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The origin of ball and bead lightning from an expanded lightning channel
Journal of Atmospheric and Solar–Terrestrial Physics 195 (2019) 105116
Contents lists available at ScienceDirect
Journal of Atmospheric and Solar-Terrestrial Physics journal homepage: www.elsevier.com/locate/jastp
Research Paper
The origin of ball and bead lightning from an expanded lightning channel R. Morrow School of Physics, The University of Sydney, New South, Wales, 2006, Australia
A B S T R A C T
It is shown how ball and bead lightning can originate from a narrow lightning channel which has been expanded by an ionizing wave. The resulting plasma structure development is followed for 10 s, and it is shown that the electric field developed can act as an electrostatic trap for negative burning particles. Previously it has been shown that a secondary lightning stroke current travelling down the expanded channel is necessary to explain the strength of the channel luminosity. This same secondary current can constrict the channel irregularly, via the magnetic pinch effect, to produce elongated segments of the channel; these become ball and bead lightning. All the plasma balls/beads will be “phantom plasma” structures unless some negatively-charged burning particles (fuel such as soil, copper wire or smoke) are trapped in the electric field distribution providing illumination.
1. Introduction Ball and bead lightning are considered to be very closely related, and certainly both phenomena are generated during a lightning strike (Uman, 1969; Rakov and Uman, 2003). However, it is not at all clear how such a stable extended plasma ball (Morrow, 2017, 2018) of diameter 0.1–3 m can originate from a filamentary discharge channel with an observed diameter of several mm. Further, it has not been possible to produce ball lightning in the laboratory in any convincing manner. There must be some feature of the lightning discharge that makes the production of extended plasma balls possible. It is suggested that the origin of ball and bead lightning lies in the expansion of lightning channels by ionizing waves, as studied in detail recently by Morrow (2019). In that modelling study a narrow lightning channel was expanded to a diameter of ~3 m in ~3 μs, and further development of the channel was followed for 1 ms. The structure pro duced resembles a cylindrical form of ball lightning (Morrow, 2017, 2018). Note that it was shown by Morrow (2019) that the diameter obtained for the lightning channel, and therefore ball and bead light ning, depended on the value of the net positive charge in the channel; a range of radii from 0.2 to 1.5 m is shown in Fig. 4 of Morrow (2019). Note that it is important to read the current paper in conjunction with the recent paper by Morrow (2019) as the current paper presents the results of the continuation of the calculations presented in (Morrow, 2019) and consequently the two papers are closely related. In this study the calculations of the previous paper (Morrow, 2019) are continued from 1 ms to 10 s, with modified equations. Previously (Morrow, 2019) after 1 ms the electron densities were three orders of magnitude less than the ion densities; thus, the equations used here are
those for the positive and negative ions only, neglecting the effects of electrons and oxygen metastable molecules. The equations are solved in cylindrical coordinates using only the radial terms, assuming no varia tions in the other coordinate directions. The results for the ion densities from the previous calculations (Morrow, 2019) after 1 ms are used as a starting point for the calculations presented here. The history and theories of ball and bead lightning have been out lined in detail previously (Morrow, 2017, 2018); however, it is worth noting that bead lightning has been observed during the artificial in duction of lightning (Uman, 1969; Rakov and Uman, 2003) and can have the form of irregular elongated luminous structures along the lightning channel (Matthias and Buchsbaum, 1962; Barry, 1980), or as a series of spheres (Barry, 1980). The author has actually seen bead lightning once and it was in the form of a regular series of luminous spheres that lasted about 1 s. Most of the features of the theory for the origin of ball and bead lightning developed in this paper are embodied in the photograph of Matthias and Buchsbaum (1962) presented in Fig. 1. The details are presented in the General Discussion, section 4. 2. Theory The equations solved simultaneously with Poisson’s equation are the continuity equations for positive ions and for negative ions, neglecting the effects of electrons and metastable oxygen molecules. Effects included are ion recombination and the drift and diffusion of the ions. The equations are solved using radial coordinates in cylindrical geom etry assuming no variations in the other coordinate directions. The de tails of the equations in cylindrical geometry are detailed below.
E-mail address: [email protected]. https://doi.org/10.1016/j.jastp.2019.105116 Received 16 April 2019; Received in revised form 10 September 2019; Accepted 11 September 2019 Available online 16 September 2019 1364-6826/© 2019 Published by Elsevier Ltd.
R. Morrow
Journal of Atmospheric and Solar-Terrestrial Physics 195 (2019) 105116
which is of the order of 100 times larger than that possible when elec trons are involved, allowing much longer calculations. The ion density at late times in a relatively uniform plasma region which is dominated by recombination was found by Morrow (2018) to be given by: Np ðtÞ ¼
1 tβ
(9)
2.2. Numerical methods The details of the numerical methods used are the same as described in detail by Morrow, 2017, 2018. The solution of Poisson’s equation [equation (5)] is the same as described previously (Morrow, 2017, 2018) with a zero-gradient boundary condition at the centre; at the outer boundary equation (6) defines the boundary condition; the length of the lightning channel chosen is L ¼ 500 m, and the outer boundary is at R ¼ 100 m. For this case involving only the ions, and no rapidly changing structures, the mesh need not be moved. Also, because the electric field decreases rapidly, a generous number of mesh points can be used without increasing the computational time too much. The region from the centre of the plasma out to a radius of 1.5 m (the plasma boundary) is spanned using 750 mesh points with a spacing of 2 � 10 3 m. The mesh then expands exponentially using 250 mesh points from the edge of the plasma region to the boundary at R ¼ 100 m. The initial positive and negative ion densities were derived from the results published previously (Morrow, 2019) by linearly interpolating the values from the old mesh onto the new mesh used in this paper. The residual electron densities were added to the corresponding negative ion densities in order to ensure the same net charge is used.
