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Electrostatic proton cyclotron harmonic waves observed with the Alouette II satellite Harvey, Robert Walter 1969

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ELECTROSTATIC PROTON CYCLOTRON HARMONIC WAVES OBSERVED WITH THE ALOUETTE II SATELLITE by ROBERT WALTER HARVEY B . S c , The U n i v e r s i t y o f B r i t i s h Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of GEOPHYSICS We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agr e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Geophysics The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada D a t e 14th May, 1969. i i ABSTRACT An ELF noise band observed by the Alouette II sa t e l l i t e has been studied. A dig i t a l power spectrum program was set up in order to investigate in detail the noise band spectra. Primary results from this analysis are that: a) for altitudes 500-3000 Km. and a l l geomagnetic latitudes, sharp lower cutoffs of the noise band occur within ~30Hz of the calculated gyrofrequency, b) upper cutoffs to the noise band occur frequently near harmonics of the gyrofrequency, and c) the noise appears to be Doppler shifted when the angle between the velocity vector of the s a t e l l i t e and the geomagnetic f i e l d is near 90°. The occurrence pattern of the noise band has been investigated using available Rayspan analyzed data for epoch 1966. This analysis indicated a pronounced daytime maximum of occurrence. It is shown that almost a l l aspects of the noise band may be interpreted in terms of the hypothesis of ambient electrostatic proton cyclotron harmonic waves in the ionosphere, and the concomitant accessibility conditions in the spatially varying geomagnetic f i e l d . i i i TABLE OF CONTENTS Page ABSTRACT .. i i LIST OF FIGURES.. i v LIST OF TABLES v i ACKNOWLEDGEMENTS v i i 1. INTRODUCTION 1 2. THE ALOUETTE II VLF EXPERIMENT 4 2.1 Receiver System 4 2.2 Sheath E f f e c t s and Antenna Gain - 4. 3. SPECTRUM ANALYSIS 8 3.1 Analog Spectrum A n a l y s i s . •••• ^ 3.1.1 The Rayspan Spectrum A n a l y z e r 8 3.1.2 The Sonograph 9 3.2 D i g i t a l Spectrum A n a l y s i s 10 3.2.1 A n a l o g / D i g i t a l Conversion.. . 11 3.2.2 D i g i t a l Spectrum A n a l y s i s 13 3.3, D i s c u s s i o n 16 3.4 Sharpness of C u t o f f s I 7 4. RESULTS OF DATA ANALYSIS 19 4.1 P a r t i c u l a r C h a r a c t e r i s t i c s of the Noise Band.... 19 4.2 General Occurrence. P a t t e r n of the Noise Band. 22 5. THEORY OF PROTON CH WAVES 37 6. APPLICATION OF THE THEORY OF PROTON CH WAVES TO THE • • OBSERVATIONS 47 6.1 E x p l a n a t i o n f o r S p e c i f i c Observations on the Noise Band... 47 6.2 Occurrence P a t t e r n o f the Noise Band 53 7. OTHER POSSIBLE EXPLANATIONS FOR THE NOISE BAND 55 7.1 Results of a F u l l Wave Treatment of the B e r n s t e i n Modes... 55 7 2 I =0 Thpnrv - « • •. ^ • 58 8. SUMMARY AND CONCLUSIONS 6 0 APPENDIX 6 2 BIBLIOGRAPHY 7 0 iv LIST OF FIGURES Figure - Page .1 Model for an electric dipole antenna in a plasma 5 2 Block diagram of the equipment setup used for digitization of the VLF'analog records 12 3 Effective f i l t e r of the d i g i t a l spectrum analysis program 15 4 A "Rayspan" spectrogram from the Alouette II VLF receiver on July 23, 1966 23 5 A "Rayspan" spectrogram from the Alouette II VLF receiver on May 13, 1966 24 6 "Sonograms" from a mid-latitude pass on May 13, 1966 25 7 Digitally computed spectra from a s a t e l l i t e pass on November 7, 1966. 26 8 Digitally computed spectra from a s a t e l l i t e pass on October 14, 1966 27 9 Digitally computed spectra from a s a t e l l i t e pass on July 30, 1966 28 10 Digitally computed spectra from a s a t e l l i t e pass on •June 13, 1966 29 11 Digitally computed spectra from a s a t e l l i t e pass on May 13, 1966 30 12 Occurrence pattern of the noise band plotted as a function of invariant latitude and local mean time 33 13a Occurrence of sharp lower frequency cutoffs 34 13b Occurrence of sharp or almost sharp upper frequency cutoffs 34 14 The percentage occurrence of the noise band versus K 35 P 15 Dispersion curves for perpendicularly propagating proton CH waves in a Maxwellian plasma 42 16a Oblique propagation of proton CH waves in a Maxwellian plasma 43 16b Dispersion curves for CH waves with collisions included 43 V 17 A cross-section of the earth's magnetic f i e l d 48 18 Schematic of generalized ion Berstein mode dispersion 57 19 Data windows and corresponding spectral windows 65 v i LIST OF TABLES T a b l e Page 1 O r b i t a l Parameters of Alouette II (January 12, 1966) 1 2 Sharpness of Cutoffs ' 18 3 Isomorphism Between El e c t r o n and Proton Dispersion Relations 40 ' v i i ACKNOWLEDGEMENTS I express my appreciation and gratitude to my thesis advisor D.r. Tomiya Watanbe who interested ma in the Alouette II s a t e l l i t e experiment, encouraged me to look for warm plasma effects in the data, and then discussed the data and theory in this thesis with myself. I thank Dr. J.A. Jacobs and Dr. R.D. Russell for encouragement in this work. I gratefully acknowledge the many helpful discussions and suggestions of Dr. R.E. Barrington, of the Defence Research Telecommunications Establishment, Ottawa. I thank also Mr. W.E. Mather for much assistance in the analog data analysis, Mr. R.W. Herring for a 'fast Fourier transform' algorithm, Mr. Scott Inrig for provision of the analog/digital control program, and other members of the DRTE for comments and suggestions. CHAPTER 1 INTRODUCTION Very low frequency (VLF) broad-band receivers have been flown in a number of ionospheric s a t e l l i t e s . Spectrum analysis of the observed signals has shown the existence of a variety of quasi-continuous bands of noise, and also discrete emissions with rapidly varying frequency-time characteristics (Taylor and Gurnett, 1968). Included i n these observations are the more well known phenomena of 'whistlers', 'polar chorus 1, and the low hybrid resonance (LHR) noise band. These signals are due to the existence of electromagnetic and electrostatic waves propagating in the ionospheric plasma. In this thesis, data from the Alouette II s a t e l l i t e w i l l be used. This space-craft was b u i l t by the Defence Research Telecommunications Establishment (D.R.T.E.), Ottawa, and launched by the National Aeronautics and Space Administration of the U.S.A. on November 29, 1965. Nominal parameters of i t s orbit are shown in Table I. -of the Alouette II VLF records, and for which the following spectral characteristics are established: 1) a sharp lower frequency cutoff (i.e., intensity decrease) which occurs very near to the proton gyrofrequency, 2) an upper frequency cutoff which is often quite sharp and near harmonics of the gyrofrequency, and 3) occasionally, an apparent harmonic structure. The chief interest w i l l be a band of noise which occurs in -50% TABLE I ORBITAL PARAMETERS OF ALOUETTE II (Jan 12, 1966) Apogee Perigee Inclination Period Argument of Perigee R.A. of Ascending node Eccentricity 2981.82 Km. 502.16 Km. 78.828 prograde,. 121.369 min. - .00006 min/day. 248.25° - 1.8922°/day. 172.422° - .7922°/day. 0.15268. 2 Examples of this noise band can be seen in Figures 4 and 5. Similar noise has been noted by Guthart et a l . , (1968) from 0G02 s a t e l l i t e data, and by Burns (1966) and Gurnett and Burns (1968) f rom Injun III data. In geophysics two different approaches can generally be taken for the analysis of a series of observations particular events, which clearly exhibit the phenomena of interest, may be singled out and analyzed in detail or, the gross s t a t i s t i c a l properties of the observations may be obtained with a view to relating these to other known processes. In this thesis, both approaches are taken. Analysis of existing spectra, obtained by use of analog equipment, has been carried out for a large sample of s a t e l l i t e passes, and the stat i s t i c s or occurrence pattern of the noise.band and certain of i t s easily identifiable spectral character-i s t i c s , computed. However, analog spectrum analysis was not considered sufficient to explore in detail the spectral characteristics of the noise band. A di g i t a l spectrum analysis system was set up for this purpose. This i s a new application of d i g i t a l 'fast fourier transform' techniques, and is proving to be quite f r u i t f u l for VLF data interpretation. In particular, high resolution determinations of the lower frequency cutoff of the noise band were obtained. Typical results are shown in Figures 7 to 11. i Guthart et a l (1968) have considered possible explanations for a noise band which has a lower frequency cutoff near the gyrofrequency, and concluded that electrostatic proton cyclotron harmonic waves (hereafter, proton CH waves; also termed in the literature 'Berstein' modes, Berstein (1958)) were involved. According to Stix (1962), three wave modes can propagate at such frequencies - — l e f t - and right-hand polarized electromagnetic (EM) waves and electrostatic proton CH waves. However, immediately above the proton gyrofrequency only the right-hand EM and proton CH waves propagate. Considering the right-hand mode, waves propagating parallel to the ambient magnetic f i e l d are unaffected by the gyrofrequency. But recent studies by Fredricks (1968) indicate some gyrofrequency effect on perpendicularly propagating EM waves; the magnitude of this effect under ionospheric conditions is not known. Further discussion on this is included in Chapter.7. On the other hand, proton CH waves in a Maxwellian plasma of electrons and protons propagate unattenuated in the direction perpendicular to the ambient magnetic f i e l d , and exist above but not immediately below the proton gyrofrequency. Thus following Guthart et a l . a band of noise in the ionosphere due to these waves could be expected to have a sharp cutoff in amplitude at the proton gyrofrequency. This explanation of the noise band w i l l be extended. The presentation of this thesis w i l l be as follows. As a background on the results, details of the Alouette II receiver and antennas, and then methods of spectrum analysis both analog and d i g i t a l are. examined. Particular spectral characteristics of the noise band, and then i t s general occurrence pattern are next presented. With a view to explaining these observations on the noise band, a chapter w i l l be devoted to a review of relevant features of proton CH waves. It is then shown that in addition to the sharp lower cutoff a l l other specific spectral character-i s t i c s and some of the general occurrence patterns of the noise band may be explained consistent with the hypothesis of ambient proton CH waves in the ionosphere. Other possible explanations for the noise band are also examined. 4 CHAPTER 2 THE ALOUETTE II VLF EXPERIMENT 2.1 Receiver System Alouette II is equipped with a set of capacitively blocked dipole antennas with tip tc tip length of 240 feet. The output of these antennas is amplified by an untuned receiver of bandwidth 50 Hz - 30 KHz. Gain of the receiver is controlled by an automatic gain control (AGC) c i r c u i t . Both receiver output and AGC voltage are telemetered to the ground. The receiver-AGC system has been calibrated in the laboratory by applying a signal to the inputs and monitoring the receiver output and AGC voltages (Henderson and Cote, 1967). Typical results at 1 KHz were that a 12 dB increase in receiver input caused a 1 dB increase in receiver output, and a 1 volt increase in the AGC voltages. This holds true quite well over a dynamic range of ~ 60 dB. So, as the receiver input increases, gain of the system is decreased. 