Role of atmospheric waves in the formation and maintenance of the Northern Annular Mode

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: ATMOSPHERES, VOL. 118, , doi: /jgrd.50709, 2013 Role of atmospheric waves in the formation and maintenance of the Northern Annular Mode Yuhji Kuroda 1 and Hitoshi Mukougawa 2 Received 8 March 2013; revised 4 August 2013; accepted 5 August 2013; published 30 August [1] We examined the roles of stationary, synoptic, and medium-scale atmospheric waves in the formation and maintenance of the Northern Annular Mode (NAM) in winter by analyses of zonal momentum and wave energy based on six-hourly reanalysis data from the European Centre for Medium-range Weather Forecasts (ECMWF ERA-Interim). Medium-scale waves have periods shorter than about two days and their effect has been overlooked in previous research on the NAM. The effect of medium-scale waves on the NAM is about 10% of that of total eddies. Although their effect on the NAM is of less importance than their effect on the Southern Annular Mode (SAM), it is still significant in the Northern Hemisphere. Our analysis also suggests that synoptic, medium-scale, and stationary waves provide positive feedback to the zonal-mean zonal wind of the NAM. In particular, energy transfer from synoptic and low-frequency transient waves to stationary waves plays a key role in sustaining the stationary waves that drive zonal winds associated with the NAM. Citation: Kuroda, Y., and H. Mukougawa (2013), Role of atmospheric waves in the formation and maintenance of the Northern Annular Mode, J. Geophys. Res. Atmos., 118, , doi: /jgrd Introduction [2] Weather and climate are controlled by various types of atmospheric wave in the troposphere. Among these, synoptic waves are the most important because they are intimately related to weather in the extratropics. Medium-scale waves have recently begun to attract the attention of the climate research community. A series of studies on medium-scale waves [Sato et al., 1993; Sato et al., 2000, and references therein] have shown that they are distinct from synoptic waves in having wavelengths of about 2100 km with barotropic structure in the vertical direction and eastward phase speeds of about 22 ms 1, twice the speed of synoptic waves. They also found that medium-scale waves coexist with synoptic waves in the extratropics of both hemispheres. Kuroda and Mukougawa [2011] showed that medium-scale waves account for almost one third of the variability of the Southern Annular Mode (SAM), which is the dominant contributor to climate variability in the extratropical Southern Hemisphere. [3] The Northern Annular Mode (NAM), sometimes called the Arctic Oscillation [Thompson and Wallace, 1998], is the most prominent contributor to climate variability at high latitudes in the Northern Hemisphere [Limpasuvan and Additional supporting information may be found in the online version of this article. 1 Meteorological Research Institute, Tsukuba, Japan. 2 Disaster Prevention Research Institute, Kyoto University, Uji, Japan. Corresponding author: Y. Kuroda, Meteorological Research Institute, 1-1 Nagamine, Tsukuba, Ibaraki, , Japan. (kuroda@mri-jma.go.jp) American Geophysical Union. All Rights Reserved X/13/ /jgrd Hartmann, 1999]. The NAM is considerably more zonally asymmetric than the SAM and is most active in the North Atlantic. Unlike the SAM, activity of the NAM is much enhanced in winter. [4] The recently established importance of medium-scale waves in the formation of the SAM [Kuroda and Mukougawa, 2011] has raised the question: How do medium-scale waves contribute to the NAM? This is the motivation for this study. Additionally, as the importance of stationary waves and wave breaking of transient waves for the NAM has been noted in previous studies [e.g., DeWeaver and Nigam, 2000a, 2000b; Limpasuvan and Hartmann, 2000; Lorenz and Hartmann, 2003; Franzke et al., 2004; Kuroda, 2005], we extended our study to cover the effect of stationary and synoptic waves as well as medium-scale waves. [5] We focused on how atmospheric waves force zonalmean zonal wind variations associated with the NAM, and how these waves acquire energy from the zonal-mean fields. Our examination is based on zonal momentum and wave energy equations and shows that the interactions of these waves with the zonal-mean field sustain NAM variability through positive feedback. Our analysis also shows that mediumscale waves contribute to NAM variability. [6] Section 2 of this paper describes the data we used and our analysis method. Section 3 presents our results, section 4 provides discussion of our results, and section 5 presents our conclusions. 2. Data Used and Analysis Methods 2.1. Data Selection [7] The data we used in this study are the same as those used by Kuroda and Mukougawa [2011]: 21 years of 9048

2 six-hourly reanalysis data from the European Centre for Medium-range Weather Forecasts (ECMWF ERA-Interim) [Dee et al., 2011] from 1 January 1989 to 31 December However, we confined most of our analysis to the 20 winters from December to March of 1989/1990 to 2008/2009, as explained below. We analyzed the finestscale horizontal version of the data available for non- ECMWF member states; the data are represented on a longitude-latitude grid with 37 pressure levels in the vertical direction. [8] Although most of our analysis was based on monthly means, second-order or higher-order quantities such as Eulerian wave forcings were calculated first at 6 h intervals and then averaged over each month. We calculated frictional forcing and diabatic heating as residuals of three-dimensional momentum and thermodynamic equations using the sixhourly reanalysis data. [9] The NAM is defined as the dominant mode of variability of sea level pressure (SLP) during the Northern Hemisphere winter. To extract the NAM pattern, we applied Empirical Orthogonal Function (EOF) analysis to the monthly averaged SLP from December to March for the region north of 20 N as specified by Thompson and Wallace [1998]. Although the NAM appears in all seasons, we restricted our analysis to winter because the NAM has greater seasonal dependence than SAM and its maximum activity is in winter. EOF analysis showed that the dominant variability, the NAM, explains 21.7% of the total variance and greatly exceeds that of the second