- Email: [email protected]
Validity of IRI electron density profiles in relation to vertical incidence absorption measurements
Journalof
Atmospheric
Pergamon 0021-9169(95)00114-X
and Terrestrial
Physics, Vol. 58, No. 1 I, pp. 1195-1200, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserwJ 0021-9169196 S15.00+0.00
Validity of IRI electron density profiles in relation to vertical incidence absorption measurements K. V. V. Ramana, K. S. R. N. Murthy, M. Indira Devi, Y. V. P. K. Raghava and D. N. Madhusudhana Rao Physics Department, Andhra University, Visakhapatnam-530 003, India (Received 9 March 1995; accepted in revisedform 30 March 1995)
Abstract-The observed discrepancies between Al absorption measurements and numerical estimation of the same using IRI electron density profiles are attributed to the assumption made in the Sen-Wyller generalized magneto-ionic theory that the momentum transfer collision frequency of electrons with neutrals is proportional to the square of the electron thermal speed. Based on Budden’s (1985) suggestion that, in the lower thermosphere and mesosphere, the momentum transfer collision frequency is proportional to the electron thermal speed, a generalized magneto-ionic theory has been outlined. The new theory brings experimental measurements of Al absorption closer to the theoretical deductions based on IRI-90 electron density profiles. Copyright 0 1996 Elsevier Science Ltd
IINTRODUCTION
Ferguson (1984a), who compared measurements of Al absorption carried out at Camden, Australia (34.1”S, 150.7”E) at a frequency of 1.91 MHz during February 1980 to January 1981 with absorption numerically calculated, employing IRI-79 electron density profiles (see Lincoln and Conkright, 1981), observed that the numerical values are systematically larger than the measurements by 10-25 dB. The numerically computed group heights are lower by 24 km. In a similar study, Oyinloye (1988) reported numerical values of absorption computed using IRI79 electron density Iprofiles at Thumba, India (8.5” N, 76.9”E) in excess of 2672% compared with measurements at a number of frequencies in the range 1.8-2.5 MHz. Oyinloye (1!>88) further compared measured absorption at Thumba with that obtained from numerical calculations employing electron density profiles obtained from simultaneous soundings by rockets. The results, covering once again sounding frequencies between 1.8 MHz and 2.5 MHz, showed that numerical values are larger than measurements by 2-34%. Singer et al. (1980) as well as Ferguson and McNamara (1986) also discussed systematic differences between theory and measurements when IRI-79 profiles were used to predict radio wave absorption. Ferguson (1984b) computed numerically the absorption as a function of sounding frequency employing IRI-79 profiles for solar maximum and minimum conditions in winter and summer at Waltair,
India (17.7”N, 83.3”E) in a study of validity of George’s (George, 1971) method of reducing Al absorption data to a common sounding frequency. Figure la of his paper clearly establishes the fact that the numerical values for Waltair are significantly larger than measurements in both seasons and at both levels of solar activity. Some previous investigators on the subject, including Ferguson (1984a) and Oyinloye (1988), suggested modifications to the IRI-79 profiles/profile parameters to bring about an agreement between theory and experiment. However, it should be noted that systematic differences of the order of 50% and more cannot be explained purely on the basis of the inadequacy of the IRI profiles. Since the absorption suffered by a radio wave in its traverse through the ionosphere depends on the product of electron density and electron-neutral collision frequency integrated over the path of propagation, it is worthwhile to consider the validity of the collision frequency profile adopted in the numerical computations. Phelps and Pack (1959), amongst others, reported laboratory measurements of collision cross-section of electrons with Nz molecules, the most abundant constituent below the thermosphere and observed that, for electron energies in the range 0.01 to 0.3 eV, the electronneutral momentum transfer collision frequency v(v) varies as the square of the electron thermal speed. Sen and Wyller (1960) as well as Budden (1965) generalized the Appleton-Hartree magneto-ionic theory on the basis of the energy dependence of collision
1195
K. V. V. Ramana et al
1196
frequency. This generalized magneto-ionic theory has since been widely used in the numerical calculations of absorption and other radio propagation studies. Budden (1985) proposed that, in the mesosphere and lower thermosphere, where the electron temperatures are less than 500 K and electron energies less than 0.1 eV, the assumed velocity square dependence of v(u) may be in error. He suggested that in this region the collision frequency is proportional to the electron thermal speed and not its square. He also observed that no strong evidence exists for a velocity squared dependence despite the measurements made by Phelps and Pack (1959) and a few others. In the following section, the generalized magnetoionic theory based on the assumption that v(v) is proportional to the electron thermal speed is outlined. The theory is analogous to that developed by Budden (1965) in his tutorial paper and consequently only those equations that are relevant in the light of the modification suggested are given. In sections 3 and 4, the results of a numerical simulation of vertical incidence absorption using the theory outlined in section 2 are compared and discussed in the light of experimental measurements.
