Current Bearing Lightning Return Strokes with Low Wave Speeds

Preston B. Galla Mentor: Dr. Mostafa Hemmati

Department of Physical SciencesArkansas Tech University Russellville, AR, 72801Phone: (479)-747-8710E-mail: pgalla@atu.edu

ObjectivesIntroduction of Breakdown Current Bearing

WavesIntroduction of the completed set of Electron

Fluid Dynamical EquationsSuccessful integration of the Electron Fluid

Dynamical (EFD) Equations through the transition region.

Presentation of the low speed wave profile of electron number density, electron gas temperature, ionization rate, electric field, and electron velocity within the “sheath” region.

BackgroundFrancis Hauksbee – In 1705 was the first to pay

detailed attention to luminous pulses in evacuated chambers. Experimental evidence consistently showed propagation speeds approaching the speed of light. The speeds were shown to be supersonic even in comparison with electron acoustic speeds and thus, were called shock waves. The phenomenon involving no mass motion was concluded to be from electron fluid action alone. Therefore they were referred to as EFD equations.

Charles Wheatstone – In 1835 first speculated that the pulses observed from the discharge tube subjected to high potential differences were actually ionizing potential waves traveling down the tube.

Background J.J. Thompson – In 1893 conducted numerous

experiments of the luminous pulses in an improved apparatus of a 15 m long evacuated discharge tube. He observed the speed of the pulse to be in the order of 108 m/s

J.W. Beams – In 1930 confirmed Thompson’s observations of the speed of the pulse and had a qualitative explanation for it. He stated that the gas behind the pulse was electrically conducting so the pulse carried the potential of the discharge electrode where the potential was applied. He theorized that electrons were the main source in the wave propagation and that high ionization occurred at the electrode to which the potential was applied.

Electron Fluid Dynamical Equations

R.G. Fowler and Paxton (1962) - under the assumption that the electron gas pressure is much greater than that of the other species, successfully derived approximate equations of conservation of mass, energy, and momentum.

Fowler and Shelton (1968) –Following the previous work of Paxton (1962) furthered the EFD equations. Considering no time variation, Maxwell’s equation reduced to Poisson’s equation.

BackgroundR.G. Fowler and Shelton – Formulated

energy and momentum loss terms from the collision between heavy particles and electrons. They also were able to derive boundary conditions at the shock front.

Fowler et al.– Completed Shelton’s earlier set of equations representing a one-dimensional, three-component (electrons, ions, and neutral particles) fluid model to describe a steady-state, ionizing electron fluid-dynamical wave propagating in the same direction as the electric field force on electrons (lightning step leader).

Electron ShockwavesPro-force waves – Shockwaves

propagating in the same direction as the direction of electric field force on electrons.

Anti-force waves – Shockwaves propagating in the opposite direction of the electric field force on electrons.

Electron Fluid Dynamical Equations

Fowler et al. (1984) – Completed the EFD equations and added two noteworthy correction terms to them. They allowed for a strong discontinuity in electron gas temperature at the shock front and added a heat conduction term in the conservation of energy equation.

Wave ComponentsShock Front – The beginning portion of the

wave followed by the transition or sheath region.

Sheath Region -Thin dynamical region in which the electric field starts from its maximum value at the shock front then reduces to a negligible value at the trailing edge of the wave and the electrons slow down to speeds comparable to the heavier ions.

Equations of Conservation of Mass, Momentum, Energy and Poisson’s

Equations for Pro-force Waves

ndxnvd

)(

VvKmnenEnkTeVvmnvdxd

dxdTe

mKTenkenvVvnkTeVvmnv

dxd 2

2 5225

23 VvKmnMmnkKTe

Mm

1

0 Vvne

dxdE

=

(1)

(2)

(3)

(4)

Variable Description E = Electric field. x = position in the wave profile. β = Ionization frequency. K = Elastic collision frequency. V = Wave velocity. M = Neutral particle mass E0 = Electric field at the wave front. Φ = Ionization potential. e = Electron charge. Te = Electron Temperature. n = Electron Number Density v = Electron Velocity. m = Electron Mass.

Dimensionless Variables for Pro-force Waves

Dimensionless Equations for Pro-Force Waves

Dimensionless Variables for Antiforce Waves

Hemmati (1999) - was able to successfully change the fluid dynamical equations and derive a new set of dimensionless variables representing anti-force waves.

