Journal of Atmospheric
and Solar-Terrestrial Physics, Vol. 265, 106374 (2024). https://doi.org/10.1016/j.jastp.2024.106374
Electron-ion model of ball and bead lightning
Sergey G.
Fedosin
Postal code 614088, Sviazeva str. 22-79, Perm, Perm Krai, Russia
E-mail: fedosin@hotmail.com
Abstract: Based on the electron-ion model,
parameters of ball and bead lightning are calculated. The model allows us to
estimate maximum size of ball lightning, its energy content, electric charge
and magnetic field, to determine equilibrium conditions between positively
charged ions located inside and outer shell containing rapidly moving
electrons. An explanation is given to the fact that shells are observed inside
ball lightning that differ from each other in color of glow. The model
describes structure of not only ball lightning, but also bead lightning. The
long-term stability of bead lightning is associated with the balance of
neighboring beads under action of magnetic force of their attraction and electric
force of repulsion, which exceed in magnitude the force of wind pressure.
Keywords: ball
lightning; bead lightning; electron-ion model.
1. Introduction
The relation between
occurrence of normal linear lightnings and ball lightnings, as well as between
occurrence of normal lightnings and bead lightnings is well-known. According to
statistics, intensity of observations of ball lightnings (BL) repeats very well
distribution of thunderstorm activity during a day and frequency of
thunderstorms by months during a year almost in all countries. The same applies
to bead lightnings, which usually occur as a result of transformation of normal
lightnings into them [1-2].
In the model described
in [3-4], it is concluded that BLs and bead lightnings appear in ionized air in
channel of linear lightning, and at the moment of the lightning current
termination they rapidly increase in size under action of an expansion wave directed
radially with respect to the channel. After that, positive ions, located mainly
in center of BL, and negative ions gradually annihilate, thereby reducing
electric field inside BL. It is assumed that particles from the soil, captured
by electric field, get inside BL and cause its subsequent glow, igniting in the
very hot air located in lightning channel. The color of glow in this case
depends on composition of the soil particles and amount
of metals in these particles. The same particles may be responsible for sparks
that appear near surface of BL. Despite the fact that fragmentation of linear
lightning into separate segments is associated with the pinch effect, this
model does not calculate the effect of magnetic field on the structure and
configuration of currents and fields inside the emerging BL and segments of
bead lightning. In addition, there is a difficulty in explaining the glow of
those BLs that, according to observations, arise inside rain clouds where there
are no soil particles.
In some models, it is
assumed that there is a magnetic field inside BL, arising from the motion of charged
particles of ionized plasma [5-6]. In [7], it is assumed that BL could be a
plasmoid containing a force-free magnetic field, in which all electric currents
in plasma are parallel to magnetic field lines. However, the magnetic field
alone is not enough for a stable BL configuration; some additional forces are
required [8-9].
There is quite a number
of models that describe internal structure of BL. For example, in [10-11] a
proton-electron model is developed, according to which the shell of BL contains
high-energy protons, and electrons are located in the center of BL. Both
protons and electrons are in rotational motion, creating a magnetic field of
the BL. Due to the magnetic field, the Lorentz force arises, which, together
with the electric force, maintains the stable rotation of charged particles
inside the BL.
A characteristic feature
of the model in [12] is that the ball lightning core contains electrons and
almost fully ionized air, while electrons and ions are in oscillatory motion
with respect to each other. Electrons oscillate mainly in the radial direction.
The core is surrounded by a layer of heated air, which isolates it from the
surrounding atmosphere. This model assumes that electrons in the ball lightning
core can emit electromagnetic radiation in X-ray and gamma-ray ranges. The
disadvantage of both models [10-11] and [12] is that there is no description of
the processes that could lead to the proposed structure of BL, to the division
of air molecules into the required composition of particles with the
corresponding motion of these particles.
In [13-14], it is
indicated that there is still no satisfactory model that describes all observed
properties and mechanism of formation of bead lightning.
In this regard, in this
paper we present general mechanism of formation and internal structure of both
BL and bead lightning, based on the electron-ion model according to [15-19]. The
charge configuration in this model is such that moving electrons and ions
interact with each other by means of both electric and magnetic fields,
ensuring long-term stability of BL.
Sections 2 and 3 will describe the mechanisms
of occurrence and parameters of BL, and Section 4 is completely devoted to the
description of occurrence of bead lightning with calculation of its physical
parameters. In Section 5 we will compare the properties of BL in the
electron-ion model with the results of BL observations, and in Section 6 we
will compare the electron-ion model with the results of experiments and
theoretical studies.
2. Features
of electron-ion model
In principle, both types of lightning, though
they differ in direction of current in them, can lead to occurrence of BL. In
case when electrons in linear lightning move towards a positively charged
cloud, first, under the influence of strong electric field, the so-called step
leaders are emitted from the ground towards the cloud, creating in the air
conducting channels filled with electrons. Most of these channels will never
connect with the cloud; rather, some of them will connect with each other and with
the main lightning channel. Under additional conditions, such a multi-way
configuration of branches with electrons in them turns out to be favorable for
the occurrence of BL.
In another case, when electrons in lightning
move from the negatively charged part of a cloud to the ground, the step leader
moves towards the ground in jumps about 50 m long, at the velocity of about 1/6
of the speed of light and with stops of about 50 μs. When distance between the leader and the ground is about 100 m,
positive charges start moving up from the ground towards the leader in the
ionized air, creating conducting channels. As a result, the leader can branch
out and connect with these channels, and the first of these connected channels
will pass the greatest current through themselves and become noticeable due to
bright glow.
In [15-17], [19], a mechanism is described in
which BL occurs in the case when secondary branches of linear lightning form an
almost closed circular configuration. If an electric current appears in main
lightning channel, then the electrons present in a secondary branch begin to
move into the main channel under influence of electric field. Due to circular
configuration of the secondary branch, the motion of electrons creates a
magnetic field that traps positive ions within the branch. When the current in
the main channel suddenly stops, the electron current in the secondary branch
can close on itself, forming outer shell of the emerging BL. Such a shell is
held by the positive charge of ions inside BL. The described situation is
illustrated in Figure 1.
a) b)
Another method of BL formation can be
associated with the pinch effect, due to which the lightning current channel
can be pinched by magnetic forces. While the electrons are moving in the
lightning channel, they create a strong magnetic field, with magnetic force
lines surrounding the channel and shaped like circles. Some electrons and ions
near the lightning channel, if their velocities are directed along the magnetic
force lines, move in circles around the channel. In the case when the motion of
electrons inside the lightning channel stops, rotational motion of the external
electrons around the channel can be maintained if there is a force that keeps
the electrons from flying away. This force can be electrical force from
positive ions inside the outer electron shell. The pinch effect is most clearly
manifested when ordinary lightning is transformed into beaded lightning.
