a. Background

Synoptic models of atmospheric phenomena and dynamical or physical processes have been used extensively to communicate the results of diagnostic research and to help develop the science and art of weather forecasting. The extratropical cyclone and its baroclinic–barotropic life cycle is the most well known subject of synoptic modeling in meteorology. It has been studied with numerical models and observed datasets for more than 70 yr (e.g., Bjerknes 1919; Rossby 1941; Shapiro and Gronas 1999). Coherent and easily recognized, the behavior of baroclinic waves is the “bread and butter” of short-term prediction models, which have very good forecast skill in the 1–3-day range, at least on average (Simmons and Hollingsworth 2002).

No comparable synoptic model exists for subseasonal (3–90 days) variability and medium-range predictions. In fact, several additions to the baroclinic life cycle model would be required to construct a “useful” medium-range synoptic model. For one, the regional domain of the short-term prediction problem is no longer adequate for the medium range (e.g., Smagorinsky 1967). Wave packets or other organized atmospheric energy pulses can spread signals from regional tropical-extratropical interactions around the hemisphere in <10 days (Chang 1993, 1999a, b). The inherent scales of motion are also larger, in particular encompassing zonal wavenumbers 0–3. In the time domain, the model should consist of multiple time scales and their interactions in time and space. In terms of coherent phenomena, the quasi-oscillatory Madden–Julian oscillation (MJO; Madden and Julian 1971, 1972, 1994) is a viable candidate to provide structure to a global subseasonal synoptic model in the same way that El Niño–Southern Oscillation (ENSO) does for interannual variability. But the MJO is far from adequate by itself. Extratropical phenomena must also be included.

Historically, the most well-known extratropical variations are baroclinic waves, “blocking,” and zonal index cycles (e.g., Namias 1954). The latter involve hemispheric variations between blocked and strong zonal flows. In modern times, teleconnection patterns (Wallace and Gutzler 1981, hereafter WG81; Branstator 1992) have been recognized as a general mode of circulation variability, of which blocking is a special case (Dole 1986, 2007). More recently, zonal mean or “annular” modes of variability are again being highlighted in diagnostic studies, especially those based on surface pressure or height (Thompson and Wallace 2000). Studies of the evolutionary behavior of zonal mean anomalies have revealed episodes of coherent propagation all the way from equatorial regions to high polar latitudes (Feldstein 2001). In the Southern Hemisphere, the MJO’s quasi-stationary wave component (Weickmann et al. 1997, hereafter WKS) and a midlatitude component forced by baroclinic waves (e.g., Lorenz and Hartmann 2003) likely contribute to such episodes.

Although spatial differences are important, these phenomena all tend to be characterized by 5–10-day-decay time scales (Feldstein 2000) and by large zonal scales. Studies have sought to explain such low-frequency phenomena in terms of multiple equilibria (e.g., Charney and DeVore 1979) or normal modes (Simmons et al. 1983), although current evidence suggests that forcing helps determine their time behavior (Newman et al. 1997). Prominent forcing is supplied by the MJO, topography, the storm tracks, and sea surface temperature (SST) variations. A representative component from this class of red noise processes will make up the intermediate time scale of the global synoptic–dynamic model (GSDM). We specifically focus on a large-scale disturbance that develops over the Pacific–North American sector in association with orographically perturbed flow over the eastern Asian and western North American mountains and will refer to the phenomenon as the “τ_F–τ_M index cycle.”¹ Regionally it includes teleconnection patterns like the Pacific–North American pattern (PNA; WG81). Weickmann (2003, hereafter W03) has studied its synoptic evolution and Weickmann et al. (2000, hereafter WRP) argue that it is the dominant mode of atmospheric angular momentum (AAM) exchange and anomaly decay on intermediate subseasonal time scales.

The fast baroclinic life cycle has been studied for various background flows and meridional shears (Simmons and Hoskins 1978, 1979, 1980). Disturbance growth, phase propagation, energy dispersion, and breaking are prominent features of such studies (Thorncroft et al. 1993; Whitaker and Sardeshmukh 1998) and of the observed atmosphere. Regional baroclinic wave breaking (Nielsen-Gammon 2001) in particular produces a rapid meridional momentum exchange that influences even the zonal mean zonal wind. The residual of breaking waves has recently been linked with the initiation of teleconnection patterns like the North Atlantic Oscillation (NAO; Franzke et al. 2004). The fast time scale of the GSDM will focus on wave energy dispersion that accompanies baroclinic developments and their interaction with the major mountain ranges.

