Polar Vortex Oscillation Viewed in an Isentropic Potential Vorticity Coordinate

Transcription

1 ADVANCES IN ATMOSPHERIC SCIENCES, VOL. 23, NO. 6, 2006, Polar Vortex Oscillation Viewed in an Isentropic Potential Vorticity Coordinate REN Rongcai 1,2 (?Jç) and Ming CAI,2 1 State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics (LASG), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing Department of Meteorology, Florida State University, Tallahassee, Florida 32306, USA (Received 16 January 2006; revised 14 August 2006) ABSTRACT The stratospheric polar vortex oscillation (PVO) in the Northern Hemisphere is examined in a semi- Lagrangian θ-pvlat coordinate constructed by using daily isentropic potential vorticity maps derived from NCEP/NCAR reanalysis II dataset covering the period from 1979 to In the semi-lagrangian θ-pvlat coordinate, the variability of the polar vortex is solely attributed to its intensity change because the changes in its location and shape would be naturally absent by following potential vorticity contours on isentropic surfaces. The EOF and regression analyses indicate that the PVO can be described by a pair of poleward and downward propagating modes. These two modes together account for about 82% variance of the daily potential vorticity anomalies over the entire Northern Hemisphere. The power spectral analysis reveals a dominant time scale of about 107 days in the time series of these two modes, representing a complete PVO cycle accompanied with poleward propagating heating anomalies of both positive and negative signs from the equator to the pole. The strong polar vortex corresponds to the arrival of cold anomalies over the polar circle and vice versa. Accompanied with the poleward propagation is a simultaneous downward propagation. The downward propagation time scale is about 20 days in high and low latitudes and about 30 days in mid-latitudes. The zonal wind anomalies lag the poleward and downward propagating temperature anomalies of the opposite sign by 10 days in low and high latitudes and by 20 days in mid-latitudes. The time series of the leading EOF modes also exhibit dominant time scales of 8.7, 16.9, and 33.8 months. They approximately follow a double-periodicity sequence and correspond to the 3-peak extratropical Quasi-Biennial Oscillation (QBO) signal. Key words: polar vortex oscillation, semi-lagrangian θ-pvlat coordinate, poleward and downward propagation doi: /s Introduction * cai@met.fsu.edu The wintertime stratospheric circulation is dominated by the cyclonic vortex centered over the polar area. It has been recognized that the annular mode, the leading climate variability pattern of large-scale circulation anomalies, is intimately related to the oscillation between a weak and strong stratospheric polar vortex (Baldwin and Dunkerton, 1999; Waugh and Randal, 1999; Thompson et al., 2002; Limpasuvan et al., 2004). In general, a quasi-zonal circulation is associated with a strong polar vortex. However, a strongly zonally asymmetric circulation may not necessarily always correspond to a weak polar vortex. For example, a strongly zonally asymmetric circulation can be associated with a strong polar vortex that is not centered over the polar area. Therefore, the regular zonal average along latitude circles may not be able to reflect the intensity of the polar vortex in all cases. To faithfully capture the intensity variability of the polar vortex, one has to do the averaging in a Lagrangian coordinate. Isentropic potential vorticity (Ω) can be approximately regarded as material lines because it can be altered only by irreversible mixing and diabatic/frictional processes (Hoskins et al., 1985; Haynes and McIntyre, 1987). Many weather phenomena, such as cyclones, cut-off lows, blocking highs, and jet streams can be vividly identified on daily isentropic Ω

2 NO. 6 REN AND CAI 885 maps (Hoskins et al., 1985). The zone of the strongest Ω gradient outlines the location of the westerly jet surrounding the polar vortex. Many basic features such as the intensity, geographical location, geometric shape, and size of the polar vortex can be simultaneously tracked using a few key Ω contours (Baldwin and Holton, 1988; Waugh and Randel, 1999). Recently, Cai and Ren (2006a, b ) examined the climate variability of atmospheric anomalies in a semi-lagrangian coordinate constructed using constant isentropic (θ) and potential vorticity (Ω) surfaces. Instead of just following a few key isentropic Ω contours to outline the polar vortex itself as in Baldwin and Holton (1988) and Waugh and Randel (1999), they constructed the semi-lagrangian coordinate system by converting the area of the spherical cap encircled by each Ω contour on an isentropic Ω map to its equivalent latitude referred to as the PVLAT. Unlike the regular Eulerian longitude-latitude coordinate system, the semi-lagrangian longitude-pvlat coordinate system itself evolves in time because it follows the Ω contours. The 2 D flow in the θ-pvlat coordinate (PVLAT as the meridional axis and θ the vertical axis) is obtained by averaging the original 3 D field along PVLAT (or Ω contours), rather than along latitude circles. Because of the conservation property of the potential temperature and potential vorticity, the grids in the θ-pvlat coordinate can be regarded as natural boundaries separating air masses of different properties. The zonal mean along the PVLAT is very close to a Lagrangian averaging, capturing both the thermodynamic and dynamic properties of the same air mass more accurately compared to the conventional zonal mean along the geographical latitudes. In this paper, we wish to examine the climate variability of the polar vortex in such a semi- Lagrangian coordinate system that constantly evolves with time. Because the variability of the polar vortex in the semi-lagrangian θ-pvlat coordinate can be solely attributed to its intensity variability, we are able to isolate the most fundamental underlying dynamic/thermodynamical processes associated with the Polar Vortex Oscillation (PVO). The paper is organized as follows. The next section describes the data used in this study and concepts of PVLAT coordinate. Reported in section 3 are the characteristics of the PVO index derived from the Empirical Orthogonal Function (EOF) analysis of Ω anomalies in the θ-pvlat coordinate. Section 4 discusses the temporal evolution of the PVO index and its association with a systematic simultaneous poleward and downward propagation of circulation anomalies of both signs from the tropics to the pole and from the stratosphere to the troposphere. We will also show that in the regular latitude coordinate, the poleward propagation signal is diluted, appearing more like a quasi-standing seesaw oscillation between the subtropics and extratropics because the averaging is not along the material coordinate. Section 5 discusses the interannual variability of the poleward and downward propagation associated with the PVO. The summary and discussion are given in section 6. The Appendix documents the algorithm for constructing the semi- Lagrangian coordinate from daily isentropic Ω maps. 2. Data and the θ-pvlat coordinate The data used in this study are derived from the daily isentropic analysis (00Z) of the NCEP/NCAR (National Centers for Environmental Prediction/National Center for Atmospheric Research) reanalysis II dataset from 1 January 1979 to 31 December 2003 (Kalnay et al., 1996). The isentropic analysis includes the zonal and meridional winds, potential vorticity, temperature, Montgomery potential, relative humidity, and Brunt-Väisälä frequency square. There are 11 isentropic surfaces (the standard NCEP/NCAR reanalysis isentropic levels: θ=270, 280, 290, 300, 315, 330, 350, 400, 450, 550, and 650 K), extending from the lower troposphere to the mid-stratosphere approximately around 20 hpa. On each of the isentropic surfaces, the data are defined on grids (or resolution) covering the entire spherical surface from the South Pole to the North Pole. Daily isentropic Ω fields are used to construct the semi-lagrangian θ-pvlat coordinate with θ representing the vertically increasing constant potential temperature surfaces and PVLAT the meridional coordinate representing northward increasing Ω by assigning individual Ω contours on an isentropic surface to a latitude value that the area of the spherical cap encircled by the Ω contour is identical to that encircled by the latitude circle (Norton, 1994). Readers may refer to the Appendix for the details of the algorithm used to convert Ω contours to PVLAT in this paper. Using the algorithm outlined in the Appendix, we have created maps of PVLAT from IPV maps on daily basis. Figure 1 shows an example of such a mapping from Ω contours (contours) to PVLAT (shadings), Cai, M., and R.-C. Ren, 2006: Meridional and downward propagation of atmospheric circulation anomalies. Part I: Northern Hemisphere Cold Season Variability. J. Atmos. Sci., in press.

