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Coleman et al. (2003) observed that lightning discharge appears to deposit opposite-polarity charges in relatively localized regions of both the main negative charge and the upper positive charge, rather than simply neutralizing the space charge near the discharge channels. The charge density of the deposited charge is comparable to that of the preexisting charge, and these charges may affect the distribution of thunderstorm electrical parameters, and thus influence the development of subsequent lightning. Moreover, Tan et al. (2007) revealed the effects of IC lightning discharge on the space charge, electric field, potential, and space electrostatic energy by computer simulations. In contrast to the bidirectional leader development process of IC and CG lightning, upward lightning is a unidirectional leader discharge. However, how does the latter affect the space charge, potential, electric field and electrostatic energy? How long does it take to recover the damage caused by upward lightning to the space potential and electric field? In this section, the discharge effects of all upward lightning cases simulated by our model are discussed, and the effects of upward lightning discharge on thundercloud electrical environment parameters are also quantitatively evaluated.
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Figure 2 shows two upward flashes. The upward flash (UL 1 from ST A) shown in Fig. 2a is an upward negative flash corresponding to the dipole charge structure. Figures 2a and b show the distribution of the space charge before and after UL 1, respectively. The upward flash (UL 2 from ST B) shown in Fig. 2c is an upward positive flash corresponding to the tripole charge structure. Figures 2c and d show the distribution of the space charge before and after UL 2, respectively. After the upward lightning discharge, a large number of induced charges are deposited in the upward leader channels. As shown in Fig. 2, UL 1 and UL 2 both enter the bottom charge region. UL 1 propagates into the main negative charge region, and after leader channels are terminated, a large number of positive charges were deposited in the original discharge channels. The maximum charge density in the bottom charge region before the discharge is −1.12 nC m−3, while the maximum positive and negative charge densities in the bottom charge region after discharge are 1.36 and −1.09 nC m−3, respectively. The results reveal that after UL 1 discharges, the induced charge on the lightning channels is of the same order of magnitude as the space charge in the discharge region. According to the treatment of the induced charge in the model, the charge is not immediately neutralized by the surrounding space charge, but with the subsequent turbulent exchange, advection transport, gravity sedimentation and electrification, the charge gradually mixes with the surrounding space charge, and then prepares for the subsequent discharge. UL 2 propagates into the bottom positive charge region, and the maximum charge density in the bottom charge region before the discharge is 1.53 nC m−3, while the maximum positive and negative charge densities in the bottom charge region after discharge are 1.45 and −1.72 nC m−3, respectively. Similar to the analysis of UL 1, the upward lightning discharge does not immediately reduce the charge density in the whole space but implants a considerable amount of induced charges into the leader channel, thus forming a more complex charge structure, which may be one of the reasons for the occurrence of the associated lightning. The above simulation results in this study are consistent with those observed by Coleman et al. (2003).
Figure 2. Lightning channels and charge distribution. (a) Upward lightning channel structure for UL 1 (from ST A) and pre-discharge charge distribution. (b) Post-discharge charge distribution. (c) Upward lightning channel structure for UL 2 (from ST B) and pre-discharge charge distribution. (d) Post-discharge charge distribution. The purple diamonds represent the initiation points of the upward lightning flashes. The red channels represent the upward positive leaders, and the blue channels represent the upward negative leaders. The black solid lines are for positive charge and the dashed lines are for negative charge, with the contour values starting at ±0.1 nC m−3 with intervals of 0.5 nC m−3 [the charge distributions are all from a coarse resolution (250 m) model].
The leader channel length, induced charge estimate, and ratio of the induced charge on the discharge channels to the total opposite-polarity charge in the discharge region (Ric) are given in Table 1. Assuming that the charge density distribution on all sides of the cloud is consistent, the estimated value of the total induced charge generated by the upward lightning discharge is equal to the product of the total charge in the discharge region, the horizontal range of the thundercloud (which is an artificially defined variable and approximately equal to πr/2) and Ric, where the horizontal radius (r) of the thundercloud is taken from the 3D thunderstorm simulation results. The cloud radius r of ST A and ST B is 10 and 15 km, respectively, which is the same as those used in Tan et al. (2007), which facilitates a comparison of the discharge effect of upward lightning in this study to the discharge effect of IC lightning in Tan et al. (2007).