Fig. 1. Photograph of “Pinched Lightning” from Matthias and Buchsbaum (Matthias and Buchsbaum, 1962), reproduced here with permission.
2.1. Basic equations
∂Np ¼ ∂t
O2 Np β
∂O2 ¼ ∂t
O2 Np β
� � � 1 ∂ rNp Wp 1 ∂ ∂Np þ rDp r r ∂r ∂r ∂r
(1)
� � � 1 ∂ rO2 Wn 1 ∂ ∂O þ rDn 2 r r ∂r ∂r ∂r
(2)
where t is time, and r is the radial position, Np andO2 are the number densities of positive ions and negative ions respectively. The gas medium is air at atmospheric pressure; the effects of positive ion drift, Wp , negative ion drift, Wn, and ion neutralization coefficient, β, are included. Dp is the positive ion diffusion coefficient and Dn the negative ion diffusion coefficient; these coefficients are evaluated as explained in (Morrow, 2017, 2018). The continuity equations above are solved simultaneously with Poisson’s equation:
ρ ε0
r2 φ ¼
3. Results The previous calculations for lightning channel expansion (Morrow, 2019) were stopped after 1 ms when the ionizing wave had stopped propagating at R ~ 1.5 m; the region within this radius is called “the plasma region” for the calculations of this paper. The ion density dis tributions from the 1 ms point, with peak number densities of Np ¼ Nn ¼ 5.0 � 1015 m 3, served as a starting point for the current calcula tions. Note that at this time, 1 ms, the electron density in (Morrow, 2019) is Ne ¼ 7.0 � 1012 m 3, and therefore electrons are not included in the calculation. The calculations presented here continue to follow the lightning channel development for a further 10 s in order to determine if the conditions for the formation of ball and bead lightning are produced.
(3)
where φ is the potential distribution, ε0 is the permittivity of free space, and e is the modulus of the electron charge. The space-charge density is given by � (4) ρ ¼ e Np O2 In cylindrical coordinates, with zero angular and axial variations, Poisson’s equation becomes � � 1 ∂ ∂φ ρ : (5) r ¼ ε0 r ∂r ∂r
3.1. Ion densities in the central plasma region The development of ion densities from the centre out to a radius of 3 m are shown in Fig. 2 for times of 1, 10, 100 ms, 1 s and 10 s. The positive and negative ion number densities are equal out to a radius of R ¼ 1.5 m at 1 ms, constituting a plasma region, and have a very flat distribution. The excess positive ions move beyond a radius of 1.5 m to about 2.3 m due to mutual repulsion. As the calculation proceeds the ion distributions in the central region remain equal and flat, and their number densities decrease as described by Equation (9). Outside the plasma region (R > 1.5 m) the positive ion number density also de creases due to ions moving away from the channel, and the distribution becomes flat; both these changes are due to positive ion mutual repulsion.
The boundary condition for the potential, φR ; at the outer boundary at r ¼ R is defined(Potential of a Uniformly Charged Rod) as that due to a line of charge of length L by � � ρ L φR � : (6) ln R 2πε0 The electric field, E, is calculated from the gradient of the potential E¼
∂φ ∂r
(7)
When the continuity equations involving only ions are solved, then the time-step limitation for a mesh spacing of Δr becomes: Δt �
Δr ; Wp
3.2. Positive ion densities out to 50 m The distribution of positive ions for times 1 ms to 1 s out to a radius of 50 m is shown in Fig. 3. The positive ion density decreases with time as the same number of ions are distributed over a larger volume as the ions
(8)
2
R. Morrow
Journal of Atmospheric and Solar-Terrestrial Physics 195 (2019) 105116
Fig. 2. The development over time of the number density distributions of positive ions (solid red lines) and negative ions (dashed blue lines) near the channel centre for times shown on the figure.
Fig. 4. The corresponding electric field distributions for the data of Figs. 2 and 3 for times shown on the figure.
3.4. Electric field distributions from 1 to 10 s
move away from the plasma region due to mutual repulsion.
Ball and bead lightning typically last for 1–10 s (Uman, 1969; Rakov and Uman, 2003). Thus, the electric field distribution around the lightning channel is studied for such times, and results for 1, 2 and 10 s are shown in Fig. 5 a for the distribution from R ¼ þ 20 m to 20 m. The electric field up to 1000 Vm-1 on a radial range of R ¼ þ5 m to – 5 m is shown in Fig. 5 b. The electric field distribution shown in Fig. 5 b closely resembles those recently found by similar modelling using spherical geometry for ball lightning (Morrow, 2018), where it was found that an electric field of 100–200 Vm-1 was sufficient to entrap burning negatively charged particles to provide the illumination for ball lightning. Thus, after 1 s the expanded lightning channel has the same structure as ball lightning, but in cylindrical form. All that is required is some means of cutting the channel into short segments that become ball and bead lightning, as outlined in the General Discussion below.
