estimating power, energy, and action integral in …estimating power, energy, and action integral in...

ESTIMATING POWER, ENERGY, AND ACTION INTEGRAL IN

ROCKET-TRIGGERED LIGHTNING

By

VINOD JAYAKUMAR

A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2004

Copyright 2004

by

Vinod Jayakumar

ACKNOWLEDGMENTS

I thank Dr. Vladimir Rakov for his infinite patience, guidance, and support

throughout my graduate studies. I would like to thank Dr. Martin Uman and Dr. Doug

Jordan for their valuable suggestions during the weekly lightning conference. I thank Dr.

Megumu Miki for responding to all my questions. I sincerely thank Jason Jerauld, Jens

Schoene, Rob Olsen, Venkateshwararao Kodali, and Brian De Carlo for helping me with

the data and software, and for other innumerable favors (without which I would not have

been able to complete my thesis). Research in my thesis was funded in part by the

National Science Foundation.

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TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................................................................................. iii

LIST OF TABLES............................................................................................................. vi

LIST OF FIGURES .......................................................................................................... vii

ABSTRACT..................................................................................................................... xiii

1 INTRODUCTION ........................................................................................................1

2 LITERATURE REVIEW .............................................................................................2

2.1 Cloud Formation and Electrification ......................................................................3 2.2 Natural Lightning Discharges.................................................................................5 2.3 Mechanism of NO Production by Lightning ........................................................10 2.4 Rocket-Triggered Lightning .................................................................................11

2.4.1 Classical Rocket-Triggered Lightning .......................................................11 2.4.2 Altitude Rocket-Triggered Lightning.........................................................13

2.5 Estimates of Peak Power and Input Energy in a Lightning Flash ........................14 2.5.1 Optical Measurements and Long Spark Experiments ................................14 2.5.2 Electrodynamic Model ...............................................................................17 2.5.3 Gas Dynamic Models .................................................................................24

3 ESTIMATING POWER AND ENERGY ..................................................................29

3.1 Methodology.........................................................................................................29 3.2 Experiment............................................................................................................31

3.2.1 Pockels Sensors ..........................................................................................31 3.2.2 Experimental Setup ....................................................................................35

3.3 Electric Field Waveforms .....................................................................................37 3.3.1 V-Shaped Signatures with ∆ERS = ∆EL.......................................................37 3.3.2 V-Shaped Signatures with ∆ERS < ∆EL and Field Flattening

within 20 µs .....................................................................................................38 3.3.3 Signatures with ∆ERS (t) < ∆EL and no Flattening within 20 µs.................39

3.4 Analysis of V-Shaped E-Field Signatures with ∆ERS = ∆EL ................................39 3.4.1 Data Processing ..........................................................................................39 3.4.2 Power and Input Energy .............................................................................42 3.4.3 Statistical Analysis .....................................................................................48

iv

3.4.4 Error Analysis.............................................................................................57 3.4.5 Channel Resistance and Radius..................................................................63

3.5 Analysis of V-shaped E-field Signatures with ∆ERS < ∆EL and Field Flattening within 20 µs............................................................................................................73

3.6 Analysis of V-Shaped E-field Signatures with ∆ERS (t) < ∆EL and no Field Flattening within 20 µs ..........................................................................................82

4 CHARACTERIZATION OF PULSES SUPERIMPOSED ON STEADY

CURRENTS ...............................................................................................................89

4.1 Initial Stage in Rocket-Triggered Lightning.........................................................89 4.1.1 Introduction ................................................................................................89 4.1.2 Statistical Characteristics of ICC Pulses ....................................................92

4.2 M-Components ...................................................................................................110 4.2.1 Introduction ..............................................................................................110 4.2.2 Statistical Characteristics of M-Components ...........................................110

5 RECOMMENDATIONS FOR FUTURE RESEARCH...........................................119

LIST OF REFERENCES.................................................................................................120

BIOGRAPHICAL SKETCH ...........................................................................................123

v

LIST OF TABLES

Table page 2-1: Lightning Energy Estimates .......................................................................................26

3-1: Summary of peak current and ∆EL statistics for 8 strokes exhibiting V-shaped electric field signatures with ∆ERS = ∆EL. ................................................................38

3-2: Summary of peak current and ∆EL statistics for 5 strokes with ∆ERS < ∆EL and flattening within 20 µs or so.....................................................................................38

3-3: Summary of peak current and ∆EL statistics for 18 strokes with ∆ERS (t) < ∆EL and no flattening within 20 µs.........................................................................................39

3-4: Power and energy estimates for strokes having V- shaped E-field signatures with ∆EL= ∆ERS. ...............................................................................................................47

3-5: Dependence of peak power and energy on errors in the value of E-field peak and its position on the time scale. ...................................................................................62

3-6: Resistance and channel radius for strokes having V- shaped E-field signatures with ∆ERS = ∆EL ...............................................................................................................72

3-7: Power and energy estimates for strokes having V- shaped E-Field Signatures with ∆ERS < ∆EL and field flattening within 20 µs or so...................................................75

3-8: Power estimates for strokes having V- shaped E-Field Signatures with ∆ERS (t) < ∆EL (t) and no flattening within 20 µs ......................................................................82

4-1: Summary of parameters (geometric means) of ICC pulses......................................108

4-2: Parameters of ICC pulses in Gaisberg tower flashes as a function of season. .........110

4-3: Geometric means of the various parameters of M-components and ICC pulses. ....117

vi

LIST OF FIGURES

Figure page 2-1: Electrical structure of a cumulonimbus........................................................................5

2-2: Natural lighting discharges for a cumulonimbus..........................................................6

2-3: Four types of discharges between cloud and ground....................................................7

2-4: Downward negative cloud-to-ground lightning ...........................................................8

2-5: Classical rocket-triggered lightning ...........................................................................12

2-6 Altitude rocket-triggered lightning..............................................................................13

2-7: Relative spectral response versus wavelength for the photodiode detector used by Krider (1966) and Krider et al. (1968). ....................................................................15

2-8: Measurement of photoelectric pulse of lightning.......................................................16

2-9: Conceptual flow of charge and energy.......................................................................19

2-10: Channel structure of lightning depicting the main channel, the branches (feeder channels) in the thundercloud, and branches below the thundercloud.....................20

2-11: Electrodynamic Model .............................................................................................23

2-12: The peak values of electric and magnetic fields produced by the return-stroke breakdown pulse for the case of rch = 0.15 cm, T=15,000ο K, ∆t = 500 ns, and Imax= 20 kA, plotted as functions of the radial distance from channel axis .............24

3-1: Illustration (not to scale) of the method used to estimate power, P(t), and energy, W(t), from measured lightning channel current, I(t), and vertical electric field, E(t), near the channel. ..............................................................................................30

3-2: Calibration of the Pockels sensor ...............................................................................32

3-3: Variation of the Pockels sensor output voltage as a function of the applied electric field...........................................................................................................................33

3-4: Comparison of the electric field waveforms simultaneously measured with a Pockels sensor and a flat-plate antenna, both at 5 m................................................34

vii

3-5: Comparison of magnitudes of the vertical electric field peaks measured with Pockels sensors and a flat-plate antenna, both at 5 m ..............................................35

3-6: Experimental setup .....................................................................................................36

3-7: V-shaped electric field signatures ..............................................................................37

3-8: Stroke S0013-1. ..........................................................................................................40

3-9: Scatter plot of screen current, IS vs. strike rod current, IR, for 2000 ..........................41

3-10: Time variation of electric field, current, power, and energy for stroke S006-4. .....42

3-11: Same as Figure. 3-10, but for Stroke S008-4 ...........................................................43

3-12: Same as Figure. 3-10, but for Stroke S0013-1. ........................................................43

3-13: Same as Figure. 3-10, but for Stroke S0013-4 .........................................................44





3-18: Histogram of peak current for strokes characterized by V- shaped electric field signatures with ∆ERS = ∆EL. .....................................................................................48

3-19: Histogram of ∆EL for strokes characterized by V- shaped electric field signatures with ∆ERS = ∆EL. ......................................................................................................49

3-20: Histogram of peak power for strokes characterized by V- shaped electric field signatures with ∆ERS = ∆EL. .....................................................................................50

3-21: Histogram for input energy for strokes characterized by V- shaped electric field signatures with ∆ERS = ∆EL. .....................................................................................51

3-22: Histogram for action integral for strokes characterized by V- shaped electric field signatures with ∆ERS = ∆EL. .....................................................................................52

3-23: Histogram of the risetime of current for strokes characterized by V-shaped electric field signatures with ∆ERS = ∆EL.................................................................53

3-24: Histogram of the 0-100 % risetime of power per unit length for strokes characterized by V-shaped electric field signatures with ∆ERS = ∆EL. ....................53

viii

3-25: Peak power vs. peak current for strokes characterized by V- shaped electric field signatures with ∆ERS = ∆EL......................................................................................54

3-26: Energy vs. peak current for strokes characterized by V- shaped electric field signatures with ∆ERS = ∆EL......................................................................................55

3-27: Peak power vs. ∆EL for strokes characterized by V- shaped electric field signature with ∆ERS = ∆EL. ......................................................................................................56

3-28: Energy vs. ∆EL for strokes characterized by V- shaped electric field signature with ∆ERS = ∆EL. ......................................................................................................56

3-29: Energy vs. Action Integral for strokes characterized by V- shaped electric field signature with ∆ERS = ∆EL........................................................................................57

3-30: Flash 0013, stroke 1; Correction factor of 1.6 is applied at the instant of negative E-field peak ..............................................................................................................59

3-31: Flash 0013, stroke 1; Correction factor of 1.6 is applied at the instant of negative E-field peak, which is shifted by 0.24 µs to the left in order to partially account for the ± 0.5 µs uncertainty in the position of the peak............................................60

3-32: Flash 0013, stroke 1; Correction factor of 1.6 is applied at the instant of negative E-field peak, which is shifted by 0.24 µs to the right in order to partially account for the ± 0.5 µs uncertainty in the position of the peak............................................61

3-33: Evolution of the various quantities for the first 0.58 µs for Flash S006, stroke 4....64

3-34: Evolution of the various quantities for the first 0.4 µs for Flash S008, stroke 4......65






3-40: Evolution of the various quantities for the first 5 µs for Flash S0023, stroke 3.......71

3-41: Evolution of channel radius for S0008-4..................................................................72

3-42: V-shaped signature with ∆EL> ∆ERS. ∆ERS represents the average electric field between 44 µs to 50 µs after the beginning of the return stroke (at 50 µs)..............76

ix

3-43: Histogram of peak current for strokes characterized by V-shaped electric field signatures with ∆ERS < ∆EL and flattening within 20 µs. .........................................76

3-44: Histogram of ∆EL for strokes characterized by V-shaped electric field signatures with ∆ERS < ∆EL and flattening within 20 µs. ..........................................................77

3-45: Histogram of peak power per unit length for strokes characterized by V-shaped electric field signatures with ∆ERS < ∆EL and flattening within 20 µs. ....................77

3-46: Histogram of energy per unit length for strokes characterized by V-shaped electric field signatures with ∆ERS < ∆EL and flattening within 20 µs. ....................78

3-47: Histogram of action integral for strokes characterized by V-shaped electric field signatures with ∆ERS < ∆EL and flattening within 20 µs. .........................................78

3-48: Peak power vs. peak current for strokes characterized by V- shaped electric field signatures with ∆ERS < ∆EL and flattening within 20 µs. .........................................79

3-49: Energy vs. peak current for strokes characterized by V- shaped electric field signatures with ∆ERS < ∆EL and flattening within 20 µs. .........................................80

3-50: Energy vs. ∆EL for strokes characterized by V- shaped electric field signatures with ∆ERS < ∆EL and flattening within 20 µs............................................................80

3-51: Energy vs. ∆EL for strokes characterized by V- shaped electric field signatures with ∆EL< ∆ERS and flattening within 20 µs............................................................81

3-52: Energy vs. Action integral for strokes characterized by V- shaped electric field signatures with ∆EL< ∆ERS and flattening within 20 µs...........................................81

3-53: Histogram of ∆EL for strokes characterized by V- shaped electric field signatures with ∆ERS (t) < ∆EL and no field flattening within 20 µs. ........................................83

3-54: Histogram of ∆EL for strokes characterized by V- shaped electric field signatures with ∆ERS (t) < ∆EL and no field flattening within 20 µs. ........................................84

3-55: Histogram of peak power for strokes characterized by V- shaped electric field signatures with ∆ERS (t) < ∆EL and no field flattening within 20 µs........................85

3-56: Histogram of action integral for strokes characterized by electric field signatures with ∆ERS (t) < ∆EL and no field flattening within 20 µs. ........................................86

3-57: Peak power vs. peak current for strokes characterized by electric field signatures with ∆ERS (t) < ∆EL and no field flattening within 20 µs .........................................87

3-58: Peak power vs. ∆EL for strokes characterized by electric field signatures with ∆ERS (t) < ∆EL and no field flattening within 20 µs .................................................88

4-1: Flash 03-31, bipolar flash ...........................................................................................90

x

4-2: Flash 03-31 .................................................................................................................91

4-3: Definitions of parameters (peak, duration, rise time, half-peak width.......................92

4-4: Illustration of the removal of the background continuous current in computing charge and action integral ........................................................................................93

4-5: Histograms of the peak of the ICC pulses for 2002. ..................................................94

4-6: Histograms of the peak of ICC pulses for 2003. ........................................................95

4-7: Histogram of the peak of ICC pulses for 2002 and 2003. ..........................................95

4-8: Histogram of the duration of ICC pulses for 2002. ....................................................96

4-9: Histogram of the duration of ICC pulses for 2003. ....................................................96

4-10: Histogram of the duration of ICC pulses for 2002 and 2003. ..................................97

4-11: Histogram of the risetime of ICC pulses for 2002. ..................................................97

4-12: Histogram of the risetime of ICC pulses for 2003. ..................................................98

4-13: Histogram of the risetime of ICC pulses for 2002 and 2003....................................98

4-14: Histogram of the half-peak width of ICC pulses for 2002. ......................................99

4-15: Histogram of the half-peak width of ICC pulses for 2003. ......................................99

4-16: Histogram of the half-peak width of ICC pulses for years 2002 and 2003............100

4-17: Histogram of the charge of ICC pulses for 2002....................................................100

4-18: Histogram of the charge of ICC pulses for 2003....................................................101

4-19: Histogram of the charge of ICC pulses for years 2002 and 2003. .........................101

4-20: Histogram of the action integral of ICC pulses for years 2002..............................102

4-21: Histogram of the action integral of ICC pulses for years 2003..............................102

4-22: Histogram of the action integral of ICC pulses for years 2002 and 2003. .............103

4-23: Histograms of the peak of ICC pulses....................................................................104

4-24: Histograms of the duration of ICC pulses ..............................................................105

4-25: Histograms of the risetime of ICC pulses...............................................................106

4-26: Histograms of the half-peak width of ICC pulses ..................................................107

xi

4-27: Flash F0213 ............................................................................................................111

4-28: Flash F0213 ............................................................................................................112

4-29: Histogram of duration of M-component pulse. ......................................................112

4-30: Histogram of peak of M-component pulse.............................................................113

4-31: Histogram of risetime of M-component pulse........................................................113

4-32: Histogram of the half-peak width of M-component pulse. ....................................114

4-33: Histogram of the charge of M-component pulse....................................................114

4-34: Action integral of M-component pulse...................................................................115

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the

Requirements for the Degree of Master of Science

ESTIMATING POWER, ENERGY, AND ACTION INTEGRAL IN ROCKET-TRIGGERED LIGHTNING

By

Vinod Jayakumar

December 2004

Chair: Vladimir A. Rakov Cochair: Martin A. Uman Major Department: Electrical and Computer Engineering

The peak power and input energy for the triggered-lightning return strokes are

calculated as a function of time, using the vertical electric fields measured within 0.1 to

1.6 m of the lightning channel and the associated currents measured at the channel base.