Charge structure Charge regions affected byupward flashes Channel length (km) Induced charge (C) Ric UL 1 Dipole Main negative 8.85 28.8 26.7% UL 2 Tri-pole Lower positive 8.85 −34.4 17.1% Ric: Ratio of the total induced charge on the leader channels to the original charge on the discharge channels. Table 1. Upward leader channel length, induced charge and Ric.
UL 1 is simulated in ST A, where the upward positive leader propagates into the main negative charge region, the channel length of the upward positive leader is 8.85 km, the induced charge is 28.8 C, and Ric is 26.7%. UL 2 is simulated in ST B, where the upward negative leader extends into the bottom positive charge region, the channel length of the upward negative leader is 8.85 km, the induced charge is −34.4 C, and Ric is 17.1%. Comparing these two cases, the results indicate that the channel length of these two cases is the same, and the induced charge of UL 1 is less than that of UL 2, but the Ric of UL 1 is larger than that of UL 2. The main reason is that the thunderstorm corresponding to UL 2 develops vigorously and the thundercloud size is larger than that corresponding to UL 1. The total cloud charge of UL 2 is much more than that of UL 1. Although the induced charge in UL 2 is considerable, it still only accounts for a small proportion of the total cloud charge.
Table 2 provides the statistical characteristics of the simulated 147 upward lightning flashes in ST A and ST B. Among them, 101 upward flashes are from the ST A, and 46 upward flashes are from the ST B. The average channel length of the simulated upward flashes in ST A is 6.26 km, which is obviously smaller than the average length (8.38 km) of the upward lightning flashes obtained in ST B. The main reason is that ST B develops more vigorously, and the average cloud-base height is larger. When upward lightning is initiated in ST B, the range and magnitude of the corresponding bottom charge are usually larger, and the resulting upward leaders fully extend after entering the bottom charge region. The induced charge in the upward positive flashes simulated in ST A ranges from −0.67 to −18.5 C, which is lower than the induced charge in the upward positive flashes simulated in the ST B. Because the upward positive flashes simulated in our model usually propagate into the bottom positive charge region, as mentioned above, the range and magnitude of the bottom charge region in ST B are usually larger, and the intensity of the bottom potential well (potential extremum region) is also stronger, which is conducive to the propagation of upward leaders and neutralization of charge. The induced charge in the upward negative flashes simulated in ST A ranges from 8.03 to 118.8 C, and the average value is slightly larger than that simulated in ST B. The reason is that most of the upward negative flashes simulated in ST A occur under the dipole charge structure, which extend into the main negative charge region. The upward negative flashes simulated in ST B usually occur under the multi-layer charge structure, which propagate into the bottom negative charge region. The range and amount of the charge in the bottom negative charge region are far smaller than those in the main negative charge region. The upward positive leaders in the main negative charge region develop more vigorously and neutralize more charges. The Ric in ST A ranges from 8% to 47.3%, while that in ST B ranges from 5.9% to 17.8%. The average induced charge in all upward flashes simulated in ST A is 18.3 C, and the average value of Ric is 15.8%. In ST B, the average induced charge in all upward flashes is 20.4 C, and the average value of Ric is 12.3%. By analyzing all upward lightning cases simulated by our model, the results reveal that the absolute value of the total induced charge ranges from 0.67 to 118.8 C, the average value of which is 19.0 C, and the Ric ranges from 5.9% to 47.3%, the average value of which is 14.7%. The above results are lower than those of IC lightning in Tan et al. (2007) and consistent with the induced charge obtained by observations in Berger (1977) and Michishita et al. (2019), which also proves that our simulations are reasonable.
