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A theoretical evaluation of the lunar tidal variations in the ionospheric F2-layer
P&net. Spocc
Scl. Vol. 23, PP. 1071 to 1080.
Permmon
A THEORETICAL VARIATIONS Department
Press. 1975.
Printed
in Northern
Ireland
EVALUATION OF THE LUNAR TIDAL IN THE IONOSPHERIC F2-LAYER M. P. K. ABUR-ROBB of Mathematics, Kuwait University,
Kuwait, Arabia
and E. DUNFORD
S.R.C., Appleton Laboratory, Ditton Park, Slough, England (Received infiM1 form 13 Jamuny 1975) Abstract-Time-varying solutions of the full continuity equation for electrons in the n-region are obtained. The effects of production, loss, diffusion and electrodynamic ‘E x B’ drift are taken into account. The ‘E X B’ drift term consists of a solar and a lunar component. The solar component of drift is assumed diurnal with 14.6 m/xc maximum upward speed at mid-day. The lunar component is assumed sinusoidal with period of half lunar day and amplitude one tenth of the solar drift; the phase is assumed to remain constant in lunar time, in accordance with Chapman’s phase law. The results show that the lunar variations in the F2-region are markedly dependent on solar time and latitude. It is also shown that the average semi-diurnal lunar variations in N,,,M and h,F2 at any particular hmar time are almost oppositz in phase to each other (i.e. out of phase by 6 hr) in thk magnetic eauatorial zone. and out of chase bv 2 hr at moderate latitudes. The phase of &.I?? is 10 hr at 1:~ latitides and 9 hrat moderate’ latitudks. The phase of 6N,F2 is 4 hr it low latit&es and 11 lunar hr at moderate latitudes. The results also show that the phase of the lunar semi-monthly oscillations in N,F2 undergoes a rapid shift of about 5 lunar hr in going from 8 to 12” and the so called phase reversal occurs at about 10” lat at which the amplitude of NJ3 becomes extremely small. These and other results are in good agreement with observations. Thus it is shown that the main features of the observed lunar tidal variations of the F2-region within 20” of the magnetic equator can
be explained satisfactorily by the superposition of a small lunar drift on a large solar drift. 1. INTRODUCTION
studies by Martyn (1947) which established the existence of a semi-diurnal lunar tide in the F2-region above Huancayo, led to the determination of lunar variations at many other stations. Detailed experimental studies of the amplitude and phase of the predominant semi-diurnal component of the lunar tide in the F2-region indicate a remarkable dependence on magnetic latitude. The maximum in lunar variations in N,F2. averaged over the whole lunation, occurs at about 10 lunar hr in middle latitudes and around 3-5 hr in low latitudes. In other words, there is a 6-hr difference (i.e. completely opposite phase) between middle and low latitudes. The maximum of the lunar tide in h,F2 occurs around 6-7 lunar hr in middle latitudes and around 8-10 hr in low latitudes. Thus, there is a time lag of about 3 lunar hr between the phases of h,F2 and N,F2 in middle latitudes and about 6 hr in the magnetic equatorial zone (see review article by Matsushita, 1967). Recent published data were collected and studied The
pioneering
by Rastogi (1961,1963, 1968), Rastogi and Alurkar (1964, 1966), Rush and Venkateswaran (1968) and Zagulyaeve and Fatkullin (1969). These studies have shown that the lunar tidal variations in the F2-region vary with both latitude and solar time. Rastogi (1961) has shown that the amplitude of the predominant semi-monthly component of the critical frequency fvF2, when plotted against magnetic latitude near solar noon, has peaks at the equator and at about f20’ magnetic lat.. and the phase, which is otherwise (almost) inde pendent of latitude, has reversal at about fl0’ lat. Rastogi’s analysis was, however, restricted to the mid-day values off,FZ. Rush et al. (1968) have shown that the maximum tidal amplitude in fOF2 is not reached at local noon at all the low latitude stations they investigated. They have also shown that the average lunar semi-diurnal variation in f,F2 undergoes an abrupt
phase-shift from about 4 to 10 lunar hr as we proceed away from the magnetic equatorial zone to higher latitudes and that the phase-shift occurs
1071
1072
M. F. K. ABUR-RO~B and E. DLJNNXD
