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In this section, along every longitude with a 1° interval over 117°−124°E, at each time in the selected period mentioned above, the atmospheric temperature, geopotential height, and winds were taken to set up a y−p plane for diagnosis. The latitudinal range of 30°−55°N was selected so that both the airflow in the monsoon front area and the balanced airflow of midlatitude westerlies could be considered simultaneously. Vertically, a total of 24 layers were taken between 975 hPa and 20 hPa. Since the values of vertical velocity of the layers at and above 70 hPa are 0, these layers were ignored when calculating the vertical velocity. The vag and ω components in Fig. 3 were used to check the non-divergence assumption in the y−p plane. From the mass continuity equation
${\partial (U+{u}_{\rm{ag}})}/{\partial x}+{\partial (V+{v}_{\rm{ag}})}/{\partial y}+{\partial \omega }/{\partial p}=0$ , geostrophic balance assumption${\partial U}/{\partial x}+{\partial V}/{\partial y}=0$ and${\partial {u}_{\rm{ag}}}/{\partial x}=0$ , and considering the β effect, it was derived thatSubstituting the vag, ω and V into the lhs of Eq. (7), and then considering time-averaging, the values of the lhs of Eq. (7) in the free atmosphere of the midlatitudes are mostly below 10−6 s−1 (Fig. 2a)—significantly less than the values near the monsoon front and in the planetary boundary layer (PBL). Therefore, the assumption [Eq. (7)] in this study is reasonable. This assumption is also found in the algorithm provided by the post-processing software of the WRF model (Stoelinga, 2013). Next, we take the Ekman pumping effect as the lower boundary condition and substitute it into the following equation:
where ZPBL is the thickness of the PBL, assumed to be 1000 m,
$ \rho $ is the air density, while Fper is the relative size of the component of true wind normal to geostrophic wind, which can be regarded as a kind of Rossby number, assumed to be 0.2; and the stream function in the bottom layer is assumed to be 0. At each diagnostic time, the lateral boundary condition is given as the form$ {\psi }_{k}-{\psi }_{k-1}=-{v}_{{\rm{ag}},k}·({p}_{k}-{p}_{k-1}) $ , where$ k $ = 2……n denotes the vertical layer from the layer just above the ground to the top layer,$ {p}_{k} $ is the pressure at kth layer and$ {v}_{{\rm{ag}},k} $ means the meridional component of the ageostrophic wind in the kth layer of the lateral boundary. Friction forcing$ \left({Q}_{\rm{f}}\right) $ includes terrain friction and the effect of Ekman pumping. Since there are no high mountains or highland terrain in the DoI in this paper, terrain friction can be set to 0. As a result, the effect of$ {Q}_{\rm{f}} $ , acting as the form of Eq. (8), is taken as a kind of boundary condition for Eq. (3).In this study, the diagnoses were conducted in each meridional panel along the latitudes with a 1° interval over a limited area, which is indicated by rectangle “A” in Fig. 1a, during the period 28 June to 12 July in 2001−13. Hence, a total of 6240 sample planes were obtained and, based on Fig. 2, the q in every grid point in each sample should be further checked. Once the q in each grid point of a sample plane is positive, Eq. (3) is resolved numerically to obtain the corresponding ψ, ω and vag in the plane. In this study, the numerical solution method was “successive over-relaxation”, with an acceleration factor of 1.63 and an iteration error threshold of 1.0. It is worth noting that not every diagnostic equation can be resolved correctly—about 36% of the iteration processes in resolving Eq. (3) are divergent and fail to obtain the resolutions. The numbers of convergence times when taking each term as an only forcing term on the rhs of Eq. (3) in this study are listed in Table 1. These numbers range from 458 to 529, which is about 64% of the total 6240 samples. Table 1 also indicates that, regardless of any kind of forcing, the proportion of convergence samples increases from west to east, which implies that the air flows in the east part of the DoI are more stable. After these manipulations, the composite fields of resolved ψ, ω and vag can be used to describe the overall characteristics of the resolved SCs and demonstrate the relative contribution to the formation and maintenance of SCs by the forcing terms listed in Eq. (3). The results are presented in following sections.
