A simple physical model for self-triggered upward leaders from high-rise buildings

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1 214 International onference on Lightning Protection (ILP), Shanghai, hina A simple physical moel for self-triggere upwar leaers from high-rise builings Ming Kit han, Mingli hen*an Yaping Du Department of Builing Services Engineering The Hong Kong Polytechnic University Hong Kong, hina *mingli.chen@polyu.eu.hk Abstract This paper presents a simple physical moel for an upwar positive leaer initiate from a tall groune object. haracteristics such as the leaer channel structure, leaer initiation criteria, leae propagation spee, leaer current an channel longituinal electric fiel are escribe an moelle step-by-step. Other parameters incluing the evolution of leaer line charge ensity, leaer core conuctance an raius, an leaer corona sheath raius are also compute an evaluate. Particularly, the minimum corona sheath raius an corona charge ensity for leaer initiation an stable propagation base on existing experiment ata are propose. The moel may serve as a virtual lab to stuy the initiation of an upwar leaer at the top of a tall groun structure for tall builing lightning protection analysis. Keywors- lightning, upwar leaer, propagation moel; I. INTRODUTION Nowaays, many high-rise builings, such as the Dubai Tower, have been built in metropolises aroun the worl. When a thunerclou is growing above a tall builing, more an more charges accumulate in the clou an the ambient electric fiel strength below the charge clou increases. If the electric fiel at the top of the builing reaches the critical breakown value, air breakown will take place. Then, an upwar leaer is initiate an propagates towars the charge clou. As the upwar ischarge can amage high objects, such as win turbines [1,2], more researches have been rapily focuse on stuying the initiation of upwar lightning for protecting humans an animals [3-5]. Lightning physics moelling is important because most etection systems still cannot preict the initial lightning strike from a certain thunerclou. Moeling of the leaer evolution of a lightning flash can help us to evaluate the physical threshols for the phenomenon of lightning strikes to tall groune objects. Recently, Oscar Diaz et al. [6] have mae a general review of the existing engineering an physical moels for positive leaer channels an compare the moels simulation results with the experimental results of the long air gap electrical ischarge. However, they have assume the leaer propagation spee is nearly constant. This paper presents a simple physical moel for stuying the continuous propagation of a self-initiate upwar positive leaer. This approach is ifferent from other existing moels. We will also propose a metho to estimate the minimum corona raius an charge ensity for leaer initiation. II. UPWARD LEADER MODEL A thorough escription of the self-triggere positive upwar leaer moel, which inclues both physical an engineering moel features, is presente in this section. In this paper, we use the sign convection that a negative charge overhea prouces a negative electric fiel on the groun. A. Moel Details Although the thunerstorm clou charge has alreay been recognize as a typical triple structure, in the present moel, the charge clou is simplifie as a conuctive plane with a potential, φ clou (e.g. -6 MV), at a height, H (e.g. 3 m), resulting in a uniform clou electric fiel between the clou an the groun, E clou, (e.g. -2 kv/m). The electric fiel ue to the corona space charge near the groun, E corona, (e.g. -3 kv/m), can be etermine base on the observe space electric fiel profile versus height uner a thunerclou, which will be iscusse later. For an upwar leaer triggere by other ischarges such as a ownwar negative leaer, it will be stuie in a separate paper. The upwar positive leaer (UPL) is simply assume to propagate along a straight line without bening an it gets energy from an ambient electric fiel environment E A, which is the uniform clou electric fiel plus the corona electric fiel, i.e. E A = E clou + E corona.. The height epenent ambient fiel can be expresse as the following empirical equation [7] z / Lc A clou groun clou (1) E z E E E e where E clou is the clou charge prouce electric fiel, E groun is the electric fiel measure on groun, an L is the characteristic ecay length of the space corona charge. In our moel, a tri-layer structure of the leaer channel shown in Fig. 1 is introuce. While insie the black line is the leaer core (R L), the blue region is calle the transition region (R T). The thir layer is the corona sheath zone (R ). The leaer core is full of quasi-neutral plasmas, positive an negative particles can move freely in the core an the lateral electric fiel, E L is nearly zero. However, insie the surrouning corona sheath zone, the electric fiel is equal to the breakown electric fiel, E. Outsie the corona zone, the electric fiel reuces to the ambient electric fiel, E A. To insert a thin non-quasi-neutral transitional layer between the core an the corona zone, the