most dominant factor (14.2%). The NAM index is defined by standardized month-to-month time coefficients. [10] Synoptic and medium-scale waves were extracted from the six-hourly data using a Lanczos band-pass filter with a window width of 31 days [e.g., Hamming, 1977]. The same filter was used by Kuroda and Mukougawa [2011]. We defined synoptic waves as those with periods of two to six days and medium-scale waves as those with periods of less than 1.75 days, according to the definition of Sato et al. [2000]. As the frequency ranges of medium-scale waves and tidal waves overlap, after time filtering, we removed the tidal waves by discarding zonal wave number components from zero to two. Stationary waves were extracted by applying a 31 day running mean to the dailymean data, which imitates monthly averaging and corresponds to a loose low-pass filter with a cutoff period of about 50 to 90 days. In the following, we use 70 days for the typical cutoff period p for stationary waves as its transfer function becomes 1= ffiffiffi 2 for a period of 70 days. Transfer functions of the filters used in this study are shown in Figure S1. [11] To examine the relationships between NAM variability and the three types of atmospheric waves, we performed regression analysis of the NAM index with various physical quantities after removing seasonal cycles Diagnostic Equations [12] The model we used to evaluate the balance of zonalmean momentum is the same as that used by Kuroda and Mukougawa [2011] and is described as follows. u t ¼ 2Ω v sin ϕ þ F 1 þ F 2 þ F n þ X ; 2Ωu sin ϕ þ 1 Φ a ϕ ¼ J; (1a) (1b) Φ p ¼ RT p ; T t Γ ω ¼ Q 1 þ Q 2 þ Q n þ S; 1 a cosϕ ϕ ðv cosϕþþ ω p ¼ 0; where overbars represent zonal means and where the terms F 1 ¼ 1 a cos 2 ϕ ϕ u v cos2 ϕ ; F 2 ¼ 1 ρ 0 z ρ 0u w ; Q 1 ¼ 1 v T cosϕ ; a cosϕ ϕ Q 2 ¼ 1 ρ 0 z ρ 0w T ; (1c) (1d) (1e) are eddy mechanical forcings F 1 þ F 2 and thermal forcings Q 1 þ Q 2 with suffix 1 (2) signifying forcing from meridional (vertical) convergence; X and S are zonal-mean zonal frictional forcing and diabatic heating, respectively; Γ = T 0 / p + κt 0 /p is the stability of the basic atmosphere at temperature T 0 ( p); Ω is the angular velocity of the Earth; z is log-pressure height; ϕ is latitude; a is the radius of the Earth; ρ 0 (z) is the basic air density; and other terms follow the usual conventions [e.g., Andrews et al., 1987]. Here, nonlinear advection termsf n andq n are defined by the following equations: F n v u a ϕ u u v ω þ p a tanϕ; Q n v T T ω a ϕ p þ κ p ω T þ T p þ κt p Γ ω: [13] The violation of balanced wind J is defined by equation (1b). We used Γ(z) as the climatological stability obtained from monthly averaged temperature north of 30 N in an extended winter from November to April. [14] As equations (1a) to (1e) can be expressed as an elliptical differential equation of ω: 1 cosϕ ω cosϕ ϕ sin 2 þ 4Ω2 a 2 p 2 ω ϕ ϕ RΓ p 2 "!# ¼ 2Ωap cosϕ J RΓ cosϕ ϕ sinϕ p F 1 þ F 2 þ F n þ X 2Ω sinϕ 1 cosϕ Γ cosϕ ϕ sin 2 ϕ ϕ Q 1 þ Q 2 þ Q n þ S ; the meridional circulation can be calculated from equations (4) and (1e) with suitable boundary conditions. As equation (4) is completely linear with respect to the forcings F i, Q i, X, S, and J, the meridional circulation due to each forcing is evaluated separately. Acceleration of zonal winds due to each forcing can be directly evaluated from equation (1a) using the obtained meridional wind. [15] To evaluate zonal-mean zonal wind acceleration due to each wave forcing, it is also necessary to evaluate nonlinear advection terms F n and Q n, because the meridional circulation induced by wave forcing inevitably creates (2) (3) (4) 9049

3 (a) (b) (c) (d) (e) (f) (g) Figure 1. December to March means of (a) total zonal-mean zonal wind, and zonal-mean amplitudes of (b) stationary, (c) synoptic, and (d) medium-scale waves, each shown as height-latitude sections; and December to March means of 300 hpa level amplitudes in the Northern Hemisphere of (e) stationary, (f ) synoptic, and (g) medium-scale waves. Zonal-mean amplitudes in Figures 1b to 1d are the square root of the squared mean of the geopotential height anomaly at a given parallel of latitude. Amplitudes at the 300 hpa level in Figures 1e to 1g are the square root of the squared mean of the geopotential height anomaly for a given spatial point. Contour intervals are 5 m s 1 in Figure 1a, 20 m in Figure 1b, 10 m in Figures 1c and 1d, 50 m in Figure 1e, and 10 m in Figures 1f and 1g. Shading indicates the month-to-month standard deviations of deseasonalized variations: light, medium, and dark shading indicates standard deviations of 1, 2, and 3 m s 1 in Figure 1a; 5, 10, and 20 m in Figure 1b; 1, 2, and 5 m in Figures 1c and 1d; 10, 20, and 50 m in Figure 1e; and 5, 10, and 15 m in Figures 1f and 1g. nonlinear terms. Preliminary analysis showed that meridional circulation and zonal wind acceleration due to nonlinear terms alone are smaller than those from Coriolis and wave forcings, so we evaluated them by an iterative method as follows. First, only wave forcing terms were included in equation (4) to provide a first guess of the meridional circulation. Then, a first guess of nonlinear terms was calculated from the observed wind and temperature with the first-guess meridional circulation. These nonlinear terms, as well as wave forcings, were again put into equation (4) to provide more accurate meridional circulation. These iterations were repeated three times to achieve sufficient convergence. These calculations were performed for every day using daily averaged wave forcings. Note that the effects of nonlinear terms were ignored in the previous study of Kuroda and Mukougawa [2011]. [16] This model has the same resolution as that of Kuroda and Mukougawa [2011]: 100 vertical levels at intervals of 10 hpa and 121 horizontal grids of equal intervals of the sine of latitude. The Appendix of Kuroda and Kodera [2004] presents more details on solving these equations Energy Transfer Between Zonal Fields and Eddies [17] The model we used to evaluate energy transfer from zonal fields to atmospheric waves was the same as that used by Kuroda and Mukougawa [2011]. We used the equation for energy conversion between the zonal-mean field and an eddy by Holton [1975]: d n o n o ðk þ P Þ ¼ K; K þ P; P þ W þ D; (5) dt where K and K are the eddy and zonal-mean kinetic energies, P and P are the eddy and zonal-mean available potential energies, {K, K } is the energy conversion from K to K, {P, P } is the energy conversion from P to P, W is the surface contribution, and D is the external forcing or dissipation term. [18] The integrands ε(k, K ) andε(p, P ) of{k, K } and {P, P } are explicitly written as and ε K; K ¼ ρ0 u v 1 a þv w v z u ϕ þ u w u z þ v 21 v a ϕ tanϕ þ u u v v u 2 tanϕ ; a a (6) 9050