Thunderstorm case Frequency of upward flashes Average channel height (km) Induced charge of upward positive flashes (C) Induced charge of upward negative flashes (C) Ric ST A 101 6.26 −0.67 to −18.5 8.03–118.8 8%–47.3% ST B 46 8.38 −15.92 to −76.39 7.75–21.84 5.9%–17.8% Table 2. The Ric and induced charge of all upward lightning.
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It is generally believed that the energy released during lightning discharge comes from the electrostatic energy stored in the form of the electrostatic field in thunderstorms. How much, though, does an upward flash influence the space potential, electric field and electrostatic energy in thunderclouds? So far, there has been no relevant research. In this section, the effects of an upward flash on the space potential, electric field and electrostatic energy are quantitatively discussed.
Figure 3 shows the distributions of the space potential and vertical electric field before and after UL 1. Comparing Figures 3a and b, it is found that a negative potential well exists in the bottom space before the upward lightning discharge, and the potential extremum at the center of the potential well is −55.2 MV. After UL 1 enters the negative potential center, the original negative potential well is destroyed, a new potential well with a smaller range is formed at the original position, and the potential at the center of the new potential well changes to −17 MV. Comparing Figures 3c and d, it is found that the vertical electric field distribution is positive, negative and positive from top to bottom before the discharge, and the maximum electric field is located at the junction of the positive and negative charge regions. The maximum electric field in the discharge region is −47.5 kV m−1. After the discharge, the positive electric field area in the bottom space is greatly reduced, and the electric field extremum changes to 14.1 kV m−1.
Figure 3. Upward lightning channels and space potential and vertical electric field of UL 1. (a) Upward lightning channel structure and pre-discharge space potential. (b) Post-discharge space potential. (c) Upward lightning channel structure and pre-discharge space electric field. (d) Post-discharge space electric field. In (a, b) the black solid lines represent positive potential and the dashed lines represent negative potential, with the contour values starting at 0 MV. In (c, d) the black solid lines are for positive electric field (physics sign convention) and the dashed lines are for negative electric field, with contour values starting at ±0 kV m−1. The details are the same as those in Fig. 2.
Figure 4 shows the distributions of the space potential and vertical electric field before and after UL 2. Similar to UL 1, a positive potential well exists in the bottom space before the discharge, and the potential extremum at the center of the potential well is 65.9 MV. After the discharge, the original positive potential well is destroyed, a new negative potential well is formed at the original position, and the potential at the center of the new potential well changes to −17.7 MV. Moreover, the vertical electric field extremum in the bottom space before the discharge is −22.4 kV m−1. After the discharge, the direction of the vertical electric field in the discharge area changes from downward to upward, and the electric field extremum changes to 11.4 kV m−1. UL 1 and UL 2 both enter the bottom charge region, thereby destroying the bottom potential wells and even causing the disappearance of the bottom potential wells. Similarly, the space distribution ranges of the bottom electric fields decrease significantly after the discharges, and even the polarity of the electric field reverses.
Figure 4. Upward lightning channels and space potential and vertical electric field of UL 2. (a) Upward lightning channel structure and pre-discharge space potential. (b) Post-discharge space potential. (c) Upward lightning channel structure and pre-discharge space electric field. (d) Post-discharge space electric field. The details are the same as those in Fig. 3.
Selecting UL 1 and UL 2 as examples, the maximum variations of the space potential (ΔU) and vertical electric field (ΔE) before and after the discharges are listed in Table 3, where ΔU and ΔE are equal to the value after discharge minus the value before discharge. The ΔU and ΔE caused by UL 1 are 70.2 MV and −26.4 kV m−1, respectively. The ΔU and ΔE caused by UL 2 are −107.4 MV and 29 kV m−1, respectively. In regard to UL 1 and UL 2, the changes of the potential and electric field in the discharge area caused by these two upward flashes are comparable. Compared to the effects of IC lightning reported by Tan et al. (2007), the effects of one upward flash on the space potential and electric field are much smaller [the ΔU and ΔE caused by IC lightning discharge are −215 MV and 139 kV m−1 respectively; these data are from Table 4 in Tan et al. (2007)].