obtained by Budden (1965) for v(u) cc v2 except that c,, e2 and Q now have different values as given below. In equation (2), n is the complex refractive index, 0 is the co-dip angle and c,, t2 and 6) are the principal axis components of the dielectric tensor given by
c’ = l-
4XW a s312exp(- s) o (I+ y)W4pds 5 s
(4)
1, s”‘exp( - s) o (1-y)W_jsl’2dS
(5)
4xw t2= 1-q
s
s”‘exp( - s) W-is”2
In equations (4) to (6), X and Y have their usual meanings in magneto-ionic theory. W is the ratio of the angular sounding wave frequency, o, to v,. The integrands of equations (4) to (6) may be separated into real and imaginary parts. After integrating, equations (4) to (6) assume the form 6, = 1-X[w2(1+Y)u,,,(w2(1+Y)2) J!Y
rr2(W2(1 +
+l3& OUTLINE
OF THE THEORY
8W
+Yyz
~2W2U
( ) ST
vv, = s”2vv,,
s=G > i
36
(1)
(7)
r)‘)l (8) (9)
where the functions
where m, T and v are the electron’s mass, temperature and thermal speed, respectively. k is Boltzmann’s constant and v, the mono-energetic collision frequency. Considering the radio wave field in the unbounded, ‘homogeneous’ ionospheric medium to be a continuous super-position of plane electromagnetic waves, the dispersion equation of the medium can be obtained from the linearized Maxwell’s equations applicable and the constitutive relations of the medium as t,t2sin28+ (1/2)~~(t, + e2)(1 + cos%) f Sr/* n2 = (6, f t,)sin2fI + 2t,cos2B
-
1-X[W2U~,2(W2)+i~ 8 w ~2W2)1,
&3 =
v(v) =
Y)‘)]
t2= 1-X[w2(1-Y)U3,,(w2(1-Y)Z)
The electron-neutral momentum transfer collision frequency v(v), assumed proportional to the electron thermal speed, is written, following Shkarofsky et al. (1966), as l/2
(6)
ds
(2)
with
Up(x)= j
(3) Equation (2) is identical to the dispersion equation
0
s+x
ds
(10)
are a type of integral discussed and evaluated by Dingle et al. (1957). In equation (lo), if x --f 0, U,(x) --* p-’ and if x + co, U,(x) -+ x-‘. For intermediate values of x, U,(x) can be obtained from the recursive relations pU*(x)+xUp_,(x)
U,_,(x) = -
1
P+x
( l+
+ P@--4
S = [sin40(w, - (1/2)~(t, + t2))2+ cos2&, 2(e,- c~)~].
s
msPexp( - s)
(P+xY
= 1
(11)
~
@IX)2
+ P(P’--Px+w (P+xY
+ p(p’ - 22$x + 58px2 - 24x3) @+x)”
+ . . . (12) >
Validity of IRI electron density profiles
1197
However, if x >>p, then U,(x) may be written as (13)
Writing the complex refractive index n as n = p-_ix = p-ick, 0
where the number density of the electrons is given by:
where p is the real part of the complex phase refractive index, c is the speed of the light in vacuum and k, the linear absorption coefficient, the absorption suffered by a radio wave in a single vertical traverse through the ionosphere is given by L=
s
L=2
c“kdh, Jo
reflection at a
(19)
8
--
v,
3
1: -
v,
2
for (0 f wB) >>v,~
(20)
If (w + wa) < v,e, v,~ is deduced as
WI
x
3fi 4 = Pvm2L3vm 4
(21)
The inter-relationships between the various collision frequencies defined at various stages are given
b:v) = (&)‘:‘%
=~-3& 8
= ~(&)?
m
‘12vveg
( 2kT >
for (w f WJ >>v,e
(l+ Y-Z)’
,c2=: 1 -
t3=1-
x
(1-Y-iZ)’
(16)
x
for (w * oB) < vCff
(22)
(l-iz)’
where Z = kff
NUMERICAL
w
Shkarofsky et al. (1966) defined v,* as the average of [I& $(v)v3] over the unperturbed (w * ws) >> v,~, i.e.,
(18)
On integration, equation (17) gives
efl-3fi
where h, is the altitude at which the radio wave undergoes reflection. It is possible to define effective collision frequencies v,= for (w + oa) >> reffand (w + oB) < v,~, where or, is the angular electron gyrofrequency in the geomagnetic field. The v,~, defined thus, and for the limits specified, is independent of the electron thermal speed or energy and hence may be used in the Appleton-Hartree magneto-ionic refmctive index formula, i.e. equation (2) with t,, ~~and 6) given by 6, =: l-
h = N(ir2exp(s).
