1. Radar Detection of Lightning
Williams et al. (1989, J. Atmos. Sci.)
Observations of lightning by radar has a history as long as radar itself.
Generally accepted that radar echoes from lightning are reflections from thin highly ionized channels created by the intense heating of air due to a lightning discharge. The temperature of the plasma determines the electron density of the plasma which then dictates the EM backscattering characteristics of the lightning channel.
A key feature here is
how the plasma
frequency fp compares
to the frequency of
the incident radar
wave
n is the electron density (in this case produced by
intense heating in the lightning channel)
e is the charge of an electron (-1.6 x C)
m is the mass of an electron (9 x kg)
ε0 is the permittivity of free space (8.85 pF m-1)
n on order of 1018 – 1020
m-3; for n = 1018 m-3,
fp = 9 x 109 Hz (9 GHz)
This is the frequency that
an X-band radar (3 cm) wavelength operates at.

2. Williams et al. (1989), J. Atmos. Sci.
The figure at left shows the plasma
frequency as a function of temperature. Also
illustrated are frequencies of various
radar bands. For two background pressures, 100
and 1000 mb. Spans range of pressures of interest.
The key physical process here is the
relationship between the plasma
frequency and the frequency of the
incident (radar) radiation impinging on
it. The plasma frequency is basically
determined by how hot the plasma is,
i.e., its temperature. The temperature
of the plasma determines the level of
ionization, which is measured by the
concentration of electrons (or ionized
species). For T > 5000K, the plasma
frequency is greater than the frequencies
used for weather radars. This will be a key point
for this discussion
Plasma frequency refers to the natural
oscillation frequency of the plasma. A plasma
at frequency f can readily “respond” to any
EM wave at a frequency < f. The plasma frequency
is also referred to as Langmuir waves which are
rapid oscillations of the electron density in a plasma.

3. A little on the temperature and physics of lightning channels...
The electrical resistance of air is quite large. So when a high current lightning
channel passes through air, the air is rapidly heated (via Joule heating, I2R, where
I is current and R is resistance). Joule heating results when ionized particles and
electrons in the channel are accelerated to large speeds by the strong E field (or
voltage gradient at the leading edge of the channel). These particles slam into
molecules and gives them large kinetic energies. These motions are random, which
is how we describe the temperature, or thermal motion in this “plasma”. We use the
term plasma when the air is sufficiently ionized to have low resistance.
This heating continues until there are sufficient electrons and ions in the channel,
which reduces the resistance and therefore Joule heating. In about 10 μs, the air is
heated to > 20,000 K.
The channel then cools by radiation and heat conduction. Quickly the channel
becomes non-conducting, typically about 50 ms after the high current input. The
concentration of electrons and ionized molecules quickly decreases as the channel
cools. The channel becomes non conducting when the temperature falls to 2000-
4000K. It is during this brief time when the channel is conducting that there is a chance
to observe the channel with microwave radar.
Lightning channel temperatures have most often been measured by spectroscopic
techniques. The previous figure is very much on the low side for channel temperatures.

4. Radar detection of lightning has a long historyλ

5. Overdense Plasma
For an overdense plasma, the characteristic frequency of the
plasma is greater than the frequency of the EM wave that impinges on it.
That is,
fP > fEM
Recall that the period of an EM wave (or oscillation) is inversely proportional to
frequency.
So for an overdense plasma, Tp < TEM
That is, the oscillation period of the plasma is short compared to the EM wave period.
This means that the plasma can respond to the EM wave as electrons on its outer
sheath oscillate at the frequency of the incident wave. These oscillations can
be viewed as oscillations in the density of free charge (electrons) in time. So the
plasma acts as a conductor since the incident wave only interacts with the outer
portion of the sheath. This is precisely the physics that causes an EM wave
to “reflect” from a metallic conductor. Metals have many free electrons making it
“overdense” at all but highest frequencies, such as X-ray frequencies, which are
1016 to 1020 Hz.

6. Underdense plasma For an underdense plasma, fP < fEM or TP > TEM
The period of oscillation of the plasma is longer than that of the incident
EM wave so in this case the response of the plasma is “sluggish”. That is, it
behaves as a dielectric. In this case the incident wave penetrates into the
plasma and the backscatter of the incident EM is greatly reduced.

7. As the lightning channel cools, the electron density will fall rapidly so the ability of the
channel to backscatter radar waves will diminish. Observations and theory show the
reflectivity falls at 0.2 db/ms (Holmes et al., 1980). So the observation of lightning
channels is a very transient thing!
Incident wave
Reflected wave
Overdense
Conductor
Underdense
Dielectric
But even with this transient nature of the conducting lightning channel one might
expect that observing lightning channels by radar is not that uncommon. But it is
indeed rare—WHY IS THIS THE CASE?

8. Reflectivities from lightning plasma.
Volume
reflectivity
can be
converted to
Z or dBZ
Broad spectrum from
lightning—due to greater
spatial complexity of the
target compared to weather
echoes.
We expect λ-4 dependence
for Rayleigh targets
(r < 0.07λ).

9. Lightning channels “emerge”
from the Rayleigh backscatter
at long wavelengths.
Illustrates masking effects of precipitation.
Histogram shows the
distribution of η for
lightning echoes. These echoes
follow a 1/λ2 dependence.

10. A quote from Williams et al. (1989)

11. Radar Detection of Lightning: Conclusions
1. Lightning plasma is generally overdense at all meteorological wavelengths. Hence, lightning channel responds like a metallic conductor for times on the order a few tens of ms (when high T is maintained).
2. The lightning echo behaves as a volume target to radar. Attributed to a 3-D dendritic structure composed of overdense channel segments, which are long and thin compared to the radar wavelength.
3. Apparent radar wavelength dependence of lightning echoes is highly variable and on average, ηlight ~1/λ2
4. Tendency of strongest lightning echoes to occur in regions of more intense precipitation. Hence the Rayleigh backscatter “masks” the return from lightning.
5. Going to long wavelengths, such as L-band (20 cm) the Rayleigh return falls below that of the lightning return allowing detection of lightning channels at these long wavelengths.
Now look at Ligda video!!

12. Ligda, 1956.
radarmet.atmos.colostate.edu/AT741/papers/Ligda_Film/