$\Delta $E (kV m−1) $\Delta $U (MV) Pre-discharge eex−z (J m−1) Post-discharge eex−z (J m−1) $\Delta $ee (J) UL 1 −26.4 70.2 2.23×105 1.93×105 4.82×108 UL 2 29.0 −107.4 1.55×106 1.41×106 3.46×109 Table 3. Pre-discharge and post-discharge maximum changes in the space potential and vertical electric field and the electrostatic energy variation of the two upward lightning cases.
As mentioned above, the space charge around the channel is not immediately neutralized after the upward lightning discharges. Instead, a large amount of induced charge accumulates in the leader channels, and is slowly neutralized by the surrounding space charge through the subsequent turbulent exchange, advection transport, gravity sedimentation, etc. However, the space potential and electric field in the discharge region are immediately destroyed after the upward lightning discharge, and the polarity of the potential and electric field in this region are even reversed. Regardless of the effects of discharge on the charge, potential or electric field, none of these considerations enable a good evaluation of the overall effect of upward lightning discharges. Therefore, the concept of electrostatic energy is introduced here. The density of the electrostatic energy at each point in space (ee) is given as follows:
where ε is the electric permittivity, and E is the electric field at grid points. First, the total electrostatic energy of the simulated surface (eex-z) is calculated, and then the total electrostatic energy in space is calculated by multiplying eex-z and the horizontal range of the cloud. The total electrostatic energy consumed by an upward lightning discharge (Δee) is equal to the total electrostatic energy in space before a discharge minus the total electrostatic energy in space after a discharge.
As indicated in Table 3, the Δee of UL 1 is 4.82×108 J, and that of UL 2 is 3.46×109 J. The average electrostatic energy consumed by upward lightning in ST A is 5.96×108 J, while the average electrostatic energy consumed by upward lightning in ST B is 2.16×109 J, and the average value of all upward lightning is 1.06×109 J. These values are one order lower than the average electrostatic energy consumed by IC lightning. The main reason is that IC lightning belongs to the bidirectional leader discharge, which simultaneously propagates in two charge regions. Therefore, in general, the longer the IC lightning channels are and the larger the action area is, then the greater the amount of neutralized charge there is, and the more significant the influence on the space potential, electric field and electrostatic energy. In addition, IC lightning discharge usually deposits a large number of positive charges in the main negative charge region, which may cause the upward positive leader to pass through the main negative charge region into the upper positive charge region along positive charge deposition channels. This issue will be discussed in further work.
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The effects of upward lightning discharges are not to neutralize the surrounding space charge immediately, but to deposit the induced charge on the discharge channel and gradually neutralize the surrounding space charge with the subsequent process. In this section, UL 1 is taken as an example to discuss the slow neutralization process of charge on the discharge channel.
In order to better describe this process, Rrem (defined as the ratio of the residual charge on the channels to the total induced charge on the channels after the discharge), Emax (the maximum vertical electric field in the discharge region), eex−z, and QL (the absolute value of the total charge in the bottom charge region) are introduced. Their changes over time are shown in Fig. 5. After the UL 1 discharge (UL 1 occurs at 68.4 min, which corresponds to the thunderstorm dissipation stage), Emax and eex-z decrease rapidly, and this process lasts tens to hundreds of milliseconds, which is consistent with the duration of a single upward lightning discharge. The subsequent slow rise process corresponds to the reconstruction of the electric field in the cloud, which depends on the turbulent exchange, advection transport, gravity sedimentation and electrification in the cloud. There are obvious differences in this process among different regions, thunderstorms and thunderstorm stages.
Figure 5. The curves of Rrem, Emax, eex-z and QL between two upward lightning flashes from ST A. (a) The black solid line represents Rrem and the dotted line represents Emax. (b) The black solid line represents eex-z and the dotted line represents QL (absolute value); t is simulation time.
