Ionisation below the night F2 layer—a global model

Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 1035 – 1052

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Ionisation below the night F2 layer—a global model J.E. Titheridge Physics Department, University of Auckland, City Campus, Private Bag 92019, Auckland, New Zealand Received 29 November 2002; accepted 8 May 2003

Abstract A full ionospheric model, including the four night-time sources identi2ed by Strobel et al. (1980), is used to calculate electron density pro2les throughout the E; F1 and F2 regions. Full allowance is made for the e4ects of secondary production, atmospheric winds, and a realistic NO model. Near midnight, results show an E-layer peak at 105 km with a density of 2– 2:6 × 103 cm−3 under most conditions. The peak thickness corresponds to a scale height of ≈ 5 km. A wide valley, with a mean density of typically 1.2–1:6 × 103 cm−3 , extends from ≈ 120 km to the sharply de2ned base of the F2 layer at 190 –225 km. This agrees with rocket and backscatter observations, but di4ers considerably from the deep, narrow valley assumed in the International Reference Ionosphere. IRI also has a night E layer that is far too thick (and too dense, near sunset). Experimental data on densities in the night E and F1 regions are available for only a limited range of conditions. Model calculations are therefore used to study the changes that will occur under di4erent conditions, due to known changes in atmospheric composition and EUV radiation. In particular we examine electron density pro2les across sunrise and sunset, when changes are rapid and observations are di:cult. A numerical model is derived to give a close approximation to the full theoretical calculations under all conditions. It begins at 80 km, to include some realistic D region ionisation, and extends into the lower F2 region. Combined with the day model described previously, it is available as a FORTRAN program that gives smooth, physically based variations at all times of day, at all heights up to at least 230 km. Results are obtained as a function of local time, height, latitude, season, solar
1. Introduction A full ionospheric modelling program was used to obtain a global model for the peak density (Nm E) and height (hm E) of the ionospheric E layer, as a function of L.T., latitude, season and solar cycle (Titheridge, 2000). For day conditions, this has been extended by theoretical calculations of pro2le shape in the E and E–F valley regions. Results were compared with experimental data, and used to derive a global, analytic model describing changes in the lower ionosphere under a wide range of conditions (Titheridge, 2002). E-mail address: [email protected] (J.E. Titheridge).

The present paper deals with the more di:cult problems at night. With no direct solar production, changes are larger and less well de2ned. Experimental data is also very limited, for the night E and F1 regions, since densities are too low for routine observations by ionosonde or most backscatter radars. Such data as are available cover only a limited number of di4erent geophysical conditions. Most backscatter results are from Arecibo, at 18◦ N, for a total of around 100 days and fewer nights. Rocket pro2les are from a few sites at speci2c times. A careful analysis of selected sweep-frequency ionograms can give an estimate of the total amount of ionisation beneath the F region, but gives little or no information on the height distribution; indeed ionogram analysis normally requires some model for the E and

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valley regions to get accurate results for the upper layers (e.g. Titheridge, 1975, 1986, 1988). Thus the present paper has three main aims: 1. To see whether current theory is adequate to explain observed densities in the lower ionosphere at night. 2. To obtain new information showing how densities in the night E and F1 regions will vary with height, local time, latitude, season and solar
rocket data (Section 3.2) con2rm the overall shape and density of the night valleys obtained for quiet mid-latitude conditions. These di4er considerably from night pro2les in the International Reference Ionosphere model, which is based on inadequate data (Section 3.2). 2. Sources of ionisation at night Strobel et al. (1974) investigated the night time ionisation that would result from EUV radiations from both terrestrial and extraterrestrial sources. The 2rst of these, the geocorona, consists of solar radiation (mainly Ly and Ly ) that is resonantly scattered through the earth’s atmosphere into the night sector. Changes with solar zenith angle were calculated using a full scattering model, and mean intensities adjusted to agree with satellite observations. The extraterrestrial source is from resonant scattering of solar radiation by interplanetary hydrogen and helium. This gives an isotropic source that is independent of , but has an annual change through a range of about 8:1 with a maximum in December. The mean intensity was again set to agree with mean satellite data. Calculations showed that these sources could account for the observed ionisation densities at night. Particle precipitation is a major source for the polar E region, but does not seem to be signi2cant at lower latitudes (Morse and Rice, 1976). A meteoric source has also been claimed, for the night E layer, but this seems both unnecessary and unlikely (Titheridge, 2001). In the current work, the intensity of the solar radiations is chosen to give mean production rates that agree with those calculated by Strobel et al. (1980). A solar cycle variation is included, corresponding to that in the appropriate lines of the solar spectrum (from the EUVAC radiation model of Richards et al., 1994). Changes with height are de2ned by Eq. (1) below. Changes with zenith angle  (for the geocorona), or with day number (for the interplanetary source), are as described in Titheridge (2000). Strobel et al. (1980) also examined the possibility of non-solar sources. Signi2cant contributions were found from starlight, and from photons produced in the F2 region. The I was total starlight intensity, at wavelengths of 911–1026 A, determined by adding the contributions from all signi2cant stars (O and B types) in the SAO catalogue. Due to neglect of some absorption e4ects, it was estimated that this result could be too high by up to 50%. For the present work the Strobel et al. intensities are assumed, with the height variation calculated from Eq. (1). Starlight produces mainly O+ 2 ionisation in the E region (Fig. 1). Recombination of O+ 2 is quadratic, so a reduction of 30% in the assumed starlight intensity would give a decrease of ¡ 15% in the calculated peak density Nm E. The 2nal source identi2ed by Strobel et al. (1980) is photons produced in the F2 region by the direct radiative recombination of O+ ions and electrons. This airglow is appreciable at tropical latitudes, below the peak of the F region

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(a)

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(b)

Fig. 1. Calculated pro2les at midnight, using only one of the night production sources. (b) is for conditions when all sources are stronger. Solid and 2ne lines are at F10:7 = 80 and 180 respectively.