3 886 POLAR VORTEX OSCILLATION VIEWED IN AN ISENTROPIC POTENTIAL VORTICITY COORDINATE VOL. 23 Fig. 1. A map of the Ω field (contours) at θ=650 K taken on 2 February The shadings are the corresponding PVLAT. The color bar on the top is for Fig. the1 Northern A map of the Hemisphere PV field (contours) (orat positive θ = 650K taken Ω) PVLAT on February and 2, that The inshadings the bottom are the is corresponding for the Southern PVLAT. Hemisphere The color bar on (or the top positive is for the Ω) Northern PVLAT. Hemisphere (or positive PV) PVLAT and that in the bottom is for the Southern Hemisphere (or positive PV) PVLAT. taken on 2 February 1996 at θ=650 K. It is seen that on this day the circulation in the Northern Hemisphere extratropics is highly zonally asymmetric, dominant by a wavenumber two pattern in high latitudes. The north pole (e.g., the maximum Ω) in the PVLAT coordinate at θ=650 K on that particular day is over the Hudson Bay where the maximum Ω is equal to 300 PVU (1 PVU=10 6 m 2 s 1 K kg 1 ). The northern polar cap in the PVLAT coordinate (as indicated by those blue shadings) on that Ω map is like a dumbbell with two centers over the North America and Western Europe. Between the two high Ω centers, there is a low Ω tongue over the Northern Pacific in which PVLAT can be as low as 30 N in the regular latitudes of N. Low Ω also appears in the Northern Atlantic. In the low latitudes, the flow is relatively more zonally symmetric where the PVLAT is very close to the actual latitude. February 2 corresponds to a summer day in the Southern Hemisphere (SH). The flow is quite zonally symmetric. As a result, the PVLAT is nearly parallel to the actual latitude for the SH flow shown in Fig. 1. It is of interest to point out that there is a systematic intrusion of the negative Ω into the Northern Hemisphere (NH) near the equator as evident from the presence of negative PVLAT on the NH side of the equator where negative Ω is recorded. It is straightforward to obtain the 2-D fields of Ω and other variables in the θ-pvlat coordinate by averaging the original 3-D daily isentropic analyses along the PVLAT according to [X] PVLAT = X cos φdl/ cos φdl (1) Ω where φ is the latitude and the generic variable X could be Ω, wind, temperature, or other variables; [ ]dl is the line integral along a PVLAT (or a Ω Ω contour). Shown in Fig. 2 are the Ω, zonal wind, temperature fields averaged along latitudes (thin curve) and PVLAT (curve with solid dots) at 650 K taken on the same day as Fig. 1. It is seen 29 that on that particular day, the stratosphere polar jet is located at about 60 N where the meridional temperature gradient is the strongest. The maximum zonal mean zonal wind is slightly less than 30 m s 1 and the corresponding meridional temperature contrast from 55 to 75 N is about 5 K. In contrast, the maximum zonal mean zonal wind averaged along the PVLAT is close to 50 m s 1 and the temperature drops as much as 18 K from 55 N to 75 N in PVLAT. The maximum meridional Ω gradient is observed between 60 N to 70 N in PVLAT within which Ω increases by 100 PVU, nearly as twice large as that in the regular latitude coordinate. The zonal average along latitudes differs little from that averaged along PVLAT in the summer stratosphere. The results above are examples showing that the θ- PVLAT coordinate can be regarded as natural bound- Ω