According to electron-ion model, BL has an axisymmetric configuration with a
strong surface electron current in the thin outer shell. Positive ions can be
under atmospheric pressure in the very hot air inside BL, left after a linear
lightning stroke. Rapidly moving electrons in outer shell generate a magnetic
field that keeps ions on orbits inside BL. Such BL will
be stable as long as it maintains an equilibrium configuration of ion and
electron currents isolated from each other.
In [21-22] we can find that the electron mean
free path with respect to elastic collisions with neutrals in the air under
standard temperature and pressure is within 1 – 1.7 μm. With the electrons’ velocity of the order of 
m/s this gives the time 
s as an estimate of the time which it takes
for one free electron to collide with an air molecule. The relative decrease in
kinetic energy of an electron during one elastic collision depends on the ratio
of masses of the electron and a molecule and is estimated as 
. This leads to the fact that after
a series of collisions in about 
s the electron loses its energy of motion and
binds to some molecule, forming an ion. In this case, how can spherical
electron currents exist for a long time, being in the outer shell of BL?
In order to answer this question, we will take
into account that electrons at the moment of BL formation acquire significant
energy through the electric and magnetic fields near the channel and branches
of linear lightning. Due to this energy, electrons moving rapidly in the outer shell of BL continuously ionize the molecules of
surrounding air, generating new fluxes of electrons that replace the losses of
electrons. On the other hand, the total positive charge of BL acts on the
positively charged air ions in such a way that it ejects them out of the outer shell
with electrons, and then repels these ions from the surface of BL. Due to the
high current density in the outer shell, electrons create their own dynamic
pressure in this shell that prevents penetration of ions and air molecules into
this shell. As a result, electrons, when moving in the outer shell, do not
often collide with ions and do not lose their energy for a long time.
The high electric field strength of BL
contributes to additional ionization of neutral air molecules. As for the
interaction of the shell electrons with the air ions inside BL, this
interaction occurs mainly through magnetic field, forcing the ions to rotate
around the common axis of rotation. Fixation of ions by the magnetic field
prevents them from interacting with the shell electrons. Although the radial
electrostatic force of positive volume charge of BL also acts on the ions, the
total pressure from this force on the shell is decreased due to the ions’
connection with magnetic field, so that the force between volume charge inside
BL and electrons in the outer shell can keep BL in equilibrium. Thus, the air
ions inside BL rather weakly reduce the energy of motion of these electrons.
All this effectively reduces the number of interactions between electrons in
the shell and the air ions and molecules, as well as the loss of electron
energy per unit time, which increases the lifetime of BL to observable values
of up to tens of seconds.
The electrons in outer
shell repel each other, but due to electrical attraction of positive ions
inside BL, the electrons do not get scattered and are retained in this shell.
In addition, the attraction of ions is the main part of centripetal force, which
is necessary to hold electrons as they rotate in the outer shell [16-17]. Based
on the approximately spherical shape of BL, the radius of rotation of outer
electron cloud about common axis decreases as it moves from equator to poles. We can assume that the electrons in outer shell form
electron liquid that uniformly fills the entire shell and is in continuous
rotation.
This relatively stable configuration makes it
possible to explain the observed lifetime of BL, which is much longer than
lifetime of homogeneous ion-electron plasma under atmospheric pressure. The
electron shell effectively isolates the air heated to high temperature inside
BL, slowing down the energy transfer to environment.
3.
Physical properties of ball lightning
Electric currents and
magnetic field inside the BL are shown in Figure 2.
Inside BL we can find
shells where thermal velocity of certain ions is equalized with the velocity of
rotation of these ions in magnetic field [17]. At this point, we can equate the
centripetal force acting on rotating ions and the Lorentz magnetic force acting
on the ions in the equatorial plane perpendicular to magnetic field:
, (1)
where 
are the masses, velocities, and radii of
rotation of ions,
is elementary electrical charge, 
is magnetic
field induction.
In the general case,
instead of (1), we should write a system of vector equations of motion of
neutral and charged particles (air molecules, ions and electrons) in
electromagnetic field, taking into account vector pressure field and vector
acceleration field similarly to [23-24]. Another approach involves the use of
equations of hydrodynamics and magnetohydrodynamics [5], [21] with corrections
for various effects of electromagnetic field’s action on the air molecules.
Instead, we isolate the leading terms of the equations to obtain estimates of
physical quantities in the first approximation.
In order to simplify
calculations, it is assumed in (1) that most of ions are ionized once, and
therefore their charge is equal to elementary electric charge. Indeed,
according to [25], the observed BLs contain singly ionized oxygen and nitrogen
atoms, as well as doubly ionized nitrogen.
On the other hand,
charged particles can freely move along magnetic field lines, passing along the
axis of BL from one pole to the other. Consequently, there could also be such
shells inside BL, where ordered mutually perpendicular ion fluxes take place,
which is accompanied by strong friction in gas and corresponding energy release
in the form of radiation. Since the air consists of several gases, then
positive ions of various masses and various degrees of ionization appear at
high temperatures. These ions would have not coincident thermal velocities and
radii of rotation in magnetic field. As a result, nested luminous shells of
different colors and brightness can appear in BL, depending on composition of
ions and on the rate of transition of kinetic energy into radiation during
collisions of ions in one shell or another.
Let us present our
calculations of parameters of BL with the total radius 
cm and the radius of isolated shell 
cm, similar to that described in [26-27]. For
the sake of simplicity, we will assume that charges and currents are mainly
concentrated near the equatorial plane or have cylindrical symmetry, and ions
are singly charged. According to [17], the electron current 
in outer shell and magnetic field 
of this current acting on the ions can be
expressed in terms of number of electrons 
and their velocity 
at the known temperature 
:
. (2)
.
(3)
.