V
mkdh. II
If, however, the wave undergoes height h, in the ionosphere,
N = 4n ^jfov2dv and s
(14)
Maxwellian distribution f0 for
RESULTS EMPLOYING
IRI-90 PROFILES
In the numerical evaluation of Al absorption using the generalized Appleton-Hartree refractive index given by equation (2) in which t,, tl and t3 are given by equations (7H9), and the Appleton-Hartree refractive index, again given by equation (2) but with E,, c*, and &3as defined in equation (16) the latest version of the International Reference Ionosphere, IRI-90 (Bilitza, 1990) has been employed. The height
K. V. V. Ramana
1198 Table 1, Measured
and numerically
Sounding frequency (MHz)
Station
Date
Waltair Waltair Waltair Waltair Waltair Camden
19 September 1978 7 April 1983 24 April 1983 18 September 1983 21 December 1983 15 July 1980
computed
Solar zenith angle (degrees)
2.40 2.40 2.40 2.40 2.40 1.91
45.4 45.5 45.4 45.7 45.4 56.2
absorption
141 55 110 31 16 136
(23)
following Thrane and Piggott (1966) and Smith et al. (1978). Since the base height of the IRI-90 electron density profiles is 65 km, the same as that for IRI-79, it can be assumed that (cu f wg) >> v,~ Consequently, the effective collision frequency height profile used in the calculation of the Appleton-Hartree index is given by 3 v,fl= -v, 2
= 9.45 x 105PSK’
with IRI-90 electron
Measured
R,
profile of the v, is assumed to be given by v, = 6.3 x 1O’P (P is pressure in Pascals) SK’
et al.
(24)
Five IRI electron density profiles, two pertaining to the month of April, two to the month of September and one to the month of December at a solar zenith angle 45.5”, are generated using IRI-90 software packages, provided by NSSDC/WDC for Rockets and Satellites, GSFC, Greenbelt, Maryland, U.S.A. These profiles represent a spread of sunspot numbers R, from 16 to 141. Experimental Al absorption measurements are available not only on these days (listed in Table 1 ) but several days before and after each of the days chosen. Absorption at 2.4 MHz and R, changed insignificantly in the course of each set of measurements. Another IRI-90 profile for Camden, noon, mid-July, is used in the numerical calculation of absorption. The experimental Al absorption value at 1.91 MHz is taken as the average of the weekly median values for July 1980 read from Fig. 3 of Ferguson’s (Ferguson, 1984a) paper. The generalized AppletonHartree linear absorption coefficients are computed as a function of height from 65 km to 90 km in steps of 1 km and from 90 km to the height of reflection in steps of 0.1 km. In the last 0.1 km or the fraction thereof, the step size has been reduced to 0.01 km. The absorption L is obtained by resorting to numerical integration using Simpson’s rule. Phase integral correction (Thorpe, 1971) has been applied to each of the
L(dB) 35.4 31.2 45.6 30.1 28.2 31.5
Value from present theory L(dB)
38.0 30.2 35.4 27.8 26.2 31.2
density profiles AppletonHartree formula
L(dB) u,~ = ; u,
(dB) u,~ = ; u,
38.3 30.3 35.5 28.0 26.3 30.0
61.7 49.2 57.5 45.6 42.8 48.8
numerical values. Table 1 presents the results of the numerical computation along with measured values and average Zurich sunspot number for each data point. In Table 2 , we present the results of numerical calculations of Al absorption at 2.4 MHz for Waltair using IRI-79 and IRI-90 Waltair noon electron density profiles at different levels of solar activity, as indicated by the R, values, and at different sounding frequencies.
