Abstract

Part I of this study examined the modulation of the monsoon trough (MT) on tropical depression (TD)-type–mixed Rossby–gravity (MRG) and equatorial Rossby (ER) waves over the western North Pacific based on observations. This part investigates the interaction of these waves with the MT through a diagnostics of energy conversion that separates the effect of the MT on TD–MRG and ER waves. It is found that the barotropic conversion associated with the MT is the most important mechanism for the growth of eddy energy in both TD–MRG and ER waves. The large rotational flows help to maintain the rapid growth and tilted horizontal structure of the lower-tropospheric waves through a positive feedback between the wave growth and horizontal structure. The baroclinic conversion process associated with the MT contributes a smaller part for TD–MRG waves, but is of importance comparable to barotropic conversion for ER waves as it can produce the tilted vertical structure. The growth rates of the waves are much larger during strong MT years than during weak MT years.

Numerical experiments are conducted for an idealized MRG or ER wave using a linear shallow-water model. The results confirm that the monsoon background flow can lead to an MRG-to-TD transition and the ER wave amplifies along the axis of the MT and is more active in the strong MT state. Those results are consistent with the findings in Part I. This indicates that the mean flow of the MT provides a favorable background condition for the development of the waves and acts as a key energy source.

1. Introduction

It has long been recognized that tropical wave activity is affected by monsoon trough variability over the western North Pacific (WNP) (Chen and Weng 1998). The observational analysis reported in Wu et al. (2015, hereafter Part I) examined the structure and evolution of westward-propagating tropical waves (WTWs) under different states of the monsoon trough. The results reveal that the WTWs tend to grow in the monsoon trough region through an interaction between the WTWs and the monsoon trough. A strong monsoon trough (S-MT) event, which is associated with an eastward extension of enhanced monsoon trough, leads to an increase in lower-tropospheric WTWs to west of the date line, but those waves become weaker and are located farther west for a weak monsoon trough (W-MT) event in association with a westward retreat of weakened monsoon trough. For the tropical depression (TD)-type–mixed Rossby–gravity (MRG) waves, an apparent transition from MRG waves to off-equatorial TD-type disturbances is identified in the region of the monsoon trough. The equatorial Rossby (ER) waves have a faster growth in amplitude than TD–MRG waves in the same monsoon trough condition, but their structures and propagation characters have no marked change (no apparent transition from ER waves to small-scale disturbances). An east–west contrast is identified between the different states of the monsoon trough, which is consistent with the eastward extension of the monsoon trough. Some previous studies have noted that the monsoon trough provides conditions favorable for the MRG-to-TD transition and the evolution of ER waves (e.g., Wu et al. 2014a; Chen and Huang 2009; Li 2006; Gall et al. 2010; Gall and Frank 2010). However, the basic mechanisms controlling the variability and the dynamic processes that establish these changes of the WTW structure along the monsoon trough are not fully understood.

To explore the physical mechanism of interactions between eddy and mean flow, Lau and Lau (1992) used energy balance equations, and diagnosed principal energy sources of the tropical disturbances. Sobel and Maloney (2000) and Maloney and Hartmann (2001) noted that the summer mean flow pattern in the WNP favors an energy conversion to wave disturbances. Through diagnosis of the barotropic energy conversion, Wu et al. (2012, 2014a) identified that the wave disturbances moving westward from the tropical eastern Pacific gain energy from the mean flow when they meet with the eastward-extending monsoon trough. They pointed out that the energy conversion is an important mechanism for the linkage between the monsoon trough variability and the growth of the precursor synoptic-scale perturbations, which results in more tropical cyclogenesis in the southeast quadrant (5°–20°N, 150°E–180°) of the WNP. Several previous studies (e.g., Sobel and Bretherton 1999; Done et al. 2011) suggested that the wave energy accumulation by a mean flow plays an important role in changes of wave structure and amplitude. Those results suggested that the developing of tropical wave disturbances may be related to the large-scale background flow such as the Madden–Julian oscillation (MJO) and monsoon trough through the energy conversion from mean flow to eddy. Here, we focus on investigation of how the monsoon trough modulates the energy conversion of mean flow to TD–MRG and ER waves. This may help to explain the wave transitions over the region of the monsoon trough in Part I.

Several idealized numerical models have been invoked to understand and explain the scale interaction processes of tropical waves with the background flow. Aiyyer and Molinari (2008) conducted idealized numerical experiment using a barotropic model and indicated that the propagation paths as well as the amplification of eddies differ substantially between different MJO environments. Using a hierarchical modeling approach, Done et al. (2011) showed that the interactions of easterly waves with the background flow reduce the longitudinal and vertical scale of the waves. Kuo et al. (2001) showed that a preexisting disturbance allows the formation of a new vortex disturbance through interaction with the monsoonlike mean flow. Their results suggest that the formation of a new vortex disturbance may be due to the scale contraction of waves by the confluent background flow as waves approach the monsoon region. Aiyyer and Molinari (2003) employed a linear shallow-water model to simulate the evolution of MRG waves in background states representative of the convective phase of the MJO. Their result suggests that off-equatorial TD disturbances may develop as a result of the reduction of the wavelength of the MRG waves due to the interaction with persistent large-scale flow.

In this paper, we examine the energy budget of TD–MRG waves and ER waves over the WNP during S-MT and W-MT years, and explore the possibility that the evolutions of WTWs are driven by the monsoon trough background flow. To further clarify the relationship between monsoon trough and WTWs, we use an idealized numerical model to understand the evolution of WTWs within monsoon trough background flow, and compare it with the observed result in Part I. The data and methods we use are described in section 2. The energy budget analysis for TD–MRG waves and ER waves during S-MT and W-MT is conducted in section 3. Section 4 discusses the evolution of idealized MRG wave and ER wave response to different monsoon trough background states in a shallow-water model. The results are summarized in section 5.