Longitude Forcing $ {Q}_{\rm{con}} $ $ {Q}_{\rm{she}} $ $ {Q}_{\rm{g}} $ $ {Q}_{\rm{ag}} $ $ {Q}_{\rm{d}} $ $ {Q}_{\rm{tot}} $ Average 117°E 458 477 482 480 477 481 61% 118°E 473 483 491 483 494 496 62% 119°E 474 488 495 487 497 491 63% 120°E 482 492 503 499 510 489 64% 121°E 504 506 496 520 503 497 65% 122°E 503 516 528 529 518 521 67% 123°E 518 527 506 512 524 501 66% 124°E 510 526 508 517 503 507 66% Average 63% 64% 64% 65% 65% 64% 64% Note: $ {Q}_{\rm{con}} $ denotes geostrophic confluence forcing, $ {Q}_{\rm{she}} $ denotes geostrophic shear forcing, and $ {Q}_{\rm{tot}}={Q}_{\rm{g}}+{Q}_{\rm{ag}}+{Q}_{\rm{d}} $. Table 1. Numbers and average percentages of convergence cases in the iterations of numerically resolving Eq. (3) along each longitude over (117°−124°E, 30°−55°N) for the period 28 June to 12 July during 2001 to 2013.
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The horizontal confluence and shear of the large-scale geostrophic wind field were the earliest-found forcing terms of the S−E equation (Sawyer, 1956; Eliassen, 1962). As shown in Fig. 4a, the composite horizontal confluence forcing term,
$2({\partial U}/{\partial p})/({\partial V}/{\partial y})$ , mainly acts on the stratosphere and higher troposphere, and has a relatively small positive center in the boundary layer around 40°N. In the stratosphere, to the north of 32°N, is positive forcing, and to the south of it is negative forcing. Figure 4b presents the averaged vectors composed of ω and vag, directly plotted from reanalysis data for the convergence cases, and these vectors can be regarded as “original” circulation components. The numerically solved stream function represents a clockwise sub-SC whose center is located at 500 hPa and 38°N, and a half of a counter-clockwise sub-SC centered at 300 hPa and south of 30°N (Fig. 4c). Comparing with other figures, it can be concluded that the latter sub-SC is related to the lateral boundary conditions. The maximum sinking speed of the sub-SC is 0.032 Pa s−1, equal to 94% of the maximum sinking speed of the original circulation (Fig. 4b). The maximum rising speed forced out is −0.093 Pa s−1, approximately 76% of that of the original circulation. The shaded area in Fig. 4c denotes that the sinking speed pattern shows an obvious difference from that of the original circulation, especially above 500 hPa. The main sinking zone is located at 40°−45°N and distributed from 975 hPa to 100 hPa in the vertical direction. At the same time, the averaged shear forcing term,$-2 ({\partial U}/{\partial y})/({\partial V}/{\partial p})$ , also acts on the higher layer of the troposphere and the lower layer of the stratosphere. The negative forcing center is on the south side of the 200-hPa high-level jet axis, and positive forcing exists on the north side of the axis and in the stratosphere above (Fig. 4d). The averaged numerical solution represents a part of a large anticlockwise SC whose center is located at 300 hPa and 28°N (Fig. 4f). The downdrafts are relatively shallow and mainly exist between 30°N and 45°N below 600 hPa. The maximum sinking and rising speeds forced out by the term are 0.031 Pa s−1 and −0.066 Pa s−1 respectively, accounting for 94% and 54% of the original value respectively (Fig. 4e). In the y−p plane, superimposition of the above two forcing terms (Fig. 4g) produces a vertical circulation structure that sinks in the center and rises