DISCUSSION
The numerical results tabulated in Table 1 clearly demonstrate that the absorption computed using v,~ = (3/2) v, in the Appleton-Hartree formula agrees with that computed using the generalized AppletonHartree formula outlined in section 2. Barring the numerical value for Camden, the agreement is to within 1%. At Camden, the difference is between 3 and 4%. The theoretical calculations thus justify the assertion made earlier that the Appleton-Hartree formula can be used in the numerical computation of absorption provided that the collision frequency in the formula is assumed to be v,~ = (3/2) v, despite the limitations imposed on the definition of v,~ Secondly, the experimental determinations of the absorption at Waltair and Camden are closer to the theoretical computations of the same when the theory is based on the assumption that v(v) is proportional to the electron thermal speed rather than to the square of the thermal speed, as assumed in the Sen-Wyller’s generalized theory. The numerically computed data presented in the last two columns of Table 1 show that the assumption of a velocity squared dependence of v(v) gives absorption values which are in excess of 16.5-23.4 dB over those computed using a velocity dependence of v(v). Ferguson (1984a) also observed that numerical computations based on the Sen-Wyller
Validity of IRI electron density profiles
1199
theory led to absorption values which are larger by 10-25 dB than the measurements. A comparative study of numerical calculations of Waltair using IRI-79 and IRI-90 electron density profiles, presented in Table 2, brings out the fact that IRI79 overestimates the absorption by a few percentage points. Ferguson (1984a) argued that the differences noted between theory and experiment should not be attributed to the assumptions relating to the collision frequency because of discrepancies in virtual height. He suggested that the IRI-79 electron densities are high above 90 km. However, the discrepancies in virtual height are of the order of a few kilometers in the altitude range 90-100 km and it does not seem plausible that the discrepancies are beyond the error limits in the measurement of the group height. On the other hand, the observed discrepancies in absorption, in most cases, are certainly beyond the error limits. It is true that Smith et al. (1978) did not notice significant differences between measurement and theory based on the Sen-Wyller formula. However, the rocket-borne determinations of electron density profiles used by Smith et al. (1978) refer to a few spot measurements. Ramanamurty (1983) compared D-region IRI-79 electron density profiles with those deduced from ELF/VLF/LF/MF measurements and concluded that there is no reason to suspect the validity of the IRI79 model above 70 km on the basis of those measurements. For sounding frequencies in this range, (o 4 os) may be assumed to be less than v,~ and v,~ is (3/2) v, for the Appleton-Hartree theory and (4/3) v, for the theory outlined in section 2. The differences between these two are not significant and, consequently, good agreement between IRI-79 profiles and those derived from ELF/VLF/LF/MF measurements can be understood.
CONCLUSIONS
The observed discrepancies in the Al absorption at medium and high frequencies and the numerical simulation of the same using IRI electron density profiles arise from the assumption made in the generalized Appleton-Hartree magneto-ionic theory that the collision frequency employed in the numerical simulation is based on the assumption that v(v) is proportional to the square of the electron thermal speed. A generalized magneto-ionic theory based on the assumption that v(v) is proportional to the electron thermal speed is developed. It is shown that the new generalized magneto-ionic formula gives results identical to the Appleton-Hartree magneto-ionic formula
1200
K. V. V. Ramana et al.
if v,~ is (3/2) v, for frequencies (w + wB) >> v,~ and v,~ is (4/3) v, for (w + wr,) << v,~ A comparison between Al measurements at Waltair and Camden with numerical results based on the new assumption and IRI-90 electron density profiles results in better agreement between theory and experiment. Vertical incidence absorption computed using IRI-79 electron
density profiles overestimates the absorption by a few percentage points relative to that computed using IRI90 profiles. Acknowledgemenrs-We thank the University Grants Commission, New Delhi and Council of Scientific and Industrial Research, New Delhi, for supporting this investigation.
REFERENCES
Bilitza D.
1990
Budden K. G. Budden K. G.
1965 1985
Dingle R. B., Arndt D. and Roy S. K. Ferguson B. G. Ferguson B. G. Ferguson B. G. and McNamara L. F. George P. L. Lincoln J. V. and Conkright R. 0.
1957
1984a 1984b 1986 1971 1981
Oyinloye J. 0. Phelps A. V. and Pack J. L. Ramanamurty Y. V. Sen H. K. and Wyller A. A. Shkarofsky I. P., Johnston T. W. and Bachynski M. P.
1988 1959 1983 1960 1966
J. atmos. terr. Phys. 50, 519. Phys. Res. Lett. 3,340. Adv. Space. Res. 2, 205. J. geophys. Res. 65, 393 1. The Particle Kinetics of Plasmas. Addison-Wesley,
Singer W., Taubenheim J. and Bremer J. Smith L. G., Walton E. K. and Mechtly E. A. Thorpe M. R. Thrane E. V. and Piggott W. R.
1980 1978 1971 1966
J. J. J. J.
International Reference Ionosphere-IRI 90, Report NSSDC/WDC-A-R and S 90-22, Greenbelt, Maryland, U.S.A. Radio Sci. 69, 19 1. The Propagation of Radio Waves. Cambridge University Press, Cambridge. Appl. Sci. Res. 6B, 144. J. atmos. terr. Phys. 46,403. J. atmos. terr. Phys. 46, 399. J. atmos. terr. Phys. 48,41. J. atmos. terr. Phys. 33, 1893.
International Reference Ionosphere-IRI 79, Report UAG-82, WDC A, Boulder, Colorado, U.S.A.
U.S.A. atmos. atmos. atmos. atmos.
terr. terr. ferr. terr.
Phys. Phys. Phys. Phys.