Specifically, synoptic-scale wave energy dispersion within the northern and subtropical waveguides over Asia and the Pacific Ocean (Hoskins and Ambrizzi 1993; Chang 2005) will represent the model’s fast time scale. The process is linked with 10–30-day oscillations of the circulation, which have been observed throughout the atmosphere. Wave retrogression at high northern latitudes (Branstator 1987), mountain torques over Asia and North America (Lott et al. 2001; Lott and D’Andrea 2005), equatorial convectively coupled Rossby and Kelvin waves (Wheeler et al. 2000), and monsoon oscillations (Hartmann et al. 1992) all display coherent behavior on this time scale. This component can be energetic for an extended time period as during the 1996–97 rainy season over central California (Mo 1999). The specific ∼25-day oscillation represented in the GSDM involves subtropical flow variations, a likely candidate for the behavior described by Mo (1999). Mo (2001) has incorporated the intermediate time scale explicitly into the week 1–3 prediction process.

The GSDM organizes these multiple subseasonal phenomena into a repeatable sequence. The relationship among the time scales is based on maximizing global and zonal AAM anomalies. The GSDM emphasizes four primary stages but the duration of the cycle and the days between stages are variable. The model is valid during persistent forcing by sea surface temperature anomalies as well as during the time-varying MJO. In fact, two of the stages represent behavior also observed during El Niño and La Niña events. The GSDM is similar to using composites of ENSO or MJO for real-time monitoring and seasonal or weekly predictions, although we seek to explain and account for more variance through the multiple time-scale approach.

A dynamical underpinning for the model is atmospheric angular momentum, which is determined by the three-dimensional distribution of atmospheric zonal wind and mass (Peixoto and Oort 1992). Its global integral is an excellent index of ENSO, the MJO, and other subseasonal variability. Moreover, processes that change global AAM, that is, the frictional torque and the mountain torque, have medium and fast time scales, respectively (WRP). Thus, indices involving these processes are used to define the components of the GSDM. WRP show that the mountain torque forces AAM anomalies and the frictional torque damps them (Egger and Hoinka 2002), and this will be evident in the GSDM. Atmospheric dynamics linked with wave growth and dispersion processes connect the mountain source regions of AAM anomalies with the subtropical frictional sink regions. The latter tend to occur near the seasonally varying zero zonal wind line. The torques are produced by a variety of known, large-scale patterns of wind and pressure as described by WRP and W03.

There are two primary motivations for the synoptic modeling effort. The first is scientific understanding of the evolving atmospheric circulation with a practical goal of attribution of weather–climate anomalies. Diagnostic tools and analysis results derived from the science and research side are used on the monitoring and prediction side to examine cause and effect or the attribution of weather climate variations. The application of diagnostics to actual forecasting is mostly neglected in meteorology because of the strong reliance on numerical forecast models and lack of a useful synoptic framework. The second motivation is daily monitoring of the circulation to evaluate numerical model predictions in the subseasonal band, including the potential predictability of transitions of the circulation and of extreme weather events. Extreme events of global AAM are used as a framework within the GSDM. The specific intent is to target the boundary between weather and climate, which encompasses the transition from deterministic to probabilistic forecasts. An evaluation of numerical forecasts based on synoptic reasoning is expected to add value because there are well-known, stubborn model errors, particularly in the prediction of tropical convection (Lin et al. 2006) and the subsequent circulation response.

Section 2 introduces aspects of the GSDM with a case study presentation of the 2001–02 northern winter season. Two rapid transitions of the circulation that involve tropical–extratropical interaction are described, as well as the predictive skill of an ensemble forecast model. The derivation of the GSDM is then presented in section 3. The evolving circulation anomalies of the different components of the synoptic model are organized using a linear superposition based on the time tendency of global relative AAM. A summary and conclusions follow in section 4.

a. Tropical convection

The two MJOs are shown in a time–longitude format in Fig. 1. MJO 1 forms over the Indian Ocean in November 2001, shifts to the west Pacific in early December 2001, and stalls west of the date line. The stalling may reflect a feedback from the eastward shift of warm ocean water cited above. MJO 2 forms over the Indian Ocean in mid-January 2002 and reaches the date line in mid-February. It contains more Kelvin and equatorial Rossby wave activity than MJO 1 and has its convective signal south of the equator for most of February 2002 (see Fig. 1). Both MJOs have significant amplitude in the filtered field, especially at ∼120°E. Using this longitude as a reference gives a recurrence interval of 65 days, which is on the long end of the MJO period range.

