Thursday Apr. 11, 2013
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There are several short subjects that we should cover but that won't fill an entire class.  So you're probably going to find a sometimes odd mixture of topics in some lectures as we near the end of the semester.  Such is the case today.  We will have a quick look at

(i) Fast time resolved lightning spectroscopy.  Spectroscopic measurements have been used to determine some interesting and important properties of the lightning channel such as temperature and pressure.  Also you should become familiar with some of the bright spectral emissions that might be used to detect and locate lightning from a satellite.

(ii) Thunder, one of the loudest sounds in nature.

Fast time resolved lightning spectroscopy
A conventional spectrometer is shown in the figure below

Light from an extended source enters the spectrometer through a narrow slit which is positioned at the focal point of a collimating lens.  The parallel rays of light then passes through a prism or diffraction grating and are refracted and dispersed.  These rays are focused onto a detector with a second lens.  The spectrum that is obtained is a multitude of images of the entrance slit, each at a different wavelength, spread out across the detector.

A schematic diagram of a somewhat simpler "slitless" spectrometer design is shown below (the lenses have been left out of the figure).

Light from a lightning channel passes through both a prism and a diffraction grating and is imaged on film (or a detector of some kind).  An entrance slit is not needed because the lightning channel is essentially a line source.  The spectrum that you would obtain in this case would be a series of images of the channel at each of the wavelengths emitted by the discharge.  In a 1961 article in Science, Leon Salanave explains that both a prism and a diffraction grating were used and arranged "so that their respective deviations counterbalance, to make a more 'straight-through' optical system."  Salanave directed much of the lightning spectroscopic work done at the University of Arizona for several years.  He is also the author of "Lightning and Its Spectrum " published by the University of Arizona Press in 1980.

To obtain a time resolved spectrum we need to add a horizontal slit to isolate just a segment of the lightning channel and then cause the film to move perpendicularly.


The film is shown moving downward in this picture (and sorry that the perspective isn't quite right).  Spectra from two separate return strokes, perhaps, have been recorded on the film.

Here are actual implementations that provide moderate and very fast time resolution (from E.P. Krider, "Lightning Spectrosocpy," Nuclear Instruments and Methods, 110, 411-419, 1973).



The rotating drum in the lower figure provides the fastest time resolution.  Rapidly rotating drums in streaking cameras are also used in photographic measurements of return stroke velocity.

The next figure shows an actual example of a very fast time resolved spectrum from a return stroke.

Several emission features are identified on this spectrum early in the return stroke discharge (NII denotes a singly ionized nitrogen atom).  Now what is usually done next is to scan across the film image using a densitometer at say perhaps the level of the green line (i.e. at a time about 10 microseconds after the beginning of the return stroke). 



Here is an example of a "digitized" spectrum.  Film's response to light is nonlinear, so the film must be calibrated.  Film also reacts different to short bright impulses of light than it does to longer duration low amplitude light signals.  So that effect must also be considered.

Six emission lines in the NII(5) multiplet spectrum above have been identified.  I am not familiar enough with spectroscopy and quantum mechanics to be able to say for sure what precise states and transitions cause these features

but I'm guessing it looks something like shown here.  Transitions from 2 slightly different energy levels from an excited state to 1 of 3 levels at a lower energy state could produce emissions of 6 slightly different wavelengths.

In any event it is the relative amplitudes of lines in a multiplet group that can be used to estimate lightning channel temperature.  The procedure is described below.  Our goal is not understand all the steps but rather to get a flavor for the procedure or method.

The first assumption made is that the number of atoms in a particular energy level can be described using a Maxwell-Boltzmann distribution




Nn is the number of atoms in energy level n, N is the total number of atoms.


The intensity of the emission produced by transitions from energy level n to r is shown above. 

Now we will look at the ratio of measured intensities from two different transitions: n to r and m to p

It is these transition intensities that can be measured on the fast time resolved lightning spectra.  Note that many of the parameters cancel.  This is fortunate because some of them (such as the geometric factor) might be difficult to determine.  Solving for temperature yields

Often what is done is that several determinations of temperature are made using different combinations of lines in the multiplet group and an average is computed.  In the case of the NII(5) multiplet 5 ratios could be computed: 4630/4601, 4630/4607, 4630/4613, 4630/4621, and 4630/4643.
 
The next several figures shown some of the results of these spectroscopic measurements and analyses.


Concentrations of NIII (doubly ionized atomic nitrogen), NII (singly ionized atomic nitrogen), and NI (neutral atomic nitrogen) early in a return stroke discharge.  NII is initially the most abundant species and is used in spectroscopic determinations of peak temperature in the return stroke channel.

Estimate of peak return stroke channel temperature.  The value approaches 30,000 K which is approximately 5 times hotter than the surface of the sun (6000 K).

Here are estimates of channel pressure.  This requires a measurement of temperature and probably density (including electron density from the ionized air molecules).

Thunder

Early in a return stroke, pressure in the channel is several times (maybe a few 10s of times) atmospheric pressure.  The lightning channel expands rapidly outward initially as a shock wave.  Much of the energy in a lightning stroke is dissipated during the formation and outward growth of the shock wave.  The shock wave expands outward a few meters in 10s of microseconds and quickly decays into the sound wave that we eventually hear as thunder.


