lightning stepped leaders
TRANSCRIPT
Lightning stepped leaders
Freddie Hendriks
July 18, 2016
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Lightning stepped leadersFreddie Hendriks
Abstract
The initial and yet fundamental process in a typical cloud-to-ground lightning strikeincludes the propagation of a very faint and charged channel which is called stepped leader.The exact mechanism for the step leaders is not understood. The reason for this is that thetemporal and/or spatial resolution of the devices exploited for observing this phenomenonhas not been sufficient. The radio interferometric array of LOFAR however is capableto measure radio signals with 1 ns temporal resolution. Thus LOFAR can measure theradio pulses emitted by stepped leaders at multiple times during the formation of the stepsand locate the positions of the pulses with sub-meter accuracy. This provides with newpossibilities to test and probe the theories explaining the propagation of a stepped leader.To interpret the measurements of LOFAR, I have developed a new simulation tool that isable to calculate the electric fields expected on the basis of current models. By comparingthe simulation and experimental data, it could be possible to determine the characteristicsof stepped leader formation.
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Contents
1 Introduction 5
2 Lightning and negative stepped leaders 5
3 Maxwell’s equations in potential formulation 7
4 Analytic solutions 94.1 Moving point charge (Lienard-Wiechert Potentials) . . . . . . . . . . . . . . . . . 94.2 straight line current with polynomially increasing current . . . . . . . . . . . . . 14
5 Simulation 185.1 The general idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2 Implementing extended sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6 Simulation tests 21
7 Simulation results 22
8 Conclusion and Discussion 27
A Summary of the simulation components 29
B Presentation 31
C Poster 34
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1 Introduction
Nearly everyone has experienced a thunderstorm at least once. It usually start with a heavydownpour from the familiar anvil shaped dark clouds, followed by the bright flashes and loudthunders of the lightning discharges. The main processes in thunderstorms are known quitewell [1]. The details of the physics behind the various processes however are not fully known.Getting a better understanding of them can be an end in itself, but it can also help to improvelightning protection methods.
The formation of a stepped leader in a negative cloud-to-ground flash is an example of one ofthese unknown processes [1], [2], [3],. This stepped leader is a conducting channel that grows insteps from the cloud to the ground. Once this channel connects to the ground, the big dischargewill take place through this channel. Stepped leaders are also observed in triggered lightning andin laboratory sparks longer than a couple of meters, which means that it is a general phenomenonoccurring in negative discharges though air over longer distances. But there are no solid theoriesto describe the formation of lightning stepped leaders. Mostly because it is difficult to measurethe processes involved in stepped leader formation accurately enough: their propagation speedcan be as high as a half the speed of light. Combine this with the fact that lightning is usuallyobserved from several kilometers away, while the length of a new step is in the order of 50meters, and it should be no surprise that it is hard to obtain both a high enough spatial andtemporal resolution with the conventional measurement equipment like high-speed cameras andVHF antennas.
Fortunately, radio telescopes such as LOFAR provide new possibilities for doing measurements[4]. These radio telescopes are able to measure the radio signals emitted by stepped leaders, andthey have a high enough resolution to be able to do several measurements during the formationof a new step, while also having a high enough spatial resolution to distinguish the start andend point of a step. My research is about finding a way to interpret the measurements of thesetelescopes in terms of charge and current distributions in a stepped leader.
To do this, I developed a simulation to calculate the electric fields produced by one-dimensionalcharge distributions. This simulation can be used to calculate the fields produced by trial sourcesthat could represent a real stepped leader. Once relations have been found between the trialsources and the electric fields they produce, these calculated fields can be compared with themeasured ones. If the measured field has the same features as the electric field from a trialsources, then the real stepped leader is likely to be similar to this trial source.
2 Lightning and negative stepped leaders
There are many different kinds of lightning. The differences are caused by the type of thunder-cloud they connect to, the sign of the net charge transfer, the place where the lightning starts,and many more factors. A common type of lightning is the negative cloud-to-ground flash,where negative charge is transported from the cloud to the ground. A complete description ofthe general processes involved in this type of lightning can be found in [1]. I will only give ashort overview to point out the main processes that could be important for the understandingof stepped leader formation in this type of lightning:
An initial discharge between the positive upper part of the cloud and the negative lower partprovides the right conditions for a stepped leader to form. This stepped leader is a conductingchannel that is negatively charged and it grows in steps from the cloud to the ground. Onceit connects to the ground (usually it connects to a small positive leader that has grown out ofthe ground during the formation of the negative stepped leader), positive charge flows from theground into the conductive channel and neutralizes most of the charge. This is called the return
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stroke, and it produces the bright flash and the loud thunder where lightning is known for. Afterthe return stroke, charge in the cloud redistributes and some of the negative charge flows intothe remnants of the ionized channel. These processes are called the J- and k-processes. Oftenanother leader grows from the cloud to the ground after the first return stroke, but this timeit propagates continuously most of the time. When this so-called dart leader connects to theground, another return stroke takes place. The process of dart leader formation followed by areturn stroke can repeat a couple of times before everything calms down.
As already mentioned in the introduction, it is difficult to do accurate measurements onlightning stepped leaders with the usual equipment like high-speed cameras and VHF antennas.Stepped leaders in laboratory sparks however can be measured much more accurately [2]. In theseexperiments, sparks are created over a distance of several meters in a controlled environment. Thestepped leaders can be observed from a much smaller distance compared to the ones occurring inlightning, and it is also possible to measure the voltage and current throughout the experiment.Therefore, most theories about stepped leaders are based on extrapolating the theory of thebetter known long spark stepped leaders to the dimensions of lightning.
An important concept in these theories is the streamer. Streamers are a type of electricdischarge that occur in a medium when the electric field is high enough. In the case of steppedleaders they start at the tip of the last step, where the electric field is the strongest(fig. 1a).This strong electric field accelerates the few free electrons that are present in the air. Whenthese electrons get accelerated fast enough, they can ionize air molecules by colliding with them.This produces more free electrons that get accelerated. They too can ionize air molecules,creating even more free electrons. This exponential increase in free electrons is called the electronavalanche (fig. 1b). However, not all collisions result in ionization. Free electron can just collideelastically with air molecules. In this case the electrons lose energy through Bremsstrahlung.Another possibility is that the electrons do not transfer enough energy to the air moleculesand only excited them. The photons emitted by the subsequent relaxation could ionize theair somewhere outside of the streamer, which could trigger another streamer. These ionizingphotons can also be produced when electrons recombine with positive ions. It is also possiblethat electrons are captured by air molecules, resulting in negative ions (fig. 1c). The electrons aremuch more mobile than the positive and negative ions in the streamer. Therefore, the ions canbe regarded as almost stationary compared to the electrons. This means that when the electronsare repelled by the negative streamer tip, the positive ions stay behind and shield the electricfield, counteracting the development of the streamer. This results in a streamer of limited size,with positive charge mainly near the tip of the leader, and negative charge at the other end ofthe streamer. When streamers are formed near positively charged objects, the electric field ofthe object is not shielded, but enhanced by the positive ions near the tip. Positive streamershave a higher average propagation velocity and in contrast to their negative counterpart, theygrows continuously most of the time [1].
The formation of a new step in a stepped leader start with the streamers at the tip of theleader. The current that flows through these streamers heat the air. At the intersection ofmultiple streamers, the current is the highest and the air gets heated the fastest. At such anintersection, a space stem can form. This is a region that is hotter, more ionized and moreluminous than the streamers. The space stem also marks the transition from positive streamersto negative ones. When the space stem is heated enough by the current flowing though it, thenegative ions formed in the streamers get unstable and release their captured electrons. Thisresults in a large increase in conductivity, and also in current and luminosity. The space stemis now called a space leader. This space leader has approximately no net charge. The steppedleader induces a positive charge at the side closed to the leader tip, and a negative charge on
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Figure 1: Various processes in a negative streamer. More information about these processes canbe found in the text.
the other side. The positive part of the space leader grows toward the stepped leader tip. Whenthey connect, charge from the stepped leader flows into the space leader and creates a coronadischarge at the end. Charge and current wave from the newly formed step to the cloud inresponse to the charge flowing into the space leader. After the formation of this new step, theprocess repeats.
The length of a step in a stepped leader can vary from several meters to about 200m, witha typical length of about 50 meters. The diameter of the core of the leader channel is probablyless than a centimeter. The time between two step formations can vary from 20 to 100µs. Thespeed at which charge flows from the stepped leader into the space leader is estimated to beabout a third of the speed of light. The conductivity of the stepped leader is in the order of104S/m, which is in the same order as the conductivity of carbon. The linear charge density isapproximately 0.7 mC/m. All these values are found in [1].
