In this lecture we will learn
how lightning return stroke current parameters such as Ipk and (dI/dt)pk might be estimated
from remote measurements of E and B fields. It
is much easier to make remote measurements of lightning
electric and magnetic fields than it is to make direct
measurements of lightning return stroke currents.
There is no first return stroke in a rocket triggered
cloud-to-ground discharge, just return strokes. We
can hope to learn something about peak I and dI/dt in
first return strokes when we use E and B field
measurements.
Radiated Fields
We must first learn more about the fields radiated by lightning.
Radiated Fields
We must first learn more about the fields radiated by lightning.
First here's the
geometry used in the E and B fields calculations that
follow. The lightning channel is
assumed to be straight and vertically oriented.
The ground is flat and a perfect conductor. Just
as we used the method of images problem earlier in the
semester to solve a problem in electrostatics, we can
match the boundary condition at the ground surface by
replacing it with an image of the lightning channel
below the ground.
In a problem of this kind you are basically trying to find E and B that satisfy Maxwell's equations.
The general approach is to find the scalar and
vector potentials φ and A
We aren't going to look at all the steps in the derivation, we'll simply write down the results. If you are interested in the details a good place to start would be the following article: "The Electromagnetic Radiation from a Finite Antenna," M.A. Uman, D.K. McLain, and E.P. Krider, Am. J. Phys., 43 33-38, 1975.
The expression below gives the electric field radiated by a short segment dz located at an altitude z above the ground. We'll limit our attention to the E field at the ground (assumed to be a perfect conductor) which has just a z component. There is interest in how lightning E and B fields couple onto electric power lines. To properly examine this problem you would need to compute E and B for points above the ground (at the level of the power lines). That becomes a more complicated problem because the E and B fields will each have 3 components (vertical, radial, and azimuthal components in cylindrical geometry).
In a problem of this kind you are basically trying to find E and B that satisfy Maxwell's equations.
We aren't going to look at all the steps in the derivation, we'll simply write down the results. If you are interested in the details a good place to start would be the following article: "The Electromagnetic Radiation from a Finite Antenna," M.A. Uman, D.K. McLain, and E.P. Krider, Am. J. Phys., 43 33-38, 1975.
The expression below gives the electric field radiated by a short segment dz located at an altitude z above the ground. We'll limit our attention to the E field at the ground (assumed to be a perfect conductor) which has just a z component. There is interest in how lightning E and B fields couple onto electric power lines. To properly examine this problem you would need to compute E and B for points above the ground (at the level of the power lines). That becomes a more complicated problem because the E and B fields will each have 3 components (vertical, radial, and azimuthal components in cylindrical geometry).
To obtain the
total E field we need to integrate from the ground
to the top of the channel
At the ground the magnetic field has just an
azimuthal component
and again we integrate over the length of the
channel to determine the total field.
I should
mention also that a recent publication points
out and corrects a small error in the
expressions above. This won't affect our
discussion, but if you would like to see the
correct expressions, click here.
Static, Induction, and Radiation field terms
We won't really be using these rather complex general expressions. Rather let's just note that the electric field expression contains terms involving a time integral of current, the current itself, and a time derivative term.
We refer to these are the electrostatic,
induction, and radiation field
components. The B field has just
induction and radiation field components.
Except for the very beginning of the discharge, the electrostatic field is generally dominant at close range. Because it decreases as 1/R, the radiation field is the only field component observed at long range (beyond a few 10s of kilometers). Also because peak dI/dt occurs very early in a return stroke discharge, the radiation field also dominates at the very beginning of a return stroke.
The sketch below provides a rough idea of the what the 3 field components would look like given the current waveform at the top of the figure.
The solid lines show typical first return
field values, the dotted lines are for
subsequent strokes. These aren't actual
waveforms. Rather they are essentially
an average of measurements of actual waveforms
(lots of waveforms).
The hump on these magnetic fields is probably coming from the induction field. We don't see it on E field waveforms because the electrostatic field component dominates. B fields don't have a magnetostatic field component.
Lightning Channel Current I(z,t)
The E and B fields at a particular distance from a lightning strike depend in a complex way on return stroke current, the current derivative, and, for E fields, on the time integral of the current. We really don't know what the return stroke current is as a function of altitude and time [ that is, I(z,t) ]. At best we can sometimes measure I(0,t) [i.e. the current at the ground in triggered lightning discharges, for example].