(4)
In (2) 
is total charge of free electrons in outer
shell of BL, the quantity 
defines period of the electrons’ rotation, so
that the current , in
accordance with its definition, is obtained as the ratio of entire moving
charge to time equal to period of the electrons’ rotation. Relation (3)
describes magnetic field 
on the axis of ring current , with the
radius of the ring equal to 
, 
is the magnetic permeability of medium inside
BL, 
denotes the vacuum permeability.
In (4) the condition for the equilibrium of
an electron during its rotation in the electron shell of the BL is presented,
while 
is a total charge of the BL, 
is the number of positively charged ions
inside the BL, 
is the mass of the electron
is a relative dielectric permittivity of the
medium inside the BL, 
is the electrical constant.
In order to use the
condition of equilibrium of forces for ions (1), it is necessary to estimate
the average mass 
of a typical air ion and velocity 
of its motion, which is equal in the order of
magnitude to velocity of thermal motion. The relation between kinetic and
thermal energies for an ion has the form
, (5)
where 
is the Boltzmann constant.
An approximate relation
for magnetic field in the center of BL follows from (1) and (5):
.
(6)
With the temperature of air inside BL of the
order of 
K and with
the average ion mass 
kg, like that of a nitrogen molecule, the
velocity of ions in (5) will be equal to 
m/s.
Taking into account that in this BL cm, cm, from (6)
and (3) at 
we find
the magnetic induction inside the BL under consideration, which is equal to 
T, and
total current of electron shell 
A.
Based on the approach considered in [17], other
parameters of BL were found depending on its radius. Thus, for BL with a radius
cm, an estimate
of number 
of
electrons in the outer shell and number of ions
inside BL gives 
; total electric charge of BL 
C;
velocity of electrons’ motion in the outer shell 
m/s;
kinetic energy of electrons 
J;
magnetic field energy 0.4 J; electrostatic energy in the volume of BL is 2.8 J;
internal energy of ionized air inside BL is 500 J; electric field strength and
electric potential near the surface of BL are 530 kV/m and 37 kV, respectively.
The pressure
from electrical repulsion of ions inside BL is equal to 
Pa, and magnetic pressure acting on the ions
is equal to
Pa. (7)
The
magnetic pressure 
and the pressure 
from the electrical repulsion of ions inside
BL do not exceed the atmospheric pressure, so the outer shell with electrons,
which is under the external atmospheric pressure, can retain the ionized and
charged air inside BL.
Let
us consider rotation of electrons in the outer shell near equator of BL. The
electrons here are under action of centripetal force in the form of electrical
attraction from the positive ions inside BL. If we take into account only this
force, then the virial theorem is satisfied in its simplest form, when the sum
of doubled kinetic energy of electrons and their potential energy in electric
field is equal to zero. However, the equation of electrons’ motion in
accordance with the plasma equilibrium condition in magnetohydrodynamics includes
two additional small forces, one of which is proportional to pressure gradient
in the electron shell, and the other is Lorentz magnetic force [21]. The
Lorentz force can be transformed into s term containing gradient and also into
a term that can be neglected in the first approximation. This is how the
relation 
appears, where 
is magnetic pressure in the shell with
electrons, 
is electron pressure across the magnetic
lines. The electrons moving in outer shell at velocity 
create a pressure equal in the order of
magnitude to
, (8)
where
is kinetic temperature of electrons, 
J is kinetic energy of one electron, 
is electron mass, while the energy 
is 22 keV in energy units. If we assume that 
,
then in view of (7-8) we obtain the thickness of BL shell which equals 
cm.
In
the case of BL formation in linear lightning channel, the electron velocity
distribution may be such that electrons will have the velocity not only in outer shell, but also in central
part of BL. In this case, such electrons will rotate in magnetic field and will be held at the center of BL. From the
equality of centripetal force and Lorentz force we obtain the following
relations:
, 
cm. (9)
The
radius 
in (9) limits central part of BL, filled with
fast moving electrons. These electrons can provide that violet glow in the
center of BL that was observed in [26-27]. Comparing volume of central part of
BL with the radius and volume of outer shell with the thickness 
,
we can conclude that energy density of electrons near the center of BL can be
by an order of magnitude higher than energy density of electrons in the outer
shell of BL. This allows us to explain the fact that when BL passes through
glass, holes with a diameter ranging from fractions of a millimeter to the size
of BL itself are usually observed [2], [28]. Sometimes holes can be melted,
sometimes not; in the latter case, it can be assumed that the glass first gets
very hot and then breaks off under pressure of BL moving under influence of
strong electric fields.
Table
1 shows calculated parameters for BL of different sizes. Designations in the table: is radius of BL, is total electron current in BL
shell, is magnetic field induction in the center of
BL, 
is total energy taking into account
the energies of magnetic and electric fields, the energy of electrons’ motion
and the plasma energy inside BL.
Table
1.
Parameters of ball lightnings of different
sizes
|
|
|
|
|
|
|
1 |
20 |
|
|
2.2 |
|
7 |
|
|
|
503 |
|
17 |
|
0.5 |
|
10600 |
The radius of the
smallest BLs is about 1 cm, while the energy of electrons’ motion and magnetic
energy can be considered approximately the same. At a radius of 7 cm, the
energy of BL is equal to 503 J, almost all of it is contained in thermal energy
of moving ions and in potential energy of ionized particles.
The main energy of
powerful BL with its radius of 17 cm is contained in energy of ionized
particles and in energy of electromagnetic field. In this case, the total
energy of 10.6 kJ falls within the range of upper values of observed energy
content of BL, determined through interaction of BL with surrounding objects
(melting of metal parts, heating and evaporation of water, etc.).
In
the model under consideration, BL in general must be positively charged so that
the ions inside BL could retain the electron shell for a long time. However,
the total charge of BL cannot exceed such a value, at which electric field
strength on its surface exceeds 
kV/m, in order to avoid the
breakdown of atmospheric air. Due to high electric field strength and rapid
electrons’ rotation in BL shell, the observed effects of hissing, crackling and
sparking occur. The observed radius of BL can be greater than 17 cm in Table 1
due to glow of electrified air. In addition, there may be burning soil
particles in the air, trapped in the BL’s electric field.
As
can be seen from Table 1, the maximum electron current in BL is quite close to
current in the channel of average linear lightning.
4. Bead
lightning
When
constructing the bead lightning model, we will rely on the data from [29].
