人工引雷上行正先导传播过程中爆发式磁场脉冲极性反转现象的观测与分析
|
摘要
为了进一步理解人工引雷上行正先导的传播过程及相关的爆发式电磁辐射脉冲,本文分析了2014年8月23日山东沾化人工触发闪电实验(SHATLE)中获得的三次上行正先导通道含有明显向下传播过程的个例。结果表明,在初始连续电流阶段观测到的爆发式磁场脉冲极性反转现象与正先导头部发展方向相对于观测位置投影方向的变化密切相关。因此可以推断,爆发式磁场脉冲是由正先导头部较小空间尺度放电过程辐射出的,就本文分析的个例而言,爆发式磁场脉冲辐射源的放电尺度约为2 m,放电强度的最大值达到2.49 k A。爆发式磁场脉冲的辐射机制不同于人工引雷先导始发阶段初始脉冲电流产生的磁场脉冲的辐射机制,可能与正先导通道头部的梯级形式发展过程相关。
In order to understand the propagating process of upward positive leader and related burst-type electromagnetic pulses, in this paper we analyze three cases of downward propagation of positive leader channels obtained by SHATLE(SHAndong Triggering Lightning Experiment) on August 23, 2014. The results show that the reversals in the polarity of magnetic pulses observed in the stage of initial continuous current are related to the variation of projection direction of positive leader head relative to the observation position. Thereby it can be inferred that the burst-type magnetic pulses are emitted by small spatial scale discharge process of positive leader head. For the cases analyzed in this paper, the discharge scale is about 2 m, and the maximum discharge intensity reaches 2.49 k A. The radiation mechanism for these magnetic pulses is different to the radiation mechanism for the magnetic pulses generated by the initial pulse current in the channel measured at the initial stage of rocket-triggered lightning, which may be related to the stepped propagation of positive leader.
In order to understand the propagating process of upward positive leader and related burst-type electromagnetic pulses, in this paper we analyze three cases of downward propagation of positive leader channels obtained by SHATLE(SHAndong Triggering Lightning Experiment) on August 23, 2014. The results show that the reversals in the polarity of magnetic pulses observed in the stage of initial continuous current are related to the variation of projection direction of positive leader head relative to the observation position. Thereby it can be inferred that the burst-type magnetic pulses are emitted by small spatial scale discharge process of positive leader head. For the cases analyzed in this paper, the discharge scale is about 2 m, and the maximum discharge intensity reaches 2.49 k A. The radiation mechanism for these magnetic pulses is different to the radiation mechanism for the magnetic pulses generated by the initial pulse current in the channel measured at the initial stage of rocket-triggered lightning, which may be related to the stepped propagation of positive leader.
引文
Biagi C J,Jordan D M,Uman M A,et al.2009.High-speed video observations of rocket-and-wire initiated lightning[J].Geophys.Res.Lett.,36(15),doi:10.1029/2009GL038525.
Biagi C J,Uman M A,Hill J D,et al.2011.Observations of the initial,upward-propagating,positive leader steps in a rocket-and-wire triggered lightning discharge[J].Geophys.Res.Lett.,38(24),doi:10.1029/2011GL049944.
陈绍东,张义军,杨少杰,等.2009.两次仅有连续电流的负极性人工引发雷电特征分析[J].中国电机工程学报,29(1):113-119.Chen Shaodong,Zhang Yijun,Yang Shaojie,et al.2009.Characteristics of analysis of two negative artificially triggered lightning discharges with only continuous current[J].Proc.CSEE(in Chinese),29(1):113-119,doi:10.13334/j.0258-8013.pcsee.2009.01.013.
Diendorfer G,Mair M,Schulz W.2002.Detailed brightness versus lightning current amplitude correlation of flashes to the Gaisberg tower[C]//26th International Conference on Lighting Protection.Cracow,Poland:ICLP,8-13.
樊艳峰,陆高鹏,张鸿波,等.2016.人工触发闪电实验中初始电流脉冲辐射磁场的观测与模拟[J].高电压技术,42(2):987-993.Fan Yanfeng,Lu Gaoping,Zhang Hongbo,et al.2016.Observations and simulations of magnetic fields radiated by initial current pulses in rocket-triggered lightning[J].High Volt.Eng.(in Chinese),42(2):987-993,doi:10.13336/j.1003-6520.hve.20160524001.