Since the diameter of the stepped leader channel is much smaller than the length of a step,and much smaller than the typical distance from which lightning is observed (usually severalkilometers), it is justified to approximate a stepped leader by a line charge.
3 Maxwell’s equations in potential formulation
To calculate the electric fields generated by stepped leaders, we need the classical theory ofelectrodynamics. There are many books about this subject, for example [5]. But to refresh your
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mind, I will give a short overview, starting with the Maxwell equations:
∇ ·E =ρ
ε0, (1)
∇ ·B = 0 , (2)
∇×E = −∂B∂t
, (3)
∇×B = µ0J + µ0ε0∂E
∂t. (4)
These equations describe how a given charge distribution ρ, and a current distribution J, generateelectric and magnetic fields. It is exactly what we need. But they still need to be solved. Oneway to do this is to use the potential formulation of Maxwell’s equation. In this formulation, onedoes not calculate the electric and magnetic field, but potentials from which these fields can bederived. The relation between these potentials and the electric and magnetic field is derived inmost books about electrodynamics, for example [5]. The relation is
E = −∇V − ∂A
∂t, (5)
B = ∇×A . (6)
V is called the scalar potential and A the vector potential. When switching to the potentialformulation of the Maxwell equations, one has the freedom to choose a convenient gauge thatfixes the divergence of A. I use the Lorentz gauge
∇ ·A = − 1
c2∂V
∂t. (7)
Maxwell’s equations can now be written in potential formulation by replacing E and B with thecorresponding derivatives of V and A as described by eq. 5 and 6. After rewriting the divergenceof A by using the Lorentz gauge (eq. 7), the resulting equations can be written in the followingform:
1
c2∂2V
∂t2−∇2V =
ρ
ε0, (8)
1
c2∂2A
∂t2−∇2A = µ0J . (9)
Unlike the original Maxwell equations, eq. 8 and 9 do not imply the continuity equation. It hasto be added to the potential formulation as an additional constraint
∇ · J = −∂ρ∂t
. (10)
Eq. 8 and 9, together with eq. 10, contain the same physics as the original Maxwell equations.The advantage of the potential formulation is that the equations are easier to work with. Forexample, V and A can be written as an integral over the (in our case known) charge and currentdistribution. When r denotes the position of the observer and r′ the position of the source (fig.2), these integrals are
V (r, t) = 14πε0
∫ ρ(r′,tr)|r−r′| dr′ =
1
4πε0
∫ρ(r′, t′)
|r− r′|δ(t− tr) dt′ dr′ , (11)
A(r, t) = µ0
4π
∫ J(r′,tr)|r−r′| dr′ =
µ0
4π
∫J(r′, t′)
|r− r′|δ(t− tr) dt′ dr′ , (12)
tr(r, r′, t) = t− 1
c |r− r′| . (13)
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The only catch here is that ρ and J are not evaluated at a given time, but at a given retarded
observerO
sourcepoint
Figure 2: the definition of the vectors r and r′
time. The physical interpretation of this is that changes in V and A travel at a finite speed: thespeed of light. The potentials at a given observer point in space and time are determined by thecharges and currents at the time they ’emitted’ those potentials. So, ρ and J at a given retardedtime are literally the charge and current distribution as you would see them at the observerpoint.
The retarded time not only depends on the time and position at which you want to evaluatethe potential. It also depends on the position of the source. Since the retarded time will frequentlyappear in big formula, I will omit its arguments to make those formula more readable. Fromnow on, I write the retarded time as tr. But do remember that it still depends on the time andposition of the source and on the position of the observer.
4 Analytic solutions
The equations describing the scalar and vector potential may not seem that complicated. Apartfrom the retarded time, they look exactly the same as the equations for V and A in electrostatics.But the retarded time makes it really hard to find analytic solutions to these equations when thecharge and current distribution depend on time. Only for the simplest charge distributions exactsolutions are known. For slightly more complicated distributions, solutions can only be obtainedby making some approximations (e.g. using multipole expansions), or by solving the equationsnumerically. In this section, I treat some of the analytic solution. They are not very useful todirectly calculate the electric fields produced by stepped leaders. But they can be useful whenthey are used in a numerical calculation to approximate more complicated problems. Or theycan be used to test numerical calculations.
4.1 Moving point charge (Lienard-Wiechert Potentials)
The electric field generated by an arbitrarily moving point charge is perhaps the most importantingredient I need my simulation. This is because every charge and current distribution can beapproximated by a finite set of moving point charges. How this can be done is explained insection 5.2. Also, points with a changing charge are needed to satisfy the continuity equation(eq. 10) at the end of a non-zero line current.
The formula describing the electric fields does not give the electric field as a function of timeat a given observer point. Instead, given a point on the trajectory of the point charge, it gives away to calculate the electric field in a given observer point in space at some later time. The usageof this result is explained in section 5.1. But let’s first find the expression of its electric field.One way to do this is to first calculate the scalar and vector potential of the point charge, andthen use eq. 5 to calculate the electric field. The derivation of the expression for the potentialscan also be found in [6].
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The charge distribution of a point charge with a time-varying charge Q(t), located at positionR(t) (fig. 3), can be written in terms of the three-dimensional Dirac delta function
ρ(r, t) = Q(t) δ3(r−R(t)) . (14)
Plugging this expression into eq. 11 to calculate the scalar potential results in the following
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Figure 3: the definition of R(t)
integral over two delta functions
V (r, t) =1
4πε0
∫Q(t′) δ3(r′ −R(t′))
|r− r′|δ(t′ − tr) dt′ d3r′ . (15)
It is easier to first integrate over space and then over time than the other way around. Afterswapping the order of integration and evaluating the integral over space using the general rulefor integrating over delta functions∫
f(x)δ(g(x)) dx =∑i
f(xi)
|g′(xi)|(16)
(the sum is over all values of xi for which g(xi) = 0) the expression for the scalar potentialbecomes
V (r, t) =1
4πε0
∫Q(t′) δ(t′ − tr)|r−R(t′)|
dt′ . (17)
The retarded time is now evaluated at r′ = R(t′). To solve the remaining integral, we need toknow how many solutions there are to the equation t′ − tr = 0. All these points are neededto evaluate this integral over a delta function. A nice way to determine this is given in [5].The reasoning is as follows: Let’s suppose there are two solutions. Then we have t′ = tr,1 =t − (r −R(tr,1))/c and t′ = tr,2 = t − (r −R(tr,2))/c. Taking the difference of these equations:tr,1 − tr,2 = (R(tr,2) − R(tr,1))/c. This actually means that the average velocity of the pointcharge is equal to the speed of light. In vacuum, this is impossible and gives rise to Cherenkovemission. In a medium with a positive index of refraction, it could be possible. But steppedleaders, the phenomenon that we want to describe with these equations, move slower than thespeed of light in air. Therefore, we must conclude that there is at most one solution to theequation t′− tr = 0 in this case. Assuming that the charge existed for a long enough time, thereis exactly one solution.Now we can use eq. 16 to evaluate this integral over time
V (r, t) =Q(tr)
4πε0
1
|r−R(tr)|1
| ddt′ (t′ − tr)|(18)
=Q(tr)
4πε0
1
|r−R(tr)|1
|1− ddt′ (tr)|t′=tr |
. (19)
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The expression for dtrdt′ in the last equation can be obtained by taking the time derivative of the
defining equation of the retarded time, eq. 13, after replacing r′ by R(t′) (this was due to theintegration over space in eq. 15)
dtrdt′
=d
dt′
(t− 1
c|r−R(t′)|
)(20)
= −1
c
d
dt′
√∑i
(ri −Ri(t′))2
(21)
= −1
c
1
2√∑
i (ri −Ri(t′))2· 2∑i
((ri −Ri(t
′)) · d
dt′(−Ri(t
′))
)(22)
=r−R(t′)
|r−R(t′)|· v(t′)
c. (23)
The vector v(t′) is just a short notation for dR(t′)dt′ . It is the velocity of the point charge at the
time t′. The fraction r−R(t′)|r−R(t′)| is the direction from the position of the point charge to the point
at which the potential is evaluated. The dot is the dot product between this direction and thevelocity. Putting this result back into eq. 19 gives
V (r, t) =Q(tr)
4πε0
1
|r−R(tr)|1
1− r−R(tr)|r−R(tr)| ·
v(tr)c
. (24)
Since |v| < c, there is no need for absolute values in the last fraction.Only an expression of tr in terms of t and r is needed to solve this equation. This means
solving eq. 13 for tr after replacing r′ by R(tr). It turns out that in many cases this is not atrivial task. One of the few non-static cases in which this equation can be solved exactly is whenthe point charge is moving with a constant velocity. When it is uniformly accelerated, one hasto solve a fourth order polynomial. In principle this can be done, but the final result will notclarify much. The solution is horribly long [7].
