Speed-Current Speed-Current Relation in Lightning Relation in Lightning

Return StrokesReturn StrokesRyan Evans, Student - Mostafa Hemmati, Advisor

Department of Physical SciencesArkansas Tech University

Russellville, AR 72801

Objectives• Introduction of characteristics of breakdown waves,

specifically lightning return strokes

• Introduction of the EFD equations for proforce and antiforce waves, as well as dimensionless sets of equations.

• Inclusion of a current behind the shock-front for antiforce waves

• Presentation of the wave profiles for solutions and speed and current relation in lightning return strokes

Lightning Return Strokes• The lightning return strokes discussed are antiforce waves – waves

propagating in the opposite direction of electric field force on electrons

• Electron gas pressure is much larger than the partial pressures of neutral particles and ions; therefore it is considered the driving force in propagating the wave

• The wave front has shown to have a strong discontinuity (shock front), followed by a thin sheath region where electron velocity and electric field reach maximum values. Electric field quickly reduces to zero and electron velocity to that of heavy particles.

• The sheath region is followed by an extended quasi-neutral region in which the electron gas cools due to further ionization.

VariablesElectron number densityIon number densityElectric fieldElectron velocityWave velocityElectron massNeutral particle mass

Electron temperaturePositionIonization potential of gasElectron chargeElastic collision frequencyIonization frequency

n-Ni-E-v-V-m-M-

Te-x- Φ-e-K-β-

Dimensionless VariablesElectric field strengthElectron number densityElectron velocity Electron gas temperature

Position within sheathIonization rateWave parameterWave parameterWave parameter

η-ν-ψ-θ-

ξ-μ-α-κ-ω-

EFD Equations for Proforce Waves

(conservation of mass)

(conservation of momentum)

(conservation of energy)

(Poisson’s Equation)

Dimensionless Variables for Proforce Waves

Dimensionless EFD equations for Proforce Waves(conservation of mass)

(conservation of momentum)

(conservation of energy)

(Poisson’s Equation)

Dimensionless Variables for Antiforce Waves

Dimensionless EFD Equations for Antiforce Waves

(conservation of mass)

(conservation of momentum)

(conservation of energy)

(Poisson’s Equation)

Current Bearing Waves

EFD Equations for Current Bearing Antiforce Waves(conservation of mass)

(conservation of momentum)

(conservation of energy)

(Poisson’s Equation)

Integration of the Electron Fluid Dynamical Equations

• To achieve solutions for the set of EFD equations, the equation set is integrated through the sheath region, by trial and error.

• Wave speeds, α, and current, ι, were chosen for integration. Then electron velocity, ψ1, electron number density, ν1, and wave constant κ were changed in integrating the EFD equations through the sheath region until solutions agreed with expected conditions at the end of the sheath.

• For 4 different wave speeds, α, solutions were found for the largest current values, ι, that maintained agreement with expected conditions (η→0 as ψ→1).

Initial Boundary Conditions• Successful solutions required the following boundary values:

• Waves speeds range from 3 x 106 m/s (α=1.0) to 9 x 107 m/s (α=0.001)

α = 0.001 , ι = 7.0 , κ = 0.144 , ψ1 = 0.4721 , ν1 = 0.2161

α = 0.01 , ι = 5.0 , κ = 1.3 , ψ1 = 0.7 , ν1 = 0.7696

α = 0.1 , ι = 1.5 , κ = 0.435 , ψ1 = 0.808 , ν1 = 0.7350

α = 1.0 , ι = 0.25 , κ = 0.18 , ψ1 = 0.75 , ν1 = 0.6726

0

0.5

1

1.5

2

2.5

3

3.5

4

0.5 1 1.5 2

η

ψ

ι = 7 ; α = 0.001

ι = 5 ; α = 0.01

ι = 1.5 ; α = 0.1

ι = 0.25 ; α = 1.0

Figure 1: Electric field, η, as a function of electron velocity, Ψ, within the sheath region of lightning return strokes for current values 0.25, 1.5, 5 and 7.

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5

η

ξ

ι = 7 ; α = 0.001

ι = 5 ; α = 0.01

ι = 1.5 ; α = 0.1

ι = 0.25 ; α = 1.0

Figure 2: Electric field, η, as a function of position, ξ, within the sheath region of lightning return strokes for current values 0.25, 1.5, 5 and 7.

Electric Field• Figure 1 shows that for these wave speeds and current values, the

solutions meet the expected conditions at the end of the sheath region (η→0 as ψ→1).

• Figure 2 shows that the electric field peaks early on in propagation and quickly drops to values below initial value and reduces to 0.

• Higher currents prove to peak earlier and even drop more rapidly than that of lower currents. This agrees with results predicted by Rakov and Dulzon as well as several models tested and measured by Uman.