樊艳峰,陆高鹏,蒋如斌,等.2017.利用低频磁场天线遥感测量人工引雷中的初始连续电流[J].大气科学,41(4):1027-1036.Fan Yanfeng,Lu Gaopeng,Jiang Rubin,et al.2017.Application of low-frequency magnetic sensor for remote measurement of the initial continuous current in rocket-triggering lightning[J].Chinese Journal of Atmospheric Sciences(in Chinese),41(4):1027-1036,doi:10.3878/j.issn.1006-9895.1702.16248.
Fuchs F,Landers E U,Schmid R,et al.1998.Lightning current and magnetic field parameters caused by lightning strikes to tall structures relating to interference of electronic systems[J].IEEE Trans.Electrom.Compatib.,40(4):444-451,doi:10.1109/15.736205.
Gao Yan,Lu Weitao,Ma Ying,et al.2014.Three-dimensional propagation characteristics of the upward connecting leaders in six negative tall-object flashes in Guangzhou[J].Atmos.Res.,149:193-203,doi:10.1016/j.atmosres.2014.06.008.
Horii K.1982.Experiment of artificial lightning triggered with rocket[J].Mem.Fac.Eng.Nagoya Univ.,34(1):77-112.
Jiang R B,Qie X S,Wang C X,et al.2013.Propagating features of upward positive leaders in the initial stage of rocket-triggered lightning[J].Atmos.Res.,129-130:90-96,doi:10.1016/j.atmosres.2012.09.005.
Jiang R B,Qie X S,Wu Z J,et al.2014.Characteristics of upward lightning from a 325-m-tall meteorology tower[J].Atmos.Res.,149:111-119,doi:10.1016/j.atmosres.2014.06.007.
Lu G P,Jiang R B,Qie X S,et al.2014.Burst of intracloud current pulses during the initial continuous current in a rocket-triggered lightning flash[J].Geophys.Res.Lett.,41(24):9174-9181,doi:10.1002/2014GL062127.
Lu G P,Zhang H B,Jiang R B,et al.2016.Characterization of initial current pulses in negative rocket-triggered lightning with sensitive magnetic sensor[J].Radio Sci.,51(9):1432-1444,doi:10.1002/2016RS005945.
Miki M,Rakov V A,Rambo K J,et al.2002.Electric fields near triggered lightning channels measured with Pockels sensors[J].J.Geophys.Res.,107(D16):ACL 2-1-ACL 2-11,doi:10.1029/2001JD001087.
Miki M,Rakov V A,Shindo T,et al.2005.Initial stage in lightning initiated from tall objects and in rocket-triggered lightning[J].J.Geophys.Res.,110(D2),doi:10.1029/2003JD004474.
Qie X S,Zhang Q L,Zhou Y J,et al.2007.Artificially triggered lightning and its characteristic discharge parameters in two severe thunderstorms[J].Sci.China Ser.D:Earth Sci.,50(8):1241-1250,doi:10.1007/s11430-007-0064-2.
Qie X S,Zhao Y,Zhang Q L,et al.2009.Characteristics of triggered lightning during Shandong artificial triggering lightning experiment(SHATLE)[J].Atmos.Res.,91(2-4):310-315,doi:10.1016/j.atmosres.2008.08.007.
郄秀书,杨静,蒋如斌,等.2010.新型人工引雷专用火箭及其首次引雷实验结果[J].大气科学,34(5):937-946.Qie Xiushu,Yang Jing,Jiang Rubin,et al.2010.A new-model rocket for artificially triggering lightning and its first triggering lightning experiment[J].Chinese Journal of Atmospheric Sciences(in Chinese),34(5):937-946,doi:10.3878/j.issn.1006-9895.2010.05.09.
Qie X S,Jiang R B,Wang C X,et al.2011.Simultaneously measured current,luminosity,and electric field pulses in a rocket-triggered lightning flash[J].J.Geophys.Res.,116(D10),doi:10.1029/2010JD015331.
郄秀书,杨静,蒋如斌,等.2012.山东人工引发雷电综合观测实验及回击电流特征[J].大气科学,36(1):77-88.Qie Xiushu,Yang Jing,Jiang Rubin,et al.2012.Shandong artificially triggering lightning experiment and current characterization of return stroke[J].Chinese Journal of Atmospheric Sciences(in Chinese),36(1):77-88,doi:10.3878/j.issn.1006-9895.2012.01.07.