2.2 Sheath E f f e c t s and Antenna Gain Gain of the antenna is not so easily estimated as for the VLF receiver. Effects of the ionospheric plasma on dipoles are not well understood. However, a few of the factors which must be considered in determinations of antenna gain w i l l be reviewed. The antennas and s a t e l l i t e body are electrically floating in the ionospheric plasma, and w i l l come to equilibrium with no net current flowing to them. Due to the high mobility of electrons relative to the heavier ions, they assume a negative potential (the floating potential) different from the potential of the local plasma (the plasma potential). This gives rise to a region surrounding the antennas and s a t e l l i t e body of" primarily positive ions, and a few electrons (the ion "sheath"). Effects of the sheath on measurements of differences in plasma potential (due to electromagnetic and electrostatic waves) are capacitive and resistive. The antennas of Alouette II, which are capacitively blocked, and insulated from the s a t e l l i t e body by 1 meter long non-conducting sleeves, may be represented to a f i r s t approximation as in the following figure. sleeve Figure 1 Model for an electric dipole antenna in a plasma where Rs, Cs = the resistance and capacitance of the sheath. V^, ^2 = different plasma potentials. Zd = the detector impedance. I =. current. Then V± - V 2 = I(Zs + Zd) Zs = (~ + jtoCg)"1 Thus the effect of the sheath impedance is to degrade the observed voltage across the detector. A more complete discussion of this model is given by Storey (1963). Simplified considerations indicate, over an effective length of ]20 f t . and at frequency 1 KHz, that the magnitude of the sheath impedance for each antenna element is 50 K°, - j 100 Kfl. How-ever, since the antennas used for the VLF experiment are also used by the "topside-sounder" experiment, low VLF receiver input impedance had to be chosen in order to insure compatibility; impedance of the receiver through the coupling network is estimated to be quite variable at the low frequency end of the VLF band, being 1 Kfl at 100 Hz to 7 Kfl at 1 KHz. Absolute electric f i e l d measurements are thus not feasible because of the uncertainty in the impedance of both the sheath and the receiver. There are also orientation effects on the sheath when the s a t e l l i t e is in a magnetized plasma. Storey has argued that since the plasma conducts best in the direction of the magnetic f i e l d , then the effective area of contact of the antenna with the ambient plasma is the sheath area projected in the magnetic f i e l d direction; this area is minimum when the antennas are aligned with the magnetic f i e l d . Thus antenna gain w i l l be greatest when the antennas are perpendicular to the ambient magnetic f i e l d and least when they are parallel to i t . Another effect leading to orientation-sensitive gain of the antenna is the induced potential due to motion of the spacecraft across magnetic amount to as much as 10 volts over each half of the antennas. Effects of this potential have been considered by Dickinson (1964). He finds that perpendicular to V x B) that the sheath on each antenna element may be distorted to a "tear-drop" shape, one end with zero sheath radius and thus at plasma potential, and the other end with a maximum sheath radius. This maximum sheath radius is ~2cm. at low latitudes and ~90cm. at high latitudes. Thus induced voltage can strongly perturb the sheath and lead to a direct coupling of an electric f i e l d in the plasma to one. end of each antenna element. The gain of the antennas would be expected to be fi e l d lines. For the Alouette II, when V x B.dJ is maximum (that i s , dit the ant enna orientation vector i s 7 increased due to this effect. Thus there are at least two phenomena which lead to minimum gain of the antennas when they are aligned parallel to the ambient magnetic f i e l d . The possible significance of this w i l l be pointed out in a later section in connection with "spin modulation". * 8 CHAPTER 3 SPECTRUM ANALYSIS The Alouette II VLF receiver signal is stored in an analog mode oh precision magnetic tapes. Interpretation of these signals is usually fac i l i t a t e d by spectrum analysis. Figures 4 and 6 il l u s t r a t e two methods commonly seen i n the literature for displaying VLF data. These figures represent the result of analog methods of spectrum analysis. In addition to this type of analysis, a system was set up employing 'fast Fourier transforms' of the digitized VLF signal; this technique gave detailed spectra of the noise band, as shown in Figures 7 to 11. In order to compare the advantages and disadvantages of the above analyses, the equipment and methods to produce these spectra w i l l be discussed. 3.1 Analog Spectrum A n a l y s i s 3.1.1 The Rayspan Spectrum A n a l y z e r " The Rayspan consists of a band of 420 band-pass (magnetostrictive) f i l t e r s spaced with center frequencies on the average 25 Hz apart. These f i l t e r s are fed i n parallel by the VLF signal. Outputs of the f i l t e r s are sampled by a high speed capacitive commutator, and the results after passing through a logrithmic amplifier are displayed by intensity modulation of synchronized sweeps on a CRT. 35 mm. film is rolled past the CRT screen at a uniform rate. Thus a frequency versus time display of the spectra is produced, where the darkness of the film i s proportional to the power density at each frequency. A typical Rayspan record is shown in Figure 4. The commutator is such that f i l t e r s are sampled in groups of three giving effectively a band of 418 inverse phased t r i p l e t s . To study the frequency range of 0-2.5 KHz, the input VLF data is played r back at four times real time. Under this condition, the nominal characteristics'of the system are: Effective bandwidth of a single f i l t e r 10 Hz Average spacing of single f i l t e r s = 6.25 Hz Attenuation outside bandwidth of single f i l t e r = 6 dB/octave bandwidth Attenuation outside bandwidth of three f i l t e r set = 18 dB/octave bandwidth Clear to black dynamic range of 35 mm. film = 12 dB. The actual operation of the instrument has been found to be far from ideal. Main sources of error are the variations in shape of the response curves of each of the f i l t e r s , unequal spacing between the center frequencies of adjacent f i l t e r s , and the limited dynamic range of the film. These problems.lead to errors, i n the measurement of noise band cutoff frequencies, as great as 40 to 50 Hz. However, the method does provide a means of obtaining, with good time resolution, a quick synoptic view of the VLF noise. 3.1.2 The Sonograph The Sonograph (made by Kay Electric Co.) also produces a display of the VLF signal in the frequency versus time format, with amplitude of noise proportional to the darkness of the paper. The principle of i t s operation consists of recording 2.4 seconds of the VLF signal on a magnetic drum; this re-recorded signal i s then played back repeatedly through a single f i l t e r i The center frequency of the f i l t e r is slowly increased by mechanically varying the resistance. Output of the f i l t e r causes electro-sensitive paper to be darkened on a linear scale. For use in analyzing relatively low frequencies ( E L F ) , the input data is played onto the magnetic drum at four times real time, giving 9.6 seconds of data on the drum and a frequency display from 0 to 1.75 KHz. The effective bandwidth of the f i l t e r is 11 Hz with skirt attenuation 10-15 10 dB/octave bandwidth. Dynamic range of the paper covers about 5-8 dB. Results of this type of analysis are shown in Figure.6. The Sonograph has several advantages over the Rayspan which are not immediately apparent from the specifications.. Chief amongst these is that a single f i l t e r is used. Thus problems of non-uniform response of different f i l t e r s , as i s the case for the Rayspan, are eliminated. Also, since the center frequency is easy to vary in a uniform manner, uneven f i l t e r spacing problems are avoided. However, the dynamic range of the instrument is significantly less than for the Rayspan. Also i t takes about 10 minutes to analyze 9.6 seconds of data, whereas the Rayspan can run at four times real time. When trying to measure the frequency of a sharp lower cutoff of a noise band, one is s t i l l limited in practice to an uncertainty of about ± 25 Hz. 3.2 D i g i t a l Spectrum A n a l y s i s In order to achieve greater frequency resolution than was previously available, and to obtain the actual shape of the power spectrum for ELF noise bands, a system for d i g i t a l spectrum analysis was set up. It is desirable to analyze the noise over the frequency range from 70 Hz to 1 KHz with a resolution of 4 Hz. This was achieved; details of the analog/digital conversion and then the d i g i t a l spectrum analysis are given in the following section. A synthesis of recent ideas in the f i e l d of d i g i t a l spectrum analysis techniques is provided in the Appendix. Typical results of the analysis are shown in Figures 7 to 11. 11 3.2.1 A n a l o g / D i g i t a l Conversion Playback of the analog VLF signal for conversion to di g i t a l form is effected using an Ampex Model FR 104 tape recorder. Frequency response of the recorder i s f l a t down to ~30 Hz and equalization problems between this recorder and the original (a Sangamo Model 4712) are negligible. y" The block diagram in Figure 2 illustrates the basic arrangement whereby the analog VLF signal i s converted to an equivalent binary representation on magnetic tape. Sampling rate of the A/D is controlled by an external pulse generator which i s 'AND'-ed with the computer. The signal from the analog tape recorder i s amplified to a level compatible with f u l l use of the A/D input range, and then band-pass fil t e r e d to reduce the effects of 'aliasing' (see Appendix) and to keep out any large quasi-D.C. components. Skirt attenuation of the f i l t e r is 20 dB per octave. Conversions are performed by a Texas Instruments Inc. Model 846E Analog-Digital-Analog Converter. Aperture time i s 0.1 sec., input dynamic range ± 8.19 volts, and output the 12-bit (including sign) binary equivalent. Sampling rates up to ~50 KHz may be achieved. A 'machine language' program for control of the A/D has been + written for the CDC 3200 computer . The following operations are performed: (1) Blocks of N samples are buffered into core memory from the A/D converter. Samples are stored i n memory un t i l the block of N is completed and then transferred to d i g i t a l magnetic tape. (2) M blocks of the above length N are taken with no intervening gaps. The program was provided by Mr. Scott Inrig of the D.R.T.E., Ottawa. PLAYBACK OF VLF RECORD DC AMPLIFIER FREQUENCY METER PULSE GENERATOR BANDPASS FILTER _5> A / D COMPUTER INTERFACE A CONTROL CONVERSIONS COMPUTER WITH CONTROL PROGRAM DIGITAL TAPE Fig. 2. Block diagram of the equipment setup used for digitization of the VLF analog records. 13 (3) Each set of M blocks is taken beginning at T sec. intervals. Clearly, M • N • (1/sample rate) < T. For analysis of the ELF noise bands, eight or twelve blocks of 2048 samples each were taken commencing each 60 seconds, and at a sample rate of 4,000 samples/second. In practice, i t is possible to increase playbacks of the VLF signal to 2 or 4 times real-time, along with corresponding adjustments to f i l t e r frequencies and sample rates. 