The data were acquired during the 2000 rocket-triggered lightning experiments at the

International Center for Lightning Research and Testing (ICLRT) at Camp Blanding,

Florida. Results were compared with estimates found in the literature, including those

based on gas-dynamic models, on electrostatic considerations, and optical measurements

and long spark experiments. We additionally examined the action integral, the variation

of resistance per unit length, and the radius of the lightning channel during the return-

stroke process. We also examined the correlation of various parameters. Our estimates for

energy and peak power are in reasonable agreement with those predicted by the gas

dynamic models found in the literature. Finally, pulses superimposed on the initial stage

xiii

current (ICC pulses) and similar pulses superimposed on the continuing current that

follows the return stroke process (M-component pulses) were analyzed for years 2002

and 2003, and compared with statistics found in the literature. The following comparisons

were made: (a) ICC pulses in triggered lightning recorded at the ICLRT in 2002 and 2003

(relatively high sampling rate) vs. their counterparts recorded earlier (relatively low

sampling rate), (b) ICC pulses in triggered lightning vs. those in object-initiated

lightning, and (c) ICC pulses in triggered lightning vs. M-component pulses in triggered

lightning. Duration, risetime, and half-peak width of ICC pulses were somewhat greater

than those of M-component pulses. Current peak, charge, and action integral of M-

components and ICC pulses were similar.

xiv

CHAPTER 1 INTRODUCTION

Lightning strikes are the cause of many deaths and injuries. Electromagnetic fields

and currents associated with lightning also can have deleterious effects on nearby

electronic devices. The energy of lightning is a fundamental quantity required, for

example, in estimating nitrogen oxide (NO) produced by lightning; which, in turn, is

needed in global climate-change studies. Trace gases produced by atmospheric electric

discharges are important to the ozone balance of the upper troposphere and lower

stratosphere. Atmospheric electric discharges might have played an important role in

generating the organic compounds that made life possible on Earth. Currently, there is no

consensus on lightning input energy. Estimates found in literature differ by one to two

orders of magnitude. Our study measured electric fields using Pockels sensors in the

immediate vicinity of the lightning channel, along with the channel base currents, to

estimate the energy and peak power in triggered lightning. We also analyzed the action

integral (energy per unit resistance at the strike point) and other parameters of return

strokes and pulses superimposed on steady currents in triggered lightning.

1

CHAPTER 2 LITERATURE REVIEW

Three approaches used to estimate lightning peak power and energy input were

found in the literature.

• The first approach is based on electrostatic considerations. The total electrostatic

energy of a lightning flash lowering charge Q to the ground can be estimated as

the product of Q and V, where V is the magnitude of the potential difference

between the lower boundary of the cloud charge source and ground (Rakov and

Uman, 2003). The typical value of Q for a cloud-to-ground flash is 20 C. The V is

estimated to be 50 to 500 MV (Rakov and Uman, 2003). Thus each flash

dissipates energy of roughly 1 to 10 GJ. Borovsky (1998), using electrostatic

considerations, estimated the energy associated with individual strokes to be

1×103–1.5×104 J/m, close to that predicted by gas dynamic models (Section

2.5.3).

• The second approach was described by Krider et al. (1968). Radiant power and

energy emitted within a given spectral region from a single-stroke lightning flash

are compared with those of a long spark whose electrical power and energy inputs

are known with fair accuracy. The value of lightning input energy per unit

channel length estimated by Krider et al. (1968) is 2.3 ×105 J/m.

The third approach involves the use of gas dynamic models proposed by a number of

researchers (Plooster. 1971; Paxton et al. 1986, 1990; Dubovoy et al. 1968, 1991; Hill

1977; Strawe 1979; Bizjaev et al. 1990). A short segment of a cylindrical plasma

2

3

column is driven by the resistive (joule) heating caused by a specified flow of electric

current as a function of time. Lightning input energy predicted by these models is one

to two orders of magnitude lower than that of Krider et al. (1968).

2.1 Cloud Formation and Electrification

The primary source of lightning is the cloud type, cumulonimbus (commonly

known as thundercloud). The process of charge generation and separation is called

electrification. Apart from the cumulonimbus, electrification can also take place in a

number of other cloud types (in stratiform clouds, for example; and in clouds produced

by forest fires, volcanic eruptions, and atmospheric charge separation in nuclear blasts.

According to Henry et al. (1994), eight types of thunderstorms are known. Among them,

some common in Florida are sea/land-breeze thunderstorms, oceanic thunderstorms, air-

mass thunderstorms, and frontal thunderstorms. Portier and Coin (1994) give other

classifications. The formation of air-mass clouds is explained next.

On a sunny day, Earth absorbs heat from the sun, causing both water vapor and air

to rise to higher atmospheric levels, forming clouds. The energy of the water vapor

decides the intensity of the thunderstorm; the hotter the air, the more water vapor it can

hold, and the more powerful the thunderstorm can be. When water vapor condenses, it

releases the same amount of energy required for heating water, to produce water vapor.

Convection causes warm, humid air to reach higher altitudes. The released energy heats

the surrounding atmosphere, which raises the cloud to higher altitudes, pulling the humid

air from below (setting a chain reaction); with an updraft velocity of round 30 m/s

forming cells. A cell is said to be in mature stage (actually this stage is related to the

cloud’s ability to generate lightning) when it reaches higher altitudes, and its top flattens,

forming an anvil. This kind of cloud formation can be divided into three stages

4

• Developing stage

-Starts with warm, rising air

-The updraft velocity increases with height

-Super-cooled water droplets are far above freezing level

-Small-scale process that electrifies individual hydrometeors takes place in this stage

• Mature stage

-The heaviest rains occur

-The downdraft occurs, due to frictional drag of the raindrops

-Evaporative cooling leads to negative buoyancy

-The top of the cloud forms an anvil

-The graupel-ice mechanism and the larger convective mechanism take place, leading

to electrical activity.

• Dissipating stage

-The downdraft takes over the entire cloud

-The storm deprives itself of supersaturated updraft air

-Precipitation decreases

-The cloud evaporates

Various measurements were made to estimate the distribution of charge within the

cloud. Initially, from ground-based measurements, it was assumed that the charge within

the cloud forms an electric dipole (positive charge region above negative charge region).

Simpson and Robinson (1941) made in-cloud measurements with balloons, and suggested

a tripole model with an additional positive charge at the base of the cloud. There is still

no consensus on the detailed distribution of charge within the cloud.

5

Figure 2-1: Electrical structure of a cumulonimbus. [Simpson, G. and Scrase, F.J.; “The

Distribution of Electricity in the Thunderclouds,” Proc. R. Soc. London Ser. A, 161: Figure. No.4. pp.315, 1937]

• Precipitation model: Heavy soft hail (graupel) with a fall speed > 0.3 m/s interacts

with lighter particles (ice crystals) in the presence of small water droplets. As a result,

heavy particles in cold regions (T<-15° C) acquire negative charge; heavy particles in

warm regions (T>-15° C) acquire positive charge. The second process (gravitational

force) separates the heavier and lighter charged particles, forming an electric tripole.

• Convection model: Charges are supplied by external sources such as corona and

cosmic rays. Separation of charges is accomplished by organized convection.

2.2 Natural Lightning Discharges

Lightning discharges can be classified as

• Cloud discharges

- Intracloud discharges

- Cloud-to-cloud discharges

- Cloud-to-air discharges

• Cloud-to-ground discharges

6

- Downward negative discharges

- Downward positive discharges

- Upward positive discharges

- Upward negative discharges

- Bipolar discharges

Figure 2-2: Natural lighting discharges for a cumulonimbus. Adapted from “Encyclopedia Britannica”

Most (47 to 75 %) discharges are cloud discharges and the rest are cloud-to-ground

discharges. Intracloud discharges are apparently most numerous in the cloud-discharge

category, compared to the intercloud and cloud-to-air discharges.

Most of cloud-to-ground discharges can be divided into four categories. They are

downward negative, downward positive, upward positive, upward negative. Upward

positive and upward negative discharges occur rarely, while 90% of the cloud-to-ground

discharges are downward negative discharges and 10% are downward positive

discharges. There are also discharges transporting both negative and positive charges to

7

ground. Such bipolar discharges are usually of upward type and constitute probably less

than 10% of all cloud-to-ground discharges.

Figure 2-3: Four types of discharges between cloud and ground.1. Downward negative 2. Upward negative 3.Downward positive 4. Upward positive. [M. A. Uman, The Lightning Discharge; Dover Publications, Minneola, New York; Figure.. 1-3, pp.9, 1987]

At t=0 ms, the thundercloud has a tripolar charge structure with positive charge in

the upper region, negative charge in the lower region, and a small pocket of positive

charge at the cloud base. Between t=0 and t=1 ms, preliminary breakdown occurs within

the cloud due to the local discharge between the pocket of positive charge at the base and

the primary negative charge. The local discharge neutralizes the positive charge at the

base and continues towards the ground as a stepped leader.

8

Figure 2-4: Downward negative cloud-to-ground lightning [V.A. Rakov, Uman, M.A, Lightning: Physics and Effects; Cambridge University Press, New York; 2003].

This leader consists of a narrow current-carrying core and a much wider radial

corona sheath. Between 1.10 ms to 19 ms, the stepped leader moves towards the ground

with an average speed of 105 to 106 m/s, exhibiting steps of some tens of meters in length

and separated by some tens of microseconds. At t= 20 ms, the stepped leader approaching

the ground causes the electric field near ground to exceed the breakdown value for air,

which in turn results in an upward positive leader extending from ground towards the

9

descending stepped leader. Between 20.0 and 20.1 ms, the downward leader attaches to

one of the upward leader branches and a return stroke is initiated with a typical peak

current of 30 kA. From t= 20.10 ms to 20.20 ms, the first return stroke propagates

upward towards the cloud in the ionized channel left behind by the stepped leader with a

speed of around 108 m/s. The return stroke neutralizes the negative charge (typically 5 C)

deposited by the leader (lowers negative charge to ground). At t= 40 ms, some in-cloud

processes called K and J processes occur inside the cloud. At t= 60 ms to 62 ms, a dart

leader propagates downward along the channel left by the first return stroke with an

average speed of 107 m/s [Uman, 1987]. The dart leader deposits a negative charge of the

order of 1C onto the channel.

When the dart leader reaches the ground, a second return stroke is initiated which

travels upward with an average speed of 108 m/s. A sequence of leader and return stroke

is called a stroke, with the average number of strokes per flash being 3 to 5 [Rakov and

Uman, 2003].

Positive cloud-to-ground discharges can originate from the upper positive charge

region or positive charge pocket at the cloud base (assuming the tripolar model of cloud

charge distribution). It starts with a downward propagating positive leader and connects

to a negative upward leader launched from the ground. Then an upward return stroke is

initiated which transfers positive leader charge to ground. Typically there are no

subsequent strokes in positive cloud-to-ground discharges. The typical values of first

stroke peak currents measured at ground for positive cloud-to-ground discharges is

35 kA, not much different from 30 kA for negative cloud to ground lightning. On the

other hand, larger currents are more probable in positive strokes that in negative ones.

10

2.3 Mechanism of NO Production by Lightning

During a return stroke, the lightning channel attains a peak temperature of 30,000° K

in a few microseconds. If the cooling of the channel takes place slowly, equilibrium

composition at a given temperature is established, i.e., the final constituents of the cold

air would be the same as the constituents prior to the return stroke. It has been shown by

Uman and Voshall (1968) and Picone et al (1981) that the residual hot channel cools

from around 10,000° K to 3,000° K in a few milliseconds. The time required by NO to

attain equilibrium concentration increases rapidly with decreasing temperature. Hence, as

the channel cools down to the ‘freeze out’ temperature, the temperature at which the

reactions that produce and destroy NO become too slow to keep NO in equilibrium

concentration, and hence NO remains at the density characteristic of the ‘freeze out’

temperature. Chemical reactions, which characterize the production of NO, are

O2 ↔ O + O

O + N2 → NO + N

O2 + N → NO + O

The reactions that compete with the NO producing reactions are shown below.

NO + N → O + N2

NO + N → O + N2

NO ↔ N + O

NO + NO → N2O + O

These equations assume importance as NO produced by natural processes decreases

the ozone (O3) concentration in the stratosphere via the dominant equation

NO + O3 → NO2 + O2

11

Ozone in the stratosphere is important to life because it shields the Earth from Sun’s

harmful ultraviolet radiation. Ozone in the troposphere acts as a greenhouse gas by

absorbing the infrared radiation.

2.4 Rocket-Triggered Lightning

The study of natural lightning is extremely difficult since it is impossible to

accurately predict its occurrence in space and time. For this reason, in order to study the

lightning properties a method to produce lightning artificially from natural thunderclouds

has been developed. The rocket and trailing wire technique is used to initiate lightning

that is referred to as rocket-triggered lightning.

Currently, rocket-triggered lightning (Rakov, 1999) can be produced in two

different ways:

• Classical rocket-triggered lightning. • Altitude rocket-triggered lightning.

2.4.1 Classical Rocket-Triggered Lightning

In classical triggering, the wire is continuous and is connected to the grounded

launcher. After the rocket is launched, it travels upward with a velocity of around 200

m/s. When the rocket reaches a height of around 200 m, an upward positive leader is

generated at the rocket tip, which travels with a velocity of around 105 m/s. The current

of the upward positive leader vaporizes the wire, and an initial continuous current (ICC)

follows for some hundred of milliseconds. During the formation of the upward positive

12

Figure 2-5: Classical rocket-triggered lightning [V. A. Rakov, “Lightning Discharges Triggered using Rocket-and-Wire Techniques," J. Geophys. Res., vol.100, pp.25711-25720, 1999]

leader, the so-called initial current variation (ICV) occurs, which is not shown in

Figure.2-5, but explained in the next paragraph. After the completion of ICC, there exists

a no current interval for a few tens of milliseconds that is followed by one or more leader/

return stroke sequences (Figure. 2-5). These leader/return stroke sequences are similar to

subsequent leader/return stroke sequences in natural lightning.