roughly at about f 10” lat. These deductions about the phase-shift are in essential agreement with Rastogi’s (1961) results. These observed features of the lunar tidal variations in the FZregion are unexplained, although several authors have put forward simplified models which are valid at the magnetic equator (Martyn, 1947 and Maeda, 1955) and at midlatitudes (Gliddon and Kendall, 1964). Dunford and Lawden (1969) have shown that the variations present over the range of latitudes within 25’ of the magnetic equator can be explained in terms of the mechanism first suggested by Martyn (1947). They found that changes in the n-layer equatorial anomaly brought about by the superposition of a small (lunar) electrostatic field on a large (solar) field could lead to changes in the FZlayer distributions similar to those observed. However, the calculations of Dunford and Lawden were based on numerical solutions of the equilibrium continuity equation and their results are therefore only applicable to the quasi-stationary conditions around mid-day. A complete understanding of the lunar tidal phenomena in the F2-layer requires solution of the full time varying continuity equation, a problem which has been studied so far by Abur-Robb and Dunford (1969), Abur-Robb (1970) and Anderson et al. (1973). These authors have either restricted their discussion to the magnetic equatorial zone (Abur-Robb and Dunford, 1969 and Abur-Robb, 1970) or to the region within 32’ of the magnetic equator (Anderson et al., 1973). All these authors have shown that the amplitudes and phases of the lunar oscillations in N,F2 are consistent with observational results. It is the purpose of the present work to present a fairly comprehensive study of the luni-solar variations in both N,,,F2 and h,F2 which result from the complex interaction between solar and lunar effects. The discussion is based on numerical solutions of the full time varying continuity equation for the F2-layer ionization. Terms representing production by photoionization, loss by recombination, diffusion and electrodynamic ‘E x B’ drift are included. A static model atmosphere is adopted and the loss coefficient is assumed to vary linearly with the concentration of molecular nitrogen. It is also assumed that the drift velocity due to the solar component of the electric field is diurnal with a maximum upward drift of 14.6 m/set at mid-day. If the electric field is assumed to drive E-region currents, the magnetic field observed at the ground would be -100 y at the magnetic equator. The lunar component of drift is assumed to be sinusoidal
with a period of half lunar day and amplitude one tenth of the solar drift amplitude. This fraction does not seem unreasonable on the evidence of magnetograms; the magnetic field observed at the ground would be -20 y at the magnetic equator. The theory of the present work is described in Section 2, parameters in Section 3 and numerical procedure in Section 4. The numerical results are presented and discussed in Section 5 ; these are compared with experimental results. The discussion is divided into two parts, for convenience. Part (A) is devoted to discussing the results for the region close to the magnetic equator and part (B) those for the range of latitudes up to 20” north and south of the equator. Conclusions are drawn in Section 6. 2. THEORY The continuity equation which governs the electron number-density, N, in the FZ-region is given by: aN/at = q - KN - div (NV). (1)
Here q is the production rate given by the Chapman function, K the linear loss coefficient and V is the velocity. The term div (NV) can be expressed in the form div (NV) = -DglN
- gzN - 9sN,
(2)
where D is the coefficient of ambipolar diffusion and .Qr, 4 and 9s denote linear differential operators representing the effects of diffusion, transport by ‘E x B’ drift and neutral air transport respectively (see Baxter and Kendall, 1968 and Abur-Robb and Windle, 1969). We shall neglect any contribution due to the motion of the neutral air so that 9sN = 0. Approximating the geomagnetic field to a dipole, 9r as derived by Kendall (1962) and Lyon (1963) is given by: 2H2g1 = sin21 + 2(15,u4 + 10,~~ - 1)H/(rA4) + 3H sin Z[l + H(3 + 5p2)/(r A2)1d+ 2H2d2, (3) where I is geomagnetic dip angle, H the scale height, r the geocentric distance. and if 0 is the geomagnetic colatitude, /J = cos 8, A = d/(1 + 3~3, Z = arctan (2 cot 0) and d=sinIalar
+cosIa/ra&
The equation of a typical field line is r = a sin2 0, where the parameter a is the value of r at the magnetic equator for a given field line.