3.2.2 D i g i t a l Spectrum A n a l y s i s In this section an'outline w i l l be given of the spectrum analysis of the digitized data (for details refer to the Appendix). Briefly, modified power spectrum estimates are constructed for each block of 2048 samples. Results of spectrum analysis of 8 (or 12) blocks are then combined. A block of samples is designated: x Q, x1,... x 2 0 4 ? . ___ This data i s Fourier transformed according to -2irik£) 2047 a r + i b r = I e ""^W k=0 r = 0,1,1,..., 1024, using a 'fast Fourier transform' routine"'". The Fourier coefficients (a^ + lb ) are transformed by 'hanning' (equation A-17, Appendix) to reduce 'leakage', thus obtaining the modified Fourier coefficients: A r + i B r , r = 0,1,2,..., 1024. Modified power spectrum estimates are then Program written by Mr. R.W. Herring of the D.R.T.E., Ottawa. 14 P r = A r + K ' r = O.1'^--.. 1024> where normalization has been neglected. Averages are then taken over the 8 (or 12) blocks in a set: P ' + P . + ... +.P M p _ r l r2 rM r M avg where, M = the number of blocks per set. This represents a time average of 4 (or 6) seconds. These power estimates have s t a t i s t i c a l s t a b i l i t y (for an approximately stationary gaussian process) corresponding to a •chi-square' distribution of 16 (or 24) degrees of freedom. Reference to tables of this distribution (c.f. Blackmail and Tukey, (1959)) shows that 80% of the estimates should deviate from the average value by no more than: ± 4 dB for 16 degrees of freedom ± 3 dB for 24 degrees of freedom. For purposes of analyzing a steady noise band, 8 blocks of data per set suffices when whistler activity is low. The whistlers represent a non-stationary perturbation of the noise speptrum. For moderate whistler activity, 12 blocks per set were used. Although the shape of the effective f i l t e r used in this type of analysis is theoretically known from the spectral window for the 'hanning' process (Figure 19 in the Appendix), an experimental determination of i t s shape was performed by generating a 400 Hz cosine wave for eight blocks of data. The cosine function for each block was displaced by 2ir/10 radians, thus allowing an estimation of the average effects"of f i n i t e length records to be included. Results of spectrum analysis of this data are shown in Figure 3. Characteristics of the effective f i l t e r are: bandwidth ~ 3.12 Hz skirt attenuation ~ 28 (dB/octave bandwidth) down to 40 dB. ~ i T r j i T i i i 1 — r — | — i — i — i — r -8 -6 - 4 -2 0 2 4 6 8 FREQUENCY MEASURED FROM 400. Hz (UNIT = 1.953 Hz) Fig. 3. Effective filter of the,digital spectrum analysis program. Bandwidth is 3.125 Hz and skirt attenuation down to -40 dB is 28 dB/octave.bandwidth. 16 Thus, the frequency at which a sharp cutoff occurs can be measured to within i 4 Hz. Other possible errors in the analysis are: flutter in tape recorders (± .5%) inaccuracy in sample rate (± .2%). Therefore, the maximum error in determining a cutoff frequency near 500 Hz is ± 7.5 Hz. A plotter "routine was written to output results of the spectrum analysis. Using a CDC 3200 computer, the time required to produce the modified power spectrum averaged over 8 blocks of data is 19 seconds. An additional 21 seconds is required for each plot routine. 3.3. D i s c u s s i o n For the morphology of the ELF noise bands, and the selection of s a t e l l i t e passes for further analysis, Rayspan analysis i s quite suitable. This method has the advantage over d i g i t a l techniques of giving spectra continuously with time, and can be performed at rates four times faster than the real-time acquisition of the data. A synoptic view of the VLF activity in each pass is easily and quickly obtained. However, estimates of cutoff frequency of a noise band obtained from Rayspan records were found to be in error by as much as 50 Hz. The Sonograph has greater frequency resolution than the Rayspan, but is s t i l l not sufficient to locate a lower frequency cutoff to better than' 25 Hz. Further d i f f i c u l t y arises from i t s relatively low dynamic range. The d i g i t a l analysis does not give the comprehensive view of the noise band obtained from continuous (analog) analysis. However, the following advantages may be pointed out by way of comparison: 17 (a) integration i n time and frequency can be widely varied to obtain the desired resolution and s t a b i l i t y . For the analyses to be presented, the frequency resolution i s adjusted to ~ 3 Hz. The analog system does not have resolution much greater than 25 Hz; (b) the d i g i t a l analysis gives a quantitative picture of the relative power density versus frequency. This power is measured from the system noise level. Analog white-to-black presentations only indicate qualitatively the shape of the spectrum. Furthermore they are limited to a 6-12 dB dynamic range; (c) the effective d i g i t a l f i l t e r has greater skirt attenuation than the analog methods. This is especially important when measuring a noise band with a steep 20 dB cutoff. 3.4 Sharpness of C u t o f f s Often in the literature, one sees references to the cutoff frequency of a noise band, and this cutoff is classified as either sharp or diffuse. When using analog analysis, this i s a visual statement, in that sharp cutoff refers to an abrupt change from black to white, representing a similar decrease in power over a small range of frequency. Thus the noise band in Figure 4 has sharp lower frequency cutoff, whereas that in the bottom Sonogram of Figure 6 has a diffuse cutoff. As part of the investigation of ELF noise bands, a large amount of Rayspan data was examined, and the upper and lower frequency cutoffs of bands near the gyrofrequency were classified as 'sharp', 'almost sharp', or 'diffuse'. Scale of the Rayspan data was 0 to 2.5 KHz "on the vertical 18 (transverse to the film) axis and 1" = 4 sec. horizontally. From observations of the Rayspan data and corresponding d i g i t a l analysis results, i t i s possible to determine what is called a 'sharp' or 'diffuse' cutoff. The definitions listed in Table 2 were used for classification of the analog data, and also may be applied to the di g i t a l spectra. TABLE 2 SHARPNES S OF CUTOFFS Sharp: > 5 dB in 25 Hz Almost Sharp: Diffuse: < 10 dB i n 100 Hz 19 CHAPTER 4 RESULTS OF DATA ANALYSIS 4.1 P a r t i c u l a r C h a r a c t e r i s t i c s o f the Noise Band Spectrum analysis results, displaying most of the particular characteristics of the ELF noise band, are shown in Figures 4 to 11. The Rayspan records (Figures 4 and 5) are typical examples of the noise band spectra, showing that the main features vary only slowly in time. Under each of these records, the corresponding AGC level i s indicated in decibals of noise into the VLF receiver. Sonograms of structure, which occasionally appears on the noise band, are displayed in Figure 6. The results of d i g i t a l spectrum analysis of five Alouette II passes are presented in Figures 7 to 11. These latter passes were preselected only on the basis that Rayspan data indicated a sharp lower frequency cutoff over part of each s a t e l l i t e pass, and that they represent a wide variety, of latitudes and altitudes. On each spectrum the Greenwich Mean Time (GMT), calculated proton gyrof requency (£2p), angle between the velocity vector of the s a t e l l i t e and the geomagnetic f i e l d (L (V,B)), height, and invariant latitude (A, as defined by O'Brien et a l . , (1964)), respectively, are indicated. Also on each spectrum, arrows locate in frequency one or two prominent features. When the cutoff is sharp, the indicated frequencies are measured at the top of the cutoff. In the absence of such cutoffs the arrows locate relatively large fluctuations in the spectrum or merely the position of the gyrofrequency. The drop in noise level below 70 Hz is due to the pre-filtering of the data mentioned in Section 3.1. Each spectrum has been independently normalized. There exists uncertainty with respect to the calculated parameters of the s a t e l l i t e and magnetic f i e l d . The absolute position of the s a t e l l i t e 20 is known to within ~ 3 Km. (G.E.K. Lockwood, personal communication). The magnetic f i e l d model used in calcuation of the proton gyrofrequency and L (V,B) is GSFC, 9/65 (Hendricks and Cain, (1966)). These authors indicated a maximum error of ±200 y in total f i e l d , corresponding to - 3 Hz in proton gyrofrequency. A more exhaustive study of possible errors in geomagnetic f i e l d models (Cain et a l . , (1967), indicates that the standard error in declination and inclination angles at altitudes of 1,000 Km. is (on a world wide basis) approximately .3° and .07°, respectively. . These standard errors w i l l probably be larger near the magnetic poles, due to the scarcity of magnetic data in these regions. Thus, errors in £(V,B) of ± 3°, as a result of uncertainty in the direction of the geomagnetic f i e l d , are quite conceivable. These sets of spectra w i l l now be individually examined, and the characteristics of the noise band along with associated orbital features, discussed. Figure 7 i s from a low altitude s a t e l l i t e pass covering low and mid-latitudes. The sharp lower frequency cutoff of the noise band follows the calculated proton gyrofrequency to within 3 Hz. During part of the pass (GMT 2128 to 2131) the noise band is not vi s i b l e . This time -fr-eer res ponds to very low latitudes, and also £ (V,B) ~ 0 (i.e., the s a t e l l i t e is travelling nearly parallel to the geomagnetic f i e l d ) . Note that the noise band has an upper frequency cutoff from GMT 2133 to 2137, and this frequency is below twice the proton gyrofrequency. Figure 8 also covers mid-latitudes but at altitudes near s a t e l l i t e apogee. Lower frequency cutoffs are approximately 20 dB in 25 Hz. The low frequency side of the cutoff occurs within 30 Hz of the calculated gyrofrequency. The upper frequency cutoff is near three times the proton gyrofrequency and there is some evidence of a change in noise level near 21 twice the gyrofrequency. At such high altitudes the s a t e l l i t e i s , most probably, well into the protonosphere (Hoffman, (1969)). The next two figures (9 and 10) cover the high latitudes at very high s a t e l l i t e altitudes. Observations on these spectra are: (a) above A = 78°, harmonic structure on the noise band is evident; (b) the noise band has a progressively less sharp lower cutoff frequency above A = 79°; (c) at s a t e l l i t e positions where Z(V,B) ~ 90°, the noise band is not well defined. Figure 11 shows spectra from a low altitude, mid- and high-latitude s a t e l l i t e pass. The lower cutoff frequency generally li e s from 5 to 15 Hz below the calculated gyrofrequency. The result of Rayspan analysis of this same pass is shown in Figure 5. By comparing these two figures, i t -*• -> i s seen that there are two effects which may be associated with Z,(V,B) going through 90°. The noise with lower frequency cutoff near the gyro--y -y frequency is increasingly smeared out as L (V,B) -*• 90°; and another noise band comes down from higher frequencies, crossing over to the very low frequencies. Similar effects are seen on the noise band in Figure 4. The smearing out of the noise band when L (V,B) ~ 90° was further investigated, by calculating i(V,B) throughout 230 s a t e l l i t e passes. Thirty passes were found where this angle passed through 90°, and Rayspan data indicated that the noise band existed. In a l l these cases, the spectral characteristics of the noise exhibited a dependence on Z,(V,B). Typical of this dependence is growing down of noise below the lower frequency cutoff, and cutoffs which become diffuse, when the "angle between the velocity vector of the s a t e l l i t e and the geomagnetic f i e l d is near 90°. Another observation on this noise band is shown in Figure 6. These Sonograms, also taken from the s a t e l l i t e pass of Figures 4 and 11, show 22 structure in the form of rising tones superimposed on the noise band. This structure i s seen only occasionally for s a t e l l i t e passes from invarient latitudes of approximately 50° to 70°, and is typically characterized by a lower frequency cutoff near the proton gyrofrequency. An additional observation on this noise band is a marked decrease in noise band intensity (~ 10-20 dB) which occurs twice per spin period when the spin axis of the s a t e l l i t e i s nearly perpendicular to the geo-magnetic f i e l d . This phenomenon is termed 'spin modulation'. Comparison of the spin modulation times with data from the three-component magnetometer aboard the s a t e l l i t e , indicated that (within an accuracy of ± 15°) the intensity nulls occurred when the VLF antennas were aligned with the geomagnetic f i e l d . 