An investigation of the reconfiguration of the electric field in the stratosphere following a lightning event

Journal of

ELECTROSTATICS ELSEVIER

Journal of Electrostatics 36 (1996) 331-347

An investigation of the reconfiguration of the electric field in the stratosphere following a lightning event M.E. B a g i n s k i * , A.S. H o d e l , M. L a n k f o r d Department c~fElectrical Engineering, Auburn UniversiO,, 200 Brown Hall, Auburn AL 36849, USA Received 14 October 1994; accepted after revision 20 September 1995

Abstract

In recent papers Baginski and Hodel (1994) and Baginski and Jarriel (1994) discussed some peculiarities in the late-time electric field and Maxwell current density signatures following a lightning event. In the present paper we extend the analysis to investigate the temporal relationship between the vertical and horizontal electric field in the stratosphere (altitudes 30 50 kin) and consider how a quasi-planar region can be defined by introducing the angular orientation (0 = tan 1(E,/E,)) between the respective field components. The geometry, constitutive parameters and assumptions used in the earlier modeling are employed here also. Special attention is given to two dimensional graphical representations of the behavior of the electric field that clearly illustrate that a relatively thin segregated region can be defined between a time-varying upper and lower region were the maximum rate of change of the orientation of the field occurs at Zp(t) = hslog(T(z)/t). The research reinforces the model presented by Greifinger and Greifinger. An empirical model is also derived from previous research that is in good overall agreement with the simulations.

Keywords: Lightning; Atmospheric electricity; Maxwell current; Return stroke; Middle atmosphere

I. Introduction

Recently, Baginski and Hodel [13, Baginski and Jarriel [2], and Baginski [33 investigated the behavior of simulated electric field waveforms following the introduction of charge perturbations into the atmosphere. The primary focus of these studies was to examine the waveform signatures that were not associated with propagating energy but rather with the late-time effects that correspond to the non-propagating component (electrostatic and magnetostatic energy) of the waveform. The charge perturbations were centered at altitudes corresponding to thunderstorm charge

* Corresponding author. 0304-3886/96/$15.00 ~ 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 3 8 8 6 ( 9 5 ) 0 0 0 5 4 - 2

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M.E. Baginski et al. /Journal of Electrostatics 36 (1996) 331 347

centers (6 and 10 km) and the induced electric fields were observed (simulated) at altitudes of ~ 30-50 km. Several observations were made that indicate that the use of the lossy full wave equation is generally necessary unless sufficient justification can be made for using the quasi-static assumption. It was further observed that the vertical electric field signatures tend to have the same fundamental characteristics throughout the region investigated (stratosphere) with the amplitude and relative duration being a function of the position of the charge perturbation and the point of observation of the electric field. An obvious trend in the field signatures in previous studies [4, 5] is that the vertical electric field tends to be compressed temporally and in magnitude as altitude is increased. The present study was initiated as a next logical step in the process of generating a concatenation of information about electric field behavior in the stratosphere that stems from late-time electromagnetic effects. In particular, the vertical and horizontal electric field signatures were analyzed for differences in temporal structure; plausible reasons for observed simulated behavior are presented with a supporting empirical model. Based on contrasts between the behavior of the horizontal and vertical electric fields as a function of position and time, the primary observations of this study are (1) The time rate of change of the angular orientation 0 = tan- 1 (E~/Er) of the simulated electric field varies as a function of position and time, and indicates that the "moving plate theory" of Greifinger and Greifinger [6] is a valid and useful assumption for some lower atmospheric (stratospheric) research (see below). (2) An assumption that the Maxwell current density is primarily upward oriented may be used to develop an empirical model whose simulated electric field data closely resembles that generated by models that explicitly include the horizontal component of the Maxwell current density [7]. (A brief discussion of the impact of this research relative to the understanding of the behavior of the Maxwell current density is presented in Section 2.) As a courtesy to the reader, due to the significant role played by the Maxwell current density in several extant models [8], a presentation of relevant background information on the physics governing the Maxwell current density is given in Section 2.2. This background is central to the development of the empirical model formulation presented in Section 4. Several plausible reasons for the behavior observed in simulated data, based on the governing physics, are discussed throughout the paper. Greifinger and Greifinger [6] describe a "time-dependent boundary" between conducting and non-conducting regions in the atmosphere; this boundary descends from the upper limit of the model to a position dictated by the dielectric relaxation time, i.e., zp(t) = hs log (z(z)/t)

(1.1)

where Zp denotes the altitude of the boundary, hs is the scale height and r = ~o/a(z) is the dielectric relaxation time at the point of observation. Zp(t) obviously tends to infinity as t tends to zero; however, this is a result of the impulsive nature of the Hale-Baginski model [7] where at t = 0 the atmosphere can be viewed as a pure dielectric (i.e., plate height assumed as z = ~ at t = 0). Our observations of simulated field orientation vs. time reinforce their concept of a segregated atmospheric model.