4 (a) All (b) (c) (d) (e) (f) (g) Figure 2. Height-latitude sections of December to March means of accelerations of zonal-mean zonal wind due to (a) all wave forcings; (b) stationary waves, (c) synoptic waves, (d) medium-scale waves; (e) frictional forcing; (f ) diabatic heating; and (g) the sum of all types of forcings. Accelerations were based on daily calculations for each forcing. Contour intervals are 0.5 m s 1 day 1 in Figures 2a and b, 0.2 m s 1 day 1 in Figure 2c, 0.1 m s 1 day 1 in Figure 2d, and 0.5 m s 1 day 1 in Figures 2e to 2g. The zero contour is shown as a thin solid line; dashed lines indicate negative values. Shading indicates the month-to-month standard deviations of deseasonalized variations: light, medium, and dark shading indicate standard deviations of 0.2, 0.3, and 0.5 m s 1 day 1 in Figures 2a and 2b; 0.05, 0.1, and 0.2 m s 1 day 1 in Figure 2c; 0.02, 0.05, and 0.1 m s 1 day 1 in Figure 2d; and 0.1, 0.3, and 0.5 m s 1 day 1 in Figures 2e to 2g. ε R 2 P; P ¼ ρ0 N 2 H 2! v T 1 T a ϕ þ w T T ; (7) z where the prime denotes the eddy component. The integrand ε(d) ofd is explicitly written as εðdþ ¼ ρ 0 u X þ v Y þ R2 ρ 0 H 2 N 2 S T ; (8) where X, Y, and S are the eddy components of zonal friction, meridional friction, and diabatic heating, respectively. [19] In computing the energy conversion rate of waves, a 31 day running average of zonal-mean temperature over the region from 30 N to 80 N was used as the basic state temperature T 1 (z). The squared buoyancy frequency N 2 was calculated from T 1. [20] To obtain accurate conversion rates, we neglected the contribution under orography as follows. To evaluate v T T= ϕ, a three-dimensional distribution of v T T = ϕmðxþis zonally averaged. Here, m(x) represents a masking function of x, defined by 0 (1) at the spatial point x which is located under (over) orography. For consistency, we applied a similar zonal averaging to the evaluation of wave forcings. 3. Results 3.1. Climatological Features [21] Figure 1 shows climatological parameters averaged over the winter season (December to March) and their 9051 month-to-month variabilities. Zonal winds attained their peak of 42 m s 1 at 200 hpa and 30 N (Figure 1a), and the stationary, synoptic, and medium-scale waves (Figures 1b to 1d) attained their peaks of about 172, 71, and 20 m at 250, 300, and 350 hpa and 50 N, poleward of the polar jet stream at 40 N. All of these peaks occurred around the tropopause and the altitudes of the peaks decreased with increasing frequency. The synoptic maps of atmospheric waves at the 300 hpa level (Figures 1e to 1g) show pronounced peaks in the amplitude of stationary waves over Europe (240 m) and the Far East (300 m). The peak over the Far East (Europe) corresponds to the trough (ridge) of stationary waves (not shown). The amplitudes of synoptic and medium-scale waves show peaks of 95 m and 25 m, respectively, over the Pacific sector and peaks of 100 m and 28 m, respectively, over the Atlantic sector. The peaks of medium-scale waves are downstream of those of the synoptic waves. [22] Figure 2 shows climatological zonal-mean zonal wind acceleration due to various forcings and their month-to-month variabilities during the winter season. The acceleration of zonal winds due to all waves (Figure 2a) shows two maxima: 1.7 m s 1 day 1 at 30 N and 400 hpa, and 2.3 m s 1 day 1 at 45 N and the surface. These peaks correspond well to the jet stream core of westerly winds extending from 30 N and 400 hpa to 45 N and the surface. Although the acceleration due to stationary waves (Figure 2b) is particularly dominant around the subtropical jet stream core and the lower troposphere, synoptic and medium-scale waves play an important

5 (a) (b) (c) Figure 3. Height-latitude sections of December to March means of totalenergy transfer from zonal-mean fields to (a) stationary, (b) synoptic, and (c) medium-scale waves (ε K; K þ ε P; P of equations (6) and (7)). Contour interval is Wm 3. Shading indicates month-to-month standard deviations of deseasonalized variations: light, medium, and dark shading indicates ,2 10 5,and Wm 3, respectively. Zero contour is plotted as a thin solid line and dashed lines indicate negative values. role in zonal wind acceleration at 40 N in the middle and lower troposphere (Figures 2c and 2d). It is interesting to note that the main contributor to acceleration of the subtropical jet stream around 30 N and 200 hpa is from stationary wave forcing (Figure 2b), and the forcing associated with diabatic heating decelerates the jet stream there (Figure 2f). Clearly, the speed of the subtropical jet stream is determined by the balance of these forcings. It is also noteworthy that a prominent quadrupole structure of zonal wind acceleration appears in the tropics due to diabatic heating. Acceleration of zonal winds due to atmospheric waves is well balanced with that associated with frictional forcing and diabatic heating (Figures 2e to 2g). [23] Figure 3 shows the climatological winter mean of total energy transfer from the zonal field to each atmospheric wave type (ε K; K þ ε P; P of equations (6) and (7)), and month-to-month variabilities. The energy transfer for stationary waves (Figure 3a) was stronger in the extratropical lower troposphere and peaked at Wm 3 at 60 N and the surface. Synoptic and medium-scale waves (Figures 3b and 3c) show the largest total energy transfer in the lower troposphere at around 40 N, with peaks of and Wm 3, respectively, decreasing with increasing altitude. Medium-scale waves have a much smaller total energy transfer rate than stationary and synoptic waves (Figure 3c). The climatological zonal-mean meridional circulation in winter induced by various forcings is shown in Figure S NAM Variability and Factors Contributing to Variability NAM Variability [24] Figure 4 shows SLP, zonal-mean zonal wind, and the Eulerian mass stream function