Using frame-by-frame reconstruction of a video film, which recorded lightning striking spire of Ostankino TV tower in Moscow on July 24, 2001, formation
of bead lightning was discovered at the site of decay of normal lightning. The
average distance between the neighboring beads (or “pearls”) of the bead
lightning was 
m, the average bead length
was 
m, and the bead width when
viewed from the side was 
m. Since the width of the
primary linear lightning channel was 0.25 m, the bead width 
exceeds the width of the
primary lightning channel.
We
can assume that each bead has the shape of a cylinder with the length 
and diameter
, and the
area of maximum lateral section of such a cylinder is 
m2. Since wind
speed at the height of the spire of 533 m on that day equaled 
m/s, and dynamic air pressure
could be expressed by the formula 
, where 
kg/m3 was the air
density, then the pressure force from the wind on each bead had the following
value:
N. (10)
The paradoxicality of
the situation with that bead lightning was that despite the hurricane wind, it
almost did not move during its lifetime of 0.625 seconds. The video recording
showed that the lightning beads remained in their places. The authors of [29]
suggested that the lightning beads could possibly be held by electric forces;
for this case, taking into account (10), they calculated the corresponding
electric field strength 
:
, 
, 
V/m, (11)
where 
is vacuum permittivity, 
is charge of a bead.
The electric field strength 
in (11) does
not exceed the air breakdown strength 
V/m.
However,
the point is that free charged objects cannot be fixed in their places by the
electrical forces alone – they will be unstable, and either rush towards each
other or scatter in different directions, depending on the signs of charges.
Let us apply to analysis
of bead lightning the data on structure of ball lightning (BL) presented in the
previous section. According to the electron-ion model, BL has a positive
charge, as well as magnetic field that arises mainly from electrons rapidly rotating
in the outer shell of BL. Let us assume that the structure of individual beads
of bead lightning is the same as that of BL, then the neighboring beads are
repelled from each other by electric forces, and at the same time are attracted
by magnetic forces. As a result, the electromagnetic forces are balanced, and
the beads remain in their places until the complete loss of energy in the
lightning.
Figure 3 shows two
neighboring beads interacting with each other. The inherent stability of each
bead is determined by the fact that internal volumetric positive charge holds
the rotating electrons in the bead shell by electric forces. At the same time,
magnetic field arising from the motion of electrons fixes positive charges
inside the bead as they rotate in this magnetic field.
Let
us now consider stability of the beads relative to each other. If each bead,
just like BL, has the total charge 
, then the
electric repulsion force of equally charged beads equals:
, (12)
where 
is distance between the neighboring beads.
The magnetic force of
attraction between the beads is approximately calculated as the force of
interaction of two identical solenoids located on the same axis at the distance
from each other:
.
(13)
In (13) 
is electron current through the bead’s
cylindrical surface, the ratio 
is current per bead’s unit length, the value 
is cross-sectional area of the bead’s
cylinder, is vacuum permeability.
If we do not take into
account the action of remaining beads, the balance of two neighboring beads is
achieved when the forces 
(12) and 
(13) are equal, which, in view of the
well-known identity 
,
gives the following:
, 
, (14)
where 
is the speed of light.
The magnitude of
electron current can be related to the number of electrons in the bead’s cylindrical shell by
corresponding formula, so that with the electrons’ velocity tending to the
speed of light, for the current,
similarly to (2), we find:
, (15)
where 
is elementary electric charge.
The fact that the
maximum velocity of electrons’ motion in (15) can be of the order of the speed
of light is due to extremely high current intensity in the lightning that
struck the spire of Ostankino TV tower in Moscow on
July 24, 2001. At the temperature in lightning of the order of 
K,
according to [30], the thermal velocity of electrons, calculated by formula (4)
taking into account electron mass instead of ion mass 
, gives the
value 
m/s.
On the other hand,
electrical forces can accelerate electrons more efficiently than thermal
heating of gas. This is apparent
from the fact that step leader during lightning formation can move at the
velocity of the order of 1/6 of the speed of light [20]. Thus, formula (15)
gives an estimate of maximum possible electron current . More
precise current values could be determined if we take into account relativistic
corrections and conditions of current flow in ionized plasma and in outer shell
of bead lightning.
Substitution of the
current (15) into relation (14) gives:
(16)
On the other hand, the
total charge of one bead is expressed as difference between all
positive and negative charges, then, in view of (16), we have:
, 
, 
, (17)
so that the number of
positive ions inside the bead is 1.16 times greater than the
number of electrons in the bead shell.
Now we will equate the wind force 
from (10),
acting on the bead, to the electric force from (12),
which will allow us to find the charge of the bead:
, 
C. (18)
Substituting the charge (18) into (16), we
determine the number of electrons in the bead shell:
,
as well as the number of uncompensated positive
ions inside the bead, according to expression (17): 
Knowing the quantity , we can estimate the electron current in the
bead shell from (15):
A.
(19)
The magnetic field induction reaches its maximum
in the center of the bead and based on the formula for solenoid is equal to:
T.
(20)
Comparison of obtained values of internal
induction of magnetic field and electron current in the bead shell with the
corresponding values for powerful BL from Table 1 shows that they are close in
magnitude. Based on the magnitude of magnetic field, we can conclude that
magnetic pressure in the center of the bead, determined by the formula 
,
does not exceed atmospheric pressure in the air
outside the bead volume. This means that due to the magnetic pressure, the
volume of the bead has an ability to change within certain limits, adjusting to
forces acting in the bead.
Apparently, formation of bead lightning from
normal lightning occurs due to the pinch effect, in which the conductive
channel of lightning in some place is pinched by the magnetic forces arising
from the flowing current. Indeed, according to Ampère law, the parallel
electron currents in lightning channel are attracted to each other. Due to the
current in the lightning, the magnetic field appears, the force lines of which
have the form of circles around the current channel. As a result, the lightning
current channel is surrounded by the shell of electrons rotating along the
magnetic lines and shielding the channel from external environment. The greater
is the current in lightning and the greater is magnetic field, the smaller is
distance from magnetic lines, at which the shell’s electrons can move off
during their motion. On the average, this distance does not exceed the radius
of Larmor rotation due to Lorentz force.