3.3. The electric field distribution The flat positive ion distribution leads to a linearly rising electric field from the edge of the plasma, at R ¼ 1.5 m, out to the edge of the positive ion distribution, as shown in Fig. 4. The plasma conditions in the central region, where the positive and negative ion densities are equal, leads to the electric field being effectively zero from the centre to R ¼ 1.5 m (Fig. 4.). Note that the peak in the electric field is always below E*, the electric field where ionization equals attachment (E* ¼ 2.7 MVm 1 (Morrow, 2018)), so there is no net ionization and no ionizing wave is involved.
4. General Discussion The expansion of a lightning channel via ionizing waves creates the necessary large-dimension plasma structure with attendant external positive ion distribution required to create ball lightning (Morrow, 2017, 2018). The structure described so far is cylindrical (Morrow, 2019) and some means must be found by which the plasma channel is segmented into balls and elongated structures to form ball and bead lightning. Note also, as stated earlier, that channels of different diameter are generated by starting with different values for the net positive charge in the initial channel (Morrow, 2019); thus, different diameters of ball and bead lightning are produced. It was found in the study of lightning channel expansion via ionizing waves (Morrow, 2019) that the light output from the wave could not explain the brightness of the channel. It was necessary to postulate that secondary lightning strokes down the broad channel caused the illumi nation of the channel. It is suggested here that this secondary stroke current causes “sausage instability” via the pinch effect (Uman, 1969; Barry, 1980; Vikhrev et al., 1993) and effectively cuts the lightning channel into plasma balls and irregular elongated segments. Some form of standing plasma wave may in some cases cause the segmentation to be regular, producing a regular string of plasma balls as observed by the
Fig. 3. The development over time of the positive ion number density distri butions (solid red lines) with radius from R ¼ 050 m for times shown on the figure. 3
R. Morrow
Journal of Atmospheric and Solar-Terrestrial Physics 195 (2019) 105116
lightning the fuel necessary to produce ~1 W of light output would be provided near the Earth by burning particles from the soil (Morrow, 2018). The evidence for ball lightning being fuelled from the soil is as fol lows. The spectra of various metal ions from the soil were measured during the ball lightning observations of Cen et al. (Cen et al., 2014). The generation of metal ions during a lightning strike has been explained by Abrahamson and Dinniss (2000) and Abrahamson (2002) as being due to the reduction of metal oxides by carbon from carbon material in the soil (as with the refining of silicon metal). These metals can then oxidise to produce the light output of ball lightning and the observed spectra. An ingenious method of getting the soil particles into the region of the plasma ball is described by Abahamson and Denniss (Abrahamson and Dinniss, 2000); the lightning strike creates a channel in the soil called a fulgurite cavity which is full of hot gas, reacting oxides and carbon particles during the lightning current flow; as soon as the current ceases a jet of hot gas and soil particle rises into the air. Abrahamson and Denniss regard this as a method of generating ball lightning as a vortex swirl; there is even an eye-witness account from Russia reported by Abrahamson et al. (Abrahamson et al., 2002) where a small ball light ning object the size of an orange originated from the place where a lightning strike left a small hole. The idea of a jet of soil particles from a fulgurite cavity is an ideal method of injecting soil particles into the plasma ball electrostatic trap, injecting the necessary fuel. The plasma regions/balls in the upper part of the lightning channel will not have any fuel and will be phantom plasma balls. The nanoparticle networks suggested by Abrahamson and Dinniss (2000) can easily be incorporated into the current model of ball light ning; however, they are probably too delicate to form the structure of ball lightning, as discussed by Morrow (2018). Such a nanoparticle structure cannot be used to describe bead lightning. For bead lightning to be produced some kind of fuel must be intro duced and ignited in the segmented plasma regions high in the lightning channel. In the case of rocket-induced lightning, bead lightning is known to occur (Uman, 1969; Rakov and Uman, 2003); the fuel is provided by the copper wire exploding, which burns providing the observed light output. In other cases, the fuel may be provided by dense smoke which can contain carbon particles and other organic materials that could be ignited. However, such fuel will in general be less dense than the ma terial from soil so the lifetime of the illuminated regions will be shorter. This was the case for the bead lightning observed by the author; the light output from a string of balls lasted ~1 s. Note that the famous drawing of ball lightning appearing down the chimney of a farmhouse [7, Figure 1.2] could have been fuelled by the smoke from the fireplace. Lowke et al. (Lowke et al., 2012) have made an extensive study of how ball lightning could be produced inside an aeroplane. They calcu lated that a plasma ball would be produced inside the front glass of the plane due to charges impinging on the outer surface of the glass. It is difficult to see how the ball would not be a phantom plasma ball (Morrow, 2018), unless the pilots were smoking and the smoke, or dust, provided the fuel. An alternative scheme proposed by Abrahamson [12, p78] may overcome this problem of the source of fuel. Abrahamson suggests that a lightning strike on the fuselage of the aeroplane runs along the surface and ruptures the rubber seal around a door. Thus, a vapour of carbon and metal will be generated which can enter the cabin, and expand into a ring vortex, and form ball lighting. From the discussion above, it is clear that most attempts to produce ball lightning in the laboratory are doomed to failure because they do not produce the expanded plasma generated by an ionizing wave from a narrow high-density column (Morrow, 2019). One way to achieve this would be to use an impulse generator to produce a narrow highly ionized channel, and to use a “trigatron” (Lucas, 2001) to short the current abruptly to zero so as to produce a “radial ionizing wave” as described in detail by Morrow (2019). If an expanded plasma channel is produced then a copper wire, smoke or suspended silicone dust could be
Fig. 5. a). The electric field distributions around the lightning channel at times 1, 2 and 10 s. b). The electric field distributions around the lightning channel at times 1, 2 and 10 s on a finer scale.