associated with NAM variability, calculated by regression with respect to the NAM index. SLP shows a seesaw pattern between the polar cap and midlatitudes with centers over the Atlantic and Pacific (Figure 4a). The polar-cap signal is oval shaped with a peak of about 7.0 hpa east of Iceland, whereas the Atlantic center extends from the Atlantic to the Middle East with a peak of about 3.7 (a) (b) (c) Figure 4. Regression patterns with respect to the NAM index of (a) sea level pressure in the Northern Hemisphere, and height-latitude sections of (b) zonal-mean zonal wind and (c) the Eulerian mass stream function. The heavier shading indicates statistical significance at the 95% level (correlation higher than 0.30) according to Student s t test with an effective sample size of 42, which is derived from a sample size of 80 with a lag-one autocorrelation of 0.31 [e.g., Trenberth, 1984]. Contour intervals are 1 hpa in Figure 4a, 0.5 m s 1 in Figure 4b, and kg s 1 in Figure 4c. The zero contour line is shown as a thin solid line and dashed lines indicate negative values. 9052

6 (a) (b) (c) (d) (e) (f) (g) (h) (i) Figure 5. Height-latitude sections showing regression patterns with respect to the NAM index for (a to c) Eulerian meridional circulation, and (d to f) acceleration of zonal-mean zonal wind. The latitudinal trends of surface pressure changes (g to i) are also shown. The left column is for eddy forcings, the central column is for frictional forcings, and the right column is the sum of all forcings. Arrowheads in Figures 5a to 5c indicate velocities on the meridional plane. The contour interval is kg s 1 for Figures 5a to 5c, 0.1 m s 1 day 1 for Figures 5d to 5f, and the ordinate units for Figures 5g to 5i are hpa day 1. Shading is as described in caption of Figure 4. Small vectors were ignored. hpa west of Portugal. The Pacific center has a smaller peak of 1.8 hpa northeast of Japan. These features are similar to those obtained in previous studies [Thompson and Wallace, 2000], although the Pacific center we identified is somewhat smaller. The zonal-mean zonal wind pattern (Figure 4b) exhibits a meridional dipole with centers on the 300 hpa level at 55 N (2.5 m s 1 )and30 N( 1.5 m s 1 ). Anomalous positive winds at higher latitudes increase with altitude up to the middle stratosphere and are observed even at 1 hpa. The Eulerian mass stream function pattern (Figure 4c) shows a clear meridional dipole structure with anticlockwise circulation at higher latitudes and clockwise circulation at lower latitudes. Upwelling over the polar cap corresponds closely to anomalous low pressure there, and downwelling at 40 N corresponds to the anomalous high pressure zone there. The Eulerian mass stream function at higher latitudes peaks at kg s 1 at 650 hpa and 60 N and at lower latitudes it peaks at kg s 1 at 700 hpa and 25 N Effect of Atmospheric Waves on Variations of the Zonal-Mean Field [25] We first examined the overall features of zonal-mean variations associated with the NAM. Figure 5 shows the results of regression analyses with respect to the NAM index of changes induced by total eddies, friction, and the sum of all forcings for Eulerian-mean meridional circulation, zonal wind acceleration, and zonally averaged surface pressure change. The contribution of all forcings was obtained by linearly adding the contribution due to each forcing, including nonlinear terms, as was done by Kuroda and Mukougawa [2011]. We included nonlinear effects in evaluating the contribution of total eddies and friction. To evaluate the effect of zonal-mean friction, we included extrapolated frictional data under the topography where they are specified the same as those just on the topography. Without this procedure, the frictional effect cannot be well reproduced by our model due to the assumption of a flat lower boundary. [26] The total meridional mass flux driven by all wave forcings is kg s 1 (Figure 5a). Frictional forcing drives the meridional mass flux with a peak of kg s 1 in the lower troposphere (Figure 5b), which counteracts the zonal wind anomaly associated with NAM variability. For diabatic heating, nonlinear and violation of balanced wind terms, meridional circulation is significant only in the tropics (not shown). The meridional circulation induced by all forcings 9053

7 (a) (b) (c) (d) (e) (f) (g) (h) (i) Figure 6. Height-latitude sections showing regression patterns with respect to the NAM index for (a to c) zonal-mean wave amplitudes, and wave amplitudes in the Northern Hemisphere (d to f) at the 300 hpa level and (g to i) at the 1000 hpa level. The left column is for stationary waves, the central column is for synoptic waves, and the right column is for medium-scale waves. Contour intervals are 1 m in Figures 6a to 6c, 5 m in Figures 6d and 6g, and 2 m in Figures 6e, 6f, 6h, and 6i. Shading is as described in caption of Figure 4. (Figure 5c) is almost the same as that obtained in Figure 4c. In fact, the peak values of kg s 1 and kg s 1 in the lower troposphere for all forcings (Figure 5c) are almost the same as those of the Eulerian mass stream function (Figure 4c). These results confirm the accuracy of our diagnostic modeling. [27] The acceleration of zonal wind due to all atmospheric waves (Figure 5d) is 0.44 m s 1 day 1 at 60 N in the lower troposphere, which is about one third of the maximum value of mechanical forcing (1.5 m s 1 day 1 at 55 N and 250 hpa level; not shown). Frictional forcing (Figure 5e) decelerates the zonal wind through the troposphere between 40 N and 70 N. The contributions of diabatic and other forcings are small (not shown). The sum of all forcings (Figure 5f) shows that zonal wind acceleration is almost negligible. [28] Atmospheric waves reduce surface pressure over the polar cap and increase it at 50 N (Figure 5g). Frictional forcing has the opposite effect (Figure 5h). Contributions from diabatic and other forcings are small at all latitudes (not shown). Consequently, the surface pressure change outside the polar cap is negligible due to the balance between eddies and friction, again confirming the accuracy of our modeling (Figure 5i). [29] We next examined the influence of stationary, synoptic, and medium-scale waves on variations of the NAM. Figure 6 shows the results of regression analyses with respect to the NAM index of zonally