M,E. Baginski et al./Journal of Electrostatics 36 (1996) 331-347

333

This result is further reinforced by Hale and Baginski's comments on flux realignment [7] based on this "moving plate" model; a discussion on the use of the moving plate model, including boundary conditions, is presented in [7]. The remainder of this paper is organized as follows. Motivation and background for the numerical simulations performed in this study are presented in Section 2, including a discussion of Maxwell current density as it relates to this study. Following this, the results of the numerical simulations are presented in Section 3. It is in Section 3 that supporting data for the moving plate theory of Greifinger and Greifinger [6] is presented. An empirical model based on observations from this and previous studies is presented in Section 4, and conclusions from this study are summarized in Section 5.

2. Simulations The model used in the numerical simulations of this study is identical to that used in [2, 3, 7, 9]. The motivating physics used to develop the model is presented in these previous studies, and thus an abbreviated treatment is given in this section. (A detailed discussion of the numerical simulations used in this study is contained in the PIER publication [3]. Source code for the simulations is available from the authors upon request.) Prior to the discussion of the simulation model itself, a discussion of the Maxwell current density and the role it plays in lightning transients in the stratosphere is presented in Section 2.1. Further physical background is described in Section 2.2. The actual simulation model and its partial differential equations is presented in Section 2.3. 2.1. Maxwell current density

The current densities generated by charge perturbations associated with lightning consist of either conduction (resulting from charge movement) or displacement (~:o~E/&) terms. The sum of these current densities are referred to as the Maxwell current density (Jm): Jm = V x H = Jp + ~oc~E/~t,

(2.1)

Jp = the sum of all conduction current densities including source (impressed) terms, ~ogE/c~t = displacement current density.

(2.2)

One of the assertions made by several research groups is that the transient event (as well as steady-state phenomena) considered in this research could best be investigated by considering the atmosphere's response in terms of the Maxwell current densities' behavior. The Maxwell current density has several characteristics that are necessary to discuss prior to the specific details of this study. The use of Maxwell currents and current densities to describe the electromagnetic response of the atmosphere has been investigated by Krider and Musser [10], Baginski [3], Hale and Baginski [7], Baginski and Jarriel [2] and Nisbet [8] among others. They have all suggested that

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M.E. Baginski et al. /Journal of Electrostatics 36 ¢I996) 331 -347

the thunderstorm is fundamentally a current source and therefore should be investigated in terms of current densities rather than electric fields. At the present time this concept is still somewhat controversial for investigating the "late time" effects [1 1]. However, in view of the fact that charge or current perturbations are the fundamental cause of the induced fields, it seems reasonable to describe the entire thunderstorm using charge and current related forcing functions that result in transient electric field phenomenon. As such, the Maxwell current density will likely play a crucial role in any study related to thunderstorm electromagnetic behavior. The Maxwell current density has several properties that may be manipulated to more accurately describe both the local and global effects of lightning transients on the atmosphere. Probably the most important of these is that the divergence of the Maxwell current density is zero ( V ' J m = 0) making it a solenoidal quantity. As a consequence of this, the lines of Maxwell current density Jm form closed loops. Along these Maxwell current density stream lines the displacement term is usually dominant at low altitudes (Jp<> eo?E/c~t). This solenoidal character of the Maxwell current density ( J m = 17x H , I7. [ V× HI = 0) implies that the electrical parameters of the entire path of circulation strongly affect its response; conversely, if the Maxwell current density along the streamline is mathematically describable, the corresponding streamline's electric fields may be formulated by simple time domain convolution [12].