1073
The E&layer lunar tidal variations The electrodynamic drift term may be derived in the following manner. We assume that the electric field is derived from a potential R and that its component in the direction of the magnetic field is zero. Furthermore, we assume that the vertical electrodynamic drift at the magnetic equator is independent of height. It may then be shown that the potential can be written in the form h2 = -gBeRs/a2,
e/A)li + 2g sin8 &,
(5)
where ii and i are unit vectors upwards and normal to the geomagnetic field B in the meridian plane I, and in the east-west direction, respectively, and f = dg/dv. Using this expression for V, we may derive the electrodynamic drift operator which is given by: --g2 = 2g a/aae + f[sin” O(cos Z a/ CJr- sin Z a/r
f, =
(4)
where g is a function of longitude (time) Q only, Be the equatorial magnitude of the magnetic field at the Earth’s surface and R is the Earth’s radius. The drift velocity is then V, =f(sin8
to the F2-region along the highly conducting geomagnetic field lines (Martyn, 1947). Dynamo theory (Tarpley, 1970) and direct observations (Balsley, 1969) show that the dominant electric field is diurnal, being eastwards during the day and westwards at night. We assume therefore a sinusoidal variation with a maximum at mid-day. The solar contribution to f is thus taken to be
ae)
+ W - r3Ka A’)l/A. (6)
w, sin (p::>g,
=
-
w, cos
Q,
(11)
where 9, is the local solar time measured in radians from sunrise. If W, = 14.6 m/see, the electric field giving rise to such drifts would also produce E-region currents whose mid-day magnetic field would be observed at the ground to be with a magnitude of about 100 y at the magnetic equator. Such a value is fairly representative for stations in the magnetic equatorial zone. The dynamo action of the Moon’s gravitational field produces an electric field which is an order of magnitude smaller than the solar generated field and has yet not been observed directly. We may, however, estimate this field from observations of the magnetic field at the ground (Dunford, 1970) using the dynamo theory as a guide. Figure 1 AP’fER
CHAPMAN 6 BARTELS
=,,, ,
THEORETICAL
The system of coordinates (r, 8, t) is changed to the system (x, a’, q’) moving with the electrodynamic drift, by means of the transformations X2 = (e-Pl2H - e-o12H)/(e-r,12= - ea/=) (7) a’ = a/(1 + 2gT/a)
(8)
ql) = q = t/r,
(9)
where x is positive in the northern hemisphere and negative in the southern hemisphere. Thus x = rt 1 at the level of maximum production r = r. and x = 0 at the magnetic equator r = a. The symbol 7 refers to the number of seconds in a radian. In exactly similar manner to that used by AburRobb and Windle (1969), Equation (2) is transformed to an equation of the form ~~ atflap, = I;,a2vjax2 + Fzavjax + F,V + F,,
(IO)
where v is proportional to N and F. to F4 are complicated functions of position and time. Further details may be obtained from Abur-Robb and Windle (1969) and references cited therein. It remains to define the form of the function f inthe expressions for drift velocity and electrostatic potential. We assume that the electric field has its origins in the dynamo layer of the E-region and that it is communicated without serious attenuation
LUNAR
TIME
FIG. 1. CONTOURSOF THE LUNAR VARIATIONS
IN THE HORIZONTAL MAGNETIC INTENSITY H AT THE MAGNETIC EQUATOR ABOVE HUANCAYO (AFIZR CHAPMAN AND BARTELS; SEE RASTOGI, 1965), AS COMPARED WITH THOSE CALCULATED FOR A SINUSOIDAL VARIATIiN OF ELECTRICFIELD USINGTHEFOR~KJLA dH= H,crE.
Ho = 2Oy, u = 41~0s (I+ Y + n-)1and E = -sin 21, where 2 is the local lunar time (0 --t 2~) and v is the local lunar age (0 at new Moon, n at full Moon). The theoretical contours are shifted in time by 1 lunar hr to get better agreement with observations. The shaded areas correspond to solar night-time.
1074
M. F. K. ABIJR-ROBBand E. DUNF~RD
(left) shows contours of the lunar daily variations of the horizontal component of the magnetic field at Huancayo (Rastogi, 1965). We may interpret the variations as being the result of a predominantly semi-diurnal electric field modulated by a conductivity which is small during solar night-time and rises to a peak value at mid-day. The phase of this semi-diurnal electric fleld remains constantin lunartime in accordance with Chapman’s phase law (Chapman and Bartels, 1940). In an attempt to reproduce these features, we have constructed the following model. We assume for simplicity that the Moon makes 29 rotations round the Earth in a period of 30 solar days, as compared with 28.984 rotations in reality. The lunar contribution to f we will then assume to be of the form fi = w, sin (WV + 6,) 3 = -w,
wg,
cos (WV + 6,),
(12)
where w = 29115, and we give W, a value one tenth of the corresponding solar value, namely 1.46 m/set. The phase 6, was assigned a fixed value in accordance with Chapman’s phase law and for simplicity the value chosen is zero. To check the reality of these assumptions, the magnetic field variations caused by such an electric field have been calculated. The conductivity is assumed to vary as 4~0s x (where x is the solar zenith angle) during the day and very small at night. The calculated variations are illustrated in Fig. 1 (right). We see that the agreement with Rastogi’s experimental results is satisfactory during the sunlit hours, but there is insufBcient evidence at night, when the variations are extremely small. We shall assume that the electric field has the same form during day and night. In this case, the functions f and g take the forms
f = W, sin cp + W, sin (29~115) g = - W,cos Q - (15/29)W,cos
(29&5).