4 NO. 6 REN AND CAI 887 PV (PVU) U (ms -1 ) T (K) Fig. Fig.2 2. The zonal mean mean of (a) of PV (a)(unit: Ω (unit: PVU), PVU), (b) zonal (b) zonal wind wind (unit: (unit: ms 1 ), mand s 1 (c) ), temperature and (c) temperature (unit: K) (unit: θ = K) at θ= K taken K taken on February on 2 February 2, The dotted The curves are with solidobtained dots are by averaging obtainedalong by averaging PVLAT and along the thin PVLAT curves and thinby curves averaging by averaging along the regular along the latitudes. regular latitudes. aries separating air masses of different properties. Particularly, the zonal mean along the PVLAT is very close to a Lagrangian averaging, capturing both the thermodynamic and dynamic properties of the same air mass more accurately compared with the conventional zonal mean along the geographical latitudes. 3. EOF analysis of Ω anomalies in the θ-pvlat coordinate The annual cycle calculated in the θ-pvlat (Lagrangian) coordinate is different from that in the conventional longitude-latitude (Eulerian) coordinate because the PVLAT coordinate itself varies in time following Ω contours. We have calculated the daily annual cycle (obtained by first averaging the daily data on each calendar day and then applying a 31-day running mean) and anomalies in both regular longitudelatitude and PVLAT coordinates. The difference between the two is small except that the amplitude of the Lagrangian anomalies is slightly smaller than that of the Eulerian anomalies because the amplitude of the Lagrangian annual cycle is stronger than that (a) (b) (c) calculated in the Eulerian sense. The daily anomalies reported below are obtained in the θ-pvlat by taking out the annual cycle from the total field in the θ-pvlat coordinate except those displayed in Figs which are obtained on the conventional longitudelatitude grids. Figure 3 shows the first two EOF modes derived from the daily Ω anomalies in the θ-pvlat coordinate. These two modes, respectively, account for 69% and 13% of the total variance of daily Ω anomalies over the entire Northern Hemisphere. The first EOF mode shows a tripolar pattern from the equator to the pole in the stratosphere and a dipole pattern in the lower portion of the atmosphere. The largest Ω anomalies are in the stratospheric polar cap, which are negatively correlated with the Ω anomalies in the mid-latitudes and positively correlated with that in the tropics. It is also apparent that the tropospheric Ω anomalies in high latitudes are negatively correlated with the Ω anomalies above. The second EOF mode exhibits a quadruple pattern along the meridian. The daily time series of the first two EOF modes are displayed in Fig. 4. It is seen that in spite of using daily data, the time series of the Ω anomalies in the θ- PVLAT coordinate have little signal at synoptic time scales. Instead, they exhibit a very rich spectrum at Potential Temperature (K) 30 (a) (b) Fig. 3. (a) The first and (b) second EOF modes of the daily Northern Hemispheric Ω anomalies (unit: PVU; 1 PVU=10 6 m 2 s 1 K kg 1 ) in the θ-pvlat coordinate. The ordinate Fig.3 The isfirst the (a) potential and second temperature (b) EOF mode (unit: of K) the and daily the abscissa Northern ishemispheric PVLAT. PV anomalies (unit: PVU; 1PVU = 10 6 m 2 s 1 Kkg 1 ) in the θ-pvlat coordinate. The ordinate is the potential temperature (unit: K) and the abscissa is PVLAT.

5 888 POLAR VORTEX OSCILLATION VIEWED IN AN ISENTROPIC POTENTIAL VORTICITY COORDINATE VOL. 23 Fig. 4. Daily time series of the first two leading EOF modes shown in Fig. 3. Red curve is for the first and blue for the second EOF mode. the time scale longer Fig.4 than Daily a fewtime weeks. series Itof isthe alsofirst verytwo leading ter season. EOF On modes top shown of the intra-seasonal Fig. 3. Red and seasonal evident that the two curve time is for series the shown first and in blue Fig. for 4 have the second timeeof scales, mode. the first mode seems to have a peak amplitude similar time scales. The power spectral analysis indicates at periods of about 8.7, 16.9, and 33.8 months. that the two time series have common peaks at These three periods, following a double-periodicity sequence, the periods of about 5 months and 107 days (Fig. 5). correspond to the 3-peak extratropical Quasi- The peak at period of 5-month indicates the seasonal Biennial Oscillation (QBO) signal found by Tung and variation of the amplitude of these two modes, which Yang (1994). The second mode also has peak amplitude at the nearby frequencies although exhibit a large amplitude primarily from late fall to 32 these lowfrequency peaks appear less dominant compared to the early spring each year and are relatively quiescent in between (Fig. 4). As to be shown later on, the periodicity first mode. of 107 days corresponds to the dominant time According to Figs. 4 5, these two modes have sim- scale of the intraseasonal variability of PVO in winilar time scales and exhibit large-amplitude fluctua-

6 NO. 6 REN AND CAI months 107days (a) PC1 Power spectral density months months months months 107days (b) PC months months months months Number of cycles per 300 months Fig. 5. Power spectral density (solid: estimated using AR400 model and dashed using AR1200 model) of the daily time series of (a) the first and (b) second leading Fig.5 Power EOF mode spectral of the density Ω anomalies (solid: estimated in PVLATusing on the AR400 Northern Hemisphere. model The and periods dashed atusing peaks AR1200 labeled model) 1, 2, of 3, the daily and 4 time are 33.8, 16.9, 8.7, series and 6.8 of the months, first respectively. (a) and second (b) leading EOF mode of the PV anomalies in PVLAT on the Northern Hemisphere. The periods at peaks labeled 1, 2, 3, tions concurrently. and 4 are This 33.8, seems 16.9, to suggest that they are 8.7, and 6.8 months, respectively. intimately coupled together. To confirm this, we have calculated the lead/lag temporal correlation between the time series of the first two leading EOF modes (Fig. 6). It is found that the two time series are positively correlated with a correlation of about 0.19 when the second mode leads the first mode by about 17 days and the lag correlation becomes negative with a peak value of 0.29 when the second mode lags the first by about 40 days. Figure 6 suggests that the time scale for a complete cycle of the oscillation between33the two modes is about 114 days. This strongly suggests that the peak spectral density at about 107 days shown in Lead days of the first principle component Fig. 5 is related to the coupling of the first two modes. With reference to Fig. 3a, the positive phase of the first mode corresponds to a stronger polar vortex (a Fig. 6. Lead/lag correlation between the first two principle components. Fig.6 Lead/lag The abscissa correlation is the between lead time the first of the two first positive Ω anomaly over the polar cap) and negative modeprinciple with respect components. to the second The mode abscissa (unit: is the day). lead time phase a weaker polar vortex (a negative Ω anomaly of the first mode with respect to the second mode (unit: day). Correlation coefficient