2. Data and methods

The data used in this study, as described in Part I, are NCEP-II datasets [Kanamitsu et al. (2002), available online at http://www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalysis2.html] used to compute the energy budget during the period 1979–2007. We use the daily datasets with a horizontal resolution of 2.5 latitude × 2.5 longitude with 17 pressure levels. The original daily fields are filtered to isolate the TD–MRG waves and ER waves using a space–time spectral filtering (Wheeler and Kiladis 1999). The details of the method and domain have been described in Part I. The analysis is performed in the WNP and during July–November for the period 1979–2007. Consistent with Part I, the wave activity is measured by the eddy kinetic energy (EKE) and the interannual variability of intensity and spatial location of the monsoon trough is measured by a monsoon trough index (Wu et al. 2012). The S-MT and W-MT years are determined according to the value of the monsoon trough index [based on Wu et al. (2012)’s criterion], which is defined as the time coefficient of the leading EOF mode to the average latitude (5°–20°N) of the 850-hPa positive relative vorticity in the 100°E–180° longitudinal band for July through November in each year. The S-MT years with the seven highest values of the monsoon trough index (MTI) include 1982, 1986, 1990, 1991, 1997, 2002, and 2004 and the W-MT years with the seven lowest values of the MTI are 1984, 1988, 1995, 1996, 1998, 1999, and 2007.

3. Impact of energy conversion in TD–MRG and ER waves

a. Energy budget of TD–MRG and ER waves

The observational analysis of Part I reveals the difference in structure and evolution of the WTWs between the S-MT and the W-MT years. To further understand how the monsoon trough–induced mean flow change modulates the activity of WTWs, we examine the energy conversion in the growth of TD–MRG waves and ER waves, respectively. As noted earlier, the singular approach of the energy conversions in the zonally averaged problem provides the interpretation as a measure of time-mean eddy energetics. The method for diagnosis of time-mean eddy energetics for the atmosphere using zonal average can be found in, for example, Lorenz (1967) and Oort and Peixoto (1974). For the present problem, the relationship between energy conversions and wave activity may be spatially inhomogeneous. In this study, the approach differs from previous studies of the energy cycle in that the calculations are based on each wave band rather than a derivation of time mean. In addition, the zonal average is not used as typical. This approach gives a more accurate estimate of wave energy cycles in which the geographical distributions of the energy conversion need to be shown at each tropospheric level so that it provides a better physical explanation for the relationship between energy conversions and wave activity.

For a given time-invariant basic state, the conversion from the basic state (e.g., monsoon trough) total energy to the eddy energy is encapsulated in the following expression:

 
formula
 
formula

where the overbar represents a time average and the prime represents a deviation, and {A, B} designates conversion of energy from form A to form B. The eddy kinetic energy () and potential energy () is represented by

 
formula
 
formula

where u, υ, and θ‬ represent zonal and meridional velocity, and the potential temperature, respectively; N is the buoyancy frequency; and g is the acceleration due to gravity.

The energy transformations from the above equations are

 
formula
 
formula
 
formula

and the source or sink terms are

 
formula
 
formula

where Φ is the geopotential, V is the horizontal velocity vector, w is the vertical velocity, cp is the specific heat at constant pressure, p0 = 1000 mb, Rd = 287 J K−1 kmol−1 is universal gas constant, Q1 is the apparent heat source, D is the drag coefficient, and is the horizontal divergence operator. We have computed the conversion from eddy available potential energy to eddy kinetic energy (), the conversion from mean to eddy kinetic energy (), and the conversion from mean available potential energy to eddy potential energy () using the NCEP-II data, respectively. The ε′ term is the dissipation of eddy kinetic energy by frictional and other subgrid-scale effects, which is not considered in our budget. Those energy conversion terms are computed based on the filtered perturbation and the time-mean fields. The overbar represents a time average over the analysis period. The prime denotes a perturbation from the time mean, and it is defined using the filtered data for TD–MRG waves and ER waves, respectively. We note that the diagnostic results based on the above provide important information of the energy conversion though with some uncertainty.

The energy cycles of July–November mean states over the southeast quadrant (5°–20°N, 150°E–180°) of the WNP, which display more significant variations in monsoon trough and its relation with WTWs (see Wu et al. 2012; Part I), are shown in Fig. 1 for S-MT and W-MT years. For the TD–MRG waves, the direction of the energy flow is consistent between the S-MT and W-MT years (Fig. 1a). The EKE is mainly maintained by the conversion from mean kinetic energy (MKE) to EKE. The conversion from eddy potential energy (EAPE) to EKE, which is a minor source of EKE, is about one-seventh of the magnitude of conversion from MKE to EKE. The EAPE is maintained by the conversion from mean available potential energy (MAPE) to EAPE, the conversion from EAPE to EKE, and the EAPE generation/loss through diabatic heating processes (R′). Those three terms have approximately the same order of magnitude. Nevertheless, there are differences in the magnitude of energy conversion between the S-MT and W-MT years. In general, the magnitude is larger in S-MT years than in W-MT years. From the viewpoint of the energy transport, EKE has an increase in both S-MT and W-MT via barotropic energy of kinetic energy from monsoon trough flow to the TD–MRG waves, implying reduced baroclinic developing of TD–MRG waves; and EAPE has a decrease in W-MT years via the energy conversion from MAPE to EAPE, as a source of barocliniticy, and diabatic heating processes. To assist the discussion, we use the correlation analysis to investigate the relationship of wave activity (EKE) to various conversion processes in the monsoon trough region and a positive (negative) significant correlation indicates that, as suggested by linear theory, wave activity (EKE) tends to be enhanced (weakened) as the energy conversion changes, possibly related to the intensity and local change of monsoon trough flow. The top row of Table 1 shows the correlation coefficients of 850-hPa EKE with various conversions for TD–MRG waves. A significant positive correlation above the 99% confidence level is present between 850-hPa EKE and the conversion from MKE to EKE. The negative correlation between 850-hPa EKE and the R′ is above the 90% confidence level.

Fig. 1.

The 850-hPa energy cycle diagram averaged over S-MT years (solid arrows) and W-MT (dashed arrows) in the southeast quadrant of the WNP for (a) TD–MRG waves and (b) ER waves. Units are 10−6 m2 s−3. The arrows indicate the direction that corresponds to positive values.

Fig. 1.

The 850-hPa energy cycle diagram averaged over S-MT years (solid arrows) and W-MT (dashed arrows) in the southeast quadrant of the WNP for (a) TD–MRG waves and (b) ER waves. Units are 10−6 m2 s−3. The arrows indicate the direction that corresponds to positive values.

Table 1.

Correlation coefficients between the energy transformations and the EKE at 850 hPa in the southeast quadrant (5°–20°N, 150°E–180°) and over all years in the analysis (1979–2007). Italics (boldface) indicate that the value exceeds the 99% (90%) confidence levels.