Under condition of compensation of electric
forces due to simultaneous presence of positive and negative charges and with
the sufficient current, magnetic pressure can exceed the gas-dynamic pressure
and lead to restructuring of the lightning structure — translational motion of
charges along the entire lightning channel becomes difficult and rotational
motion becomes significant in individual formations, that is in the beads of
lightning. The condition for occurrence of pinch effect is described in this
case by the Bennett relation [31], which takes into account equality of
densities of current’s magnetic energy and thermal energy of electrons and gas
ions:
.
(21)
If we substitute in (21) typical values of
parameters for linear lightning according to [30], in particular, radius of
lightning channel 
m, concentration of electrons 
m–3, electron and ion
temperatures 
K,
and take into account values of magnetic constant and
Boltzmann constant , then we can estimate the current magnitude: 
A. As could be expected, this current is quite strong and not less
than the current (19) in the bead shell. Current strength in lightning that
struck the spire of the Ostankino television tower in
Moscow on July 24, 2001 could really be so significant, since the length of the
lightning and accordingly electrical resistance of lightning channel decrease
due to height of the television tower.
Defragmentation of linear lightning into separate
beads begins with a decrease in current strength in lightning channel, due to
which external magnetic field decreases, and electron shell moves away from the
channel. Since motion of electrons in lightning shell creates a longitudinal
magnetic field in the channel, this field, magnetic pressure from which
counteracts the occurrence of pinch effect, also decreases. With expansion of
electron shell of lightning, the volume of lightning channel in some places increases
to the size of beads, while the channel takes form of separate beads with gaps
between them, mainly as a result of electric repulsion of the positively
charged contents of neighboring beads.
As in the case of ball lightning, the beads’ glow
can be associated with the friction of mutually perpendicular ion fluxes, one
of which is directed along magnetic field lines inside the beads, and the other
is associated with rotation of ions around these magnetic lines.
According to [32], the fraction of free electrons
in the air is close to 0.7 at the electric field strength of the order of 700
kV/m. Inside and outside the beads, the field strength can be even greater than
1700 kV/m according to (11). Such fields lead to appearance of a large fraction
of free electrons, which makes it possible to substantiate the number of
positive ions inside the bead and the number of
electrons in the bead shell in (17) within the framework of electron-ion model.
On the other hand, electric field near the beads has the same order of
magnitude as the critical electric field 
kV/m. This
leads to a rapid loss of energy by bead lightning due to discharge into the
surrounding air and to a decrease in the lifetime of bead lightning by an order
of magnitude as compared to the lifetime of ball lightning.
5. Compliance of electron-ion model
with observations
As indicated in Section
2, BL can occur in curved secondary branches of linear lightning when
sufficiently large circular electron currents appear in these branches. Such
currents generate a dipole magnetic field that retains positive ions inside the
resulting BL. The result is a stable configuration, shown in Figure 2. In this
configuration, the electron shell contains fast-moving electrons, which are
subject to an electric force from the positively charged ions inside the BL.
As an example, [33]
describes a case when one of the branches of linear lightning, passing almost
horizontally near a power transmission line, started twisting in an arc and BL
immediately appeared at this place. In a number of other cases, BL occurred
when two different lightnings or their branches crossed each other, as well as
at the places of lightning branching.
Another option for BL
formation is associated with the pinch effect, the condition of which is
presented in (21). This mechanism is supported by the fact that sometimes not
one but two BL are found at once, connected to each other by a chain of
luminous beads, similarly to connection of beads in bead lightning [30].
Occasionally, BL with a shape of an elongated cylinder are
observed [2]. In [33], the case was observed in which, after a linear lightning
strike, the lightning channel began to fade out, consistently losing its glow,
starting from the clouds towards the ground. When only a small part of the
luminous channel remained near the ground, BL appeared at this place, which
seemed to be rapidly rotating.
Based on such
observations, it is concluded that BL can be formed in the linear lightning
discharge channel at the stage of its decay through an intermediate state such
as beaded lightning. In addition, BL can also occur in secondary branches of
linear lightning, in which there can be a significant electric current even in
the absence of noticeable glow. Observations from airplanes show that many BLs
occur inside cumulonimbus clouds, which is associated with abundance of
branches of each linear lightning, and with a large number of electrical
discharges inside and between clouds.
Thus,
BL can appear even away from the brightly sparkling main channel of linear
lightning at any height from the ground. In some cases, the distance between
emerging BL and main channel of linear lightning can reach hundreds of meters [34],
which can be explained by the branching of main lightning channel into several
less bright channels. Then BL emerges near one of the secondary channels, and
really, according to observations, many BLs emerge near relatively weak
lightning with current of the order of tens of kA.
In [35], the cases are
described of even greater distances between BLs occurring during a thunderstorm
and those points on the ground where strikes of normal linear lightnings were
recorded. In order to explain this situation, it should be taken into account that
large neighboring cumulonimbus clouds can have opposite signs of their charges.
Thus, one cloud can be charged negatively in the lower part, and positively in
the upper part, and another cloud as a whole can be charged positively due to a
significant discharge of its negative charge, which occurred earlier due to
many lightning strikes. Then the first cloud will produce standard thunderstorm
activity, including intra-cloud lightnings, rain showers and normal negative
cloud-ground lightnings with electrons moving from the cloud to the ground. In
contrast, the second cloud, moving several kilometers from the first, will
likely generate positive cloud-ground lightnings with electrons moving from the
ground to the cloud, and BL accompanied by light rain. In such a case, the
probability of BL occurrence during positive lightnings significantly exceeds
the probability of its occurrence during negative lightnings.
According to [37-38], out of 67 direct
observations of BL occurrence, 31 events occurred in close proximity to linear
lightning channel, 29 events were associated with “blowing-out” of BL from
metal conductors such as electrical sockets, radio receivers, antennas and
telephones, and in the remaining 7 cases BL appeared “from nothing”. Quite
often, BL occurs in an enclosed space, while almost simultaneously or within a
short period of time about a minute earlier or later, a lightning strike occurs
somewhere very closely [2], [30], [39]. All this indicates that in the room and
close to it, under the action of strong electric field of cumulonimbus cloud,
electricity-conducting branches are formed in ionized air, which are ready to
transform into future lightning’s channel. Most commonly this should happen in
those premises that don’t have high lightning rods. Metal conductors indoors
conduct electricity better than ionized air, therefore most often BL occurs
near these conductors at the moment of electrical breakdown of air and
subsequent impulse of strong electric current. Indeed, cases have been recorded
when during opening and closing of electrical circuits of battery units and
powerful generators strong currents appeared and BLs occurred [2], [30].