(13)
It is clear that the solar drift has a period of 24 solar hr whereas the lunar drift has a period of 15 solar days, i.e. half a lunar month; one solar day representing a lunar phase of 08 lunar hr, reckoning 24 lunar hr in a lunar month. The lunar drift has also a period of half a lunar day when measured with respect to local lunar time. It can be shown that the lunar drift maximizes at 9hr local lunar time. Thus in our simple model, the lunar drift is controlled by the lunar time rather than by the lunar age, i.e. by the position of the Moon irrespective of the position of the Sun.
3. PARAMETERS OF THE PROBLEM
In solving Equation (10) the following parameters are used: K,, = 3.8 x lO-a/sec, D, = 6.1 x 106m2/ set, H=SOkm, 6=0 and I=1.75, where the s&ix zero applies to the level of maximum production at the base of the FZlayer (180 km above the ground). The loss coefficient is denoted by K, the diffusion rate by D, the scale height by H, the solar declination angle by 6 and A is the vertical gradient of In K. The values for Hand 6 correspond to sunspot minimum and equinoctial conditions, respectively. With these fixed parameters, two sets of results are obtained using the following combinations. (1) W, = 14.6 m/set, W, = 0 (2) W, = 14.6 m/set, W, = 1.46 m/set. Here W, and W, denote the amplitude and lunar drifts, respectively.
of the solar
4. NUMERICAL PROCEDURE
Equation (10) is integrated numerically on a digital computer for a series of fixed values of a’ (> r0 = 6374 + 180 km) using the Crank-Nicolson technique. On the boundaries r = r,,, we take v =
-F,/F,
at
x -
fl.
i.e. we assume photochemical equilibrium at and below the base of the F2-layer (r < r,,). The integration is performed over the two hemispheres covering the entire region -1 < x < 1 and started at q = 0 (sunrise) with u = 0 and o = 2. The integration is continued stepwise in q until the quasi-steady state is reached (normally at Q = 6~). The resulting u’s at q = 6~ are then taken as starting values and the integration is recommenced at Q = 0 with o = 29/15 and continued stepwise in q until q = 32n (i.e. for 16 Earth rotations). The steplengths used are 0.05 in x and 1.5’ (6 min) in 8. The results for 30~ < q < 32n (full Moon) were found to be identical to those for 0 < q < 2n (new Moon) and the numerical integration is regarded as satisfactory. This procedure is used to obtain the results with W, # 0 whereas the solar tidal variations with W, = 0 are obtained by terminating the integration at v = 6n, i.e. after three Earth rotations. The values of N,,,F2 and h,F2 are obtained as functions of local solar time at fixed lunar ages by interpolating the computer results. The difference between the results for W, # 0 and those for W, = 0 gives the lunar tidal variations.
1075
The 2%layer lunar tidal variations 5(A). RESULTS (FOR THE MAGNETIC EQUATORIAL ZONE) AND DISCUSSION
was found to be larger than the other harmonics by orders of magnitude. In Table 1 are shown the amplitude (P2) and phase (t2) of the calculated lunar semi-monthly oscillations, as compared with observations at the solar hours 0, 3, . . . ,21. The phases are given in terms of local lunar time measured in lunar hr. The results presented here are for the case WJ W, = 0.1, but those marked (*) are obtained with W,/ W, = 0.2. The last line in Table 1 gives the daily average values; the phases being given in terms of local lunar age measured in lunar hr from new Moon. No precise comparison with observations is attempted here because of the difference between the various determinations of the observed lunar The two sets of experimental results variations.
The diurnal variations As a typical example of the F2-layer behaviour when both solar and lunar tides are taken into account, the theoretical values of the maximum height h,F2 + 6h,F2 at the magnetic equator are plotted in Fig. 2 against local solar time for eight phases of the Moon. Figure 2 indicates the presence of strong lunar tides. The lunar semi-monthly oscillations The lunar tidal variations in N,F2 and h,F2 at a fixed solar time are analysed into different harmonics. The predominant component, being semi-monthly,
a
J-4
42
0
d‘3
x
I 0
I
I
I
24
0
12
LHT
I 24
HEIGHT
OF
THE
I
12
DAILV
WT
VARIATIONS
DF
THE
8 RlASES
Fro. 2. THE CALCULATED
I 0 F2 PEAK
DF
THE
I 24
I
12
LMT
AT THE
I 0
I
12
MAGNETIC
LMT
EQUATOR
I 24
FOR
MOON
DIURNAL VARIATIONS OF MAXIMUM HEIGHT
h,J?2 AT
THE
MAGNETIC
EQUATOR
POR 8 PHASES OF THE MOON.
drift = 14*6m/sec, lunar drift = l-I.6 m/set.