7 890 POLAR VORTEX OSCILLATION VIEWED IN AN ISENTROPIC POTENTIAL VORTICITY COORDINATE VOL. 23 Potential Temperature(K) modes evolves clockwise on the phase plane (see Fig. Fig.7 Fig. A 7. series A series of of vertical-latitude cross-section diagrams of PV anomalies 12 below for (unit: examples). The poleward and downward PVU) of Ωas anomalies a function (unit: of the phase PVU) angle as a function the phase of the plane phase spanned propagation by the first two signal can be easily seen in the series of EOF angle modes, on the corresponding phase planeto spanned one complete by the PVO firstcycle two from EOF135 vertical-latitude to 180 as time cross-section diagrams of Ω anomalies progresses modes, corresponding forward. The abscissa to one complete is the PVLAT PVO and ordinate from is the as (θ) surface a functionlevel. of φ (Fig. 7) constructed based on Eq. 135 The number to 180 on the as time progresses forward. The abscissa right hand side of each panel indicates the phase (2). angle Thisof confirms the PVO the poleward and downward propagation correspond signal derived to from the first two EOF modes is the PVLAT and ordinate is the θ-surface level. The orbit on the phase plane. Positive (negative) EOF1 and EOF2 phases number on the right hand side of each panel indicates the phase phase angle angle φ = of 0( the ( 180 ) PVO and orbit φ on = 90 the ( 90 ), phase plane. respectively. of zonal mean zonal wind anomalies (Kodera et al., Positive 2000). (negative) EOF1 and EOF2 phases correspond to phase angle φ=0 ( 180 ) and φ=90 ( 90 ), respectively. over the polar cap). For this reason, we refer to the daily time series of the first mode as the polar vortex oscillation (PVO) index hereafter. We have calculated the lead/lag correlation between PVO index and the NAM index at various levels (not shown here). It is found that the highest correlation between these two indices exists at 20 hpa (0.91) when PVO index lags NAM index about 10 days. The altitude of the maximum correlation decreases for a shorter lag time, reflecting a downward propagation signal (Cai and Ren, 2006b). From the lead/lag temporal correlation between the two modes shown in Fig. 6 and the spatial patterns shown in Fig. 3, one may easily arrive at a conclusion that these two modes are essentially a pair of poleward and downward propagation modes. In another word, the first mode describes the intensity oscillation of the polar vortex whereas the second mode represents the transition between the two extreme phases of a PVO event. Therefore, these two modes jointly describe the complete cycle of the PVO and they together explain as much as 82% of the total variances of the daily Ω anomalies over the entire Northern Hemisphere. The time scale of the PVO index, on average, is about 107 days as suggested in the spectral density function of the two modes, explaining why there are only 1 2 PVO events in each cold season. Following Kodera (2000), the propagation signal (denoted as P ) described by the two EOF modes can be represented by a periodic orbit on the phase plane spanned by the two EOF modes, P (φ = ωt) = A[cos(φ)M EOF1 + sin(φ)m EOF2 ] (2) where M EOF1 and M EOF2 are the spatial patterns of the two EOF modes, A is the amplitude of the propagation signal, φ is the phase angle of the orbit on the phase plane, and ω is the frequency of the propagation. According to Figs. 5 6, ω 2π d 1 (3) 107 the negative sign in Eq. (3) is due to the fact that time series of the second EOF mode leads the first EOF mode (Fig. 6). Therefore, the orbit of these two EOF Relation of the PVO with global circulation anomalies In the previous two sections, we have illustrated

8 NO. 6 REN AND CAI 891 that the PVO primarily evolves a pair of poleward and downward propagating modes on a time scale of about 107 days. To depict the temporal and spatial evolution of the global circulation anomalies, we have obtained a series of lead/lag regression maps of anomalies against the PVO index I PVO according to Y reg (y, θ, τ) = I PVO(t) Y (y, θ, t + τ) I PVO (t) I PVO (t) (4) where, Y is a (zonally averaged) anomaly field and ( ) stands for a temporal mean obtained by averaging data over the time t from December 1 through March 31 for all of the 25 years from 1979 to Because the dominant time scale of the coupling between the first two EOF modes is about 110 days (Fig. 5), we have carried out the regression calculation from τ = 60 days to τ = 60 days in order to get a complete cycle of the evolution. In order to compare the temporal and spatial evolution of zonally averaged circulation viewed in the semi- Lagrangian θ-pvlat coordinate with that viewed in the conventional Eulerian geostationary coordinate, we have obtained regressed anomalies in both coordinates. Obviously, for the regressed anomalies in the θ-pvlat, y in Eq. (4) is just the PVLAT. In the case of the conventional Eulerian zonal averaging, y stands for the latitude and the anomalies are defined as the departures from the seasonal cycle calculated in the regular longitude-latitude coordinate and the zonal averaging is made along the latitudes. Figure 8 shows the vertical-time cross-section diagrams of the regressed isentropic temperature and zonal wind anomalies in the θ-pvlat coordinate. First of all, it is very clear that the time scale derived Potential Temperature (K) Lead days of the PVO index Fig. 8. Regressed anomalies averaged along PVLAT as a function of potential temperature (ordinate, unit: K) and the lead time of the PVO index (abscissa, unit: day). (a) temperature (unit: K) and (b) zonal wind (unit: m s 1 ). The panels from the top to the bottom are obtained by averaging over (50 90 N), (25 50 N), (5 25 N), respectively. Fig.8 Regressed anomalies averaged along PVLAT as a function of potential temperature (ordinate, unit: K) and the lead time of the PVO index (abscissa,