42,241. 40, 1185. 33, 1597. 28, 721.
Atmospheric
Pergamon 0021-9169(95)00114-X
and Terrestrial
Physics, Vol. 58, No. 1 I, pp. 1195-1200, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserwJ 0021-9169196 S15.00+0.00
Validity of IRI electron density profiles in relation to vertical incidence absorption measurements K. V. V. Ramana, K. S. R. N. Murthy, M. Indira Devi, Y. V. P. K. Raghava and D. N. Madhusudhana Rao Physics Department, Andhra University, Visakhapatnam-530 003, India (Received 9 March 1995; accepted in revisedform 30 March 1995)
Abstract-The observed discrepancies between Al absorption measurements and numerical estimation of the same using IRI electron density profiles are attributed to the assumption made in the Sen-Wyller generalized magneto-ionic theory that the momentum transfer collision frequency of electrons with neutrals is proportional to the square of the electron thermal speed. Based on Budden’s (1985) suggestion that, in the lower thermosphere and mesosphere, the momentum transfer collision frequency is proportional to the electron thermal speed, a generalized magneto-ionic theory has been outlined. The new theory brings experimental measurements of Al absorption closer to the theoretical deductions based on IRI-90 electron density profiles. Copyright 0 1996 Elsevier Science Ltd
IINTRODUCTION
Ferguson (1984a), who compared measurements of Al absorption carried out at Camden, Australia (34.1”S, 150.7”E) at a frequency of 1.91 MHz during February 1980 to January 1981 with absorption numerically calculated, employing IRI-79 electron density profiles (see Lincoln and Conkright, 1981), observed that the numerical values are systematically larger than the measurements by 10-25 dB. The numerically computed group heights are lower by 24 km. In a similar study, Oyinloye (1988) reported numerical values of absorption computed using IRI79 electron density Iprofiles at Thumba, India (8.5” N, 76.9”E) in excess of 2672% compared with measurements at a number of frequencies in the range 1.8-2.5 MHz. Oyinloye (1!>88) further compared measured absorption at Thumba with that obtained from numerical calculations employing electron density profiles obtained from simultaneous soundings by rockets. The results, covering once again sounding frequencies between 1.8 MHz and 2.5 MHz, showed that numerical values are larger than measurements by 2-34%. Singer et al. (1980) as well as Ferguson and McNamara (1986) also discussed systematic differences between theory and measurements when IRI-79 profiles were used to predict radio wave absorption. Ferguson (1984b) computed numerically the absorption as a function of sounding frequency employing IRI-79 profiles for solar maximum and minimum conditions in winter and summer at Waltair,
India (17.7”N, 83.3”E) in a study of validity of George’s (George, 1971) method of reducing Al absorption data to a common sounding frequency. Figure la of his paper clearly establishes the fact that the numerical values for Waltair are significantly larger than measurements in both seasons and at both levels of solar activity. Some previous investigators on the subject, including Ferguson (1984a) and Oyinloye (1988), suggested modifications to the IRI-79 profiles/profile parameters to bring about an agreement between theory and experiment. However, it should be noted that systematic differences of the order of 50% and more cannot be explained purely on the basis of the inadequacy of the IRI profiles. Since the absorption suffered by a radio wave in its traverse through the ionosphere depends on the product of electron density and electron-neutral collision frequency integrated over the path of propagation, it is worthwhile to consider the validity of the collision frequency profile adopted in the numerical computations. Phelps and Pack (1959), amongst others, reported laboratory measurements of collision cross-section of electrons with Nz molecules, the most abundant constituent below the thermosphere and observed that, for electron energies in the range 0.01 to 0.3 eV, the electronneutral momentum transfer collision frequency v(v) varies as the square of the electron thermal speed. Sen and Wyller (1960) as well as Budden (1965) generalized the Appleton-Hartree magneto-ionic theory on the basis of the energy dependence of collision
1195
K. V. V. Ramana et al
1196
frequency. This generalized magneto-ionic theory has since been widely used in the numerical calculations of absorption and other radio propagation studies. Budden (1985) proposed that, in the mesosphere and lower thermosphere, where the electron temperatures are less than 500 K and electron energies less than 0.1 eV, the assumed velocity square dependence of v(u) may be in error. He suggested that in this region the collision frequency is proportional to the electron thermal speed and not its square. He also observed that no strong evidence exists for a velocity squared dependence despite the measurements made by Phelps and Pack (1959) and a few others. In the following section, the generalized magnetoionic theory based on the assumption that v(v) is proportional to the electron thermal speed is outlined. The theory is analogous to that developed by Budden (1965) in his tutorial paper and consequently only those equations that are relevant in the light of the modification suggested are given. In sections 3 and 4, the results of a numerical simulation of vertical incidence absorption using the theory outlined in section 2 are compared and discussed in the light of experimental measurements.