This figure (this and the next two figures are from an article on Thunder by A.A. Few that appeared in "The Physics of Everyday Phenomena", Readings from Scientific American, W.H. Freeman & Co, San Francisco, 1979) shows the transition from shock wave to sound wave.  The size of the channel when this occurs (a few meters) is referred to as the relaxation radius.  Channel tortuosity features that are smaller than the relaxation radius are swallowed up by the shock wave and don't really affect properties of the thunder such as frequency.


Lightning channels consist first of mesotortuous segments that are 5 to a few 10s of meters long.  Several of these positioned end-to-end and oriented in about the same direction form a macrotortuous segment which is a 100 or a few 100s of meters long.  The variation of sounds that are heard during thunder are determined primarily by your orientation relative to the mesotortuous elements.  Sounds emitted in a direction that perpendicular to the channel are louder than sounds emitted more nearly parallel to the channel.

Microphone A is located more nearly perpendicularly to the 4 mesotortuous segments highlighted in the figure.  Four relatively loud sound pulses arrive in quick succession at microphone A and form a clap of thunder.  Microphone B is positioned more nearly end.  Weaker impulses arrive at the second microphone over a longer period of time and produce more of a rumble.




Another, more realistic, illustration of how the pattern of sounds in thunder and the duration of the thunder depend on your location and the geometry of the lightning channel. 

It is generally not possible to separate the sounds of the separate return strokes in a thunder record (either by ear or on thunder recordings).  A sound that resembles "tearing cloth" has often been reported a split second before the loud clap of thunder from a close lightning strike.  When I mention this in the lecture version of this class, most students agree that they have heard something like this.  The cause of that sound is not known.

At the start of today's notes I mentioned that thunder is one of the loudest sounds in nature.  Because the intensity of sound can vary and is audible over a very wide range it is common to express the intensity of sound in decibel (dB) units.

Intensity = 20 log10 (P/Pref)

Pref  in the equation above is 2 x 10-10 bar (normal atmospheric pressure is approximately 1 bar).

The  chart below (compiled from a variety of sources) lists the intensities of some common sounds.

Intensity
comments
 approximate
allowed exposure time
140
 deck of an aircraft carrier
130
 jet plane at 100 feet
sound intensity above this level can cause immediate ear damage
120
 thunder, chain saw
110
 car horn at 1 meter
 1 min
100
 lawn mower, jack hammer
 15 min
95
1 hour
90
 sounds above this level regularly cause hearing damage
 2 hours
80
 garbage disposal, dishwasher, electric shaver, alarm clock
70
 radio or TV audio
60
 normal conversation
0 dB
 threshold of hearing


Thunder duration is examined is a bit more detail in the next figure (from: M.A. Uman, The Lightning Discharge, Ch. 15, Academic Press, Orlando, 1987).




The points on the graph shows actual measurements of thunder duration from discharges at different distances from the observer.  The solid line is a calculation of thunder duration from a vertical lightning channel 7 km tall.  Points A and B highlighted above refer to the calculations below.

Person A in the figure above will hear the thunder coming from the bottom of the channel instantaneously.  Sound travels about 340 m/sec in 20 C air (1 km in 3 seconds, 1 mile in approximately 5 seconds), so the sound from the top of the channel will arrive about 21 seconds later.  That's a duration of 21 seconds.

The observer at Point B will hear the thunder from the bottom of the channel 30 seconds after the lightning strike.  The bottom of the channel is 10 km away from the observer.  The top of the channel is 12.2 km away, so thunder coming from the top of the channel will arrive about 6.6 seconds later than the sound coming from the bottom of the channel (3 seconds/km x 2.2 km).  The thunder only lasts for about 6.5 seconds at Point B.

Wind can affect the sound of thunder and refraction can mean there is a distance beyond which thunder is not heard





These are pictures of a microphone that has been used to make thunder measurements in the infrasonic to low audio frequency range (a Globe Model 100-B Capacitor Microphone).  The microphone consists of a flexible diaphragm and a metal back plate which act as plates of a capacitor.  The microphone is normally set up right as shown in the left most figure (the metal housing is sometimes covered with a "rubberized hair" wind screen (similar to the "rat's hair" used as a packing material).  The microphone has been turned on its side, in the center picture, to shown the opening that exposes the microphone to the pressure changes caused by thunder.  The right most picture shows the microphone electronics.


Calibrated response of the Globe capacitor microphone.  The microphone response is flat from about 0.5 Hz to about 300 Hz.
Here are three plots of the measured thunder frequency spectra



This first plot is actually a distribution of peak values in sound spectra.  Interestingly a portion of the distribution is below 20 Hz and is inaudible; this is infrasound. 



This is the spectrum from a single discharge.  The peak in the spectrum is near 100 Hz.


An example of a sound spectrum (lower right corner) that actually peaks in the infrasound. 

Up to this point we have explained thunder as being due to the explosive expansion of a hot lightning channel.  This may not be the cause of all of the emissions at infrasonic frequencies.  Some researchers have suggested that a sudden change in the cloud electrostatic field caused by a lightning discharge might be the cause of the very low frequency infrasonic emissions.