1 Introduction

In boreal summer, the most pronounced circulation system in the upper troposphere over South Asia is the South Asian high (SAH), covering vast area of South Asia. The SAH is also called the Tibetan Plateau (TP) high, because its main body often locates over the neighboring region of the TP and its formation has long been attributed to elevated strong diabatic heating over the summer TP (Flohn 1960; Mason and Anderson 1963; Ye and Gao 1979; Reiter and Gao 1982; Yanai et al. 1992; Hoskins and Rodwell 1995; Wu et al. 1997; Ye and Wu 1998; Liu et al. 2004, 2013; Liu and Wu 2004). The intensity and location of the SAH exhibit multi-timescale variabilities (e.g., Tao and Zhu 1964; Krishnamurti 1973; Krishnamurti and Bhalme 1976; Tao and Ding 1981; Zhang et al. 2005; Zhao et al. 2009), and so their associated weather and climate anomalies over South Asia and across the Asia–Pacific region (Luo et al. 1982; Zhang and Wu 2001; Zhang et al. 2002; Duan and Wu 2005; Duan et al. 2008; Wu et al. 2009; Liu et al. 2013 and the references there in). Collectively, variabilities of the SAH also represent one of the most important factors influencing the transport of water vapor and chemical constituents between the troposphere and stratosphere in the Asian monsoon region (e.g., Randel et al. 2006; Park et al. 2008; Garny and Randel 2013). Therefore, investigating the formation, variability, and climatic impacts of the SAH has long been an important aspect of our understanding of the Asian monsoon and climate.

In addition to variabilities of the SAH operating at interannual or longer timescales (Zhang and Qian 2000; Jiang et al. 2011), significant variabilities are found at sub-seasonal (30–60 days), and sub-monthly [10–20 days, or quasi biweekly (QBW)] timescale in observational (Tao and Zhu 1964; Krishnamurti and Ardanuy 1980; Nitta 1983; Annamalai and Slingo 2001; Popovic and Plumb 2001; Fujinami and Yasunari 2004; Randel and Park 2006; Garny and Randel 2013; Ren et al. 2015; Wei et al. 2014, 2015; Yang and Li 2016; Wang and Ge 2016; Ortega et al. 2017) and modeling (Hsu and Plumb 2000; Liu et al. 2007) studies. The 30–60-day variation of the SAH is often associated with northward propagation of convection during the onset phase of the Asian summer monsoon (Yasunari 1981), while its QBW variation exhibits a zonal propagation feature that is prominent during the mature phase of the monsoon (Krishnamurti and Ardanuy 1980). Moreover, the variation at the QBW timescale has been recognized as a recurrent mode involving with almost all the elements of the Asian summer monsoon system (Krishnamurti and Bhalme 1976; Ortega et al. 2017). In particular, the QBW mode has been identified in the monsoon rainfall over East Asia (Yang et al. 2014, 2016), and in variations of the column-integrated apparent heat source over the summer TP (Wang and Duan 2015; Zhu et al. 2018).

It has been recognized that sub-seasonal variations of the SAH manifests not only in its intensity (Krishnamurti and Ardanuy 1980; Yasunari 1981; Nitta 1983; Annamalai and Slingo 2001; Fujinami and Yasunari 2004; Randel and Park 2006; Garny and Randel 2013; Wang and Duan 2015), but also in its zonal location (Tao and Zhu 1964, Hsu and Plumb 2000; Ren et al. 2007, 2015; Liu et al. 2007; Yang and Li 2016; Wang and Ge 2016; Ortega et al. 2017). Several studies also emphasize the role of potential vorticity (PV) shedding resulting from the zonally asymmetric instability induced by the zonally asymmetric PV distribution in leading to the periodic zonal migration of the SAH center (Hsu and Plumb 2000; Popovic and Plumb 2001; Liu et al. 2007). The occurrence of PV shedding in boreal summer (July–August) during the mature phase of the South Asian summer monsoon results in mainly westward propagation. In other seasons, PV shedding is related mainly to eastward propagation (Ortega et al. 2017). This is consistent with the recurrent westward displacement of the SAH core center from the TP region in boreal summer. Zhang et al. (2002) indicates a bimodality of the SAH center at 100-hPa from the pentad mean circulation patterns in boreal summer, reflecting the TP and IP phase of the SAH. They report that the TP phase is mainly connected to the strong diabatic heating over the TP, while the IP phase is connected to the adiabatic heating by the prevailing descending motion in the free atmosphere and the surface sensible heating over the IP. They also show drastically different circulation and climate patterns in eastern Asia and China between the TP and IP phases of the SAH. Yang and Li (2016) finds a similar bimodality of the 200-hPa SAH at the sub-seasonal time scales (30–60 days) using daily reanalysis data.