9 892 POLAR VORTEX OSCILLATION VIEWED IN AN ISENTROPIC POTENTIAL VORTICITY COORDINATE VOL. 23 Lead days of the PVO index Fig. 9. Regressed anomalies averaged along PVLAT and between 300 and 650 K as a function of latitude (ordinate) and the lead time of the PVO index (abscissa; unit: day). (a) Temperature (unit: K) and (b) zonal wind (unit: m s 1 ). Fig.9 Regressed anomalies averaged along PVLAT and between 300 and 650K as a function of latitude (ordinate) and the lead time of the PVO index (abscissa; unit: day). (a) Temperature (unit: K) and (b) zonal wind (unit: ms 1 ). Potential Temperature (K) Lead days of the PVO index Fig. 10. As Fig. 8, but the zonal average is done along regular latitudes. 37

10 NO. 6 REN AND CAI 893 Lead days of the PVO index Fig. 11. As in Fig. 9, but the zonal average is done along regular latitudes. Fig.11 As in Fig. 9, but the zonal average is done along regular from the regression calculation is highly consistent latitudes. with the power spectral analysis and correlation calculation shown in Figs Perhaps, the most striking feature shown in Fig. 8 is that the anomalies of both signs appear to peak at high elevation first and then gradually propagate downward at all latitudes. The downward propagation time scale is about 20 days in high latitudes (the top panels of Fig. 8), consistent with the results of Baldwin et al. (1999) and Kodera et al. (1990). The downward propagation in low latitudes (the bottom panels of Fig. 8) is nearly as fast as that in high latitudes. However, it takes a longer time (about 30 days) for the anomalies to propagate downward in mid-latitudes (the middle panels of Fig. 8). It is also evident from Fig. 8 that the downward propagation of the anomalies of the same polarity appears in low latitudes first and then takes place in mid-latitudes, and then progressively appears in high latitudes at a later time. Therefore, there exists a simultaneous poleward and downward propagation of anomalies of both signs from the equator to the pole and from the stratosphere to the troposphere. Figure 9 shows that the poleward propagation of anomalies of both signs is very fast in tropics and polar cap. The poleward propagation is slower from 20 to 60 N. On average, it takes about 53 days for the anomalies of each sign to arrive at the pole, or about 107 days to complete a whole PVO cycle, consistent with the 4- month time scale found in Kodera et al. (2000). Another important feature is that the poleward and downward propagating zonal wind anomalies lag the temperature anomalies of the opposite sign at all latitudes [panel (a) versus panel (b) in Figs. 8 9]. The lag of the zonal wind anomalies with respect to the temperature anomalies of the opposite sign is longer in mid-latitudes (about 20 days) than that in low and high latitudes (about 10 days). Figure are the counterparts of Figs. 8 9 showing the results obtained by averaging along regular latitudes (e.g., the Eulerian anomalies). In general, the temporal and spatial evolution of the anomalies averaged along latitudes is very similar to that in the θ- PVLAT coordinate. Particularly, the downward propagation of both temperature and zonal wind anomalies of both signs is vividly apparent in high latitudes. In the tropics, only the temperature anomalies averaged along the latitudes show a noticeable downward propagation. The temporal phase relation between the temperature and zonal wind anomalies in the Eulerian coordinate is also similar to that in the θ-pvlat. The main difference lies in the mid-latitudes. It is very clear that the downward propagation of the Eulerian zonal averaged anomalies in mid-latitudes is much faster than that in the θ-pvlat coordinate. Also the apparent poleward propagation signal appears more like a seesaw oscillation pattern between the low latitudes and the extratropics for the Eulerian anomalies. 5. Interannual variability of the temporal evolution of PVO events There is considerable low-frequency interannual variability of PVO events as indicated from the time series of the two EOF modes (Fig. 4) and their power spectral density function (Fig. 5). Following Kodera et al. (2000), we have made the orbit diagrams of individual PVO events from the time series of the two EOF modes for each winter season from 1979 to 2003 (Fig. 12). Table 1 lists the time span of each orbit as well as the major positive and negative peak values and their dates of these PVO events. Below we attempt to infer the interannual variability of PVO events from the 39

11 894 POLAR VORTEX OSCILLATION VIEWED IN AN ISENTROPIC POTENTIAL VORTICITY COORDINATE VOL. 23 Time series of EOF2 Time series of EOF1 Fig. 12. The orbits of the state vector representing EOF1 (abscissa) and EOF2 (ordinate) for the PVO events in winters from 1979 to 2003 based on the 15-day running mean daily time series of the two EOF modes. The time interval between two adjacent dots is 3 days. Red-colored portion of the orbit corresponds to the first 30 days and blue are for the last 30 days of each PVO event. The exact time span of each PVO orbit is given in Table 1. Fig.12 The orbits of the state vector representing EOF1 (abscissa) and EOF2 (ordinate) for the PVO events in winters from 1979 to 2003 based on the 15-day variation of the geometry properties of their orbits on ) and on the right side a stronger polar vortex the phase plane running spanned mean by the daily first time two leading series of EOF the two (e.g., EOF ). modes. When The the time orbit interval center is near the origin of the phase portion plane, of the the season orbit mean polar vortex is modes (Fig. 12). between two adjacent dots is 3 days. Red-colored As indicated corresponds by the size of to the orbits first 30 ondays the phase and blue closer are to for the the climatological last 30 days polar of each vortex PVO strength (e.g., plane, there isevent. a considerable The exact interannual time span variability of each PVO ). orbit is The given time in scale Table of1. an individual PVO event in the PVO strength. Particularly, it appears that in is indicated by the total number of points of its orbit the early years of 1980s and 1990s ( , 1990 (also see Table 1 for the time span of each event). 93), the PVO events are considerably weaker than the It is also interesting to point out that there are fewer late years of the same decades, suggesting a stronger points along the orbit of a PVO event in the neighbourhood decadal time scale of the PVO strength variation. Another of its negative peak (the maximum negative interesting feature is that the center locations value of EOF1) than around the positive peak. This of these PVO orbits vary greatly from year to year, graphically illustrates the faster transition towards a reflecting the interannual variability of the seasonal warming event than towards a cold event. mean strength of the polar vortex. Specifically, the orbits whose center is located on the left side correspond to a winter that has a weaker polar vortex (e.g., A circular orbit or a horizontally/vertically orientated elliptic orbit implies that the time series of the 40 two EOF modes are nearly orthogonal (or no instan-