author. Most of the points above are demonstrated in the results of Matthias and Buchsbaum (1962) in their photograph presented in Fig. 1. They estimate that the channel is 1–5 m in diameter. They state that the channel is much brighter than a normal lightning channel; this is un doubtedly due to a large current flowing down a broad lightning channel. The evidence for the large current, besides the brightness, is the magnetic pinching of the channel into segments. This pinching is the means by which ball and bead lightning are created. If no fuel is present to produce illumination via burning negative particles (Morrow, 2018) then all the plasma segments will be “phantom plasma balls” and go unnoticed (Morrow, 2017). In the case of ball 4
R. Morrow
Journal of Atmospheric and Solar-Terrestrial Physics 195 (2019) 105116
used to inject fuel into the plasma ball to produce illumination and hence ball lightning.
continued support with the provision of an office, computer and library facilities. The author also thanks Dr. John Lowke for advice on publi cation. The author thanks Dr. Vivienne M. Bowers Morrow for editing the manuscript. I also thank Zorba for his constant support and encouragement to take regular exercise.
5. Conclusions Ball and bead lightning both originate from a lightning channel that has expanded into a broad plasma channel by a radial ionizing wave. A secondary lightning stroke down the broad plasma channel causes it to break up via the pinch effect into a series of plasma regions or balls along its length. Due to escaping excess positive ions, space-charge fields are produced external to each plasma region; this electric field rises with distance from the plasma region creating an electrostatic trap for negatively-charged burning particles, as was found for ball lightning. Bead lightning is produced when fuel, such as an exploding copper wire or smoke, is ignited and charged by a lightning stroke. The nega tively charged burning particles are trapped by the positive ion spacecharge field and provide illumination for many plasma balls along the lightning channel. Because the fuel is rather unsubstantial the lifetime of bead lightning will be short. Ball lightning is generally produced when particles from the soil are charged and ignited with the negative particles trapped inside a plasma ball to produce illumination. Thus, only one or two plasma balls near the Earth can be fuelled in this way and the plasma balls higher up along the lightning channel will go unnoticed as phantom plasma balls. The fuel from the soil will be more substantial and burn for longer (>1 s) for the case of ball lightning, compared with smoke or exploded copper wire for bead lightning (<1 s).
References Abrahamson, J., 2002. Phil. Trans. R. Soc. Lond. A 360, 61–88. https://doi.org/10.1098/ rsta.2001.0919. Abrahamson, J., Dinniss, J., 2000. Nature 403, 519–521. https://doi.org/10.1038/ 35000525. Abrahamson, J., Bychkov, A.V., Bychkov, V.L., 2002. Phil. Trans. R. Soc. Lond. A 360, 11–35. https://doi.org/10.1098/rsta.2001.0917. Barry, D.A., 1980. Ball Lightning and Bead Lightning: Extreme Forms of Atmospheric Electricity. Plenum Press, New York and London. Cen, J., Yuan, P., Xue, S., 2014. Phys. Rev. Lett. 112 (035001), 1–5. https://doi.org/ 10.1103/PhysRevLett.112.035001. DOI: 10.1029/2012JD017921. Lowke, J.J., Smith, D., Nelson, K.E., Crompton, R.W., Murphy, A.B., 2012. J. Geophys. Res. Atmos. 117, 1–14. Lucas, J.R., 2001. High Voltage Engineering SCRIBD (Sri Lanka). Matthias, B.T., Buchsbaum, S.J., 1962. Nature 194, 327. https://doi-org.ezproxy1. library.usyd.edu.au/10.1038/194327a0. Morrow, R., 2017. J. Phys. D Appl. Phys. 50, 395201. http://doi.org/10.1088/1361-64 63/aa8398. Morrow, R., 2018. J. Phys. D Appl. Phys. 51, 125205. http://doi.org/10.1088/1361-64 63/aaaf8d. Morrow, R., 2019. J. Atmos. Sol. Terr. Phys. 189, 18–26. https://doi.org/10.1016/j.jastp .2019.04.003. Potential of a Uniformly Charged Rod. http://web.mit.edu/viz/EM/visualizations/cou rsenotes/modules/guide03.pdf. Rakov, V.A., Uman, M.A., 2003. Lightning: Physics and Effects. Cambridge University Press, Cambridge, UK, ISBN 0 521 58327 6. Uman, M.A., 1969. Lightning. McGraw Hill, New York. Library of Congress Catalog Card Number 68-8036-65745. Vikhrev, V.V., Ivanov, V.V., Rozanova, G.A., 1993. Nucl. Fusion 33, 311. https://doi. org/10.1088/0029-5515/33/2/10.