The ICV occurs when the triggering wire is replaced by the upward positive leader

plasma channel. The upward positive leader produces current in the tens to hundreds of

amperes range when measured at ground, and this current vaporizes the wire. At that

time, the current measured at ground goes to nearly zero since there is no well-

conducting path for the current to travel to the ground. Then a downward leader-like

process bridges the resultant gap and initiates a return stroke type process from the

13

ground. The latter leader/return stroke type sequence serves to re-establish the interrupted

current flow to ground.

Figure 2-6 Altitude rocket-triggered lightning [V. A. Rakov, “Lightning Discharges

Triggered using Rocket-and-Wire Techniques," J. Geophys. Res., vol.100, pp.25711-25720, 1999]

2.4.2 Altitude Rocket-Triggered Lightning

The altitude triggering technique uses an ungrounded wire in an attempt to

reproduce some of the features of the first stroke of the natural lightning which is not

possible using classical, grounded-wire triggering. Generally, the rocket extends three

sections, a 50 m long copper wire connected to the grounded launcher, a 400 m long

insulating Kevlar cable in the middle, and a 100 to 200-m long copper wire connected to

the rocket. The upper, floating wire is used for triggering and the lower, grounded wire

for intercepting the descending leader as discussed below. When the rocket reaches a

height of around 600 m an upward positive leader and a downward negative leader

14

(forming a bi-directional leader) are initiated, each propagating at a speed of 105 m/s. The

electric field produced by the downward negative leader initiates an upward connecting

positive leader from the grounded 50-m wire, which connects to the downward negative

leader. Finally, a return stroke is initiated which travels with a speed of 107-108 m/s and

catches up with the upward positive leader tip. After this stage, the processes are similar

to those of the classical rocket- triggered lightning.

2.5 Estimates of Peak Power and Input Energy in a Lightning Flash

There are various methods to estimate the peak power and energy dissipated in

lightning discharges. Some of them are described in the following sections.

2.5.1 Optical Measurements and Long Spark Experiments

Krider et al. (1968) estimated the average energy per unit length and peak power

per unit length to be 2.3×105 J/m and 1.2×109 W/m. Their optical measurements were

similar to those performed by Krider (1966) and are described below. A calibrated silicon

photodiode and an oscilloscope were used as a fast-response lightning photometer

covering the visible and near-infrared regions of the spectrum from 0.4 to 1.1 µm.

Simultaneous still photographs of the discharge channels were taken to determine the

dependence of the photoelectric pulse profile on the type of lightning and the geometry of

its channel. The photodiode detector consisted of an Edgerton, Germeshausen and Grier

model 560-561 ‘lite-mike’ and ‘detector head’ (Krider, 1966). The photodiode and

associated circuitry were linear over a wide range of incident light levels (within 5% over

7 decades) and had a response time of less than 1 µs. The calibrated relative response of

the detector is shown in Figure. 2-7. Photographs of the lightning channels were taken

with an Ansco Memar 35-mm camera, which had a focal length of 45 mm. Using the

15

same experimental setup optical measurements have been performed on a 4-meter air

spark produced by the Westinghouse 6.4 MV impulse generator.

Figure 2-7: Relative spectral response versus wavelength for the photodiode detector

used by Krider (1966) and Krider et al. (1968).

The basic principle of power and energy estimation is as follows. The spark current

and voltage were recorded as functions of time, enabling the calculation of electrical

power and total energy input. The radiant power reaching the detector is proportional to

the voltage measured at the output of the optical detector. Considering the spark channel

to be straight (to avoid taking the dependence of the radiant power on the azimuth of the

channel), the radiant power output from the light source is determined from equation (1).

It has been demonstrated experimentally using the long spark that the distance

dependence of the radiant flux is 1/R2. Hence, the total radiant power emitted in all

directions within the detector bandwidth is given by

P= (V/K) × (4πR2/ A) (1)

where, V is the optical detector output voltage, K is the pulse calibration factor, R is the

distance from the light source to the detector, and A is the sensitive area of the detector.

The radiant power vs. time curve can be integrated to obtain the total radiant energy

emitted in a given bandwidth. The radiative efficiency is calculated by comparing the

value of total radiant energy to that of measured electrical energy input. Making a

16

simplifying assumption that the radiative efficiencies for laboratory spark and lightning

are the same, power and input energy can be found for a lightning stroke whose total

radiant energy is known from measurements. In this experiment, knowing the

approximate location of the lightning, cloud base height was obtained from the U.S.

Weather Bureau. Using the cloud base height, which determined the length of the

lightning channel that was visible, the size of the photographic image, and knowledge of

the camera focal length one can estimate the distance to the channel. This method

assumes that the channel is vertical. Figure 2-8 b shows the photoelectric voltage pulse

corresponding to the lightning whose still picture is shown in Figure 2-8a.

Figure 2-8: Measurement of photoelectric pulse of lightning. a) Still photograph of a typical cloud-to-ground lightning at a distance of 6 km. b) the corresponding photoelectric voltage pulse [E.P. Krider, “Some photoelectric observations of Lightning”, J.Geophys. Res., Figure. 2-3, pp.3096-3097, 1966].

For the lightning stroke which was under study in Krider, (1968), the cloud base

height was 1.8 km, and the distance calculated from the photographic image size was 8.2

km. At this distance and maximum signal at the optical detector, the peak power radiated

17

from the lightning stroke is calculated to be 1.1×1010 W, or dividing by the channel

length, to be as 6.2×106 W/m. The corresponding measurements for the spark were made

at a distance of 23 m, and the peak radiant power within the detector bandwidth was

obtained to be 1.0×1010 W/m. The peak electrical power dissipated in the spark was

obtained accurately from the direct traces of current recorded as a function of voltage. At

the time of peak power, the current was 4.2×103 A and the voltage

1.8×106 V, yielding a peak electrical power input of 7.6×109 W, or 1.9×109 W/m

Comparison of the radiant and electrical peak powers for the long spark yields a radiative

efficiency of 0.52%. Assuming the same radiative efficiency for the lightning at the

instant of peak radiant power, the peak electrical power dissipated in the lightning stroke

is 2.1×1012 W. Dividing this value by the channel length, peak electrical power dissipated

per unit length is obtained as 1.2×109 W/m.

The electrical energy per unit length dissipated in the long spark is obtained by

integrating over time the product of the current and voltage values obtained from the

traces taken during the experiment. Comparing the radiant and electrical energy values,

the average radiative efficiency of 0.38% is obtained for the spark. Applying the same

radiative efficiency to lightning, the total average energy dissipation per unit length is

estimated to be 2.3×105 J/m. This value is in agreement with the thunder theory data of

Few et al. (1969), but is one to two orders of magnitude larger than the values predicted

by gas-dynamic models (Section 2.5.3).

2.5.2 Electrodynamic Model

We present here the electrodynamic model proposed by Borovsky (1995), which

describes dart leaders and return strokes as electromagnetic waves that are guided along

18

conducting lightning channels. The downward propagating dart leader deposits negative

charge onto the channel and deposits electrostatic energy around the channel. The

subsequent upward-propagating return stroke drains the negative charge off the channel

and heats the channel by expending the stored electrostatic energy. The net result is that

the negative charge is lowered from the cloud to the ground and the energy is transferred

from the cloud to the channel. This electrodynamic model also accounts for the flow of

energy associated with the flow of charge. In this model energy dissipated per unit length

in lightning channels is calculated as a relation to the linear charge density on the channel

and not to the cloud-to-ground electrostatic potential difference.

This model serves as a tool to visualize the dynamics of lightning during the dart-

leader and return stroke phases. Figure 2-9 illustrates the concept of the model.

The amount of energy deposited on the lightning channel can be estimated based on

electrostatic considerations, from the following expression (Uman, 1984, 1987)

W/L = (Qlow × ∆Φtot)/ Lmain (2)

Where, Qlow is the amount of charge lowered from the cloud to ground, ∆Φtot is the

electrostatic potential difference between the thundercloud charge region and the ground,

and Lmain is the length of the main channel.

19

Figure 2-9: Conceptual flow of charge and energy. a) Flow of charge during a dart leader

and return stroke. b) Flow of energy during the dart leader and return stroke. [J. E. Borovsky, “Lightning Energetics: Estimates of energy dissipation in channels, channel radii, and channel-heating risetimes”, J.Geophys. Res., vol.103, Figure.1, pp.11538, May. 1998]

The above relation for W/L is unreliable due to the following reasons,

• The amount of energy expended in the branch channels is unknown.

• Difficulty in estimating ∆Φtot (requires integration of the height varying electric field from the ground level to the cloud charge source).

• As seen in Figure 2-10, which gives a sketch of the structure of lightning channel, the energy expended in creating the “feeder” channels in the cloud and branches

20

(for first strokes only) has to be included. The total length of the channel network is very difficult to estimate.

Cloud base

Figure 2-10: Channel structure of lightning depicting the main channel, the branches (feeder channels) in the thundercloud, and branches below the thundercloud.

Borovsky (1995) gives a more accurate estimate of the energy dissipation by considering

the stored electrostatic energy density around the channel. Electrostatic energy density

(ED) is given by (3).

ED = ε0 E2/2 (3)

Where, ε0 = 8.85× 10-12 F/m and E is the electric field at a distance r from the channel. E

is given by (4).

E = ρL / (2π ε0 r) (4)

Where, ρL is the charge per unit channel length.

21

Note that equations (3) and (4), employ SI units, while the corresponding equations given

by Borovsky (1995) are in CGS metric unit system. The original equations and

conversion can be found in the Appendix. The stored energy per unit length equation is

derived as follows. If all the charge resides on the channel, the electric field very close to

the channel exceeds the air-breakdown limit Ebreak. Ebreak is approximately equal to 2×106

V/m (Cobine, 1941). At locations along the lightning channel where E exceeds Ebreak,

conductivity increases rapidly, facilitating the movement of free charge, thus reducing the

electric field E. So, around the channel the electric field will be approximately equal to

Ebreak, up to a radius of rbreak.

rbreak= ρL / (2π ε0 Ebreak ) (5)

Beyond rbreak, the electric field falls off as 1/r. Thus, the radial dependence of the

electrostatic energy density residing around the channel is given by,

ED = ε0 Ebreak 2/2 if r ≤ rbreak (6)

ED = ρL 2/8π2 r2 ε0 if r ≥ rbreak (7)

Total amount of electrostatic energy per unit length stored around the channel is given by

Wstored /L = ∫ (ED) 2πr dr (8) ∞

Because of the radial dependence of elect

broken into

Wstored /L = ∫ (ED

Substituting the appropriate expressions f

Wstored /L = ρL2/(4π ε0 rbreak)+

Where rcut is the cutoff radius that is intro

logarithmically diverging as r → ∞. The

0
rostatic energy, the above integral can be
) 2πr dr +r

or ED in e

ρL2 /(4π ε

duced to p

physically

∞
∫ (ED) 2πr dr (9) break
0

rbreak

quation (9),

0) ln( rcut/rbreak) (10)

revent the integral from

reasonable choice for rcut is the radius

22

at which the electric field of the channel equals the background electric field. Therefore,

rcut is given by the expression

rcut = ρL/ (2π ε0 Ecloud ) (11)

Thus the total electrostatic energy per unit length is given by

Wstored /L = ρL2/4π ε0 [1/ rbreak + ln( Ebreak / Ecloud)] (12)

Where,ρL = charge per unit length of the channel.

Ebreak = breakdown electric field.

Ecloud = background electric field.

In Borovsky (1998), the electric field under the thundercloud is taken to be the

background electric field. Ebreak value is taken to be 2.0 × 10 6

V/m. For Ecloud, two

limiting values taken are 1×104 V/m and 4×105 V/m.

The value of ρL is typically chosen in the range 1×10-4 C/m and 5×10-4 C/m, with

the dart-leader loaded channel being at the lower end of this range and stepped-leader

loaded channel being at the upper end of this range. This model estimates the energy per

unit channel length to be about 1×103 and 1.5×104 J/m for the dart-leader and stepped

leader respectively. These values are in good agreement with estimates of gas-dynamic

models of lightning [Rakov and Uman, 2003] considered in Section 2.5.3.

Borovsky (1998), whose electrodynamic model is illustrated in Figure 2-11, also

estimates the initial and final channel radii. Taking the charge per unit channel length

ρL = 4×10-4 C/m and number density of atoms in unexpanded channel, ηatomic = 5.0×1019

cm-3, the initial radius of the return stroke channel can be estimated to be 0.32 cm. The

values chosen for ηatomic and ρL are appropriate for a stepped–leader channel. In the

calculation of final radius (channel radius after expansion) the values of channel

23

parameters chosen are εdisso =9.8 eV, εioniz= 14.5 eV, Tinit= 30000 °K, Tatmos= 300 °K,

where Tinit is the temperature of the channel before expansion, Tatmos is the temperature of

the ambient air outside the channel, εdisso and εioniz are the dissociation and ionization

constants. The final channel radius is estimated to be about 4.7 cm. Similarly, for the

dart-leader channel, the initial and final radii are found to be 0.26 cm and 3.8 cm. In this

latter case, the values chosen for ρL and ηatomic are 1×10-4 C/m and 5.0×1018 cm- 3.

(a) (b)

Figure 2-11: Electrodynamic Model. (a) Downward propagating dart leader that loads charge and electrostatic energy onto a lightning channel, (b) upward-propagating return stroke that drains charge off the channel and uses up the stored electrostatic energy [J. E. Borovsky, “An electrodynamic description of lightning return strokes and dart leaders: Guided wave propagation along conducting cylindrical channels”, J.Geophys. Res., Figure. 10, pp. 2717, Feb. 1995].

According to this model, the vertical (longitudinal) electric field outside the channel

decreases with increasing the distance r as –loge (0.9 γout r), which is a slowly varying

24

function of r, where γout is the external wave number (Borovsky, 1995). γout is chosen to

be (8.8 + i 9.4)×10-5 cm-1 for the return stroke break-down pulse used in the illustration of

the variation of the horizontal and vertical components of the electric field shown in

Figure 2-12.

Figure 2-12: The peak values of electric and magnetic fields produced by the return-

stroke breakdown pulse for the case of rch = 0.15 cm, T=15,000ο K, ∆t = 500 ns, and Imax= 20 kA, plotted as functions of the radial distance from channel axis. rch is the channel radius, T is the channel temperature, ∆t is the rise time of the wave e-iwt, where w is the angular frequency(ω = 1/∆t, i e. ∆t is around one sixth of the time period of the sine wave), and Imax is the peak current . [J. E. Borovsky, “An electrodynamic description of lightning return strokes and dart leaders: Guided wave propagation along conducting cylindrical channels”, J.Geophys. Res., Figure. 8, pp.2712, Feb. 1995]

2.5.3 Gas Dynamic Models

Gas dynamic models consider a short segment of a cylindrical plasma column

driven by the resistive (Joule) heating caused by a specified flow of electric current as a

function of time. Rakov and Uman (1998) review essentially all the gas dynamic models

25

found in the literature. The basic assumptions in the most recent models are: 1) the

plasma column is straight and cylindrical; 2) the algebraic sum of all the charges is zero;

3) local thermodynamic equilibrium exists at all times. The initial conditions of the

lightning channel are temperature of the order of 10000°K, channel radius of the order of

1 mm, and pressure equal to ambient (1 atm) or mass density equal to ambient (of the

order of 10-3 g/cm3), the latter two conditions representing, respectively, the older and

newly created channel sections. The initial condition assuming the ambient pressure best

represents the upper part of the of the leader channel, since that part had sufficient time to

expand and attain equilibrium with the surrounding air. The initial condition of ambient

density is most suitable for the recently created, bottom part of the leader channel. At

each time step: 1) electrical energy sources; 2) the radiation energy sources; 3) Lorentz

force are computed and the gas dynamic equations are solved for the thermodynamic and

flow parameters of the plasma.