Rrem reflects the degree of neutralization between the induced charge and the space charge around the discharge channels. The lower the Rrem, the more sufficient the induced charge neutralization and the smaller the residual effect of the previous lightning discharge. The time taken to reduce Rrem to 30% is defined as the relaxation time of charge neutralization (τ), and the τ of UL 1 is 48 s. This process is very slow compared to the time of a single upward lightning discharge, which also conforms to the previous analysis. The charge density in the discharge region does not decrease immediately, but with the turbulent exchange, advection transport, gravity sedimentation and electrification, these charges gradually neutralize with the surrounding space charges. It is revealed by the change of QL, as shown in Fig. 5. QL exhibits a large drop after the discharge of UL 1, but in contrast to the rapid rise after Emax and eex−z decline, the decline process of QL lasts 36 s. First, the charge deposited in the discharge channel rapidly neutralizes the surrounding space charge, and then the neutralization process decelerates. Finally, due to the gravity sedimentation and electrification, QL increases gradually, and a new space charge distribution is formed. UL 2 occurs at 46.8 min, which corresponds to the thunderstorm mature stage. The τ of UL 2 is 40 s, which is shorter than that of UL 1. This occurs because the UL 2 discharge is simulated in the plain thunderstorm (ST B) and occurs in the thunderstorm mature stage. The electrification and activity of plain thunderstorms are much higher than those of mountain thunderstorms (UL 1 is simulated in ST A, which belongs to a mountain thunderstorm), and the neutralization process of the induced charges of opposite polarity in the channels for UL 2 is faster. Figure 6 shows the post-discharge space charge distributions at nine different times for UL 1, which reflects the neutralization process of the induced charges and surrounding space charges and the redistribution of the space charges. Combining the changes of Rrem and QL, as shown in Fig. 5, it is found that the neutralization process of the induced charges and surrounding space charges exhibits a sharp deceleration. Rrem decreases from 100% to 65%, which takes 14 s; Rrem decreases from 65% to 30%, which takes 34 s; while Rrem decreases from 30% to 5%, which takes 50 s. Subsequently, it takes nearly 3 min for the positive charge deposited in the discharge channel to be completely neutralized, as shown in Fig. 6. The reason why the positive charge on the discharge channel has decreased and no subsequent increase process occurs is that UL1 occurs at the thunderstorm dissipation stage, and the electrification process can be neglected. Therefore, the induced positive charge deposited in the negative charge area continues to decrease until it is completely neutralized. As shown in Fig. 5, eex−z reaches 4.73×105 J m−1 before the subsequent lightning (the subsequent lightning occurs at 73.3 min), which is twice the electrostatic energy before UL1. This occurs because the subsequent advection transport, gravity sedimentation etc. result in the gradual concentration of the original scattered charges. Although the total charge does not increase, the charge distribution becomes more concentrated and the average charge density increases, which enhances the space electric field and increases the electrostatic energy. Comparing the charge structures shown in Figs. 2a and 6, the charge distribution shown in Fig. 2a as a whole indicates a dipole charge structure. The charge distributions from 68.53 to 71.80 min shown in Fig. 6 exhibit multiple-layer and complex charge structures, and the charge distributions at 73.10 min return to the dipole structure. Coleman et al. (2003) presented that recurring lightning charge deposition increases the complexity of storm charge structures. The induced charge neutralization process shown in Fig. 6 partly verifies the above conclusion. Moreover, the complexity of the charge distribution gradually decreases with the induced charge neutralization process and eventually returns to the classical dipole or tripole charge structure.
Figure 6. Post-discharge space charge distributions at nine different times for UL 1. The details are the same as those in Fig. 2.
In summary, after a lightning discharge, the impact of upward lightning discharge does not immediately disappear. The induced charges deposited in the discharge channel will gradually neutralize with the surrounding space charges. This process usually lasts tens of seconds or even several minutes, thus forming a new charge distribution and imposing a lasting effect on the subsequent lightning discharge.