For the primary origins of the sub-seasonal oscillation of the SAH, as well as its QBW oscillation and related rainfall anomalies over the TP, previous studies fall into two basic categories. One concerns whether the SAH oscillation is thermally regulated by the changes in the diabatic heating over the TP and the other relates its oscillation to the nearby circulation perturbations related to the TP. The thermal forcing proponents relate the SAH oscillations to variations of the atmospheric diabatic heating over the TP region (Nitta 1983; Randel and Park 2006; Nützel et al. 2016; Wang and Ge 2016; and references there in), as the existence of the SAH itself is thought to be a response to the diabatic heating associated with convection over the TP (Hoskins and Rodwell 1995; Liu et al. 2004). The proponents of the dynamical influence emphasize that the SAH oscillation is related to a southward movement of the intra-seasonal perturbation centers originating from the midlatitudes along the subtropical jet (Fujinami and Yasunari 2004; Yang and Li 2016; Ortega et al. 2017), or to the intra-seasonal oscillation signals propagating from the tropics (Wang and Duan 2015). Ren et al. (2015) finds evidence showing both mechanisms are at work. Liu et al. (2007) reproduces the QBW oscillation of the SAH using idealized model experiments forced by a steady large-amplitude TP heating. Currently, consensus is still lacking on the primary causes of the SAH variations at various timescales.

The main objective of this study is to examine the relationship of day-to-day variations of the SAH with the TP heating. In particular, we will reveal how the SAH oscillation is connected to the QBW variation of the TPUHS and explore to what extent the SAH bimodality or the SAH oscillation between the IP and TP phases can be attributed to as a dynamic response to diabatic forcing over the TP. This would advance our understanding of the primary origin of the SAH QBW oscillation, the nature of the SAH bimodality, and their coupling with the TP thermal forcing. In the next section, we describe the data and methods used in this study. Section 3 reports the main results and Sect. 4 summarizes the key findings of this study.

2 Data and methods

2.1 Data

We derive the daily circulation fields and atmospheric diabatic heat field from the National Centers for Environmental Prediction and National Center for Atmospheric Research (NCEP/NCAR) reanalysis dataset (Kalnay et al. 1996), covering the period from 1 January 1979 to 31 December 2016. The spatial resolution of the NCEP/NCAR reanalysis product is 2.5° latitude × 2.5° longitude with 17 pressure levels (1000–10 hPa). The daily outgoing longwave radiation (OLR) fields are from the National Oceanic and Atmospheric Administration, and have similar horizontal resolution and temporal coverage (1 January 1979–31 December 2013) to the NCEP/NCAR data (Liebmann and Smith 1996). The daily anomaly fields of all variables are obtained by removing their daily annual cycles from the original fields. Following Zhu et al. (2018), the analysis here is mainly confined to the boreal summer period from 1 July to 31 August in regard of the relatively stable meridional position of the SAH, although the data from June and September have also been used in temporal filtering and lead/lag composites.

2.2 Methods

2.2.1 TPUHS index

The daily field of the atmospheric heat source over the TP is obtained by first calculating the apparent heat source (Q1) at each pressure level based on the heat budget equation (Yanai et al. 1973):

$${Q_1}={c_p}\left[ {\frac{{\partial T}}{{\partial t}}+\vec {V} \cdot \nabla T+{{\left( {\frac{p}{{{p_0}}}} \right)}^\kappa }\omega \frac{{\partial \theta }}{{\partial p}}} \right],$$
(1)

where T is the air temperature; V is the horizontal wind vector; p is pressure; p0 = 1000 hPa; κ = R/cp, with R being the gas constant and cp the specific heat capacity of dry air at constant pressure; ω is the vertical velocity in the pressure (p) coordinate; and θ is the potential temperature. We then obtain the column-integrated atmospheric heat source (\(\left\langle {{Q_1}} \right\rangle\)) by vertically integrating Q1 from the tropopause pressure level (pt = 100 hPa) to the lowest pressure level pb that is about 100 hPa away from the surface, namely,

$$\left\langle {{Q_1}} \right\rangle =\frac{1}{g}\int\limits_{{{p_t}}}^{{{p_b}}} {{Q_1}dp.}$$
(2)