Acknowledgements The author is particularly indebted to Professor David R. McKenzie and the Physics Department of The University of Sydney for their
5
Contents lists available at ScienceDirect
Journal of Atmospheric and Solar-Terrestrial Physics journal homepage: www.elsevier.com/locate/jastp
Research Paper
The origin of ball and bead lightning from an expanded lightning channel R. Morrow School of Physics, The University of Sydney, New South, Wales, 2006, Australia
A B S T R A C T
It is shown how ball and bead lightning can originate from a narrow lightning channel which has been expanded by an ionizing wave. The resulting plasma structure development is followed for 10 s, and it is shown that the electric field developed can act as an electrostatic trap for negative burning particles. Previously it has been shown that a secondary lightning stroke current travelling down the expanded channel is necessary to explain the strength of the channel luminosity. This same secondary current can constrict the channel irregularly, via the magnetic pinch effect, to produce elongated segments of the channel; these become ball and bead lightning. All the plasma balls/beads will be “phantom plasma” structures unless some negatively-charged burning particles (fuel such as soil, copper wire or smoke) are trapped in the electric field distribution providing illumination.
1. Introduction Ball and bead lightning are considered to be very closely related, and certainly both phenomena are generated during a lightning strike (Uman, 1969; Rakov and Uman, 2003). However, it is not at all clear how such a stable extended plasma ball (Morrow, 2017, 2018) of diameter 0.1–3 m can originate from a filamentary discharge channel with an observed diameter of several mm. Further, it has not been possible to produce ball lightning in the laboratory in any convincing manner. There must be some feature of the lightning discharge that makes the production of extended plasma balls possible. It is suggested that the origin of ball and bead lightning lies in the expansion of lightning channels by ionizing waves, as studied in detail recently by Morrow (2019). In that modelling study a narrow lightning channel was expanded to a diameter of ~3 m in ~3 μs, and further development of the channel was followed for 1 ms. The structure pro duced resembles a cylindrical form of ball lightning (Morrow, 2017, 2018). Note that it was shown by Morrow (2019) that the diameter obtained for the lightning channel, and therefore ball and bead light ning, depended on the value of the net positive charge in the channel; a range of radii from 0.2 to 1.5 m is shown in Fig. 4 of Morrow (2019). Note that it is important to read the current paper in conjunction with the recent paper by Morrow (2019) as the current paper presents the results of the continuation of the calculations presented in (Morrow, 2019) and consequently the two papers are closely related. In this study the calculations of the previous paper (Morrow, 2019) are continued from 1 ms to 10 s, with modified equations. Previously (Morrow, 2019) after 1 ms the electron densities were three orders of magnitude less than the ion densities; thus, the equations used here are
those for the positive and negative ions only, neglecting the effects of electrons and oxygen metastable molecules. The equations are solved in cylindrical coordinates using only the radial terms, assuming no varia tions in the other coordinate directions. The results for the ion densities from the previous calculations (Morrow, 2019) after 1 ms are used as a starting point for the calculations presented here. The history and theories of ball and bead lightning have been out lined in detail previously (Morrow, 2017, 2018); however, it is worth noting that bead lightning has been observed during the artificial in duction of lightning (Uman, 1969; Rakov and Uman, 2003) and can have the form of irregular elongated luminous structures along the lightning channel (Matthias and Buchsbaum, 1962; Barry, 1980), or as a series of spheres (Barry, 1980). The author has actually seen bead lightning once and it was in the form of a regular series of luminous spheres that lasted about 1 s. Most of the features of the theory for the origin of ball and bead lightning developed in this paper are embodied in the photograph of Matthias and Buchsbaum (1962) presented in Fig. 1. The details are presented in the General Discussion, section 4. 2. Theory The equations solved simultaneously with Poisson’s equation are the continuity equations for positive ions and for negative ions, neglecting the effects of electrons and metastable oxygen molecules. Effects included are ion recombination and the drift and diffusion of the ions. The equations are solved using radial coordinates in cylindrical geom etry assuming no variations in the other coordinate directions. The de tails of the equations in cylindrical geometry are detailed below.
E-mail address: [email protected]. https://doi.org/10.1016/j.jastp.2019.105116 Received 16 April 2019; Received in revised form 10 September 2019; Accepted 11 September 2019 Available online 16 September 2019 1364-6826/© 2019 Published by Elsevier Ltd.
R. Morrow
Journal of Atmospheric and Solar-Terrestrial Physics 195 (2019) 105116
which is of the order of 100 times larger than that possible when elec trons are involved, allowing much longer calculations. The ion density at late times in a relatively uniform plasma region which is dominated by recombination was found by Morrow (2018) to be given by: Np ðtÞ ¼
1 tβ
(9)
2.2. Numerical methods The details of the numerical methods used are the same as described in detail by Morrow, 2017, 2018. The solution of Poisson’s equation [equation (5)] is the same as described previously (Morrow, 2017, 2018) with a zero-gradient boundary condition at the centre; at the outer boundary equation (6) defines the boundary condition; the length of the lightning channel chosen is L ¼ 500 m, and the outer boundary is at R ¼ 100 m. For this case involving only the ions, and no rapidly changing structures, the mesh need not be moved. Also, because the electric field decreases rapidly, a generous number of mesh points can be used without increasing the computational time too much. The region from the centre of the plasma out to a radius of 1.5 m (the plasma boundary) is spanned using 750 mesh points with a spacing of 2 � 10 3 m. The mesh then expands exponentially using 250 mesh points from the edge of the plasma region to the boundary at R ¼ 100 m. The initial positive and negative ion densities were derived from the results published previously (Morrow, 2019) by linearly interpolating the values from the old mesh onto the new mesh used in this paper. The residual electron densities were added to the corresponding negative ion densities in order to ensure the same net charge is used.