Correlation coefficients between the energy transformations and the EKE at 850 hPa in the southeast quadrant (5°–20°N, 150°E–180°) and over all years in the analysis (1979–2007). Italics (boldface) indicate that the value exceeds the 99% (90%) confidence levels.
Correlation coefficients between the energy transformations and the EKE at 850 hPa in the southeast quadrant (5°–20°N, 150°E–180°) and over all years in the analysis (1979–2007). Italics (boldface) indicate that the value exceeds the 99% (90%) confidence levels.

For the ER waves (Fig. 1b), the direction of the energy flow is the same as the TD–MRG waves except for the R′ during the W-MT years. Nevertheless, there are differences in the magnitude of energy conversion between the two types of WTWs. The baroclinic conversion from MAPE to EAPE is much larger for ER waves than for TD–MRG waves during both S-MT and W-MT years and the barotropic conversion from MKE to EKE is substantially larger for ER waves. The bottom row of Table 1 shows that 850-hPa EKE has a significant positive correlation with the barotropic conversion from mean to eddy kinetic energy as well as the baroclinic conversion from mean to eddy available potential energy.

b. Eddy kinetic energy balance

The barotropic energy conversion from the basic state (environment) to the EKE of WTWs (both the TD–MRG waves and ER waves) is consistent and closely linked with the 850-hPa EKE of WTWs. This suggests that barotropic energy conversion processes play an important role in the WTW activity. The eastward extension of the monsoon trough in response to an S-MT event leads to an increase in barotropic energy conversion over the southeast quadrant of the WNP. To further understand how the monsoon trough–induced mean flow change modulates the wave activity, we diagnose the effect of the barotropic energy conversion in the TD–MRG and ER wave growth, respectively.

Figures 2a,b show the composite longitude–vertical structure of the barotropic energy conversion term averaged over 5°–20°N during S-MT and W-MT years, respectively, for the TD–MRG waves. During the S-MT years (Fig. 2a), the maximum EKE growth rate, which indicates the energy conversion from the mean flow to the TD–MRG waves, appears in the lower troposphere (1000–700 hPa) over the tropical western Pacific with a pronounced maximum at 850 hPa over the longitude 140°–150°E. During the W-MT years (Fig. 2b), the maximum energy conversion appears in the upper troposphere (400–100 hPa) over the tropical central Pacific. In the lower troposphere (1000–800 hPa), the large energy conversion is located in the tropical western Pacific with a pronounced maximum at 850 hPa near 120°E. Figure 2c shows that the most significant difference of the barotropic energy conversion between S-MT and W-MT years occurs over the western Pacific (130°E–180°) in the lower levels (1000–600 hPa), which is consistent with the difference in the location of monsoon trough in the lower troposphere. At 850 hPa (Figs. 2d and 2e), in both types of years, a tongue of large positive energy conversion values extends southeastward along the monsoon trough from the Philippines to the equatorial central Pacific, indicating that the summer monsoon trough flow always provides energy for the TD–MRG waves to grow. In comparison, the barotropic energy conversion is much more strengthened during S-MT years than during W-MT years. The difference field shows a similar northwest–southeast-oriented pattern (Fig. 2f). The anomalous barotropic energy conversion term is significant in the southeast quadrant of the WNP where the monsoon trough expands eastward (retreats westward) during the S-MT (W-MT) years. The result implies that more (less) rapid growth of the TD–MRG wave during the S-MT (W-MT) years is attributed to more (less) efficient low-level barotropic energy conversion.

Fig. 2.

(top) Longitude–height cross section of the EKE time change rate due to the barotropic energy conversion (unit: 10−5 m2 s−3) associated with TD–MRG waves averaged over the latitudes of 5°–20°N for (a) S-MT years, (b) W-MT years, and (c) their difference (S-MT composite minus W-MT composite). (bottom) The horizontal distribution of the 850-hPa EKE time change rate for (d) S-MT composite, (e) W-MT composite, and (f) their difference (S-MT composite minus W-MT composite). Light and dark shades in (c) and (f) indicate areas where the difference exceeds the 90% and 95% confidence levels, respectively.

Fig. 2.

(top) Longitude–height cross section of the EKE time change rate due to the barotropic energy conversion (unit: 10−5 m2 s−3) associated with TD–MRG waves averaged over the latitudes of 5°–20°N for (a) S-MT years, (b) W-MT years, and (c) their difference (S-MT composite minus W-MT composite). (bottom) The horizontal distribution of the 850-hPa EKE time change rate for (d) S-MT composite, (e) W-MT composite, and (f) their difference (S-MT composite minus W-MT composite). Light and dark shades in (c) and (f) indicate areas where the difference exceeds the 90% and 95% confidence levels, respectively.

The pattern of the barotropic energy conversion for the ER waves, which is shown in Fig. 3, is similar to that for the TD–MRG waves. The positive maxima of the conversion rate are located in the low troposphere (1000–600 hPa) over the western Pacific, and the negative values are found in the upper troposphere (300–100 hPa) (Figs. 3a,b). The conversion rate in the lower troposphere is larger during the S-MT years than during the W-MT years (Fig. 3c). The largest difference in the energy conversion appears in the lower troposphere over the WNP (120°E–180°). At 850 hPa, large positive energy conversion of the ER waves extends southeastward from the Philippines to the equatorial central Pacific along the monsoon trough location (Figs. 3d,e). This conversion is stronger during S-MT years, and the difference is greater around the monsoon trough (Fig. 3f). Compared to that of the TD–MRG waves (Fig. 2), the barotropic energy conversion of the ER waves has a similar pattern and difference between the two monsoon trough states, but with a larger magnitude.

Fig. 3.

As in Fig. 2, but for the ER waves.

Fig. 3.

As in Fig. 2, but for the ER waves.