Once the scalar potential is known, the vector potential can be calculated fairly easily. Thedefinition of current density is
J(r, t) = ρ(r, t) · v(r, t) (25)
where J is the current density, ρ the charge density and v the velocity field of the of the chargedistribution. Using this definition and the expression for the charge density of a point charge(eq. 14), eq. 12 can be written as
A(r, t) =µ0
4π
∫Q(t′) δ(r′ −R(t′)) · v(r′, t′)
|r− r′|δ(t− tr) dt′ dr′ . (26)
Performing the integration over space and time in exactly the same way as done for the scalar
potential, one arrives at the expression for V (r, t) (eq. 24), multiplied by µ0ε0v(tr) = v(tr)c2 :
A(r, t) =µ0Q(tr)v(tr)
4π
1
|r−R(tr)|1
1− r−R(tr)|r−R(tr)| ·
v(tr)c
=v(tr)
c2V (r, t) . (27)
From the scalar and vector potential, the electric and magnetic field can be calculated using eq.5 and 6. For a constant charge Q, This calculation is done in, for example, [5]. The resulting
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electric and magnetic fields for a constant charge are
E(r, t) =Q
4πε0
|r−R(tr)|((r−R(tr)) · u)
3
((c2 − v2)u + (r−R(tr))× (u× a)
), (28)
B(r, t) =1
c
r−R(tr)
|r−R(tr)|×E(r, t) , (29)
u = cr−R(tr)
|r−R(tr)|− v . (30)
To obtain the results for a time-varying charge, we can use the product rule in the derivativesof the potential
∇V (r, t) = ∇
Q(tr)
4πε0
1
|r−R(tr)|1
1− r−R(tr)|r−R(tr)| ·
v(tr)c
(31)
= Q(tr) · ∇
1
4πε0
1
|r−R(tr)|1
1− r−R(tr)|r−R(tr)| ·
v(tr)c
+∇Q(tr) ·
1
4πε0
1
|r−R(tr)|1
1− r−R(tr)|r−R(tr)| ·
v(tr)c
(32)
= (result for constant charge evaluated at tr)
+ Q(tr)∇(tr)
1
4πε0
1
|r−R(tr)|1
1− r−R(tr)|r−R(tr)| ·
v(tr)c
. (33)
The dot above the Q denotes the derivative with respect to t. The explicit expression of the firstterm in eq. 33 is not needed. In the end it will only contribute to the already know expressionfor the electric fields of a moving point charge with a constant charge (eq. 28), with the onlydifference that the charge is must be evaluated at the retarded time. Similar for the vectorpotential:
∂A
∂t=
∂
∂t
µ0Q(tr)v(tr)
4π
1
|r−R(tr)|1
1− r−R(tr)|r−R(tr)| ·
v(tr)c
(34)
= (result for constant charge evaluated at tr)
+ Q(tr)∂tr∂t·
v(tr)/c2
4πε0
1
|r−R(tr)|1
1− r−R(tr)|r−R(tr)| ·
v(tr)c
. (35)
Filling in these derivatives in eq. 5 to calculate the electric field results in the electric field ofa point charge with a constant charge (evaluated at tr), plus an additional term. This termcontains both a time derivative and a gradient of tr. They can be calculated by taking therespective derivative on both sides of the defining equation of the retarded time (eq. 13) and
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solve for it. For the time derivative it goes as follows:
dtrdt
=d
dt
(t− 1
c|r−R(tr)|
)(36)
= 1− 1
c
dtrdt
d
dtr(|r−R(tr)|) (37)
= 1 +1
c
dtrdt
r−R(tr)
|r−R(tr)|· v(tr) . (38)
Solving for dtrdt gives the result
dtrdt
=1
1− r−R(tr)|r−R(tr)| ·
vc
. (39)
The gradient of tr can be obtained in the same way. It is given by the following expression
∇tr = −1c
r−R(tr)|r−R(tr)|
1− r−R(tr)|r−R(tr)| ·
vc
. (40)
Now that we have the derivatives of the retarded time, we can write down the additional termin the electric field of a moving point charge that is due to the change of the charge. Using eq.5 to calculate this field from eqs. 33 and 35, this additional term is
Eadditional =Q(tr)
4πε0
|r−R(tr)|((r−R(tr)) · u)2
u . (41)
The total field is the sum of this additional field and the field of a point charge with a constantcharge (eq. 28)
E(r, t) =1
4πε0
r−R(tr)
((r−R(tr)) · u)2 ·(
Q(tr)
((r−R(tr)) · u)
((c2 − v2)u + (r−R(tr))× (u× a)
)+ Q(tr)u
). (42)
This equation gives the exact electric field generated by a point charge with a time dependentcharge, moving along an arbitrary trajectory. It can be used to estimate the order of magnitudeof the electric field produced by a point-like source, or as a source or sink of charge at the end ofa (possibly moving) current to comply with the continuity equation. But as already mentionedin the beginning of this section, the main use of this equation in my research is to calculate theelectric field of a source once it is approximated by a set of moving point charges.
When the point charge is stationary, then eq. 42 reduces to the well-known Coulomb field.When it is moving with a constant velocity, the electric field can be written explicitly as afunction of time instead of retarded time [5]. The expression for this field is
E(r, t) =q
4πε0
1− v2/c2(1− v2
c2 sin2(θ))3/2 R
R2(43)
where R is the vector from the present position of the particle to the field point r, and θ is theangle between R and v .
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4.2 straight line current with polynomially increasing current
Another problem than can be solved analytically is the potential generated by a straight linecurrent. In particular, a line current that is initially zero, but from a given time onwards can bedescribed by a polynomial. This means that the current can be described by the formula
I = Θ(t− t0) · P (t) (44)
where I is the current, Θ the Heaviside step function and P a polynomial. Since the electricfield is linear in the current, we can calculate the electric field due to each separate term of thepolynomial, and then add all fields together. This means we only have to consider currents ofthe form ΘI(t) = I0 · tn to describe an electric field due to a polynomial current. To simplify theproblem a bit further, we set t0 = 0. The solutions for t0 6= 0 can be obtained by replacing t byt− t0 in the solutions for t0 = 0. A convenient choice of basis is to choose the line current alongthe z-axis, and the observation point on the x-axis. The line current is on the interval betweenz = a and z = b. The observation point is at x = d. The electric field in any other basis canbe obtained by a applying the appropriate basis transformations on the electric fields found inour basis. Note that the continuity equation (eq. 10) says that the charge at the begin and endpoint of the line current changes. But to keep the calculations simpler,the electric field due tothis change in charge is added only at the very end of the calculation.
To calculate the electric field of the line current only, without the changing charge at its endpoints, we need the scalar potential and the vector potential of the line current. The former isthe easiest to calculate and it turns out that it is the same for all line currents. Therefore wewill first calculate the scalar potential. Putting a uniform linear charge density q in eq. 11, weobtain
V (r, t) =1
4πε0
∫q
|r− r′|d3r′ . (45)
This is the same formula as for the static case, because the charge distribution does not changein time. This is the reason why the scalar potential is the same for all line currents we willconsider. In our basis, we can write this integral as an integral over the z-axis. Furthermore wecan write the denominator of the integrand in terms of d and z. We can also move q outside ofthe integral because it is a constant:
V (d, t) =q
4πε0
∫ b
a
1√d2 + z2
dz . (46)
The solution of this integral is
V (d, t) =q
4πε0ln
(b+√d2 + b2
a+√d2 + a2
). (47)
To calculate the electric field due to this scalar potential, we need to calculate the partial deriva-tives with respect to x and z. The former is calculated by first rewriting eq. 47 as the differenceof two logarithms using the rule ln(a/b) = ln(a)− ln(b), and then taking the derivative:
∂V
∂x=∂V
∂d=
q
4πε0
(1
b+√d2 + b2
· d√d2 + b2
− 1
a+√d2 + a2
· d√d2 + a2
)(48)
=q
4πε0
(d
d2 + b2 + b√d2 + b2
− d
d2 + a2 + a√d2 + a2
). (49)
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To calculate the derivative with respect to z we again write eq. 47 as the difference of two loga-rithms. Furthermore, notice that taking the derivative to z is the same as taking the derivativeto −a with the additional condition that b = a+ l, where the constant l is the length of the linecurrent. This means that differentiating to a is the same as differentiating to b.