The energy input is determined as follows. The plasma channel is visualized as a

set of concentric annular zones, in which the gas properties are assumed constant. For a

known temperature and mass density, plasma composition can be obtained from the Saha

equation (Paxton et al. (1986, 1990), Plooster (1971)) or from tables of precompiled

properties of air in thermodynamic equilibrium (Hill (1971), Dubovoy et al. (1991,

1995)). The plasma conductivity can be computed from the plasma composition,

temperature and mass density. The current is distributed among the annular zones as if

they were a set of resistors connected in parallel. Using the cross-sectional distribution of

current and plasma conductivity, the amount of electrical energy input can be computed

for each of the annular zones.

26

The energy is deposited at the center of the channel in the form of heat, which is

transported to cooler outer regions in the form of radiation. The radiative properties of air

are complex functions of frequency and temperature. Radiation at a given frequency can

be absorbed and re-radiated at different frequencies while traversing the channel in the

outward direction. Paxton et al. (1986, 1990) and Dubovoy et al. (1991, 1995) used

tables of radiative properties of hot air to determine absorption coefficients as a function

of temperature for a number of selected frequency intervals to solve the equation of

radiative energy transfer in the diffusion approximation.

The pinch effect due to the interaction of electric current with its own magnetic

field was included in the gas dynamic model of Dubovoy et al. (1991, 1995). This

phenomenon counteracts the channel’s gas dynamic expansion, resulting in 10-20%

increase in input energy for the same input current because of reduced channel size.

Table 2-1,which is found in Rakov and Uman (1998), summarizes predictions of

the various gas dynamic models for the input energy and percentages of this energy

converted to kinetic energy and radiated from the channel. Additionally included in Table

2-1 are energy estimates based on experimental data (Krider et al; see Section 2.5.1 and

on electrostatic considerations (Uman, 1987; Borovsky, 1998; see Section 2.5.2). Brief

comments on each of these estimates follow the table.

Table 2-1: Lightning Energy Estimates [Rakov and Uman, 1998]. Source Current

Peak, kA

Input Energy, ×103 J/m

% Converted to Kinetic Energy

% of Energy Radiated

Hill (1971,1977) 21 15 (~3)

9+ (at 25 µs) 2*+ (at 25 µs)

Plooster (1971) 20 2.4 4 (at 35 µs) 50 (at 35 µs) Paxton et al. (1986,1990)

20 4 2 (at 64 µs) 69 (at 64 µs)

27

Continued Table 2-1. Source Current

Peak, kA

Input Energy, ×103 J/m

% Converted to Kinetic

Energy

% of Energy Radiated

Dubovoy et al. (1991,1995)

20 3 - 25 (at ≥55 µs)

Borovsky (1998) - 0.2-10 - Krider et al. (1968) Single-

stroke flash

230 - 0.38#

Uman (1987) (200-2000) -

+ Incorrect due to a factor of 20-30 error in electrical conductivity. * Estimated by subtraction of the internal and kinetic energies from the input energy shown in figure 1 of Hill (1977). # Only radiation in the wavelength range from 0.4 to 1.1 µm.

Hill (1971, 1977) overestimates the input energy by a factor of 5 or so due to the

underestimation of electrical conductivity. The corrected value is given in the

parentheses. Plooster’s (1971) model gives a crude radiative transport mechanism

adjusted to the expected temperature profile. Paxton et a/. (1986, 1990) gives individual

temperature dependent opacities for several wavelength intervals. Dubovoy et al.’s

(1991, 1995) model is in principle the same as the previous one, except for the fact that

the pinch effect was taken into account. Uman (1987) estimates the input energy by

assuming that tens of coulombs are lowered from a height of 5 km to ground. An

assumption made is that the potential difference between the ground and charge center

inside the cloud is 108-109 V.

Krider et al. (1968) estimated the average energy per unit length and peak power to

be 2.3×105 J/m and 1.2×109 W/m (see Section 2.5.1). In this experiment the radiative

efficiencies of the long spark and the lightning channel are assumed to be constant in the

wavelength range of 0.4 to 1.1 µm. Since the input energy for long spark energy is

known, the radiative efficiency can be determined (0.38%) and applied to the lightning

28

return stroke. This estimate appears to be consistent with the thunder theory of Few

(1965, 1995).

Borovsky (1995, 1998) describes the dart leaders and return strokes as

electromagnetic waves that are guided along the conducting channels (see Section 2.5.2).

In this electrodynamic representation of lightning, the stored electrostatic energy Wstored

around a charged channel is the source of power for a return stroke. Borovsky, based on

electrostatic considerations, estimates the energy per unit channel length to be around

1×103–1.5×104 J/m, which is consistent with that predicted by the gas dynamic models.

Hence, there are one to two orders of magnitude differences in the estimates of

energy per unit length. The higher end of the energy range is likely to have included a

significant fraction of the energy dissipated by processes other than the return stroke.

These include the in-cloud discharge processes like the one in which charges are

collected from isolated hydrometeors in volumes measured in cubic kilometers and

transported into the developing leader channel. Additional experiments are required to

resolve the up to two orders of magnitude uncertainty in the estimate of lightning energy

input. In chapter 3, we will attempt to estimate lightning energy using recently acquired

experimental data for rocket-triggered lightning.

: CHAPTER 3 ESTIMATING POWER AND ENERGY

3.1 Methodology

Power per unit length and energy per unit length, each as a function of time are

estimated from the vertical (longitudinal) component of the electric field in the immediate

vicinity of the triggered-lightning channel and associated lightning return stroke current.

Additionally, channel resistance per unit length and channel radius are estimated. The

vertical electric field was measured by Miki et al. (2002) using a Pockels sensor placed at

a radial distance of 0.1 m from, and at a height of 0.1 m above the tip of the 2-m vertical

rod. The measured field was assumed to be equal to the longitudinal electric field inside

the channel. Indeed, according to Borovsky (1995), the longitudinal electric field at radial

distances of 10 cm and 1.6 m from the channel axis differs from the field at the channel

axis only by 2.1 ×10-4 % and 18 ×10-4 %, respectively (see Ez in Figure 2-12). The

average values of leader electric field changes (approximately equal to return stroke field

changes) at 0.1 to 1.6 m, 15 m, and 30 m from the lightning channel are 577 kV/m,

105kV/m, and 60 kV/m, respectively (Miki et al., 2002; Schoene et al., 2003, JGR).

Lightning current was measured at the base of the 2-m strike rod. We will assume that

this current is representative of the current flowing in the lightning channel at a height of

the Pockels sensor. Under these assumptions, the power and energy per unit length can be

estimated as P(t) = I(t) E(t) and W(t) = ∫ P(τ)dτ , respectively (Figure. 3-1). This energy is

associated with joule heating of the lightning channel and can be viewed as the input

energy for the return-stroke process that is spent for ionization, channel expansion, and

t

0

29

30

production of electromagnetic (including optical) and acoustical radiation from the

channel.

Figure. 3-1: Illustration (not to scale) of the method used to estimate power, P(t), and

energy, W(t), from measured lightning channel current, I(t), and vertical electric field, E(t), near the channel.

Very close (0.1 to 1.6 m) vertical electric fields and associated channel-base

currents were obtained for 36 strokes in nine triggered lightning flashes (see section 3.2

for the experimental setup). Out of 36 strokes, only 31 strokes in 12 flashes were suitable

for the analysis presented here. For the remaining strokes, though the current records

were available, the corresponding electric field records were saturated. All the acquired

electric field signatures can be divided in three types: 1) “classical” V-shaped signature

with return-stroke electric field change ∆ERS being approximately equal to the leader

electric field change ∆EL (∆ERS = ∆EL); 2) V-shaped signature with ∆ERS being

appreciably smaller ∆EL; 3) same as 2, but with the return stroke portion exhibiting no

flattening that is expected to occur within 20 µs or so of the beginning of the return

stroke. These three types of waveforms are illustrated in Figure 3-7. The reason for the

31

residual electric field some tens of microseconds after the return stroke for Types 2 and 3

is apparently due to the fact that the return stroke fails to neutralize all the leader charge

in the corona sheath surrounding the channel core (Kodali et al., 2003). The statistics for

peak power and energy are produced separately for the three types of electric field

signatures (no energy estimates for Type 3). Since Type 1 represents the “classical”

leader/return stroke sequence, while Types 2 and 3 indicate the presence of an additional,

slower process involved in the removal of charge from the channel (not all the

electrostatic energy deposited along the channel by the leader is tapped by the return

stroke), all the analysis concerning the channel resistance per unit length and channel

radius is presented only for Type 1. Further, the power and energy estimates for Type 2

were performed after adjusting the E- field waveforms to account for the residual field.

Events of Type 3 were used for estimating peak power only.

3.2 Experiment

Experimental data used in Section 3 have been acquired at the International Center for

Lightning Research and Testing (ICLRT) at Camp Blanding, Florida, in 2000. The

experiment was a joint University of Florida / CRIEPI, Japan, project, which is described

by Miki et al. (2002).

3.2.1 Pockels Sensors

The Pockels sensors used in this experiment had a stated dynamic range of 20 kV/m

to 1 MV/m (Miki et al., 2002). The lower measurement limit was determined by noise. In

this study we applied filtering (moving time averaging; see section 3.4.1) that allowed us

to significantly reduce this noise and, hence, to lower the measurement limit. The residual

noise translated to noise in power waveforms but did not materially influence energy

estimates.

32

During the calibration in laboratory in Japan, the 2/50 µs voltage waveform from a

1-MV impulse generator was applied across a plane-plane gap formed by two electrodes

separated by 2-3 m for creating fields less than 1 MV/m and 0.1-0.2 m for creating fields

between 1 and 2 MV/m (Miki et al., 2002). 1.2-kV and 18 kV generators were also used.

The calibration setup is shown in Figure 3-2. The electric field was obtained by dividing

the voltage by the gap length, h (Figure. 3-2). The Pockels sensor was placed in this gap

and its output voltage was measured. The variation of the sensor output voltage as a

function of the external electric field is shown in Figure 3-3. The sensor output voltage

varies linearly with the E-field, and this linear relationship was applied to all

measurements analyzed here, even when the field values were less than the lowest field

used in the calibration process.

Field calibration of the Pockels sensors was performed at the ICLRT and

accomplished by comparing the outputs of Pockels sensors with that of a flat-plate

antenna, both installed 5 m from the triggered-lightning channel. Figure 3-4 shows

1.2-kV 18-kV, or 1-MV Impulse Voltage Generator

Figure 3-2: Calibration of the Pockels sensor. Courtesy Megumu Miki of CRIEPI, Tokyo, Japan. h=2 or 3 m for creating fields less than 1 MV/m and h=0.1 or 0.2 m for creating fields between 1 and 2 MV/m.

examples of the two types of observed electric field waveforms, termed slow and fast,

measured simultaneously with a Pockels sensor and a flat-plate antenna.

33

The flat-plate antenna was calibrated theoretically [e.g., Uman, 1987], and the

Pockels sensors were calibrated (up to about 2 MV/m) in plane-plane gaps by CRIEPI

Figure 3-3: Variation of the Pockels sensor output voltage as a function of the applied electric field: (a) sensor No.6 (used to measure vertical electric field component) (b) sensor No.7 (used to measure horizontal electric field component). Courtesy Megumu Miki of CRIEPI, Tokyo, Japan.

34

personnel (see above). Figure 3-5 shows a scatter plot of the magnitude of the vertical

electric field due to lightning measured with the Pockels sensor versus that measured

with the flat-plate antenna. Figures 3-4 and 3-5 show that the magnitudes of slow

waveforms are essentially the same for the flat-plate antenna and the Pockels sensor

records. However, the magnitudes of the relatively fast waveforms measured with the

Pockels sensor are on average about 60% of those measured using the flat-plate antenna.

This implies that electric field peaks measured using Pockels sensors may be

underestimates by 40% or so, provided that the frequency content of the electric field in

the immediate vicinity of the channel is not much different from that of relatively fast

waveforms at 5 m. The difference in the response of the Pockels sensors to slow and fast

waveforms is presumably caused by the insufficient upper frequency response of 1 MHz

of the Pockels sensor measuring system. If the frequency content is higher very close to

Figure 3-4: Comparison of the electric field waveforms simultaneously measured with a Pockels sensor and a flat-plate antenna, both at 5 m.

35

the channel than at 5 m, the field peaks measured by the Pockels sensors may be

underestimated by more than 40%.

Figure 3-5: Comparison of magnitudes of the vertical electric field peaks measured with

Pockels sensors and a flat-plate antenna, both at 5 m. Pockels sensors No.6 and No.7 were subsequently used for measuring the vertical and horizontal electric field components, respectively, in the immediate vicinity of the lightning channel. [M. Miki, V.A. Rakov, K.J. Rambo, G.H. Schnetzer, and M.A. Uman; "Electric Fields Near Triggered Lightning Channels Measured with Pockels Sensors," J. Geophys. Res., vol.107 (D16), Figure. 4, pp. 4, 2002]

3.2.2 Experimental Setup

Pockels sensors were installed on the underground rocket launching facility at the

ICLRT [Rakov et al., 2000, 2001; Crawford et al., 2001], as shown in Figure 3-6.

36

Figure 3-6: Experimental setup. [M. Miki, V.A. Rakov, K.J. Rambo, G.H. Schnetzer, and M.A. Uman; "Electric Fields Near Triggered Lightning Channels Measured with Pockels Sensors," J. Geophys. Res., vol.107 (D16), Figure. 3, pp. 3, 2002]

The vertical field sensor was placed at a radial distance of 0.1 m from, and at a

height of 0.1 m above the tip of the 2-m vertical strike rod, and the horizontal field sensor

was placed directly below it. A metal ring having a radius of 1.5 m was installed around

the strike rod. The ring was connected to the base of the strike rod, which was grounded.

Since the lightning channel could attach itself either to the strike rod or the ring, the

horizontal distance between the channel and the Pockels sensor varied between 0.1 m to

1.6 m. The corresponding lightning currents are measured using a current viewing

resistor (shunt), placed at the base of the strike rod. Currents were also measured using a

different method as discussed in Section 3.4.1. There are two types of current records: 1)

low-current records, whose duration is about 250 ms and the measurement range is from

–2 kA to 2 kA. The amplitude resolution of low-current records is about 1.8 A. The

sampling interval was 100 ns; 2) high-current records, whose duration is 50 µs and the

measurement range is from –31 kA to 18 kA. The resolution of high-current records was

37

about 450 A. The current sampling interval was 20 ns. The sampling interval for electric

field records was 0.5 µs.