Fig. 1. Photograph of “Pinched Lightning” from Matthias and Buchsbaum (Matthias and Buchsbaum, 1962), reproduced here with permission.
2.1. Basic equations
∂Np ¼ ∂t
O2 Np β
∂O2 ¼ ∂t
O2 Np β
� � � 1 ∂ rNp Wp 1 ∂ ∂Np þ rDp r r ∂r ∂r ∂r
(1)
� � � 1 ∂ rO2 Wn 1 ∂ ∂O þ rDn 2 r r ∂r ∂r ∂r
(2)
where t is time, and r is the radial position, Np andO2 are the number densities of positive ions and negative ions respectively. The gas medium is air at atmospheric pressure; the effects of positive ion drift, Wp , negative ion drift, Wn, and ion neutralization coefficient, β, are included. Dp is the positive ion diffusion coefficient and Dn the negative ion diffusion coefficient; these coefficients are evaluated as explained in (Morrow, 2017, 2018). The continuity equations above are solved simultaneously with Poisson’s equation:
ρ ε0
r2 φ ¼
3. Results The previous calculations for lightning channel expansion (Morrow, 2019) were stopped after 1 ms when the ionizing wave had stopped propagating at R ~ 1.5 m; the region within this radius is called “the plasma region” for the calculations of this paper. The ion density dis tributions from the 1 ms point, with peak number densities of Np ¼ Nn ¼ 5.0 � 1015 m 3, served as a starting point for the current calcula tions. Note that at this time, 1 ms, the electron density in (Morrow, 2019) is Ne ¼ 7.0 � 1012 m 3, and therefore electrons are not included in the calculation. The calculations presented here continue to follow the lightning channel development for a further 10 s in order to determine if the conditions for the formation of ball and bead lightning are produced.
(3)
where φ is the potential distribution, ε0 is the permittivity of free space, and e is the modulus of the electron charge. The space-charge density is given by � (4) ρ ¼ e Np O2 In cylindrical coordinates, with zero angular and axial variations, Poisson’s equation becomes � � 1 ∂ ∂φ ρ : (5) r ¼ ε0 r ∂r ∂r
3.1. Ion densities in the central plasma region The development of ion densities from the centre out to a radius of 3 m are shown in Fig. 2 for times of 1, 10, 100 ms, 1 s and 10 s. The positive and negative ion number densities are equal out to a radius of R ¼ 1.5 m at 1 ms, constituting a plasma region, and have a very flat distribution. The excess positive ions move beyond a radius of 1.5 m to about 2.3 m due to mutual repulsion. As the calculation proceeds the ion distributions in the central region remain equal and flat, and their number densities decrease as described by Equation (9). Outside the plasma region (R > 1.5 m) the positive ion number density also de creases due to ions moving away from the channel, and the distribution becomes flat; both these changes are due to positive ion mutual repulsion.
The boundary condition for the potential, φR ; at the outer boundary at r ¼ R is defined(Potential of a Uniformly Charged Rod) as that due to a line of charge of length L by � � ρ L φR � : (6) ln R 2πε0 The electric field, E, is calculated from the gradient of the potential E¼
∂φ ∂r
(7)
When the continuity equations involving only ions are solved, then the time-step limitation for a mesh spacing of Δr becomes: Δt �
Δr ; Wp
3.2. Positive ion densities out to 50 m The distribution of positive ions for times 1 ms to 1 s out to a radius of 50 m is shown in Fig. 3. The positive ion density decreases with time as the same number of ions are distributed over a larger volume as the ions
(8)
2
R. Morrow
Journal of Atmospheric and Solar-Terrestrial Physics 195 (2019) 105116
Fig. 2. The development over time of the number density distributions of positive ions (solid red lines) and negative ions (dashed blue lines) near the channel centre for times shown on the figure.
Fig. 4. The corresponding electric field distributions for the data of Figs. 2 and 3 for times shown on the figure.
3.4. Electric field distributions from 1 to 10 s
move away from the plasma region due to mutual repulsion.
Ball and bead lightning typically last for 1–10 s (Uman, 1969; Rakov and Uman, 2003). Thus, the electric field distribution around the lightning channel is studied for such times, and results for 1, 2 and 10 s are shown in Fig. 5 a for the distribution from R ¼ þ 20 m to 20 m. The electric field up to 1000 Vm-1 on a radial range of R ¼ þ5 m to – 5 m is shown in Fig. 5 b. The electric field distribution shown in Fig. 5 b closely resembles those recently found by similar modelling using spherical geometry for ball lightning (Morrow, 2018), where it was found that an electric field of 100–200 Vm-1 was sufficient to entrap burning negatively charged particles to provide the illumination for ball lightning. Thus, after 1 s the expanded lightning channel has the same structure as ball lightning, but in cylindrical form. All that is required is some means of cutting the channel into short segments that become ball and bead lightning, as outlined in the General Discussion below.