Solar
TABLE1.
~EFFICIENTS
AND PHASES OF THE TEIBGRBTICAL AND OBSRRVBD LUNAR SEMI-MONTHLY OF THE F2-LAYER AT THE MAGNETIC EQUATOR
Experimental fa2, Rastogi Local solar time
(1968)
MHz
Theoretical
Rush et al. (1968) 2F,I Mean (%)
h,,,FZ, km
J’s
ts
PI
ts
5.9 :.;*
0.10 0.04 0.11
11.6 0.0 5.8
3.39 5.64 1.90
0.22 0.22 0.13
1.5 6.5 1.8
7.46 11.3 6.19
2.36 0.33 0.82
0.258 0.056 0.037
0.9 0.8 1.2
10.9 11.3 6.83
1;
8.4 7.9
0.33 0.17
2.7 3.7
4.05 a.35
0,33 0.18
3.3
4.28 8.35
4.64 3.91
0.205 0.098
4.4 4.3
4.42 2.51
12 15 18
88:;
aTo 6.12
0;7 0.23
8;10 5.41
5*29
0.40* 0.194 0.244
4*2*
75 5.6
co 5.2
f.?
oY4 0.26
;..i
8.62* 4.07 4.61
21
7.5
0.17
8.0
4.53
0.16
9.1
4.27
4.21
0.167
9.1
Mean
6.27
0.11
3.51
-
-
-
3.125
0.11
10.5
0 :
10
results
100 N,,,F2/qTse
2FsI Mean (%)
Mean foF2
OSCIUATIONS
Fsl Mean
Mean
P,
t8
(%I
Fsl Mean
Mean
PS
t,
(%)
9.9 10.3 10.2 9.2 7.2 7.4. 7.9
0.74 1.47 I.57 1.32 1.74 3.08’
2.25 4.50 5.20 4.50 7.00 12.4*
3.97
306 305 331 341 403 403* 389 383 351
66
1’0.“9
1.96 2.73
3.52
355
2.5
7.5
0.71
The theoretical results are for the case WC/W, = 0.1, but those marked (*) are for the case W,/W, variations (last line) are given in terms of local lunar age measured in lunar hr from new Moon.
7’;’
= 0.2.
.
1.41
Phases of the mean
1076
M. F. K.
ABLJR-ROBB
given in Table 1 give an illustration of this variability; in these two cases the same methods were used but the data were from different periods. However, the main features of the observed variations are fairly well reproduced theoretically here; the phases are in fair agreement and the amplitudes are in better agreement with the results of Rush et al. than with Rastogi’s results. We note here that, in spite of the noticablediscrepancies between theory and experiment at particular solar hours, the observed daily average variation in N,,,F2 is predicted almost exactly (see last line in Table 1). Table 1 shows that the amplitude of the lunar semi-monthly variations in N,F2 ranges from 251 per cent at 09:Ofl hr to 11.3 per cent at 06:OO hr solar time. The amplitude at Huancayo during the period 1942-1944 was found by Martyn (1947) to vary from 2 per cent at 07:OO hr to 14.6 per cent at 05:OO hr; this is in fair agreement with our theoretical predictions. Table 1 also shows the amplitude of the lunar tide in h,F2 ranging from 2~25kmat00:OOhrtoasmuchas10kmat21:OOhr. The determination of the lunar variations in N,,,F2 and h,F2 near solar noon have received the attention of many workers. We mention here Martyn’s analysis for Huancayo; he obtained an 8.6 per cent change from the noon mean value of N,,,F2 and a phase of 4 lunar hr, together with 8 km and 7 lunar hr for the amplitude and phase of the semi-monthly variation in h,F2. It can be seen from Table 1 that, for the case W,/WS = 0.2, the daytime theoretical results are in close agreement with observations. During the night, however, the calculated variations in this case were much larger than those observed. It should be noted that the value of N,,,F2 or h+,,F2 averaged over all hours of a lunar day is found to depend on the lunar age. The daily average value, when plotted against lunar age, exhibits a pure sinusoidal wave having two peaks of the same magnitude within one lunar month. The amplitudes and phases are given in Table 1 (last line) as 352 per cent and 10.5 lunar hr for NmF2 and 2.5 km and 7.5 hr for h,F2. The results for N,,,F2 are in close agreement with the experimental results of Rastogi (1968) who obtained 351 per cent and 10 hr, respectively, for the amplitude and phase of the lunar semi-monthly variation at Huancayo. The lunar daily variations The lunar tidal variations in N,,,F2 and h,F2 at the magnetic equator can also be analysed as