anomaly. The photons come from an e4ectively isotropic source at a mean height (about 400 km) near the centre of the night F2 region. Downgoing photons will produce O+ ions in the F1 region, at heights above 150 km (Fig. 1). Strobel et al. used a 2xed intensity obtained from satellite observations of the photon
the F2 layer density is much greater than that at normal latitudes. For the current work the intensity of the isotropic source is set equal to the total number of direct radiative recombinations that occur in a unit column extending through the F2 region. This number is obtained from the full ionospheric model, which provides a good representation of the F2 layer throughout the night (Titheridge, 1993). It gives a well-de2ned value (or possibly an upper limit) for the strength of the e4ective source, and is generally much less than 15 R. The height of the source is taken as 400 km (as in Strobel et al.), but this is not critical since absorption of I radiation is small at heights above 250 km. the 911 A For each omnidirectional source, height variations are determined by calculating the vertical absorption of each wavelength in a horizontally strati2ed atmosphere. The downcoming radiation is assumed to be spread equally over a hemisphere of unit radius. This is sliced with horizontal planes at distances of 0.2, 0.4, 0.6, 0.8 and 1.0 below the central plane. Each slice includes 20% of the area of the hemisphere, and hence 20% of the downcoming radiation. The central rays in each Section are at a downcoming angle  (to the vertical) such that cos  = 0:1, 0.3, 0.5, 0.7 and 0.9. If Ao is the absorption integral for a vertical ray, the corresponding integrals for the oblique rays (over the same height range) are then Ao =0:1; Ao =0:3; Ao =0:5; Ao =0:7, and Ao =0:9. Thus the radiation intensity at a height h is Ih = 0:1I400 [exp(−Ao =0:9) + · · · + exp(−Ao =0:1)];

(1)

where I400 is the assumed (omnidirectional) source intensity at 400 km. For a spherical atmosphere the values of cos  should be replaced by 1=Ch(), where Ch() is the Chapman grazing incidence function, giving slightly larger values of Ih . The di4erence is negligible for cos  ¿ 0:3 ( ¡ 72:5◦ ), and very little radiation reaches the F1 region at zenith angles larger than this.

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The relative strength of the di4erent sources, at 40◦ N, is illustrated in Fig. 1(a). This shows the electron density pro2le that would be produced by each source, acting alone, at two di4erent levels of solar activity. Starlight dominates at all heights up to 180 km. There is a slight decrease near solar maximum, due to changes in the neutral atmosphere. Next important are the airglow sources (terrestrial and then planetary). These show some increase with solar activity, although this is reduced by changes in the neutral atmosphere. The F2 region source is generally negligible, except for low latitudes in summer, solar maximum. The major change near solar maximum is an increase of 25 km in the height of the base of the F2 layer; this height is sharply de2ned at all times and does not depend on the night production. Fig. 1(b) shows the ionisation produced for summer conditions in the southern hemisphere, when each source is near its maximum. Starlight intensities are ≈ 3 times larger in the south, at mid-latitudes, giving a large increase in the calculated values of Nm E. Airglow production increases in summer, when solar zenith angles are smaller. Scattering from the interplanetary gas increases by a factor of 2–3 in December (and decreases in June) because of the changing angle to the interstellar wind. The F2 recombination source is also enhanced in summer, when F2 densities are large, and can become a major source in the F1 region for low latitudes near solar maximum (Fig. 4a). Seasonal variations in NE are reduced by increased NO densities in summer, giving an increase in loss rates. 3. Densities at midnight 3.1. Calculated pro2les Fig. 2 shows theoretical pro2les calculated for equinox conditions at latitudes of 20 –70◦ N. All show an E region peak at 105 km, with a rapid decrease below and a somewhat slower decrease above this height. A low-density region extends from the 2rst valley minimum, near 133 km, to a second minimum at 200 –210 km. There is then a rapid increase in density corresponding to the base of the F2 layer, with a similar gradient at all latitudes from 20◦ to 60◦ . The pro2les change little with latitude, apart from increased E region densities (caused by the increase in starlight radiation) and a lower F2 layer at 20◦ N. Results for midlatitude conditions at summer, winter, solar minimum and solar maximum are shown in Fig. 3. Dotted lines are from the analytic model of Section 5; this uses a scale constant HF ≈ 21 km in Eq. (6) giving F-layer gradients in good agreement with the theoretical results (solid and broken curves). The agreement continues to heights of over 280 km, giving a realistic start to the night F2 layer. Changes at other latitudes are illustrated in Fig. 4. Seasonal e4ects are generally small (as at Arecibo, Titheridge, 2001). Most pro2les show a wide valley extending up to

Fig. 2. Model results for equinox conditions at local midnight, using winds from the HWM90 model and a solar
the base of the F2 layer, where densities increase rapidly with a gradient that is almost independent of latitude, season and solar
J.E. Titheridge / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 1035 – 1052

(a)

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(b)

Fig. 3. Calculated pro2les at midnight, for typical summer, winter, solar maximum and solar minimum conditions at a latitude of 40◦ N. Continuous and broken lines are from the full ionospheric model. Dotted curves show the best-2t variations obtained using Eqs. (4)–(5).

(a)

(b)

(c)

Fig. 4. Pro2les at midnight for latitudes of 25◦ (dashed) and 55◦ (chain). Long and short segments are for solar minimum (F10:7 = 80) and solar maximum (F10:7 = 180) respectively.

night, but arises from the subsequent change to hE =110 km with no corresponding correction at lower heights. Above hE , IRI-95 gives a rapid decrease to a minimum at hV = 138 km, with NV = 0:19NE . The valley width hT − hE is 2xed at 57 km for a dip latitude of 20◦ , and 66 km at 50◦ , giving the deep, narrow valley shown in Fig. 5. The default F2 layer in IRI-95 gives pro2les shown by the chain lines. A more recent option (due to Gulyaeva, 1987) gives the dotted lines, in better agreement with present results for the F region. Results from 59 rocket measurements in middle latitudes, under generally quiet conditions, are reviewed by Chasovitin and Nesterov (1976). They plotted densities at 8 2xed heights as a function of zenith angle . Results were roughly constant from  = 140◦ in the evening to  ≈ 110◦ before sunrise, and mean values over this interval are shown

as solid triangles in Fig. 5. These means are only approximate, since results from di4erent rocket
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Fig. 5. Comparison of midnight pro2les from IRI-95 (chain lines) and results from the present model, for equinox conditions at latitudes of 20◦ N and 50◦ N at F10:7 = 120. Dotted lines are from IRI-95 using the alternative Gulyaeva values for the bottomside thickness parameter B0. Triangles are mean results from a study of rocket data by Chasovitin and Nesterov (1976).