4.2 General Occurrence P a t t e r n o f the Noise Band Rayspan data for approximately 75 hours of Alouette II VLF data has been examined. The s a t e l l i t e passes, each about 20 minutes long, were primarily from the northern hemisphere and of epoch 1966. At 4 to 5 minute intervals during each pass the occurrence or non-occurrence of the noise, bands near the proton gyrofrequency was noted, giving a total of 971 observations. When the band occurred the upper and lower frequency cutoffs were classified as sharp, almost sharp, or diffuse according to Table 2.. There exists some question as to whether the noise band s t i l l exists although i t is not seen in the Rayspan data. If very intense noise occurs at frequencies other than in the ELF band, the AGC w i l l desensitize the receiver thus making i t more d i f f i c u l t to see the ELF noise band in the Rayspan (0-2.5 KHz) record. It has been-noted in section 2.1 that a 12 dB change i n receiver input only effects a 1 dB Fig. 4. A 'Rayspan' spectrogram from the Alouette II VLF receiver on July 23, 1966. AGC level, time, angle between the velocity vector of the satellite and the geomagnetic field, and invariant latitude are indicated below and each portion of the spectrogram. Digital spectrum analysis of this pass indicated a sharp lower frequency cutoff from 2328 CMT. These cutoff lay <20 Hz below the calculated proton gyrofrequency. ___ 1.5-ISI — o Q; 0.5-$. "3'! 0 4 4 -C0 22-2020 2 0 2 2 T 2024 2 0 2 6 TIME (GMT) 2028 2 0 3 0 7 2 1^ 79 8 5 r 9 0 9 4 Z.(V,B), DEGREES 9 7 52.5 60.1 67.2 73.8 79.4 A, DEGREES 83.5 Fig. 5. A 'Rayspan' spectrogram from the Alouette II VLF receiver on May 13, 1966. Digital spectra for the same pass are presented in Figure 11. 1500-1000- _ . „ 5 0 0 -x 2021/22 2021/26 2021/30 g , &p=669 Hz Z ( V , B) = 7 7 ° HEIGHT = 614 Km A=58.25° -z. SI o LU - • - - - .-• •  - .. ... . ; r-2: I 5 0 0 ~ . • • • : ; * . > . f - .. ' 1000- .... ,r ^ '• ..... ' 2023/22 2023/26 2023/30 &p=669 Hz Z ( V , " B ) = 84 0 HEIGHT = 684 Km A = 65.44° Fig. 6. 'Sonograms' from a mid-latitude pass on May 13, 1966. Structure in the form of rising tones is seen superimposed on a noise band which has a lower frequency cutoff near the proton gyrofrequency. Spectra from the same pass are shown in Figures S and 11. 275 2127/00 Op'277 Hi Z.(V,B) = I4' HEIGHT * 967 Km A = 2334* -25 500 460 - 1 2135/00 Qp = 46l Hi Z ( V , B) = 4I' HEIGHT =612 Km A = 32.84' -254 V 2129/00 flp=300Hl £(V,B)=4« HEIGHT = 859 Km A=23.02* 500 1000 0-/ ****** 2137/00 Op = 531 Hi £ (V ,B> = 52* HEIGHT =560 Km A = 38.45 1000 -25 2131/00 Op=339 Hi /.{7,'B') = I6' HEIGHT =763 Km A=24.26' 500 -25 597 1 I 'A \ , i lViA 2139/00 np = 595 Hi z (V ,B) = 6l* HEIGHT = 526 Km A = 45.09 1000 2133/00 fip = 395Hi <UV,B) = 29» HEIGHT =680 Km A=27.84* 500 —t i 1000 FREQUENCY (Hi) 645 2141/00 Ap>644 Hi ^(V.Bl'eg* HEIGHT «5II Km'A = 5l.9l* 900 1000 Fig. 7. Digitally computed spectra for a low altitude, low and mid-latitude pass on November 7, 1966. 8. Digitally computed spectra from a high altitude, mid-latitude pass on> October 14, 1966. 19(8/30 Op « 340 H i l ( V , B ) * 6 2 * HEIGHT .2327 Km A ' 8 4 . 3 0 * 900 1000 1926/30 O p . J 8 0 H i Z(7,B*)=90* j HEK3HT.«2I04 Km A=75.22 800 1000 1920/30 fip-330 H i H ( V , B ) = 84* HEIGHT » 2 4 3 3 Km A = 85.62* 800 1000 -28 383 H. i • 1928/30 ftp = 388 H i /(V,B)=93* HEIGHT = 1982 Km A =70.71* 300 - 2 3 r A.. •388 1922/30 O p » 360 H i /(7,B>)=86* HEIGHT 2330 Km A=83.26* 900 1000 -25 385 % ' 1930/30 Op=395 H i / ( V , B * ) = 9 € * HEIGHT =1857 A='66J02* 900 ~ r ' ' ' •woo -29 355 it 1924/30 flp« 370 H i 1 < V , B ) = 8 3 ° I HEIGHT « 2 2 2 0 Km A = 79.42* 800 + f 1000 -23 390 4. 1932/30 flp«400Hi l ( V , B ) * 9 9 * HEIGHT • 1728 Km' A = 61.22* 500 1000 FREQUENCY (Hi) Fig. 9. Digitally computed spectra from a high altitude, high latitude pass on July 30, 1966. 316/00 fip = 376Hl /(V,B)=9I* HEIGHT « 2160 Km A =70.4* 500 S22/0O flp=342 Hi /<V,B) = 96* HEIGHT = 2496 Km A = 79.17* 500 1000 25 517/00 fip=369 Hi Z<V,B) = 92* HEIGHT = 2237 Km A = 72.I3* -25 324/00 Op = 3J3 Z(V,B) = 97* HEIGHT = 2656 Km A = 00.28* 500 1000 500 1000 318/00 fip = 363 Hi /(V, B) = 93* HEIGHT = 2292 Km A = 73.80 Km 500 1000 -25 329 4^ 326/00 ftp=323 Hi /(7,B*)=98* HEIGHT = 2668 Km A=79.93* 500 1000 -29 320/00 fip=352 Hi /(V,B) = S4* HEIGHT • 2398 Km A=7631* 500 -254 328A» Op = 3i5Hl Z(V,B)=99* HEIGHT = 2741 Km A=70.40* WOO FREQUENCY (Hi) 500 1000 Fig. 10. Digitally computed spectra from a high altitude, high latitude pass on June 13, 1966. 659 2021/00 np-666 Hi i(V.S)-76" HEIGHT-600 Km A-56.41* 500 WOO 2022/00 flp-671 Hi *(V,B)-79» HEIGHT-630 Km A-60.12* 500 WOO 659 2023/00 np-671 Hi Z(V,B)-82* HEIGHT.665 A-63.68" - * 1 ' 500 ' ( - 1060 1 2024/00 flp'683 Hi *(V,B)-B5' HEIGHT-704 Km A-67.20" 500 WOO 2025/00 flp-665 Hi £<V,BI'63' HEIGHT.746 Km A-7056' 500 642 2026/00 Op-642 Hi iW.h-W HEIGHT -791 Km A'7371* 500 2027/00 np-642 Hi /(V,B)-92* HEIGHT >B39 Km A'76.77» 500 25 2028/00 flp-610 Hi W.B)-94* HEIGHT -890 Km A.79.54' 500 WOO 1000 1000 1000 FREQUENCY (Hi) 11. Digitally computed spectra from a low altitude, high latitude pass on May 13, 1966. 31 change in receiver output. If i t is the ELF noise band which controls the AGC, then the Rayspan (which has a 12 dB clear to black range) is capable of covering an input range of ~ 72 dB. However, i f large signals, at frequencies other than in the ELF band, enter the receiver system, then receiver gain may be sufficiently reduced so that the band is no longer v i s i b l e in the Rayspan record. Thus a region of observed non-occurence of the noise band may only represent a region of high signal strength at frequencies other than in the ELF range. A saving feature is that the distribution of AGC levels in invariant latitude and local m.ean time (Barrington and Harvey, (1969)) shows high AGC levels only at times when the ELF noise-band occurrence is high. There-fore, one may infer that AGC degredation of receiver gain cannot have substantially influenced the f i n a l ELF noise band occurrence pattern. In analyzing occurrence patterns of a phenomenon observed by s a t e l l i t e , some care must be taken in choosing the independent variables. A peculiar problem which arises from the orbital characteristics of Alouette II is that (for epoch 1966) the s a t e l l i t e is generally at high altitudes on the night-time side of the earth and at lower altitudes on the day-side. That this can be so, for a f u l l year, can be understood by examining the orbital elements of Table 1. It is seen that the argument of the perigee varies by 1.8922 degrees/day and this is closely matched by the variation of the longitude of the ascending node in the earth-sun system ( - ,7922°/day -360°/year = - 1.7778°/day). Thus there exists a strong correlation between the local mean time (LMT) at the s a t e l l i t e and i t s altitude, and the concomitant ambiguity in the occurrence data being analyzed cannot be resolved. In the analysis which follows, LMT is taken as an independent variable, but the connection with altitude should properly be remembered. 32 Analysis of the data on occurrence and sharpness of cutoffs, was performed by entering this data into storage in the computer along with the corresponding world map data for the s a t e l l i t e (geomagnetic, and > invariant latitude, local mean time, and the three hourly planetary magnetic index, Kp). The morphology results shown in Figures 12, 13 and 14 were obtained by sorting this merged data. The percentage occurrence of the noise band as a function of local mean time and invariant latitude (Figure 12), i s computed by counting the number of times the noise band i s seen in each LMT (3 hr.) -. invariant latitude (10°) block, and dividing by the number of times the s a t e l l i t e is in the LMT — i n v a r i a n t latitude block. These results were then contoured. Percentage occurrence of the sharp lower (and upper) frequency cutoffs in Figures 13a and 13b is calculated by obtaining the number of times there was a sharp cutoff of the band in each LMT (6 hr.) — invariant latitude (10°) block and dividing by the number of times a noise band was observed i n the block. The results shown i n Figure 14 have been derived in a similar manner except that only Kp was subdivided. Since the s a t e l l i t e generally goes through 10 degrees of invariant latitude within 4-5 minutes, then the results of each observation on the noise band, for any given pass, w i l l be tabulated in a different c e l l of LMT — invariant latitude space. Thus, Figures 12 and 13 are derived from 971 independent observations of the noise band. This is not the case for the Kp variations of Figure 14. Since Kp is constant for each pass, this figure represents averages over ~ 230 s a t e l l i t e passes. Summarizing the results of Figures 12 to 14: (a) The noise band appears, on the average, 57% of the time. There is a marked diurnal variation in the occurrence. Near P E R C E N T A G E OCCURRENCE OF ELF NOISE BAND LOCAL NOON 0 LOCAL MIDNIGHT 12. Occurrence pattern of the noise band plotted as a function of invariant latitude and local mean time. Contours are in steps of 10% and give the percentage of the time the noise band was observed. IOO-I UJ 80-O < 60-2 UJ o or 40-UJ Q. 20-0 — L M T 1200-2000 L M T 2000-0400 L M T 0400-1200 ~ l I — I — I — r — I — 1— I — I 10 20 30 40 50 60 70 80 90 INVARIANT L A T I T U D E Fig. 13a. Occurrence of sharp lower frequency cutoffs as a percentage of the times for which the noise band was observed. This is plotted versus invariant latitude with local mean time as a parameter. IOO - i UJ 80-< 1- 60-2 UJ <_> cc 40-UJ 0_ 20-4 L M T 1200-2000 L M T 2000-0400 L M T 0400-1200 1 1 1 1 [—1 1 1 1 10 20 30 40 50 60 70 80 90 I N V A R I A N T L A T I T U D E Fig. 13b. Occurrence of sharp or almost sharp upper frequency cutoffs as a percentage of the times for which the noise band was observed. 