and E. Dunroan functions of local lunar time at fixed lunar ages. It is found that each curve shows two waves of unequal amplitude indicating the presence of luni-solar diurnal and lunar semi-diurnal oscillations of comparable magnitude. The average whole lunation curve is almost a pure sinusoidal wave having two peaks of the same magnitude within one lunar day. The diurnal component seems to have been largely cancelled out in the averaging over the whole lunation. The coefficients and phases of the calculated semi-diurnal lunar tides in N,F2 and h,F2 when averaged over the whole lunation are given in Table 2. Experimental results are also given for N,F2 at Ibadan (after Brown, 1956), for N,F2 and h,F2 at Huancayo (after Martyn, 1947) and for N,F2 at Huancayo (after Rastogi, 1968). In their analysis, these authors used the annual mean values and the data were taken during periods of low solar activity. The theoretical mean value of N,F2 is found to be O.O3125rq,e and the experimental mean value of f0F2 is given by 6.154 MHz (after Martyn, 1947) a.ndby6.27MHz(after Rastogi,1968). It can be seen from Table 2 that the calculated semi-diurnal oscillations agree fairly well with observations at Huancayo. The theoretical oscillations in N,,,F2 is 3.1 per cent with a phase of 4 lunar hr which compares well with Martyn’s results for Huancayo, viz. 3.25 per cent with phase 4.33 hr. The theoretical results also compare well with those of Rastogi who obtained 3.03 per cent and 4 hr for the amplitude and phase of the lunar tide at Huancayo. The calculated amplitude of h,F2 is 4.6km with a phase of 1Ohr which compares well with Martyn’s results, 5.2 km and 8.43 hr. It is worth pointing out that the discrepancy between the observed results for Ibadan and Huancayo may be due to the fact that these stations lie in different ionospheric zones. 2.
TABLE AND
COEFFICIENTS
OBSERVED WHOLE
AND
SEMI-DIURNAL LUNATION
PHASES LUNAR
AT THE w,/w,
PI
=
OF TIiJ3 THEORETICAL VARIATIONS
MAGNETIC
FOR
THE
EQUATOR.
0.1
tl, lunar hi-
Reference
Theoretical N,,,F2 hmF2 Huancayo N,m
h,F2 N,,,F2 Ibadan N,F2 &J2
3-l % 4.6km
4 10
Present work
3.25% 5.2 km 3.03 %
4.33 8.43 4.00
Martyn 1(1974) Rastogi (1968)
3.83 % 7.7 km
3.80 8.10
Brown (1956)
The F2-layer lunar tidal variations Contours of lunar tidal variations In Figs. 3 and 4 are shown contours of constant values of 6N,,,F2 and 6h,F2 on the coordinates of local lunar time and local lunar age. The contours of the lunar variations are taken to be free from any solar tidal component. However, their values are governed by the position of both the Sun and the 10F2 AmER
00
Nm F2
(EX~~IWENTAL)
(THEORY)
RASTOQI AN0 ALURKAR
_ 06
1.2
24
..._-z.
LUNAR
00
06
1.2
12
24
TIME
FIG. 3. tiNTOuRS OF CONSTANT VALUES OF THE LUNAR VARIATIONS IN f,F2 (EXPERIMENTAL RESULTS AFTER RASTOGIAND ALLJRKAR; SEE RASTOGI, 1965)~~~ CALCULA~D N,,,F2 AT THE MAGNETIC EQUATOR As FUNCTIONSOPLOCALLUNARTIMEANDLUNARAGE.
Solar drift = 14.6 m/w, lunar drift = 1.46 m/set.
Lunar Time,L.hr.