To understand how the monsoon trough flow modulates the wave activity via the barotropic energy conversion, we compute the contribution from each term in the barotropic energy conversion and construct the composite (Figs. 4 and 5). The results indicate a dominance of the generation by and due to large zonal and meridional variations in the zonal wind over the monsoon trough region for both TD–MRG and ER waves. So, the dominant EKE generation is caused by wave disturbances associated with longitudinal zonal wind convergences ( > 0) and latitudinal zonal wind gradients ( > 0) over the WNP, consistent with the speculation in Maloney and Hartmann (2001). It is interesting to note that the barotropic energy conversion is mainly contributed by the and terms during S-MT years, and the term is much smaller likely due to weak lower-level shear () flow in regions to west of 150°E with a broad monsoon trough (Figs. 4a,b and 10b) so that it can be ignored during W-MT years. In other words, the effect of the monsoon trough confluent flow (zonal gradients of the zonal wind) is greater in contributing to the barotropic energy conversion than that of the monsoon trough shear flow (meridional gradients of the zonal wind) when the monsoon trough retreats westward during W-MT years. They do, however, show a comparable magnitude when the monsoon trough extends eastward during S-MT years. During W-MT years, the weak energy conversion and westward retreat following the monsoon trough lead to reduced wave growth and a westward shift in the location of wave growth. Meanwhile, the relatively weak barotropic energy conversion occurs in the midlatitudes during W-MT years due to the weak lower-level convergence () flow north of the monsoon trough in association with the recurve flow in eastern part of the Pacific subtropical high (Fig. 4b). This weak convergent flow leads to weak positive barotropic energy conversion through the interaction of large-scale ER waves with the convergent flow to the north of the monsoon trough during W-MT years, but it does not appear in small-scale TD–MRG waves (Fig. 2e).

Fig. 4.

The 850-hPa vector winds (m s−1), meridional shear of zonal wind (, 10−6 s−1, contours), and zonal convergence of zonal wind (, 10−6 s−1, shaded) for the average of (a) S-MT and (b) W-MT years. (top) Longitude–height cross section and (bottom) 200-hPa horizontal distribution of barotropic energy conversion terms (c),(g) ; (d),(h) ; (e),(i) ; and (f),(j) associated with TD–MRG waves for (c)–(f) S-MT years and (g)–(j) W-MT years. Otherwise, as in Fig. 2.

Fig. 4.

The 850-hPa vector winds (m s−1), meridional shear of zonal wind (, 10−6 s−1, contours), and zonal convergence of zonal wind (, 10−6 s−1, shaded) for the average of (a) S-MT and (b) W-MT years. (top) Longitude–height cross section and (bottom) 200-hPa horizontal distribution of barotropic energy conversion terms (c),(g) ; (d),(h) ; (e),(i) ; and (f),(j) associated with TD–MRG waves for (c)–(f) S-MT years and (g)–(j) W-MT years. Otherwise, as in Fig. 2.

Fig. 5.

As in Figs. 4d–j, but for the ER waves.

Fig. 5.

As in Figs. 4d–j, but for the ER waves.

Further, note that a large positive barotropic energy conversion near 160°E–180° at the upper levels appears for TD–MRG waves (Fig. 2b) due to the increase of upper-tropospheric trough. This is mainly contributed by the and terms at upper levels during W-MT years (Figs. 4g–j). In other words, the effect of the upper-level meridional wind confluence ( < 0) and zonal wind divergence ( > 0) flow in association with the deep equatorward intrusion of the increase of upper-tropospheric trough during W-MT years (not shown) is greater contributing to the upper-level barotropic energy conversion. Moreover, for TD–MRG waves, upper-level barotropic energy conversion associated with the meridional wind confluence ( > 0) appears more important than that associated with the zonal wind divergence ( < 0) over the upper trough region (Fig. 4c–f), suggesting a large barotropic energy conversion from MKE to EKE at the upper levels. Compared with the barotropic energy conversion terms of TD–MRG waves above, the increased zonal eddy wind magnitude () and decreased meridional eddy wind magnitude () of ER waves would lead to a barotropic energy conversion from EKE to MKE at the upper levels (Figs. 3b and 5).

How then does the wave energy growth above influence the change of horizontal structure? As noted in numerous studies (e.g., Webster and Chang 1988; Sobel and Bretherton 1999; Maloney and Hartmann 2001), a zonal convergence of zonal wind of the monsoon trough provides a favorable environment for the wave growth via the positive feedback between wave growth () and the barotropic energy conversion term (), which can be described as wave accumulation associated with variations of the low-level zonal flow. Moreover, many studies (e.g., Kuo et al. 2001; Done et al. 2011) have identified that the wave reduces its scale via wave energy accumulation in confluent background flow. Sobel and Bretherton (1999) suggested that the wave convergence mechanism in general is consistent with the idea that the barotropic conversion of mean to eddy kinetic energy is important in the disturbance energetics. However, few studies have focused on how various background flows modify the tilted wave structure. As illustrated in Fig. 6, a meridional shear of zonal wind of the monsoon trough systematically alters the horizontal structure of tropical waves through the barotropic energy conversion process. For an initial cyclonic or anticyclonic eddy, uυ′ > 0 in the northwestern (NW) and southeastern (SE) quadrants of the eddy, and uυ′ < 0 in the southwestern (SW) and northeastern (NE) quadrants of the eddy. Thus, the positive (negative) barotropic energy conversion due to the eddy–MT mean flow interaction occurs at the NW and SE (SW and NE) quadrants of the eddy where the eddy increases (decreases) its energy and reduces (enlarges) its horizontal scale. This leads to a systematic NE–SW tilting in eddy. Such a horizontal tilt of eddy structure favors the further extraction of barotropic energy from the basic flow because the positive uυ′ can increase and extend to most parts of the eddy. This positive feedback between the wave growth and horizontal structure through barotropic energy conversion processes in the monsoon trough region may play a critical role in the development and maintenance of the NE–SW-tilted wave train along the monsoon trough over the WNP. This wave structure change (scale shrink and horizontal tilt) tends to lead to convergence in the cyclonic vortex region of the lower troposphere (Part I). Its sustainable development can lead to wave transitions that enhance the potential for development of small-scale waves (such as TD disturbances). In W-MT years, the monsoon trough retreats westward, and the low-level anticyclonic shear anomalies amplify ( decreases), which is unfavorable for an increase in the barotropic energy conversion over the WNP. Thus, the wave amplitude reduces and the wave structure changes in the eastern part. In other words, the monsoon trough-induced mean flow change modulates the energy conversion from the mean flow to the WTWs and determines the local growth rate of the wave disturbances and, thus, the reorganization of wave structure. The growth rates of the ER waves are much larger than the TD–MRG waves, consistent with the difference in the amplitudes of those waves.

Fig. 6.

Schematic diagram for illustrating the development and maintenances of horizontal tilted wave by barotropic energy conversion processes in the monsoon trough region. (from left to right) A zonal wind anomaly along the monsoon trough, the eddy and the mean anomalous eddy momentum flux of the initial tropical waves in the sheared zonal flow of monsoon trough, and the mean eddy momentum flux and NE–SW tilting of eddy generated and development, when the eddy interacts with the sheared zonal flow of the monsoon trough by barotropic energy conversion, which accelerates to extract barotropic energy from the monsoon trough.