2.2. Physical background

The method used to simulate the vertical and horizontal electric fields is based on the finite element routine used in prior investigations conducted by Baginski and Hodel [1], Baginski and Jarriel [2], and Baginski [3]. A contrast of the vertical and horizontal electric fields generated by the charge redistribution immediately following a simulated lightning return stroke is shown in Fig. 2. The electric field recovery shows the horizontal electric field to be decaying exponentially in time and the vertical electric field "peaking" as discussed by Baginski [3]. Since it is known that both components of the electric field would decay exponentially in time [ 12] if a temporally and spatially invariant conductivity was used, it is obvious that the departure from purely exponential decay was caused by the spatially varying conductivity. The equation that gives the most insight into why this is the case is the continuity equation: I7. J + Op/~t = 0, where J = aE, a = fro ez/h (commonly used atmospheric approximation; 1 see [13]), h = scale height (6 km), V.E = p/e,, with the permittivity

The use of more complexconductivityprofilesa(z) was investigatedby Baginski in [3]; while a change in the conductivityprofilehas a quantitative impact on the electricfield waveformsignatures generatedvia simulation, it does not change the qualitative information and trends in these signatures. Therefore,since several other investigations have used the simple exponential conductivity a(z) = a0e~/h, an exponential conductivitywas selectedfor simplicityin implementationand to allow for comparison to previous work.

M.E. Baginski et al. /Journal of Electrostatics 36 (1996) 331-347

335

being equal to that of free space t0 = 8.854×10-12F/m and hence, Op/c~t + aP/eo + E~a/h = 0. By inspection, it is obvious that E~ generally behaves in a nonexponential manner. This was discussed by Baginski [3], Hale and Baginski [7] and Baginski and Jarriel [2]. In order to investigate the probable physics responsible for this behavior (that 17a dictates the Maxwell currents orientation to a large extent) a two-step procedure was used. (1) The relevant literature was reviewed and found to reinforce the above assertion that the behavior of the electric field, following the introduction of a charge perturbation, was strongly dictated by the gradient of the conductivity. More specifically, due to the Va being vertically oriented, the vertical field component of both the Maxwell current density and the electric field are more severely affected than the horizontal components. The literature search also revealed that the Maxwell current density is generally upward oriented in the region of interest to this study, with very little flux tube expansion [7]. This trait has been used in the development of models that depict the Maxwell current behavior, and will be so used in this study to develop the empirical model. The simulated data also provide the necessary information to determine the Maxwell current density. (2) Based on this assumption, a simple empirical model that shows good overall agreement with trends in the simulated data (see Section 4) was developed. This empirical model comprises a mathematical description of the dominant physics that govern the trends and behaviors observed in simulated data. The relative magnitudes of the E-field components at a given point of observation in space are contrasted as well as the rate of change of the relative components, i.e., the change in "angular orientation" 0 = tan-l(EJE,). (Indeed it is the time rate of change of angular orientation dO/dt that most strongly supports the "moving plate" concept put forth by Greifinger and Greifinger [6].) Details of the model's development are presented in the next subsection.

2.3. Simulation description In this study, the numerical model is identical to models used by Baginski [9], Hale and Baginski [7], Baginski and Jarriel [2] and Baginski [3], and hence only a brief overview will be given; see [1, 3] for further discussion. The region selected (Fig. 1) is contained within a perfectly conducting right circular cylinder with radius of 80 km and height of 110 km. The earth's surface is modeled electrically as a perfect conductor (lower plate). Typical values of 0.001 to 0.01 mho/m are given for the conductivity [14] of the earth's surface, while 10-1, to 10-13 mho/m is typical for the adjacent atmosphere's conductivity. This is a difference of more than 10 orders of magnitude, making the earth's surface appear to be an electrically perfect conductor with respect to the atmosphere. The conditions at the other boundaries are somewhat arbitrary. The simulations of interest were found to be insensitive to increases in either the radial (80 km) or vertical (110 km) limit. The radial boundary condition (80 km) was tested as both a perfect conductor and as a perfect dielectric; for both conditions the simulations results were approximately the same and the former case was chosen arbitrarily. Special consideration was given to the possible effect on

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M.E. Baginski et aL /Journal of Electrostatics 36 (1996) 331 347 z

conducting p

l

a

t

~

/ 110 km

/ Fig. I. Simulation region.