Let's remember that what we are hoping to do is to measure E and/or B and go back and figure out what the return current must have been. In order to do that we will need to make some assumptions or put some constraints on the return stroke current. We will need a lightning return stroke "current model."
We can roughly classify current models into 3 groups:
1. Sophisticated plasma-fluid-dynamics models that try to realistically determine the actual physical conditions in a lightning channel (temperature, electron density, pressure, and other electrical and fluid properties). This type of approach is beyond the scope of this class. (see M.N. Plooster, "Numerical Simulation of Spark Discharges in Air" and "Numerical Model of the Return Stroke of the Lightning Discharge" for examples of this type of approach)
2. Because the lightning return stroke current begins at (or near) the ground and travels upward along a conducting channel, many researchers treat the return stroke as a voltage or current pulse traveling along a transmission line with distributed resistance, inductance, and capacitance. (see for example P.F. Little, "Transmission Line Representation of a Lightning Return Stroke" and G.H. Price and E.T. Pierce, "The Modeling of Channel Current in the Lightning Return Stroke")
We will adopt more of an engineering or empirical approach
3. We will assume a functional form for I(z,t), calculate the E and B field that such a current would produce, and then compare the calculations with field measurements (often made at two distances from the return stroke). We will then adjust I(z,t) or other model parameters such as propagation speed to get the best agreement between calculated and measured fields.
Bruce Golde Model
In the Bruce Golde model the return
stroke current is assumed to be uniform
along the length of the channel. The
current amplitude can change with time but
it does so along the entire length of the
channel simultaneously. This is
illustrated further in the figure below
which shows the height of the
return stroke channel and the current
distribution along the channel at evenly
spaced time intervals t1
through t5.
The current waveform measured at the ground is shown at the bottom of the figure. The top part of the figure shows the upward development of the channel. The height and current distribution is shown at evenly spaced time intervals. The width of each vertical bar indicates the current amplitude. As the current amplitude is changing at the ground with time, current amplitude also changes simultaneously along the entire length of the channel. This is physically unreasonable as it would require that information propagate along the length of the channel at infinite speed.
Static, Induction, and Radiation field terms
We won't really be using these rather complex general expressions. Rather let's just note that the electric field expression contains terms involving a time integral of current, the current itself, and a time derivative term.
Except for the very beginning of the discharge, the electrostatic field is generally dominant at close range. Because it decreases as 1/R, the radiation field is the only field component observed at long range (beyond a few 10s of kilometers). Also because peak dI/dt occurs very early in a return stroke discharge, the radiation field also dominates at the very beginning of a return stroke.
The sketch below provides a rough idea of the what the 3 field components would look like given the current waveform at the top of the figure.
More
realistic shapes of the E and B waveforms
that you might expect to see at various
distances from a CG stroke are shown below
(source: Lin,
Y.T.,
M.A.
Uman,
J.A.
Tiller,
R.D.
Brantley,
W.H. Beasley, E.P. Krider, C.D. Weidman,
"Characterization of Lightning Return
Stroke Electric and Magnetic Fields from
Simultaneous Two-Station Measurements,"
J. Geophys. Res., 84-6307-6314, 1979).
.
.
The hump on these magnetic fields is probably coming from the induction field. We don't see it on E field waveforms because the electrostatic field component dominates. B fields don't have a magnetostatic field component.
Lightning Channel Current I(z,t)
The E and B fields at a particular distance from a lightning strike depend in a complex way on return stroke current, the current derivative, and, for E fields, on the time integral of the current. We really don't know what the return stroke current is as a function of altitude and time [ that is, I(z,t) ]. At best we can sometimes measure I(0,t) [i.e. the current at the ground in triggered lightning discharges, for example].
Let's remember that what we are hoping to do is to measure E and/or B and go back and figure out what the return current must have been. In order to do that we will need to make some assumptions or put some constraints on the return stroke current. We will need a lightning return stroke "current model."