∂V
∂z= −∂V
∂a=
q
4πε0
(
1 + a√d2+a2
)a+√d2 + a2
−1 + b√
d2+b2
b+√d2 + b2
(50)
=q
4πε0
(1√
d2 + a2
(√d2 + a2 + a
a+√d2 + a2
)− 1√
d2 + b2
(√d2 + b2 + b
b+√d2 + b2
))(51)
=q
4πε0
(1√
d2 + a2− 1√
d2 + b2
). (52)
Now that we have found the scalar potential and its derivatives, we move on to calculate thevector potential. Putting a current of the form I(t) = Θ(t)I0t
n into eq. 12 and only integratingover the current results in
An(r, t) =µ0
4π
∫ b
a
Θ(tr)I0 · tnr|r− r′|
zdz . (53)
Again, we can write the denominator of the integrand in terms of d and z, and move the constantI0 out of the integral. Also replacing tr with its definition given in eq. 13 (and writing it interms of d and z) gives the integral
An(r, t) =µ0I04π
∫ b
a
Θ(t− 1
c
√d2 + z2
)·(t− 1
c
√d2 + z2
)n√d2 + z2
zdz . (54)
The integrand, and hence the integral, is zero when the argument of the step function is smallerthan zero, i.e. when t − 1
c
√d2 + z2 < 0. This is the case when t < 1
c
√d2 + a2, because the
smallest value of z is a. At slightly later times, the step function is zero when z >√
(ct)2 − d2.
This means that we can replace the upper bound of the integral with√
(ct)2 − d2, as long as itis smaller than b. When it is larger than b, the upper bound is replaced again by b. Definingχ = min{b,
√(ct)2 − d2}, the integral at times later than t =
√(ct)2 − a2 can be written as
An(r, t) =µ0I04π
∫ χ
a
(t− 1
c
√d2 + z2
)n√d2 + z2
zdz . (55)
This integral can be solved by making the substitution z = d · sinh(φ). Using the substitutionrule, the integral becomes
An(r, t) =µ0I04π
∫ z=χ
z=a
(t− 1
c
√d2 + (d · sinh(φ))2
)n√d2 + (d · sinh(φ))2
d · cosh(φ) zdφ . (56)
Factoring out the d in both square roots, using the identity 1 + sinh2(φ) = cosh2(φ) and simpli-fying a bit gives
An(r, t) =µ0I04π
∫ z=χ
z=a
(t− d
ccosh(φ)
)nzdφ . (57)
15
Using the binomial theorem to expand the power, and taking the summation and the constantsout of the integral results in
An(r, t) =µ0I04π
n∑k=0
(n
k
)tn−k
(−dc
)k ∫ z=χ
z=a
cosh(φ)k dφ z . (58)
A nice way to solve these integrals is to rewrite the cosine hyperbolic in terms of exponentials,then use the binomial theorem, and then use the symmetry of the binomial coefficient to rewritethe exponentials back as hyperbolic cosines
cosh(φ)k =
(1
2(eφ + e−φ)
)k(59)
=
m=k∑m=0
(k
m
)(1
2
)ke(k−m)φe−mφ (60)
=
m=k∑m=0
(k
m
)(1
2
)kemφe−(k−m)φ . (61)
Simplifying the exponentials in eq. 60 and eq. 61, adding them and dividing the result by 2 (thisis still equal to cosh(φ)k), results in
cosh(φ)k =1
2
(m=k∑m=0
(k
m
)(1
2
)ke(k−2m)φ +
m=k∑m=0
(k
m
)(1
2
)ke−(k−2m)φ
)(62)
=1
2
m=k∑m=0
(k
m
)(1
2
)k (e(k−2m)φ + e−(k−2m)φ
)(63)
=
m=k∑m=0
(k
m
)(1
2
)kcosh((k − 2m)φ) . (64)
Using this, we can calculate the integral in eq. 58:
An(r, t) =µ0I04π
n∑k=0
(nk)tn−k
(− d
2c
)k k∑m=0m 6= 1
2k
[(k
m
)sinh((k − 2m)φ)
k − 2m
]{+
(k12k
)φ
}z=χ
z=a
z
(65)
The term in the curly braces is only used when m = 12k. The integration bounds are given in
terms of z to keep the equation a bit shorter. To switch between z and φ, use the definition ofφ used for the substitution: z = sinh(φ).This is a very messy equation. It is not a very appealing to take a time derivative of this tocalculate the electric field. Fortunately, it is not necessary to this explicitly. It turns out thatthe expression for dA/dt is very closely related to the expression for A, as I will show now.
Instead of taking the time derivative of eq. 65, it is nicer to calculate the time derivative ofeq. 55. Using the Leibniz integral rule rule
d
dt
∫ f(t)
a
g(t, z) dz =
∫ f(t)
a
∂g(t, z)
∂tdz + g (t, f(t)) · ∂f(t)
∂t(66)
16
the time derivative of eq. 55 can be written as
d
dtAn(r, t) =
µ0I04π
(d
dt
∫ χ
a
(t− 1
c
√d2 + z2
)n√d2 + z2
zdz
)(67)
=µ0I04π·
∫ χ
a
n(t− 1
c
√d2 + z2
)n−1√d2 + z2
dz +
(t− 1
c
√d2 + χ2
)n√d2 + z2
· dχ(t)
dt
z (68)
= n ·An−1 +µ0I04π·
(t− 1
c
√d2 + χ2
)n√d2 + χ2
· dχ(t)
dtz . (69)
Filling in χ = min{b,√
(ct)2 − d2} results in a simple equation:
d
dtAn(r, t) =
{0 if t < 1
c
√d2 + a2
n ·An−1(r, t) if t ≥ 1c
√d2 + a2
(70)
This means that we can use the results for the vector potential almost directly for the electricfield: only multiply by an integer. Finally we have everything to calculate the electric field of astraight line current of which the current is a polynomial in time. Combining eqs. 5, 49, 52 and70 gives for the electric field in our coordinate frame
Ex =q
4πε0
(d
d2 + a2 + a√d2 + a2
− d
d2 + b2 = b√d2 + b2
)(71)
Ez =q
4πε0
(1√
d2 + b2− 1√
d2 + a2
)− n ·An−1 (72)
where An−1 is given by eq. 65. Now we only have to add the electric fields due to the changingcharge at the end of the line current. To find the change in charge, you can integrate thecontinuity equation (eq. 10) over an infinitesimal region B around the desired endpoint:∫
B
∇ · J dV = −∫∂ρ
∂tdV (73)∫
∂B
JdA = − ∂
∂t
∫ρdV (74)
Iout = −∂Q∂t
. (75)
To go from the first equality to the second, I used the divergence theorem to rewrite integralof the divergence of J over the infinitesimal volume B as an integral over the boundary of thisregion, ∂B. The resulting integral is the total current going out of the endpoint, Iout. Theintegral over the charge density at the endpoint, ρ, is the total charge Q at this point. Applyingthis result to our current, we get
Q(t) =
{−I(t) at z = a
I(t) at z = b(76)
When setting the charge to zero at t = 0, these equations can be integrated to find the chargeat the endpoints as a function of time:
Q(t) = ±∫ t
0
I(t′) dt′ = ±∫ t
0
Θ(t′)I0t′n dt′ =
{0 if t < 0
nI0tn−1 if t > 0
(77)
17
where the plus-sign is for the point z = b, and the minus-sign for z = a. Now we can put theQ(t) and Q(t) of both points in eq. 42, together with the position, and with zero velocity andacceleration. The retarded time can be calculated explicitly this time, because the distance ofthe endpoints to the observer point stays constants. Filling in eq. 13 gives the retarded times:tr,a = t− 1
c
√d2 + a2, tr,b = t− 1
c
√d2 + b2. Putting thes in eq. 42 as well gives
Eendpoints(r, t) =1
4πε0
1
d2 + a2
(Q(t− 1
c
√d2 + a2)
√d2 + a2
− I(t− 1
c
√d2 + a2)
)(dx− az)
+1
4πε0
1
d2 + b2
(Q(t− 1
c
√d2 + b2)
√d2 + b2
+ I(t− 1
c
√d2 + b2)
)(dx− bz) . (78)
The total electric field is the sum of eqs. 71, 72 and 78, where the charge at the endpoints isgiven by eq. 77.