obtained by Budden (1965) for v(u) cc v2 except that c,, e2 and Q now have different values as given below. In equation (2), n is the complex refractive index, 0 is the co-dip angle and c,, t2 and 6) are the principal axis components of the dielectric tensor given by
c’ = l-
4XW a s312exp(- s) o (I+ y)W4pds 5 s
(4)
1, s”‘exp( - s) o (1-y)W_jsl’2dS
(5)
4xw t2= 1-q
s
s”‘exp( - s) W-is”2
In equations (4) to (6), X and Y have their usual meanings in magneto-ionic theory. W is the ratio of the angular sounding wave frequency, o, to v,. The integrands of equations (4) to (6) may be separated into real and imaginary parts. After integrating, equations (4) to (6) assume the form 6, = 1-X[w2(1+Y)u,,,(w2(1+Y)2) J!Y
rr2(W2(1 +
+l3& OUTLINE
OF THE THEORY
8W
+Yyz
~2W2U
( ) ST
vv, = s”2vv,,
s=G > i
36
(1)
(7)
r)‘)l (8) (9)
where the functions
where m, T and v are the electron’s mass, temperature and thermal speed, respectively. k is Boltzmann’s constant and v, the mono-energetic collision frequency. Considering the radio wave field in the unbounded, ‘homogeneous’ ionospheric medium to be a continuous super-position of plane electromagnetic waves, the dispersion equation of the medium can be obtained from the linearized Maxwell’s equations applicable and the constitutive relations of the medium as t,t2sin28+ (1/2)~~(t, + e2)(1 + cos%) f Sr/* n2 = (6, f t,)sin2fI + 2t,cos2B
-
1-X[W2U~,2(W2)+i~ 8 w ~2W2)1,
&3 =
v(v) =
Y)‘)]
t2= 1-X[w2(1-Y)U3,,(w2(1-Y)Z)
The electron-neutral momentum transfer collision frequency v(v), assumed proportional to the electron thermal speed, is written, following Shkarofsky et al. (1966), as l/2
(6)
ds
(2)
with
Up(x)= j
(3) Equation (2) is identical to the dispersion equation
0
s+x
ds
(10)
are a type of integral discussed and evaluated by Dingle et al. (1957). In equation (lo), if x --f 0, U,(x) --* p-’ and if x + co, U,(x) -+ x-‘. For intermediate values of x, U,(x) can be obtained from the recursive relations pU*(x)+xUp_,(x)
U,_,(x) = -
1
P+x
( l+
+ P@--4
S = [sin40(w, - (1/2)~(t, + t2))2+ cos2&, 2(e,- c~)~].
s
msPexp( - s)
(P+xY
= 1
(11)
~
@IX)2
+ P(P’--Px+w (P+xY
+ p(p’ - 22$x + 58px2 - 24x3) @+x)”
+ . . . (12) >
Validity of IRI electron density profiles
1197
However, if x >>p, then U,(x) may be written as (13)
Writing the complex refractive index n as n = p-_ix = p-ick, 0
where the number density of the electrons is given by:
where p is the real part of the complex phase refractive index, c is the speed of the light in vacuum and k, the linear absorption coefficient, the absorption suffered by a radio wave in a single vertical traverse through the ionosphere is given by L=
s
L=2
c“kdh, Jo
reflection at a
(19)
8
--
v,
3
1: -
v,
2
for (0 f wB) >>v,~
(20)
If (w + wa) < v,e, v,~ is deduced as
WI
x
3fi 4 = Pvm2L3vm 4
(21)
The inter-relationships between the various collision frequencies defined at various stages are given
b:v) = (&)‘:‘%
=~-3& 8
= ~(&)?
m
‘12vveg
( 2kT >
for (w f WJ >>v,e
(l+ Y-Z)’
,c2=: 1 -
t3=1-
x
(1-Y-iZ)’
(16)
x
for (w * oB) < vCff
(22)
(l-iz)’
where Z = kff
NUMERICAL
w
Shkarofsky et al. (1966) defined v,* as the average of [I& $(v)v3] over the unperturbed (w * ws) >> v,~, i.e.,
(18)
On integration, equation (17) gives
efl-3fi
where h, is the altitude at which the radio wave undergoes reflection. It is possible to define effective collision frequencies v,= for (w + oa) >> reffand (w + oB) < v,~, where or, is the angular electron gyrofrequency in the geomagnetic field. The v,~, defined thus, and for the limits specified, is independent of the electron thermal speed or energy and hence may be used in the Appleton-Hartree magneto-ionic refmctive index formula, i.e. equation (2) with t,, ~~and 6) given by 6, =: l-
h = N(ir2exp(s).