averaged wave amplitudes and horizontal variations of wave amplitudes at the 300 and 1000 hpa levels. Comparison of zonal-mean wave amplitudes shows that the behavior of stationary waves is quite different from the others. Both synoptic and medium-scale waves exhibit barotropic dipole structures with centers at 60 N and 30 N (Figures 6b and 6c), whereas stationary waves show a quadrupole meridional structure in the middle troposphere that is out of phase with those of synoptic and medium-scale waves (Figure 6a). The peak value of the correlation coefficient of zonal-mean amplitude for mediumscale waves with the NAM index is 0.78 (not shown), which is higher than those of both synoptic waves (0.63) and stationary waves (0.60). Regressed zonally averaged peak wave amplitudes are 1.9 m at 350 hpa and 60 N for medium-scale waves (Figure 6c), 3.3 m at 250 hpa and 60 N for synoptic waves (Figure 6b), and 9 m at 1000 hpa and 70 N for stationary waves (Figure 6a). Stationary waves show another prominent negative peak of 7 m at 300 hpa and 70 N (Figure 6a). These higher latitude peaks reside poleward of 9054

8 (a) (b) (c) (d) (e) (f) Figure 7. Height-latitude sections showing regression patterns with respect to the NAM index for Eulerian wave forcings: (a to c) mechanical and (d to f) thermal forcings. The contour interval is 0.1 m s 1 day 1 in Figures 7a to 7c and 0.02 K day 1 in Figures 7d to 7f. Shading is as described in caption of Figure 4. the climatological peaks at around 50 N (Figures 1b to 1d). The horizontal amplitude variations of synoptic and medium-scale waves (Figures 6e, 6f, 6h, and 6i) associated with the NAM reach maxima over Europe, downstream of the climatological peak over the Atlantic (Figures 1f and 1g), and their variations are almost barotropic. [30] We next examined the role of each type of wave forcing on variations of the NAM. Figure 7 shows the results of regression analyses with respect to the NAM index for Eulerian mechanical forcings and thermal forcings for each type of atmospheric wave. Eulerian mechanical forcings have peaks of 0.66, 0.53, and 0.19 m s 1 day 1 at hpa and 50 N to 60 N for stationary (Figure 7a), synoptic (Figure 7b), and medium-scale waves (Figure 7c), respectively. The latitude of the peak of each mechanical forcing corresponds well to that of the regressed zonal wind anomaly (Figure 4b). Although the contributions from thermal forcing of synoptic and medium-scale waves (Figures 7e and 7f) are very small, that for stationary waves is very large (Figure 7d), exhibiting a tripole pattern with prominent peaks of 0.1, 0.16, and 0.15 K day 1 at 40 N, 55 N, and 75 N, respectively, in the lower troposphere, and another prominent peak of 0.12 K day 1 above the tropopause at 65 N (Figure 7d). [31] The meridional circulation, zonal wind acceleration, and surface pressure changes produced by each type of wave associated with variations of the NAM are shown in Figure 8. The induced meridional mass flux is kg s 1 for stationary waves (Figure 8a), kg s 1 for synoptic waves (Figure 8b), and kg s 1 for medium-scale waves (Figure 8c). [32] The acceleration of zonal-mean zonal wind due to stationary waves (Figure 8d) has peaks of 0.29 and 0.25 m s 1 day 1 in the lower and upper troposphere, respectively. Peaks due to synoptic and medium-scale waves are 0.13 m s 1 day 1 (at300hpa)and0.06ms 1 day 1 (at 650 hpa) at 50 N (Figures 8e and 8f), respectively. Comparison of the acceleration of zonal wind (Figure 8d) with mechanical forcing (Figure 7a) for stationary waves shows the former to be relatively larger (~1/2.5) due to the additive contribution from thermal forcing (Figure 7d). In contrast, the accelerations of zonal wind for synoptic and medium-scale waves (Figure 8e and 8f) are about one fourth to one third of their mechanical forcings (Figures 7b and 7c). The zonal wind acceleration due to all wave types (Figure 5d) shows that each of the three wave types has an important role in the acceleration of the zonal wind associated with NAM variability. In fact, the peak of the westerly acceleration due to stationary, synoptic, and medium-scale waves at 50 N to 60 N (Figures 8d and 8f) coincides well with the high-latitude center of the zonal wind variation (Figure 4b). [33] The contribution from stationary and synoptic waves (Figures 8g and 8h) explains most of the wave-induced surface pressure changes between 40 N and 60 N (Figure 5g). It is also clear that about two third of the surface pressure change poleward of 60 N (Figure 5g) comes from stationary and synoptic waves (Figures 8g and 8h) Effects of Waves of Various Periods and Horizontal Scales [34] We examined the effects of atmospheric waves on the NAM not only for specific wave types but also considering 9055

9 (a) (b) (c) (d) (e) (f) (g) (h) (i) Figure 8. Height-latitude sections showing regression patterns with respect to the NAM index for (a to c) Eulerian meridional circulation and (d to f) acceleration of zonal-mean zonal wind. The latitudinal trends of surface pressure changes (g to i) are also shown. The left column is for stationary waves, the central column is for synoptic waves, and the right column is for medium-scale waves. The contour interval is kg s 1 for Eulerian meridional circulation and 0.03 m s 1 day 1 for zonal wind acceleration. Shading is as described in caption of Figure 4. differences in wave periods and horizontal scales (wave number) to develop a more comprehensive examination as had been done by Kuroda and Mukougawa [2011]. We applied a set of high-pass Lanczos band-pass filters to the six-hourly data set to isolate wave components with periods ranging from less than 1 day to 70 days. [35] Figure 9 shows the regressed Eulerian mechanical forcing, induced meridional circulation, zonal wind acceleration, and surface pressure changes associated with the Eulerian wave forcings (mechanical plus thermal forcings) for various wave periods. Mechanical forcing tends to increase with increasing period up to 6 days (Figures 9a to 9d). The peak value for waves with a period of less than 1 day is only 0.06 m s 1 day 1 (Figure 9a), but it increases from 0.17 to 0.39 m s 1 day 1 as wave period increases from 1 to 6 days (Figures 9b to 9d). For waves with periods of 6 10 days, wave forcing and induced circulation are very small (not shown). For the wave components