We can roughly classify current models into 3 groups:
1. Sophisticated plasma-fluid-dynamics models that try to realistically determine the actual physical conditions in a lightning channel (temperature, electron density, pressure, and other electrical and fluid properties). This type of approach is beyond the scope of this class. (see M.N. Plooster, "Numerical Simulation of Spark Discharges in Air" and "Numerical Model of the Return Stroke of the Lightning Discharge" for examples of this type of approach)
2. Because the lightning return stroke current begins at (or near) the ground and travels upward along a conducting channel, many researchers treat the return stroke as a voltage or current pulse traveling along a transmission line with distributed resistance, inductance, and capacitance. (see for example P.F. Little, "Transmission Line Representation of a Lightning Return Stroke" and G.H. Price and E.T. Pierce, "The Modeling of Channel Current in the Lightning Return Stroke")
We will adopt more of an engineering or empirical approach
3. We will assume a functional form for I(z,t), calculate the E and B field that such a current would produce, and then compare the calculations with field measurements (often made at two distances from the return stroke). We will then adjust I(z,t) or other model parameters such as propagation speed to get the best agreement between calculated and measured fields.
We'll
mostly just discuss two well know current
models: the Bruce-Golde and the
Transmission Line models.
Bruce Golde Model
The current waveform measured at the ground is shown at the bottom of the figure. The top part of the figure shows the upward development of the channel. The height and current distribution is shown at evenly spaced time intervals. The width of each vertical bar indicates the current amplitude. As the current amplitude is changing at the ground with time, current amplitude also changes simultaneously along the entire length of the channel. This is physically unreasonable as it would require that information propagate along the length of the channel at infinite speed.
Here
is first an expression for the
radiation field component of the
electric field (Erad)
at a distance D from the lightning
stroke. There are a few tricks
involved in this derivation (that we
won't worry about) because the current
is discontinuous at the tip of the
upward propagating return stroke
current. The expression for E
can be inverted to give the channel
base current as a function of the
field (the last equation in the figure
above).
Transmission Line Model
In the transmission line model the current waveform measured at the ground is assumed to propagate up the channel without changing shape and at constant speed. Here's how that is expressed in equation form.
Basically the current value observed at height z at time t is the same as the current seen at the ground at a time (t - z/v) earlier (i.e. the time it took to travel from the ground up to height z). Again this might be clearer in picture form.
The current waveform measured at the ground propagates up the channel without changing shape or speed.
The radiation field produced by a segment dz of channel located at height z above the ground is shown below
Transmission Line Model
In the transmission line model the current waveform measured at the ground is assumed to propagate up the channel without changing shape and at constant speed. Here's how that is expressed in equation form.
Basically the current value observed at height z at time t is the same as the current seen at the ground at a time (t - z/v) earlier (i.e. the time it took to travel from the ground up to height z). Again this might be clearer in picture form.
The current waveform measured at the ground propagates up the channel without changing shape or speed.
The radiation field produced by a segment dz of channel located at height z above the ground is shown below
Because
of the functional form of
I(z,t), we can replace the
partial derivative with
respect to time with a
partial with respect to z.
This
makes
it easy to integrate over
z.
H is the height of the return stroke channel. The first term in brackets is zero at times less than H/v (i.e. before the return stroke tip reaches the top of the channel).
Experimental Test
H is the height of the return stroke channel. The first term in brackets is zero at times less than H/v (i.e. before the return stroke tip reaches the top of the channel).
You
couldn't ask
for a simpler
relationship
between E and
I (or dE/dt
and dI/dt).
Erad has the same shape as the current waveform measured at the ground. Ditto for dErad/dt and dI/dt.
These expressions are widely used to estimate Ipeak and peak values of dI/dt from measured Erad and dErad/dt.
Note both the peak I and peak dI/dt occur when the tip of the upward moving return stroke current waveform is close to the ground. This assumption that the return stroke channel is straight and vertical might not be too bad at this point.
Experimental Test
Now
we'll look at an
experimental test
of the Bruce Golde
(BG) and
Transmission Line
(XL) models.
Measurements
of electric and
magnetic fields
were measured at
2 stations: one
close to and the
other far from
the strike
point. The
far field
measurements
(which are just
radiation fields
and don't
contain any
induction or
electrostatic
fields) were
used to
determine the
return stroke
channel current,
I(t), using of
the two
equations
above.
Then both near
and far fields
were calculated
and compared
with the actual
measurements at
the two
sites.
Here are some of the results from the tests. In this case the distant station (the Kennedy Space Center, Florida) was 200 km from the lightning strike point, the close station (Gainesville, Florida) was only 2 km away. These data are from "Lightning Return Stroke Models," by Y.T. Lin, M.A. Uman, and R.B. Standler.
Point 1. The experimental tests used only the fields from subsequent strokes because, without branches, they are closer to the model assumptions that the lightning channel is straight and vertical. A constant return stroke propagation speed was assumed and the value was adjusted to give the best fit between measured and computed fields.