2 lateral electric fiel can increase from nearly zero to the breakown electric fiel continuously. Accoring to hen et al. [8], the critical breakown space electric fiel is proportional to the air ensity. This means it is a function of height above sea level: z / H, (2) E z E e where E equals 5 kv/m for the positive polarity is the critical breakown value at groun level, z is the longituinal irection above groun, an H is a constant of 84 m. In aition to the leaer structure moel an the critical breakown fiel, the leaer charge istribution is calculate by the charge simulation metho (SM) [8] an the metho of mirror charges. The builing structure is simply assume to be a regular structure which can be equivalent to a groune thin metal ro. For a long channel, the breakown electric fiel is irecte raially from it, the corona raius of the leaer channel can be given by [9] R z L z, (3) 2 E z where λ L(z) is line charge ensity istribution along the leaer channel. To simulate the leaer propagation spee, we assume the riving force of the moving leaer tip is ue to the average of the corona sheath electric fiel, E, an the electric fiel insie the channel core, E L, that may either increase or ecrease when the leaer grows steaily. Besies this, insie the streamer-leaer transition region, the air heating is assume to be isobaric ( P = ) an the total mass of the tip is conserve ( M = ). Base on conservation of energy, the gain in kinetic energy of charge particles insie the leaer tip is equal to the ifference of the injecte electrical energy an the energy losses ue to heat, vibration an friction. Therefore, the spee of the leaer tip can be express as v Tip F E EL 2EDr l, (4) M where η represents the heat an vibrational energy lost, F is the Faraay constant equals /mol, M = i(w im i) is the effective molar mass among ions composition, an E Dr (n elnλ)/t e, is the Dreicer fiel [1], lnλ is the oulomb logarithm, an T e is the electron temperature. Note: Q/M = en/m = en A/M = F/M, where Q is the total injecte charges. Then, the leaer propagation spee equals vtip v, (5) L 1 / a where τ a an τ are the three-boy attachment time scale, an elay time for a new leaer segment to cross the stream-toleaer transition region, respectively. τ is also efine as the time neee to heat the channel up to 5 K. Figure 1. R The channel structure of an upwar positive leaer (not in scale) From [11, 12], the gas ynamic moel simulation result showe that τ ρ a -2, where ρ a is the ambient air ensity. In our moel, we assume. (6) e 2 z/ H Furthermore, we have applie the semi-empirical equation propose by M. Bazelyan an P. Raizer [13] to estimate the average leaer electric-fiel: EI E v b, (7) L L L L L B. Minimum initial channel length of a steay upwar leaer When the leaer is just triggere (t = ), E c >> E L an E Dr, an the leaer spee is given by lf E vl ( ) / 1 / a M Experimental results have showe the minimum starting leaer spee is nearly 1 kms -1 [15,16]. By using the parameter value shown in Table I, we can estimate the minimum initial leaer length at groun level as l =.1463 m R. Here, we have assume the minimum corona raius (R ) is equal to the minimum initial leaer length at groun level. RT RL

3 TABLE I. PARAMETERS FOR SIMULATION OF AN POSITIVE UPWARD LEADER. Symbol Table column subhea Value Unit M Effective molar mass 73 g/mol H Simulation height 3 m Hb Length of the builing height 15.5/33 m Eclou Electric fiel ue to the clou charge -2 kv/m E Breakown electric fiel at groun level 5 kv/m Te Electron temperature 2 K τa Three-boy attachment time scale at groun level.1 µs τ Delay time at groun level 1 µs η Heat an vibrational energy lost.175 Not applicable B. Space charge layer inclue In case B, the ambient fiel is no longer constant but changing with height. The minimum self-triggere height is estimate as H b = 33 m which is higher than case A. The evolutions of leaer channel line charge ensity, corona raius are presente in Figs. 9-1 respectively. Although the shape of the figures is very similar to case A, the amount of charge along the channel is relatively smaller As a result, the minimum initial leaer line charge ensity at groun level is equal to λ L = 2πε E R = µ/m. 15 The minimum height (H b) of a builing to trigger an upwar leaer uner E A is then estimate base on 1 ) Hb ( E z z / E R. A 5 III. SIMULATION RESULTS AND DISUSSION Base on the moels presente in the previous sections, simulations were one for a self-triggere upwar positive leaer on a builing with an without a space charge layer. Backgroun conitions: E clou = -2 kv /m, E groun = -3 kv/m, H = 3 km, L = 25 m. Step length: l =.3 m, t i = l/v Li A. onstant amibent fiel In case A, the ambient fiel remains constant an the minimum self-triggere height is estimate as H b = 15.5 m. The evolutions of leaer channel line charge ensity, corona raius, propagation spee, current, electric fiel, leaer conuctance an core raius with time an height are presente in Figs. 2-8 respectively. As shown in Fig. 2, the charge along the channel is increasing. This is because the leaer electric fiel is ecreasing with time an the potential ifference between the leaer channel an the environment keeps increasing. Simulation results also show the leaer propagation spee reaches its maximum value at a certain tip height, an then starts to ecrease Line charge ensity (/m ) 1-4 Figure 2. Line charge ensity evolution of the leaer for ase A Leaer sheath raius (m) Figure 3. orona sheath raius evolution of the leaer for ase A