36 dawn, occurrence is highest (100%) at the low latitudes. The zone of maximum occurrence broadens to higher latitudes by noon, and has a maximum at 60° - 70° invariant latitude and LMT 12 to 16 hours. Occurrence abruptly decreases near dusk. The occurrence of sharp lower cutoffs as a percentage of noise bands observed follows a pattern similar to (a). Occurrence is generally higher during the local daytime (associated with low altitudes). Occurrence of sharp (or almost sharp) upper cutoffs occurs most often at very high latitudes, and shows l i t t l e dependence on LMT (or, what is the same thing, on altitude). \ 37 CHAPTER 5 THEORY OF PROTON CH WAVES Quite comprehensive numerical studies have been made for the dispersion of electrostatic electron CH waves. To date though, no extensive numerical results have been reported for the proton CH waves. This i s partly due to the fortunate circumstance that, for waves propagating perpendicular to the ambient magnetic f i e l d , results derived for electron CH waves at frequencies of the order of the electron gyro-frequency (fie) can be applied almost directly to proton CH waves at frequencies « fie. The following w i l l include a derivation of the isomorphism between the proton and electron CH dispersion relations with considerations of the effect on this isomorphism of small deviations from perpendicular (i.e., oblique) propagation. The literature on perpendicular and oblique propagation, and effects of collisions, for electron CH waves w i l l then be reviewed, and the results applied to proton CH waves. A few remarks on in s t a b i l i t y in these modes w i l l be added. To show the equivalence that can be set up between proton and electron CH waves, the general dispersion relation for electrostatic, waves (Stix, (1962)) is used. In a magnetized by-Maxwellian plasma, this is k? (5.1) where D. J A n T i + i (cu - k.. V.' + nfi)T, - nfiT-k. 11 2KT m 11 v. 38 F (an) o h k l l - a 2 j n < c/ \ TT e n + 2 i S(an) S (an) 5 - a2 e n J ran 2 e dt , the error function I n ( Z ) tl\ Z COS v _1 e cos (nO) d8, the modified Bessel f c t n TT an 0) - k n V u + nfi "11 m l i e 2K T 11 Q2m (gyroradius) 2/(wavelength) ' Z-eB 3 o mc , the gyrofrequency 4TT n. Z-e 2  -1 J , the plasma frequency ; and B i s the ambient magnetic f i e l d ; k _-, and k i are the p a r a l l e l and o \^ t h e co perpendicular wavenumbers and^frequency of a wave of the type E(u),k) e i ( - ' r " W t ) ; m. , T,VJ/, T./ J / , ix, , • V,., , and Z 4e (j) T (J) 3 ' .4- ' ' l l » "j» " l l j are r e s p e c t i v e l y , the mass, perpendicular and p a r a l l e l temperature, th number density, p a r a l l e l d r i f t v e l o c i t y , and charge, of the j type of charged p a r t i c l e i n the plasma; K and c are the Boltzmann constant, and v e l o c i t y of l i g h t . Units are cgs e l e c t r o s t a t i c . In p a r t i c u l a r i z i n g the general dispersion r e l a t i o n Eq. (5.1) to proton CH waves, the following assumptions are made: CO « fie (and therefore, an g >> 1 for |n| _> 1) , X << 1 (that is, the electron gyroradius is small with respect to a wavelength), 39 V.., =0 (there are no parallel d r i f t velocities), 11 p,e N v ' ~ TT..^ (the electron and proton parallel temperatures are of the same order). The electron portion of the dispersion relation can then be calculated to f i r s t order in X : 2 ' 2 e 2 TT m e e K T (e) n i / \ • " . h -aoe' S1 (aoe) + x TT aoe e 11 _ TT2 m , e e . + T (e) Xe K i t T, (e) (e) + T (e) T (e) ' l l " 'J-11 [ S1 (aoe) + iTT% aoe e -aoe T, (e) r ( e ) 11 (5.2) For Xp < 1 , a similar expansion could be made for the proton portion of u -aoe 2 the dispersion relation. The factor {S'(aoe) + i i r 2 aoe e } in the f i r s t order term of Eq. (5.2) may be neglected in comparison with the zeroth order term. Also, 2 TT m e e K T (e) TT2 m P P 11 K T (p) 11 and aoe = OJ 11 m 2 K T-(e) 11 m ~ • ( -J*. ) aop . The behaviour of S'(z) for real z is S'(z) = -1.0 for z = 0 0.0 for z = .924 0.285 for z = 1.502 1 2z' for z > 2 . From this, i t i s seen for small deviations of from zero, that 1 < < aoe < < aop '. AO Thus, for small and < 1 , the effect of the {S'(aoe) + iTT aoe e a ° e } term w i l l be even greater than the corresponding proton term. As a result, i t is expected that the effects of non-zero k ^ (to be discussed) on proton CH waves w i l l be more pronounced than i n the corresponding electron CH case. Proceeding now to take the limit as k ^ ->• 0 , Jj_ ~ , then Eq. (5.2) becomes: TT2 TT2 m X TT2 k 2 ~ D = e  e . r£ = (5.3) ft2 e K T ( e ) fl2 e e Then Eqs. (5.1) and (5.3) may be combined to give k , 2 ( l + ^e/ft 2) . 3J1 = _ D (5.4) 6 TT 2 P P The corresponding dispersion relation for electron CH waves (neglecting a l l ion effects) i s ft2, 2 k 2 = - D (5.5) e/TT 1 e e Note that the terms D and D contain no reference to particle density. e. p Then i t is seen that an isomorphism between Eqs. (5.4) and (5.5) may be set up as shown in Table 3. The frequency ^ H ^ S re^erre^ t o i - n t n e literature as the lower hybrid resonance frequency. TABLE 3 ISOMORPHISM BETWEEN ELECTRON AND PROTON DISPERSION RELATIONS Electron CH waves Proton CH waves < T ( E ) K T ^ -< = : — m m e P « e < • % TTe <- ~ > (1 + 1 ) ~ \ # H > \ " p V (for TT2»fi2) V r p 41 Dispersion curves for perpendicularly propagating electron CH waves have been computed by Crawford (1965). These curves, particularized to proton CH waves, are shown in Figure 15. These curves have been confirmed for electron CH waves by a number of experimenters using laboratory plasmas (Harp (1965); Mantie (1967)). Any small non-Maxwellian component w i l l slightly alter the dispersion. The values of T r 2 / f i 2 to L. H. p be expected at s a t e l l i t e altitudes are in the range 100 to 300. Summarizing these curves, i t is seen that propagation can occur in modes between each harmonic of the proton gyrofrequency. The solutions of Eq. (5.4) for co and k are pure real and therefore propagation is unattenuated in the perpendicular direction. Phase velocities in the f i r s t pass band (Q p _< w < 2Qp) are of the order of the proton thermal velocity _ rK TJh% v ol , which for s a t e l l i t e altitudes is in the range of 3-5 Km/sec. These curves also indicate that group velocity (dto/dk__) w i l l approach zero at the bounding harmonics of the gyrofrequency. Tataronis, (1967) has presented an extensive study of perpendicular and oblique propagation of electron CH waves under Maxwellian and non-Maxwellian conditions. Figure 16a shows the dispersion for oblique propagation of proton CH waves (from Tataronis via Table 3). The important results here are that another set of modes is introduced (mode 2), and that both sets of modes are heavily attenuated for non-zero ^11" decrease in energy for the f i r s t set of modes (directly related to the perpendicularly propagating mode) is ~ 10 dB per cyclotron period for propagation at angles only 5° off perpendicular. Attenuation of the second set of modes is even greater (~ 25 dB/cyclotron period). For this reason, CH wave propagation in the ionosphere most likely w i l l occur only with wave normals nearly perpendicular to the geomagnetic f i e l d . It is i — r 2 3 4 Fig. 15. Dispersion curves or perpendicularly propagating proton CH waves in a - Maxwellian plasma. These are based on results of Crawford (1965) and particularized to frequencies less than the electron cyclotron frequency. A sequence of pass bands are defined, the n t J l mode being confined to frequencies between fi and (n + l ) f i p . In the ionosphere T T £ H / p 2 w i H usually be in the range 100 to 300. Fig. 16a. 0 0.25 Q50 0 0.23 (feAR)-REAL PART (kjR) IMAGINARY PART ^ Fig. 16b. ... Fig. 16a. Oblique propagation of proton CH waves in a Maxwellian plasma, TT£ „ /ft2 - 20.0, Li .ii, p and (kj v Q l/fi ) = 3.0. Mode 1 corresponds to the perpendicularly propagating modes, whereas Mode 2 is a new set which exists only for oblique propagation. Curves are based on Crawford (1967). Fig. 16b. Dispersion curves for perpendicularly propagating proton CH waves in a Maxwellian plasma including collisions, IT, 2 H /ft2 = 5.0 and the collision Li .n • p frequency v is v/ftp = .05. Th.es2 curves are based on results of Tataronis and Crawford (1965). 44 also to be noted that attenuation for oblique propagation w i l l be, in fact, greater than indicated by Figure 16a, since the electron non-co l l i s i o n a l (Landau) damping represented by the term { + irr 2aoe g. } in Eq. (5.2) has been neglected. If a small number of collisions are included in the dispersion relation for perpendicularly propagating CH waves in a Maxwellian plasma, then damping as indicated in Figure 16b is obtained. This figure, sketched from Tataronis and Crawford (1965), is for the case of V/fip (i.e., c o l l i s i o n a l frequency normalized to the proton gyrofrequency) equal to .05. Ionospheric value of .001 are quite r e a l i s t i c , and thus this damping can be expected to be important. The c o l l i s i o n a l damping is an inverse function of the group velocity (dto/dk|), and, as a result, i t increases for the higher harmonics. The perpendicular and oblique propagation of ion CH waves under a variety of non-Maxwellian conditions has been investigated by Hall, Heckrotte and Kammash, (1965). The i n s t a b i l i t i e s found by these authors were classified into two groups: (1) type-A i n s t a b i l i t i e s which occur when propagation is almost exactly perpendicular and the velocity distribution fj_(vj_) has at least one maxima at other than zero velocity; (2) type-B in s t a b i l i t i e s which require that the propagation vector have a significant parallel component, but the transverse velocity distribution can decrease monotonically. The relations developed by Hall et a l are complicated; no detailed numerical computations of the dispersion is made by these authors. However, i t appears that the type-B i n s t a b i l i t i e s are related to the previously mentioned additional modes for oblique propaga-tion of CH waves. 45 A general.requirement derived by Hall et al for type-A insta-b i l i t i e s in the f i r s t pass band is that < P1(3) > = 2TT /•ft > CM P r 6ft ^  dp J 2(3) f; [--^j > o where, J-^ (3) = the Besel function of 1st order, ki vt P .f,(v f) = the transverse velocity distribution. Considerations of an elementary physical model for perpendicularly propagating electrostatic waves in a p r o t o n gas indicated the physical meaning of < P^(3) > to be a proportionality constant determining the i n i t i a l direction of energy flow from the particles to the wave. A necessary condition for i n s t a b i l i t y is clearly f_{(v,) > 0 for some v,. This, of course, is not true for a Maxwellian distribution. However, f,(v,) is expected to have positive derivative at some velocities in the ionosphere, where a high velocity suprathermal distribution of particles is superimposed on the background Maxwellian distribution. This w i l l be further discussed in the next chapter. In fact, one of the non-Maxwellian situations discussed by Hall et a l consists of such an approximation to the ionosphere. A group of high energy (hot) ions with anisotropic velocity distribution i s superimposed on a lower energy (colder) istropically distributed group of the same ion species. The hotter particles can then give up energy to the colder back-ground particles via the mechanism of electrostatic CH waves, The transverse energy of the hot particles is assumed to be much greater than that of the colder particles. As a result, a type-A instability is found. For the idealized situation of a delta function for the transverse distribution 46 of hot ions, then, at frequencies slightly above the gyrofrequency TT 2 i n s t a b i l i t y occurs even at very low plasma densities, ( p/fip < < ; 1), and for small fractions of the total particle density in the hot distribution. 