To start the process of telescoping to the two transitions and their link to MJO-related convective forcing, seven numbers have been marked on Fig. 1. These numbers “sample” rapid changes within the two MJOs’ convective envelope and in the ∼25-day quasi oscillation along 160°E. They will be referred to as “tropical convective flare-ups.” The dates for the flare-ups were determined from a time series of the first 2 EOFs of 20–100-day filtered outgoing longwave radiation (OLR) anomalies (shown below). The two GSDM transitions are associated with flare-ups 3 and 5. Here, T1 represents the transition to stage 3 (i.e., like El Niño) when MJO 1 is located over the west-central Pacific, and T2 is the transition to stage 1 (i.e., like La Niña) when MJO 2 is over the Indian Ocean. Table 1 gives the calendar dates corresponding to the numbered events and the two transitions. Although only flare-ups 3 and 5 are studied in what follows, the other cases also produced regional and downstream transient impacts on the atmospheric circulation and on weather-producing disturbances.

b. Midlatitude 250-mb meridional wind

The midlatitude circulation anomalies that accompany the tropical OLR signal are depicted using a daily time-longitude plot. Figure 2 shows the 250-mb meridional wind anomalies averaged between 30° and 60°N and depicts (i) signatures of storms in the storm-track regions, (ii) energy dispersion in midlatitude regions, and (iii) slow, quasi oscillations in the midlatitude meridional wind pattern. The latter will be discussed in terms of the implied geopotential height anomalies, which are marked on the figure. These height patterns involve ridging near 150°W (Hs) during the first half of December 2001, troughing in the same location (Ls) during the rest of December into early January 2002, and a return to ridging (Hs) in mid-January–early February 2002. Each persistent episode includes 3–5 synoptic-scale events whose ridges and troughs tend to develop or amplify at the same longitude. The synoptic events have been numbered for each of the three persistent “regimes.”

The boundaries of T1 and T2 are marked on Fig. 2 and a wave train that extends from 120°E to 90°W is highlighted during each transition. The wave trains are linked with the retrogression of existing anomalies (red arrows) and are followed by a series of similar synoptic events. After T2 and the return of ridging to the eastern North Pacific, the synoptic activity is more spatially variable than during stage 3. It also includes a slow westward drift of the circulation anomalies (light dashed lines) at a time when positive tropical convection anomalies are over the Indian Ocean and Indonesia. The final synoptic regime seen in Fig. 2 is a progressive one (light solid lines), which coincides with the return of convection to the date line around 12 February 2002.

c. Low-frequency indices and their forcing

Figure 3 shows daily indices that capture the low-frequency (>10 day) variations described in Fig. 2 and illustrate the atmospheric response to tropical diabatic heating and orographic forcing to be detailed later. The indexes include (i) the global integral of AAM, (ii) the first EOF of the combined 200-/850-mb vector wind field (with the zonal mean removed), and (iii) the PNA teleconnection pattern. These represent global, hemispheric-subtropical, and regional anomaly patterns, respectively. All have similar low-frequency behavior, that is, minima in early December 2001 (day 30), maxima in late December–early January 2002 (day 60), and back to minima in late January 2002 (day 80). The center of transitions T1 and T2 are again marked on the curves. The PNA slightly lags the other curves, which is consistent with a signal that starts in the Tropics, expands to the subtropics, and eventually affects the regional circulation over the midlatitude Pacific Ocean.

The time series of processes that can force these changes are depicted in Fig. 4. Figure 4a shows the first EOF of the zonal mean mountain (red) and frictional torque (blue) over the Northern Hemisphere. EOF 1 of the mountain torque (not shown) is a monopole pattern between 20° and 60°N, and EOF 1 of the frictional torque (not shown) is a north–south shift in the winter mean boundary of the surface westerly flow. Figure 4b shows coefficients of the first two EOFs of OLR, which describe an eastward movement of the MJO. The convective flare-ups numbered in Fig. 1 (see also Table 1) were derived from these OLR EOFs and are also marked.