12 NO. 6 REN AND CAI 895 Table 1. A list of the time span of all PVO events from 1979 to The dates and values of positive peak PVO index are given in the 2nd and 3rd columns and the negative peak dates and values in the 4th and 5th columns, The dominant trajectory rotation direction is indicated in the 6th column. The peak warming dates of these major warming events identified in Limpasuvan et al. (2004) are given in the 7th column. Year Event DD/MM/YYYY Positive PVO Negative PVO PVO-Index/PC2 Limpasuvan Peak Date Index Peak Date Index phase space /02/ /05/ /01/ /03/ clockwise 22/02/ /11/ /03/ /02/ clockwise 20/03/ /09/ /04/ /01/ /02/ clockwise 07/03/ /01/ /05/ /03/ anticlockwise /10/ /05/ /01/ /04/ clockwise 25/03/ /11/ /05/ /02/ /03/ anticlockwise- 10/03/1984 clockwise /09/ /05/ /01/ clockwise 19/01/ /12/ /04/ /03/ /04/ clockwise /01/ /04/ /02/ clockwise 10/02/ /10/ /03/ /03/ /12/ clockwise 15/12/ /11/ /05/ /02/ /03/ clockwise 15/03/ /12/ /05/ /02/ anticlockwise 09/04/ /12/ /03/ /02/ anticlockwise- 12/02/1991 clockwise /01/ /03/ /01/ clockwise 10/02/ /11/ /04/ /02/ anticlockwise /11/ /04/ /02/ clockwise /12/ /05/ /02/ /01/ anticlockwise /10/ /04/ /01/ /02/ clockwise- 31/01/ /10/ /04/ /01/ /02/ anticlockwise 16/03/ /10/ /04/ /02/ /12/ clockwise 04/12/ /11/ /05/ /03/ clockwise /12/ /04/ /03/ /01/ anticlockwise 05/01/ /12/ /05/ /01/ clockwise 28/12/ /03/ /03/ /03/ /10/ /03/ /01/ anticlockwiseclockwise /10/ /04/ /12/ clockwise 18/12/ /02/ /10/ /02/ /01/ anticlockwise 21/03/ /09/2002 9/05/ /12/ /01/ clockwiseanticlockwise taneous correlation), corresponding to the case of a relatively stronger propagation signal. A sloped orbit or an elongated orbit along a diagonal direction suggests that the two time series have a strong correlation with the polarity of the slope indicative of the sign of the correlation. This corresponds to the case of a weak propagating PVO event. The extreme case in which the orbit becomes a straight line corresponds to a pure standing PVO event. The fact that none of the orbits are close to a straight line suggests that all of the PVO events have a propagation signal (note that the and orbits appear like a straight line because their amplitude is much smaller. The propagation signals have been confirmed from the Hovemöller

13 896 POLAR VORTEX OSCILLATION VIEWED IN AN ISENTROPIC POTENTIAL VORTICITY COORDINATE VOL. 23 diagram for each of the PVO events). According to Eq. (2) Eq. (3) and Fig. 7, a clockwise rotating trajectory orbit on the phase plane spanned by the two EOF modes corresponds to a simultaneous poleward and downward propagating Ω anomalies. As summarized in Table 1, 14 of total 25 PVO events have a clockwise rotating trajectory orbit (note that a small fraction of some orbits may still exhibit a counter-clockwise trajectory), and therefore correspond to a simultaneous poleward and downward propagation signal. An interesting point is that nearly 13 of these 14 clockwise-rotating-trajectory cases have a major warming event as identified by Limpasuvan et al. (2004) and the remaining clockwise-rotatingtrajectory case ( ) also has a noticeable warming phase. The orbit has two loops corresponding to the two major negative PVO peaks (or two major warming events). Both loops have a clockwise rotating trajectory. Therefore, it appears that other than few exceptions (4 of them), most of the PVO events that have a major breakdown of the stratospheric polar vortex are always associated with a simultaneous poleward and downward propagating Ω anomalies. In the winter , the orbit can be divided into two portions. The first portion is for the evolution towards the positive PVO event which has a counter-clockwise rotating trajectory. The second portion represents the evolution from the positive peak to the negative peak and back to the climatology, showing a clockwise rotating trajectory. Therefore, at least during the warming event, the Ω anomalies in still propagate poleward and downward. However, the entire orbit of the other three exceptions ( , , and ), that have a major warming event as identified by Limpasuvan et al. (2004), has a counter-clockwise rotating trajectory. We have confirmed that the Ω anomalies represented by the first two EOF modes indeed exhibit an equatorward propagating signal in these three cases. However, the total Ω anomalies still have a dominant poleward propagation signal (top panel of Fig. 13b). This seems to suggest that other EOF modes would have a significant contribution to these three PVO events as far as the poleward propagation is concerned. As in other cases, the poleward signal is more noticeable in temperature anomalies (bottom panel of Fig. 13b). Fig. 13. Hovemöller diagrams of 15-day running mean daily Ω (top panels; unit: PVU) and temperature (bottom panels; unit: K) anomalies on θ=650 K in the winter of (a) and (b) Fig.13 Hovemöller diagrams of 15-day running mean daily PV (top panels; unit PVU) and temperature (bottom panels; unit K) anomalies on