the model's accuracy of assuming a perfectly conducting layer as the 110 km boundary condition. A series of simulations was made with the upper boundary increased in 5 km increments from 80-110km; no appreciable difference was observed in the field signatures produced in these cases. The Hall and Pederson components of the conductivity (present above ~ 70 km) are neglected in the formulation. These components tend to prolong the atmosphere's transient response (i.e., reduce damping caused by the conductivity) and therefore simulations based on a scalar conductivity in this region are likely to underestimate the duration of the transient event at high altitudes, but would have a negligible effect on the simulations of interest to this study [31 One may use Maxwell's equations to derive a single equation in which the electric field is dependent on the source charge and current densities [13] Vx

V × E = - p o ( a O E / O t + 3 J ~ / ~ t ) - l~oeoOE2/~t 2

or 17p/~o = 172E - - p o ( a S E / S t

+ 8Js/Ot) - poeo8E2/St:

(2.3)

where Js = source current density associated with return stroke current and p = charge density. The resulting second-order partial differential equation is analytically solvable for only the simplest cases. To pursue this problem further either simplifying assumptions must be made or a numerical solution must be developed (the latter approach is used here). The continuity equation as implemented in this study is derived by taking the divergence of the Maxwell current density [10] 0 = V.(V×H)

= 17.(¢E + J, + %t3E/at)

= ¢Plso + Va. E + ~?p/at + a,

(2.4)

M.E. Baginski et al./Journal qf Electrostatics 36 (1996) 331 347

337

where 17. J~ = Gs = source of charge perturbation (deposition of return stroke current at charge centers). This neglects any charge accumulation or current related phenomena associated with the lightning return stroke channel. Eq. (2.4) describes the electrodynamic response of the atmosphere to charge perturbations. The simulation is thus characterized by Eqs. (2.3) and (2.4) subject to the boundary conditions given above. A detailed discussion of the finite element code employed in the simulations is given by Baginski [3]. It is important to underscore the fundamental difference in the forcing function (charge-related) used here relative to alternative approaches presented in previous work (typically current-related forcing function). Much of the existing literature focuses on the propagating phenomenology associated with the return stroke current. The simulations in the present paper neglect the electromagnetic field transients caused by the lightning return stroke itself; rather, only the effects of the charge perturbation caused by the return stroke are used. It should be noted that a currentrelated transient (e.g., current loop) is allowable in a charge neutral region but charge movement through space does not necessarily imply a zero-volume charge density. Furthermore, a charge perturbation may be considered separately from the currents that cause them (charge perturbation implies charge generation at a point in space). In the latter case, movement of charge is required (conservation of charge), but the relative contribution of each (charge vs. current) mechanism to the resulting total electric field may be analyzed separately. Initially (prior to the first iterative time step), no charge is assumed displaced and the electric field everywhere is assumed zero. In lightning research, fields resulting from the return stroke current I are usually considered the source of propagating electromagnetic energy [15]. The late-time (quasi-static) electromagnetic fields that are considered here are generated by charge redistribution following the return stroke.

3. Results of the simulations The graphical representations of the data generated by the simulations described in Section 2 are presented in this section. It should be noted that these simulations explicitly neglect the propagating field components generated by the lightning return stroke current for t ~< 2 ms. Also neglected in the simulations are the possibility of the creation of secondary sources stemming from the initial Maxwell current "spike" into the middle and upper atmosphere that may be a cause of red sprites and blue jets [16]. However, the effects of these fields and secondary sources are negligible for the time scales involved with the late-time (quasi-static) field effects addressed in this study; that is, for times t > 2 ms, the waveforms presented provide a correct representation of the behaviors and trends simulated in this study. Fig. 2 identifies the spatial and temporal variation of the horizontal and vertical electric fields at points of observation of 0 < r < 35 km and altitudes of 30 k m < z < 50 km. Baginski and Hodel [1] have considered the error between numerical solutions based on only a solution of the continuity equation and those that include the full set of Maxwell's equations. That study indicates that, for the schema considered in the present

338

M.E. Baginski et al./Journal o f Electrostatics 36 (1996) 331 347

horizontal field data - z=3Okm • ....

4

i

3:

horizontal field data - z=5Okm .

i

i

•~

i

i ..... i

0.6-

0.4'ID

E

....... _

1

/lO

0 1

~ 2

..