on the south and north sides (Fig. 4i). Between 30°N and 35°N the position and scope of the upward motion zone are basically the same as those of the upward motion of the original wind field, but the intensity is −0.070 Pa s−1, about 61% of the original value (Fig. 4h). The downward-motion zone north of 35°N is very close to that of the original SC, with a maximum sinking speed of 0.033 Pa s−1, accounting for 94% of the original SC’s value (Fig. 4h).Figure 4. (a, d, g) Large-scale dynamical forcing terms (lines at 4 × 10−7 m s−2 hPa−1 intervals) including (a) averaged geostrophic confluence forcing, (d) averaged geostrophic shear forcing, as well as (g) the sum of (a) and (d). (b, e, h) The averaged ageostrophic flow vectors of (vag, ω) (black arrows) plotted directly from the convergence data samples, and (b), (e) and (h) correspond to (a), (d) and (g), respectively. The maximum rising (AM) and sinking speeds (DM) are also denoted in the top of each sub-figure. (c, f, i) The averaged stream function
$ \psi $ (red lines at intervals of 50 m hPa s−1; dotted lines denote negative values) and the ageostrophic flow vectors of (vag, ω) (black arrows) of the numerical solutions of the S−E equation over each meridional panel at every time level with the forcing term settings of (a), (d) and (g), respectively. The shaded areas denote the sinking velocity.Though the forcing effect of the large-scale geostrophic deformation determines the basic structure and position of SC, its observed rising velocity cannot be fully represented yet. When the S−E equation is used in observational diagnosis, such a phenomenon usually occurs: As early as in the 1950s, Sawyer (1956) found in his study of frontogenesis that the intensity of the upward branch of the SC diagnosed was smaller than the observed value; and the maximum sinking speed diagnosed in the jet−front system by Shapiro (1981) using the S−E equation was only about 50% of the observation. Bui et al. (2009) and Lagouvardos et al. (1992) drew similar conclusions. This may imply that, during the formation of the SC, there are other forcing terms at work. These forcing terms may include not only ageostrophic dynamical forcing, latent heat forcing and boundary layer friction forcing, which explicitly appear on the RHS of Eq. (3), but also other thermodynamic effects hidden in the environment.
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It is the inherent requirement of the primitive-equation form of the S−E equation, i.e., Eq. (3), to introduce the ageostrophic dynamical forcing term Qag, which enables the forcing terms on the rhs of the S−E equation to contain ageostrophic forcing. Thus, some effects of the response circulation are present on the lhs of the equation (Keyser et al., 1985a).
Based on Eq. (3), we can obtain the streamfunction response and the vertical velocity component corresponding to ageostrophic forcing. As seen in Fig. 5, at the higher level of the troposphere on the north side of 45°N, the ageostrophic dynamical forcing term forces out a very weak direct circulation whose intensity is obviously less than the original flow in Fig. 5b; near 30°N there is a dominant upward motion that is also determined by the lateral boundary condition.
Figure 5. As in Fig. 4 except for the ageostrophic forcing at intervals of 4 × 10−7 m s−2 hPa−1 in (a) and response stream function at intervals of 50 m hPa s−1 in (c).