14 NO. 6 REN AND CAI 897 The other 7 orbits that have a counter-clockwise trajectory are in the winters of , , , , , , and , respectively. It is observed that none of these 7 cases has a major warming event as identified by Limpasuvan et al. (2004). We found that except the case, the total Ω anomalies in other 6 cases still propagate poleward although the Ω anomalies represented by the first two EOF modes do not show a poleward propagating signal, suggesting that other EOF modes would have to play an important role in these PVO events that do not have a major warming event. As shown in the top panel of Fig. 13a, in the winter of , even the total Ω anomalies do not exhibit a clear poleward propagating signal. However, the temperature anomalies still display a clear poleward propagating signal (bottom panel of Fig. 13a). As explained in Cai and Ren (2006, see footnote before), the Ω anomalies are due to the dynamic response to the heating anomalies, reflecting the changes in both circulation and static stability anomalies. In other words, a poleward propagating heating anomaly always leads to a change in the polar vortex although it does not always imply a poleward Ω anomaly. The results presented in this section suggest that the PVO, regardless of whether it has a major warming event or not, is always associated with the poleward propagating heating anomalies, consisting with the global mass circulation paradigm proposed in Cai and Ren (2006). 6. Summary This paper examines the temporal variability of the polar vortex oscillation (PVO) in the semi-lagrangian θ-pvlat coordinate from daily isentropic potential vorticity maps. The θ-pvlat coordinate, which has been proposed as an alternative choice of the coordinate system for diagnosing and understanding climate variability of the general circulation (Cai and Ren, 2006a, b), is made by converting the area of the spherical cap encircled by each Ω contour on an isentropic potential vorticity map to its equivalent latitude (or PVLAT). The longitude-pvlat coordinate system made from isentropic Ω contours evolves in time. The 2-D flow in the θ-pvlat coordinate is obtained by averaging the original 3-D field along PVLAT (or isentropic Ω contours), capturing both the thermodynamic and dynamic properties of the same air mass (or same isentropic Ω) more accurately compared with the conventional zonal mean along the geographical latitudes. The semi-lagrangian property ensures that the polar vortex variability in the θ-pvlat coordinate is only due to its intensity variability because the changes in its location and shape would be naturally absent by following isentropic Ω contours. As a result, its temporal variation has little signal at the synoptic time scales. It is found that the polar vortex oscillation in the θ-pvlat coordinate essentially consists of a pair of poleward and downward propagating modes. These two modes together account for about 82% variance of the daily Ω anomalies over the entire Northern Hemisphere. These two modes exhibit large variability only in cold season from late fall to early spring and are quite inactive in the rest of the year. The power spectral analysis reveals that the time series of these two modes have common dominant time scales. Particularly, both of them have a peak amplitude at 107 days, corresponding to the dominant oscillation time scale of the PVO in cold season. The time scale of about 107 days has also been independently obtained by the lead/lag correlation between the time series of the two leading EOF modes. The time series of the first EOF mode has been used as the PVO index. The temporal relation between the PVO and global circulation anomalies is examined by regressing anomalies in the θ-pvlat coordinate against the PVO index. The regression calculation confirms the association of the PVO with a systematic simultaneous poleward and downward propagation of circulation anomalies of both signs. The poleward propagation time scale is about 53 days, corresponding to one half of the time scale of a complete PVO cycle (about 107 days, which is very similar to the 4-month time scale found in Kodera et al., 2000). The downward propagation time scale is faster in low and high latitudes (about 20 days) and slower in midlatitudes (about 30 days). The zonal wind anomalies lag the poleward and downward propagating temperature anomalies of the opposite sign by about 10 days in low and high latitudes and by 20 days in mid-latitudes. The regression analysis using the anomalies zonally averaged along regular latitudes (Eulerian averaging) yields a similar temporal relation between the PVO and global circulation anomalies. The main difference is that the signal of the poleward propagation is much less noticeable in the lower stratosphere and appears like a seesaw pattern between the subtropics and extratropics when viewed in the Eulerian coordinate. In the upper stratosphere where the circulation is highly zonal symmetric, the poleward propagation is very visible even in the Eulerian coordinate (Kodera et al. 1990, 2000; Dunkerton 2000). On top of the intraseasonal time scale of about 107 days in winter, the time series of the leading EOF modes also exhibit dominant time scales of 8.7, 16.9, and 33.8 months. They approximately follow a doubleperiodicity sequence and correspond to the 3-peak ex-

