Download atu_p_galla - Arkansas Space Grant Consortium

Current Bearing Lightning Return
Strokes with Low Wave Speeds
Preston B. Galla
Mentor: Dr. Mostafa Hemmati
Department of Physical Sciences
Arkansas Tech University
Russellville, AR, 72801
Phone: (479)-747-8710
E-mail: [email protected]
Objectives




Introduction of Breakdown Current Bearing
Waves
Introduction of the completed set of
Electron Fluid Dynamical Equations
Successful 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.
Background
Francis 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.
Background
R.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 Shockwaves
Pro-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 Components

Shock 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
d ( nv )
 n
dx
(1)
d
mnvv  V   nkTe  enE  Kmnv  V 
dx
(2)
d 
5nk 2Te dTe 
2
mnvv  V   nkTe5v  2V   2env 

dx 
mK dx =

m
m
2
 3 nkKTe    Kmnv  V 
M 
M 
dE e  v 
 n  1
dx  0  V 
(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 Proforce Waves
Dimensionless Equations for ProForce 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 antiforce waves.
Dimensionless Variables for Anti-force Waves
𝐸
η=
𝐸0
2𝑒Φ
ν=
2 𝑛
ε0 𝐸𝑜
𝑣
ψ=
𝑉
𝑇𝑒 𝑘
θ=
2𝑒Φ
𝑒𝐸0 𝑥
ξ=−
𝑚𝑉 2
2𝑒Φ
α=
𝑚𝑉 2
β
μ=
𝐾
𝑚𝑉
κ=−
𝐾
𝑒𝐸0
2𝑚
ω=
𝑀
Description of Dimensionless
Variables








ν
ψ
θ
κ
η
μ
ξ
ι
= Electron concentration.
= Electron velocity.
= Electron Temperature.
= Wave Parameter.
= Electric field.
= Ionization rate.
= Position inside the wave.
= 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 ι =
𝐼1
, the dimensionless EFD equations

ε0 𝐾𝐸0
with current included become:
Dimensionless Equations for
Antiforce Current Bearing Waves
𝑑
νψ = κμν
𝑑ξ
𝑑
νψ ψ − 1 + ανθ = νη − κν(ψ − 1)
𝑑ξ
𝑑
5α2 νθ 𝑑θ
2
νψ ψ − 1 + ανθ 5ψ − 2 + ανψ −
+ αη2
𝑑ξ
κ 𝑑ξ
= 2ηκια − ωκν 3αθ + 𝜓 − 1 2
𝑑𝜂
𝜈
= κι − (1 − 𝜓)
𝑑𝜉
𝛼
(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.
Experimental References







Initial conditions: Ƞ1 = 1; ψ1 < 1
Final conditions required: η2 = 0; ψ2 → 1
α =0.25 corresponds to a wave speed of
5.93*106 m/s
ι = 1 corresponds to approximately 10 kA
current
Rakov (2000) observed 10 kA currents
Wang et all (1999) observed a range of 12kA –
21 kA
Nakahori et al observed a return stroke current
pulse with a peak of 31 kA.
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.
Experimental References

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.
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.
Experimental References

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.
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.
Experimental References

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.
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.
Conclusions

For 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.
Acknowledgements

Arkansas Space Grant Consortium

Dr. Hemmati
References








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).