Fig. 6.

Schematic diagram for illustrating the development and maintenances of horizontal tilted wave by barotropic energy conversion processes in the monsoon trough region. (from left to right) A zonal wind anomaly along the monsoon trough, the eddy and the mean anomalous eddy momentum flux of the initial tropical waves in the sheared zonal flow of monsoon trough, and the mean eddy momentum flux and NE–SW tilting of eddy generated and development, when the eddy interacts with the sheared zonal flow of the monsoon trough by barotropic energy conversion, which accelerates to extract barotropic energy from the monsoon trough.

c. Eddy available potential energy balance

EAPE is also important for WTWs. The analysis here focuses on the EAPE generation through diabatic heating processes (R′) of TD–MRG waves and baroclinic energy conversion of ER waves at 850 hPa that display year-to-year fluctuations consistent with wave activity. The EAPE generation through diabatic heating processes (R′) of TD–MRG waves at 850 hPa is shown in Fig. 7. Although the generation of EAPE by perturbation diabatic heating processes (Fig. 7) is rather weak compared to the other conversion terms in both monsoon trough states, it plays an important secondary role in the interannual variability of TD–MRG wave activity. Negative values are generally seen in the monsoon trough region. This R′ term describes the generation of EAPE by diabatic effects as external heating (cooling) is applied to warm (cold) air parcels. This term is negative in lower levels, suggesting the destruction of EAPE due to diabatic processes, and implying ascending cold air (T′ < 0) with the release of latent heat ( > 0) or descending warm air (T′ > 0) with evaporative cooling ( < 0). This diabatic effect appears to extract wave energy in lower levels, consistent with the results of Norquist et al. (1977), Lau and Lau (1992), and Hsieh and Cook (2007). This suggests that the diabatic heating tends to develop in lower-tropospheric ascending cold air over the monsoon trough region with the loss of EAPE (Fig. 9 in Part I). Meanwhile, these processes reduces EAPE accumulation that tends to disfavor the EAPE conversion to EKE by baroclinic instability processes and reduce baroclinic wave developing. The negative value of the generation of EAPE by perturbation diabatic heating, largely balancing the conversions of EAPE to EKE and MAPE to EAPE, may contribute to the maintenance and increase of barotropics (weakened conversions of EAPE to EKE can produce baroclinic decay) associated with TD–MRG waves though the development of the barotropic structure driven by enhanced barotropic conversion, which is proportionally more important for TD–MRG waves in the monsoon trough region.

Fig. 7.

As in Figs. 2d–f, but for the EAPE generation through diabatic heating associated with TD–MRG waves at 850 hPa.

Fig. 7.

As in Figs. 2d–f, but for the EAPE generation through diabatic heating associated with TD–MRG waves at 850 hPa.

For ER waves, compared to the barotropic conversion, the energy conversion from the MAPE to the EAPE is relatively large and the maximum conversion is located at the middle latitudes in the lower troposphere (Figs. 8a,b). The baroclinic energy conversion of ER waves is displaced significantly with increases in the southeast quadrant of WNP from W-MT to S-MT years (Fig. 8c). This pattern of increase in baroclinic energy conversion over the WNP is accompanied by an eastward extension of the monsoon trough during S-MT years. To understand this result, we have examined the contribution from each term in the baroclinic energy conversion from the MAPE to the EAPE (not shown). Conversions due to the term contribute about equally to the total as the baroclinic energy conversion from the MAPE to the EAPE in association with the strong mean meridional temperature gradient ( < 0) at lower levels. However, accompanying the change of monsoon trough, the mean meridional temperature gradient does not significantly change and appears limited in latitude. Furthermore, we examined meridional heat fluxes (υT′) of ER waves at 850 hPa (Figs. 9a–c). At 850 hPa, positive υT′ is found in tropical central to western Pacific, implying conversion from the MAPE to EAPE as the temperature perturbation tends to be in phase with the meridional wind perturbation in the lower troposphere. Associated with the difference in the location of the monsoon trough, there is significantly enhancement in poleward heat fluxes (υT′ > 0) over the southeast quadrant of the WNP from W-MT to S-MT years (Figs. 9a–c). This suggests that thermal advection, in association with correlations between poleward heat fluxes and strong meridional temperature gradients, converts MAPE to EAPE. Unlike ER waves, the meridional heat flux is insignificant for TD–MRG waves at 850 hPa (Figs. 9d,e), suggesting that the baroclinic energy conversion from the MAPE to the EAPE cannot significantly contribute to TD–MRG wave activity. The strong loss of EAPE through conversion to EKE is mostly maintained by the baroclinic energy conversion from MAPE to EAPE (Lau and Lau 1992). Therefore, the stretching baroclinic energy conversion and its accompanying moist convection is one of the important energy sources for the baroclinic disturbances, such as baroclinic mode of ER waves. This development of baroclinic mode leads to an eastward tilt of the perturbation heating () and upward motion. This tilted vertical structure of ER waves is similar to that observed, especially in the lower troposphere (Figs. 14 and 15 in Part I). Therefore, this baroclinic conversion process may play an important role in the increase of the baroclinicity and reorganizing of the vertical structure of ER waves resulting from the interaction of ER waves with the monsoon trough.

Fig. 8.

As in Fig. 3, but for the baroclinic energy conversion associated with ER waves at 850 hPa.

Fig. 8.

As in Fig. 3, but for the baroclinic energy conversion associated with ER waves at 850 hPa.

Fig. 9.

(a)–(c) Values of the covariance between the 850-hPa eddy υ wind and temperature (υ′T′) for ER waves during July–November for S-MT years, W-MT years, and their difference (the S-MT years minus the W-MT years). Units: m s−1 K. (d)–(f) As in (a)–(c), but for TD–MRG waves.

Fig. 9.

(a)–(c) Values of the covariance between the 850-hPa eddy υ wind and temperature (υ′T′) for ER waves during July–November for S-MT years, W-MT years, and their difference (the S-MT years minus the W-MT years). Units: m s−1 K. (d)–(f) As in (a)–(c), but for TD–MRG waves.