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Though early diagnostic studies on frontal zone SC were carried out based on an adiabatic assumption (Namias and Clapp, 1949; Sawyer, 1956; Eliassen, 1962; Hoskins and Bretherton, 1972; Shapiro, 1981; Keyser et al., 1985a, b), water phase change associated with the SC near the frontal zone will alter the thermal structure of the atmosphere and lead to adjustment in mass, which in turn causes changes in horizontal and vertical air motion. Prior studies (Thorpe and Nash, 1984; Huang and Emanuel, 1991) indicated that condensation may intensify upward motion in the frontal zone, and evaporation enhances downward motion. However, since only a small part of condensed water vapor can evaporate in the downdraft, it is possible that including the evaporation effects on the latent heating processes might lead to an overestimation of the magnitude of downdrafts. Xu (1989a) linked the latent heating only to the upward motion in his formulations (2.4)−(2.5), and following these expressions, in this study we assume
where the expression of
$ {N}^{*2} $ is different from Eq. (4) and required the addition of an another constraint for saturated cases as follows:Substituting the vertical velocity
$ {\omega }_{1} $ , which is diagnosed through Eq. (3) without Qd, into Eq. (10) and Eq. (9), and then into Eq. (3) again, by using the diagnostic method used in previous sections and with Qd as the only forcing term, we can obtain an estimate of sub-forced SC and the forced vertical velocity field,$ \omega $ , excited by latent heating. An obvious uncertainty exists in this procedure in that Eq. (9) is applicable only for saturated wet mass, and therefore 95% relative humidity is defined as the criterion for saturation in the local zonal-mean sense. Figure 6 shows that the weak downdrafts are located at around 32°N, 35°N, 45°N and 50°N, and mostly within the PBL. In Fig. 6, both the maximum sinking and rising speeds in the SC forced out by the heating term are about 0.068 Pa s−1 and −0.031 Pa s−1, as in Fig. 5c, are determined by the lateral boundary conditions. In fact, diabatic heating includes not only condensation heating, but also cumulus heating, radiative heating, and so on. Therefore, more experiments using different parameterization schemes should be conducted in future SC diagnosis studies.Figure 6. As in Fig. 5 except for the diabatic heating forcing at intervals of 0.4 × 10−7 m s−2 hPa−1 in (a) and response stream function at intervals of 50 m hPa s−1 in (c).
According to the above analyses, between the mei-yu front zone and midlatitude high-level jet, on average, there exists an SC that rises in the warm area and sinks in the cold area, and its position is dictated by the large-scale geostrophic deformation. In addition, the SC intensity and structure are adjusted by both the diabatic heating effects of the mei-yu front zone precipitation and the Ekman pumping effect of the boundary layer.
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SC is a response circulation forced out by the basic dynamical and thermal fields near the frontal zone, but it also reacts on these basic fields. According to Keyser and Pecnick (1985a), in the derivation procedure of the S−E equation, the momentum equation and heat flux conservation equation are transformed together into the frontogenesis diagnostic equation. Using the streamfunction components as in Figs. 4c, f and i to calculate the frontogenetical function, we can obtain the results shown in Fig. 7. Along the high-level jet axis that tilts northwards with altitude appears an obvious frontogenesis area, which increases in intensity and tilts northwards with altitude. Most of the frontogenesis area is within the substantial updraft area, thus being conducive to the occurrence of monsoon convection and precipitation. South of 32°N and north of 38°N are frontolysis areas; the area between 36°N and 38°N is a downdraft area. Judging from the structure of the dynamic frontogenesis region, the main area that is conducive to convective precipitation is restricted to a narrow zone between 32°N and 36°N.
Figure 7. Geostrophic dynamical forcing frontogenesis function (lines at intervals of 5 × 10−6 m−1 hPa−1 s−2) and areas with Richardson number < 0.25 (shading) over 117°−124°E averaged during the period from 28 June to 12 July. The cyan dotted line denotes the axis of the jet and the green dotted line indicates the northern edge of upward motion of the mei-yu area.
To further verify the effect of the forced SC on the evolution of jet−front systems and the occurrence of monsoon convection, the frontogenesis characteristics of 34 typical deep convection events that occurred over the DoI from 28 June to 12 July during 2004 to 2013 were analyzed. Based on the 0.05° × 0.05° gridded multi-satellite datasets from Kochi University, Japan, the threshold of deep convection was set to be the maximum gridded brightness temperature of less than −52 °C . The results indicated that every single deep convection case experienced a horizontal potential temperature contraction process (i.e., frontogenesis) with a tilting of the high-level front region, and most of them (24 of 34 cases) had substantial cold advection in the cyclonic shear region for time spans of about 12−30 h. All of these frontal characteristic parameters are listed in Table 2.