3.3 Electric Field Waveforms

3.3.1 V-Shaped Signatures with ∆ERS = ∆EL

In this type, the return stroke apparently neutralizes all the charge deposited by the

leader and thus the entire waveform exhibits a V-shaped signature in which the leader

and return stroke field changes are nearly equal to each other. The rise time of the return

stroke electric field is of the order of 1 µs.

∆EL ∆ERS = ∆EL

∆EL ∆ERS < ∆EL

∆EL ∆ERS (t) < ∆EL

Time, µs

Figure 3-7: V-shaped electric field signatures with a) the return stroke field change, ∆ERS, being equal to the leader field change, ∆EL. b) ∆ERS < ∆EL, field flattening within 20 µs or so of the beginning of the return stroke (of the bottom of the V), c) ∆ERS (t) < ∆EL, no flattening within 20 µs.

38

Table 3-1: Summary of peak current and ∆EL statistics for 8 strokes exhibiting V-shaped electric field signatures with ∆ERS = ∆EL.

Peak Current, kA

∆EL, kV/m

Min Max Mean GM Min Max Mean GM

9.85

21.5

16.6

15.5

52.5

305

122

109

3.3.2 V-Shaped Signatures with ∆ERS < ∆EL and Field Flattening within 20 µs

These electric field waveforms are characterized by residual electric fields (Kodali

et al., 2003) and, hence, residual charge (and associated electrostatic field energy) that is

apparently dissipated via a slower process lasting in excess of some hundreds of

microseconds, other than the return stroke. Therefore, such waveforms cannot be used

with confidence for estimating the input energy of a lightning return-stroke using the

method illustrated in Figure. 3-1, which is based on the assumption that all the

electrostatic field energy of the leader is converted to the Joule heating of the channel by

the return stroke. However, these waveforms can be used to estimate the peak power,

which is expected to occur within the first few hundred nanoseconds, long before the

flattening takes place. We also used these waveforms for computing the input energy

after adjusting them to eliminate the residual field. The statistics for the peak current and

∆EL for this category of strokes are given in Table 3-2.

Table 3-2: Summary of peak current and ∆EL statistics for 5 strokes with ∆ERS < ∆EL and flattening within 20 µs or so.

Peak Current, kA ∆EL, kV/m Min Max Mean GM Min Max Mean GM 15.4 26.3 20.6 19.6 105 227 160 155

39

3.3.3 Signatures with ∆ERS (t) < ∆EL and no Flattening within 20 µs

In these E-field signatures, the electric field after the beginning of the return stroke

continues to increase during a time interval of the order of a few milliseconds. Such

behavior is indicative of a residual charge (and associated electrostatic field energy)

located near the attachment point and a process other than the return stokes being at work

to neutralize this residual charge. E-field waveforms with ∆ERS (t) < ∆EL and no

flattening with 20 µs were used for computing only the peak power, which occur before

the “abnormal” behavior of the return-stroke E-field begins. The statistics of the peak

current and ∆EL associated with this type of waveforms are given in Table 3-3.

Table 3-3: Summary of peak current and ∆EL statistics for 18 strokes with ∆ERS (t) < ∆EL and no flattening within 20 µs

Peak Current, kA

∆EL, kV/m

Min Max Mean GM Min Max Mean GM

5.1

26.4

11.4

10.5

175

1150

554

474

3.4 Analysis of V-Shaped E-Field Signatures with ∆ERS = ∆EL

The product of channel-base current and close longitudinal electric field, each as a

function of time, yields the power per unit channel length vs. time waveform. Since we

have the current record for the return stroke only, the following results represent

processes following the initiation of the return stroke (leader/return stroke transition). The

energy per unit length is obtained by the integration over time of the power waveform, as

discussed in Section 3.1.

3.4.1 Data Processing

E-field waveforms are typically noisy (see Figure 3-8 a) and hence some sort of

filtering (averaging) has to be performed to make the electric field tractable. Only the

40

portion of the electric field record following the initiation of the return stroke was

filtered, since only this portion was needed for estimating power and energy input. A

moving-averaging window of 100 data points, which acts as a low-pass filter, was used

for this purpose. Averaging was done after a suitable time interval after the beginning of

the return stroke, so that the initial (fast-varying) portion of the return stroke is not

modified, as illustrated in Figure. 3.8 b. In this example the E-field waveform is averaged

2.5 µs after the start of the return stroke. One can see that the main features of the

waveform are preserved, while the noise is significantly reduced. Such filtering was

performed for all the strokes analyzed here. The resultant electric fields are shown in

Figures 3-10 to 3-17.

Leader Return Stroke A

Time-AveragedOriginal

2.5µs B

Figure 3-8: Stroke S0013-1. A) Original E-field record. B) Filtered (100-µs moving-window time averaged) version of the E-field waveform shown in A).

Time, µs

In 2000, lightning currents were measured using two methods. In the first method,

the total lightning current was measured using a current viewing resistor, CVR (shunt),

41

placed at the base of the strike rod. In the second method, currents entering the launcher

grounding system (ground screen and ground rod) were measured and summed to obtain

the total lightning current. In this latter case, the current into the 70 × 70m2 buried

metallic grid (ground screen) was measured using two CVR’s and the ground rod current

was measured using two P 110A’s current transformers (CT). The current range of P

110A’s is from a few amperes to 20 kA, when terminated with a 50-ohm resistor. A

passive combiner was used to sum the two signals from the ground rod CT’s to a total

ground rod current. The ground screen current was measured by two separate

instrumentation systems IIS-S (south ground screen current) and IIS-N (north ground

screen current). The sum of the ground rod current and north and south screen currents

gives the total screen

IR = 0.85+1.02 IS R2 = 0.8 n = 36

Figure.3-9: Scatter plot of screen current, IS vs. strike rod current, IR, for 2000. [V. Kodali, “Characterization and analysis of close lightning electromagnetic fields,” Master’s thesis, University of Florida; 2003].

current. A scatter plot of ground screen current (the sum of the ground screen and ground

rod currents to be exact) vs. strike rod current is shown in Figure 3-9. With a few

42

exceptions, the two current values are very close to each other. In the following sections

(also in Tables 3-1 to 3-3), we used the strike-rod current, although in Table 3-4 the

power and energy were also computed using the ground-screen current, when available.

3.4.2 Power and Input Energy

Power as a function of time, obtained as the product of longitudinal E-field and

strike-rod current, and energy, the integral of the power curve, are shown in Figures 3-10

to 3-17, for the eight strokes having the V-shaped E-field signatures with ∆ERS = ∆EL.

The estimated peak power and energy values are given in Table 3-4. Histograms of the

various quantities and associated scatter plots are found in Section 3.4.3. Error analysis is

presented in Section 3.4.4.

Figure 3-10: Time variation of electric field, current, power, and energy for stroke

S006-4.

µs

43

µs

Figure 3-11: Same as Figure. 3-10, but for Stroke S008-4. Negative values in the variation of power with time are due to residual noise in the electric field waveform.

µs Figure 3-12: Same as Figure. 3-10, but for Stroke S0013-1.

44

Time, µs


Time, µs


45

µs

Figure 3-15: Same as Figure. 3-10, but for Stroke S0015-4.

Power and energy are estimated for both strike-rod and ground-screen currents,

when both currents are available. As seen from Table 3-4, the peak power varies from 2.2

×108 W/m to 25.1×108 W/m and input energy at 10 to 50 µs from 0.9 ×103 J/m to 6.35

×103 J/m. The peak power values are consistent with 12 ×108 W/m reported by Krider et

al. (1968), and the energy values are in agreement with predictions (of the order of 103

J/m) of gas-dynamic models (Sections 2.5.1 and 2.5.3, respectively).

46

Figure 3-16: Same as Figure. 3-10, but for Stroke S0015-6. The time scale here is different from that in Figures. 3-10 to 3-15 and 3-17 (50 µs vs. 55 µs). The energy is obtained at 10 µs after the return stroke, because at later times the electric field becomes positive causing the power waveform to change polarity (become negative), which is physically unreasonable.

Figure 3-17: Same as Figure. 3-10, but for Stroke S0023-3.

Time, µs

Time, µs

Leader Return Stroke

47

Table 3-4: Power and energy estimates for strokes having V- shaped E-field signatures with ∆EL= ∆ERS.Peak current,

kA

Peak power, ×108 W/m

Energy, ×103 J/m

Date FlashID

Stroke order

Termination point

Rod Screen

∆EL, kV/m

Rod Screen Rod Screen

Remarks

6/13

S0006

4

Rod

14.3

10.2

52.5

2.35

2. 9

1.8 (at 45.7 µs)

1.7(at 41.3 µs)

Classical trigger, 5

RSs

6/17 S0008 4 Ring 20.9 19.3 60.0 2.2 4.8 0.9 (at45.6 µs)

0.8 (at 45.5 µs)

Classical trigger, >8

RSs 6/18 1 Rod 11.6 - 125.0 5.2 - 2.57 (at

45.7

µs)

-

6/18

S0013 4 Ring 11.6 - 122.5 8.6 - 1.34 (at

45.5 -

µs)


RSs

6/23 2 Rod 19.3 Noisy 105.0 9.9 - 6.35 (at45.5 µs)

-

6/23 4 Rod 21.5 Noisy 112.5 8.7 - 5.0 (at45.5 µs)

-

6/23

S0015

6 Rod 20.0 Noisy 92.5 14.5 - 1.3 (at 10.0 µs)*

-


RSs

7/11 S0023 3 Ring 9.85 16.3 305.0 25.1 22.0 6.2 (at 45.7 µs)

5.4 (at 47.3 µs)


RSs

* For this stroke, after 10 µs ∆ERS > ∆EL causing the power waveform to change polarity (become negative) which is physically unreasonable.

48

3.4.3 Statistical Analysis

Histograms displaying the distributions of peak current, ∆EL, peak power, energy,

and action integral for the V-shaped E-field waveforms with ∆ERS = ∆EL are shown in

Figures 3-18 through 3-22. The means and standard deviations are given separately for

different lightning channel termination points (rod or ring) and for all data combined.

Additionally presented in Figure 3-23 and Figure 3-24 are histograms of risetimes for

current and power.

Num

ber

Peak Current, kA ring n = 3 Mean = 14.1 kA St. Dev. = 5.9 kA

rod n = 5 Mean = 18.1 kA St. Dev. = 3.8 kA

n = 8 Mean = 16.1 kA St. Dev. = 4.8 kA Min = 9.9 kA Max = 21.5 kA

Figure 3-18: Histogram of peak current for strokes characterized by V- shaped electric

field signatures with ∆ERS = ∆EL.

49

Num

ber

∆EL, kV/m ring n = 3 Mean = 162.5 kV/m St. Dev = 127.3 kV/m

rod n = 5 Mean = 97.5 kV/m St. Dev = 27.8 kV/m

n = 8 Mean = 121.9 kV/m St. Dev = 78.8 kV/m Min = 52.5 kV/m Max = 305 kV/m

Figure 3-19: Histogram of ∆EL for strokes characterized by V- shaped electric field

signatures with ∆ERS = ∆EL.

50

Num

ber

Peak Power, × 108 W/m ring n = 3 Mean =11.9 × 108 W/m St. Dev. = 11.7 × 108 W/m

rod n = 5 Mean =8.2 × 108 W/m St. Dev. = 4.4 × 108 W/m

n =8 Mean =9.6 × 108 W/m St. Dev. = 7.4 × 108 W/m

Min = 2.2 × 108 W/m St. Dev = 25 × 108 W/m Figure 3-20: Histogram of peak power for strokes characterized by V- shaped electric


51

Num

ber

Energy, × 103 J/m ring n = 3 Mean = 2.8× 103 J/m St. Dev. = 2.9 × 103 J/m

rod n = 5 Mean = 3.6 × 103 J/m St. Dev. = 2.5 × 103 J/m

rod n = 8 Mean = 3.6 × 103 J/m St. Dev. = 2.5 × 103 J/m Min = 0.9 × 103 J/m St. Dev. = 6.2 × 103 J/m

Figure 3-21: Histogram for input energy for strokes characterized by V- shaped electric


52

Num

ber

Action Integral, × 103 A2s

ring n = 3 Mean = 2.14 × 103 A2s St. Dev. = 1.63 × 103 A2s rod n = 5 Mean = 1.89 × 103 A2s St. Dev. = 1.37 × 103 A2s

n = 8 Mean = 2.04 × 103 A2s St. Dev = 1.44 × 103 A2s Min = 0.56 × 103 A2s Max = 4.68× 103 A2s

Figure 3-22: Histogram for action integral for strokes characterized by V- shaped electric field signatures with ∆ERS = ∆EL.

53

Mean = 0.85 µs Std. dev = 0.37 µs Min = 0.40 µs Max = 1.6 µs

Num

ber

Risetime, µs Figure 3-23: Histogram of the risetime of current for strokes characterized by V-shaped

electric field signatures with ∆ERS = ∆EL. The risetime is defined as the time taken by current to rise from 0% to 100% of the peak value.

Mean = 0.43 µs Std.dev = 0.12 µs Min = 0.28 µs Max = 0.60 µs

Num

ber

Risetime, µs Figure 3-24: Histogram of the 0-100 % risetime of power per unit length for strokes

characterized by V-shaped electric field signatures with ∆ERS = ∆EL.

54

Scatter plots showing correlation between the various parameters are presented in

Figures 3-25 through 3-29.

P = -0.49 I + 17.6 R2 = 0.1

Peak

Pow

er, ×

108 W

/m

Peak Current, kA

Figure 3-25: Peak power vs. peak current for strokes characterized by V- shaped electric field signatures with ∆ERS = ∆EL. The filled circles and the hollow circles represent the strokes for which the lightning channel terminated on the rod and ring, respectively.

It can be observed from Figures 3-25 and 3-26 that the determination coefficient,

R2, which is the square of the correlation coefficient, R, in both cases is close to zero.

Two possible reasons for the low correlation coefficients are the following: 1) the

influence of the electric field is more significant than that of current and 2) the electric

field decays to a negligible value before the current attains its peak magnitude. Indeed,

the mean risetime to peak current is 0.85 µs (see Figure. 3-23), and the average risetime

of power per unit length to its peak 0.43 µs (see Figures. 3-23 and 3-24). Hence, electric

55

field has a more pronounced effect on the peak power value, since the current attains its

peak value only after the power does, resulting in the lack of dependence of the peak

power on the peak current value. It is likely that both reasons listed above can contribute

to the observed lack of correlation between the peak power and peak current. There

appears to weak positive correlation between the energy per unit length and action

integral. The latter was computed as the integral of the square of current over the same

time interval (typically between 40 and 50 µs) as the corresponding power per unit

length. The unit for action integral is A2s, which is the same as J/Ω.

W = -0.02 I + 3.6 R2 = 2.3 × 10-3

Ener

gy, ×

103 J/

m

Peak Current, kA Figure 3-26: Energy vs. peak current for strokes characterized by V- shaped electric field

signatures with ∆ERS = ∆EL. The filled circles and the hollow circles represent the strokes for which the lightning channel terminated on the rod and ring, respectively.