These forcing time series have some distinct low-frequency behavior in addition to the rapid convection changes seen in Fig. 4b and the highly variable synoptic component seen in the mountain torque time series of Fig. 4a. From Fig. 1, a 65-day recurrence interval for MJO convection was estimated for 120°E; this is marked at the bottom of Figs. 4a,b. Negative values of OLR EOF 2 are seen at the endpoints of the interval in the lower panel and signify that convection is indeed located over Indonesia at these times. More generally, OLR EOF 1 (red) leads EOF 2 (blue), consistent with the eastward propagation of anomalies.

Another low-frequency behavior marked on Fig. 4a is a 50-day recurrence interval in the midlatitude mountain torque (Lott et al. 2001). As mentioned previously, the frictional torque (Fig. 4a) acts to damp angular momentum anomalies forced by the mountain torque, but it is also involved in producing the MJO AAM signal. In the GSDM, the MJO, intermediate ∼6-day decay time scale, and 10–30-day oscillation can contribute to circulation anomalies during stages 1 and 3. These phenomena are difficult to disentangle since in all, a positive mountain torque is followed by a negative friction torque and vice versa. A section of each time series has been highlighted with a dotted line that shows this lag relationship for a ∼50-day oscillation. It is perhaps clearest after T2. Namely, as the large negative mountain torque removes westerly momentum from the atmosphere, easterly wind anomalies and a positive friction torque develop. This slowly removes easterly momentum from the atmosphere, thereby bringing AAM back into temporary balance. It is during this damping process that circulation anomalies of interest to monitoring and prediction occur, in this case a negative PNA. Another example but with opposite sign is also highlighted during December 2001 on Fig. 4a with two black arrows.

The locations of T1 and T2 are marked with shading on Fig. 4. The transitions follow closely the very large positive and negative maxima in the mountain torque anomaly, which are embedded in strong daily variability as synoptic systems move over the topography. Asian–European topography is a strong contributor to both the positive and negative anomaly while North American topography contributes primarily to the later negative anomaly (not shown). The mountain torque events are linked with the amplification of a baroclinic wave downstream of the Asian topography over the western half of the North Pacific. During T1, the amplifying wave leads to a cyclonic wave break over the east Pacific that brings westerlies south and ushers in stage 3. The opposite happens during the transition to stage 1.

Figure 4b shows that the transitions occur 5–10 days after flare-up 3 and 1–3 days after flare-up 5. This suggests a loose relationship between rapid transitions in the circulation and the sudden shift of convection within the MJO convective envelope. In this case, it is the midlatitude mountain torque that is more directly linked with both transitions. However, the persistence of tropical forcing, as represented by the heavy horizontal bars on the EOF 1 OLR time series, undoubtedly contributed to the persistence of the resulting large-scale circulation anomalies.

d. Synoptic maps during circulation transitions

Figure 5 shows selected daily maps of the 150-mb vector wind anomaly during T1. The key anomalies from an MJO perspective are suppressed convection over the Indian Ocean and active convection near the date line (not shown, but see Fig. 1). Early in the sequence (Figs. 5a,b), a wave train is evident across the Indian Ocean/south Asia with a trough near 20°N, 70°E. The trough is quasi-stationary and eventually evolves into a large cyclonic anomaly covering the entire region (Fig. 5f). The MJO at this stage helps force the trough and sets the phase of a wavenumber 5–6 wave train through which energy flows eastward. Another more stationary wave train farther north (60°N) combines with the subtropical wave train over the midlatitude North Pacific in Fig. 5b, which precedes the wave amplification over the west Pacific seen in Figs. 5d–f. The phase of the amplifying couplet west of the date line is consistent with the orographic response to a positive mountain torque (W03). Baroclinic processes and upper-level outflow from the tropical convection west of the date line help the amplification process. In Fig. 5f, the downstream low over the northeast Pacific has developed a negative tilt, and a full cyclonic breaking event is in progress. This behavior also becomes linked with a retrograding anticyclonic anomaly across Canada and the North Atlantic, which allowed for a strong projection onto the positive phase of the PNA and the negative phase of the NAO. This pattern of strong tropical–subtropical westerlies stays in place for ∼15 days.