The TPUHS index is defined as a spatial average of \(\left\langle {{Q_1}} \right\rangle\) over the TP region (25°–40°N, 70°–105°E; above the altitude of 1500 m). In addition, the diabatic heating rate (K/day) at each pressure level is also calculated, from dθ/dt (Wu et al. 1999), to relate the diabatic heating over the TP with the temperature changes at each pressure level.

It should be noted that previous studies have validated the reliability of the NCEP/NCAR reanalysis in representing the TP diabatic heating and its variabilities (e.g., Nigam et al. 2000; Rodwell and Hoskins 2001; Wang et al. 2011; Duan 2003). Here we have also confirmed the general consistency of the NCEP/NCAR reanalysis with other data sets (e.g., JRA-55).

2.2.2 Bimodality index of the SAH

To describe daily variations of the SAH core center quantitatively, we define a bimodality index of the SAH (SAHI) as the difference in the 200-hPa geopotential height anomalies between the TP (25°–35°N, 72.5°–92.5°E) and the IP (25°–35°N, 45°–65°E) regions based on the composite SAH patterns for strong and weak TPUHS days. Positive/negative SAHI then represents a relatively stronger/weaker SAH core center over the TP region than that over the IP region.

2.2.3 Composite QBW cycle

To obtain the composite QBW cycles of the TPUHS index, the SAHI, and other variables, we first extract the QBW (10–20 days) signals through the Lanczos band-pass filter (Duchon 1979). Based on a criterion of one standard deviation (STD) of the band-pass-filtered TPUHS index, there are a total of 92 anomalously strong (positive) and 93 anomalously weak (negative) TPUHS events in the 38 summers of 1979–2016. We then construct a ± 3-days lead–lag composite of all of the positive/negative TPUHS anomaly events against their peak/valley amplitudes. By joining the composite evolution of negative and positive events, we obtain the composite QBW cycle of TPUHS events, which consists of 14 daily phases with the peak at phase 4 and the valley at phase 11. Thus, phases 1–7 represent times when anomalies of TRUHS are negative and phases 8–14 positive anomalies. Similarly, we construct QBW cycles of other variables according to the timings of the composite TPUHS cycle.

3 Results

Figure 1 shows composite maps of daily anomalies in TP upper heating \(\left\langle {{{\text{Q}}_1}} \right\rangle\), outgoing longwave radiative fluxes, and 100 hPa geopotential height fields for days when the TRUHS index is below its normal by one standard deviation (STD, left column) and above one standard deviation (right column), It is seen that the anomalously stronger daily TPUHS coincides with anomalously lower OLR and stronger \(\left\langle {{{\text{Q}}_1}} \right\rangle\) over almost the entire TP region and the reverse is found during days of weaker TPUHS. The core center of the SAH is over the IP region when TPUHS is stronger and it shifts to the TP region when TPUHS is weaker. This is different from the known relationship between the SAH IP/TP mode and a weaker/stronger TP heating in monthly and longer timescale (Zhang et al. 2002 and references therein). The vertical profiles of spatial means of composite heating rate anomalies (Fig. 2a) exhibit a two-peak pattern: one at 400 hPa and the other at 150 hPa. In particular, the peak at 150 hPa is visible when the amplitude of TPUHS exceeds 0.5 STD. The vertical profiles of spatial means of composite geopotential height and potential temperature anomalies over the TP region (Fig. 2b) show shallow cold low pressure anomalies near surface and warm low pressure anomalies above for positive TPUHS anomalies and the reverse can be said for negative TPUHS anomalies. The temperature anomalies aloft are associated with TPUHS anomalies, namely that positive TPUHS anomalies coincide with warm temperature anomalies aloft and vice versa. The longitudinal profiles of geopotential anomalies at both 200 hPa and 500 hPa show a similar seesaw pattern between the IP and TP regions (Fig. 3). Specifically, for positive TPUHS anomalies, low pressure anomalies are found at both low and upper levels over the TP region and high pressure anomalies in both levels over the IP region and the reverse can be said for negative TPUHS anomalies.