Dimensionless Variables for Anti-force Waves

η= 𝐸𝐸0

ν=( 2𝑒Φε0 𝐸𝑜

2 )𝑛ψ=

𝑣𝑉

θ=𝑇𝑒𝑘2𝑒Φ

ξ=−𝑒𝐸0 𝑥𝑚𝑉 2

α=2𝑒Φ𝑚𝑉 2

κ=−𝑚𝑉𝑒𝐸0

𝐾

μ= β𝐾

ω=2𝑚𝑀

Description of Dimensionless Variablesν = Electron concentration.ψ = Electron velocity.θ = Electron Temperature.κ = Wave Parameter.η = Electric field. μ = Ionization rate.ξ = Position inside the wave.ι = Current

Inclusion of Current Behind the Shock Front

Including the current behind the shock front in the set of EFD equations alters the equation of conservation of energy and Poisson’s equation.

Defining the dimensionless current as , the dimensionless EFD equations with current included become:

Dimensionless Equations for Antiforce Current Bearing Waves

𝑑𝑑 ξ [νψ ]=κμν𝑑𝑑 ξ [νψ (ψ −1 )+ανθ ]=νη−κν (ψ −1)

𝑑𝜂𝑑𝜉 =κι− 𝜈𝛼 (1−𝜓)

(5)

(6)

(7)

(8)

Numerical Solution Method

Select values for α, ι, κ, ψ1 , ν1.Integrate the EFD equations, changing the

values of κ, ψ, and ν until the conclusion agrees with expected conditions at the end of the dynamic transition region of the wave.

Figure 1. Electric field, η, as a function of electron velocity, Ψ, within the sheath region of current bearing antiforce waves for a wave speed value of α=0.25 and for current values 0, 0.5, 1 and 2.

Initial conditions: Ƞ1 = 1; ψ1 < 1Final conditions required: η2 = 0; ψ2 → 1α =0.25 corresponds to a wave speed of 5.93*106

m/sι = 1 corresponds to approximately 10 kA currentRakov (2000) observed 10 kA currentsWang et all (1999) observed a range of 12kA – 21

kANakahori et al observed a return stroke current

pulse with a peak of 31 kA.

Experimental References

Figure 2. Electric field, η, as a function of position, ξ, within the sheath region of current bearing antiforce waves for a wave speed value of α=0.25 and for current values 0, 0.5, 1 and 2.

Figure 3. Electron velocity, Ψ, as a function of position, ξ, within the sheath region of current bearing antiforce waves for a wave speed value of α=0.25 and for current values 0, 0.5, 1 and 2.

ξ = 1 corresponds to a sheath thickness of 2mm.

Sanmann and Fowler reported a total sheath thickness of 5 cm.

Fujita et al. report a wave thickness of approximately 5 cm.

Experimental References

Figure 4. Electron number density, ν, as a function of position, ξ, within the sheath region of current bearing antiforce waves for a wave speed value of α=0.25 and for current values 0, 0.5, 1 and 2.

Our average non-dimensional electron number density of 0.7 represents an electron number density of 7.7*1015 / m^3.

Hagelaar and Kroesen report an average electron number density of 7*1015 / m^3.

David Graves reports electron number density values between 5*1015 /m^3 and 2*1016 / m^3.

Experimental References

Figure 6. Electron temperature, θ, as a function of position, ξ, within the sheath region of current bearing antiforce waves for a wave speed value of α=0.25 and for current values 0, 0.5, 1 and 2.

Sanmann and Fowler reported that the electron temperature increases rapidly away from the wave front until it reaches a peak value of about 3.17*107 K at a distance of 5.4cm behind the wave front.

Experimental References

Figure 5. Ionization rate, µ, as a function of position, ξ, within the sheath region of current bearing antiforce waves for a wave speed value of α=0.25 and for current values 0, 0.5, 1 and 2.

ConclusionsFor current values for which solutions

were possible, the electric field and electron velocity values met the expected conditions at the end of the sheath region.

Our results are in agreement with previous experimental and theoretical work done on anti-force waves.

AcknowledgementsArkansas Space Grant

Consortium

Dr. Hemmati

Fujita K, Sato S, and Abe T.: Journal of Thermodynamics and Heat Transfer 17, (2003)

Graves DB.: J. Appl Phys. 62, 1 (1987) Hemmati, M. et al. “Antiforce Current Bearing Waves” Hagelaar GJM and Kroesen GMW: Journal of

Computational Physics 159, (2000) Sanmann E and RG Fowler: The Physics of Fluids 18

11 (1975) K Nakahori, T Egawa, H Mitani: “Characteristics of

winter lightning currents in Hokuriku district” IEEE Trans. Power Apparatus System, vol. 101, 4407-4412, 1982.

Rakov VA. 2000. Positive and bipolar lightning discharges: a review. In: Proceedings of the 25nd International Conference on Lightning Protection. Pp 103-108.

Wang D, VA Rakov, MA Uman, N Takagi, T Watanabe, DE Crawford, KJ Rambo, GH Schnetzer, RJ Fisher and ZI Kawasaki. 1999. J. Geophys. Res. 104(D2).

References