Almost the opposite behavior was observed during T2, whose synoptic evolution to GSDM stage 1 is shown in Fig. 6. Now instead of a trough, there is a ridge at 20°N, 70°E (Figs. 6a,b), which is being forced by the strong convection over the Indian Ocean. The phase of the wave train is reversed compared to Fig. 5, and the Indian Ocean ridge grows into a large anticyclonic anomaly over south Asia as the transition matures (see Fig. 6f). The downstream amplification over the west North Pacific also occurs, but with the phase reversed (Figs. 6b–d) and little evidence of a preexisting wave train farther north. This leads to a wave break over the east Pacific (Figs. 6e,f) that becomes anticyclonic along the U.S. west coast. The stage 1 pattern was maintained by additional meridional–zonal flow variations leading to more anticyclonic wave breaking along the U.S. west coast (not shown). The final wave break occurred during late January 2002 and linked up with an anomalously strong subtropical jet, forced by tropical convective flare-up 6 near the date line, to produce an extreme winter weather event (snow and freezing rain) over the Great Plains. The GSDM stage 1 characteristics of the circulation finally broke down in mid-February 2002 (see Fig. 2).

e. Predictive skill of circulation transitions

The abrupt onset of the stage 3 and stage 1 circulation regimes has forecast implications for the medium range. Figure 7 shows the ensemble mean forecast skill out to day 15 during the 2001–02 winter season using a 1997 version of the National Centers for Environmental Prediction (NCEP) Medium-Range Forecast model. This model has a 25-yr reforecast dataset available and real-time predictions are posted on the Web (Hamill et al. 2004; see http://www.cdc.noaa.gov/people/jeffrey.s.whitaker/refcst/week2/). The skill measure used here is the spatial correlation between verifications and predictions of Northern Hemisphere 150-mb streamfunction anomalies. There are 3 periods where the 11-day ensemble-mean forecast skill of the Northern Hemisphere 150-mb streamfunction anomaly is relatively poor (<0.5). The common thread in these low-skill regimes is that a major change in tropical convective forcing is observed (and presumably not captured) during the forecast cycle. The major convective changes are highlighted with arrows in the top panel of Fig. 7 (i.e., flare-ups 3, 5, and 7). Note that the 5 and 7 transitions occur near the end of the forecast cycle, implying a rapid impact on the circulation anomalies of the Northern Hemisphere, if indeed they are the cause for the low forecast skill.

For T1 and T2, the amplification of a wave disturbance over the western North Pacific is a key synoptic event. On 23–24 December 2001, the “low–high” couplet west of the date line (Fig. 5d) was predicted 11 days in advance but the “high–low” couplet in the same place on 15–16 January (Fig. 6d) was not. A clue about what may be happening is derived from the general pattern of forecast skill described above for the 2001–02 winter. The forecast for 23–24 December 2001 is initialized after flare-up 3 while the forecast for 15–16 January 2002 is initialized before event 5. Failure to capture the start of flare-up 5 and the large negative mountain torque in the forecast cycle presumably contributed to the low forecast skill during the second transition. Many more cases need to be examined to confirm these assertions.

f. Discussion

Several atmospheric phenomena and processes were introduced during the case study description. Tropical convective forcing involved multiple time scales including the 30–60-day MJO, 20–30-day convective flare-ups, and other convectively coupled equatorial waves. Topographic forcing from synoptic-scale disturbances was clustered on a 50-day time interval and produced zonal momentum anomalies that were removed by the frictional torque. The PNA teleconnection pattern was involved in the removal process. Finally, the phase of synoptic-scale wave trains over the Asian–Pacific sector was related to tropical convective forcing over the Indo-Pacific warm pool region and to subsequent anticyclonic and cyclonic wave breaking over the eastern North Pacific.

The 50-day “oscillation” in the mountain torque is consistent with the 6-day decay time scale of the τ_F–τ_M index cycle. A red noise process with a 6-day decay time has its variance concentrated in a band with periods at around 6 days × 2π = ∼38 days. Plotting its spectrum with the log of frequency as the abscissa in a variance-conserving format shows this feature of red noise most clearly (see, e.g., Fig. 2 of WRP). A 50-day oscillation is a representative example from such a process, but of course no coherent forcing exists at 50 days or any other period in the red noise background.

In the case study, these and other processes are seen as working together to produce the evolving atmospheric circulation anomalies. The indices in Fig. 3 show a slow evolution while Fig. 2 has the transition occurring rapidly in association with synoptic-scale wave trains. The latter are seen as a mechanism for the spatial spreading of information about the changes in forcing from key regions around the hemisphere (e.g., western ocean baroclinic zones, tropical convective zones, and centers of low-frequency variability). Certainly, a fortuitous or random nature must be assumed about these fast (∼30 m s⁻¹) eastward-moving disturbances and their role in the transitions of the circulation; that is, there is always another synoptic system or wave train that can play the transition role. After the transition, repeatable synoptic events were superimposed on the quasi-stationary flow, and these represent the opportunities for skillful extended forecasts of synoptic-scale behavior.