Fig. 1
figure 1

Composite a, b daily anomalies of \(\left\langle {{{\text{Q}}_1}} \right\rangle\) (units: W m−2; contours; interval: 30) and OLR (units: W m− 2; shaded); and c, d daily geopotential height (contours; interval: 10; contours above 12,520 gpm) and its zonal deviation (shaded) at 200 hPa. The left column is for days when TPUHS are below its normal by one standard deviation (STD) and the right for positive TPUHS anomalies above one STD. The rectangles mark the TP and IP regions for the definition of the SAH index

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Fig. 2
figure 2

a Vertical profiles of the composite heating rate (units: K/day) for anomalously positive (red) and negative (blue) TPUHS anomalies at different intensity of the TPUHS index (in unit of its STD), and b the composite vertical profiles of potential temperature (K, shaded) and geopotential height anomalies (units: gpm; contours) over the TP region at different intensity of the TPUHS index (in unit of its STD). Dotted areas in b mark the 90% confidence level of the composite geopotential height anomalies

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Fig. 3
figure 3

Composite longitude distributions of a, b 200–hPa and c, d 500–hPa geopotential height anomalies (units: gpm) averaged within 25°–40°N. The left column is for negative TPUHS anomalies and the right for positive TPUHS anomalies

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One of the key features shown in Fig. 3 is that the magnitude of geopotential height anomalies at upper level is weaker than that at low level over the TP region but over the IP region, the magnitude of geopotential height anomalies in the upper level is stronger than that at low level. According to the hydrostatic balance relation, the configuration of positive TPUHS anomalies over surface low pressure anomalies would imply a vertical decreasing amplitude profile of negative geopotential height anomalies. Similarly, the configuration of negative TPUHS anomalies over surface high pressure anomalies implies a vertical decreasing amplitude profile of positive geopotential height anomalies. In this sense, the day-to-day relationship between TPUHS and upper level height anomalies is the same as that at other time scales, namely, positive TPUHS anomalies favors positive height anomalies and vice versa. However, the relative strength of geopotential height anomalies over the TP region in comparison with that over the IP region is also determined by the strength of surface pressure anomalies. In other words, the day-to-day variability of TPUHS is strongly coupled with surface pressure anomalies over the TP region. Stronger surface low pressure anomalies favor deep convections, resulting in strong positive TPUHS anomalies and thus a warmer upper troposphere over the TP region. Although the warmth of the upper troposphere over the TP region tends to favor positive geopotential height anomalies in upper levels, the higher surface geopotential height (or pressure) anomalies over the IP region (Fig. 3d) would imply higher upper level geopotential height anomalies there (Fig. 3b). This explains why at daily scales, the SAH center is located over the IP instead of the TP region when TPUHS anomalies are positive.

Following Zhang et al. (2002), we examine the bimodality of the day-to-day variability of the SAH center. Shown in Fig. 4 is the longitudinal distribution of the daily frequency of the SAH core center in the 38 summers of 1979–2016 during days of negative (panel a) and positive (panel b) TPUHS anomalies. It is seen that the bimodality of the SAH center is observed regardless of the polarity of the TPUHS index. Furthermore, the TP (87.5°–97.5°E) phase clearly becomes relatively more frequent than the IP (50°–60°E) phase when the TPUHS index is negative. When the TPUHS index is positive, the IP phase is more preferred than the TP phase. The power spectral analysis of the daily SAH and TPUHS indices reveals that both of them have a dominant time scale in the range of 10–20 days or the QBW scale (Fig. 5a). In respect of the close relationship between the daily TPUHS and the SAH center demonstrated above, we anticipate a coupled temporal variation between the SAH and TPUHS indices. To prove this, we display in Fig. 5b the composite variation of the SAH index (SAHI) against the composite QBW cycle of the TPUHS index according to the procedures outlined in Sect. 2. The QBW variation of the SAHI is clear and indeed is coupled with the TPUHS index. Specifically, the positive peak of the SAHI (SAH TP mode) tends to closely follow the negative peak of the TPUHS and negative peak of the SAHI (SAH IP mode) follows positive peak of the TRUHS with a delay of about one day.