3.3. The electric field distribution The flat positive ion distribution leads to a linearly rising electric field from the edge of the plasma, at R ¼ 1.5 m, out to the edge of the positive ion distribution, as shown in Fig. 4. The plasma conditions in the central region, where the positive and negative ion densities are equal, leads to the electric field being effectively zero from the centre to R ¼ 1.5 m (Fig. 4.). Note that the peak in the electric field is always below E*, the electric field where ionization equals attachment (E* ¼ 2.7 MVm 1 (Morrow, 2018)), so there is no net ionization and no ionizing wave is involved.
4. General Discussion The expansion of a lightning channel via ionizing waves creates the necessary large-dimension plasma structure with attendant external positive ion distribution required to create ball lightning (Morrow, 2017, 2018). The structure described so far is cylindrical (Morrow, 2019) and some means must be found by which the plasma channel is segmented into balls and elongated structures to form ball and bead lightning. Note also, as stated earlier, that channels of different diameter are generated by starting with different values for the net positive charge in the initial channel (Morrow, 2019); thus, different diameters of ball and bead lightning are produced. It was found in the study of lightning channel expansion via ionizing waves (Morrow, 2019) that the light output from the wave could not explain the brightness of the channel. It was necessary to postulate that secondary lightning strokes down the broad channel caused the illumi nation of the channel. It is suggested here that this secondary stroke current causes “sausage instability” via the pinch effect (Uman, 1969; Barry, 1980; Vikhrev et al., 1993) and effectively cuts the lightning channel into plasma balls and irregular elongated segments. Some form of standing plasma wave may in some cases cause the segmentation to be regular, producing a regular string of plasma balls as observed by the
Fig. 3. The development over time of the positive ion number density distri butions (solid red lines) with radius from R ¼ 050 m for times shown on the figure. 3
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Journal of Atmospheric and Solar-Terrestrial Physics 195 (2019) 105116
lightning the fuel necessary to produce ~1 W of light output would be provided near the Earth by burning particles from the soil (Morrow, 2018). The evidence for ball lightning being fuelled from the soil is as fol lows. The spectra of various metal ions from the soil were measured during the ball lightning observations of Cen et al. (Cen et al., 2014). The generation of metal ions during a lightning strike has been explained by Abrahamson and Dinniss (2000) and Abrahamson (2002) as being due to the reduction of metal oxides by carbon from carbon material in the soil (as with the refining of silicon metal). These metals can then oxidise to produce the light output of ball lightning and the observed spectra. An ingenious method of getting the soil particles into the region of the plasma ball is described by Abahamson and Denniss (Abrahamson and Dinniss, 2000); the lightning strike creates a channel in the soil called a fulgurite cavity which is full of hot gas, reacting oxides and carbon particles during the lightning current flow; as soon as the current ceases a jet of hot gas and soil particle rises into the air. Abrahamson and Denniss regard this as a method of generating ball lightning as a vortex swirl; there is even an eye-witness account from Russia reported by Abrahamson et al. (Abrahamson et al., 2002) where a small ball light ning object the size of an orange originated from the place where a lightning strike left a small hole. The idea of a jet of soil particles from a fulgurite cavity is an ideal method of injecting soil particles into the plasma ball electrostatic trap, injecting the necessary fuel. The plasma regions/balls in the upper part of the lightning channel will not have any fuel and will be phantom plasma balls. The nanoparticle networks suggested by Abrahamson and Dinniss (2000) can easily be incorporated into the current model of ball light ning; however, they are probably too delicate to form the structure of ball lightning, as discussed by Morrow (2018). Such a nanoparticle structure cannot be used to describe bead lightning. For bead lightning to be produced some kind of fuel must be intro duced and ignited in the segmented plasma regions high in the lightning channel. In the case of rocket-induced lightning, bead lightning is known to occur (Uman, 1969; Rakov and Uman, 2003); the fuel is provided by the copper wire exploding, which burns providing the observed light output. In other cases, the fuel may be provided by dense smoke which can contain carbon particles and other organic materials that could be ignited. However, such fuel will in general be less dense than the ma terial from soil so the lifetime of the illuminated regions will be shorter. This was the case for the bead lightning observed by the author; the light output from a string of balls lasted ~1 s. Note that the famous drawing of ball lightning appearing down the chimney of a farmhouse [7, Figure 1.2] could have been fuelled by the smoke from the fireplace. Lowke et al. (Lowke et al., 2012) have made an extensive study of how ball lightning could be produced inside an aeroplane. They calcu lated that a plasma ball would be produced inside the front glass of the plane due to charges impinging on the outer surface of the glass. It is difficult to see how the ball would not be a phantom plasma ball (Morrow, 2018), unless the pilots were smoking and the smoke, or dust, provided the fuel. An alternative scheme proposed by Abrahamson [12, p78] may overcome this problem of the source of fuel. Abrahamson suggests that a lightning strike on the fuselage of the aeroplane runs along the surface and ruptures the rubber seal around a door. Thus, a vapour of carbon and metal will be generated which can enter the cabin, and expand into a ring vortex, and form ball lighting. From the discussion above, it is clear that most attempts to produce ball lightning in the laboratory are doomed to failure because they do not produce the expanded plasma generated by an ionizing wave from a narrow high-density column (Morrow, 2019). One way to achieve this would be to use an impulse generator to produce a narrow highly ionized channel, and to use a “trigatron” (Lucas, 2001) to short the current abruptly to zero so as to produce a “radial ionizing wave” as described in detail by Morrow (2019). If an expanded plasma channel is produced then a copper wire, smoke or suspended silicone dust could be
Fig. 5. a). The electric field distributions around the lightning channel at times 1, 2 and 10 s. b). The electric field distributions around the lightning channel at times 1, 2 and 10 s on a finer scale.