with periods between 10 days and that of stationary waves (low-frequency transient waves; LFT), mechanical forcing reaches 0.29 m s 1 day 1 around 70 N (Figure 9e). The thermal forcing for each wave component (not shown) is very small, and it reaches 0.05 K day 1 for LFT. [36] These results show that the meridional circulation centered at 50 N due to all wave types (Figure 5a) is produced mainly by waves with periods of 1 6 days (Figures 9g to 9i). Moreover, the correlation between mechanical forcing and the NAM index exceeds 0.5 for each type of wave for periods of less than 6 days. Thus, the small contribution to meridional circulation of waves of period less than 1 day reflects the relatively small climatological variability of such waves. [37] The peak value of zonal wind acceleration for wave components with periods of 3 6 days (0.11 m s 1 day 1 ; Figure 9n) is larger than that for wave components with periods of 1 2 days (0.06 m s 1 day 1 ; Figure 9l) and 2 3 days (0.06 m s 1 day 1 ; Figure 9m), and less than those with periods of 1 day (0.02 m s 1 day 1 ; Figure 9k). Note that the effect of LFT is to decelerate the zonal-mean zonal wind at 55 N ( 0.1 m s 1 day 1 ; Figure 9o), which is similar to the effect of frictional forcing (Figure 5e). [38] The surface pressure changes at 40 N come primarily from wave components with periods of 1 6 days (Figures 9q to 9s), whereas LFT contribute a positive peak at 60 N and a negative peak in the polar-cap region (Figure 9t). It is noteworthy that the contribution of atmospheric waves to surface pressure changes in the polar region increases with 9056

10 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) (s) (t) Figure 9. Height-latitude sections showing regression patterns with respect to the NAM index for different wave periods for Eulerian mechanical forcing (first row), induced meridional circulation (second row), and zonal wind acceleration (third row). Regressions of thermal forcing are not shown. The latitudinal trends of surface pressure changes (bottom row) are also shown. Wave periods range from less than 1 day (left column) to between 10 days and stationary-wave period (right column). Contour intervals are 0.05 m s 1 day 1 in Figures 9a to 9e, kg s 1 in Figures 9f to 9j, and 0.02 m s 1 day 1 in Figures 9k to 9o. Shading is as described in caption of Figure 4. increasing period (note that the vertical scales of Figures 9s and 9t are different from those of the other panels). [39] We also examined the contributions of atmospheric waves with different zonal wave numbers. Figure 10 shows the contributions to Eulerian mechanical and thermal forcings and the induced meridional circulation and surface pressure changes for waves of different wave number. All wave components except those of wave numbers 3 and 4 make large contributions to mechanical forcing (Figures 10a to 10d). For thermal forcing, only wave components with wave numbers 1 (Figure 10e) and 2 (Figure 10f) make significant contributions. Wave components with wave numbers 1 and 2 show quasi-stationary behavior (not shown) and they will be forced by geographical distribution around the Arctic. Wave components with wave numbers 1 and 2 drive a larger meridional circulation than the other wave components because both mechanical and thermal wave forcings contribute to the formation of the meridional circulation of these waves (not shown). These wave components also force large surface pressure changes (Figures 10m and 10n). Wave components with wave numbers 5 9 show wave forcings (Figures 10c and 10g), meridional circulations (Figure 10k), zonal wind accelerations (not shown), and surface pressure changes (Figure 10o) that are similar to those due to synoptic waves (Figures 7b and 7e, 8b, 8e, and 8h), but the contributions of wave numbers 5 9 are slightly larger. This is because wave numbers 5 9 and synoptic waves are represented by similar wave components (not shown). The contribution of wave components with wave numbers larger than 10 (Figures 10d, 10h, 10l, and 10p) is about twice that due to medium-scale 9057

11 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) Figure 10. Height-latitude sections showing regression patterns with respect to the NAM index for selected wave numbers (WNs) for mechanical forcing (first row), thermal forcing (second row), and induced meridional circulation (third row). Regressions for zonal wind accelerations are not shown. The latitudinal trends of surface pressure changes (bottom row) are also shown. WNs range from 1 (left column) to greater than 10 (right column). Contour intervals are 0.1 m s 1 day 1 in Figures 10a to 10d, 0.05 K day 1 in Figures 10e to 10h, and kg s 1 in Figures 10i to 10l. Shading is as described in caption of Figure 4. waves (Figures 7c and 7f, 8c, 8f, and 8i). This is because waves with longer periods but smaller horizontal scales are beyond the parameters that define medium-scale waves Feedback Between Zonal Fields and Eddies [40] Our analyses show that the three types of atmospheric waves considered here amplify the zonal-mean zonal wind anomaly associated with NAM variability. Previous studies [DeWeaver and Nigam, 2000a, 2000b; Lorenz and Hartmann, 2003] have also suggested that the NAM (or North Atlantic Oscillation) is sustained by positive feedback between atmospheric waves and zonal winds such that the waves reinforce westerly zonal winds, which in turn amplify the wave forcings. The addition of energy to each wave amplifies its amplitude and wave forcings. To examine how the waves are amplified by their interactions with mean zonal-mean fields, wave-energy conversion (equations (6) and (7)) for stationary, synoptic, and medium-scale waves has been evaluated. Figure 11 shows the regressed total ε K; K þ ε P; P and potential energy conversion ε P; P with the NAM index. The patterns of total energy transfer for all wave types (Figures 11a to 11c) show a 9058