15 898 POLAR VORTEX OSCILLATION VIEWED IN AN ISENTROPIC POTENTIAL VORTICITY COORDINATE VOL. 23 tratropical QBO signal reported in Tung and Yang (1994). The interannual variability of the PVO events can be succinctly summarized by their trajectory orbits on the phase plane spanned by the first leading EOF modes. The size of an orbit reflects the amplitude of an individual PVO event. We found that the PVO events are considerably weaker in the early years of 1980s and 1990s than the late years of the same decades, suggesting a stronger decadal time scale of the PVO amplitude variation. Not all of the PVO events are associated with a major warming event. Those PVO events that have a warming event are always associated with a poleward propagation of Ω anomalies. However, those PVO events that do not have a major warming event display a less noticeable poleward propagating signal in the Ω field. Nevertheless, we found that a PVO event, regardless of whether it has a major warming event or not, is always associated with the poleward propagating heating anomalies. The positive phase of the PVO is associated with cooling anomalies over the polar stratosphere and negative phase with heating anomalies. Acknowledgments. The authors thank Dr. Kim at the Florida State University for kindly providing the power spectral analysis program. The insightful comments and suggestions from two anonymous reviewers are greatly appreciated. This study has been supported by a grant from the NOAA Office of Global Programs GC Ren Rongcai is also supported by the National Natural Science Foundation of China Grant No APPENDIX The algorithm of mapping Ω contours to PVLAT We here briefly document the algorithm used for mapping Ω contours to PVLAT. Specifically, we first choose 2N prefixed latitudes denoted as φ Ω,j (j = 1, 2,..., 2N) and φ Ω,j > φ Ω,i for j < i. The first N prefixed latitudes, {φ Ω,j, j = 1,..., N}, are for the Northern Hemisphere (NH) and the remaining N prefixed latitudes, {φ Ω,j = φ Ω,2N+1 j, j = N + 1, N + 2,..., 2N}, are for the Southern Hemisphere (SH). The same prefixed latitudes are used at every isentropic surface and every day. Let us denote the spherical cap area encircled by these prefixed latitudes as A(φ Ω,j ). By definition we have A(φ Ω,j ) < A(φ Ω,i for N + 1 > j > i and A(φ Ω,j = A(φ Ω,2N+1 j ) for j = N + 1, N + 2,..., 2N. In the remaining part of the discussion, we will focus only on the mapping for the positive Ω contours since the mapping for negative Ω contours is similar except that the sign of Ω and PVLAT are reversed. To do the mapping, we need to choose a proper value for N and {φ Ω,j, j = 1, 2..., 2N}. In general, a larger value of N implies a higher resolution. However, to ensure that each PVLAT grid point can be easily matched with a Ω contour everyday for all seasons, all years, and all levels, it is expected that N should be larger than the number of the meridional grid points in the original data. In this study, we choose N to be 59 larger than 36, the total number of meridional grid points over one hemisphere. Table A1 lists the 59 values of {φ Ω,j } and the corresponding {A(φ Ω,j )} used in this study. Because the meridional gradient of Ω tends to be much stronger in high latitudes than in lower latitudes, the increment between two adjacent A j,φ should be smaller in high latitudes and gradually increase towards the equator in order to capture the feature that the Ω contours are more densely packed in high latitudes and become more sparsely packed towards low latitudes. It should be mentioned that the results are actually not very sensitive to the choice of PVLAT grids as long as the distribution of PVLAT grids is no coarser than the original latitude grids. For a given daily (positive) Ω field on one isentropic surface, we bin the Ω field into 1000 equally spaced intervals, covering from 0 to the maximum of the Ω on that day. Let us denote those 1000 Ω values as {Ω m }, which are defined as Ω m = 1000 m Ω max, m = 0, 1, 2,..., 1000 (A1) 1000 where Ω max is the maximum Ω on the Ω map under consideration. For each Ω m, we can calculate the area occupied by the Ω exceeding Ω m according to A(Ω m ) = λφ 2π cos φ k H(Ω(λ i, φ k ) Ω m ) i,k (A2) where λ = φ = 2.5/180π, λ i = (i 1) λ is the longitude and φ k = (k 37) φ is the latitude, and the summation in Eq. (A2) is over all possible grid points over the entire globe. H(x) in (A2) is defined as H(x) = It follows that we have { 1, if x 0 0, otherwise A(Ω m ) A(Ω n ) if m > n. (A3) (A4) The equality in (A4) is applied only in one of the two conditions: (i) the Ω field is extremely densely packed between Ω m and Ω n such that there are no grid points whose Ω values are between Ω m and Ω n ; (ii) none of

16 NO. 6 REN AND CAI 899 Values of Northern Hemispheric PVLATs and the corresponding encircled area normalized by the hemi- Table A1. sphere area. No. Encircled Area PVLAT( N) No. Encircled Area PVLAT( N) the grid points has Ω values smaller than Ω n which typically takes place at the lowest isentropic surfaces and in winter time when part of the isentropic surface is below the surface. In this case, A(Ω m ) A(Ω n ) for all m > n. Next, for each j starting from j = 1, we loop m from m = 1 to 1000 to find the value m j such that A(Ω mj ) is closest to A(φ Ω,j ). Then we assign φ Ω,j to all of the grid points (λ i, φ k ) whose Ω values are between Ω mj and Ω mj 1 as their PVLAT (note that Ω m0 Ω max ). The above procedure is repeated to the next value j = j + 1 till one of the three conditions is met: Condition A: A(Ω m ) A(Ω mj 1 ) for all m > m j 1. In this case, no PVLAT will be assigned to the remaining grid points in NH. Condition B: A(Ω 100 ) is the closest to A(φ Ω,j ) and A(Ω 1000 ) 1. In this case, we assign φ Ω,j to all of the grid points (λ i, φ k ) whose Ω values are between Ω 1000 = 0 and Ω mj 1 as their PVLAT. Condition C: A(Ω m ) > 1 for 1000 > m > m j 1 = m 59. This has to correspond to the situation in which the positive Ω invades into the Southern Hemisphere. For this case, we find the value of [2 A(Ω m )] that is closest to A(φ Ω,j ) and assign φ Ω,j to those grid points (λ i, φ k ) whose Ω values are between Ω mj and Ω mj 1 as their PVLAT (note that in this case, j > 59 so that φ j = φ 2N+1 j < 0. The special procedure for this case is repeated by letting j = j + 1 till m = It is of importance to point out that the algorithm can assign a positive φ Ω,j to a SH grid point when positive Ω invades into SH. The algorithm can also assign a negative φ Ω,j to a NH grid point that has a positive Ω, which is possible only when the Condition C is met. Obviously, the situation of the presence of the negative PVLAT in NH for positive Ω reflects an even