0.2"

ILl

O-

20 3

4 30

r (kin)

t (sec)

t (sec)

vertical field data

vertical field data

! i ...... '

2.5 .!"

"

:

"l

1

32 ~ ' ~ 3 0

'."

2

"

i

-

~

...... i . . . . . i. . . . . . . . . . .

~" 1.5 "

0

~~ 00.51

r (km)

0

0

l l 0

1

~

0

r (km)

4

t (sec)

t (sec)

Fig. 2. Contrast: vertical and horizontal field waveforms.Fields from 30 to 50 km follow trends shown above. study, the difference between these two methods is insignificant relative to information of interest, and therefore the figures shown in this study could have been generated by either method. The former method (continuity equation) was selected to generate the figures in order to simplify extraction of the simulated data from program output.

M.E. Baginski et al./Journal qf Electrostatics 36 (1996) 331 347

339

It is a general observation that following the onset of the peak electric field, both field components show a more rapid decay as altitude is increased. This trait has been mentioned and is in keeping with conventional wisdom [12]. There are two additional focal points of interest that arise after considering the results. The first is that in all cases simulated, the horizontal component of the electric field showed a quasiexponential decay at a rate corresponding to the dielectric relaxation time of the atmosphere at the point of observation followed by behavior that has been referred to as "long tails" I-7]. A second point, also cited earlier, is that the vertical electric field in all cases studied here shows a "peaking" phenomenon discussed in detail by Baginski and Jarriel [2]. This will be considered later in the formulation of the empirical model (Section 4). The "peaking" phenomenon in the vertical electric field is attributed to the vertical orientation of Va (see discussion in Section 2 and [2]). The behavior of the electric field orientation (Fig. 3) 0 = tan- ~(EJEr) and its time rate of change dO/dt were investigated; it was observed that the region of greatest rate of change dO(r, z, t)/dt corresponded to region of the "moving plate" identified by Greifinger and Greifinger I-6] (see Eq. 1.1). This altitude corresponds to a region separating the conduction and displacement current density as discussed in Section 2.1. Figs. 4 6 show behavior that supports the moving plate model. These figures show the progression of the spatial variation of the horizontal and vertical E-fields during the initial 3 s of field recovery following the introduction of the charge perturbation. The plate-like behavior may be observed in the lower-right surface plot of each figure. The plot displays the time rate of change dO~dr of the E-field's angular orientation as a function of position. At t = 23.98 ms, dO/dt is roughly homogeneous over all radial distances and altitudes, indicating that for this time the region of interest can be viewed as a dielectric. However, at t = 439 ms, the "leading edge" of the plate, that is, the onset of a region of rapid change of the angular orientation 0 of the electric field, may be observed at an altitude of 50 km. The plate-like region, i.e., points of observation where dO/dt shows rapid change, may be observed descending through altitudes 43 and 35 km in Figs. 4 and 5, respectively. A "trailing edge" of the plate, i.e., a termination of rapid change in the angular orientation of the electric field, can be observed at ~ 47 km (t = 1.484 s), 45 km (t = 2.049 s), 43 km (t = 2.504 s), and 37 km (t = 3.011 s). This observation strongly reinforces the idea that a segregated model will be useful tool in analysis of thunderstorm transients. Such an approach should be effective in the reduction of the computational burden associated with numerical techniques, and would involve examination of each region of interest on the basis of either a purely displacement or a conduction current density. It is apparent from Fig. 2 that horizontal E-fields achieve their peak values nearly instantaneously following the charge injection (within the margin of error incurred by numerical integration techniques) which suggests an impulsive Maxwell current in the horizontal direction. This observation is employed in the empirical model presented in Section 4. In contrast, peak values of the vertical E-field occur progressively later as altitude is decreased from 50 to 30 kin. This phenomenon is apparently caused primarily by the dielectric relaxation time at the point of observation of the field and by the upward orientation of 17a. However, it should be noted that if the charge perturbation was introduced at an altitude above the point of observation then its

340

M.E. Baginski et al./Journal of Electrostatics 36 (1996) 331-347

Orientationof E:altitudez=3Okm

OrientationofE:altitudez=4Okrn

:!:::.

ili

!