Base on the above analysis, the WTWs extract the energy from the basic flow of monsoon trough such that the perturbations grow and become favorable for tropical cyclogenesis. The current study suggests that the barotropic conversion processes play a more important role in the influence of the monsoon trough on the evolution of WTWs. As these waves propagate westward, they extract the barotropic energy from the monsoon trough flow through the positive feedback, and gain a more NE–SW tilt and barotropic structure with increased amplitudes. Generation of EAPE through baroclinic conversions can produce the tilted vertical structure of ER waves, especially in the lower troposphere. Those energy conversions may offer a physical explanation of why the growth of WTWs is accelerated in the southeast quadrant of the WNP during the years when the monsoon trough extends eastward. In the following section, an idealized model is used to further show the evolution of WTWs in the monsoon trough environment.

4. Evolution of idealized WTWs in the monsoon trough environments

In this section, we focus on the evolution of WTW characteristics in different background flows using a linear shallow-water model to further understand tropical wave interactions with the monsoon trough via the dynamics of energy conversion and wave accumulation in the WNP. We derive an analytic solution for an idealized MRG or ER wave on the circulation of the monsoon trough during its two extreme states.

a. Model description

Numerical experiments are conducted using the shallow-water model for a single homogeneous layer of fluid on an equatorial β plane. We separate the flow field into a time-invariant background state and a perturbation state. Following the work of Aiyyer and Molinari (2003), the shallow-water equations are specified to maintain a balance between the higher-order terms after neglecting the nonlinear terms and the perturbation forcing term by physical mechanisms. The rationale for choosing this value is based on the fact that the influence of the perturbation forcing term and nonlinear term on the growth of WTWs is rather small, which has been detected from the contribution of energy transfer in section 3. The governing equations for the perturbation state on an equatorial β plane can be written as (Aiyyer and Molinari 2003)

 
formula
 
formula
 
formula

where the overbar represents a time-invariant background state field and the prime indicates a perturbation state field; u, υ, and h represent zonal and meridional velocity, and the free surface height, respectively; and β is the meridional gradient of the Coriolis parameter; g is the acceleration due to gravity; and H is the mean fluid depth that is set at 25 m in the model after neglecting orography.

Our approach is similar to the method used in several past studies (e.g., Aiyyer and Molinari 2003; Shapiro 1980; Kuo et al. 2001). All equations described here are cast in finite-differential form and integrated on an equatorial β plane with periodic boundary conditions along the zonal direction and rigid walls at the north and south boundaries. The computational domain is a channel centered on the equator with a zonal extent of 40 000 km, which is roughly the circumference of the earth along the equator, and a meridional extent of 8000 km. A grid spacing of 50 km and a time step of 600 s are used for all numerical experiments in this section. The discretization of the model is based on a leapfrog method with a time filter that damps high-frequency waves. Although those idealized experiments are based on a linear shallow-water model with quasi-stationary large-scale state, the key energy transfer process is still being retained to confirm the mechanisms of the change of a wave structure and generate smaller-scale disturbances within the slowly varying large-scale state (e.g., monsoon trough).

b. Generation of the background states

The evolution of WTW structure exhibits a good relationship with the circulation of the monsoon trough (in Part I). There is a significant change in the three-dimensional structure as those waves propagate westward to the east of the monsoon trough. Differences in the location of the monsoon trough may lead to an east–west contrast in the WTWs. To study the monsoon trough effect, a sensitivity test has been done by switching the background flow patterns between the S-MT and W-MT years. The observed 850-hPa horizontal flows and geopotential height in the WNP for S-MT and W-MT years are used as initial background flow conditions for the shallow-water model experiments. All data fields have been interpolated to give grid points (50 km). The experimental domain is shown in Fig. 10 with latitudinal boundaries at 10°S and 30°N and lateral boundaries at 100°E and 180°. The wind and height values outside of this domain are identically set to zero. To avoid a boundary problem, the wind and height values decrease toward the exterior in a boundary zone covering about 4.5° (10 grid points) from the boundaries. Two different background flows (S-MT and W-MT) representative of monsoon trough flow configurations are constructed, and the parameters u, υ, and h are shown in Figs. 10a and 10b, respectively. The winds were given by composite S-MT and W-MT fields of the 850-hPa wind vectors, and the height fields were constructed by composite S-MT and W-MT fields of the 850-hPa geopotential height deviation from the region-averaged geopotential height during the 1979–2007 periods. Note that the monsoon trough flow used here is a quiescent environment that does not evolve with the time. In other words, the waves can extract energy from the monsoon trough background flow without limits.

Fig. 10.

The monsoon background flow at 850 hPa for (a) S-MT and (b) W-MT years used as initial conditions for the shallow-water experiments. The shadings denote the geopotential height deviation (gpm) from the region (10°S–30°N, 100°E–180°) average.

Fig. 10.

The monsoon background flow at 850 hPa for (a) S-MT and (b) W-MT years used as initial conditions for the shallow-water experiments. The shadings denote the geopotential height deviation (gpm) from the region (10°S–30°N, 100°E–180°) average.

c. Generation of MRG and ER waves

The structures of the initial MRG and ER waves are created following the method of Zehnder (1991) and Aiyyer and Molinari (2003). The horizontal winds and perturbation height associated with the MRG (ER) waves are shown in Fig. 11a (11b). The wavenumber of MRG waves is chosen as 8 and the wave period is about 6.3 days. The maximum wind speed of 0.7 m s−1 associated with this wave occurs at the equator in the meridional component of the wind. The ER waves are often labeled as a meridional mode number 1 (n = 1) ER waves. For comparison with the MRG waves, the wavenumber of ER waves is chosen as 8 and the wave period is about 20.2 days. The maximum wind speed of 0.7 m s−1 associated with this wave occurs at the equator in the zonal component of the wind. Those initial MRG (ER) waves are substituted in Eqs. (10)(12). To do this initialization procedure, we insert MRG (ER) waves into a numerical model that is able to simulate faithfully the evolution of waves.

Fig. 11.

Fields of wind vectors (m s−1) and height (contours, m) associated with (a) the wavenumber-8 MRG wave and (b) the ER wave in an equatorial β plane.

Fig. 11.

Fields of wind vectors (m s−1) and height (contours, m) associated with (a) the wavenumber-8 MRG wave and (b) the ER wave in an equatorial β plane.

d. Simulation of MRG wave and ER wave response to MT environments

The first simulation uses a quiescent basic state ( = = = 0) to examine the evolution of the MRG waves and ER waves without monsoon background flow, respectively. It is noted that the waves propagate westward after 7 days of simulation without any change in the wave structure (not shown). This implies that the change of MRG and ER waves does not occur as a result of westward propagation without the effect of mean flow.