5 Simulation
The method I use to find relations between the charges and currents inside a stepped leaderand the electric field it produces, is to calculate the electric field of many different sources thatall could somehow represent a stepped leader. When some characteristics of these trial sourcescan be linked to characteristic in the produced electric field, then these calculated fields can becompared to the measured field of a real stepped leader. When the measured field has one ofthese characteristics, then it is likely than the real stepped leader has the same characteristicsas the trial sources that produces these calculated fields.
Analytic solutions alone do not help very much when you want to calculate the electric fieldsproduced by these trial sources. These solutions only exist for very special, and most of the timeextremely simple, sources. But when the real source can be approximated accurately enoughby a set of these special sources, these analytic solutions in combination with the principle ofsuperposition can be very useful. Let a computer approximate the real source by a set of yourspecial sources to the desired accuracy, and let it add the fields from each individual source. Thisis the approach I use in my simulation.
5.1 The general idea
The special source I use in my simulation to approximate the real source is the point charge. Asmentioned earlier in section 4.1, an arbitrarily moving point charge has no closed expression forthe electric field as a function of observer time. But given a time, position, velocity, accelerationand charge of a point source, you can calculate the electric field and the corresponding arrivaltime at any observer point. There are several ways to use point charges. One way is to choosea time grid at the observer points. Then you have to calculate the retarded position of thepoint charge (which requires you to know the charge and current distribution at the retardedtime), and finally you can calculate the electric fields at the observer points. The advantageof this method is that you calculate the electric fields only at the times you want to observethem. This comes in handy when you need a regular time grid, for example if you want to doa Fast Fourier Transform of the fields. But this method has its downsides. For every observerpoint, you have to calculate the retarded charge and current distribution for every point in yourtime grid. Furthermore, when given the source, it is not trivial to choose a suitable time gridthat captures all the details of the electric fields produced by the sources. This is because rapidchanges in the source results in rapidly electric field changes at later times at the observer point.
18
Moreover, since this time delay depends on the position of the observer point, this time grid hasto be chosen separately for every observer point.
I have chosen another way to use the point charges. Instead of choosing a time grid at theobserver point, I choose a time grid at which I evaluate the charge and current distribution. Atthese times I pick the point charges to approximate the source, and then calculate for each pointcharge the fields it produces at the observer points and at what time they arrive. When there ismore than one point charge, the arrival times of the electric fields of different point charges doin general not coincide. Therefore, you have to interpolate the fields of each point charge on asingle time grid at the observer point before you can add the fields.When you want to calculate the fields produced by a known source, one of the big advantagesof choosing the time grid at the source is that it is relatively easy to construct a time grid anda spatial grid that capture all the details of its dynamics, and therefore also in the electric fieldsit produces. Therefore, these grids can be used for all observer points. Only some refinementsmay be needed depending on the position of the observer point. Another advantage is that oncethe position, velocity, acceleration and charge of each point charge is calculated for one observerpoint, these values can be reused for all other observer points. You may need some additionalvalues, but you don’t have to evaluate the whole source again for each observer point, becausethe time grid and the position of the point sources and already captured most of the details ofthe source. Advantages specific to the calculation of the fields of extended sources are discussedin section 5.2. A more detailed explanation of the simulation is given in section A
5.2 Implementing extended sources
When you use point charges which have a constant charge to approximate a charge and currentdistribution, then, by the principle of superposition, the electric field of the original source isapproximately the sum of the electric fields of the point charges. In the limit of infinitely manypoint charges with an infinitesimal charge, this sum becomes an integral:
Esrc(r, t) =
∫dEpoint(r, t) (79)
After changing the integration variable from Epoint to Q, and then from Q to V (V is the volumeat a given ’normal’ time, not at a constant retarded time):
Esrc(r, t) =
∫dEpoint
dQ
dQ
dVdV (80)
The derivative of the electric field of a point charge to its charge can be calculated from eq. 28:just leave out the Q in this equation. The derivative of the charge to the volume is the chargedensity ρ. Filling this in gives:
Esrc(r, t) =
∫ρ(r′, tr)
4πε0
|r− r′|((r− r′) · u)
3
((c2 − v2)u + (r− r′)× (u× a)
)dV (81)
where the integration is over the volume of the charge distribution to sum the contribution of allthe point charges. The position, velocity, acceleration and charge are evaluated at the retardedtime. To approximate the integral, one could divide the source in a finite number of point charges,calculate for each point charge the electric field as a function of time in the observer point, andthen add the fields as described in section 5.1. But this way of solving converges only linearlywith the number of point charges. To get a faster convergence, and a better estimate for theerror due to the discretization, I rewrote the integral such that a better numerical integration
19
scheme can be used to solve it. By changing the integration variable from V to Vobs = V (tr),the volume of the source as the observer sees it at a given retarded time, the integral becomes:
Esrc(r, t) =
∫dEpoint
dQ
dQ
dV
dV
dVobsdVobs (82)
where the derivative of V to Vobs is given by
dV
dVobs(tr) =
dV (t)dt
∣∣∣tr
dV (tr)dtr
dtrdt
=1dtrdt
= 1− r− r′
|r− r′|· vc. (83)
Eq. 39 was used to fill in the expression for the time derivative of the retarded time. The integralI actually use in my simulation is
Esrc(r, t) =
∫ρ(r′, tr)
4πε0
|r− r′|(
1− r−r′|r−r′| ·
vc
)((r− r′) · u)
3
((c2 − v2)u + (r− r′)× (u× a)
)d3r′ (84)
where the integration is this time over the volume of the source evaluated at a constant retardedtime. My simulation only uses line sources, where the line can be any, possibly extending,parameter curve. This means that the current density at a particular point in the source cannow be represented by a number. The direction is then given by the tangent vector at that point.
To determine the integration interval in eq. 84, my simulation takes the trajectory of theboundary points of the source as input. For each observer point, the electric field as a function oftime is determined for these point charges. Then the fields are interpolated on a new time grid atthe observer. The positions of the point sources at these particular observer times are calculatedby performing a linear interpolation. Once the positions of the boundary points are know, morepoints on the parameter curve can be added in order to use a numerical integration scheme toapproximate the integral to the desired accuracy. The corresponding time can be calculated byfilling in t, r and r′ in eq. 13. My simulation uses a recursive implementation of the adaptiveSimpson’s method.
It is also possible to add points in the charge distribution that you always want to use inthe numerical integration. This is useful when the charge or current distribution has sharppeaks, or steep steps in it. The points can be chosen at or close to these regions to be sure thatthey are integrated in the right way. This improves the numerical stability and the speed ofthe calculations. Adding these points is done by specifying the trajectory it follows along theparameter curve and the time interval in which you want to use this point. These points aretreated in the same way as the two boundary points.
The downside of this numeric integration is that it is harder to reuse the source evaluations ofthe extra points that are added for the numerical integration. Also, the position of the boundarypoints must be calculated iteratively for each point in the observer time grid. But when thereare only a few observer points, it does not really matter that many source evaluations are notreused. And the higher order integration makes up for the time needed to calculate the positionof the boundary points, and for the time needed for the additional source evaluations needed bythe other approach to archive the same accuracy.
In order to use either approach, an expression for the velocity and the acceleration of a pointcharge in terms of the source parameters is needed. To do this, you have to write the source asa sum of a source consisting entirely of positive charge, and one consisting entirely of negativecharge. For each of these two sources, the velocity of the point charge can be obtained fromρ(r, t) and J(r, t) of the respective source. Rewriting the definition of the current distribution,
20
eq. 25, gives the velocity in the general three dimensional case:
v(r, t) = J(r, t)/ρ(r, t). (85)
The acceleration for each of these two sources can be obtained from v, ρ and J of the respectivesource by taking the material derivative of this velocity to time. In the general three-dimensionalcase this is:
a =Dv
Dt=∂v
∂t+∇vv (86)
where ∇vv is the covariant derivative of v, along v. Since my simulation only works with onedimensional charge and current distributions, it needs the expressions for v and a in terms of ρ,the magnitude of the current density J , and the tangent vector of the parameter curve. All thesequantities are functions of the curve parameter s. For simplicity we will only consider curvesthat are parametrized by arc length.