Fig. 4
figure 4

Longitude distributions of the number of days of 200-hPa SAH center for a negative and b positive TPUHS anomalies in boreal summer

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Fig. 5
figure 5

a Power spectrum (thick solid line) of the normalized daily SAHI (black) and the THI (gray) derived from data in the 38 summers (1 July to 31 August) of 1979–2016. The short and long dashes mark the 95% confidence level and the red noise spectrum respectively. b Composite evolution of the SAHI (units: gpm; black) against the composite QBW cycle of the THI (i.e., TPUHS index; units: W m−2; gray). The abscissa in a is the period in days, and in b the time line of the composite QBW cycle of the TPUHS (in days)

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Shown in Fig. 6 are pressure–longitude cross-section diagrams of circulation anomalies along the SAH ridgeline at the time when the TPUHS is in its valley (P4, panel a) and peak (P11, panel b) phases, with the former corresponding to the weakest TPUHS and the latter the strongest TPUHS. It is seen at the phase 4 (Fig. 6a), descending motion anomalies prevail over the TP and the East Asian region, which correspond to anomalously weaker TP convection and shallower diabatic heating. As a result, warm temperature anomalies over the TP lie mainly in the lower layer of troposphere and near surface. Associated with the suppressed TP convection are descending motion anomalies over most the TP region and weak ascending motion anomalies over the IP. The cold anomalies in the lower troposphere over the IP are resulted from the anomalously weaker adiabatic heating associated with the ascending motion anomalies. In contrast, the strongest TPUHS at P11 (Fig. 6b) corresponds to anomalously strong and deep TP convection, which are manifested by the pronounced ascending motion anomalies and the elevated warm anomalies over the TP. Associated with the anomalously strong convection over the TP are descending motion anomalies over the IP. The adiabatic heating associated with anomalously descending motion anomalies gives rise to warm anomalies in the troposphere over the IP.

Fig. 6
figure 6

Pressure–longitude cross-sections of the composite anomalies of potential temperature (units: K; shading), geopotential height (units: gpm; contours), and vertical motion ω (units: 10−3 Pa s−1; arrows) anomalies along the SAH ridgeline (27.5°–37.5°N), in P4 and P11 of the QBW cycle of the TPUHS. Dotted areas mark the 95% confidence level of the composite potential temperature anomalies

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Figure 7 displays the composite time–longitude evolution of the geopotential height fields (both the total and anomaly fields) at 100 hPa (panel b) and 200 hPa (panel c) along the SAH ridgeline in the context of the QBW cycle (panel a) of the TPUHS index. SAH center oscillates between the IP and TP region with the center over the TP when the TPUHS is in the valley phase and over the IP region at the peak phase of the TPUHS. Accompanied with the oscillation of the SAH center between the two regions is a systematic westward propagation of the geopotential height anomalies from East Asia to the Mediterranean Sea in the zonal direction. The development of the TPUHS from its valley (P4) to peak (P11) phases corresponds to the westward propagation of positive height anomalies from the TP–East Asia region to the IP region while the decaying of the TPUHS from its peak (P11) back to its valley (P4) is accompanied with the westward propagation of negative height anomalies. The systematic westward propagation of the circulation anomalies of the SAH at the QBW time scale coincides with the periodic westward movement of 200-hPa streamfunction centers in the model simulation forced by idealized TP heating (Liu et al. 2007), as well as the periodic westward PV-shedding in case studies (Hsu and Plumb 2000; Popovic and Plumb 2001). Liu et al. (2007) provides evidence suggesting that the westward movement of streamfunction centers is related to the zonally asymmetric instability induced by strong diabatic TP heating in summer. Alternatively, the westward propagation can also be understood based on the basic Rossby wave propagation theory (Hoskins et al. 1985).

Fig. 7
figure 7

Composite QBW evolution of a the TPUHS index itself, and b 100–hPa (contours above 16,750 gpm) and c 200–hPa (contours above 12530gpm) geopotential height (units: gpm; interval: 10) and their anomalies (shaded) along the SAH ridge (27.5°–37.5°N). The portion from P15 to P21 is a repeat from P1 to P7 to show a complete temporal evolution