• The electric field reducing to 0 in Figures 1 and 2 signifies the end of the sheath region and the start of the quasi neutral region, where electrons come to rest relative to heavy particles, fitting the definition of the two-region electron fluid-dynamical wave established by Shelton and Fowler [8].

0.7

0.9

1.1

1.3

1.5

1.7

1.9

0 1 2 3 4 5

ψ

ξ

ι = 7 ; α = 0.001

ι = 5 ; α = 0.01

ι = 1.5 ; α = 0.1

ι = 0.25 ; α = 1.0

Figure 3: Electron velocity, ψ, as a function of position, ξ, within the sheath region of lightning return strokes for current values 0.25, 1.5, 5 and 7.

Wave Velocity

• This data set shows wave velocities reaching 9 x 107 m/s which agrees to ranges found by Idone [4] and Wang [10].

• Data gathered by Idone et al. [4] yielded speeds between 1.6 x 108 m/s and 9.0 × 107 m/s.

• Wang et al. [10] triggered return strokes with speeds ranging from 4 x 107 to 1.5 x 108 m/s.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5

ν

ξ

ι = 7 ; α = 0.001

ι = 5 ; α = 0.01

ι = 1.5 ; α = 0.1

ι = 0.25 ; α = 1.0

Figure 4: Electron number density, ν, as a function of position, ξ, within the sheath region of lightning return strokes for current values 0.25, 1.5, 5 and 7.

Electron Number Density• Our electron number densities range from 4 x 1015/m3 to 2 x

1016/m3, which match the values recorded by Hagelaar and Kroeson [2] as well as Graves [1].

• David Graves [1] reported electron number density values ranging from 5 x 1015/m3 and 2 x 1016/m3, showing very high agreement with our calculated values.

• Hagelaar and Kroeson [2] report an average electron number density of 7 x 1015/m3 within the sheath region of the wave, also falling within our range.

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5

μ

ξ

ι = 0.25 ; α = 1.0

Figure 5: Ionization rate, μ, as a function of position, ξ, within the sheath region of lightning return strokes for current values 0.25 and wave speed 1.0.

• The ionization rates represented in Figure 5 remain considerably constant within the sheath region as predicted by Shelton and Fowler [8].

Figure 6: Electron temperature, θ, as a function of position, ξ, within the sheath region of lightning return strokes for current values 0.25, 1.5, 5 and 7.

0.1

1

10

100

1000

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00

θ

ξ

ι = 7 ; α = 0.001ι = 5 ; α = 0.01ι = 1.5 ; α = 0.1ι = 0.25 ; α = 1.0

• Figure 6 shows a noticeable relation for electron temperature as a function of position. For all currents, electron temperature increases as position within the sheath increases, as reported by Sanmann and Fowler [7].

• Figure 6 also shows that higher wave speeds allow for larger currents and higher temperatures.

• The EFD equations were able to be integrated through the sheath region of the wave for a range of current and speed values behind the wave front.

• For all sets of waves, our solutions showed a moderate to high level of agreement with expected conditions at the end of the sheath region.

• Several theoretical and experimental results show agreement with our results.

• These results once again confirm the validity of the EFD model of current bearing antiforce waves to describe lightning return strokes.

Conclusions

Acknowledgements

• Dr. Hemmati

• Arkansas Space Grant Consortium

References1. Graves D.B.: J. Appl. Phys. 62 (1987).2. Hagelaar G.J.M. and Kroesen G.M.W.: Journal of Computational Physics, 159, (2000).3. Hemmati, M., et al. "Antiforce current bearing waves." 28th International Symposium

on Shock Waves. Springer Berlin Heidelberg, (2012).4. Idone, V.P., et al., “The propagation speed of a positive lightning return stroke”

Geophys. Res. Lett., col. 14, 1150-1153, (1987).5. Norman et. al. "Ionization Rate, Temperature, and Number Density for Breakdown

Waves with a Large Current Behind the Shock Front." Journal of the Arkansas Academy of Science 62 (2008).

6. Rakov, V. A., and A. A. Dulzon. "A modified transmission line model for lightning return stroke field calculations." Proc. 9th Int. Symp. Electromagn. Compat. (1991).

7. Sanmann E and RG Fowler: The Physics of Fluids. 18, 11. (1975).8. Shelton GA and RG Fowler. “Nature of electron fluid dynamical waves.” The Physics of

Fluids 11(4):740-746. (1968).9. Uman et. al. "Comparison of lightning return stroke models." ‐ Journal of Geophysical

Research: Atmospheres. 98.D12 (1993).10. Wang, D., et al. "Observed leader and return stroke propagation characteristics in the ‐

bottom 400 m of a rocket triggered lightning channel." ‐ Journal of Geophysical Research: Atmospheres, 104.D12 (1999).