2 ".i.''i

1.6 " i"

.i

~1.5 i,

i

~ 1 t

i

~ 1.4

:r

o

)

: ~

.......

o.o

(kin) 0

(see)

-...... '" " "": i/: (

3~~430 2 t (sec)

1o

/20

r (km)

Orientatioof n E:altitudez=5Okrn (:

......

. .....

....

4

2,

.~c1.5

~_

.....

~- 0.5. ~ O0

0

0 1

2 3 t (sec)

4 30 r (km)

Fig. 3. E-field angular orientation: z = 50 km. recovery time would dictate the peak field behavior [5]. Since there is a substantial delay between the time that charge injection ceases and the time that the vertical E-field magnitude reaches its peak, it is clear that the Maxwell current density will tend to remain upward oriented (i.e., eoOE/Ot +crE in the positive z direction). The

M.E. Baginski et al./Journal of Electrostatics 36 (1996) 331-347 d (angular orientation)/dt at t--0.0006708 sec

341

d (angular odentation)/dt at t=0.02398 sec

80~

6040. 73

20. O,

O~

-203O

r (km)

30

altitude (kin)

a

~

•

•

~i~

.

:

d (angular orientation)/dt at t=0.6783 sec

0

~

0

~

35

40

r (km)

•

0

30

30 altitude (km)

d (angular orientation)/dt at t=0.439 sec

i

~

45

altitude (km)

5O 3O

r(km)

, -0.5.

"

0

10

;0

r(km) altitude (km)

Fig. 4. Time rate of change of the angular orientation O(t) of the E-field: t = 670.8 las-0.6783 s.

horizontal component of the Maxwell current density is small relative to the vertical component since the aE term is positive when the eo~E/Ot term is negative (see Eq. 2.1), thereby causing a type of cancelation between the two terms. Therefore, as briefly alluded to by Baginski and Jarriel [2], the net Maxwell current density has

M.E. BaginsM et al./Journal o f Electrostatics 36 (1996) 331 347

342

d (angular orientation)/dt at t=0.7471 sec

....

..

.

.................

d (angular orientation)/dt at t=1.017 sec

:~

o.4 •i.~

.

i

0.2

i

°"~ -0.2 t!

o.1

Oo

'

-0.4

....o

"/~

-0~o

~

~

•*v

45

" . . . . 2010 rkm

50 JU

(

)

3

3~

altitude (kin)

d (angular orientation)/dt at t=l. 147 sec

40

45 altitude (km)

50

30

,(kin)

d (angular orientation)/dt at t=1.484 sec

0.1 " 0.1-

~

0

0

-0.1

~ m 0- .i. -o

0

"

20 3

" '0

35

40

45

altitude (km)

50 30

-0.2r (km)

30

r (km)

altitude (km)

Fig. 5. Time rate of change of the angular orientation O(t) of the E-field: t = 0.7471 1.484 s.

a primarily upward profile at least to the point where the peak in the vertical electric field is observed. Discussion is presented in the next section of the development of an empirical model based on these observations of the simulated Maxwell current density.

M.E. Baginski et al./Journal of Electrostatics 36 (1996) 331 347

d (angularorientation)/dtat t=2.049sec

,ii •

•

d (angularorientation)/dtat t=2.504 s e c

i

f~

0.1

343

i

~

i

0.05 "

.

o

o l\/,,\\l/Hlf~.,.....

~

::

-_o.o~t:~~.~.. I........... /oo,f~ V ..... ...... /,o 0

1

5

0

~

;

0

~-o.o5J~-~'/

............

i,:,~/.. o, 1 ...... ..... ......

r(km) 0

1

;

0

altitude (krn)

~

3

;

/o /,o 0

r(km)

altitude (kin)

d (angularorientation)/dtat t=4 sec

d (angularorientation)/dtat t=3.011 s e c

0.05 -

0.02 -

0-

0-

-0.05-

-0.02 : i

"

.

"

/o

0

l/,O

.

-0.1 -0.15 30 r

. ......

i. . . . . •

10 20

-0.04 -0.06(km) 30

altitude (km) Fig. 6. T i m e r a t e o f c h a n g e o f t h e a n g u l a r o r i e n t a t i o n

20

35

40 45 altitude (km)

O(t) o f

50 30

r(km)

t h e E-field: t = 2 . 0 4 9 M s.