12 (a) (b) (c) (d) (e) (f) Figure 11. Height-latitude sections showing regression patterns with respect to the NAM index of total wave energy transfer (a to c) and potential energy transfer (d to f) from zonal-mean fields to eddies. Left column is for stationary waves, center column is for synoptic waves, and right column is for medium-scale waves. Contour interval is Wm 3 in Figures 11a and 11d and Wm 3 in Figures 11b, 11c, 11e, and 11f. Shading is as described in caption of Figure 4. meridional dipole structure with a positive peak at high latitude and a negative peak at lower latitude. The positive (negative) peak of ( )Wm 3 for stationary waves is at 65 N (45 N) in the lowermost troposphere (Figure 11a). The corresponding peaks for synoptic and medium-scale waves are ( )Wm 3 at 55 N (40 N) in the middle troposphere, and ( )Wm 3 at 50 N (40 N) in the lower troposphere, respectively. Our analyses also show that, except for stationary waves, the kinetic energy conversion rate is much smaller than the potential energy conversion rate. Most of the potential energy conversion is derived from the term that is proportional to v T T = ϕ in equation (7). The meridional patterns of total energy conversion rate for each wave type (Figures 11a to 11c) are very similar to those of their zonal wind accelerations, especially in the lower and middle troposphere (Figures 8d to 8f). Thus, our analysis shows that potential energy conversion is the main contributor to the amplification of each wave type, and there is a positive feedback process between the zonal-mean zonal wind field and each wave type. [41] We also performed lagged regression analyses of the NAM index with 31 day running averaged zonal wind acceleration (Figure 12a) and total energy conversion (Figure 12b) for each wave type. The specific part of the lower troposphere covered in these analyses (950 to 800 hpa and 55 N to 60 N) was selected because the highest correlation (>0.9) between zonal-mean zonal wind and the NAM index (Figure 4b) exists there. A positive lag means precedence of the NAM variation. The zonal-mean zonal wind in this part of the troposphere has the highest correlation with the NAM index, with a lag of +1 day (not shown). Here, zonal wind acceleration was calculated using the lagged regression of wave forcings, as for Figure 8. The peaks of zonal wind acceleration of 0.23, 0.08, and 0.02 m s 1 day 1 appear at 11, 0, and +20 days for stationary, synoptic, and medium-scale waves, respectively (Figure 12a). For medium-scale waves, though the peak time does not precede the NAM index, there is a marked increase when the time lag approaches day 0. These results show that each of the three wave types has a role in driving NAM variations in this part of the troposphere, and that the contribution from stationary waves is largest. The number of days of precedence for each wave type should correspond to its time scale (period). The peak values for energy conversion from the zonal field of , , and Wm 3 appear at +6, +2, and +3 days for stationary, synoptic, and medium-scale waves, respectively (Figure 12b). The peak for each wave type is lagged with respect to the NAM index, which indicates that wave generation tends to peak after the NAM has matured. Because the generation of atmospheric waves accelerates the zonal-mean zonal wind, the life span of the NAM is prolonged by this positive feedback process. [42] We conducted the same regression analyses for the entire extratropical region (Figures 12c and 12d). The part of the troposphere we selected for these analyses (poleward of 45 N and from the surface to the 100 hpa level) corresponds to the region of anomalous westerly winds associated with the NAM (Figure 4b). The zonal wind acceleration for each 9059

13 (a) (b) (c) (d) Figure 12. Lagged regressions with respect to the NAM index of (a) zonal-mean zonal wind acceleration due to each wave type and (b) total energy transfer from zonal fields to each wave type averaged between 55 N and 60 N and between the 950 and 800 hpa levels. (c) Estimated power transferred from eddies to zonal wind and (d) from zonal fields to eddies averaged for the area north of 45 N and below the 100 hpa level. Curves are blue for stationary waves, black for synoptic waves, and red for medium-scale waves. Abscissas show the lag (days) against the NAM index. Ordinate units are 1 m s 1 day 1, Wm 3, W, and W for Figures 12a, 12b, 12c, and 12d, respectively. wave type was converted to power density P (W m 3 )by the expression u P ¼ ρ 0 u ; (9) t e where u= t is estimated zonal wind acceleration for each e wave type (equation (1a)); its integration is shown in Figure 12c. The peak values of , , and W appear with lags of +2, 10, and +4 days for stationary, synoptic, and medium-scale waves, respectively. However, the peaks for stationary and medium-scale waves are very flat at around day 0. The zonal wind acceleration due to synoptic waves in the entire extratropical region (Figure 12c) is more important than that in the smaller region (Figure 12a) and occurs very close to the time of the peak for stationary waves in the smaller region (Figure 12a). Precedence of the peak zonal wind acceleration with respect to the NAM index is observed for only synoptic waves in the extratropical region. Thus, synoptic waves have a primary role in driving the zonal wind anomaly for the entire extratropical region. Our analysis of the integrated energy transfer rate from the zonal-mean zonal field to each wave type in the entire extratropical region (Figure 12d) shows peak values of , , and W with lags of +11, +2, and 1 days for stationary, synoptic, and medium-scale waves, respectively. The energy transfer from the zonal-mean field to synoptic waves is much larger than those to other wave types (Figure 12d). Thus, synoptic waves play a dominant role in the interaction of atmospheric waves with zonal-mean zonal wind flow in the entire extratropical region. The overwhelming predominance of synoptic waves is because the energy transfer to stationary waves occurs mainly near the surface (Figure 11a), whereas those for synoptic and medium-scale waves occur over a broader range of heights (Figures 11b and 11c). The large negative energy transfer south of 55 N for stationary waves (Figure 11a) also contributes to these differences. [43] The small energy conversion from the zonal-mean field to stationary waves (Figure 12d) suggests that stationary waves do not significantly amplify and hence do not accelerate it greatly. However, the acceleration that is present is controlled mostly by stationary waves (Figure 12c), which suggests that stationary waves obtain energy from other sources. Because equation (5) is exactly satisfied when all wave components are included