4 Tip Leaer spee (m/s) Time (s) Leaer spee (m/s) 1 5 Figure 4. Leaer spee evolution for ase A onuctance per unit length (S/m) Time (s) 1-3 Tip onuctance per unit length (S/m) Figure 7. Leaer conuctance per unit evolution for ase A Leaer current (A) Tip Leaer core raius (mm) 2 4 Time (s) Leaer current (A) Figure 8. Leaer core raius evolution for ase A Leaer Fiel (V/m) Figure 5. Leaer current evolution for ase A Tip Time (s) Leaer Fiel (V/m) Line charge ensity (/m ) 1-4 Figure 6. Leaer electric fiel evolution for ase A Figure 9. Line charge ensity evolution of the leaer for ase B

5 Leaer sheath raius (m) Figure 1. orona sheath raius evolution of the leaer for ase B IV. ONLUSION A simple physical moel for a self-initiate upwar leaer from tall groune object was presente an moel etails were thoroughly iscusse. Two ifferent cases: a clouprouce ambient electric fiel of -2 kv/m with an without a corona space charge layer near the groun were simulate respectively. Some physical parameters such as the evolution of leaer spee, leaer charge an current, leaer conuctance an leaer core, an corona sheath raius were presente as well. Simulation results are reasonable compare to observations in literature. AKNOWLEDGMENT Works leaing to this paper is supporte by the Research ommittee of The Hong Kong Polytechnic University an the Research Grant ouncil of Hong Kong Government (Grant No.: PolyU 5123/12E). REFERENES [1] J. Montanyà, O. van er Vele, an E. R. Williams, Lightning ischarges prouce by win turbines, J. Geophys. Res. Atmos., vol. 119, pp , 214. [2] D. Wang an N. Takagi, haracteristics of Winter Lightning that Occurre on a Winmill an its Lightning Protection Tower in Japan, IEEJ Trans. Power Energy, vol. 132(6), pp , 212 [3] A. Smorgonskiy, A. Tajalli, F. Rachii, M. Rubinstein, G. Dienorfer, H. Pichler, An analysis of the initiation of upwar flashes from tall towers with particular reference to Gaisberg an Säntis Towers, J. Atmos. Sol. Terr. Phys., vol. 136, pp , 215. [4] R. Jiang, X. Qie, Z. Wu, D. Wang, M Liu, G. Lu, D. Liu, haracteristics of upwar lightning from a 325-m-tall meteorology tower, Atmos. Res., vol. 149, pp , 214. [5] M. Guimarães, L. Araujo,. Pereira,. Mesquita, S. Visacro, Assessing currents of upwar lightning measure in tropical regions, Atmos. Res., vol. 149, pp , 214. [6] O. Diaz, L. Arevalo, V. ooray, Leaer channel moels for long air positive electrical ischarges, J. Electrost., vol. 76, pp , 215. [7]. J. Biagi, M. A Uman, J. Gopalakrishnan, J. D. Hill, V. A. Rakov, T. Ngin, an D. M. Joran, Determination of the electric fiel intensity an space charge ensity versus height prior to triggere lightning, J. Geophys. Res., vol. 116, D1521, 211. [8] M. hen, X. Gou, Y. Du, The effect of groun altitue on lightning striking istance base on a bi-irectional leaer moel, Atmos. Res., vol.125, pp , 213. [9] Y. Xu an M. hen, A 3D self-organize leaer propagation moel an its engineering approximation for lightning protection analysis, IEEE Trans. Power Del., vol. 28, 4, pp , 213. [1] T. Fülöp an M. Lanreman, Ion Runaway in Lightning Discharges, Phys. Rev. Lett., vol. 111, 156, 213. [11] Riousset et al., Air ensity epenent moel for analysis of air heating associate with streamers, leaers, an transient luminous events, J. Geophys. Res., vol.115, A12321, 21. [12]. L. a Silva an V. P. Pasko, Dynamics of streamer-to-leaer transition at reuce air ensities an its implications for propagation of lightning leaers an gigantic jets, J. Geophys. Res.-Atmos., vol. 118, pp. 13,561 13,59, 213. [13] M. Bazelyan an P. Raizer, Lightning Physics an Lightning Protection, Bristol: Institute of Physics Publishing, 2. [14] M. hen, S. ai an Y. Du, Dynamic moelling of lightning return stroke an its optical an electromagnetic raiations base on Maxwell's integral-equations, th Asia-Pacific onference on Environmental Electromagnetics, pp , 215. [15] E. M. Bazelyan, Yu. P. Raizer, N. L. Aleksanrov, The effect of reuce air ensity on streamer-to-leaer transition an on properties of long positive leaer, J. Phys. D: Appl. Phys., vol. 4, pp , 27. [16] X. Wang, J. He, Z. Yu, R. Zeng an F. Rachii, Influence of groun wire on the initiation of upwar leaer from11 to 1 kv A phase line, Electr. Power Syst. Res., vol. 13, pp13-112, 216.