height range 130 –190 km, and an approximate mean density for this interval is shown as Nval in Table 1. The sharp corner at the top of the valley region is shown as hU in Table 1 (and Fig. 11). The incoherent scatter radar at Arecibo (18:5◦ N geographic, 30◦ N magnetic) can measure night densities with reasonable accuracy. Published results show a valley with a
the valley is similar to that obtained from the present theoretical model, at latitudes of 30 –50◦ . Only two exceptions were found to this picture, both in pro2les derived from the analysis of sweep-frequency ionograms. This analysis requires some assumptions about the valley region, which is not directly observed, and can give little more than the total amount of ionisation below the F2 layer (e.g. Titheridge, 1975, 1985). Knight (1972) used a collection of about 60 night time pro2les, from ionogram, rocket and incoherent scatter, to derive idealised mean pro2les for di4erent intervals. Between 1 and 9 h after sunset, increased mean densities near 180 km were caused entirely by the inclusion of ionosonde pro2les (about 1=3 of the total); all but 1 of the non-ionosonde pro2les show low valley densities extending to heights over 200 km. Some pro2les also show the intermediate layer, at a height of 130 –150 km, with a peak density similar to or greater than the night E peak. Many experimental pro2les are similar in shape to the IRI-95 model (Fig. 5) at heights up to 150 km. This occurs when there is an intermediate peak, produced by vertical drifts which also give low densities near 120 km. In all cases, a peak near 140 –150 km is followed by a trough extending up to the bottom of the F2 layer (near 200 km). The V-shaped valley assumed in the IRI-95 model is based on a careful study of rocket pro2les by Maeda (1971). All the data considered were, however, limited to heights below 150 km, so the assumed upper limit of ≈ 160 km for the valley (in IRI-90) is not an observed feature. More recently Mahajan et al. (1997) used 330 pro2les from Arecibo to derive parameters for the night E region. For  ¿ 120◦ , valley widths were nearly all in the range 80 –140 km. Thus most valleys extended to 190 –230 km, near midnight, in full agreement with the present theoretical results. The valley minimum was generally in 120 –130 km, giving an overall shape similar to the pro2les in Figs. 1–5. 3.3. Changes with magnetic activity The mean valley density (Nval ) in Table 1 shows no clear variation with solar
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Table 1 Experimental data on densities in the E–F1 region, near midnight, as plotted in Fig. 6 Reference

Method

Location latitude

Date

F10:7

KP

Nval 103 cm−3

hU km

Notes

Chasovitin and Nesterov (1976) Geller et al. (1975)

Rocket

Mid-lat (north)

1968–71

≈ 140

≈2 8

1.6 11

(200)

Mean from 59
Rocket

Wallops 38◦ N

1968

77 80 140 173 102 84 100

0.3 1.0 2.3 3.3 8 1.5 5.7

0.55 1.3 1.6 2 14 2.1 3

¿ 180

High resolution 9
¿ 180

High resolution 2
Smith and Gilchrist (1984) Aikin and Blumle (1968) Rowe (1974)

Rocket

Wallops 38◦ N

1977

Rocket, Huancayo ISR

Dip-equator

March 1965

73

2.5

1.7

210

(1
Arecibo 19◦ N

1972

≈ 100

ISR

Arecibo 19◦ N

1972

≈ 120

210 220 200 220 —

Shen et al. (1976)

ISR

Arecibo 19◦ N

April 1974

≈ 86

Taylor (1974)

ISR

1968–1970

≈ 150

Wakai (1967)

Low-freq. ionograms

Malvern 52◦ N Boulder 40◦ N

0.5 0.8 1.7 7 1.1 1.1 1.6 0.4 1.2 1.5 1.45 ≈2

4 nights, at di4erent KP

Trost (1979)

1 2 4 8 1.5 1.8 4 0.8 4 4.5 5 ≈2

1960

Ionogram, Rocket, ISR

0 –52◦ N

1958–1970

0.6 3.8 6 ≈ 1:8

0.6 3.5 5 1.85

≈ 220

Knight (1972)

165 210 160 ≈ 120

200 –220

Smoothed mean pro2les

Model Model

IRI-95 Current theory

Low to mid-lat 20 –60◦ N

— —

Any 80 180

Any Any

( ≈ 1:5) 1.35 1.45

(150) 195 220

All seasons Mean equinox

200 200 210 200 210

4 nights, to 140 km only 4 nights, low resolution Mean, approximate

Nval is the mean density in 130 –190 km, and hU the height of the upper valley corner.

activity. This suggests some consistent bias in Arecibo results, which are near the detectable limit, or lower mean densities at Arecibo (latitude 18◦ N) compared with the mid-latitude rocket sites (mostly near 40◦ N). A search for low-latitude rocket data found two
mean variation with KP is well approximated by the solid line, corresponding to Nval = 0:7 exp(0:33KP ):

(2)

Model calculations depend on KP only through small changes in the MSIS86 atmospheric model (Hedin, 1987). An increase of KP from 0.5 to 6.5 has little e4ect at heights below 140 km, but gives an increase of 5 –25% in density at the top of the valley region. There is also an increase of ≈ 15 km in the height of the base of the F2 layer. All calculations in this paper are for typical quiet conditions, with Ap = 10 (KP ≈ 1:7). Mean valley densities at latitudes of 25 –60◦ (Fig. 2) are shown by the horizontal line ‘Model’ in Fig. 6. This intersects the experimental result (heavy line) at KP ≈ 2:0. Model results are therefore adjusted to match the mean experimental data, and show the observed

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Fig. 6. Experimental values for the mean density in the valley region (130 –190 km), as a function of magnetic activity, from the data of Table 1. Sets with 3 or more points are connected, and solid symbols indicate the more reliable data from rocket pro2les. The solid line is a mean variation from Eq. (2).

changes with magnetic activity, using a scaling factor FKP = exp[0:33(KP − 2)]:

(3)

Most observations show that a change in KP has little effect at heights below 120 km, but is increasingly important at greater heights. Under very active conditions the minimum near 130 km almost disappears, and densities are large and fairly constant above 160 km. To include these changes in model calculations, variations with FKP are reduced by 85% at the lower valley height (hV ≈ 130 km) and increased by 40% at the upper limit (hU ≈ 210 km) as described in Section 5.2. hV is also varied slightly with KP , to give the pro2les shown in Fig. 7. These have the same general shape as mean rocket data. Valley densities become fairly constant above 160 km, with a mean value that almost doubles for each increase of 2 in KP (as required by Fig. 6). There is little change in the upper valley height hU , in agreement with the data of Table 1. Reasons for the variations in Fig. 7 are not known, since they are not reproduced by theoretical calculations. Possible causes include large changes in neutral composition (not included in the MSIS86 model), large vertical movements of ionisation in the F1 region, or some additional source of ionisation (corpuscular?) during disturbed periods. 4. Changes across sunset and sunrise Pro2le changes across sunset and sunrise, for typical mid-latitude conditions, are shown in Fig. 8. For near-day

Fig. 7. Pro2les calculated from the equations in Section 5.2, for mid-latitude equinox conditions at di4erent values of KP .

conditions ( ¡ 85◦ ), NE is slightly larger at sunrise, when decreased NO densities give decreased loss rates. At larger zenith angles pro2les depart strongly from the normal daytime shape. There is a smooth progression for zenith angles up to about 95◦ , as the O+ 2 production peak rises and the night E layer appears. The ionospheric time constant (proportional to 1=N ) increases as the ionisation decays, so that densities are generally smaller after midnight (the 2ne lines). The base of the F layer also rises by 20 –50 km after midnight. During the main sunset transition, at  ≈ 80–95◦ , the pro2le shape is governed by local production and loss and varies little with latitude or with season. Fig. 9 shows calculated pro2les for three seasons, three latitudes and two levels of solar activity, at a 2xed solar zenith angle  = 90◦ . The top curve in each plot (broken line) is at 55◦ latitude in summer, and shows the e4ect of increased F2 region heights. Otherwise the results for di4erent latitudes and seasons are reasonably similar, particularly near solar maximum. There is some seasonal separation at solar minimum, with increased loss rates producing lower densities in summer. Apart from some vertical expansion, the pro2les at F10:7 = 180 correspond closely to a 2xed increase of 36% on the densities at F10:7 = 80. Thus the sunset densities vary approximately as (F10:7 + 40)0:5 , in agreement with the change found at noon (Titheridge, 1997). A single mean pro2le, scaled for changes in F10:7 , can therefore give a good representation of electron density pro2les at  = 90◦ under a wide range of conditions. The analytic pro2le used at night (Fig. 11 below) allows a good 2t to calculated pro2les throughout the night. There are no true valleys for  near 90◦ , but comparison with a linear increase (dotted in Fig. 9) shows well-de2ned ‘minima’

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shown by the dotted line in Fig. 10(a). At larger values of , use of a single, mean pro2le for all latitudes and seasons is adequate up to about  = 90◦ . From this point there is a signi2cant seasonal change (apparent in Fig. 9a), with higher densities in winter; this is very marked at  = 98◦ (Fig. 10d). The change is due to decreased loss rates in winter, and the faster variation of  with local time. The winter increase is largely absent from corresponding plots for the sunrise period. There are also small seasonal changes in the densities NV and NB near sunset, and in the gradient at the base of the F region. Equations in Section 5.3 incorporate these variations, to de2ne analytic pro2les (as plotted in Fig. 10) which include all main features of the model results. 5. An analytic model 5.1. The form of the pro2le Fig. 8. Calculated pro2les near sunset (heavy lines) and sunrise (thin lines), for equinox conditions at 40◦ N; F10:7 = 120.

at hV ≈ 112 km and hU ≈ 170–180 km. The bulge between these heights is well represented by the modi2ed sine variation of Eq. (5), using a ¡ 1 to give an asymmetrical peak. The analytic model can then match theoretical pro2les to good accuracy, as shown by the plotted points in Fig. 9. Pro2les calculated for other values of zenith angle are shown in Fig. 10. At  = 78◦ there is still a normal daytime valley, with a small depth but a width W ≈ 20 km. This pro2le is well 2tted by the daytime equations derived previously (Titheridge, 2002). The night equations of Section 5 can also give a reasonable, smoothed approximation as

(a)

Theoretical pro2les were calculated for a wide range of di4erent conditions, and used to derive an analytic model for the variation of electron density with height. The model has three sections that join smoothly to give an overall pro2le as shown in Fig. 11. The 2rst section begins at 80 km and includes the E layer up to the 2rst valley minimum. It is de2ned fully by the valley point (hV ; NV ), and a parameter Nn that determines the size of the night production peak (at a 2xed height hn = 104:5 km). The second section is a linear increase in density from (hV ; NV ) to the upper valley point (hU ; NU ), plus a smooth “bulge” of amplitude NB . Above hU we have the start of the F2 region, which is well represented by a single term in (h − hU )3 .

(b)

Fig. 9. Sunset pro2les at  = 90◦ , for low and high levels of solar activity. Each plot has 9 curves, for summer, equinox and winter at latitudes of 25◦ N; 40◦ N and 55◦ N. Plots shown as • (for summer) and + (winter) are from the equations of Section 5.

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(a)

(b)

(c) (d)

Fig. 10. Calculated pro2les across sunset, for summer (broken), equinox (solid) and winter (chain lines), at latitudes of 25◦ N; 40◦ N and 55◦ N with F10:7 = 120. Plots shown as •; + are summer, winter results from the numerical model of Eqs. (8)–(11), which allow for some additional D region ionisation.