Rakov V A,Uman M A.2003.Artificial initiation(triggering)of lightning by ground-based activity[M]//Rakov V A,Uman M A.Lightning:Physics and Effects.Cambridge:Cambridge University Press,265-307.
Rakov V A,Uman M A,Rambo K J.2005.A review of ten years of triggered-lightning experiments at Camp Blanding,Florida[J].Atmos.Res.,76(1-4):503-517,doi:10.1016/j.atmosres.2004.11.028.
Schoene J,Uman M A,Rakov V A,et al.2003.Statistical characteristics of the electric and magnetic fields and their time derivatives 15 m and 30 m from triggered lightning[J].J.Geophys.Res.,108(D6),doi:10.1029/2002JD002698.
Schoene J,Uman M A,Rakov V A.2010.Return stroke peak current versus charge transfer in rocket-triggered lightning[J].J.Geophys.Res.,115(D12),doi:10.1029/2009JD013066.
Thottappillil R,Rakov V A.2007.Review of three equivalent approaches for computing electromagnetic fields from an extending lightning discharge[J].J.Lightn.Res.,1:90-110.
Uman M A,Rakov V A,Schnetzer G H,et al.2000.Time derivative of the electric field 10,14,and 30 m from triggered lightning strokes[J].J.Geophys.Res.,105(D12):15577-15595,doi:10.1029/2000JD900046.
Uman M A,Schoene J,Rakov V A,et al.2002.Correlated time derivatives of current,electric field intensity,and magnetic flux density for triggered lightning at 15 m[J].J.Geophys.Res.,107(D13):ACL 1-1-ACL 1-11,doi:10.1029/2000JD000249.
王彩霞,郄秀书,蒋如斌,等.2012.一次人工触发闪电上行正先导的传输特征[J].物理学报,61(3):039203.Wang Caixia,Qie Xiushu,Jiang Rubin,et al.2012.Propagating properties of an upward positive leader in a negative triggered lightning[J].Acta Phys.Sinica(in Chinese),61(3):039203,doi:10.7498/aps.61.039203.
Wang D,Rakov V A,Uman M A,et al.1999.Characterization of the initial stage of negative rocket-triggered lightning[J].J.Geophys.Res.,104(D4):4213-4222,doi:10.1029/1998JD200087.
Wang D,Takagi N,Watanabe T,et al.2005.A comparison of channel-base currents and optical signals for rocket-triggered lightning strokes[J].Atmos.Res.,76(1-4):412-422,doi:10.1016/j.atmosres.2004.11.025.
Wang Z C,Qie X S,Jiang R B,et al.2016.High-speed video observation of stepwise propagation of a natural upward positive leader[J].J.Geophys.Res.,121(24):14307-14315,doi:10.1002/2016JD025605.
Yang J,Qie X S,Zhang G S,et al.2010.Characteristics of channel base currents and close magnetic fields in triggered flashes in SHATLE[J].J.Geophys.Res.,115(D23),doi:10.1029/2010JD014420.
Yoshida S,Biagi C J,Rakov V A,et al.2010.Three-dimensional imaging of upward positive leaders in triggered lightning using VHF broadband digital interferometers[J].Geophys.Res.Lett.,37(5),doi:10.1029/2009GL042065.
张义军,刘欣生,肖庆复.1997.中国南北方雷暴及人工触发闪电特性对比分析[J].高原气象,16(2):113-121.Zhang Yijun,Lin Xinsheng,Xiao Qingfu.1997.A analysis of characteristics of thunderstorm and artificially triggered lightning in the north and south of China[J].Plateau Meteor.(in Chinese),16(2):113-121,doi:10.3321/j.issn:1000-0534.1997.02.001.
张义军,言穆弘,郄秀书,等.1998.人工引发雷电对雷暴特性可能影响的观测与分析[J].高原气象,17(4):412-419.Zhang Yijun,Yan Muhong,Qie Xiushu,et al.1998.Observation and analysis of influences of artificially triggered lightning on characteristics of thunderstorm[J].Plateau Meteor.(in Chinese),17(4):412-419,doi:10.3321/j.issn:1000-0534.1998.04.010.