V
mkdh. II
If, however, the wave undergoes height h, in the ionosphere,
N = 4n ^jfov2dv and s
(14)
Maxwellian distribution f0 for
RESULTS EMPLOYING
IRI-90 PROFILES
In the numerical evaluation of Al absorption using the generalized Appleton-Hartree refractive index given by equation (2) in which t,, tl and t3 are given by equations (7H9), and the Appleton-Hartree refractive index, again given by equation (2) but with E,, c*, and &3as defined in equation (16) the latest version of the International Reference Ionosphere, IRI-90 (Bilitza, 1990) has been employed. The height
K. V. V. Ramana
1198 Table 1, Measured
and numerically
Sounding frequency (MHz)
Station
Date
Waltair Waltair Waltair Waltair Waltair Camden
19 September 1978 7 April 1983 24 April 1983 18 September 1983 21 December 1983 15 July 1980
computed
Solar zenith angle (degrees)
2.40 2.40 2.40 2.40 2.40 1.91
45.4 45.5 45.4 45.7 45.4 56.2
absorption
141 55 110 31 16 136
(23)
following Thrane and Piggott (1966) and Smith et al. (1978). Since the base height of the IRI-90 electron density profiles is 65 km, the same as that for IRI-79, it can be assumed that (cu f wg) >> v,~ Consequently, the effective collision frequency height profile used in the calculation of the Appleton-Hartree index is given by 3 v,fl= -v, 2
= 9.45 x 105PSK’
with IRI-90 electron
Measured
R,
profile of the v, is assumed to be given by v, = 6.3 x 1O’P (P is pressure in Pascals) SK’
et al.
(24)
Five IRI electron density profiles, two pertaining to the month of April, two to the month of September and one to the month of December at a solar zenith angle 45.5”, are generated using IRI-90 software packages, provided by NSSDC/WDC for Rockets and Satellites, GSFC, Greenbelt, Maryland, U.S.A. These profiles represent a spread of sunspot numbers R, from 16 to 141. Experimental Al absorption measurements are available not only on these days (listed in Table 1 ) but several days before and after each of the days chosen. Absorption at 2.4 MHz and R, changed insignificantly in the course of each set of measurements. Another IRI-90 profile for Camden, noon, mid-July, is used in the numerical calculation of absorption. The experimental Al absorption value at 1.91 MHz is taken as the average of the weekly median values for July 1980 read from Fig. 3 of Ferguson’s (Ferguson, 1984a) paper. The generalized AppletonHartree linear absorption coefficients are computed as a function of height from 65 km to 90 km in steps of 1 km and from 90 km to the height of reflection in steps of 0.1 km. In the last 0.1 km or the fraction thereof, the step size has been reduced to 0.01 km. The absorption L is obtained by resorting to numerical integration using Simpson’s rule. Phase integral correction (Thorpe, 1971) has been applied to each of the
L(dB) 35.4 31.2 45.6 30.1 28.2 31.5
Value from present theory L(dB)
38.0 30.2 35.4 27.8 26.2 31.2
density profiles AppletonHartree formula
L(dB) u,~ = ; u,
(dB) u,~ = ; u,
38.3 30.3 35.5 28.0 26.3 30.0
61.7 49.2 57.5 45.6 42.8 48.8
numerical values. Table 1 presents the results of the numerical computation along with measured values and average Zurich sunspot number for each data point. In Table 2 , we present the results of numerical calculations of Al absorption at 2.4 MHz for Waltair using IRI-79 and IRI-90 Waltair noon electron density profiles at different levels of solar activity, as indicated by the R, values, and at different sounding frequencies.