To test the effects of the monsoon trough on the waves, we examine the evolution of the MRG waves in different constant background flows. Figures 12a and 12b show wind and height fields of the MRG waves at the 7th day, and the barotropic conversion term averaged over 7 days of the simulation. Maximum barotropic conversion in the model is observed in the monsoon trough region, similar in structure and intensity to those in observation (see section 3b). Following the eastward extension of the monsoon trough, maximum barotropic conversion values extend more eastward during the S-MT years than during the W-MT years. While there are some differences in the monsoon trough mean flow, a large area of positive and statistically significant increase in the barotropic conversion term covers the tropical central Pacific to the WNP during the S-MT years though the initial MRG waves have the same intensity. Those monsoon trough–induced mean flow changes such as enhanced lower-level confluence () and shear () flow modulate the energy conversion between the mean flow and the small-scale eddies and determine the local growth rate of the wave disturbances. The MRG waves propagate along the equator toward the WNP. When the waves reach the eastern part of the monsoon trough during their westward movement, their structures experience a change due to their interactions with the monsoon trough where the barotropic conversion is larger. The change of wave structures in such a region allows the waves to extract efficiently the barotropic energy from the mean flow to sustain their growth in the region of the monsoon trough because the positive u′υ′ increases and extends to most parts of the eddy. After 7 days, following those changes in wave structures with depression (height) in cyclonic vortex center, the depression fills the cyclonic vortex center and off the equator along the axis of monsoon trough, and the waves appear to have a mixed feature of two types of waves: the MRG gyre centered on the equator over the central Pacific (east of the monsoon trough) and off-equatorial smaller-scale eddies (TD waves). The TD waves then move northwestward along the monsoon trough. The longitudinal zone for transition is shifted to the east, consistent with the eastward extension of the monsoon trough during S-MT years. In contrast, such a transition is vague during W-MT years. Accompanying westward retreat of the weakened monsoon trough, the monsoon trough significantly modulated the atmospheric structure surrounding the WNP region, leading to significant weakening of lower-tropospheric cyclonic shear around the monsoon trough to the west of ~140°E, and, hence, weakened barotropic energy conversion from the monsoon trough to the small-scale eddy. In addition, the easterly flow developed ahead of the monsoon trough to the east of ~140°E, which leads to weaker wave amplitude through the interaction of waves with the easterly flow during W-MT years when the monsoon trough retreats westward. Therefore, the TD waves become much weaker and ultimately dissipate to the west of ~140°E. The evolution of the MRG waves during the different monsoon trough states is consistent with the observation in Part I.

Fig. 12.

The wind vectors (m s−1) and height (contours shown are ±0.2, 0.4, 0.8, 1.6, and 3.2 m) for the linear shallow-water model with the evolution of the MRG wave in the monsoon trough basic state at day 7 for (a) S-MT and (b) W-MT years. The shaded area denotes the barotropic conversion terms (10−5 m2 s−3) averaged over the 7 days.

Fig. 12.

The wind vectors (m s−1) and height (contours shown are ±0.2, 0.4, 0.8, 1.6, and 3.2 m) for the linear shallow-water model with the evolution of the MRG wave in the monsoon trough basic state at day 7 for (a) S-MT and (b) W-MT years. The shaded area denotes the barotropic conversion terms (10−5 m2 s−3) averaged over the 7 days.

Figure 13 is the same as Fig. 12 but for the ER waves. The main result demonstrates significant interactions between wave and mean flow, but there are some differences in the MRG waves though the background flow is the same. Consistent with the barotropic conversion found previously for the ER waves, a stronger monsoon trough produces a stronger wave amplitude through a more vigorous wave–mean flow interaction. As such, the ER waves extract more kinetic energy from the monsoon trough than the MRG waves. The influence of the barotropic conversion on wave structures seems different in different waves. After 7 days, the wave circulations display a clear tilt of their axes in a northwest–southeast direction with an increase in the amplitude for westward ER waves. It can also be seen that there is scale contraction during their propagation along the monsoon trough, but there is no signal for the development of smaller-scale off-equatorial disturbances such as the TD waves. Similar changes occur in W-MT years except that the wave amplitude is weaker. Unlike the MRG wave, the ER wave does not show a significant shift in the structure, as there is no apparent transition from ER waves to small-scale disturbances, but it has a significant increase in the wave amplitude. The evolutions of the ER waves during the different states of the monsoon trough are also consistent with the observations in Part I.

Fig. 13.

As in Fig. 12, but for the ER wave.

Fig. 13.

As in Fig. 12, but for the ER wave.

To further clarify the dependence of the wave growth on initial conditions, we carry out several experiments to examine briefly the sensitivity of those results to factors such as the monsoon trough intensity and wave eddy intensity. Figure 14a shows the barotropic conversion and EKE of MRG wave averaged from days 1 to 7 as a function of monsoon trough intensity. The barotropic conversion and EKE are averaged over the region 5°–15°N, 150°E–180°, which is the region of maximum interannual variation in monsoon trough activity. The barotropic conversion and EKE accelerate the growth when the monsoon trough intensity increases. The resulting eddy structures are qualitatively similar although the strength of the eddies differs for different monsoon trough and initial eddy intensity (not shown). The evolution of wave characteristics in the weak monsoon trough environment also indicates a qualitative similarity in W-MT years (Fig. 12b), but the location of wave–flow interaction shifts westward when the monsoon trough retreats westward. In addition to the remarkable difference in growth rate, the initial eddy intensity also significantly impacts the wave growth (Fig. 14a) and wave structure evolution, but is weaker compared with the monsoon trough effect.

Fig. 14.

The barotropic conversion terms averaged from days 1 to 7 (blue line, unit is 10−5 m2 s−3) and EKE at day 7 (red line, unit is m2 s−2) as a function of monsoon trough (solid line) and eddy (dashed line) for the linear shallow-water model with the evolution of the (a) MRG wave and (b) ER wave in the monsoon trough basic state. The barotropic conversion and EKE are averaged over the region 5°–15°N, 150°E–180°. The horizontal axis shown is a multiple of amplitude initial condition of the strong monsoon trough condition in Fig. 10a and the eddy in Fig. 11.

Fig. 14.