47 CHAPTER 6 APPLICATION OF THE THEORY OF PROTON CH WAVES TO THE OBSERVATIONS 6.1 E x p l a n a t i o n f o r S p e c i f i c Observations on the Noise Band In this section, i t w i l l be shown that most of the particular observations on the noise band may be explained as due to proton CH waves propagating in a primarily Maxwellian ionosphere. The sharp, lower frequency cutoffs .and harmonic structure are interpreted in terms of waves propagating perpendicularly to the geomagnetic f i e l d and both inward and outward with respect to the earth, and the concomitant accessibility conditions. This model is f i r s t developed, and then applied to the spectra of Figures 4 to 11. Waves propagating within the f i r s t pass band (i.e., with reference to Figure 15, in the frequency range of one to two times the proton gyro-frequency, Qp) w i l l f i r s t be considered. Since waves propagate nearly unattenuated in the perpendicular direction, locally observed waves may have been produced and amplified at some distance from the s a t e l l i t e . With reference to Figure 17, consider the s a t e l l i t e at a low or mid-latitude position such as A. A l l waves propagating upwards to A (i.e., excited in a zone of larger gyrofrequency than at A) may reach A except those produced at frequencies greater than 2 f i ^ . Waves of frequency 2fi ^  w i l l tend to zero group velocity as they approach the level of A. Similarly, waves propagating downwards and excited above A can propagate to A i f they were produced at frequencies greater than ^p^« Waves at frequency fip^ w i l l tend to zero group velocity as they approach A. This situation would lead to an observed spectrum'of power between fip^ and 2fip^ with cutoffs (i.e., decreases in noise intensity) on each side of the band. The sharpness of Fig. 17. A cross-section of the earth's magnetic field, reproduced to help visualize the effects of accessibility on proton CH waves. 49 the cutoffs may be l a r g e l y c o n t r o l l e d by the rate at which c o l l i s i o n a l damping increases at frequencies near the gyrofrequency harmonics. The above comments on propagation of CH waves i n the f i r s t mode may also be applied to waves i n the second mode (2fT <_ oJ <_ 3Qp) , leading to a band of noise l i m i t e d by the second and t h i r d harmonic of ftp. As mentioned i n Chapter 5, the higher modes s u f f e r i n c r e a s i n g l y from c o l l i s i o n a l damping, and thus may not be observed as frequently as f i r s t pass band noise. At very high l a t i t u d e s such as point B i n Figure 17, a c c e s s i b i l i t y conditions cannot be expected to play such an important part i n the shape of the observed spectrum. Waves that propagate perpendicular to the ambient magnetic f i e l d w i l l not accumulate at flpg and 2ftpfi since they are propagating i n a d i r e c t i o n of approximately zero gradient i n the magnetic f i e l d . Under these circumstances, i t i s expected that the power spectrum obtained at a point such as B w i l l represent the t o t a l spectrum of waves produced i n the v i c i n i t y and not appreciably a l t e r e d by a c c e s s i b i l i t y conditions. C o l l i s i o n a l damping may play a more important part i n determining the f i n a l shape of the spectrum. The above discussion of noise band cutoffs i s based on the assumption that propagation e f f e c t s dominate. This i s reasonable regardless of the type of i n s t a b i l i t y that excites the waves. For i f the i n s t a b i l i t y mechanism i s absolute ( i . e . , temporal growth of the waves at every point to the non-linear l i m i t ) , then there may be am p l i f i c a t i o n over only a small portion of the frequency rang'e covered by a pass band of the waves; i n which case, propagation effects-may dominate. The other a l t e r n a t i v e , convective i n s t a b i l i t y (growth of p a r t i c u l a r wave fronts as they propagate), i s also compatible with the cutoff explanations. 50 It should also be noted that the cutoffs need not occur exactly at the calculated gyrofrequency, since, as previously mentioned, the dispersion curves of Figure 15 hold s t r i c t l y only for a Maxwellian plasma, and thus minor deviations could possibly be due to the small non-Maxwellian component of the ionosphere. Upper frequency cutoffs (used in the phenomenological sense) may also be below 2fip for low altitude s a t e l l i t e passes, since, i f production and amplification occurs primarily near the gyrofrequency, then the maximum frequency that may be produced w i l l occur near the gyrofrequency at the base of the ionosphere, and this frequency may be less than twice the gyrofrequency at the s a t e l l i t e . Another factor that' may be expected to play an important part in observations of CH waves with a s a t e l l i t e is significant Doppler shifting of the spectrum. From the dispersion relation depicted in Figure 15, the wave phase velocity (W^k,) near the center of the f i r s t pass band is of the order of the thermal velocity (v0-r_) . For a typical ionospheric temperature of 1500° K., vQj_ is 4.3 Km/sec, while the s a t e l l i t e velocity is 6-7 Km/sec. Thus, significant Doppler shifting would be expected for a l l waves that have wave normals with appreciable components along the s a t e l l i t e velocity vector V. Since the waves must propagate in the plane perpendicular to the magnetic f i e l d , the most pronounced effects are expected to occur when the angle between the velocity vector of the s a t e l l i t e and the geomagnetic f i e l d (Z,(V,B)) is equal to 90°. For a uniform azimuthal distribution of wave normals, only a small percentage of these waves would not be significantly shifted, and thus the noise band would be effectively 'smeared' out. The spectral features of the noise in Figures 4 to 11 as noted 51 in Chapter 4 (4.1) w i l l now be interpreted. Each of the predictions from the above model w i l l be considered and the evidence in the Figures for occurrence of the phenomena enumerated: (1) Lower frequency cutoffs near ftp. A l l the noise bands in the Figures, covering a broad range of latitude and s a t e l l i t e altitudes, have lower frequency cutoffs within 30 Hz of the calculated value. These cutoffs cover the range of about 200 to 650 Hz. This result i s expected for CH waves in the lowest pass band. (2) Harmonic structure. Figure 8 shows a noise band with upper frequency cutoff near 3ftp. A change in noise level near 2ftp is also evident. These observations may be explained as CH waves in the f i r s t two pass bands.' Figure 9 — GMT 1918/30-1922/30, and Figure 10 — GMT 322/00-326/00, indicate more clearly an upper cutoff to the noise band near 2ftp and then further noise i n the second pass band. These observa-tions are for relatively high s a t e l l i t e altitudes. (3) Upper frequency cutoff below 2Qp. The noise bands from s a t e l l i t e passes at altitudes below 1000 Km generally have upper frequency cutoffs below 2fip (Figures 7 and 11). This may be the maximum frequency of waves produced below the s a t e l l i t e . (4) Doppler shifting. Figure 11, GMT 2024/00-2025/00, exhibits a smearing out of the noise band near the calculated position where L (V,B) 90°. The situation is complicated by the unexplained noise which comes down from higher frequencies near i .(V,B) = 90° (seen clearly in Figure 5). Figure 9 . — GMT 1926/30 - 1928/30, and Figure'10 -- GMT 316/00 - 318/00, also show evidence of 'smearing'. This phenomena may be interpreted as due to Doppler shifting. 52 (5) Shape of spectra changes above A = 80°. Both Figure 9 — GMT 1913/30 - 1922/30, and Figure 10 -- GMT 324/00 - 326/00, show evidence of a rounding off of the lox^er frequency cutoff relative to the spectra at lower latitudes. This may be due to the decreased importance of accessi-b i l i t y conditions at these high latitudes. An additional phenomena of the noise band which may be explained as due to perpendicularly propagating CH waves is "spin modulation". When the antennas are aligned with the magnetic f i e l d , the wave electric f i e l d and i t s gradiant w i l l be perpendicular to the antenna and thus no signal w i l l be observed. This agrees with the observations. However, there is a definite possibility that the spin modulation is due to:the orientation-sensitive gain'of the receiver antennas, as discussed i n Section 2.2. An observation, which is not f u l l y explained by the simple model of noise bands due to steady production and amplification of noise, is the structure shown in the Sonograms of Figure 6. Swift (1968) has suggested that similar dispersion (i.e., rising tones) may be due to a convective instability of CH waves propagating from regions of smaller to larger magnetic fields, and is possibly associated with micro-bursts (Parks, (1967); Oliven and Gurnett, (1968)). Another consideration on this noise band is that i t is very d i f f i c u l t to argue conclusively that the observed noise band is in fact an ambient phenomena and not merely excited by the motion of the s a t e l l i t e through the ionospheric plasma. The necessity for "considera-tions of this nature has been discussed by Scarf et a l , (1968). The pronounced altitude and latitude dependence of the shape of the spectrum, and also the ~ 50% non-occurrence of the noise band, support the assumption that the waves are ambient. 53 6.2 Occurrence P a t t e r n o f the Noise Band Without a d e t a i l e d knowledge of the energy spectrum of the non-thermal p a r t i c l e and the generation mechanism possibly operative for proton CH waves, the occurrence pattern of the noise cannot be f u l l y explained. However, a few remarks may be pertinent. From Eq. (5.6) i t has been noted that a general requirement for type-A i n s t a b i l i t i e s i s < P^(3) > > 0, and the rate of energy t r a n s f e r from the p a r t i c l e s to the waves i s proportional to t h i s quantity. This may account for the pronounced d i u r n a l v a r i a t i o n of occurrence, i . e . , occurrence increases markedly at dawn and continues high throughout the day. In t h i s connection, a number of remarks have been made by Brice (1964), who discusses a d i u r n a l peak occurrence of chorus between dawn and noon. This d i u r n a l peak i s suggested to be due to adiabatic compression of the magnetic f i e l d l i n e s by the s o l a r wind and the sub-sequent increase i n the transverse energy of the trapped p a r t i c l e s . This process would thus increase < P^(3) > during the day, and the proton CH waves would have the observed d i u r n a l occurrence v a r i a t i o n . Another factor which could contribute to daytime observation of the noise band i s the a s s o c i a t i o n of daytime LMT with low a l t i t u d e of the s a t e l l i t e . < P^(3) > i s expected to be larges t at low a l t i t u d e s , since t h i s i s where the perpendicular energy of the suprathermal p a r t i c l e s maximizes. Since the second adiabatic i n v a r i a n t i s approx-imately conserved, then for each charged p a r t i c l e — = = constant D where v, = perpendicular v e l o c i t y of a p a r t i c l e gyrating around the magnetic f i e l d l i n e , B = the geomagnetic f i e l d induction. 54 Since B maximizes a low a l t i t u d e , so does v, . In regard to the occurrence pattern of sharp lower cutoffs (Figure 13a), i t i s seen that these are also most probable during the daytime (or, at the associated low a l t i t u d e s ) . Considering the model for lower frequency cutoffs due to a c c e s s i b i l i t y conditions on inward propagating waves (Section 6.1), then the occurrence pattern may be due to the p o s s i b i l i t y of wave production over a wide range of a l t i t u d e s above the s a t e l l i t e . I f the waves can propagate over large distances, the p r o b a b i l i t y of lower frequency cutoffs may besa function of how low the s a t e l l i t e i s i n the region of wave production. This gives a corresponding maximum occurrence during the daytime (low a l t i t u d e ) passes. The v a r i a t i o n of upper frequency cutoff (Figure 13b) and noise band occurrence with Kp (Figure 14) are somewhat abstruse. 