The basic perspective is that forcing is of primary importance for determining the evolution of the atmospheric circulation and that different sources of forcing produce circulation responses that combine linearly to produce the circulation anomalies at any particular time. Anomalies may be maintained by feedbacks but eventually they disperse as the forcing waxes and wanes, even when coupling from persistent SST anomalies is involved. This perspective is supported by Sardeshmukh et al. (1997), Newman and Sardeshmukh (1998), and Newman et al. (1997), who demonstrate the importance of forcing for the evolution of low-frequency circulation anomalies. The success of the Winkler et al. (2001, hereafter WNS) linear inverse model (LIM) of weekly circulation anomalies is based on the importance of tropical and other forcing as a source for week 1–3 predictive skill.

WNS’s linear inverse model of 7-day averages has features in common with the GSDM introduced in the next section. The LIM has three optimal structures that undergo nonmodal growth over a wide range of truncations and maximized growth rates. Two optimals are linked with forcing from the MJO and ENSO while the third is a PNA pattern linked to eddy feedbacks in midlatitudes. The LIM is objective, so it can be used for dynamical feedback studies (WNS) as well as predictability studies (Newman et al. 2003).

The GSDM is subjective but includes a synoptic component and a submonthly oscillation in addition to the MJO and the PNA, arguably four “modes” versus three for LIM. Its dynamics are based on the terms in the global and zonal AAM budget. Meridional and vertical transports help produce the torques at the surface. A linear phasing of GSDM components is assumed, and this unites several time and space scales within the model. The model does not explicitly include ENSO, but knowledge of the ENSO cycle is critical when applying it in real time. Both models can be further extended and are tools for monitoring, attribution, and prediction. The GSDM is introduced next.

a. Atmospheric angular momentum

In the case study, global AAM (Fig. 3) undergoes a relatively smooth evolution despite the complicated forcing depicted in Fig. 4. A careful examination of the AAM time series can discern the responses to the fast and medium excursions of the global mountain and frictional torques seen in Fig. 4, while the overall ∼60-day oscillation implies a forcing by the weaker but more persistent torques associated with the MJO. Different time scales are clearly working together to produce the observed behavior. We now illustrate these building blocks and suggest why and how they may be related.

Figures 9b–d show the global and zonal AAM anomalies obtained when regressing onto the three indices of the GSDM in Figs. 8a–c. The heavy solid lines on Figs. 9a–e confirm that all three indices have a global AAM signal. The lag at which the AAM anomaly crosses 0 is ∼+2 days (Fig. 9b), +7 days (Fig. 9c), and 0 days (Fig. 9d) for the 3 GSDM components. This is the key assumption made in constructing our model.³ The point is that a linear combination of the three time scales, when shifted by these amounts, would result in a large relative AAM time tendency. We are specifically interested in large anomalies because synoptic experience indicates that extreme events consist of multiple time scales “working together.” Such a linear combination would produce torques whose sum resembles the time series seen in Fig. 4a of the case study. Maximum AAM tendency would occur at stage 2 of the GSDM and would have contributions from all model components.

The zonal mean AAM anomalies when added depict a coherent signal of AAM anomalies moving out of equatorial regions toward the subtropics (Fig. 9a) and a weaker signal of anomalies moving equatorward around 50°N. The intermediate time scale τ_F–τ_M index cycle (W03) imposes an asymmetry around day 0. Momentum transports by different types of eddies (e.g., subtropical wavenumber 1, midlatitude teleconnections, and midlatitude synoptic transients) play an important role in forcing all three of the zonal mean anomaly fields and provide a dynamical link between the time scales.