in the eddy, when we consider a particular wave type, additional terms representing the energy exchange between the other waves should be included [Saltzman, 1957]. In the present case, the additional terms representing energy exchange with other wave components include many terms (see equation (S3.6)). However, we decided to evaluate the net effect of these terms as the residual of the energy conservation equation for a specific wave component because direct computation of all the terms in equation (S3.6) would accumulate numerical errors due to the high number of computational operations. We checked the accuracy of our numerical method by computing the energy budget for all waves, which does not include any energy exchange terms among the wave components. Table S4, listing each component of the climatological energy budget for the region north of 45 N and below 100 hpa, indicates that the discrepancy from energy conservation is negligible. The magnitude of the discrepancy inferred from the residual (SUM) is only 5% of the smallest forcing term (DIAB; diabatic heating). 9060

14 (a) (b) wave-wave interactions between stationary and other waves, especially synoptic waves and LFT. However, further work is needed to obtain a more detailed understanding of this phenomenon. (c) Figure 13. Lagged regressions of changes of energy of (a) stationary, (b) synoptic, (c) medium-scale waves, and (d) LFT with respect to the NAM index over the entire extratropical region (north of 45 N and below the 100 hpa level). Curves are black for change of total energy, red for energy input from zonal fields, blue for energy flow from the boundary of the area analyzed, and green for energy input from other waves. Abscissas show the lag (days) against the NAM index. Ordinate units are W. [44] Figure 13 shows the results of lagged regressions with respect to the NAM index of energy changes for each wave type integrated over the entire extratropical region according to equation (5), thus deriving the time derivative of wave energy d(k + P )/dt, energy outflow at the boundary of the area analyzed W, total energy conversion rate from the zonalmean field K; K þ P; P, and the energy exchange rate between wave types estimated by Δ ¼ d n o ðk þ P Þ K; K dt n o P; P W D: (10) [45] Contributions from diabatic heating and frictional forcing D are relatively smaller and are not shown in Figure 13. These analyses show that stationary waves receive a very large energy input from other waves (Figure 13a), whereas synoptic waves and LFT provide a substantial output of energy to other waves. For mediumscale waves, there is a relatively small energy input from other waves (note that the vertical scale of Figure 13c is different from others). Although LFT tends to reduce the zonal-mean zonal wind anomaly associated with NAM variability (Figure 9o), it also contributes energy to stationary waves, especially during the growing stage of the NAM. It is also noteworthy that energy conversion from the zonal-mean field is quite small for LFT. [46] The results of our analyses suggest that a positive feedback between stationary waves and the zonal-mean flow is established through the energy conversion of (d) 4. Discussion [47] We found that stationary, synoptic, and medium-scale waves all tend to accelerate the zonal-mean zonal wind anomaly associated with the NAM, and that the zonal-mean field feeds energy to these waves. Thus, these two processes constitute a two-way feedback mechanism that sustains NAM variability. Previous studies [e.g., DeWeaver and Nigam, 2000a, 2000b; Lorenz and Hartmann, 2003] have suggested that both stationary and synoptic waves sustain NAM variability through a positive feedback mechanism. Here, we have shown that medium-scale waves (of shorter period than synoptic waves) also sustain NAM variability through a positive wave-mean flow interaction, although their contribution is smaller than those of synoptic and stationary waves and constitutes about 10% of the zonal-mean zonal wind acceleration. Our analysis also shows that although stationary waves are the most important drivers of the zonal-mean zonal wind anomaly associated with NAM variability, synoptic waves also play a significant role. Synoptic waves not only directly drive zonal-mean flow, but also feed energy to stationary waves, which in turn accelerates the zonal-mean zonal wind anomaly. [48] We used the NAM to represent the dominant variability of the zonal-mean field in the Northern Hemispheric winter. However, whether the NAM is really a dynamical mode of variability is still controversial, which is not the case for the SAM [e.g., Deser, 2000; Feldstein and Franzke, 2006; Itoh, 2008]. We have undertaken the same analysis using the North Atlantic Oscillation [e.g., Hurrell et al., 2003] index, but the overall results remain the same. Hence, our results are not crucially distorted as far as we discuss the zonal-mean zonal wind variation in the Northern Hemisphere. [49] We found that the contribution of stationary waves to NAM variability is very important. Stationary waves tend to accelerate the zonal-mean zonal wind anomaly associated with the NAM, which is very different from the situation for the SAM. Stationary waves in the Southern Hemisphere have a climatological peak amplitude of 107 m, which is about two third of that of stationary waves in the Northern Hemisphere winter. Moreover, it has been shown that both stationary waves and LFT tend to decelerate the zonal-mean zonal wind anomaly associated with the SAM (not shown). To explain the relationship between stationary eddies and the zonal-mean flow anomaly associated with the NAM, Kimoto et al. [2001] proposed a tilted-trough mechanism, in which the anomalous zonal wind changes the meridional structure of stationary waves, which in turn causes a wave forcing to accelerate the anomalous zonal wind. This mechanism explains well why the stationary wave amplitude at 60 N in the tropopause does not increase for the NAM (Figure 6a). However, from the energetics point of view, acceleration of zonal wind accompanies energy transfer from stationary waves to the zonal-mean field. So some energy input is needed to maintain the energy of stationary waves. Amplification of waves and the associated energy transfer associated with NAM variability in the troposphere 9061