4. Formulation of the empirical model 4.1. Peaking effect at early times It is known that mathematical [5, 6] and numerical [13] solutions are available for transient or pseudo-transient events (charge injected rapidly but not necessarily

344

M.E. Baginski et al./Journal of Electrostatics 36 (1996) 331 347

assumed to be impulsive in nature) that are representative of the scenario considered here. They do not, however, provide the scientist with information about the actual mechanisms that determine the field's behavior. Consider now that the Maxwell current is almost entirely vertically oriented in the stratosphere [7] for the time frames of interest to this work. This allows the use of an assumption that, to first order, the Maxwell current will be assumed in the vertical direction (only for the transients in the stratosphere considered here). After considering several researcher's work in the area of Maxwell current densities /-8] a simple signature was derived based on investigations by Greifinger and Greifinger/-6] and Hale and Baginski [7]. Using the moving plate theory it was observed that the Maxwell current decayed temporally approximately as lit. Since the actual lightning event is not an ideal impulse of charge, as assumed in Hale and Baginski's model, a more realistic model was developed. The Maxwell current was assumed to be given simply by (1 -

Im--

e -t/t) t

where ~ -- rise time of the injected current waveform (5 x 10-s s) and hence, the vertical Maxwell current density is given by

-t/~,l

Jm= VxH=k(l-e t

where k is proportional to the magnitude of the charge perturbation and point of observation of the field. The vertical electric field can then be found by solving the equation: Jm =

~7× H = ~E + eoOE/?t

which has a solution given by Ez = A exp ( - t/r) f ( Vx H)exp (t/T)dt + B exp ( -- t/v) where A and B are constants related to the relative position of the point of observation from the charge perturbation. Due to variations in atmospheric parameters, it would not be advantageous to derive the constants A and B from the simulated data for the entire stratosphere considered here. Sample waveforms of this empirical model are shown in Fig. 7. An alternative way of viewing the peaking effect would be that, provided that the "moving plate" proposed by Greifinger and Greifinger [6] did not pass through the region of interest (r < t), the decaying electromagnetic energy is being "squeezed" into a smaller region (via the collapsing plate) and, therefore, even though the majority of the transient behavior is already past, the residual electric field would still exist and

M.E. Baginski et al./Journal of Electrostatics 36 (1996) 331 347

345

Sample forms of empirical model l

I

I

I

I

I

I

Waveform 1: A=10, B=10

>

-o3 I.kl

I-

Waveform 2: A=I 0, B=2

I

I

°o

I

o.,

]

I

Time (sec)

Fig 7. Sample forms of empirical waveform:~ = 2 s.

decay in a non-exponential manner. After the imaginary "plate" [6] has been lowered through the region the late time Maxwell current density would continue, becoming predominately a conduction current. 4.2. Late-time effect

The late-time tails observed on all of the simulated electric field signatures are due solely to the continuing conduction current density aE. A likely reason for this phenomenon is shown as follows. The displacement current density ~gE/Ot is negligible at the time scales when these late-time tails are observed, since the time constant eo/a<
V a . E + a V . E = Vtr. E + ap/co.

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M.E. Baginski et al. /Journal of Electrostatics 36 ?1996) 331 347

For the conductivity model used, Va = (a/h)d~, and hence aEz -

ap/to

Jz = - ahp/eo

~

Ez =

where

-

hp/eo,

P = f l V. J dt.

That is, the conduction current is maintained by an accumulation of charge in the region, and the late time E-field tails observed in the simulations may likely be attributed to this behavior.

5. Conclusions The primary purpose of this study has been the characterization of vertical and horizontal electric fields in terms related to their relative magnitudes (angular orientation) that occur in the stratosphere as a result of charge perturbations related to lightning. In the simulations, the vertical electric field exhibits a significant time delay prior to the onset of its peak value where the horizontal field shows relatively immediate decay. This behavior has been observed in prior investigations involving both simulated and measured behavior. Plausible reasons for this behavior are discussed. It was found that the behavior could be viewed in at least two ways that lead to a more basic understanding of the underlying physics. The first method assumed that the Va dictates to a large extent both the electric field and the Maxwell current density orientation for early times (times that are of the order of 5r, where r = e/a). The second method employs the "moving plate" proposed by Greifinger and Greifinger to "squeeze" the electromagnetic energy in the transient response into an increasingly smaller region so that transient behavior is compressed temporally as altitude increases.

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