author. Most of the points above are demonstrated in the results of Matthias and Buchsbaum (1962) in their photograph presented in Fig. 1. They estimate that the channel is 1–5 m in diameter. They state that the channel is much brighter than a normal lightning channel; this is un doubtedly due to a large current flowing down a broad lightning channel. The evidence for the large current, besides the brightness, is the magnetic pinching of the channel into segments. This pinching is the means by which ball and bead lightning are created. If no fuel is present to produce illumination via burning negative particles (Morrow, 2018) then all the plasma segments will be “phantom plasma balls” and go unnoticed (Morrow, 2017). In the case of ball 4
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Journal of Atmospheric and Solar-Terrestrial Physics 195 (2019) 105116
used to inject fuel into the plasma ball to produce illumination and hence ball lightning.
continued support with the provision of an office, computer and library facilities. The author also thanks Dr. John Lowke for advice on publi cation. The author thanks Dr. Vivienne M. Bowers Morrow for editing the manuscript. I also thank Zorba for his constant support and encouragement to take regular exercise.
5. Conclusions Ball and bead lightning both originate from a lightning channel that has expanded into a broad plasma channel by a radial ionizing wave. A secondary lightning stroke down the broad plasma channel causes it to break up via the pinch effect into a series of plasma regions or balls along its length. Due to escaping excess positive ions, space-charge fields are produced external to each plasma region; this electric field rises with distance from the plasma region creating an electrostatic trap for negatively-charged burning particles, as was found for ball lightning. Bead lightning is produced when fuel, such as an exploding copper wire or smoke, is ignited and charged by a lightning stroke. The nega tively charged burning particles are trapped by the positive ion spacecharge field and provide illumination for many plasma balls along the lightning channel. Because the fuel is rather unsubstantial the lifetime of bead lightning will be short. Ball lightning is generally produced when particles from the soil are charged and ignited with the negative particles trapped inside a plasma ball to produce illumination. Thus, only one or two plasma balls near the Earth can be fuelled in this way and the plasma balls higher up along the lightning channel will go unnoticed as phantom plasma balls. The fuel from the soil will be more substantial and burn for longer (>1 s) for the case of ball lightning, compared with smoke or exploded copper wire for bead lightning (<1 s).
References Abrahamson, J., 2002. Phil. Trans. R. Soc. Lond. A 360, 61–88. https://doi.org/10.1098/ rsta.2001.0919. Abrahamson, J., Dinniss, J., 2000. Nature 403, 519–521. https://doi.org/10.1038/ 35000525. Abrahamson, J., Bychkov, A.V., Bychkov, V.L., 2002. Phil. Trans. R. Soc. Lond. A 360, 11–35. https://doi.org/10.1098/rsta.2001.0917. Barry, D.A., 1980. Ball Lightning and Bead Lightning: Extreme Forms of Atmospheric Electricity. Plenum Press, New York and London. Cen, J., Yuan, P., Xue, S., 2014. Phys. Rev. Lett. 112 (035001), 1–5. https://doi.org/ 10.1103/PhysRevLett.112.035001. DOI: 10.1029/2012JD017921. Lowke, J.J., Smith, D., Nelson, K.E., Crompton, R.W., Murphy, A.B., 2012. J. Geophys. Res. Atmos. 117, 1–14. Lucas, J.R., 2001. High Voltage Engineering SCRIBD (Sri Lanka). Matthias, B.T., Buchsbaum, S.J., 1962. Nature 194, 327. https://doi-org.ezproxy1. library.usyd.edu.au/10.1038/194327a0. Morrow, R., 2017. J. Phys. D Appl. Phys. 50, 395201. http://doi.org/10.1088/1361-64 63/aa8398. Morrow, R., 2018. J. Phys. D Appl. Phys. 51, 125205. http://doi.org/10.1088/1361-64 63/aaaf8d. Morrow, R., 2019. J. Atmos. Sol. Terr. Phys. 189, 18–26. https://doi.org/10.1016/j.jastp .2019.04.003. Potential of a Uniformly Charged Rod. http://web.mit.edu/viz/EM/visualizations/cou rsenotes/modules/guide03.pdf. Rakov, V.A., Uman, M.A., 2003. Lightning: Physics and Effects. Cambridge University Press, Cambridge, UK, ISBN 0 521 58327 6. Uman, M.A., 1969. Lightning. McGraw Hill, New York. Library of Congress Catalog Card Number 68-8036-65745. Vikhrev, V.V., Ivanov, V.V., Rozanova, G.A., 1993. Nucl. Fusion 33, 311. https://doi. org/10.1088/0029-5515/33/2/10.
Acknowledgements The author is particularly indebted to Professor David R. McKenzie and the Physics Department of The University of Sydney for their
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