张义军,张阳,郑栋,等.2016.2008-2014年广东人工触发闪电电流特征[J].高电压技术,42(11):3404-3414.Zhang Yijun,Zhang Yang,Zheng Dong,et al.2016.Current Characteristics of triggered lightnings in Guangdong from 2008 to 2014[J].High Volt.Eng.(in Chinese),42(11):3404-3414,doi:10.13336/j.1003-6520.hve.20160412005.
周恩伟.2010.触发闪电放电过程的光电同步观测与分析[D].中国科学技术大学硕士学位论文.Zhou Enwei.2010.Simultaneous optical and electrical observations and analysis of rocket-triggered lightning[D].M.S.thesis(in Chinese),University of Science and Technology of China.
周忠华,刘欣生.1993.人工触发闪电放电电流的间接测量及其特征分析[J].高原气象,12(1):67-76.Zhou Zhonghua,Lin Xinsheng.1993.Indirect measurement of artificially triggered lightning discharge current and its characteristic analysis[J].Plateau Meteor.(in Chinese),12(1):67-76.
祝宝友,陶善昌,马明,等.2004.用远场辐射场反演云闪放电参数[J].气象学报,62(4):498-503.Zhu Baoyou,Tao Shanchang,Ma Ming,et al.2004.Evaluation of in-cloud lightning discharge parameters from remote electromagnetic fields[J].Acta Meteor.Sinica(in Chinese),62(4):498-503,doi:10.11676/qxxb2004.051.
Biagi C J,Uman M A,Hill J D,et al.2011.Observations of the initial,upward-propagating,positive leader steps in a rocket-and-wire triggered lightning discharge[J].Geophys.Res.Lett.,38(24),doi:10.1029/2011GL049944.
陈绍东,张义军,杨少杰,等.2009.两次仅有连续电流的负极性人工引发雷电特征分析[J].中国电机工程学报,29(1):113-119.Chen Shaodong,Zhang Yijun,Yang Shaojie,et al.2009.Characteristics of analysis of two negative artificially triggered lightning discharges with only continuous current[J].Proc.CSEE(in Chinese),29(1):113-119,doi:10.13334/j.0258-8013.pcsee.2009.01.013.
Diendorfer G,Mair M,Schulz W.2002.Detailed brightness versus lightning current amplitude correlation of flashes to the Gaisberg tower[C]//26th International Conference on Lighting Protection.Cracow,Poland:ICLP,8-13.
樊艳峰,陆高鹏,张鸿波,等.2016.人工触发闪电实验中初始电流脉冲辐射磁场的观测与模拟[J].高电压技术,42(2):987-993.Fan Yanfeng,Lu Gaoping,Zhang Hongbo,et al.2016.Observations and simulations of magnetic fields radiated by initial current pulses in rocket-triggered lightning[J].High Volt.Eng.(in Chinese),42(2):987-993,doi:10.13336/j.1003-6520.hve.20160524001.
樊艳峰,陆高鹏,蒋如斌,等.2017.利用低频磁场天线遥感测量人工引雷中的初始连续电流[J].大气科学,41(4):1027-1036.Fan Yanfeng,Lu Gaopeng,Jiang Rubin,et al.2017.Application of low-frequency magnetic sensor for remote measurement of the initial continuous current in rocket-triggering lightning[J].Chinese Journal of Atmospheric Sciences(in Chinese),41(4):1027-1036,doi:10.3878/j.issn.1006-9895.1702.16248.
Fuchs F,Landers E U,Schmid R,et al.1998.Lightning current and magnetic field parameters caused by lightning strikes to tall structures relating to interference of electronic systems[J].IEEE Trans.Electrom.Compatib.,40(4):444-451,doi:10.1109/15.736205.
Gao Yan,Lu Weitao,Ma Ying,et al.2014.Three-dimensional propagation characteristics of the upward connecting leaders in six negative tall-object flashes in Guangzhou[J].Atmos.Res.,149:193-203,doi:10.1016/j.atmosres.2014.06.008.
Horii K.1982.Experiment of artificial lightning triggered with rocket[J].Mem.Fac.Eng.Nagoya Univ.,34(1):77-112.
Jiang R B,Qie X S,Wang C X,et al.2013.Propagating features of upward positive leaders in the initial stage of rocket-triggered lightning[J].Atmos.Res.,129-130:90-96,doi:10.1016/j.atmosres.2012.09.005.