The expression for v stays almost the same. Since the direction of J is given by the tangentvector T, it can be rewritten as:
v(s, t) =J(s, t)
ρ(s, t)T (87)
The expression for the acceleration can be simplified. The covariant derivative can be replacedby a derivative with respect to the arc length s:
a =∂
∂t
(JT
ρ
)+ v
∂
∂s
(JT
ρ
)(88)
=J
ρT +
J
ρ
∂
∂tT− J
ρ2ρT + v
(∂J
∂s
1
ρT +
J
ρ
∂T
∂s− J
ρ2∂ρ
∂sT
)(89)
The second term, the time derivative of the tangent vector, is zero because the parameter curvedoes not change shape. Furthermore, using eq. 87, every occurrence of J/ρ can be replaced bythe magnitude of the velocity v. By applying the continuity equation, The derivative of J to sis equal to −ρ. Filling this in gives:
a =J
ρT + 0− v
ρρT + v
(− ρρT + v
∂T
∂s− v
ρ
∂ρ
∂sT
)(90)
=
(J
ρ− 2v
ρ
ρ+v2
ρ
∂ρ
∂s
)T− v2 ∂T
∂s(91)
This means that the point charge accelerates along the curve if the current changes in time (firstterm), if the charge density changes in time or the current changes along the curve (second term),and if the charge density changes along the curve (third term). The last term in this equation isthe centripetal acceleration due to the curvature of the curve at the position of the particle.
6 Simulation tests
Before the simulation can be used for calculating electric fields for real complicated charge dis-tributions, it needs to be tested. This testing is done by letting the simulation calculate the
21
electric field produced by simple sources for which there exist an exact solution. Only when thesimulation gives the right result within the specified tolerance, it passes the test. Each source Iuse tests a different part of the simulation.
First the implementation of the point source was tested. The simulation calculated theright field for a stationary point charge, a stationary dipole, a point charge moving with aconstant velocity and an oscillating dipole. The test of the stationary point charge shows thatthe simulation gives an accurate result when there is just one point charge with zero velocity andacceleration. The test of the moving point charges shows that it also gives the right result whenthe point charge has a nonzero velocity, and the test of the static dipole shows that the field ofdifferent point charges are added in the right way. The test of the oscillating dipole shows thatthe simulation also works properly when there are several point charges with a nonzero velocityand acceleration.
Second, the implementation of line sources was tested. The simulation calculated the rightfield for a stationary line source, a very short line source moving with a constant velocity (thisfield closely resembles the field of a point charge moving with a constant velocity), a straight linecharge with a current flowing trough, and a ring charge that rotates along its symmetry axis.The test of the stationary line charge shows that the simulation can approximate a simple staticline source accurate enough by stationary point charges, and that this approximation gives theright electric field. The test of the line charge with a current flowing through indicates thatthe simulation also gives the right result when these point charges have a nonzero velocity, andthe test of the rotating ring shows that the simulations works properly if these charges have anonzero acceleration. The test of the very short moving line charge tells that the simulationgives the desired accuracy when the end points of the source are moving.
7 Simulation results
Now that the simulation has been tested, it is time to calculate the fields of some sources thatcould represent a stepped leader. For the stepped leader model, I assume that the leader channelhas a constant charge density, and that charge flows from the cloud into the stepped leaderchannel when a new step is formed to keep this charge density constant. This means thata current flows throughout the channel when a new step is formed. I will only calculate theelectric fields of a simple source to show the capabilities of the simulation.
Instead of calculating the electric fields for a whole stepped leader all at once, I split up thestepped leader in the individual old steps and the new step. By the principle of superposition,the total electric field of the stepped leader is the sum of the electric field of the individual oldsteps and the new step. Each old step is modeled by a straight line segment with a constantuniform charge density. The current that flows through it is equal to its charge density timesthe extension velocity of the tip of the leader (assuming there are no branching points betweenthis step and the leader tip). The new step is modeled by a superposition of an extending linecharge with a constant uniform charge density, and a peak of charge density at the tip with atail only in the region where the leader has already been. Just like the old steps, this new stepis a straight line segment. For the peak, I take one half of the Gaussian function
ρ(s, t) =2q
σ√
2πe−
12 ( s−smaxσ )
2
. (92)
In this formula, q is the charge contained in one half of this Gaussian peak, σ is the standarddeviation, smax(t) is the position of the top of the peak (which is the same as the position of thetip of the leader) and s is the distance from the starting point of the new step, measured along
22
the step. s is zero at the point where the new step is connected to the rest of the leader, and ispositive on the interval between this starting point and the tip of the leader. Since the electricfields are proportional to the charge of the source when the dimensions of the source are muchsmaller than the distance to the observer, I will ignore the fields that could be generated by theconnections between the old steps. These connections are much shorter than both the old stepsand the new step, and therefore have a relatively small charge.
A schematic of the velocity of the tip of the leader as a function of time can be seen in fig.4. The velocity is zero until tstart. From this time on, it increases in trise seconds to vmax. Inthis time interval, the velocity is described by a 3rd degree polynomial that is zero at tstart andis vmax at (tstart + trise). Its derivative is zero at the end points of this time interval. From(tstart + trise) to (tstart + tpulse − trise), the velocity stays constant. In the time interval from(tstart+tpulse−trise) to (tstart+tpulse), the velocity goes to zero, again according to a 3rd degreepolynomial whose derivative is zero at the end points of this time interval. When the velocity iszero, it stays zero until the next step of the leader is forms. The total duration of this velocitypulse is tpulse. The distance traveled during this pulse can be found by integrating it over time.It is given by a simple expression: stotal = vmax(tpulse − trise).
Figure 4: v(t)-diagram of the tip of the trial stepped leader
Since the simulation uses point charges to approximate the source, the electric field due tothe increase and decrease of charge at the end of the steps (both the old and new ones) isautomatically included in the calculations. The field due to the accumulated charge is ignored.These charges will produce a dipole field that decreases as r−3. Not only will this contribution benegligible in the electric field of the individual steps, but in a real stepped leader the accumulatedcharge of one step will cancel with that of the neighboring step. Only at the beginning of thevery first step of the leader this charge does not cancel. If necessary, it is possible to take thischarge into account. But I have omitted it.
The steps generate both an electric field that has mainly a radial component, and an electricfield which has a its main component in the direction transversal to the direction of the step.The field with the radial component is produced by the change of charge at the end of a step.But at one end of the step the charge increases, and on the other end it decreases in the samerate. Therefore the electric field due to this change in charge does not decreases as r−1, butas r−2 when the observer distance is much larger than the length of a step. The field with thecomponent in the transversal to the direction of the step is generated by the acceleration ofcharge. Since all charges accelerate in the same direction, this electric field decreases as r−1.This means that for large observer distances, this field is dominant over the radial electric fieldproduced by the change of charge at the end of the steps. Therefore, the electric field produced
23
by the steps is expected to be the larges in the directions transversal to the direction of the step.Also, the field strength is expected to be the largest when the step is perpendicular to the lineof sight of the observer, and the lowest when the step is along the line of sight of the observer.
Figure 5 shows the simulated setup. The step can be an old step or a new step. The time atwhich the first signal arrives is defined as t = 0. In this setup, the electric field is expected to bethe largest along the x-axis according to the previous argument.
Figure 5: simulated setup
Figure 6 shows the x-component of the electric field as a function of time generated by anold step for various values of θ. The values of the other parameters are given in table 1. Again,as expected, the y-component of the electric field is much smaller than the x-component, abouta factor 80. The y-component is therefore not shown in this figure.
Figure 7 shows the x-component of the electric field as a function of time generated by theextending line charge part of the new step for various values of θ. The values of the otherparameters are given in table 1. As expected, the y-component of the electric field is muchsmaller than the x-component, about a factor 60. The y-component is therefore not shown inthis figure.
Figure 8 shows the x-component of the electric field as a function of time generated by thepeak part of the new step for various values of θ. The values of the other parameters are givenin table 1. Again, The y-component of the electric field is much smaller than the x-component,about a factor 70. The y-component is therefore not shown in this figure.
24
parameter old step extending line peaklstep 50m 50m 50mdobs 10km 10km 10kmvmax
13c
13c
13c
trise 200ns 200ns 200nsρ 1C/m 1C/m n/aq n/a n/a 1Cσ n/a n/a 0.033
Table 1: Values of the parameters used in the calculations for the electric field of the old step,the extending line part of the new step and the peak part of the new step
Figure 6: x-component of the electric field generated by an old step. Calculated for variousvalues of θ.
25
Figure 7: x-component of the electric field generated by an extending straight line with a constantuniform charge density. Calculated for various values of θ.
Figure 8: x-component of the electric field generated by the peak part of the new step. Calculatedfor various values of θ.