No. Date and time
(yyyymmddhh)Deepconvection
frequencyMax. $\dfrac{ {{\partial} } {{\theta } } }{ {{\partial} } {{x} } }$
(10−12 K m−1)Ave. $\dfrac{ {{\partial} } {{\theta } } }{ {{\partial} } {{x} } }$
(10−12 K m−1)Max. $\dfrac{ {{\partial} } {{\theta } } }{ {{\partial} } {{y} } }$
(10−12 K m−1)1 2004070606 0.28 7.7 0.9 −28.8 2 2004070712 0.34 8.3 0.6 −18.7 3 2004071012 0.29 11.2 −0.3 −30.3 4 2005062906 0.29 24.1 2 −33.4 5 2005063012 0.33 16.1 0.6 −33.8 6 2005070212 0.31 7.7 0.4 −41.2 7 2005070618 0.57 3.9 −1.5 −35.7 8 2005070800 0.36 5.5 0 −37.5 9 2006063018 0.42 5.3 0 −32.7 10 2006070312 0.52 7 0.2 −27 11 2006070506 0.33 13.5 −0.1 −28.2 12 2006071106 0.28 10.1 0.7 −47.7 13 2007070512 0.27 3.1 −0.3 −48.6 14 2007070706 0.59 6 −0.2 −40.9 15 2007070818 0.4 4.5 −0.3 −48.7 16 2008070406 0.39 15.3 0.2 −18.5 17 2008071106 0.39 13.2 0 −24.9 18 2009062812 0.31 16.7 1.1 −43.7 19 2009070700 0.27 9.8 0.4 −30.1 20 2009071206 0.3 8.8 0.3 −38.3 21 2010070212 0.45 2.3 0.1 −25.6 22 2011070518 0.32 14.6 0.4 −41.3 23 2011071218 0.31 4.7 −0.1 −34.4 24 2012062912 0.33 18.4 0.9 −50.4 25 2012070212 0.47 10.3 −0.5 −39 26 2012070412 0.39 8.6 −0.1 −41.4 27 2012070706 0.36 7.1 0.1 −23 28 2012070918 0.59 9.7 0.5 −51.7 29 2012071012 0.46 13.5 0.5 −51.7 30 2012071218 0.38 8.6 0.1 −33.5 31 2013062812 0.49 12.9 −0.1 −39.2 32 2013070118 0.64 11.3 0.9 −40.5 33 2013070412 0.28 8.8 0 −56.8 34 2013070912 0.39 10.8 0.5 −35.3 Average 0.38 10.0 0.2 −36.8 Table 2. Horizontal gradients of potential temperature in the upper-level frontal region and the corresponding deep convection frequencies in the warm region of the front for the period of 28 June to 12 July during 2004 to 2013.
The above results may also indicate that the process of the mei-yu zone precipitation is probably triggered in the upper or lower layer, and that the precipitation system, by coupling, is capable of developing into a deep precipitation system throughout the entire troposphere.
Keyser and Pecnick (1985a, b) pointed out that the cyclonic shear with the along-jet cold advection could laterally displace the subsidence branch of the direct transverse SC into the warm region of the high-level front, thus causing the tilting of the front, and then resulting in frontogenesis and further enhancement of the direct SC, all of which forms a positive feedback mechanism. Ding (2005) argued that the enhancement of the SC’s updraft is one of the triggers of deep convection in the warm regions of monsoon fronts. Figure 8 illustrates the concurrence of the frontogenesis of the monsoon high-level front and deep convection activities in a case that occurred from 0600 UTC 6 to 0000 UTC 7 July 2007. When the shear forcing is relatively weak (Fig. 8e), the convection is not vigorous (Fig. 8a). With the enhancement of the shear forcing of the cold advection (Fig. 8f), deep convection begins to occur in the region near the common updraft axis of the direct and indirect circulations (Fig. 8b). After another 6−12 h, the frontogenesis area becomes obviously tilted and extends downwards to the mid-troposphere. The subsidence branch of the direct cell moves under the jet axis and into the warm region of the front (Figs. 8c and d), and the intensity of the convection reaches its peak. By that time, the shear forcing has weakened (Figs. 8g and h).