In [26-27], the
researcher Dmitriev provided a detailed description of a BL observed by him
with the diameter of about 14 cm in horizontal direction and with the vertical
dimension of up to 16 cm. According to this description, the inner luminous
shell (with brightness similar to that of plasma heated in a plasmatron to
16000 C) was elongated in the vertical direction, and all the shells were
clearly visible only in the horizontal direction. Separate bright convective
jets were indicative of rotation of particles in the shells at a differential
velocity. The central part of the lightning was surrounded by an area 1 – cm
thick with dense violet glow, very similar to the air glow at a pressure of 0.1
mm Hg, bombarded by electrons with an energy of several tens of electronvolts.
The next outer shell about 2 cm thick was also inhomogeneous, resembling in
color a quiet electric discharge at atmospheric pressure or a peripheral glow
of an electron beam with an energy of several tens of kiloelectronvolts, which comes
into the air from a vacuum tube at ordinary pressure. The light blue glow of
this part of the lightning quickly decreased with increasing of distance from
the central ball, gradually fading away. The presence of high electric field
strength near BL and high-energy particles was confirmed by observation of
their hissing, crackling and sparking as during an electric discharge. In
addition, the air samples taken after the passage of BL showed an increased
content of ozone and nitrogen oxides (approximately 50-100 times higher than
the norm). The required concentration ratio of ozone and nitrogen oxides can be
achieved with an electric discharge in the air with a strength of up to 400
kilovolts per 1 meter, and an estimate of the required electric energy in such
an equivalent discharge for the entire lifetime of BL gives a value of 530 J.
In the electron-ion
model, it becomes clear why in Dmitriev’s ball lightning all the shells were
clearly visible only in the horizontal direction. Apparently, in this
lightning, the axis of electrons’ rotation in outer shell and internal magnetic
field were directed vertically, so that there was a flux of ions moving along
the magnetic field lines. This flux, when interacting with ions rotating in
magnetic field, formed the shells observed in horizontal direction.
The parameters presented in Section 3 describe
BL, which was observed in [26-27], quite well. The presence of strong electric
current in the shell of BL allows us to explain observation described in [33].
During a thunderstorm, BL of about 6 mm in diameter was formed on contacts of
transformer of overhead telephone line with a crackling and hissing sound. BL
approached one of the two telephone line switchboards located in the room and
discharged to its relay. At the same time, all 160 calling lamps of both
switchboards flashed brightly, despite the fact that the operating voltage of
the lamps was 24 volts. Apparently, when BL discharged to the contacts, it
could become a source of electromotive force in the circuit connecting the
lamps. In another example, BL discharged between the TV-set and the heating
battery, and the kinescope of the TV-set suddenly lit up, although the TV-set
was turned off.
In [14], [40-41] almost half a thousand cases of
BL observations were analyzed, according to which the diameter of the most
frequently occurring BL is in the range of 25-30 cm. At the same time, in
almost 50 % of cases the BL diameter was less than 25 cm. Statistical
dependencies showing the distribution of several thousand observations
depending on the diameter of BL can be found in [42-44].
According to [45], the correlation
parameter for observations of BL diameter in large databases reaches the value
of 0.99. Description of observations using the log-normal distribution gives
the standard deviation from the average value equal to 16 cm for the diameter and
11 seconds for the lifetime of the BL. In this case, the average value of the
BL diameter for 4219 observations is equal to 
cm. The most probable
energy value of the observed BLs lies in the range from 4.4 to 11 kJ. The above
distributions are in good agreement with the fact that in the electron-ion
model the maximum diameter of the BL in Table 1 does not exceed 34 cm and may
be slightly larger taking into account the halo. The energy of such BL is equal
to 10.6 kJ.
6. Correspondence of electron-ion
model with theoretical calculations and experiments
The occurrence of BL in
the place where the current lines have a bend is confirmed by the calculation
in [46] based on the interaction of the current in the linear lightning branch
with the magnetic field of this current. Under the influence of electromagnetic
forces, a curved branch of ionized air with current will tend to turn into a
ring, which then turns into a spherical shell of the emerging BL.
The initial stage of BL
formation as a result of the pinch effect, using the equations of
magnetohydrodynamics [13], [21], is described in [5], [47]/
Due to such a magnetic field, positive ions,
which appear at the breakdown site as a result of ionization by current or
voltage, start rotating around the field symmetry axis, while an electron shell
quickly grows around them. Electrons are moving along the current lines in
crossed fields, one of which is the electric field from ions, and the other is
the dipole magnetic field. At the moment when the main current stops, motion of
electrons around the region with positive ions continues due to confining electric
force, and BL is formed with a closed spherical electron current in its shell.
One example of mathematical description of BL
formation in a high-voltage discharge near two contacts, one of which is
cathode and the other is anode of an electrical circuit, is article [48]. At
the initial moment of discharge, a streamer appears, which can branch both near
the cathode and near the anode. Next, the streamer leads to formation of a
plasmoid, which can turn into BL. The calculations use a streamer model
modified with account of diffusion of electric field and ions. For the purpose
of 3D modeling of the process, a typical unit of length is selected (as the
mean free path of electrons in nitrogen under normal conditions), as well as
typical units of time, electric field, electrons’ density and velocity. A
dispersion relation is derived that relates the wave numbers, electron
diffusion coefficient, electric field strength and streamer velocity. This
makes it possible to plot a graph of electron density distribution in the
streamer and in the plasmoid, the potential and the electric field distributions,
and other quantities depending on time. The typical time of formation of a 20
cm plasmoid ranges from 150 to 500 ms, while the
minimum required electric field strength reaches the value of 35-40 kV/m [49].
Electric fields would have the same order of magnitude in BLs if they occur
near electrical sockets, radio receivers, antennas and telephones located in
closed rooms and exposed to high voltage from lightning.
In [50], it was experimentally shown that the
luminous low-temperature plasmoids created during a discharge in a laboratory
have a thin shell containing negative charges and preventing mixing of the
plasmoid contents with atmospheric air. When plasmoids come into contact with conductors, the
latter, as a rule, melt, and interaction with dielectrics is much weaker.