Jiang R B,Qie X S,Wu Z J,et al.2014.Characteristics of upward lightning from a 325-m-tall meteorology tower[J].Atmos.Res.,149:111-119,doi:10.1016/j.atmosres.2014.06.007.
Lu G P,Jiang R B,Qie X S,et al.2014.Burst of intracloud current pulses during the initial continuous current in a rocket-triggered lightning flash[J].Geophys.Res.Lett.,41(24):9174-9181,doi:10.1002/2014GL062127.
Lu G P,Zhang H B,Jiang R B,et al.2016.Characterization of initial current pulses in negative rocket-triggered lightning with sensitive magnetic sensor[J].Radio Sci.,51(9):1432-1444,doi:10.1002/2016RS005945.
Miki M,Rakov V A,Rambo K J,et al.2002.Electric fields near triggered lightning channels measured with Pockels sensors[J].J.Geophys.Res.,107(D16):ACL 2-1-ACL 2-11,doi:10.1029/2001JD001087.
Miki M,Rakov V A,Shindo T,et al.2005.Initial stage in lightning initiated from tall objects and in rocket-triggered lightning[J].J.Geophys.Res.,110(D2),doi:10.1029/2003JD004474.
Qie X S,Zhang Q L,Zhou Y J,et al.2007.Artificially triggered lightning and its characteristic discharge parameters in two severe thunderstorms[J].Sci.China Ser.D:Earth Sci.,50(8):1241-1250,doi:10.1007/s11430-007-0064-2.
Qie X S,Zhao Y,Zhang Q L,et al.2009.Characteristics of triggered lightning during Shandong artificial triggering lightning experiment(SHATLE)[J].Atmos.Res.,91(2-4):310-315,doi:10.1016/j.atmosres.2008.08.007.
郄秀书,杨静,蒋如斌,等.2010.新型人工引雷专用火箭及其首次引雷实验结果[J].大气科学,34(5):937-946.Qie Xiushu,Yang Jing,Jiang Rubin,et al.2010.A new-model rocket for artificially triggering lightning and its first triggering lightning experiment[J].Chinese Journal of Atmospheric Sciences(in Chinese),34(5):937-946,doi:10.3878/j.issn.1006-9895.2010.05.09.
Qie X S,Jiang R B,Wang C X,et al.2011.Simultaneously measured current,luminosity,and electric field pulses in a rocket-triggered lightning flash[J].J.Geophys.Res.,116(D10),doi:10.1029/2010JD015331.
郄秀书,杨静,蒋如斌,等.2012.山东人工引发雷电综合观测实验及回击电流特征[J].大气科学,36(1):77-88.Qie Xiushu,Yang Jing,Jiang Rubin,et al.2012.Shandong artificially triggering lightning experiment and current characterization of return stroke[J].Chinese Journal of Atmospheric Sciences(in Chinese),36(1):77-88,doi:10.3878/j.issn.1006-9895.2012.01.07.
Rakov V A,Uman M A.2003.Artificial initiation(triggering)of lightning by ground-based activity[M]//Rakov V A,Uman M A.Lightning:Physics and Effects.Cambridge:Cambridge University Press,265-307.
Rakov V A,Uman M A,Rambo K J.2005.A review of ten years of triggered-lightning experiments at Camp Blanding,Florida[J].Atmos.Res.,76(1-4):503-517,doi:10.1016/j.atmosres.2004.11.028.
Schoene J,Uman M A,Rakov V A,et al.2003.Statistical characteristics of the electric and magnetic fields and their time derivatives 15 m and 30 m from triggered lightning[J].J.Geophys.Res.,108(D6),doi:10.1029/2002JD002698.
Schoene J,Uman M A,Rakov V A.2010.Return stroke peak current versus charge transfer in rocket-triggered lightning[J].J.Geophys.Res.,115(D12),doi:10.1029/2009JD013066.
Thottappillil R,Rakov V A.2007.Review of three equivalent approaches for computing electromagnetic fields from an extending lightning discharge[J].J.Lightn.Res.,1:90-110.
Uman M A,Rakov V A,Schnetzer G H,et al.2000.Time derivative of the electric field 10,14,and 30 m from triggered lightning strokes[J].J.Geophys.Res.,105(D12):15577-15595,doi:10.1029/2000JD900046.