Figure 8. (a−d) Cross sections of potential temperature (contoured with intervals of 10 K, solid) and its meridional gradient (shaded, intervals of 1 × 10−12 K m−1) and downward vertical velocity (contoured with intervals of 0.5 Pa s−1, dotted): (a) 18 hours; (b) 12 hours; (c) 6 hours prior to, and (d) at 0000 UTC 7 July 2007. (c−h) Cross sections of potential temperature and its zonal gradient (red contours with intervals of 0.5 × 10−12 K m−1) and zonal wind velocity component (shaded, intervals of 12 m s−1) and vector arrows of air flow in the transverse plane (v, ω) over the same time period as in (a−d). In the lowest portion in every panel, the zonally averaged frequency of deep convection events (mean value > 0.25) over 117°−124°E is illustrated using gray lines and scales at the location of 35°N.
The S−E equation gives a direct and simplified way to clarify the relationship between horizontal geostrophic deformation and the transverse SC, which is in general linked to the frontogenesis process.
Following the investigation of Keyser and Pecnick (1985b), in order to further clarify the frontogenesis mechanism due to horizontal shear forcing in monsoon front zones, the S−E equation with only geostrophic shear forcing terms is utilized and the diagnosis results are shown in Fig. 9. It is clear that, with the enhancement of cyclonic shear on the cyclonic side of the upper-level jet (Figs. 8a−d), the circulation response to the shear terms in the mei-yu front region shifts from an indirect cell (Figs. 9a and b) to a direct cell (Figs. 9c and d). An obviously tilted-with-altitude subsidence branch has been laterally “placed” into the warm region of the front zone (Fig. 9d), strengthened and has extended downwards to the low levels over 34°−35°N, similar to the synergistic interaction between an upper-level front and a surface front diagnosed and simulated by Mak et al. (2017), in which the downward motion of upper- and lower-level ageostrophic circulation cells were merged together along a tilted-with-altitude surface. At the same time, a strengthened downward velocity maximum appears in the mid-to-upper-level frontogenesis region associated with vigorous convections (Fig. 9d vs. Fig. 8h), implying a link between the changing SC, frontogenesis, and intensifying convection. As Shapiro (1981) and Keyser and Pecnick (1985b) indicated, with the cold advection along the high-level jet axis, the shear forcing on the cyclonic shearing side of the jet can intensify the transverse SC; then, the enhanced subsidence in the SC promotes the high-level frontogenesis through the “tilt” effect, and the frontogenesis in turn promotes the SC’s vertical motion. This positive feedback process can persist until the high-level cold advection exists and can provide a favorable condition for deep convection in the warm regions of the frontal zone (Ding, 2005; Mak et al., 2017).
Figure 9. As in Fig. 4f except for 0600 UTC 6 July 2007 to 0000 UTC 7 July 2007 at 6-h intervals.
Longitude | Forcing | ||||||
$ {Q}_{\rm{con}} $ | $ {Q}_{\rm{she}} $ | $ {Q}_{\rm{g}} $ | $ {Q}_{\rm{ag}} $ | $ {Q}_{\rm{d}} $ | $ {Q}_{\rm{tot}} $ | Average | |
117°E | 458 | 477 | 482 | 480 | 477 | 481 | 61% |
118°E | 473 | 483 | 491 | 483 | 494 | 496 | 62% |
119°E | 474 | 488 | 495 | 487 | 497 | 491 | 63% |
120°E | 482 | 492 | 503 | 499 | 510 | 489 | 64% |
121°E | 504 | 506 | 496 | 520 | 503 | 497 | 65% |
122°E | 503 | 516 | 528 | 529 | 518 | 521 | 67% |
123°E | 518 | 527 | 506 | 512 | 524 | 501 | 66% |
124°E | 510 | 526 | 508 | 517 | 503 | 507 | 66% |
Average | 63% | 64% | 64% | 65% | 65% | 64% | 64% |
Note: $ {Q}_{\rm{con}} $ denotes geostrophic confluence forcing, $ {Q}_{\rm{she}} $ denotes geostrophic shear forcing, and $ {Q}_{\rm{tot}}={Q}_{\rm{g}}+{Q}_{\rm{ag}}+{Q}_{\rm{d}} $. |