26
8 Conclusion and Discussion
The simulation can be used to calculate the electric field produced by one-dimensional trialsources that represent stepped leaders. The results for a simple trial source show that only theradiation part of the electric field matters for realistic observations, unless the stepped leadergoes straight towards the observer. This can be concluded from the fact that the peaks infigs. 7, 6 and 8 start when the electric fields arrive that are emitted at the moment that thecharges in the source start to accelerates. Also, the sign of the electric field corresponds withthe direction of the acceleration, and that the width of the peaks corresponds with the durationof the acceleration. The broadening of the peaks for lower values of θ is due to the increase ofthe difference in distance from the endpoints of the step to the observer. The first peak in fig.7 is lower than the second peak because during the acceleration the line charge is shorter thanduring the deceleration. It has therefore less total charge during the acceleration and hence emitsa weaker electric field.
The charges and charge densities are all taken equal to one in the calculation of the electricfield. To obtain the fields due to a more realistic charge density or charge density, just multiplythe results by this other charge or charge density (expressed in SI units). The electric fieldgenerated by the new step is a linear combination of the electric field generated by the extendingline charge part and the peak part. The ratio between the two depends on the ratio betweenthe charge density of the line charge and the charge of the peak. A realistic charge density issomewhere between 10−4 and 10−3 C/m [1]. To make an estimate of the tip charge, I assumethat it is a small sphere of charge and that only this charge causes the dielectric breakdown atthe tip of the streamer. The electric field of the tip charge is just the coulomb field
E =qtip4πε0
1
r2. (93)
The electric field at which air breaks down is approximately 106V/m at an altitude of 6km whensome hydrometeors are present [1]. Filling this in, solving for qtip, and assuming that the electricfield of the tip charge causes the air to break down up to a meter away from the tip (i.e. r= 1m), gives qtip = 10−4C. This is the same as the charge in 1m of channel up to an order ofmagnitude. This means that the ratios of the electric fields in figs. 6, 7 and 8 could be a realisticratio within an order of magnitude. In the used model, the electric field from all the individualold steps together are thus much larger than the electric field generated by the new step. In othermodels however, this could be totally different. For example, when the you assume that mostof the charge is concentrated in the leader tip, that some of this tip charge spreads out over theleader channel as the leader grows, and that no charge flows from the cloud into the leader whenit is not connected to the ground. Then the old steps emit only a very faint electric field. Themeasured electric fields are then only generated in the region around the leader tip. To find outwhat models are realistic, calculate the electric fields they predict and compare them to actualmeasurement.
Other models can be tested by trying different or more complicated charge and currentdistributions as trial sources. For example, another velocity profile can be used. And instead ofusing a uniform charge density in the old and new steps, a non-uniform charge density can beused, or a a wave of charge and current that travels through the stepped leader. By trying varioussources, it is possible to determine what characteristics of the source could be measured by radiotelescopes and how accurate the parameters describing these characteristics can be determined.
Before the results of the simulation can be compared to real measurements, the calculatedelectric field has to be filtered in the same way as the observing equipment filters the electricfield (e.g. due to band filters). Since the simulation does not make any assumptions about this
27
filtering in its calculation of the electric field, any filter can be used.The simulation can be optimized to make it more efficient. Obvious improvements are higher
order interpolation and integration. Another improvement could be using Jefimenko’s equations[5] (in conjunction with source and sink charges for continuity) in stead of point charges. Also,it could be desirable to reuse more of the point source evaluations.
References
[1] V. Rakov and M. Uman, Lightning: Physics and Effects. Cambridge University Press, 2003.
[2] S. Xie, R. Zeng, J. Li, and C. Zhuang, “Theoretical and experimental study of the formationconditions of stepped leaders in negative flashes,” Physics of Plasmas, vol. 22, no. 8, 2015.
[3] C. J. Biagi, M. A. Uman, J. D. Hill, and D. M. Jordan, “Negative leader step mechanismsobserved in altitude triggered lightning,” JOURNAL OF GEOPHYSICAL RESEARCH-ATMOSPHERES, vol. 119, JUL 16 2014.
[4] O. Scholten and A. van den Berg, “Lightning research.” Olaf Scholten internal note.
[5] D. Griffiths, Introduction to Electrodynamics. Pearson international edition, Prentice Hall,1999.
[6] Wikipedia, “Lienard-Wiechert potential — Wikipedia, the free encyclopedia,” 2016. Online;accessed 06-July-2016.
[7] Wikipedia, “Quartic function — Wikipedia, the free encyclopedia,” 2016. Online; accessed13-July-2016.
28
A Summary of the simulation components
The simulation is written in C++ using an object-based approach. This appendix contains a listof all the classes with their functionality that are used in the simulation
• Parameters
– Read parameters from a text file and store them in a Parameters object.
– All objects that need at least one of these parameters have a reference to this objectas a data member.
• Grid
– Stores the electric field and the corresponding observer times
– Can save its contents to and load contents from a file.
– Can make a vector plot of the electric field at a given observer time
– Can make a plot of one of the electric field components as a function of time in asingle observer point
• Source
– Contains all the separate sources of which the total source consists of (pointsources,linesources, etc.)
– Sources can be added
– Can calculate the electric field of the whole collection of sources:
∗ The electric fields and the corresponding observer times are calculated for eachindividual source
∗ A new time grid is generated in such a way that it still captures all the details ofthe electric fields of the individual sources once they are interpolated on this newtime grid.
∗ The electric fields of the individual sources are interpolated on this new time gridand are added.
• AuxPointsource
– Calculate the electric field of a moving point source. It is only used in the classLinesource.
– Store the charge, position, velocity and acceleration each time they are calculatedsuch that they can be reused when possible.
– Time points at which the point charge certainly must be evaluated can be addedmanually.
• Linesource
– Calculates the electric field of a one-dimensional, time-varying charge distributionthat is defined along a parameter curve.
– As a constructor argument, it needs the parameter curve along which the line source isdefined, and the positions of the (possibly moving) boundary points of the line source.It also needs the charge distribution along this curve.
29
– More moving points can be added between the two boundary points to mark theposition and time of large and/or fast changes in the charge distribution. For exampleat the top of a peak, or at the upper and lower part near a steep step, or around awave front.
– It can calculate the electric field from the line source in the way described in section5.2
– The charge distribution evaluations at the boundary points and the manually addedpoints are stored, such that they can be used when possible. They are not stored forthe additional points needed for the adaptive integration.
• ParametercurveBase derivatives
– Specifies the position and the first and second derivative of this position to arc lengthof the parameter curve, all as a function of arc length. These functions have to beimplemented in each derived class.
– Objects from these classes can be passed other object to indicate the curve alongwhich the line source exists.
• ChargedistParamCurveBase derivatives
– Specifies all the interesting quantities of the charge distribution, like the charge densityand current density (and their time derivatives) as a function of time and positionon the line source. The functions describing these quantities must be implementedfor each derived class. Be aware that the continuity equation should be obeyed, i.e.charge must be conserved. This is not needed for the construction of the derivedclasses (it will not give a warning or an error if theis is not the case), but is absolutelynecessary to get sensible results.
– A parameter curve along which the charge distribution exists must be specified. Thereis no need to indicate its boundaries. They are added at the construction of a Line-source object.
– Objects from these classes can be passed to a Linnesource object to indicate the chargedistribution along the curve along which the line source exists.
• TrajectoryParmCurveBase derivatives
– Specifies the position, velocity and acceleration of a point moving along a parametercurve as a function of time. These functions must be implemented for each derivedclass.
– A the parameter curve which describes the shape of the trajectory must be specified.
– Object from these classes can be passed to Linesource objects to indicate the positionof the boundaries of the line source as a function of time.
30
1
Lightning Stepped Leaders
Freddie Hendriks
2
Introduction
● LOFAR● Interpret measurements
3
Lightning
● Thundercloud● Initial breakdown● Stepped Leader● Attachment ● First return stroke● K- and J-processes● Repeated: Dart leader + return stroke + ...
4
Stepped leaders
● Observations● Lab experiments● Theory and models
Biagi et al. Journal of Geophysical Research – Atmospheres, 2014
Xie et al. Physics of Plasmas, 2015
Stepped leader in triggered lightning
Laboratory spark
5
The theory
● Basic electrodynamics
● Is very difficult
6
My simulation
● The general idea● Implementation● Test results
B Presentation
31
7
General idea
● Calculate fields of trial sources● Superposition of point charges:
– Can approximate any source
– Exact solution known
8
Implementation
● Pick important points● Calculate their fields● Add points in between● Use numerical integration
observer
9
The integral
11
Constant velocity
● Simulation ● Analytic
12
Uniform acceleration
● Uniform acceleration for a finite time
● Integrated field strength: Same as delta pulse!