Plasmoids also react to a laser beam, have an uncompensated electric charge,
while the air temperature inside such artificially created plasmoids is about
200°C. However,
the color temperature of plasmoids can reach 2000°C,
which indicates the non-thermal nature of radiation.
It was found in [51] that there is a sharp mass
density gradient in the outer shell of plasmoids. According to observations in
[25], natural BL turned out to be sensitive to high-voltage (35 kV)
transmission lines located close to it, since the frequency of changes in the
glow intensity of BL coincided with the frequency of 50 Hz of electric current
in power transmission lines. It is also well known that appearance of BL leads
to appearance of noise in a working radio receiver.
All these facts can be explained as follows. In
the electron-ion model, positive ions inside BL are weakly attracted by
electrons from the outer shell, since electric field of this shell inside the
sphere is close to zero because of balancing of all the electric forces due to
the sphere’s symmetry. Indeed, a charge uniformly distributed over the surface
of a hollow sphere does not create an electric field inside the sphere, as
follows from the Gauss theorem. Therefore, ions can be fairly freely distributed
in the volume inside BL, reaching the state of equilibrium rotation in the
internal magnetic field, while recombination of ions and electrons in BL is
significantly slowed down due to spatial separation of charges. Electrons
moving rapidly in the shell of BL create a strong current capable of melting
conductors, burning wooden objects, heating and evaporating water. Interaction
of electrons with the air surrounding BL leads to its ionization and is
accompanied by radio radiation, which is perceived by radio receivers as noise.
Being an object with electrically connected spatially separated positive and
negative charges, BL reacts to the laser beam, since the photons act on
electrons in the outer shell. The presence of charge in BL ensures interaction
of this charge with the alternating electric field near power transmission
lines, which is expressed in a change in brightness of BL glow with the
frequency of current in these lines.
According to [11] and in accordance with [52-53],
plasma BLs with a maximum diameter of up to 94 cm were created in the
laboratory in crossed alternating electric and magnetic fields. Such BLs, as
measurements showed, had spatially separated volumetric electric charges of
different signs, with the motion of charges in circles, the electric currents
of these charges and the corresponding magnetic field of dipole type. The
distribution of radiation intensity, depending on the current radius from the
center of the BL, is close in shape to the intensity distribution observed in
natural BL. In this case, the maximum intensity is achieved near the BL shell,
outside of which there is a rather quickly decaying luminous halo. Since the
probes are positively charged upon contact with plasma BL, it is assumed that
the BL shell contains relativistic protons, and the BL center contains
electrons.
In electron-ion model, BL is actually a small
piece of linear lightning, twisted into a ball with the typical size of 10 – 40
centimeters. Consequently, in both types of lightnings the currents and
magnetic fields can be close in magnitude. Since the BL
in this model has an overall positive charge, this may explain the fact that in
[11] plasma BLs positively charged the electric charge probes. However, in the
electron-ion model, the BL shell contains not protons, but electrons.
Confirmation
of the electron-ion model of BL can be found in [54], where possible shape of
magnetic field of BL was studied, which could ensure the motion of BL along
sagging conductors, including power transmission lines. Taking into account
properties of BL motion in a viscous air flow, it is concluded that electrostatic
field of positively charged BL should be centrally symmetrical. The magnetic
field of BL must have a symmetry axis, and the currents creating this magnetic
field must be azimuthal. The equation of BL motion along a wire used in the
calculations takes into account the wind pressure and interaction of the BL’s
magnetic field with electric current in the wire, which occurs due to charge
leakage from BL surface into the wire. It is shown that such motion along the
wire is possible only in the case when the currents generating the BL’s
magnetic field are concentrated in its surface layer. Indeed, the currents in
volume of BL would interact with magnetic field in such a way that BL should
increase its volume all the time, which is not observed. For configuration with
the surface current, the expressions are found for components of vector
potential and magnetic field of BL in spherical coordinates.
In [55-56] the hydrodynamic model of BL is considered
in order to describe its motion through small hole. Outside the hole, equations
of motion of an undeformed liquid ball in an ideal fluid flow of a point
hydrodynamic source are used, which are written in spherical coordinates. These
equations are modified into the Euler equation taking into account the
continuity equation used to describe motion of BL in a hole, the diameter of
which is by an order of magnitude smaller than the diameter of BL, which leads
to transformation of the shape of BL into a jet. Solutions to these equations
are found by numerical simulation under reasonable assumptions on the time of
passage through the hole and on behavior of BL in air flow. One of the
conclusions is that the shell of BL must have zero viscosity with respect to
environment, and the matter of BL is an ideal incompressible fluid.
In [57], experiments were conducted with charged
soap bubbles with a diameter of 3 cm, carrying an electric charge of the order
of 10-15 nC. With such a charge, soap bubbles tend to
be attracted by induced charges to a nearest grounded conductor. According to
conclusions in [57], this casts doubt on possibility of charged BLs to move in
horizontal direction and near grounded conductors if the BL’s charge is large
enough.
What can be said then regarding the possibility
of horizontal motion of the BL, described in [26-27], which had the radius of 7
cm ? Based on the electron-ion model, the electric
charge of such a BL is equal to 
C. According
to calculations in [17], BL is subject to Archimedes’ lifting force from the
hot air inside BL, as well as to electric force acting between the BL’s charge
and induced charges on the ground. The equality of these forces is achieved
when BL is at the height of about 10 cm above the ground, in which case BL can
be attached to some object on the ground. If BL has the radius of 17 cm and
corresponding charge 
C, then
the equilibrium state will be achieved at the height of 90 cm.
Over time, the charge of BL can change due to
interaction with environment or due to partial decay, leading to a change in
equilibrium state. As the air inside BL cools down, its volume decreases, and
with the loss of electrons from the outer shell charge Q may also increase. Therefore, BL can rise above the ground and
then move smoothly depending on the terrain configuration, the wind and
electric fields from cumulonimbus clouds. Quite often, when transitioning from
attached BL to free BL, it soars upwards and then goes along an inclined line
towards the clouds. The latter can be explained by strong electric field acting
on positive charge of BL from the negatively charged lower part of cloud. From
equality of Archimedes’ force and electric force, we can estimate the minimum
electric field strength that can influence motion of BL:
.