56

P = 0.08 ∆EL – 0.66 R2 = 0.78

Peak

Pow

er, ×

108 W

/m

∆EL, kV/m Figure 3-27: Peak power vs. ∆EL for strokes characterized by V- shaped electric field

signature with ∆ERS = ∆EL.

W = 0.02 ∆EL + 1 R2 = 0.37

Ener

gy, ×

103 J/

m

Figure 3-28: Energy vs. ∆EL for strokes characterized by V- shaped electric field

signature with ∆ERS = ∆EL.

∆EL, kV/m

57

Energy, J/m

Act

ion

Inte

gral

, kJ/Ω

AI = 348 W + 936.21 R2 = 0.31

5 4 3 2 1 0

0 1000 2000 3000 4000 5000 6000 7000

Figure 3-29: Energy vs. Action Integral for strokes characterized by V- shaped electric field signature with ∆ERS = ∆EL.

3.4.4 Error Analysis

In this section, three sources of uncertainty involved in power and energy estimates

are examined: 1) as noted in Section 3.2.1, electric fields measured using Pockels sensors

may be underestimated by 40% or so due to the insufficient upper frequency response of

1 MHz of the measuring system (Miki et al., 2002), (2) the sampling interval for electric

field records was 0.5 µs vs. 20 ns for current records (see section 3.2.2), and (3) electric

field and current records were aligned manually using the bottom of the V-shaped electric

filed signature and the beginning of the current waveform. There is little ambiguity in

selecting the return-stroke current starting point, while the bottom of the V is somewhat

uncertain within ± 0.5 µs due to insufficient sampling rate for electric field. Sensitivity of

power and energy estimates to changes in the electric field peak magnitude and its

58

position on the time scale are examined using the following procedure. The peak power

and energy are calculated after applying a correction factor of 1.6 at the instant of peak

electric field (to take into account the 40% potential error due to the insufficient upper

frequency response of the measuring system). Then the electric field waveform is

modified in three different ways: (1) keeping the position of the negative field maximum

intact (2) moving the negative maximum (the bottom of the V) 0.24 µs to the left, and (3)

moving the maximum, 0.24 µs to the right from its original position, in order to partially

account for the ± 0.5 µs uncertainty noted above. These three steps are illustrated in

Figures 3-30 to 3-32 for stroke S0013-1. Similar procedure was applied to all the strokes

analyzed here, and results are summarized in Table 3-5, along with the peak power and

energy values estimated from the original records without correction.

59

E-Fi

eld,

kV

/m

µs

Figure 3-30: Flash 0013, stroke 1; Correction factor of 1.6 is applied at the instant of negative E-field peak. The filled circles and the arrows indicate the original data points (the sampling interval of the electric field record was 0.5 µs). The hollow symbols represent fictitious data points based on linear interpolation so as to match the sampling interval of 0.02 µs of the current record. The circles represent the original E-field waveform and the triangles represent the E-field waveform after the correction factor of 1.6 is applied. Calculated values of peak power and energy before and after correction are given.

60

E-Fi

eld,

kV

/m

µs

Figure 3-31: Flash 0013, stroke 1; Correction factor of 1.6 is applied at the instant of negative E-field peak, which is shifted by 0.24 µs to the left in order to partially account for the ± 0.5 µs uncertainty in the position of the peak. The filled circles and the arrows indicate the original data points. The hollow symbols represent fictitious data points based on linear interpolation so as to match the sampling interval of 0.02 µs of the current record. Calculated values of peak power and energy before and after correction are given.

61

Figure 3-32: Flash 0013, stroke 1; Correction factor of 1.6 is applied at the instant of

negative E-field peak, which is shifted by 0.24 µs to the right in order to partially account for the ± 0.5 µs uncertainty in the position of the peak. The filled circles and the arrows indicate the original data points. The hollow symbols represent fictitious data points based on linear interpolation so as to match the sampling interval of 0.02 µs of the current record. Calculated values of peak power and energy before and after correction are given.

igure 3-32: Flash 0013, stroke 1; Correction factor of 1.6 is applied at the instant of negative E-field peak, which is shifted by 0.24 µs to the right in order to partially account for the ± 0.5 µs uncertainty in the position of the peak. The filled circles and the arrows indicate the original data points. The hollow symbols represent fictitious data points based on linear interpolation so as to match the sampling interval of 0.02 µs of the current record. Calculated values of peak power and energy before and after correction are given.

Time, µs

62

Table 3-5: Dependence of peak power and energy on errors in the value of E-field peak and its position on the time scale. Original record

E-Field peak multiplied by 1.6

E-Field peak multiplied by 1.6 and shifted by 0.24 µs to the left

E-Field peak multiplied by 1.6 and shifted by 0.24 µs to the right

Flash ID

Stroke order

P.P, ×108, W/m

E, ×103, J/m

P.P, ×108, W/m

∆ %

E, ×103, J/m

∆ %

P.P, ×108, W/m

∆ %

E, ×103, J/m

∆ %

P.P, ×108, W/m

∆ %

E, ×103, J/m

∆ %

S006 4 2.3 6.4 2.6 13 1.8 0.0 4.5 96 1.92 6.7 1.74 -24 1.77 -1.7

S008 4 1.97 0.8 2.83 44 0.82 2.5 6.58 234 0.96 20 1.96 -0.5 0.76 -5.0

S0013 1 5.2 2.6 6.1 17 2.6 0.0 8.0 54 2.8 7.7 4.1 -21 2.4 -7.7 S0013 4 8.6 1.34 8.6 0 2.3 -3.0 13 51 1.7 27 5.9 -31 1.2 -10

S0015 2 9.9 6.35 12.7 28 6.43 1.0 14 41 6.2 -2.0 8.1 -18 6.1 -4.0 S0015 4 8.7 5.0 11.9 37 5.1 2.0 13.9 60 5.38 8.0 5.9 -32 4.86 -3.0

S0015 6 14.5 1.3 14.5 0.0 1.33 2.3 17.5 21 1.7 31 10.7 26 0.94 -28

S0023 3 25.1 6.2 31.5 25 6.4 3.0 26.9 7.0 6.9 11 23.8 -5.0 5.6 -10

Mean 9.53 3.75 11.3 21 3.35 1.0 13 71 3.5 14 7.8 -20 2.9 -9.0

St. deviation 7.53 2.5 9.3 16.1 2.3 1.96 7.1 71.2 2.3 11.3 7.1 19.4 2.2 8.4

63

Table 3-5 suggests that the peak power estimates are sensitive to the considered

uncertainties, the maximum mean error being 71%. Intuitively it makes sense, because

the peak power occurs in the first few microseconds or less of the beginning of the return

stroke. On the other hand, energy estimates are relatively insensitive to the uncertainties

examined here, the variation not exceeding 31% (14% on average for the 8 strokes

analyzed).

3.4.5 Channel Resistance and Radius

The expression, R (t) = E (t) / I (t), gives the evolution of resistance per unit channel

length with time. Since we cannot measure leader currents, R (t) can be evaluated only

for the return stroke. The evolution of channel radius can be estimated from the channel

resistance using the expression, r (t) = [σ π R (t)] –0.5, where σ is the electrical

conductivity of the channel, assuming σ = 104 S/m. In reality, σ increases with time (as

the channel temperature increases), but this variation is rather weak for the expected

temperature range (≥ 20,000° K or so) (e g., Rakov, 1998). The assumption of σ =

constant implies that R (t) decreases only due to expansion of channel (increase in r (t)).

In principle, the channel radius and resistance can be evaluated for the entire length

of the field record. But the results become dominated by noise once the electric field

magnitude decreases below 20 kV/m because of the limitations on the dynamic range of

the Pockels sensor. The evolution of resistance and channel radius along with

corresponding E-field, current, and power profiles, for eight strokes exhibiting V-shaped

E-field signatures with ∆ERS = ∆EL for the time interval when the E-field magnitude is

greater than 20 kV/m is shown in Figures 3-33 to 3-40. Table 3-6 shows the resistance

and channel radius at the instant of peak power. The time scale over which the evolution

64

is shown differs from that of other Figures in Section 3.4.2 because as the magnitude of

E(t) falls below about 20 kV/m, it attains small values during zero crossings forcing R(t)

to very small values, which in turn causes r (t) to go to unreasonably high values

(Figure 3-41).

a) b) c) d) e)

S006-4

Figure 3-33: Evolution of the various quantities for the first 0.58 µs for Flash S006, stroke 4. a) E-field; b) current; c) power per unit length; d) channel resistance per unit length; e) channel radius.

65

a) b) c) d) e)

Figure 3-34: Evolution of the various quantities for the first 0.4 µs for Flash S008, stroke 4. a) E-field; b) current; c) power per unit length; d) channel resistance per unit length; e) channel radius.

66

Figure 3-35: Evolution of the various quantities for the first 1.4 µs for Flash S0013,

stroke 1. a) E-field; b) current; c) power per unit length; d) channel resistance per unit length; e) channel radius

67

a) b) c) d) e)

S0013-4

Figure 3-36: Evolution of the various quantities for the first 1.2 µs for Flash S0013, stroke 4. a) E-field; b) current; c) power per unit length; d) channel resistance per unit length; e) channel radius

68

a) b) c) d) e)


69

a) b) c) d) e)


70

a) b) c) d) e)


71

a) b) c) d) e)

S0023-3

Figure 3-40: Evolution of the various quantities for the first 5 µs for Flash S0023, stroke 3. a) E-field; b) current; c) power per unit length; d) channel resistance per unit length; e) channel radius

72

Cha

nnel

Rad

ius,

cm

Time, µs

Figure 3-41: Evolution of channel radius for S0008-4. The peaks are formed due to small values attained by the noisy electric field waveforms.

Table 3-6: Resistance and channel radius for strokes having V- shaped E-field signatures

with ∆ERS = ∆EL

Flash ID

Stroke order

Resistance*, Ω/m

Channel radius*, cm

Termination point

S006 4 0.67 0.69 Rod S008 4 1.3 0.49 Ring

1 5.1 0.25 Rod S0013 4 8.0 0.22 Ring 2 4.5 0.31 4 5.1 0.25

S0015

6 4.3 0.27

Rod

S0023

3

30.8 0.10

Ring

Mean value 7.5 0.32

* Resistance and channel radius are evaluated at the instant of peak power

73

As expected, the channel resistance decreases and channel radius increases with time as

the return-stroke current heats the channel.

In Rakov (1998), the propagation mechanisms of dart leaders and return strokes are

analyzed by comparing the behavior of traveling waves on a lossy transmission line and

the observed characteristics of these two lightning processes. The R in the transmission

line model is assumed to be a constant but different ahead of and behind either the dart-

leader or the return-stroke front with any nonlinear effects occurring at the front. The

channel radius and resistance ahead of return-stroke front are estimated to be around 0.3

cm and 3.5 Ω/m. The channel radius and resistance in Table 3-6 are obtained at the

instant of peak power which occurs at around 0.4 µs. The mean values for channel radius

and resistance in Table 3-6 are 0.32 cm and 7.5 Ω/m. In Rakov (1998), the channel radius

and resistance behind the return-stroke front are estimated to be around 3 cm and 0.035

Ω/m. For the pre-dart-leader channel, these two quantities are estimated to be around 3

cm and 18 kΩ/m, respectively.

3.5 Analysis of V-shaped E-field Signatures with ∆ERS < ∆EL and Field Flattening within 20 µs

An example of such a waveform exhibiting the residual electric field, ∆EL - ∆ERS, is

shown in Figure. 3-5 b. The product of channel-base current and close vertical

(longitudinal) electric field, each as a function of time, yields the power per unit channel

length vs. time waveform. The energy per unit length is obtained by integration over time

of the power waveform, as discussed in section 3.1. The electric field waveform due to

the return stroke was adjusted by subtracting the residual electric field, ∆EL- ∆ERS. ∆ERS

is defined as the average value of the electric field from 44 µs to 50 µs after the return

stroke. This time frame is selected since most of the energy estimates for most of the

__ __

74

events analyzed in Section 3.4 was obtained at around 45 µs after the beginning of the

return stroke. The adjustment was needed (see Section 3.3.2) to eliminate the electrostatic

energy not involved in the return stroke process. Figure 3-42 shows this procedure for the

stroke S0008-3. This methodology enables us to reasonably compare the energy estimates

for the two classes of electric field waveforms. Table 3-7 gives the peak power and

energy values for the 5 strokes exhibiting V-shaped E-field signatures with ∆ERS < ∆EL

and field flattening within 20 µs. Histograms giving the distribution of peak current, ∆EL,

peak power, energy, and action integral are shown in Figures 3-43 to 3-47.

75

Table 3-7: Power and energy estimates for strokes having V- shaped E-Field Signatures with ∆ERS < ∆EL and field flattening within 20 µs or so.

Peak current, kA


Energy, ×103 J/m

(∆EL- ∆ERS)*

kV/m Date

FlashID

Stroke order

Termination point

Screen Rod

∆EL, kV/m

Screen Rod Screen Rod Screen Rod

6/13

S0008

3

Ring

13.9

17.8

180.5

8.69

12.7

1 (at 20 µs)

8.1 (at 50 µs)

20.0

18.1

6/18 S0012 1 Ring - 21.0 189.9 - 14.9 - 2.9 (at50 µs)

- 36.7

2 - 12.3 124.9 - 9.40 - 1.9 (at47.5 µs)

17.9-

3 - 26.3 219.9 - 16.8 - 3.66 (at50 µs)

62.0-

6/18

S0013

5

Ring

- 22.7 277.3 - 20.2 - 3.3 (at45.5 µs)

- 47.5

* ∆ERS in (∆EL- ∆ERS) is the average value of electric field from 44 µs to 50 µs after the return stroke.

76

∆ERS

__ ∆EL

Figure.3-42: V-shaped signature with ∆EL> ∆ERS. ∆ERS represents the average electric field between 44 µs to 50 µs after the beginning of the return stroke (at 50 µs). The electric field waveform due to the return stroke was adjusted by subtracting the residual electric field, ∆EL- ∆ERS.

__ __ Time, µs

Min = 12.3 kA Max = 26.3 kA Mean = 20 kA Std. Dev. = 5.3 kA n =5

Num

ber

Peak Current, kA

Figure 3-43: Histogram of peak current for strokes characterized by V-shaped electric field signatures with ∆ERS < ∆EL and flattening within 20 µs.

77

Min = 107.0 kV Max = 229.8 kV Mean = 162.1 kV Std. Dev. = 43.90 kV n = 5

Num

ber

∆EL, kV Figure 3-44: Histogram of ∆EL for strokes characterized by V-shaped electric field

signatures with ∆ERS < ∆EL and flattening within 20 µs.

Min = 9.4 × 108 W/m Max = 20.2 × 108 W/m Mean = 14.8 × 108 W/m Std. Dev. = 4.08 × 108 W/m n = 5

Num

ber

Peak Power, × 108 W/m

Figure 3-45: Histogram of peak power per unit length for strokes characterized by V-shaped electric field signatures with ∆ERS < ∆EL and flattening within 20 µs.