FIG. 4. CoNTow
1077
Moon relative to the Earth. The lunar age Y is indirectly representing the local solar time t and is related to the local lunar time I by the equation: t = 1 + v. The solar night-time period is indicated by the shaded portion of the diagrams. Figure 3 shows contours of the calculated lunar tidal variations in N,,,F2 as compared with contours of foF2 quoted by Rastogi (1965). The numbers on the theoretical curves are normalized values, five units representing approximately a 15.2 per cent change from the mean value of N,,,F2. The experimental value of 0.6 MHz for the lunar variation in foF2 gives a variation of 18 per cent in NmF2 compared with a variation of 14.6 per cent obtained by Martyn (1947). It can be seen from Fig. 3 that during daylight hours, the observed variations are quite well reproduced; the amplitude and phase agreeing well. During the night however, the theoretical variations are far too large. Figure 4 shows contours of the lunar variations at Huancayo (after Rastogi, 1970) obtained for a period of moderate solar activity. The section on the 1.h.s. indicates that large changes in NmF2 do exist during the night-time and that the maximum amplitude tends to occur near both solar noon and mid-night; these features are in good agreement with the theoretical results. Figure 4 (right) shows the observed variations in h,F2 with a maximum amplitude of 10 km which is almost identical to the calculated maximum amplitude (see Table 1). Again, a feature which is common in both observed
LunarTime,L.hr.
OF THE OBSERVED LUNAR VARL~TIONS AT HUANCAYO OBTAINED FOR A PERIOD OF MODERATE SGLARACTIVITY(AFTER RASTOGIAND ALURKM,~~~~).
M. F. K. ABUR-ROBBand E. DUNPORD
1078
5(B). RESULTS (FOR THE LATITUDES UP TO &20°) AND DISCUSSION
and theoretical results is that the maximum amplitude tends to occur near solar noon and mid-night. We note here that the theoretical phases are arbitrary and if the correct ‘E X B’ drift phase is chosen, a better agreement with observations may be obtained.
The lunar semi-monthly variations In Table 3 the phases of the lunar ‘E X B’ drift, 6N,F2 and 6h,,,F2 are given at fixed solar times for the magnetic latitudes 48,. . . , 20°. The phase measured in lunar hr represents the local lunar time when the maximum positive variation occurs. Clearly, the lunar drift (and so the electric field) is controlled by the local lunar time, i.e. by the position of the Moon irrespective of the position of the Sun. It can be seen that the response of the F2-layer to small perturbations in the electric field is not immediate, and there is a phase Jag whose magnitude depends on solar time. Since the height of the F2-peak is controlled more by the solar drift than by the lunar drift, the magnitude of the phase lag depends on the current height of the peak. The phase of 6N,,,F2 is seen to vary more rapidly with solar time than does the phase of 6h,F2. However, during the 6-hr period centred at solar mid-day, the phase of 6NmF2 remains almost constant. During the same period the drift and 6N,,,F2 are out of phase at low latitudes and are in phase at moderate latitudes with a maximum delay of about 3 lunar hr at 15 hr solar time. Figure 5 shows the amplitudes and phases of the semi-monthly oscillations in N,F2 and h,F2 plotted against magnetic latitude at solar mid-day. This figure shows that the amplitude of N,F2 increases slowly in going from the equator to higher latitudes and reaches a maximum value at about 5’, then it decreases rapidly and reaches a minimum value at about 10”. Beyond 10” lat., the amplitude has a single maximum developed at about 14’. Otherwise constant, the phase of N,,,F2 undergoes a rapid shift of about 5 lunar
Physical interpretations According to observations, the phase of the lunar variations in N,,,F2 averaged over the whole lunation is about 6 hr earlier than the phase of h,F2. This feature is in good agreement with the theoretical results (see Table 2). To explain this phenomenon we note that when 6h,F2 is positive, the F2-layer is raised above the daily mean value of h,F2, and when it is negative the n-layer is lowered below this level. In the former case the ionization is transported upwards from the region near the magnetic equator and gravitates along the geomagnetic field lines to settle at the crests of the equatorial anomaly; this leads to the negative values of 6N,F2. In the latter case the ionization is transported back to settle at the magnetic equator and very low latitudes giving rise to positive values of dN,,,F2. It seems that the assumed form of the ‘E x B drift provides an explanation as to why the maximum positive variation in h,F2 occurs at 10 lunar hr. The lunar drift has been shown to maximize at 9 hr local lunar time. Thus the response of h,F2 to the drift is not immediate; the dissipative processes cause a delay of about one hour. It should be noticed that while changes in the ‘E x B’ drift are felt in h,F2 with a delay of an hour, N,,,F2 at the magnetic equator responds almost immediately to any changes in h,F2.
TABLZ3.