(22)
For BL with radius of 7 cm and charge C, the
electric field is equal to 
V/m. For comparison, field strength
in stormy weather may well reach the value of 
V/m or more, based on potential
difference between the clouds and the ground reaching 
V.
The comparison of charged soap bubbles and BL is
not entirely correct, because these bubbles do not contain very hot air and are
not subject to corresponding Archimedes force. In addition, we should consider
interaction between the bubble’s electric charge and massive conductor on the
ground in more detail. In order for a bubble to be attracted to the conductor,
the bubble’s electric charge must separate the charges in conductor by its
electric field. This requires some work to be done, which in this case is
possible due to the gravity force acting on the bubble. The closer the bubble
is to conductor, the more charges in it are separated and the stronger will be
attraction to the conductor. In the absence of Archimedes’ force, the balance
of forces is such that the bubbles would be attracted to the conductors on the
ground until they collide with them. In contrast, the motion of BL is
controlled not only by the gravity force, but also by Archimedes’ force, as
well as by electrical forces from the charges on the ground and in the clouds
and by the wind pressure. Therefore, the motion of BL can be more diverse,
including various types of horizontal and vertical motion.
In [58], the electric charge and field potential
near the BL’s surface were determined in a very original way. For this purpose,
the fact was used that BL, in accordance with the eyewitness evidence, was
moving along a curved line approximately 1.5 m above the river surface along
rafts made of logs with the initial angle of 30° to the wind direction. When
describing one-dimensional motion of BL, the Navier-Stokes equation took into
account the wind pressure, which depends on the height, and the electrostatic attraction,
found using the image method, while the rafts made of logs were considered as
dielectric protrusion above the water. As a result, with the radius of BL equal
to 10 cm, the BL’s charge was 
C, and the
electric potential was 42 kV.
According to [59], the BL with a radius of 15 cm
has a charge of the order of several microcoulombs and an electrical energy
density of the order of one hundredth of a joule per cubic centimeter.
The BL parameters in [58-59] are in good
agreement with our calculations of BL parameters in the electron-ion model in
Section 3 and Table 1. In addition, according to calculations in [17], the
electric energy of a BL with a radius of 17 cm is equal to 1.3 kJ. If we divide
this energy by the volume of the BL, we obtain an electrical energy density of
the order of 0.06 J/cm3, which is consistent with the value of the
electric energy density in [59].
7. Conclusion
The novelty of the electron-ion model compared to
the proton-electron model in [10-11] lies in the fact that the outer shell of
the BL contains not protons, but relativistic electrons, the currents of which
generate strong magnetic field. This magnetic field captures the positive air
ions inside the BL and causes them to rotate around the magnetic field lines.
In turn, the internal electric charge of the BL keeps electrons as they rotate
in the BL shell. This ensures long-term stability of the BL, which is generally
positively charged. Charge separation in a BL occurs at the stage of its
formation, with the key role played by the dipole magnetic field in the curved
channel in the linear lightning branch. In case of strong current in linear
lightning, the pinch effect becomes important.
The BL parameters found in Section 3, which
depend on the BL radius, are confirmed by the observations described in Section
5, as well as by theoretical calculations and experiments presented in Section
6.
One of the advantages of the electron-ion model
is that with its help we were able to explain not only the structure of ball
lightning (BL), but also the structure of bead lightning, as well as
occasionally observed structures of two coupled BLs. Unlike a single ball of spheroidal shape, in bead lightning there
are several beads of almost spherical or cylindrical shape, interconnected by
electric repulsive forces and magnetic forces of attraction. These forces,
together with atmospheric pressure and hot plasma pressure, form the sizes of
the emerging beads, in many cases exceeding diameter of primary linear
lightning channel.
According to [1-2],
beads often have a shape close to spherical. We can relate this to action of
magnetic pressure, which arises due to electrons’ rotation in the shell of each
bead and generates a strong magnetic field inside the beads. According to observations,
the greater are electron currents and magnetic field in the beads, the larger
are sizes of the beads and energy stored in them, and the slower such beads
cease to glow.
The beads occur in
channel of normal linear lightning and therefore lose energy due to through
currents in still conductive hot air of lightning channel. The heated air
inside the bead is isolated by electron shell from the side surface, but can
exit from the edges of the bead into lightning channel. As a rule, the lifetime
of bead lightning significantly exceeds the glow time of primary linear
lightning channel. It follows from observations that the beads can be seen for
up to 2.5 seconds. As shown by calculations presented in Section 4, the
electromagnetic forces holding the beads are so great that the beads can even
withstand the pressure of a hurricane wind.
The electron-ion model can
be useful for analyzing and explaining the results of experiments [60], in
which luminous objects in the form of balls with dimensions of order of several
millimeters were observed during an electric discharge in aqueous medium.
During the discharge, water dissociates into hydrogen and oxygen ions that get
inside these balls, which is followed by hydrogen burning. The fact that
closely spaced balls merge with each other can be associated with the fact that
magnetic force of attraction (13) at small distances between the balls is
significantly greater than electric force of repulsion (12). When the balls
collided with an obstacle, they exploded, which can be explained by stopping of
charged particles moving rapidly in the shells of the balls, followed by ion
recombination and corresponding rapid release of energy.
It
should be noted that at the micro-level there could also be objects similar in
their properties to BL and beaded lightning. Thus, in [61-62] we can find a
review of properties of the so-called Exotic Vacuum Objects (EVO), which
represent negatively charged spherical charge clusters ranging in size from 1
to 20 microns. The ratio of electrons to ions in EVO reaches . EVOs were created in a low voltage
/ low power micro-arc discharge, they had velocities up to 1/10 of the speed of
light and existed for up to 
s. An extraordinary property of EVO is their
ability to create the similar chains as in beaded lightning. Just like BLs,
EVOs can explode while producing X-ray emission. When encountering obstacles,
EVOs make deep tunnels through them. To explain this property, in [63] EVOs are
considered as objects in pulsed deposition systems and as shaped-charge
munition. Among a number of questions that remain to be answered within the
analogy between BL and EVO, the following can be mentioned: What forces cause
EVO to form and maintain a spherical shape, despite the significant
predominance of electrons over ions and repulsion of electrons from each other?
How can EVOs be combined in chains considering their strong electrostatic
repulsion from each other?
Data availability
The data that supports the findings of this
study are available within the article.
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