At heights below hV , the model is de2ned by
2 cos[0:01 (129 − h)] N (h) = NV cos[0:01 (129 − hV )]   hn − h + Nn cos2 gn at 80 ¡ h ¡ hV ; 2Hn gn = 1 +

hn − h hn − h z



(4a)



Hn −1 |hn − hz |

(4b)

with hz = 80 km at h ¡ hn , and hz = hV at h ¿ hn . The 2rst term in Eq. (4a) gives an underlying cos2 variation, shown as a dashed line in Fig. 11. This reaches a maximum at 129 km, which is near hV for all midnight pro2les. The second term adds a sinusoidal peak with scale height Hn . The shape of the night E layer changes little with season, latitude or solar
at hV ¡ h ¡ hU : (5)

The 2rst two terms de2ne a linear increase with gradient GU =(NU −NV )=(hU −hV ), shown by the dashed line in Fig. 11. The last term adds a “bulge” between zeros at hV and hU , similar to the cos2 variation in Eq. (4a). Midnight pro2les use a = 1:0, for a symmetrical bulge, but lower values give improved 2ts near sunrise and sunset (Section 5.3). Above hU , the F2 layer gives a rapid increase in density. The gradient here depends primarily on di4usion and loss rates in the neutral atmosphere, and is approximately constant at all latitudes from 20◦ to 60◦ . The increase is well 2tted by a cubic in (h−hU )3 , using a scale factor HF =21 km. Thus the 2nal section of the pro2le is N (h) = NV + GU (h − hV ) + NV [(h − hU )=HF ]3

at h ¿ hU :

(6)

The analytic model is 2tted to pro2les obtained from the full ionospheric modelling program, adjusting the di4erent parameters to obtain the best overall 2ts. This was done for summer, equinox and winter conditions at latitudes of 10 –70◦ , for solar
(7)

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1045

these changes are approximated by hU = 174 + 0:01(s − 1:7)(&d − 46)2 + 0:22F10:7 km: (10) This reproduces calculated values of hU , which vary over a range of 40 km, to within ±8 km. The corresponding density NU is de2ned using the gradient GU = (NU − NV )=(hU − hV ): GU varies from about −10 to +6, in units of cm−3 km−1 , and is approximated (with an rms error of 1.1 and a maximum error of 2.2) by GU = (1:37F10:7 − 222)=&d + 0:075&d (1:7 + s + s2 ) + 1:4s(s − 2) − 7:4 cm−3 km−1 :

(11)

The bulge amplitude (NB ) in Eq. (5) ranges from −100 to +300 cm−3 , with the largest seasonal change at low latitudes. The changes are reproduced to within 10% by NB = 220 + 1:6[F10:7 (0:7 + s)=&d ]2 −0:39(&d − 47)2 cm−3 : Fig. 11. The construction of the model pro2le for night conditions. Dashed lines omit the last terms in Eqs. (4a) and (5). The dotted pro2le is obtained from the ionospheric modelling program, for equinox conditions at 40◦ N and F10:7 = 160. Triangles are from the 2xed IRI-95 model for the night D region.

where Dy is the day number. To obtain a continuous variation between the northern and southern hemispheres, the value of s is multiplied by a factor sin|2&| at latitudes of less than 45◦ .

Above hU , the start of the F2 region is well represented by the simple cubic of Eq. (6) using a 2xed scale factor HF = 21 km. Pro2les obtained using these equations are shown as dotted lines in Fig. 3. They give a good overall representation of the calculated pro2le shapes near midnight under all conditions, for heights up to at least hU + 50 km. Experimental data show that densities in the upper valley region increase rapidly under disturbed conditions (Section 3.3). To include these changes, parameters are adjusted using the factor FKP of Eq. (3). Using a prime to denote values adjusted for changes in KP , we have Dk = exp[0:33(KP − 2)] − 1;

5.2. Model parameters at midnight



The lower section of the theoretical pro2les is 2tted by Eq. (4), using 2xed values of peak height hn = 104:5 km and scale height Hn = 5:0 km. The corresponding density Nn is nearly constant, at 1650 (±200) cm−3 , since at this height the night ionisation sources show little change with latitude, season or solar cycle. The small changes are well approximated by

× (0:4&d + 10s + 5s2 + 0:08F10:7 − 40) cm−3 :

(8)

The lower valley minimum (hV ; NV ) also shows little variation. Thus we use a 2xed value hV = 133 km at midnight, with Nv = 2350 − 0:4&d (100 − &d ) − 180s2 cm−3 :

(9)

The second pro2le section (Eq. (5)) is a linear increase from hV to the upper valley point (hU ; NU ), plus a ‘bulge’ of amplitude NB . Expansion of the neutral atmosphere causes hU to increase in summer, particularly at low latitudes, and near solar maximum (Fig. 4). For conditions at midnight

(13)

Nn = Nn (1 − 0:05Dk );

(14)

NV  = NV (1 + 0:15Dk );

(15)

NU  = NU max(1 + 1:4Dk ; 0:4);

(16)

NB  = max(NB + 0:5Dk NU ; 0);

(17)



(18)



(19)

hV = hV − 2:5Dk ; hU = hU + Dk :

Nn = 1700 + (&d − 30)

(12)

The decrease in Nn combines with the increase in NV to give a slight increase in the density of the night peak near 105 km. The upper densities NU are used to obtain the gradient GU = (NU − NV )=(hU − hV ) used in Eqs. (5) and (6). 5.3. Changes with zenith angle Plots such as those shown in Fig. 10 were determined for zenith angles  from 75◦ to 140◦ . For each plot, the mean pro2le was approximated by the night Eqs. (4)–(5) of Section 5.1, adjusting the parameters hV ; NV ; hU ; GU ; NB and a to obtain the best overall 2t (the dotted lines in Figs. 9–11). Most parameters showed signi2cant changes

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J.E. Titheridge / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 1035 – 1052

with zenith angle, but not with latitude. Changes are slower at large zenith angles, so we use an independent variable

The bulge shape parameter a is the same for sunrise and sunset, and is equal to 1.0 for X ¿ 0:036. This changes to

X = (1 − cos )0:5 − 1

a = 0:2 − 0:035=(X − 0:08)

(20)

◦

at X ¡ 0:036:

(25)

giving X =0 at =90 . Analytic approximations for each parameter are derived, as a function of X , for summer and winter conditions and values of  from 75◦ to 160◦ (X = −0:14 to 0.39). Pro2le shapes change rapidly across the sunset period, so many parameters use di4erent approximations above and below a zenith angle  ≈ 96◦ (X ≈ 0:05). Sunset and sunrise periods are distinguished using a variable r = 1 before midnight and r = 0 after midnight. To eliminate diurnal variations at the poles, r is scaled as cos(3& − ) for latitudes & ¿ 60◦ . It is also decreased when the solar zenith angle  at noon is less than 60◦ , so that normal day e4ects are reduced. Most parameters show little change with season, other than the pre-midnight increase described by Eq. (29); this is also reduced for high latitudes (and large daytime zenith angles) using the variable r. All heights are in km, and densities in 103 cm−3 . The night production term gives a 2xed density Nn = 1:8 × 103 cm−3 , at hn = 104:5 km and Hn = 5:0 km in Eq. (4). The height of the lower valley point is given (to within ≈ 1 km) by

Before midnight, calculated densities show a signi2cant seasonal variation at heights of 95 –195 km. This is caused by changes in NO giving decreased loss rates and higher densities in winter, at zenith angles of about 90 – 130◦ (Fig. 10). To allow for this, when r ¿ 0 and  ¿ 84◦ the densities N (h) from Eqs. (4) to (6) are increased (at 95 ¡ h ¡ 195 km) by an amount

hV = 118:5 + 52X − 53X 2

hU = 155 + 325X 2

at X ¿ 0:028;

hV = 103:4 + 16X + 9:5(1 + 5X )4

(21a)

at X ¡ 0:028: (21b)

The density at this height (in 103 cm−3 ) is NV = 2:3 − 5:2X + 7X 2

at X ¿ 0:049;

NV = 6:1 + 1:7r − (83 − 17r)X

at X ¡ 0:049:

(22a) (22b)

For r ¿ 0, and 0:0 ¡ X ¡ 0:049, the quantity −44rX is added to Eq. (22b). The density at the upper valley point (hU ) is de2ned by the mean gradient GU = (NU − NV )=(hV − hU ). This shows little change between the sunrise and sunset periods, and is obtained (in cm−3 km−1 ) from ln(GU + 8:2 + 1:8r) = 0:4=(X − 0:15) + 8:78

at X ¡ 0:0444;

(23a)

at X ¿ 0:0444:

(23b)

The density between hV and hU is increased by the 2nal ‘bulge’ term in Eq. (5), with an amplitude NB = 2X − 0:6

at X ¿ 0:0316;

NB = 7:5 − 245X − 300X 2

at X ¡ 0:0316:

(24a) (24b)

After midnight (r = 0), or at r ¡ 1, these values of NB are increased to NB = NB + 10(1 − r)(0:3 − X )2 :

(24c)

(26)

where Bs is the seasonal amplitude given by Bs = 2:376 − 12X + 15X 2 Bs = 0:66 + 12X

at X ¿ 0:075;

at − 0:05 ¡ X ¡ 0:075:

(27a) (27b)

More complex equations were developed to give a good 2t to the parameters which de2ne the lower F2 region: the upper valley height hU and the height scale factor HF . Thus the variations of hU , over a range of 40 km, are 2tted to within a few km using several di4erent expressions. For near-day conditions, we have at X 6 − 0:11:

(28a)

After midnight (r = 0), and for X ¿ − 0:11, we use hU = 165 + 240X + 1700X 2 at − 0:11 ¡ X ¡ 0:066: hU = 183 + 83X − 115X 2

at X ¿ 0:066;

(28b) (28c)

Before midnight (r = 1), and X ¿ − 0:11, hU = 165 + 123X − 700(X − 0:11)X 3 :

(28d)

At high latitudes, where 0 ¡ r ¡ 1 before midnight, we use a weighted average of the results for r = 0 and 1. Optimum values for the F layer parameter HF in Eq. (6) increase by about 10 km after midnight, and increase at lower values of . These changes are described well by HF = 27 − 8:8r − 20(5 − r)X − 100X 2 at X 6 0:113 − 0:06r 2 ;

ln(GU + 8:2 + 1:8r) = 0:06=(X − 0:025) + 1:9

VN (h) = Bs r(1 − s) sin[( =100)(h − 95)]2 ;

(29a)

HF = 14 + r(3:8 − 88X ) + (476 + 174r)X 2 at X ¿ 0:113 − 0:06r 2 :

(29b)

The height scale also has an appreciable seasonal change, varying with the temperature of the neutral atmosphere. The decrease in winter is de2ned by a factor SF , varying from about 0.5 to 0.9. Thus we replace HF by HF SF in winter, and HF =SF in summer. After midnight, SF is given by SF = 0:9 − 0:8X SF = 0:83 + 1:75X

at X ¿ 0:028; at X ¡ 0:028:

(30a) (30b)

J.E. Titheridge / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 1035 – 1052

For conditions before midnight, when r ¿ 0, the seasonal factor is increased to SF + r{[0:918 − 2(X + 0:1)2 ]2 + 1:1X }:

(30c)

The above results are derived from theoretical calculations at F10:7 =120. Pro2les were also calculated at F10:7 =80; 160 and 200. These showed mean densities varying approximately as (F10:7 + 40)0:5 , in agreement with the changes in daytime densities (Titheridge, 2000). To include this e4ect, densities above hE are corrected using a factor FF = [(F10:7 + 40)=160]0:5 :

(31)

The correction is smallest at hV and largest at hU . There is also an increase in F layer heights, particularly after midnight, towards solar maximum. These changes are reproduced well by the adjustments NV = NV FF 0:5 ;

(32)

NB = NB FF 0:5 + 0:16(FF − 1);

(33)

GU = GU FF + 140(FF − 1);

(34)

hU = hU + (13 + 26r)(FF − 1):

(35)

5.4. A full diurnal model At  ¿ 95◦ , theoretical pro2les show increasing variations with season and with latitude which are not fully included in the results of 5.3 (apart from the pre-midnight winter increase Bs ). The midnight equations of Section 5.2 do include appreciable changes with latitude, season, solar cycle and magnetic activity. Both sections use the same basic model, so parameters can be merged to give a smoothly varying series of pro2les with a realistic dependence on geophysical conditions throughout the night. The merging is carried out over a range  ¡  ¡ 00 , where 00 is the solar zenith angle at midnight. The starting point  must decrease when 00 is small, and is de2ned by  = 35 + 0:500 deg:

(36)

We will use -() to represent any of the pro2le parameters calculated as a function of  from the equations of Section 5.3. At midnight, this becomes -(00 ). Let -00 be the corresponding result from the midnight equations of Section A.2. Then -() is modi2ed by an amount that increases from zero at  =  to -00 − -(00 ) at  = 00 . Thus at zenith angle  we use a corrected parameter - = -() + [-00 − -(00 )]( −  )=(00 −  ):

(37)

 varies slowly near midnight, so Eq. (37) gives a smooth variation in which pro2les depend signi2cantly on latitude, season, solar
1047

valley depth drops to zero at  ≈ 82◦ under most conditions. The ‘day’ pro2le is satisfactory to larger zenith angles, however, since the model used retains a shallow valley for  up to 88◦ . The night model works well with or without a true valley minimum, and with slight adjustments to some parameters it gives a reasonable 2t to theoretical pro2les for values of  down to ≈ 70◦ (as in Fig. 10). Thus for  in 70 – 85◦ , theoretical pro2les can be represented by either model. For isolated pro2les it is adequate to use the day model at  ¡ 80◦ , and the night model at  ¿ 80◦ . When full continuity is required across sunrise and sunset, a smooth, continuous variation is obtained using a weighted mean for 70 ¡  ¡ 85◦ . In this region, the density N at each height is taken as N = w Nday + (1 − w )Nnit ;

(38)

where Nnit is from the equations in Section 5.3, merged with Section 5.2 across midnight, and w = ( − 70)=15: Nday is obtained from the daytime equations in Titheridge (2002). These include a small quadratic term above the valley to give a rough approximation to the F1 region. The present work adds a further factor at heights above 170 km, to simulate the F2 layer increase. Thus Eq. (38) uses a daytime density Nday that is equal to the value Nd from the equations in Titheridge (2002), for h ¡ 170 km, but increased to Nday = Nd {1 + [(h − 170)=50]2 }

at h ¿ 170 km:

(39)

6. Model results and comparison with IRI Changes in the strength of the neutral wind have a large e4ect on the F layer, with considerable day to day variations. The present calculations use an approximation to the HWM90 wind model (Hedin et al., 1991), which seems best for general use at mid-latitudes (Titheridge, 1995). For typical mid-latitude conditions, near midnight, the equatorward wind gives an e4ective upwards drift of 30 –45 m=s for all values of F10:7 . The drift will vary appreciably from day to day, and changes of 10 m=s alter the height of the lower F2 layer by 5 –12 km. Increased upward drifts also increase densities in the night valley region, above about 150 km. The e4ect is largest near solar maximum, when there is appreciable production in the F1 region; an increase of 10 m=s in the e4ective vertical drift then gives an increase of 30 – 45% in density at heights of 170 –220 km. Near solar minimum the change is less than 10%. With unpredictable variations of this order, the mean accuracy given by the numerical model seems all that is justi2ed at present. The reliability of the results is, however, born out by the good agreement between calculated and observed values of the upper valley limit (hU in Table 1). Pro2les from noon to midnight, for typical mid latitude conditions, are shown in Fig. 12(a). This displays the full transition from day conditions, when there is a small valley between the E and F1 layers, to the night pro2le with a wide,
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J.E. Titheridge / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 1035 – 1052

(a)

(b)

Fig. 12. Variations in electron density throughout the D; E and F1 regions, for conditions from noon ( = 40◦ ) to midnight ( = 140◦ ). All results are for equinox conditions at 40◦ N, with F10:7 = 120. Heavy lines in (a) are from the equations of Section 5, while dotted lines show results from the full theoretical model. (b) Shows corresponding results from the IRI-2001 model.

with N = dN=dh = 0 at 80 km, and give a fully continuous variation at all heights. Solid lines are from the numerical model of Section 5, for  ¿ 85◦ . Results at  6 70◦ use the daytime model (Titheridge, 2002), modi2ed as in Eq. (39). Results are similar from both approaches at 70◦ ¡  ¡ 85◦ , when a weighted average is obtained from Eq. (38). The analytic model then gives an adequate approximation to full theoretical results (the dotted lines in Fig. 12(a)) at all times. Fig. 12 shows clearly the rise and decay of the day production peak after sunset, with the gradual emergence of the night E layer near 105 km. Valley widths are in accord with the experimental data of Section 3.2, while the model reproduces the sharp upper boundary and the rate of increase generally observed at the base of the F2 layer. Reasonable agreement with theoretical calculations extends across the F1 region and to heights over 230 km, where simple adjustments to the quadratic representation can be made to match F2 layer pro2les. There is a consistent di4erence at low heights, where Eq. (4) gives increased ionisation at

h ¡ 100 km. This is done to approximate the D region ionisation in the IRI model, as in Fig. 11, since no D region is included in the current theoretical calculations. The peak height of the E layer is also increased by ≈ 0:5 km, compared with the theoretical curves, for better agreement with the data discussed in Titheridge (2000). The International Reference Ionosphere is a purely empirical model, designed to give an adequate representation of mean experimental data. It is widely used to determine electron densities at heights from 65 km (day) or 85 km (night) and extending into the topside F2 region. Di4erent mathematical functions have been used, by di4erent workers, to approximate data over di4erent height ranges. As a result there are some continuity problems. The latest version (IRI-2001, replacing IRI-95) claims to give an improved representation of the electron density in the region from the F peak down to the E peak (Bilitza, 2001). Results in the E and F1 regions, shown in Fig. 12(b), are reasonable only for zenith angles  of 70 –80◦ . The base of the F1 layer is

J.E. Titheridge / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 1035 – 1052

1049

(a)

(b)

Fig. 13. Model electron density pro2les at 18 h L.T., for latitudes of 90◦ N to 90◦ S. All calculations are for June solstice conditions near solar maximum (F10:7 = 200). (a) Shows pro2les from the equations in Section 5, with the zenith angle at each latitude shown in italics. (b) Gives corresponding results from the IRI model.