Point 2. Note first the measured distant E radiation field is labeled "Data" on the graph. This field is used in the Bruce Golde (BG) and Transmission line (XL) models to derive the return stroke current I(z,t) (the derived currents are shown at Point 4). Then the E radiation fields are computed using both the BG and TL currents. You can see for the distant fields, the agreement is pretty good but not perfect. In particular the XL calculated field never goes below zero.
Point 3. The derived currents were then used to compute the E and B fields at the closer station. The calculated fields were then compared to the measured fields. The agreement wasn't particularly good.
Point 4. The derived currents are plotted. The XL model current waveform is too narrow and is unrealistic.
Here are some of the results from the tests. In this case the distant station (the Kennedy Space Center, Florida) was 200 km from the lightning strike point, the close station (Gainesville, Florida) was only 2 km away. These data are from "Lightning Return Stroke Models," by Y.T. Lin, M.A. Uman, and R.B. Standler.
Point 1. The experimental tests used only the fields from subsequent strokes because, without branches, they are closer to the model assumptions that the lightning channel is straight and vertical. A constant return stroke propagation speed was assumed and the value was adjusted to give the best fit between measured and computed fields.
Point 2. Note first the measured distant E radiation field is labeled "Data" on the graph. This field is used in the Bruce Golde (BG) and Transmission line (XL) models to derive the return stroke current I(z,t) (the derived currents are shown at Point 4). Then the E radiation fields are computed using both the BG and TL currents. You can see for the distant fields, the agreement is pretty good but not perfect. In particular the XL calculated field never goes below zero.
Point 3. The derived currents were then used to compute the E and B fields at the closer station. The calculated fields were then compared to the measured fields. The agreement wasn't particularly good.
Point 4. The derived currents are plotted. The XL model current waveform is too narrow and is unrealistic.
An
additional set
of test
results (the
lightning
strike was 9
km from the
measuring site
at the Kennedy
Space Center
and 200 km
from the
measuring
station at the
University of
Florida in
Gainesville).
A return stroke velocity was determined for each set of near and far E and B field measurements (the velocity value was the one that provided the best fit between measured and calculated fields). The velocity values derived for the BG model are shown in the top graph and appear to be distance dependent (distance from the close station to the lightning strike point). That is a physically unreasonable result. The velocities derived for the TL model appear in the bottom plot. The values are more reasonable; perhaps a little lower than the 1 x 108 m/s commonly assumed for return strokes but at least they do not appear to vary with distance.
Here are
the peak
current values
derived for
both the TL
and BG
models.
In both cases
peak current
values appear
to be distance
dependent
which is not
realistic.
Many of the TL
peak currents
are too
large.
First return
stroke peak
currents are
typically
about 30
kA.
Subsequent
stroke peak
currents are
usually
less.
Some of the TL
peak currents
in the plot
above exceed
100 kA.
A return stroke velocity was determined for each set of near and far E and B field measurements (the velocity value was the one that provided the best fit between measured and calculated fields). The velocity values derived for the BG model are shown in the top graph and appear to be distance dependent (distance from the close station to the lightning strike point). That is a physically unreasonable result. The velocities derived for the TL model appear in the bottom plot. The values are more reasonable; perhaps a little lower than the 1 x 108 m/s commonly assumed for return strokes but at least they do not appear to vary with distance.
Neither
the transmission
line or the
Bruce-Golde did a
very good job of
reproducing the
measured fields,
particularly at
close range.
The researchers
that conducted
these
experimentals
tests made some
changes to the
assumed return
stroke
current. In
particular they
found that 3
current components
were needed to
better reproduce
the near fields: a
breakdown current,
a corona current
and a uniform
current.
(source: Uman,
M.A., The
Lightning
Discharge,
Academic Press,
Orlando, 1987, see
also the Lin,
Uman,
and Standler
(1980)
reference
mentioned
earlier).
We won't discuss this further in this class as we'll mostly be interested in estimating peak I and peak dI/dt values from measurements of radiation fields. And in that case it looks like the transmission line model does a pretty good job. We'll look at this further in our next class.
We won't discuss this further in this class as we'll mostly be interested in estimating peak I and peak dI/dt values from measurements of radiation fields. And in that case it looks like the transmission line model does a pretty good job. We'll look at this further in our next class.