78

Min = 1.9 × 103 J/m Max = 8.1 × 103 J/m Mean = 3.9 × 103 J/m Std. Dev. = 2.4 × 103 J/m n = 5

Num

ber

Energy, × 103 J/m Figure 3-46: Histogram of energy per unit length for strokes characterized by V-shaped

electric field signatures with ∆ERS < ∆EL and flattening within 20 µs.

Min = 2.05 × 103 A2s Max = 6.18 × 103 A2s Mean = 3.92 × 103 A2s Std. Dev. = 1.79 × 103 A2s n = 5

Num

ber

Action Integral, × 103A2s

Figure 3-47: Histogram of action integral for strokes characterized by V-shaped electric field signatures with ∆ERS < ∆EL and flattening within 20 µs.

79

The mean values of peak power and energy for this category of strokes are similar

to their counterparts for the first category (see Section 3.4). Scatter plots showing

correlation between the different quantities are presented in Figures 3-48 to 3-52.

P = 0.56 I + 3.17 R2 = 0.72

Peak

Pow

er, ×

108 W

/m

Figure 3-48: Peak power vs. peak current for strokes characterized by V- shaped electric

field signatures with ∆ERS < ∆EL and flattening within 20 µs.

Peak Current, kA

80

W = 0.02 I + 3.6 R2 = 1.6 × 10-3

En

ergy

, × 1

03 J/m

Peak Current, kA Figure 3-49: Energy vs. peak current for strokes characterized by V- shaped electric field


P = 0.08 ∆EL + 0.98 R2 = 0.83

Peak

Pow

er, ×

108 W

/m

Figure 3-50: Energy vs. ∆EL for strokes characterized by V- shaped electric field


∆EL, kV/m

81

100 120 140 160 180 200 220 2401

2

3

4

5

6

7

8

9

W = 0.01 ∆EL + 2.28 R2 = 0.04

Ener

gy, ×

103 J/

m

Figure 3-51: Energy vs. ∆EL for strokes characterized by V- shaped electric field signatures with ∆EL< ∆ERS and flattening within 20 µs.

∆EL, kV/m

AI = -0.05 W + 4.13 R2 = 5.2 × 10-3

Act

ion

Inte

gral

, kJ/Ω

Energy, J/m 1000 2000 3000 4000 5000 6000 7000 8000 9000

Figure 3-52: Energy vs. Action integral for strokes characterized by V- shaped electric field signatures with ∆EL< ∆ERS and flattening within 20 µs.

82

In contrast with the strokes for which ∆ERS = ∆EL, there appears to be moderate

positive correlation between the peak power and peak current (or ∆EL) and essentially no

correlation between the energy and action integral, although the sample size is small.

3.6 Analysis of V-Shaped E-field Signatures with ∆ERS (t) < ∆EL and no Field Flattening within 20 µs

An example of such a signature is shown in Figure 3-5 (c). Table 3-8 gives the

values of the strike-rod peak current (no usable ground-screen currents are available),

∆EL, and peak power for the 18 strokes of this type. Since there is no field flattening

within 20 µs (in fact, the field varied on the millisecond time scale) of the start of the

return stroke, energy computation was not performed for the events of this type.

Table 3-8: Power estimates for strokes having V- shaped E-Field Signatures with ∆ERS (t) < ∆EL (t) and no flattening within 20 µs

Date Flash ID

Stroke order

Termination point

Peak current,

kA

∆EL, kV/m


Remarks

1 11.8 492 4.4 7/11 S0022

3 Rod

8.9 486.6 3.7 Classical

trigger, 3 RSs

1 11.5 451.5 3.4 7/11 S0023

2 Ring

15.2 308.2 3.9 Classical

trigger, 3 RSs

1 15.0 743.4 11.0 2 9.4 763 5.9

3 7.1 432 2.7 7/16 S0025

4

Ring

26.4 1149 25.1

Classical trigger, 4 RSs

1 11.4 865.1 9

2 17.0 1102 17.1 7/16 S0027

3

Ring

15.3 1108 12.8


83

Continued Table3-8.

Date Flash ID

Stroke order

Termination point

Peak current,

kA

∆EL, kV/m

Peak power,

×109 W/m Remarks

1 6.7 256.8 1.6

2 5.1 358.8 1.8

3 5.9 175.7 0.64

4 11.3 446.1 4.8

5 8.2 209.5 1.3

6 12.0 364.9 4.1

7/20 S0029

7

Rod

6.9 263.6 1.8


Num

ber

Peak Current, kA

Ring

Rod

n = 9 Mean = 14.3 kA Std. Dev. = 5.6 kA n = 9 Mean = 8.5 kA Std. Dev. = 2.63 kA

Mean = 11.4 kA Std.Dev = 5.1 kA Min = 5.1 kA Max = 26.4 kA

n = 18

Figure 3-53: Histogram of ∆EL for strokes characterized by V- shaped electric field

signatures with ∆ERS (t) < ∆EL and no field flattening within 20 µs.

84

Num

ber

∆EL, kV n = 9 Mean = 769.1 kV Std. Dev. = 317.6 kV

n = 9 Mean = 339.3 kV Std. Dev. = 119.2 kV

n = 18 Mean = 554.2 kV Std. Dev. = 321 kV

Min = 175.7 kV Max = 1149 kV

Figure 3-54: Histogram of ∆EL for strokes characterized by V- shaped electric field signatures with ∆ERS (t) < ∆EL and no field flattening within 20 µs.

85

Num

ber

Peak Power, × 109 W/m Ring n = 9 Mean = 10.1 × 109 W/m Std.Dev = 7.4 × 109 W/m

Rod n = 9 Mean = 2.7 × 109 W/m Std.Dev = 1.6 × 109 W/m

Mean = 6.4 × 109 W/m Std.Dev = 6.4 × 109 W/m Min = 0.64 × 109 W/m Max = 25.1 × 109 W/m

n = 18 Figure 3-55: Histogram of peak power for strokes characterized by V- shaped electric

field signatures with ∆ERS (t) < ∆EL and no field flattening within 20 µs.

86

Num

ber

Action Integral, × 103A2s Ring n = 9 Mean = 2.14 × 103A2s St. Dev. = 1.63 × 103A2s Rod n = 9 Mean = 1.89 × 103A2s St. Dev. = 1.37 × 103A2s

n = 18 Mean = 2.41 × 103A2s St. Dev = 2.06 × 103A2s Min = 0.3 × 103A2s Max = 7.02 × 103A2s

Figure 3-56: Histogram of action integral for strokes characterized by electric field signatures with ∆ERS (t) < ∆EL and no field flattening within 20 µs.

As seen in Figure. 3-53, the mean peak power for this stroke category is several

times larger than for the first two categories considered in Sections 3.4 and 3.5. Scatter

plots showing correlation between the different quantities are presented in Figures. 3-55

and 3-56.

87

P = 1.1 I – 6.23 R2 = 0.83

Peak

Pow

er, ×

109 W

/m

Peak Current, kA

Figure 3-57: Peak power vs. peak current for strokes characterized by electric field signatures with ∆ERS (t) < ∆EL and no field flattening within 20 µs. The filled circles and the hollow circles represent strokes for which the lightning channel was attached to the rod and the ring, respectively.

88

P = 0.02 ∆EL – 3.5 R2 = 0.83

Peak

Pow

er, ×

109 W

/m

∆EL, kV

Figure 3-58: Peak power vs. ∆EL for strokes characterized by electric field signatures with ∆ERS (t) < ∆EL and no field flattening within 20 µs. The filled circles and the hollow circles represent the strokes for which the lightning channel was attached to the rod and the ring, respectively.

Similar to the strokes considered in Section 3.5, there appears to be moderate

positive correlation between the peak power and peak current (or ∆EL).

CHAPTER 4 CHARACTERIZATION OF PULSES SUPERIMPOSED ON STEADY CURRENTS

4.1 Initial Stage in Rocket-Triggered Lightning

4.1.1 Introduction

Rocket-triggered lightning is initiated by an upward leader propagating from the

upper end of a vertical grounded wire extended below the charged cloud by a small

rocket. The upward-leader stage, including explosion of the triggering wire and its

replacement by an upward-leader plasma channel, is followed by an initial continuous

current (ICC). ICC has duration of some hundreds of milliseconds and amplitude of some

tens to some thousands of amperes. The upward leader and the ICC constitute the initial

stage (IS) of rocket-triggered lightning. ICC pulses are the current pulses superimposed

on the slowly varying continuous current of the initial stage. After the cessations of the

ICC, one or more downward leader/upward return stroke sequences may occur. The

magnitudes of ICC pulses are smaller than those of return-stroke pulses. Overall, the

initial stage in rocket-triggered lightning is apparently similar to that of lightning initiated

from tall structures (e.g., Rakov, 1999). This chapter analyzes the action integral (energy

per unit resistance at the channel termination point) and other characteristics of the ICC

pulses, including duration, rise-time, pulse peak, half-peak width, and charge in rocket-

triggered lightning The following comparisons are made: (a) ICC pulses in triggered

lightning recorded at the ICLRT in 2002 and 2003 (relatively high sampling rate) vs.

their counterparts recorded earlier (relatively low sampling rate), (b) ICC pulses in

triggered lightning vs. those in object-initiate lightning, and (c) ICC pulses in triggered

89

90

lightning vs. M-component (pulses superimposed on the continuing current that follows

the return strokes) pulses in triggered lightning.

A typical initial stage current in rocket-triggered lightning is shown in Figure 4-1.

RS2

Wire explosion

ICC pulses Initial stage

RS1

Figure 4-1: Flash 03-31, bipolar flash. The negative initial stage is followed by 2 return strokes (RS1 and RS2) of opposite polarity. In the inset the instant at which wire explosion takes place is shown.

In Figure 4-1, the initial stage contains a number of ICC pulses superimposed on a

slowly varying current waveform. Figure 4-2 illustrates ICC pulses with relatively short

(ICC1 and ICC2) and relatively long (ICC3 and ICC4) risetimes, as well as a typical

return-stroke pulse (RS1). Definitions of the characteristics of ICC pulses analyzed here

are illustrated in Figure 4-3.

91

ICC4 ICC3

ICC2

ICC1

RS1

Figure 4-2: Flash 03-31. Comparison of ICC pulses with relatively short (ICC1 and ICC2) and relatively long (ICC3 and ICC4) risetimes and a typical return-stroke pulse (RS1).

While computing the charge and action integral for the ICC pulses, the background,

slowly-varying current is approximated by an imaginary line (dashed line in Figure 4-4),

the background is subtracted, and charge and action integral are computed using the

modified current pulse shown in the inset of Figure 4-4.

92

4.1.2 Statistical Characteristics of ICC Pulses

The ICC pulses analyzed here occurred in 11 flashes triggered at the ICLRT at

Camp Blanding, Florida, in 2002 and 2003. Histograms of the distributions of the

parameters are shown in Figures 4-5 to 4-22. These are compared with statistics, found in

Miki et al. (2004), of the ICC pulses in rocket-triggered lightning analyzed previously

and ICC pulses in object-initiated lightning derived from current measurements on 1) the

Gaisberg tower (100 m, Austria), 2) the Peissenberg tower (160 m, Germany), and 3) the

Fukui chimney (200 m, Japan). Miki et al. (2004) found that the characteristics of ICC

pulses in object-initiated lightning are similar within a factor of two, but differ more

Charge = ∫i(t) dt Action Integral = ∫ i(t)2dt

Flash 03-31

Figure 4-3: Definitions of parameters (peak, duration, rise time, half-peak width; additionally shown is the preceding continuous current level) of ICC pulses. Integration in evaluating charge and action integral is over the duration of the pulse. Peak= 0.31 kA, duration=1.9 ms, risetime=137 µs, half-peak width=0.43 ms, charge = 86 mC, Action integral = 6.1 A2s.

93

Cur

rent

, A

Cur

rent

, A

Flash F029

Time, s

Time, s Figure 4-4: Illustration of the removal of the background continuous current in computing

charge and action integral. The dashed line represents the imaginary line, which is used as reference. The charge and action integral are computed by integrating the modified current (or the square of current) as shown in the inset. In this example, charge = 80 mC, action integral = 7.8 A2s.

significantly from their counterparts in triggered lightning. The triggered-lightning data

analyzed by Miki et al. (2004) were acquired at the ICLRT in 1996, 1997, 1999, and

2000. The ICC pulses in object-initiated lightning exhibit larger peaks, shorter rise times,

and shorter half-peak widths than do the ICC pulses in 1996, 1997, 1999, and 2000

rocket-triggered lightning. The rocket-triggered lightning currents were recorded on a

magnetic tape and later digitized. The sampling interval was 40 µs for the years 1996 and

1997, 80 µs for the year 1999, and few microseconds for 2000. In contrast, sampling

intervals for the years 2002 and 2003 considered here were 1 µs and 0.5 µs, respectively.

The original objective of the analysis whose results are presented in Figure 4-5 through

94

Figure 4-22 was to examine the dependence of the statistics of the parameters of ICC

pulses in rocket-triggered lightning on the sampling interval. This is particularly

important for the rise time that can be smaller than the sampling intervals used in 1996

and 1997 (40 µs) and 1999 (80 µs).

GM=232 A Max=1076 A Min=6.3 A Sample size=66

0 32 64 128 256 512 1024 2048 Pulse Peak, A

18 16

14 12

10 N

umbe

r

8 4 2 0

6

Figure 4-5: Histograms of the peak of the ICC pulses for 2002.

95

GM=76.8 A Max=490.6 A Min=17.7 A Sample size=50

0 32 64 128 256 512 1024 2048

Num

ber

18 16 10 8

14 12

4

6

2 0 Pulse Peak, A

Figure 4-6: Histograms of the peak of ICC pulses for 2003.

5

25 20 15 10

GM=144.1 A Max=2082 A Min=6.3 A Sample size=116

Num

ber

2002 2003

0 32 64 128 256 512 1024 2048 0

Pulse Peak, A

Figure 4-7: Histogram of the peak of ICC pulses for 2002 and 2003.

96

GM=3.7 ms Max=16.3 ms Min=0.8 ms Sample size=66

0.5 1 2 4 8 16 32

25 20 15 10 5 0

Duration, ms

Num

ber

Figure 4-8: Histogram of the duration of ICC pulses for 2002.

30 GM=4.6 ms Max=15.4 ms Min=0.76 ms Sample size=50

Num

ber

25 20 15 10 5 0

F 0.5 1 2 4 8 16 32 Duration, ms

Figure 4-9: Histogram of the duration of ICC pulses for 2003.

97

2002 2003


0.5 1 2 4 8 16 32

50 45 40 35 30 25 20 15 10 5 0

Duration, ms

Num

ber

Figure 4-10: Histogram of the duration of ICC pulses for 2002 and 2003.

Risetime, µs


4

Num

ber

20 18 16 14 12 10 8 6 4 2 0

8 16 32 64 128 256 512 1024 2048 4096

Figure 4-11: Histogram of the risetime of ICC pulses for 2002.

98

Num

ber

20 18 16 14 12 10 8 6 4 2 0


Risetime,µs 8 16 32 64 128 256 512 1024 2048 4096 4

Figure 4-12: Histogram of the risetime of ICC pulses for 2003.