55 CHAPTER 7 OTHER POSSIBLE EXPLANATIONS FOR THE NOISE BAND 7.1 R e s u l t s of a F u l l Wave Treatment of the B e r n s t e i n Modes In Chapter 5, theoretical studies of electrostatic waves, propagating in the vici n i t y of the proton gyrofrequency, were reviewed. Recently, a f u l l wave (i.e., the electrostatic approximation, E = - V(j), is not made) study of waves propagating perpendicular to the ambient magnetic f i e l d in a collisionless Maxwellian plasma and at frequencies much less than fie, has been completed (Fredricks, (1968a) and (1968b); Fredricks and Scarf, (1968)). These authors termed the resulting wave 'generalized ion Berstein modes'. Under the condition k V V fi I p ) > > 1, (7.1) i t was shown that the electrostatic dispersion results are quite valid i f p +" ^ < < 1, (7 .2) (1 + 23+) k 2 R2 where R, E V o i / f i . p In the ionosphere at s a t e l l i t e altitudes 2 P + E V f i 2 103 P e •_ 8TT n K T _q 6+ = P . , 10 * B 2 2 *6 -1 For the range of frequencies of current interest, co ~ 10 sec ., and 56 therefore Eq. (7.2) reduces to a 2 < < 1. The electrostatic results of Figure 15 w i l l necessarily be accurate i f k-L v O J ft P 3 . The experimental studies of electron CH waves by Mantie (1967) indicated, under similar conditions of 3 and P for electrons, an excellent agreement between experiment and electrostatic theory for k v ,/ft as small as 0.1. o- p For a low 3+, collisionless Maxwellian plasma of protons and electrons, Fredricks and Scarf (1968) suggested the qualitative dispersion curves shown in Figure 18. It is seen that for any particular mode, there are three regions of k_t_ where the group velocity ~ approaches zero. The explanation for noise bands with upper and lower frequency cutoffs, as outlined in Chapter 6, is based on the zeros of the groups velocity for f i n i t e values of k^ (i.e., on the electrostatic portion of the curves). However, the possibility of noise bands with lower and upper frequency cutoffs could also be supported by the primarily electromagnetic portions of the curves. Again there is a range of k^ bounded by zeros in the group velocity. The same conditions on accessibility conditions may be applied as for the electrostatic waves. A number of d i f f i c u l t i e s are associated with such an electro-magnetic origin to the observed noise bands. Possibly, collisions w i l l lead to interaction between waves in the different electromagnetic modes causing the cold plasma treatment to be a good approximation to the dispersion. Further study is needed in order to estimate the / COLD PLASMA k EXTRAORDINARY WAVES Fig. 18. Schematic of 'generalized ion Bernstein mode' dispersion. These curves are based on the results of FredricJks and  Scarf (1968), and are for a low 3+ plasma. Note the change in scale on the horizontal axis. 58 co l l i s i o n a l effects. In addition, a basic problem with an electro-magnetic origin of the noise is the observation of dependence of the ->• -> • spectral shapes on L (V,B). Phase velocities for electromagnetic . waves in the ionosphere are ~ IO1* km/sec; this is much larger than the s a t e l l i t e velocity, and thus such waves cannot be significantly Doppler shifted. Thus, i t appears that at least some of the waves observed must be electrostatic. 7.2 L=0 Theory Noise bands of a nature very similar to those investigated in this paper have been reported by Burns (1966) and Gurnett and Burns (1968) from Injun III data. These authors reported a lower frequency cutoff . which usually occurs below the proton gyrofrequency, and noted particularly the systematic decrease of this frequency with increasing altitude. Gurnett and Burns proposed an explanation for such a noise band based on propagation effects on right-hand polarized (whistler) electro-magnetic waves. The existence of whistler mode waves of appropriate frequency and propagating down magnetic f i e l d lines from high altitudes, was postulated. The effects of increasing proton gyrofrequency and decreasing relative abundance of hydrogen seen by the downcoming wave may be understood in terms of the theory outlined by Gurnett et a l (1965); these authors considered propagation of ELF electromagnetic waves in a plasma of electrons, H +, He +", and D +. Whistler mode (right-hand polarized) waves propagating parallel to the geomagnetic f i e l d and at frequencies slightly less than the local proton gyrofrequency can 'cross-over' to the left-hand polarized mode. -These waves are reflected at a lower altitude corresponding to the cutoff frequency of left-hand waves 59 (L=0 frequency, in the notation of Stix, (1962)). As i t turns out, L=0 frequency is a decreasing function of altitude. Thus, higher frequency energy is reflected at lower altitudes and vice versa. These conditions lead to a noise band with a lower frequency cutoff at the local L=0 frequency near the s a t e l l i t e . Noise seen at the cutoff is due to waves being reflected at the L=0 frequency near the s a t e l l i t e . Noise seen in the band at higher frequencies would be due to waves which propagate to altitudes below the s a t e l l i t e and then are reflected. However, the L=0 frequency varies almost linearly in the frequency range (ft = proton gyrofrequency) to ft for relative abundances of 4 P V hydrogen ion from ~ 1.0 to ~ 0.0, respectively. This variation i s f a i r l y independent of the relative abundance of 0+. Gurnett and Burns explain observed noise band cutoffs which occur near ft in the protonosphere P v v (where relative abundance of H is - 1 and consequently the L=0 frequency is near ^p/4) as possibly due to waves propagating downwards i n i t i a l l y at an appreciable angle to the magnetic f i e l d , and thus being reflected before reaching the L=0 level for each frequency component. A sharp cutoff of such a noise band would require considerable order in the i n i t i a l (non-zero) wave normal angles. On the basis of the above theory i t is thus d i f f i c u l t to interpret the observations presented in this thesis. Besides the problem of order in the wave-normal angle distribution, apparent Doppler shifting and the upper frequency cutoff of the noise band are not accounted for. 60 CHAPTER 8 SUMMARY AND CONCLUSIONS ELF noise bands i n the ionosphere which have a .lower frequency cutoff near the l o c a l gyrofrequency have been investigated. Data from the Alouette II s a t e l l i t e was spectrum analyzed, using both e x i s t i n g analog methods and a d i g i t a l s p e c t r a l analysis program. The d i g i t a l s p e c t r a l analysis of the noise band indicated that, at a l l s a t e l l i t e a l t i t u d e s and l a t i t u d e s , the sharp lower frequency cutoffs conform to the calculated proton gyrofrequency to within ~ 30 Hz. This r e s u l t i s taken, as strong support for the hypothesis that the noise band was due to proton CH waves. In addition, t h i s hypothesis i s supported by the consistent manner i n which the following aspects of the noise band may be interpreted: (1) the sharp upper and lower cutoffs of the noise band, as due to a c c e s s i b i l i t y conditions and accumulation of waves where the group v e l o c i t y goes to zero, (2) the change from sharp to d i f f u s e lower cutoff of the noise band when the s a t e l l i t e t r a v e l s to very high l a t i t u d e s , as due to changing a c c e s s i b i l i t y conditions, (3) smearing out of the spectrum at s a t e l l i t e positions where L (V,B) ~ 90, as due to Doppler s h i f t i n g of a large portion of the wave spectrum, (4) the upper frequency cutoff of the band below 2Q^ which occurs at low l a t i t u d e s and a l t i t u d e s , as due to the region of production of the waves being p r i m a r i l y at higher a l t i t u d e s . The occurrence patterns of the noise band were derived from 971 observations on the noise band spaced at 4-5 minute i n t e r v a l s , using 61 available Rayspan data for epoch 1966. The diurnal variation with maximum occurrence on the dayside of. the earth was interpreted in terms of the regions where transverse energy of suprathermal particles maximizes, and therefore probability of excitation of perpendicularly propagating CH waves is greatest. The cyclotron harmonic wave hypothesis thus provides a consistent explanation for a large portion of the observations on the noise band. Future work is indicated, particularly in regard to studying CH wave ins t a b i l i t i e s using r e a l i s t i c models of the suprathermal velocity distributions. It would also be useful to investigate the effects of collisions on the f u l l wave treatment of Bernstein modes in order to adjudge the importance of gyrofrequency effects indicated by Fredricks (1968). The work presented in this thesis also directs attention to the continuing need in s a t e l l i t e VLF experiments of both a magnetic loop antenna and electric antenna on the same vehicle, in order to sort out electromagnetic and electrostatic waves in the ionospheric plasma. 62 APPENDIX SPECTRAL ANALYSIS FROM THE DIGITIZED RECORDS The power spectrum estimation was carried out using methods obtained from Blackman and Tukey (1959) and Bingham, Godfrey, and Tukey (1967). This section w i l l aim at tying together some of the ideas in these references, and at outlining the practical analysis as performed. It is known from continuous record analysis that C+T/2 -T/2 where P(f) = the power spectrum of the stationary random process, f = frequency T = length of the record. Define the functions D_^(t) subject to the restrictions that i t be piece-P(f) = limit 1 x ( t ) . J f i 2 l T f t dt (A.I) wise continuous and D. (t < 0) = D. (t > T ) = 0 , D. l — i — n ' l = i , 2 J for some value of T^ representing the length of record to be analyzed. Such functions w i l l be called "data windows" on account of the following definition. Let X ±(t) E D ±(t) • X(t) (A. 2) Also, define P.(f) = — T n n D.(t) • X(t)-£~ 2 T r i f t dt (A. 3) G(t) X(t) T 2 • n 0 0 < t < T — — n t < 0 or t > T (A. 4) 63 The Fourier transforms of D. and G are then 1 Q ±(f) ,4 v „-i2TTf t . D^t) I dt , (A.5) and . - ( - C O S(f) + a G(t) r 1 2 7 T f t dt (A.6) where the subscript 'a' stands for apparent. Considering ergodicity, the ensemble average of S^(f) (over an i n f i n i t e set of processes identical to that which produced X(t)) is avg ' { S a ( f ) } " where S(f) limit 1 X •+ OO i s(f) f+Tn/2 = P(f) x(t) i T i 2 7 T f t dt (A.7) (A. 8) n J-Tn/2 Using the convolution theorem with Eqs. (A.3), (A.5), and (A.6) .-(-OO W - f n J Q'.(f.