In the next subsection, the synoptic patterns related to the global and zonal AAM changes seen in Fig. 9 are presented. Since the <30-day tendency and the frictional torque regressions are performed on global indices, the results will emphasize those patterns that contribute to global AAM changes. For the <30-day tendency, the three main topographic regions are represented while for the global frictional torque, the Northern Hemisphere surface wind over both ocean basins will be important. During individual cases of subseasonal variation, not all regions will always contribute to the forcing. For example, in the case study, Asia was the primary contributor to the positive mountain torque before T1 while Asia and North America both contributed to the negative torque before T2. Nonetheless, a simultaneous forcing from the three regions is not unusual and can be organized by the poleward propagation of zonal mean wind anomalies or by the dispersion of Rossby waves into each hemisphere from west Pacific tropical heating. In general, however, the phasing of regional patterns is influenced by the use of a global measure on which to base the analysis and can distort important regional behavior.

b. Lagged regression sequences

Lagged linear regressions of global, 2D fields onto each index are used to define an evolution of circulation anomalies and will be referred to as a lagged regression sequence. Figures 10 –12 show sequences 1–3 of 200-mb vector wind and sea level pressure (SLP) anomalies when regressed onto the 3 indices of Fig. 8. The duration and time interval of each sequence emphasize its time scale. Sequence 1 is 40 days long with a 10-day interval, sequence 2 is 18 days long with a 4-day interval, and sequence 3 is 8 days long with a 2-day interval. Global fields are shown because real-time monitoring suggests that coherent subseasonal interactions occur across broad latitudinal bands, often centered on either side of the equator. The sequences illustrate the well-known observation that large spatial scales coincide with long time scales. Beyond this, there are other features familiar to weather and climate scientists.

In sequence 1 (Fig. 10), the subtropical cyclones/anticyclones of the MJO are seen as they slowly propagate eastward from day −20 through day 0 to day 20. At day 0 (Fig. 10), convection is active over the equatorial western Pacific and global AAM has a large positive time tendency, at which time equatorial westerlies are already present over the Western Hemisphere. In sequence 2 (Fig. 11), a negative Pacific–North American teleconnection pattern (W03) develops by day −2 as Rossby waves disperse across Asia and then North America during the 18-day sequence; this contributes to a large positive frictional torque anomaly at day 0 (not shown). In sequence 3 (Fig. 12), rapid energy dispersion events (30 m s⁻¹) are seen centered over the East Asian, North American, and South American mountains at day 0 and contribute to a positive global mountain torque anomaly. The events are embedded in a pattern of wavenumber 0–2 zonal wind anomalies, some of which are highlighted by arrows. Four days after the positive torque, westerly flow anomalies extend from the equatorial east Pacific eastward and poleward to the midlatitude central Pacific. These anomalies represent the ∼20-day quasi-oscillatory component of the GSDM (see also Fig. 9d).

Interestingly, sequences 1–3 all have local statistically significant wind and SLP anomalies in tropical regions, even the two faster modes of variability. The features are characterized by wavenumber 0–1 zonal wind anomalies at upper levels and provide a synoptic pattern that links the different types of subseasonal variability. This complements and extends the results for the global and zonal AAM relationship that is the basis for the model.

c. The synoptic–dynamic model (GSDM)

Figure 13 shows the 4 stages of the GSDM nominally centered about 12 days apart with an ∼50-day oscillation due to the MJO. These numbers have a wide range since the subseasonal band is broad, and transition between phases can occur rapidly. A complete temporal evolution of anomalies is shown only for the MJO. Twenty-two maps representing 11 lags for both OLR EOF 1 and EOF 2 were examined and the 4 lags in Fig. 13 were chosen to represent the MJO. The EOFs are in quadrature and show similar patterns when shifted by ∼10 days. Table 2 shows the correspondence between the stages and the lags for the two EOFs. The Hs and Ls for the other components are placed with a specific stage based on the “large AAM tendency” criterion discussed in relation to Fig. 9. Specifically, the colored Hs and Ls shown on sequences 2 (Fig. 11) and 3 (Fig. 12) are used to complete the construction of the stages.

d. The combined AAM cycle

Much is known about the subseasonal budget of global and zonal AAM based on recent diagnostic studies using reanalysis data. For the global budget, Weickmann et al. (2000) and Egger and Hoinka (2002) show that the mountain torques force AAM anomalies and the frictional torque damps them. In the semantics of Fourier analysis, this means the friction torque leads the mountain torque. All subseasonal variability has this characteristic. The zonal budget brings in the role of AAM transports, which are zero in a global integral. Meridional transports provide a link back to the structure and orientation of troughs and ridges within the spatial patterns shown in Fig. 13. Subseasonal zonal AAM anomalies have well-developed poleward (Feldstein 2001) and downward propagation that extends from the equatorial tropopause to the subtropical solid earth (Egger and Weickmann 2007; Egger and Hoinka 2005). Dynamically, many questions remain about the contribution from different mechanisms in this process. For example, Moncrieff (2004) proposes a role for the vertical transport of zonal momentum due to the structure of organized mesoscale convective complexes. Unfortunately, the zonal AAM budget cannot provide even rudimentary answers because it is not balanced in either the NCEP–National Center for Atmospheric Research (NCAR) (WKS) nor the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalyses (Egger and Hoinka 2005). The zonal mean flux convergence of relative AAM is large compared to the sum of zonal mean torques, even when the Coriolis and gravity wave drag torques are included. This contributes to a systematic dipole error in the zonal AAM budget as shown by W03 (see his Figs. 3c,d).

