well as eddy forcing, is important in explaining the formation of the AO-like SLP change.
Figure 14. Zonal-mean zonal wind (top), meridional circulation (middle), and sea level pressure (bottom) calculated by integrating the observed eddy forcings in the mechanistic model. Contour interval is 2 ms−1 for zonal wind and 5 × 108 kgs−1 for the mass stream function, and the unit is hPa for sea level pressure. The horizontal (vertical) reference arrow indicates 6 × 10−2 ms−1 (3 × 10−4 ms−1). Small vectors are neglected. See text for details.
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