-f) • S (f) df l 1 a (A.9) for every frequency f^. This relation exhibits p ^ ( f ^ ) a s the result of looking at a Fourier transform of the truncated process X(t) through a f i l t e r (called a 'spectral window') Q (f -f) °f variable transmission properties. In most cases i t is desired to look at S (f) over a narrow band of frequencies surrounding f^. Two particular data windows are ' 1 , | t | < T D'(t) = o . I t | n/2 C to n/2 (A.10) 0 , | t | > T n and Dl ( t ) - | 64 i _i_ 2-rrt 1 + cos ^ ~ n Their Fourier transforms are: (A.11) 0 n TT f T and, Q'(f) = ~ Q ^ ( f ) + ~ i tt Q l ( f + Q'(f •) n 0 n (A. 12) Use of e i t h e r D^ or the a l t e r n a t i v e i s c a l l e d 'hanning'. The transform pairs are shown i n Figure 19. It is. recognized that applying a data window of the type D Q ( t ) i s equivalent to taking a s i g n a l of f i n i t e length T . Reference to the f i g u r e shows that this 'chopping-off' of the s i g n a l corresponds to a s p e c t r a l window which has r e l a t i v e l y large side lobes. Thus, any p a r t i c u l a r s p e c t r a l estimate w i l l be considerably influenced by the rest of the spectrum. This s i t u a t i o n can be improved by applying a hanning window. This i s done by m u l t i p l y i n g into the s i g n a l or by transforming the unmodified data D Q ( t ) • X(t) and convolving S^Cf) with Q 2 ( f ) using the ' s h i f t e d ' Eq. (A.13). From the f i g u r e , i t i s seen that Q ^ f ) w i l l lead to s p e c t r a l estimates integrated over a larger i n t e r v a l around a p a r t i c u l a r frequency than would r e s u l t from Q Q ( f ) . However, side lobe e f f e c t s are reduced s u f f i c i e n t l y for present purposes. I t i s convenient that the concepts and equations of spectrum t So subscripted to conform with Blackman and Tukey (1959). ft The D! and Q! may be converted to the corresponding unprimed functions using^'the r e l a t i o n s : D!(t) = D.(t + T n / 2 ) , Q!(f) = £ 1 7 T f T n • Q . ( f ) . 1 1 This gives, Q 2 ( f ) = f Q D ( f ) l 4 Q o ( f , 1_) + Q j f _ 1_) n n -1 Fig. 19. Data windows and corresponding spectral windows, adapted from Blackman and Tukey (1959). 66 analysis on continuous records may be converted directly into forms appropriate for d i g i t a l spectral analysis of discrete time series. The f i n i t e discrete time series corresponding to a signal X(t) is obtained by sampling X(t) at intervals At for a time T , the length n of record to be studied. The time series w i l l be denoted by X0 ' X l » X2 ' ' " > ^S-l where T N = = the number of samples taken. A power density spectrum i s then defined as _ 1 N-l „ ..fry I x k r 2 T r i k y k=0 r = 0, 1 , N - l (A.14) in analogy with Eq. (A.13), and where frequency is normalized to l/N«At. The expression within the absolute value signs is called the d i g i t a l Fourier transform, and may be quickly computed using the 'fast Fourier transform' algorithm. If f denotes the Fourier coefficients from the d i g i t a l Fourier r transform, that is f E a + ib = Y X, £ " 2 7 T i k © r k ^ Q ~k _ W , r = 0, 1, N-l, (A.15) then for N even, and real ^ , s > i t c a n D e shown that (a) f is real o (b) i s real (c) f = f„ N 'N . f ° r 1 1 ' 2 ' 2 -1' ' 2 * 2 " * N Thus, the coefficients need only be calculated for r = 0, 1, 2, — These cover the frequency range from 0 to f N - ^ T — . f V T is called the 2At N (A.16) 'Nyquist' or 'f o l d i n g ' frequency, and w i l l be discussed further. A hanning window may be applied, as previously indicated, on the or the Fourier c o e f f i c i e n t s of the transformed data. That i s , for hanning i n the frequency domain, the following operation i s performed: 4 r - l 2 r r+1 - 7- b . + i b - 1/4 b ^ 4 r - l 2 r r+1 r - l " * - 1 2 ( a 0 ~ a l * (A.17) 2 B 2 2 BN = ° where the l a s t three expressions are derived with the help of (A.16). A r and B r are then the hanned Fourier c o e f f i c i e n t s . The modified power spectrum estimates are P r " f - ( A r + B r > n (A.18) An a d d i t i o n a l problem i n d i g i t a l Fourier analysis which doesn't appear i n the continuous analysis i s ' a l i a s i n g ' . This concept i s explained i n d e t a i l by Blackman and Tukey (1959). In a d i g i t a l power spectrum analysis, power at frequencies above f ^ w i l l be folded back into the estimates f o r power below f ^ . The manner i n which this happens i s i l l u s t r a t e d i n the following schematic: f. 0 + 2 f N " f l 2 f N + f l 4 f N - f l 4 f N + f l C—j—6 U 2f N 3f N -0—\—°-5f frequency N Power i s d i s t r i b u t e d i n frequency above and below f N before f o l d i n g . 68 H^i j Exploded view 0 f M Completely folded N A l l power at frequencies f N ± f^, n = 0, 2, 4, w i l l appear in our estimate of power at f-_ in the frequency interval [0, f^j]. The practical way to avoid this problem is to use an analog f i l t e r which passes only those frequencies below f^. The desired Nyquist frequency is chosen by the sampling rate. In the case at hand i t was desired to study the real-time frequency range of 0-1000 Hz. In order to save computer time, the playback tape speed of the signal was twice the real-time record rate. The signal out of the tape recorder was fed through a f i l t e r , set to have an upper pass limit of .2 KHz. Attenuation above this frequency was 20 dB per octave. The sampling frequency, chosen to be 8 K S ' g ^ — e S , gave Nyquist frequency of 4 KHz. Thus, for a reasonably f l a t spectrum, any noise folded back into the frequency range of prime interest (0-2000 Hz for twice real-time playback tape speed), would come from frequencies > 5 KHz. This folded back noise would thus be > 30 dB down. Another important consideration in spectral;analysis is some estimation of the s t a t i s t i c a l s t a b i l i t y of the spectral estimates. This sta b i l i t y can be understood in terms of the chi-square distribution. "If y^, y^, y^ are independently distributed according to a .standard normal distribution, that i s , according to a Gaussian distribution with average zero and unit variance (and, consequently, unit standard deviation), then \ = ?l + y 2 + - + » 69 which i s obviously p o s i t i v e , follows, by d e f i n i t i o n of a chi-square d i s t r i b u t i o n with k degrees of freedom. The c o e f f i c i e n t of v a r i a t i o n 2 V of X^ , the r a t i o of RMS deviation to average value i s (2/k) 2, so that as k increases, X^ becomes r e l a t i v e l y l e s s v a r i a b l e . This, s t a t e -ment also applies to any multiple of X^." t From equation (A.18), P r " T~ ( A r + n where A r and B r are the r e a l and imaginary part of the hanned Fourier th c o e f f i c i e n t for the r power spectrum estimate. I f the noise i s random (phase and amplitude at any frequencies are not st a b l y L r e l a t e d ) , then the A r and B r w i l l be independently d i s t r i b u t e d . Use of the hanning window leads to s p e c t r a l estimates which do not greatly overlap i n the frequency domain; each s p e c t r a l estimate should follow a chi-square d i s t r i b u t i o n with k=2 degrees of freedom. To increase the s t a b i l i t y of the estimates (increase k), i f the noise source i s stationary, then i n d i v i d u a l analyses on m successive records of length T R may be made and the re s u l t s combined. The estimate at each frequency i s taken to be the arithmetic mean of the corresponding estimates i n each of the analyses. As a r e s u l t , each s p e c t r a l estimate (which i s e f f e c t i v e l y averaged over a time period of m ' ^ n a n d a bandwidth of ~ ) now has k=2«m degrees of freedom, n Degrees of freedom may also be gained by taking the arithmetic mean of neighbouring s p e c t r a l estimates and using them as s i n g l e estimates of power at mean frequencies; i . e . , P» = P 2 r " 1 + P 2 r r = 1, 2, N/4 r 2 The estimates P 1 w i l l have twice as many degrees of freedom as the P . + Blackman and Tukey (1959) 70 BIBLIOGRAPHY Barrington, R.E. and R.W. Harvey. Alouette II Observations of VLF Electric Field Amplitudes. To be presented at Spring URSI, Washington, D.C. (1969). Bernstein, I.B. Waves in a Plasma in a Magnetic Field. Phys. Rev. 109, 10-21 (1958). Bingham, C., M.D. Godfrey and J.W. Tukey. Modern Techniques of Power Spectrum Estimation. IEEE Trans, Audio and Electro-acoustics, AV-15, 56 (1967). Blackman, R.B. and J.W. Tukey. The Measurement of Power Spectra, Dover, New York (1959). Brice, N. A Qualitative Explanation of the Diurnal Variation of Chorus. J. Geophys. Res., 69, 4701 (1964). Burns, T.B. A Study of VLF Noise Bands with the Injun 3 Satellite. Paper presented at the F a l l URSI Meeting, Palo Alto, Calif. (1966). Cain, J.C, S.J. Hendricks, R.A. Langel and W.V. Hudson. A Proposed Model for the Internation Geomagnetic Reference Field, 1965. J. Geomag. Geoelect., 19, 335 (1967). Crawford, F.W. A Review of Cyclotron Harmonic Phenomena in Plasmas. J. Nucl. Fusion, _5, 73 1965. Dickinson, P.G.H. Positive Ion Sheaths Due to VxB. • I.M. 123. Department of Scientific and Industrial Research, Radio Research Station, U.K. (1964). Fredricks, R.W. Structure of the Bernstein Modes for Large Values of the Plasma Parameter. J. Plasma Phys., _2, 197 (1968a). Fredricks, R.W. Structure of Generalized Low Frequency Bernstein Modes from the F u l l Electromagnetic Dispersion Relation. J. Plasma Phys., 2, 365 (1968b). Fredricks, R.W., and F.L. Scarf. Electromagnetic and Electrostatic Bernstein Modes in a Warm Collisionless Maxwellian Plasma. Presented at NATO Advanced Study Institute, "Plasma Waves in Space and in the Laboratory", Roros, Norway, (1968). Gurnett, D.A., S.D. Shawhan, N.M. Brice and R.L. Smith. Ion Cyclotron Whistlers. J. Geophys. Res., 70, 1665 (1965). Gurnett, D.A. and T.B. Burns. The Low Frequency Cutoff of ELF Emissions. J. Geophys. Res., _73, 7437 (1968). 71 Gurhart, H., T.L. Crystal, B.P. Fic k l i n , W.E. Blair and T.J. Yung. Proton Gyrofrcquency Band Emissions Observed Aboard 0G0 2. J. Geophys. Res., 73, 3592 (1968). Hall, L.S., W. Heckrotte, and T. Kammash. Ion Cyclotron Electrostatic Instabilities. Phys. Res., 139, A1117 (1965). Harp, R.S., The Dispersion Characteristics of Longitudinal Plasma Oscillations near Cyclotron Harmonics. Proc. 7th Intemat. Conf. on Phenomena in Ionized Gases. Belgrade (1965). Henderson, D.P. and O.L. Cote. Technical Details and Elec t r i c a l Evaluation of the VLF Receivers Mk I and Mk II for S27A. TM No. 482. Defence Research Telecommunications Establish-ment, Ottawa, Ontario, (1964). Hoffman, J.H. Ion Mass Spectrometer on Explorer XXXI Satellite. IEEE Special Issue on ISIS Program. In Press (1969). Mantle, T.D. Cyclotron Harmonic Wave Phenomena. SUIPR Report No. 194, Institute of Plasma Res., Stanford U., Stanford, Calif. (1967). O'Brien, B.J., CD. Laughlin, J.A. Van Allen, and L.A. Frank. Measurements of the Intensity and Spectrum of Electrons at 1000 - kilometer Altitude and High Latitudes. J. Geophys. Res., 67_, 1209 (1962). Oliven, M.N. and D.A. Gurnett. Microburst Phenomena. 3. An Association between Microbursts and VLF Chorus. J. Geophys. Res., 73, 2355 (1968). Parks, G.K. Spatial Characteristics of Auroral-Zone X-Ray Microbursts. J. Geophys. Res., 72, 215,(1967). Scarf, F.W., R.W. Fredricks and CM. Crook. Detection of Electromagnetic and Electrostatic Waves on 0V3-3. J. Geophys. Res., 73, 1723 (1968). Stix, T.H. The Theory of Plasma Waves. McGraw-Hill, New York (1962). Storey, L.R.O. The Design of an Electric Dipble Antenna for VLF Reception within the Ionosphere. Tech. Rept. 308 TC, Centre National d'Etudes des Telecommunications, Dept. 'Telecommande', Paris (1963). Swift, Daniel W. A new Interpretation of VLF Chorus. J. Geophys. Res., 73, 7447 (1968). Tataronis, J.A. Cyclotron Harmonic Wave Propagation and Instabilities. SUIPR Report No. 205, Institute for Plasma Research, Stanford U., Stanford, California (1967). Hendricks, S.J. and Cain,J.C. Magnetic Field Data for Tra'ppsd-Particle Evaluations. J. Geophys. Res., 7_1, 346(1966). 72 Tataronis,J.A. and F.W. Crawford. Cyclotron and Collision Damping of Propagating Waves in a Magnetoplasma. Proc. 7 th Internat. Conf. on Phenomena in Ionized Gases, Belgrade (1965). Taylor, William W.L. and Donald A.-Gurnett. Morphology of VLF Emissions Observed with the Injun 3 Satellite. J. Geophys, Res., 73, 5615 (1968). 

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