2e–111e–11 3e–11 4e–11 5e–11 6e–11 7e–11 8e–11 9e–11 1e–100
t (s)
E (
N/C
)
13
Static Line
● Simulation ● Analytic
14
Moving line
● Simulation ● Analytic
15
Rotating ring
● Rotating ● Static
16
To simulate
● Step formation– Charge flowing into the space leader
– Current through previous steps
17
Possibilities
● Lightning research● Other simulations involving retarded time
C Poster
For the international conference ARENA 2016 in Groningen, I made a poster together withDanny Sardjan to present our research progress on lightning stepped leaders. The abstract isshown below. The poster is on the next page.
Lightning stepped leaders; LOFAR data and simulations
The initial and yet fundamental process in a typical cloud-to-ground lightning strike includesthe propagation of a very faint and charged channel which is called stepped leader. The exactmechanism for the step leaders is not understood. The reason for this is that the temporaland/or spatial resolution of the devices exploited for observing this phenomenon has not beensufficient. The radio interferometric array of LOFAR however is capable to measure radio signalswith 1 ns temporal resolution. Thus LOFAR can measure the radio pulses emitted by steppedleaders at multiple times during the formation of the steps and locate the positions of the pulseswith sub-meter accuracy. This provides with new possibilities to test and probe the theories ex-plaining the propagation of a stepped leader. We are currently processing the data measured byLOFAR, and in addition, are preparing a simulation tool to calculate the radio signals expectedon the basis of current models. By comparing the simulation and experimental data, we aim todetermine the characteristics of stepped leader formation.
34
Lightn
ing
step
ped
lead
ers; LO
FAR
data
and
simu
latio
ns
In th
e le
ft curta
in p
lot w
e se
e th
e d
ata
as m
easu
red a
nd se
e th
at th
e p
ulse
s we re
ceiv
e
arriv
e a
t diff
ere
nt sta
tions a
t diff
ere
nt tim
es.
In th
e rig
ht cu
rtain
plo
t we sy
nch
ronize
d th
e tim
e o
f the p
ulse
s betw
een
diff
ere
nt
statio
ns. S
ince
we a
re a
ble
to sy
nch
ronize
the tim
es w
e a
re a
ble
to u
se a
time o
f arriv
al
meth
od in
ord
er to
dete
rmin
e th
e tim
e a
nd lo
catio
n o
f the e
mitte
d p
ulse
s. As o
f now
this
has n
ot y
et b
een
done a
nd still is a
work in
pro
gre
ss.
LOFA
R M
easu
rem
ents
Ste
pped le
ader th
eory
a. S
tream
ers fo
rm a
t the le
ader tip
. The a
ir gets
heate
d a
nd b
eco
mes m
ore
condu
ctive. A
space
ste
m m
ight a
ppear a
t the b
ranch
ing o
f a stre
am
er.
b. W
hen th
e sp
ace
stem
gets h
ot e
nough
, it can
transfo
rm in
to a
bid
irectio
nal le
ader. It is n
ow
m
ore
ionize
d, m
ore
condu
ctive a
nd m
ore
lum
inou
s.c. T
he le
ader tip
and th
e b
idire
ctional le
ader g
row
to
ward
s each
oth
er.
d. A
new
step is fo
rmed w
hen th
ey co
nn
ect.
Charg
e fl
ow
s in, a
nd a
coro
na b
urst fo
llow
s.e. T
he p
roce
ss repeats.
a. T
he v
ery
hig
h e
lectric fi
eld
at th
e tip
acce
lera
tes
free e
lectro
ns.
b. a
ccele
rate
d e
lectro
ns co
llide w
ith a
ir mole
cule
s a
nd io
nize
som
e o
f them
. Th
is resu
lts in m
ore
fre
e e
lectro
ns th
at a
re a
ccele
rate
d: a
n e
lectro
n
avala
nch
e fo
rms.
c. Ele
ctrons m
ay in
stead b
e ca
ptu
red b
y a
ir mole
cule
s, e
xcite a
ir mole
cule
s or re
com
bin
e w
ith p
ositiv
e
ions. P
hoto
ns p
roduce
d b
y re
laxatio
n o
r re
com
bin
atio
n m
ight io
nize
air m
ole
cule
s else
where
.
+
-
+
++
++
--
--
-
-
Initia
l bre
akd
ow
n in
thund
erclo
ud
----
----
----
--
--
+
-
+
-
-
-
-- - - -
Bid
irectio
nal le
ader
----
Ste
pp
ed le
ader tip
-
Positiv
e stre
am
er
Ste
p fo
rmatio
n in
stepped le
ader
ac
de
Stre
am
er
+-
-+
ele
ctron
positiv
e io
n
neutra
l particle
photo
n
+-
--negativ
e io
n
+-
+-
+-
+-
+-
+-
+-
+- +
-
--
---
-
+-
+-
+-
+-
+-
+-
+-
+-
+
+-
-
--
---
-
-
--+-
+-
+-
+-
+-
+-
+
++
--
-
--
---
-
+--
+-
--
ab
c
b
Sim
ula
tion
time
source nr.
evalu
atio
n tim
es o
f p
oin
t charg
es
1 2 3
ob
serv
er
ele
ctric field
s at o
bse
rver
Ex per source
time
Ex total
time
A se
ries o
f con
necte
d
straig
ht lin
e cu
rrents is
use
d to
model th
e
stepped le
ader.
Usin
g th
e p
rincip
le o
f super-
positio
n a
nd a
num
erica
l in
tegra
tion sch
em
e, th
e lin
e
curre
nts a
re a
pprox
imate
d b
y
a se
t of p
oin
t charg
es.
Choose
for e
ach
poin
t charg
e a
su
itable
time g
rid to
evalu
ate
the
ele
ctric field
it pro
duce
s. Take
sm
all tim
e ste
ps in
inte
rvalls
where
the p
roduce
d e
lectric fi
eld
ch
an
ges ra
pid
ly
For e
ach
poin
t charg
e, ca
lcula
te th
e
ele
ctric field
and th
e co
rresp
ondin
g
arriv
al tim
e a
t the o
bse
rver
Com
bin
e th
e o
bse
rvatio
n tim
e g
rids o
f th
e in
div
idual p
oin
t charg
es in
to a
new
tim
e g
rid.
Then in
terp
ola
te th
e e
lectric fi
eld
s of th
e
poin
t charg
es to
obta
in th
e fi
eld
s at th
e
new
time p
oin
ts. A
dd th
ese
field
s to o
bta
in th
e to
tal
ele
ctric field
on th
e n
ew
time g
rid.
The e
lectric fi
eld
s travel a
t the sp
eed o
f light.
There
fore
the o
bse
rver re
ceiv
es th
em
at so
me
late
r time, d
ependin
g o
n th
e d
istance
the
field
s have to
travel. W
hen th
e p
oin
t charg
e is
movin
g, th
is means th
at th
e tim
e g
rid sp
acin
g
chan
ges.
evalu
atio
n tim
e
observer time
ob
serv
er tim
e v
s evalu
atio
n tim
e
In th
e to
p le
ft plo
t we se
e th
at LO
FAR
sta
tions th
at w
ere
use
d fo
r lightn
ing
measu
rem
ents. T
he sta
tions in
the
mid
dle
are
core
statio
ns, w
hile
the o
ute
r sta
tions a
re re
mote
statio
ns. Fo
r our
measu
rem
ents w
e u
sed th
e lo
w-b
an
d
ante
nnas.
Danny S
ard
jan, Fre
ddie
Hendrik
s
Ste
pp
ed
lead
ers m
ake
the fi
rst co
nd
uctin
g p
ath
that co
nnects th
e
thun
derclo
ud
to th
e g
round
in a
neg
ativ
e
cloud
-to-g
round
disch
arg
e. It sta
rts from
an in
itial d
ischarg
e in
the clo
ud
.
~1
0 m
Here
we se
e th
e p
ulse
shape o
f a sin
gle
pulse
(the stro
ngest
pulse
) as m
easu
red b
y LO
FAR
. The sa
me sta
tions a
re u
sed
here
as in
the cu
rtain
plo
t. Fo
r a ste
pped le
ader w
e e
xpect a
sharp
peak w
ith little
stru
cture
besid
es th
is. In th
e p
lot w
e se
e so
me stru
cture
and
since
this is co
nsiste
nt in
alm
ost e
very
statio
n w
e k
now
that
this is in
trisic to th
e sig
nal a
nd n
ot ju
st noise
. At th
e
mom
ent it is u
ncle
ar w
hy th
ere
is this stru
cture
in th
e p
ulse
sh
ape a
nd w
hat th
is means.
time (m
icro se
conds)
normalized pulse amplitude
35