DISCUSSION
The numerical results tabulated in Table 1 clearly demonstrate that the absorption computed using v,~ = (3/2) v, in the Appleton-Hartree formula agrees with that computed using the generalized AppletonHartree formula outlined in section 2. Barring the numerical value for Camden, the agreement is to within 1%. At Camden, the difference is between 3 and 4%. The theoretical calculations thus justify the assertion made earlier that the Appleton-Hartree formula can be used in the numerical computation of absorption provided that the collision frequency in the formula is assumed to be v,~ = (3/2) v, despite the limitations imposed on the definition of v,~ Secondly, the experimental determinations of the absorption at Waltair and Camden are closer to the theoretical computations of the same when the theory is based on the assumption that v(v) is proportional to the electron thermal speed rather than to the square of the thermal speed, as assumed in the Sen-Wyller’s generalized theory. The numerically computed data presented in the last two columns of Table 1 show that the assumption of a velocity squared dependence of v(v) gives absorption values which are in excess of 16.5-23.4 dB over those computed using a velocity dependence of v(v). Ferguson (1984a) also observed that numerical computations based on the Sen-Wyller
Validity of IRI electron density profiles
1199
theory led to absorption values which are larger by 10-25 dB than the measurements. A comparative study of numerical calculations of Waltair using IRI-79 and IRI-90 electron density profiles, presented in Table 2, brings out the fact that IRI79 overestimates the absorption by a few percentage points. Ferguson (1984a) argued that the differences noted between theory and experiment should not be attributed to the assumptions relating to the collision frequency because of discrepancies in virtual height. He suggested that the IRI-79 electron densities are high above 90 km. However, the discrepancies in virtual height are of the order of a few kilometers in the altitude range 90-100 km and it does not seem plausible that the discrepancies are beyond the error limits in the measurement of the group height. On the other hand, the observed discrepancies in absorption, in most cases, are certainly beyond the error limits. It is true that Smith et al. (1978) did not notice significant differences between measurement and theory based on the Sen-Wyller formula. However, the rocket-borne determinations of electron density profiles used by Smith et al. (1978) refer to a few spot measurements. Ramanamurty (1983) compared D-region IRI-79 electron density profiles with those deduced from ELF/VLF/LF/MF measurements and concluded that there is no reason to suspect the validity of the IRI79 model above 70 km on the basis of those measurements. For sounding frequencies in this range, (o 4 os) may be assumed to be less than v,~ and v,~ is (3/2) v, for the Appleton-Hartree theory and (4/3) v, for the theory outlined in section 2. The differences between these two are not significant and, consequently, good agreement between IRI-79 profiles and those derived from ELF/VLF/LF/MF measurements can be understood.
CONCLUSIONS
The observed discrepancies in the Al absorption at medium and high frequencies and the numerical simulation of the same using IRI electron density profiles arise from the assumption made in the generalized Appleton-Hartree magneto-ionic theory that the collision frequency employed in the numerical simulation is based on the assumption that v(v) is proportional to the square of the electron thermal speed. A generalized magneto-ionic theory based on the assumption that v(v) is proportional to the electron thermal speed is developed. It is shown that the new generalized magneto-ionic formula gives results identical to the Appleton-Hartree magneto-ionic formula
1200
K. V. V. Ramana et al.
if v,~ is (3/2) v, for frequencies (w + wB) >> v,~ and v,~ is (4/3) v, for (w + wr,) << v,~ A comparison between Al measurements at Waltair and Camden with numerical results based on the new assumption and IRI-90 electron density profiles results in better agreement between theory and experiment. Vertical incidence absorption computed using IRI-79 electron
density profiles overestimates the absorption by a few percentage points relative to that computed using IRI90 profiles. Acknowledgemenrs-We thank the University Grants Commission, New Delhi and Council of Scientific and Industrial Research, New Delhi, for supporting this investigation.
REFERENCES
Bilitza D.
1990
Budden K. G. Budden K. G.
1965 1985
Dingle R. B., Arndt D. and Roy S. K. Ferguson B. G. Ferguson B. G. Ferguson B. G. and McNamara L. F. George P. L. Lincoln J. V. and Conkright R. 0.
1957
1984a 1984b 1986 1971 1981
Oyinloye J. 0. Phelps A. V. and Pack J. L. Ramanamurty Y. V. Sen H. K. and Wyller A. A. Shkarofsky I. P., Johnston T. W. and Bachynski M. P.
1988 1959 1983 1960 1966
J. atmos. terr. Phys. 50, 519. Phys. Res. Lett. 3,340. Adv. Space. Res. 2, 205. J. geophys. Res. 65, 393 1. The Particle Kinetics of Plasmas. Addison-Wesley,
Singer W., Taubenheim J. and Bremer J. Smith L. G., Walton E. K. and Mechtly E. A. Thorpe M. R. Thrane E. V. and Piggott W. R.
1980 1978 1971 1966
J. J. J. J.
International Reference Ionosphere-IRI 90, Report NSSDC/WDC-A-R and S 90-22, Greenbelt, Maryland, U.S.A. Radio Sci. 69, 19 1. The Propagation of Radio Waves. Cambridge University Press, Cambridge. Appl. Sci. Res. 6B, 144. J. atmos. terr. Phys. 46,403. J. atmos. terr. Phys. 46, 399. J. atmos. terr. Phys. 48,41. J. atmos. terr. Phys. 33, 1893.
International Reference Ionosphere-IRI 79, Report UAG-82, WDC A, Boulder, Colorado, U.S.A.
U.S.A. atmos. atmos. atmos. atmos.
terr. terr. ferr. terr.
Phys. Phys. Phys. Phys.
42,241. 40, 1185. 33, 1597. 28, 721.