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Figure 8 shows the 200-hPa geopotential height and divergent wind anomaly fields over the Eurasian region during each successive phase of the QBW cycle of the TPUHS. Also shown is the subtropical jet core that is zonally elongated along the northern fringe of the SAH from the Mediterranean Sea to East Asia. It is seen that positive geopotential anomaly center over the TP at the phase P4 (the valley phase) coexists with convergent wind anomalies, while the negative geopotential anomaly center over the IP coexists with divergent wind anomalies. This indicates the presence of anomalously weaker convection over the TP and anomalously weaker compensatory dynamical convergence and descending motion over the IP during P4 when the TP SAH center is anomalously stronger. The opposite can be seen at the phase P11. Furthermore, one can easily identify a westward propagation signal in the field of geopotential anomalies along the SAH ridgeline (around 30°N). Specifically, the transition between the positive/negative geopotential anomaly center over TP/IP in the P4 valley phase and the negative/positive geopotential anomaly center over the TP/IP in the P11 peak phase takes place through successive westward displacements of the anomaly centers (Fig. 8a–n).

Fig. 8
figure 8

Composite anomalies of 200-hPa geopotential height (units: gpm; shaded) and divergent wind (units: 10−1 m s−1; arrows) anomalies in each phase of the QBW cycle of the TPUHS. The red contours denote the westerly jet that is stronger than 25 m s−1. Dotted areas and black arrows respectively indicate that the composite anomalies of geopotential height and divergent wind are statistically significant above a 95% confidence level

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4 Summary

This study investigates the summertime day-to-day variability of the South Asian high (SAH) driven by the upper atmospheric heating over Tibetan Plateau (TP). Following Yanai et al. (1973), we derive the daily 3D fields of the atmospheric heat source (Q1) over the TP using the NCEP/NCAR reanalysis dataset covering the summer months (1 July–31 August) in the period from 1 January 1979 to 31 December 2016. The daily TPUHS (Tibetan Plateau upper-atmospheric heat source) index is constructed using daily values of the vertical-column integrated and spatially averaged Q1 over the TP region (25°–40°N, 70°–105°E). We then examine the relationships between the TPUHS index and circulation anomalies associated with the day-to-day variability of the South Asian high (SAH) in summertime as well as their temporal and spatial patterns. The key findings of this study are

  1. 1.

    The dominant time scale of the daily TPUHS index is at quasi biweekly (QBW) timescale. Associated with the QBW oscillation of the TPUHS index is a seesaw oscillation pattern in both geopotential height and surface pressure anomalies over the Tibetan Plateau (TP) and Iranian Plateau (IP) regions.

  2. 2.

    Positive anomalies of the daily TPUHS index correspond to elevated diabatic heating at the upper troposphere over the TP whereas negative TPUHS anomalies correspond to suppressed diabatic heating. Associated with positive anomalies of the daily TPUHS index are low/high geopotential anomalies throughout the troposphere over the TP/IP region, and associated with negative TPUHS anomalies are high/low geopotential height anomalies over the TP/IP region. Therefore, the enhanced convection over the TP region corresponds to the situation in which the center of the SAH is over the IP region while the center of the SAH is over the TP region when TP convection is suppressed.

  3. 3.

    The QBW oscillation of the TPUHS index is characterized with a systematic westward propagation of upper level geopotential anomalies along the SAH ridgeline (around 30°N) from East Asia to the Mediterranean with positive/negative geopotential height anomalies over the TP/IP at the valley phase of the TPUHS index and negative/positive geopotential height anomalies over the TP/IP at the TPUHS peak phase.

  4. 4.

    It is such association of geopotential anomalies along the SAH ridgeline with the QBW oscillation of the TPUHS index that gives rise to the SAH bimodality with negative TPUHS index corresponding to the TP phase or center of the SAH over the TP region and positive TPUHS index the IP phase of the SAH in which center of the SAH is over the IP region.

Although the QBW signal can be clearly reproduced through the composite day-to-day TPUHS variation, we may need modeling evidence to confirm the possible central role of the TPUHS and to identify the forcing-and-response relationship between the variations of the TPUHS and those of the surrounding circulation systems. We intend to seek this evidence in a separate study to further demonstrate the linkage of the QBW signals in boreal summer. Moreover, further investigations may also be needed to show if other factors may play important role in giving rise to the QBW variations of the TPUHS and SAH as well as their surrounding circulation anomalies, since the QBW signals themselves may change from year to year (e. g. Wang and Ge 2016), or be influenced by propagation signals from the tropics (Wang and Duan 2015).