4

5

10

0

40 35 30 25 20

15


2002 2003

Num

ber

Risetime, µs 8 16 32 64 128 256 512 1024 2048 4096

Figure 4-13: Histogram of the risetime of ICC pulses for 2002 and 2003.

99

GM=0.83 ms Max=4 ms Min=0.18 ms Sample size=66

16 32 64 128 256 512 1024 2048 4096 8192

25 20 15 10 5 0

Num

ber

Half-peak width, µs

Figure 4-14: Histogram of the half-peak width of ICC pulses for 2002.



16 32 64 128 256 512 1024 2048 4096 8192

25 20 15 10 5 0

Num

ber

Figure 4-15: Histogram of the half-peak width of ICC pulses for 2003.

100



2002 2003

16 32 64 128 256 512 1024 2048 4096 8192Half-peak width, µs

Num

ber

40 35 30 25 20

15 10 5 0

Figure 4-16: Histogram of the half-peak width of ICC pulses for years 2002 and 2003.

Charge, mC

Num

ber

20 18 16 14 12 10 8 6 4 2 0

0 50 100 150 200 250 300 350 400 450 500 550

GM=100 mC Max=550 mC Min=4 mC Sample size=66

Figure 4-17: Histogram of the charge of ICC pulses for 2002.

101


Num

ber

0 50 100 150 200 250 300 350 400 450 500 550 Charge, mC

Figure 4-18: Histogram of the charge of ICC pulses for 2003.


2002 2003

Num

ber

Figure 4-19: Histogram of the charge of ICC pulses for years 2002 and 2003.

0 50 100 150 200 250 300 350 400 450 500 550 Charge, mC

102

GM=15.2 A2s Max=378.8 A2s Min= 0.08 A2s Sample size=66

Num

ber

0 50 100 150 200 250 300 350 400

Action Integral, A2sFigure 4-20: Histogram of the action integral of ICC pulses for years 2002.


Num

ber

0 50 100 150 200 250 300 350 400

Action Integral, A2s Figure 4-21: Histogram of the action integral of ICC pulses for years 2003.

103


2002 2003

Num

ber

Action Integral, A2s 0 50 100 150 200 250 300 350 400

Figure 4-22: Histogram of the action integral of ICC pulses for years 2002 and 2003.

Histograms of the peak, duration, rise time and, half-peak width of ICC pulses in

lightning triggered from tall objects and in rocket-triggered lightning reported by Miki et

al., (2004) are shown in Figures 4-23 to 4-26. They did not present any charge and action

integral statistics for ICC pulses. As stated earlier, the statistics for the rocket-triggered

lightning were obtained from experiments conducted at the ICLRT in 1996, 1997, 1999,

and 2000.

104

Figure 4-23: Histograms of the peak of ICC pulses. The geometric mean (GM), maximum (MAX), and minimum (min) values are indicated on each histogram. The rocket-triggered lightning data presented in this figure were obtained at the ICLRT at Camp Blanding, Florida, in 1996, 1997, 1999, and 2000.

105

Figure 4-24: Histograms of the duration of ICC pulses. The geometric mean (GM), maximum (MAX), and minimum (min) values are indicated on each histogram. The rocket-triggered lightning data presented in this figure were obtained at the ICLRT at Camp Blanding, Florida, in 1996, 1997, 1999, and 2000.

106

Figure 4-25: Histograms of the risetime of ICC pulses. The geometric mean (GM), maximum (MAX), and minimum (min) values are indicated on each histogram. The rocket-triggered lightning data presented in this figure were obtained at the ICLRT at Camp Blanding, Florida, in 1996, 1997, 1999, and 2000.

107

Table 4-1 summarizes the ICC pulse parameters (geometric mean values) for 2002 and

2003 rocket-triggered lightning experiments and compares those with their counterparts

for 1996, 1997, 1999, and 2000, analyzed by Miki et al. (2004).

Figure 4-26: Histograms of the half-peak width of ICC pulses. The geometric mean (GM), maximum (MAX), and minimum (min) values are indicated on each histogram.

108

Table 4-1: Summary of parameters (geometric means) of ICC pulses. Type of

Lightning Experimental

Site Source Year

Sample

Size Peak,

A Duration,

ms Risetime, µs

Half-peak

width, us

Charge, mC

Action Integral,

A2s

1996,1997,1999

247-296 113N=296

2.59 N=254

464 N=267

943 N=247

- - Miki et al. (2004)

2000 110 76.6 3.18 517 1079 - -

2002 66 232 3.7 360 800 100 15.2

2003 50 76.8

4.6

460 1400 90 4.5

2002+2003 116 144 4.1 400 1000 96 8.9

Rocket- Triggered Lightning

ICLRT

This Study

1996-2003 473-522 111.4 3.0 461 987 - -

Gaisberg Tower

2000 344 377 1.2 10 276 - -

Peissenberg Tower

1996-1999 124 512 0.83 61 153 - -

Object- Initiated Lightning

Fukui Chimney

Miki et al. (2004)

1996-1999 231 781 0.51 44 141 - -

109

As seen in Table 4-1, the geometric means of the parameters of the ICC pulses in

rocket-triggered lightning from 2002 and 2003 are consistent with those for 1996, 1997,

1999, and 2000, suggesting that they are not influenced by different sampling intervals.

ICC pulses in object-initiated lightning exhibit larger peaks, shorter risetimes, and shorter

half-peak widths than do the initial-stage pulses in rocket-triggered lightning.

Two possible reasons are proposed in Miki et al. (2004). First, multiple upward

branches could have facilitated the simultaneous occurrence of a continuous current in

one branch and a downward leader in another branch in object-initiated flashes, as

observed for Monte San Salvatore and Ostankino tower flashes (Berger, 1967; Gorin et

al., 1975). Second, the charge sources for initial-stage current in thunderclouds over the

tall objects in Austria, Germany, and Japan might be located closer to the lightning

attachment point than the sources of initial-stage current pulses in Florida. Hence,

because of the shorter propagation path between the in-cloud source and the lightning

attachment point, the fronts of the downward-propagating current waves in object-

initiated flashes might have suffered less degradation due to dispersion and attenuation

than their counterparts in Florida rocket-triggered flashes. Table 4-2 which compares the

parameters of ICC pulses in Gaisberg tower flashes in winter and summer supports the

latter hypothesis. It is known that the cloud charge sources in winter are lower than those

in summer.

Hence, it is expected that the ICC pulses in winter flashes should exhibit larger

peaks, shorter rise times, and shorter half-peak widths (HPW) than the ICC pulses in

summer flashes. As seen in Table 4-2, all parameters except for the risetime are similar.

More data are needed to arrive at a more decisive conclusion. Charge and action integral

110

for ICC pulses are considerably smaller than their counterparts for return strokes (e.g.,

Rakov, 1999).

Table 4-2: Parameters of ICC pulses in Gaisberg tower flashes as a function of season. Adapted from Miki et al. (2004)

Season Sample size Peak, A Duration, ms Risetime, µs HPW, µs

Winter 36 319 1.03 74.9 298

Summer 38 368 1.25 134 269

4.2 M-Components

4.2.1 Introduction

M-components are impulsive processes that occur during the continuing current

following the return strokes. In this chapter, statistics are compiled for the following

parameters of the M-component pulse: magnitude, rise time, duration, half-peak width,

charge, and action integral. The purpose of this is to compare these statistics to their

counterparts for the ICC pulses occurring during the initial stage. The same triggered-

lightning current records as in Section 4.1 are used here. More information on the return

stroke and M-component pulses can be found in Fisher et al. (1993) and Rakov and

Uman (2003).

4.2.2 Statistical Characteristics of M-Components

An example of a triggered-lightning flash, which contains the initial stage, return

strokes, and M-components, is shown in Figure 4-27. Definitions of the characteristics of

M-components analyzed here are illustrated in Figure 4-28. A total of 72 M-components

in 14 flashes triggered at the ICLRT at Camp Blanding, Florida, in 2002 and 2003 are

used for analysis here.

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Initial stage

Cur

rent

, A

Cur

rent

, A

M

RS1

Time, s RS1 RS2 RS3

Time, s Figure 4-27: Flash F0213. The initial stage is followed by 3 return strokes (RS1, RS2 and

RS3). In the inset, 3 M-components (labeled M) following return stroke RS1 are shown.

The histograms of duration, half peak width, peak and risetime of the M-

components are shown in Figures 4-29 to 4-34. Table 4-3 compares the characteristics of

M-components obtained at ICLRT, NASA Kennedy Space Center, Florida, and in

Alabama with those of ICC pulses obtained at ICLRT during 1996 to 2003.

112

Duration

Half-peak width

Puls

e Pe

ak

Cur

rent

, A

Charge = ∫i(t) dt Action Integral = ∫ i(t)2dt

Risetime (10-90%)

Time, ms

Figure 4-28: Flash F0213. Definitions of parameters (peak, duration, rise time, half-peak width) of M-components. Peak= 70.3 A, duration=15.2 ms, rise time=0.9 ms, half-peak width=1.8 ms.

0 2 4 6 8 10 12 14 16 18 Duration, ms

Num

ber

30 25 20 15 10 5 0

2002 GM = 2.2ms N =32 2003 GM = 2.4 ms N = 40

GM = 2.3 ms Min = 0.06 ms Max = 17.2 ms N = 72

Figure 4-29: Histogram of duration of M-component pulse.

113

0 16 32 64 128 256 512 1024 2048 4096

Num

ber

14 12 10 8 6 4 2 0

2002 GM = 198 A N = 32

2003 GM = 125 A N = 40 GM = 154 A Min = 10.8 A Max = 3578 A N = 72

Peak, kA Figure 4-30: Histogram of peak of M-component pulse.

Num

ber

10 8

16 14 12

6 4 2 0

0 64 128 256 512 1024 2048 4096 Risetime, µs

2002 GM = 220.3 µs N =32 2003 GM = 252 µs N = 40

GM = 237 µs Min = 23 µs Max = 2640 µs N = 72

Figure 4-31: Histogram of risetime of M-component pulse.

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0 128 512 1024 2048 4096 Half-peak width, µs

Num

ber

35 30 25 20 15 10 5 0

2002 GM = 426 µs N =32 2003 GM = 452 µs N = 40 GM = 440 µs

Min = 44 µs Max = 3600 µs N = 72

Figure 4-32: Histogram of the half-peak width of M-component pulse.

0 100 200 300 400 500 600 700 800 900 1000 Charge, mC

35 30 25 20 15 10 5 0

Num

ber

30 2002 GM =70 mC N = 32

2003 GM = 117 mC N = 40

GM =91.5 mC Min = 1 mC Max = 909 mC N =72

Figure 4-33: Histogram of the charge of M-component pulse.

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0 200 400 600 800 1000 1200 1400 1600 1800 2000 Action Integral, A2s

Num

ber

70 60 50 40 30 20 10 0

2002 GM = 11.0 A2s N = 32

2003 GM =13.0 A2s N = 40

GM = 12.1 A2s Min = 0.03 A2s Max = 1940 A2s N = 72

Figure 4-34: Action integral of M-component pulse.

The geometric means of M-component current peak and duration obtained here are

similar to those previously reported by Thottappillil et al. (1995). The geometric means

of risetime and half-peak width are smaller by a factor of about two compared to those

obtained by Thottappillil et al. (1995). The probable reason for this discrepancy might be

the fact that in Thottappillil et al. (1995), some overlapping M components which do not

allow unambiguous measurement of such parameters as risetime, duration, and half-peak

width were not used while compiling the statistics. These overlapping M components

usually occur during the first 5 ms following the beginning of the return stroke and have a

faster rise time. Added to this, sample sizes are very small compared to those used in

116

Thottappillil et al. (1995). Table 4-3 gives the statistics of the characteristics of M

components along with those of ICC pulses.

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Table 4-3: Geometric means of the various parameters of M-components and ICC pulses. M- components

Experimental Site Source Year

Sample

Size Peak,

A Duration,

ms Risetime,µs

Half-Peak

width, us

Charge, mC

Action Integral,

A2s

Lightning triggering

sites in Florida and Alabama

Thottappillilet al. (1995) 1990,1991 113-

124 117

N=24 2.10

N=114 422

N=124 800

N=113129

N=104 -

2002 32 198 2.2 220 426 70 112003 40 125 2.4 252 452 117 13ICLRT This

Study 2002-2003 72 154 2.3 237 440 92 12

ICC pulses

1996, 97,99

247-296

113 N=296

2.6 N=254

464 N=267

943 N=247 - -

2000 110 76.6 3.2 517 1079 - -

Miki et al. (2004)

2002 66 232 3.7 360 800 100 15.22003 50 76.8 4.6 460 1400 90 4.52002-2003 116 144 4.1 400 1000 96 9.0

ICLRT

This study (Table 4.1) 1996-

2003 473-522 111 3.0 461 987 - -

118

As seen Table 4-3, the duration, risetime, and half-peak width of the ICC pulses are

approximately twice those of M-components, suggesting a frequent occurrence of slower

pulses in the initial stage compared to the pulses superimposed on the continuing currents

following return strokes. The charge and action integral of M-components and ICC

pulses are similar.

CHAPTER 5 RECOMMENDATIONS FOR FUTURE RESEARCH

1. The frequency range of the Pockels sensor measuring system used in 2000 was relatively narrow, from 50 Hz to 1 MHz. This led to underestimation of the peaks of the fast electric field waveforms by about 40%. Pockels sensors with higher upper frequency response should be used to overcome this limitation.

2. The sampling interval of the vertical electric field was 0.5 µs compared to 20 ns of the high current records. As seen in Section 3.4.4, peak power estimates are sensitive to the uncertainties related to the relatively large electric field sampling interval. Hence, smaller sampling intervals (higher sampling rates) should be used in future experiments to get more accurate estimates of the peak power.

3. The high-current record length was 50 µs. Longer current records are needed to obtain the power and energy curves for later times.

4. The estimation of power and energy in this study assumes that the lightning channel is vertical, but in reality the lightning channel is tortuous and drifts because of wind. Hence, optical sensors should be placed around the strike rod to identify those flashes for which the channel is relatively straight and vertical, so that more accurate peak power and energy estimates could be obtained.

5. Along with the vertical electric fields, horizontal electric fields were also measured for eight strokes. Unfortunately, the unavailability of high-current records for those flashes prevented the estimation of the Poynting vector associated with the upward-moving wave. It would be interesting to obtain such estimates in the future.

6. Electric and magnetic field measurements at 15 and 30 m should be used to compute the electromagnetic power and energy radiated from the channel for comparison with the input power and energy.

7. Sample sizes in this study were rather small. Additional measurements are needed to obtain a larger sample that would allow one to draw more statistically significant conclusions.

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BIOGRAPHICAL SKETCH

Vinod Jayakumar was born in Vellore, India, in 1980. He graduated with a

bachelor’s degree in electronics and communications from P.S.G college of technology at

Coimbatore, Tamil Nadu, India in 2002. In 2002, he went to the USA to pursue graduate

studies at the University of Florida.

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estimating power, energy, and action integral in …estimating power, energy, and action integral in...

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