In this section, we are not so much interested in the AAM budget as in using it to bring out additional features of the GSDM and to relate its spatial patterns in Fig. 13 with the combined AAM cycle. Some of the features of the cycle are summarized in Table 3. Figure 14 shows the lag regression of the global and zonal AAM, the flux convergence of AAM, and the mountain and frictional torque onto daily time series of the global relative AAM tendency for November–March 1979–95. The approximate locations of the GSDM stages are shown at the bottom of the figure.

In Fig. 13, the Hs and Ls from different time scales were placed on the four stages without considering the actual time behavior that might ensue. Figure 14 shows this in an average sense for the global and zonal mean fields by regressing on an index that contains all the components of the GSDM (i.e., the daily global relative AAM tendency). The time behavior is clearly asymmetric with respect to zero lag, as would be expected, if only because of the intermediate time scale τ_F–τ_M index cycle. There appears to be a buildup phase during stage 1 (see Fig. 14 prior to day −5) where several processes are persistent, and there is a gradual decrease (increase) in relative AAM (frictional torque) anomaly due to a negative mountain torque. This occurs early in the MJO’s life cycle when positive convection anomalies are over the Indian Ocean and the intermediate time scale contributes a negative PNA as positive frictional torques increase (Fig. 14c).

There then appears an abrupt reversal or weakening in all anomalies around day −2, arguably led by a reversal in the meridional AAM transport across 30°N, from poleward before to equatorward after. This occurs just as the zonal mean easterly flow anomalies in the subtropics reach their peak values at 25°N. At the same time, the dominate wave train in the GSDM shifts from the intermediate time-scale teleconnection type (stage 1) to the wave-packet-dominated fast time-scale type (stage 2). The latter helps reverse the mountain torque and possibly initiates the “flip” in meridional transports. Figure 14b has a momentum source due to transports that extends from the northern to southern subtropics. The MJO is also undergoing a transition at this time when tropical convection moves out of the Indian Ocean and into the west Pacific Ocean region. A further forcing of the torques by the MJO could then lead to a continued oscillation. Figure 14a shows more damped anomalies after day 0 again reflecting the τ_F–τ_M index cycle.

During an oscillation, the vertically integrated AAM anomaly (Fig. 14a) alternates between easterly and westerly flow anomalies that propagate from equatorial regions to the subtropics. The oscillation represents the MJO along with a sharpening of the curve by 10–30-day variations. Global angular momentum is at a minimum at stage 1 when negative AAM anomalies (easterly wind anomalies) are moving off the equator and tropical sea level pressure is low. By stage 2, the easterly anomalies are approaching the subtropics and weakening on the equator. Tropical SLP is on the rise and MJO convection is active over the western Pacific. By stage 3, westerly flow anomalies are lifting off the equator, tropical SLP is high, and global AAM reaches a maximum. Convection is near the date line, but it is strongly suppressed over the Indian Ocean from both the MJO and 10–30-day oscillations. The cycle ends with stage 4 as westerly wind anomalies move off the equator and convection is active in the three separate centers.

The frictional torque tends to be large during stages 1/2 and 3, when anomalies are more persistent than during stages 2 and 4. Synoptically, stage 1 is represented by strong trade winds and a northward-shifted westerly flow, especially over the Atlantic and Pacific Oceans. Sometimes, blocking can contribute to the torque through easterly flow anomalies that are large and farther north. The mountain torque tends to be large during stages 2 and 4, which also tend to be more transient with faster time scales involved. The positive mountain torque in the Southern Hemisphere is due to the Andes and develops as convection shifts into the western Pacific between stages 1 and 2 (Madden and Julian 1972; Milliff and Madden 1996). In the Northern Hemisphere, the positive mountain torque is produced by high pressure anomalies east of the Tibetan Plateau and cold air outbreaks that accompany the shift of convection to the west Pacific.