Uman M A,Schoene J,Rakov V A,et al.2002.Correlated time derivatives of current,electric field intensity,and magnetic flux density for triggered lightning at 15 m[J].J.Geophys.Res.,107(D13):ACL 1-1-ACL 1-11,doi:10.1029/2000JD000249.
王彩霞,郄秀书,蒋如斌,等.2012.一次人工触发闪电上行正先导的传输特征[J].物理学报,61(3):039203.Wang Caixia,Qie Xiushu,Jiang Rubin,et al.2012.Propagating properties of an upward positive leader in a negative triggered lightning[J].Acta Phys.Sinica(in Chinese),61(3):039203,doi:10.7498/aps.61.039203.
Wang D,Rakov V A,Uman M A,et al.1999.Characterization of the initial stage of negative rocket-triggered lightning[J].J.Geophys.Res.,104(D4):4213-4222,doi:10.1029/1998JD200087.
Wang D,Takagi N,Watanabe T,et al.2005.A comparison of channel-base currents and optical signals for rocket-triggered lightning strokes[J].Atmos.Res.,76(1-4):412-422,doi:10.1016/j.atmosres.2004.11.025.
Wang Z C,Qie X S,Jiang R B,et al.2016.High-speed video observation of stepwise propagation of a natural upward positive leader[J].J.Geophys.Res.,121(24):14307-14315,doi:10.1002/2016JD025605.
Yang J,Qie X S,Zhang G S,et al.2010.Characteristics of channel base currents and close magnetic fields in triggered flashes in SHATLE[J].J.Geophys.Res.,115(D23),doi:10.1029/2010JD014420.
Yoshida S,Biagi C J,Rakov V A,et al.2010.Three-dimensional imaging of upward positive leaders in triggered lightning using VHF broadband digital interferometers[J].Geophys.Res.Lett.,37(5),doi:10.1029/2009GL042065.
张义军,刘欣生,肖庆复.1997.中国南北方雷暴及人工触发闪电特性对比分析[J].高原气象,16(2):113-121.Zhang Yijun,Lin Xinsheng,Xiao Qingfu.1997.A analysis of characteristics of thunderstorm and artificially triggered lightning in the north and south of China[J].Plateau Meteor.(in Chinese),16(2):113-121,doi:10.3321/j.issn:1000-0534.1997.02.001.
张义军,言穆弘,郄秀书,等.1998.人工引发雷电对雷暴特性可能影响的观测与分析[J].高原气象,17(4):412-419.Zhang Yijun,Yan Muhong,Qie Xiushu,et al.1998.Observation and analysis of influences of artificially triggered lightning on characteristics of thunderstorm[J].Plateau Meteor.(in Chinese),17(4):412-419,doi:10.3321/j.issn:1000-0534.1998.04.010.
张义军,张阳,郑栋,等.2016.2008-2014年广东人工触发闪电电流特征[J].高电压技术,42(11):3404-3414.Zhang Yijun,Zhang Yang,Zheng Dong,et al.2016.Current Characteristics of triggered lightnings in Guangdong from 2008 to 2014[J].High Volt.Eng.(in Chinese),42(11):3404-3414,doi:10.13336/j.1003-6520.hve.20160412005.
周恩伟.2010.触发闪电放电过程的光电同步观测与分析[D].中国科学技术大学硕士学位论文.Zhou Enwei.2010.Simultaneous optical and electrical observations and analysis of rocket-triggered lightning[D].M.S.thesis(in Chinese),University of Science and Technology of China.
周忠华,刘欣生.1993.人工触发闪电放电电流的间接测量及其特征分析[J].高原气象,12(1):67-76.Zhou Zhonghua,Lin Xinsheng.1993.Indirect measurement of artificially triggered lightning discharge current and its characteristic analysis[J].Plateau Meteor.(in Chinese),12(1):67-76.
祝宝友,陶善昌,马明,等.2004.用远场辐射场反演云闪放电参数[J].气象学报,62(4):498-503.Zhu Baoyou,Tao Shanchang,Ma Ming,et al.2004.Evaluation of in-cloud lightning discharge parameters from remote electromagnetic fields[J].Acta Meteor.Sinica(in Chinese),62(4):498-503,doi:10.11676/qxxb2004.051.