far too high near noon, when it does not match the top of the E–F valley and gives unrealistic variations at  ¡ 70◦ . The valley, F1 and F2 pro2les have been matched at midnight, to give a reasonably smooth variation (at  = 140◦ , in Fig. 12b). However the matching disappears at  ¡ 140◦ , giving unacceptable pro2les with large gradient discontinuities at most times. IRI-2001 gives some increase in the width of the valley at night, compared with previous models, but the start of the F2 layer is still too low by 30 –60 km. The deep, V-shaped valley does not match the wide,
Pro2les at latitudes from 90◦ N to 90◦ S are shown in Fig. 13(a). These are calculated for June conditions at 18 h L.T., near solar maximum. Most pro2les depend primarily on the value of  (shown in italics), and are similar to those obtained at F10:7 = 120 in Fig. 12(a). Exceptions occur in the F1 region, at night, where densities show an appreciable variation with solar
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starlight at heights below 150 km (Fig. 1), and the loss process is quadratic so we expect little change in the E region (as discussed in Titheridge, 2000). At all times the E region has a deformed peak, and there is a large gradient discontinuity at the top of the valley. Across sunset (at latitudes of 10◦ N–20◦ S in Fig. 13), the density is too high near 110 km and the resulting narrow valley does not seem possible at this time (when direct solar radiation arrives horizontally). 7. Conclusions Night-time ionisation, in the E and F1 regions, is produced mainly by starlight and by solar radiation resonantly scattered round the earth (or from the interplanetary gas). The strengths of these night sources has been obtained from calculations by Strobel et al. (1980). There is also some contribution from photons produced by recombination of O+ ions and electrons in the F2 region; this is calculated directly within the ionospheric model, and can be important in the F1 region for low-latitude, summer conditions near solar maximum. The production and loss processes are close to equilibrium at all times, and can be modelled with reasonable accuracy. The situation is less well de2ned than during the day, and with longer time constants there may be some e4ect from more complex ions that are not considered in the present analysis. Changes in the neutral wind will also cause some variations, particularly near the base of the F2 layer. Current results do, however, agree well with rocket observations of the night valley and the upper valley limit hU (Table 1). For the E region, theoretical calculations reproduce closely the observed values of height and density at the peak of the night E layer (Titheridge, 2000), and give a close match to variations of Nm E observed at Arecibo (Titheridge, 2001). Pro2les are presented for latitudes from 20◦ to 70◦ , in all seasons and for values of F10:7 from 80 to 200. Near midnight, results show a symmetrical E layer with a scale height of ≈ 5 km and a peak near 105 km. Densities change little with latitude, season or solar activity at heights below 140 km. For nearly all conditions, at latitudes of 25 –60◦ , the peak density is NE ≈ Nn +0:5NV =2:28 (±0:2) × 103 cm−3 , where Nn ; NV are as in Section 5.1. Thus the lower ionosphere near midnight can be represented, to within ≈ 10%, by a 2xed pro2le N (h) = 1:3 cos2 (4:05 − 0:0314h) +1:6 cos2 (6:7 − 0:065h)

at

80 ¡ h ¡ 130 km: (40)

Above the E layer there is a wide,
and increases by 25 –35 km over the solar cycle (from F10:7 = 80 to 200). Densities increase rapidly above hU , giving a sharply de2ned lower boundary to the F2 layer. All available experimental data support this form, showing the general reliability of the theoretical pro2les. Equations are derived giving hU as a simple function of latitude, season and F10:7 . These provide a well-de2ned starting point for models of the night F2 layer, giving a much more realistic result (under all conditions) than use of values from the IRI model. Calculated densities in the night valley region are ≈ 20% greater than mean experimental data from rocket studies, at all latitudes. Results from the Arecibo radar are ≈ 40% below corresponding rocket data, suggesting some systematic bias in the radar densities (which are near the lower observable limit) or lower mean densities at Arecibo. All observations show a regular variation with magnetic activity, such that the mean valley density (in 130 –190 km) approximately doubles for each increase of 2 in KP . This change occurs mainly at heights above 150 km, and is not reproduced by the theoretical calculations. It may be caused by an overall downwards drift, or by increased production under disturbed conditions (perhaps from particle precipitation). Many experimental pro2les also show an intermediate layer, with a density that can exceed Nm E, at heights near 150 km. This could result from varying winds, and does not appear in model results. With good agreement between theoretical and observed pro2les, near midnight, theory can be used to predict the probable variations under other conditions. Some of these results must be somewhat tentative, since they are not well tested by observations. They should, however, give a good 2rst order estimate of changes expected from known variations in atmospheric composition and ionising radiations. This is valuable since reliable data are scarce, coming mostly from direct rocket measurements or from the large backscatter radar at Arecibo. Very few published results include the day/night transition, so the present study gives the 2rst detailed picture of expected changes in this period. Current models, such as the International Reference Ionosphere and the work of Mahajan et al. (1990, 1994), assume a simple variation with continuity between the day and night peaks and valleys. This is not born out by theoretical pro2les. Towards sunset, a bulge corresponding to the O+ 2 production peak remains visible as it rises to a height of ≈ 150 km (at  ≈ 95◦ ). At  ¿ 90◦ the night production peak becomes visible near 105 km. Thus for 85 ¡  ¡ 100◦ , the gradient of the electron density pro2le shows two maxima and two minima in 100 –160 km. Changes at sunrise generally mirror those at sunset, apart from slightly reduced densities at values of  ¿ 85◦ (as a result of overnight decay). For  ¡ 85◦ the peak density Nm E is slightly larger at sunrise, because of lower loss rates caused by the decreased density of NO.

J.E. Titheridge / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 1035 – 1052

Theoretical pro2les under di4erent conditions are summarised in a numerical model. The lower limit is extended to 80 km, giving some D region ionisation similar to that included in the IRI model. All pro2les start with N =dN=dh=0 at 80 km, and give smooth, physically realistic variations at all heights. Model parameters allow for two peaks at  ¿ 70◦ , giving a smooth and physically correct transition between the day production peak (which rises to merge with the F1 region) and the separate night E layer near 105 km. Over this dynamic period the model pro2les vary only with , since calculated changes with latitude and season are small. At night the E layer model has some variation with season, latitude and solar
1051

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