The barotropic conversion terms averaged from days 1 to 7 (blue line, unit is 10−5 m2 s−3) and EKE at day 7 (red line, unit is m2 s−2) as a function of monsoon trough (solid line) and eddy (dashed line) for the linear shallow-water model with the evolution of the (a) MRG wave and (b) ER wave in the monsoon trough basic state. The barotropic conversion and EKE are averaged over the region 5°–15°N, 150°E–180°. The horizontal axis shown is a multiple of amplitude initial condition of the strong monsoon trough condition in Fig. 10a and the eddy in Fig. 11.

For the ER waves, the barotropic conversion and EKE are a function of monsoon trough or eddy intensity (Fig. 14b). It supports the idea that the enhanced monsoon trough and the initial eddy favors wave growth. For the TD–MRG waves, both barotropic conversion and EKE have a slower growth rate as a function of monsoon trough or eddy intensity except for a faster enhancement of barotropic conversion with increased eddy intensity. Note that the amplitudes of the barotropic conversion and EKE for ER waves are much larger than for MRG waves in a weaker monsoon trough with initial eddy intensity, but are smaller in a stronger monsoon trough with initial eddy intensity.

The results of idealized simulations in this section are consistent with the observations in Part I. Both observations and simulations show that the mean flow of the monsoon trough has a large effect on the structure of the waves. The monsoon trough background flow provides a favorable environment for the growth of WTWs within the monsoon trough region through the energy conversion, similar to those seen in the energy budget analysis in section 3. While the waves are near the eastern part of the monsoon trough, they experience some changes in the horizontal structure and propagation characteristic through interaction with the monsoon trough. The stronger monsoon trough background flow affects the WTWs in a more eastward location of the WNP during S-MT years compared to the W-MT years and the effects of monsoon trough last long enough, leading to a stronger amplification of waves. The monsoon trough background flow has a larger impact on the structure and propagation of the MRG waves. In contrast, the ER waves experience a faster growth but with no marked changes in their structures and propagation characteristics. The results of the observational studies of Part I and the theoretical studies of section 3 strongly suggest that TD–MRG and ER waves interact with the monsoon trough forced circulations in the region, implying that the low-level flow may influence the scales of these wave disturbances along the monsoon trough.

5. Summary and discussion

Part I of this study suggested that the interannual variability of WTW activity is closely related to the monsoon trough location. There is a significant change in the three-dimensional structure as those waves propagate westward to the eastern part of the monsoon trough. For the TD–MRG waves, an apparent transition from MRG waves to off-equatorial TD disturbances is identified in the region of the monsoon trough. For the ER waves, faster growth is observed, but no apparent change is seen in their structures and propagation characteristics. Differences in the location of the monsoon trough may lead to the east–west contrast of the characteristics of the WTW.

In this study, we examined the mechanisms of WTW developments during the course of mean flow–wave interaction from an energetic point of view. We aimed to clarify how the monsoon trough state affects the structure and evolution of WTWs. The impacts of energy conversion from the monsoon trough to TD–MRG and ER waves are diagnosed separately. The results reveal that the lower-tropospheric barotropic conversion associated with the monsoon trough is the most important mechanism for the growth of eddy energy in the WTWs. In terms of barotropic dynamics, large divergent () and rotational () flows of the monsoon trough help maintain the rapid growth and tilted horizontal structure of the lower-tropospheric TD–MRG and ER perturbation through a positive feedback between the wave growth and horizontal structure based on the barotropic energy conversion processes. Although barotropic conversion is proportionally more important for the development of TD–MRG waves, the loss of EAPE by perturbation diabatic heating processes provides some help in the maintenance and growth of barotropics. The baroclinic conversion processes favors the ER waves to extract EAPE from the basic flow and induces the tilted vertical structure of ER waves, especially in the lower troposphere. Those processes contribute to the development of WTWs. Note that those waves display a larger increase in their amplitude in the lower levels during S-MT years than during W-MT years. As such, more energy is transferred from the monsoon trough to waves, and the horizontal structure of waves shrinks more and is tilted in the NE–SW direction. The westward phase speed could be reduced as the horizontal scale decreases and the diabatic heating increases (Xie and Wang 1996). The growth rates of the waves are also much larger during S-MT years than during W-MT years, consistent with differences in the magnitude of the energy conversions. As the waves continue to develop, an apparent transition from MRG waves to TD disturbances and a faster growth for ER waves are identified in the region of the monsoon trough.

In support of this view, idealized modeling experiments are conducted to examine the interaction between idealized WTWs (MRG or ER waves) and the monsoon trough. The results confirm that the wave transition associated with westward MRG waves in a monsoon background flow can lead to the activity of TD waves along the axis of the monsoon trough, in good agreement with observations. While mean flow intensification on either side of the monsoon trough increases the low-level divergent and rotational flows, the amplitude and number of the small-scale disturbances (such as TD disturbances) increase significantly with the strength of the barotropic conversion. For ER waves, the wave gyres are also modified through their dynamical response to the monsoon trough, and they shrink, become more horizontally tilted, more energetic, and more active in the S-MT state. It is interesting to note that the structure and propagation characters of the ER waves have no marked change in either the observation or in the model. This is likely because the ER waves are dynamically different from the MRG waves. The results are similar to those in observational and analytic studies. In addition, further experiments are performed to examine the sensitivity of the results to monsoon trough intensity and eddy intensity. The barotropic energy conversion and EKE accelerate growth when the monsoon trough (eddy) intensity increases, indicating that strong monsoon trough (eddy) can induce accelerated barotropic conversion growth, creating conditions that favor energetic eddies.

This sequence of two papers indicates that the monsoon trough favors the growth of synoptic-scale disturbances over the WNP through the barotropic energy conversion. The monsoon trough–dependent wave activity may offer a physical explanation of why more frequent tropical cyclones (TCs) form in the region of the monsoon trough (Wu et al. 2012, 2014b). However, the origin and initiation mechanism of the TCs in the wave train remain uncertain. It will be left to a future study to investigate the effect of the TD–MRG and ER waves in regulating the development of TCs over the tropical Pacific Ocean through their dynamical response to the monsoon trough.

Acknowledgments

The authors thank Prof. Ronghui Huang and Prof. Tim Li for their helpful discussions. This work is jointly supported by the National Basic Research Program of China under Grant 2014CB953902, and the National Natural Science Foundation of China Grants 41205052, 41475077, 41461164005, 41230527, and 41375065.

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