CALCULATRD DENSITY
PHASES OF THE SEMI-MONTHLY
N&Z
AND MAXIMUM
HRIGHT
m
OSCILLATIONS w,/w,
Solar
time
Lunar drift
0”
Nm 0.9 0.8 1.2 4.3 4.4 4.4 ;:;’
IN THE LUNAR
6 MAGNRTIC
POR
=
4”
12O
80
Nm
hm
Nm
hm
9.9
o-9
9.9
O-4
0.8
9.9 10.3 10.2
0.9
10.3 10.2 9-2 7.2 7.9 10.9 9.6
O-8 1.2
10.3 10.0 *i: ’
0.8 l-2 11.4 11.6
4.3 :I: 4.7 6.1
P-8
9.2 8.3 7.7 . I;.:
44
6.2 7-6 11.2
DRIFT,
MAXIMUM
ELECTRON
AT FIXED SOLAR HOURS.
o-1
hm
1.2
‘E X B’
LATITUDES
Nm
109.: .
11.9
10.8
10.6 0.5
20”
16”
h, 9.9 9’:; 9.4 10.3 11-s 11.5 10.0
N,
h,
N,
h,
9.9 8.9
0.4 0.8
9.2 8.5
lA.3
95 9.6
10.0 1.2
;:;
11.4 11.9 1-l 0.5
10.1 10.2 10.0 9.8
11.2 o-1 0.4 0.4
10.0 10.0 9.8 9.5
0.4 0.8
The n-layer
lunar tidal variations
1079
the individual curves for 6N,,,F2 and 6h,,J2 at fixed magnetic latitudes. The amplitudes and phases of the predominant semi-monthly components plotted against magnetic latitude are shown in Fig. 6. The resulting variations are similar to those obtained at solar mid-day. Here we have smaller amplitudes and the phase shift occurs at about 13-14’lat as compared with 10” for mid-day 6N+,,F2. The lunar daily average variations
i
0'
0
I
t
4 6 hlognetic
I
I
I
12 16 20 Latitude , Degs.
24’
FIG.
5. THE LATITUDINAL VARIATION OF TI-IBAMPLITUDE (BOITOMCURVES)ANDPHASE f2 (TOP CURvBS)OPTHE LUNARSEMI-MONTHLYOSCILLATIONSIN MAXIMUMELFCTRON DBNSITY N,,,F2 AND MAXIMUM HEIGHT hz2 AT SOLAR MID-DAY.
P2
Solar drift =
These are the lunar variations in NmF2 and h,F2 obtained for the whole hmation by averaging over all lunar ages at fixed magnetic latitudes. Figure 7 shows the amplitudes and phases of the predominant semi-diurnal oscillations plotted against magntic latitude. It can be seen that the latitude of phase reversal is 9’ and the amplitude of SN,,,F2 achieves maximum values at the equator and 14” lat. The amplitude of Sh,F2 develops a single maximum at about 9’ which coincides with the latitude of phase reversal. The top curves show that the maximum of lunar daily average variations in N,,,F2 occurs at about 11 lunar hr in moderate latitudes and at 4 hr in low latitudes. In other words, there is a 5-hr difference between low and moderate latitudes. The maximum of the lunar variations in
146 m/set, lunar drift = 146 m/set.
hours in going from 8 to 12’ and the so-called phase reversal occurs at about 10” lat. These theoretical details are in fair agreement with Rastogi’s (1961) experimental results. The same Fig. 5 shows that the amplitude of 6h,F2 is a minimum at the equator and increases by 3 kmlin going from the equator to 6’ lat. Beyond 6” it decreases with increasing latitude. Figure 5 (top curves) show that 8N,,,F2 and Sh,F2 are out of phase by 3 hr at the equator and by 1 hr at moderate latitudes. The one hour phase difference at moderate latitudes can be interpreted as being the time constant for the ionization to re spond to changes in h,F2. There is also a phase lag of about 1 hr between h,F2 and the electric field. Thus near solar mid-day, the maximum height responds to the applied electric field with a delay of about 1 hr whilst the maximum concentration responds to the same field with a delay of about 2 lunar hr at moderate latitudes. It must be pointed out here that N,,,F2 responds to the electric field via h,F2. The solar dairy average variations The average semi-monthly variations for the whole solar day may be obtained by averaging
(Doily-Average)
0'
0
I
I
I
I
4
I
20 hiognk
24
IZitude YDegs.
FIG. 6. THE LATITUDINAL VARIATIONS OF TIIBAMPLIT~E P2 (BOlTOh CUBvBS)AND PHASE V2 (TOPCURvBs)OPTHE LUNAR SBMI-MONTHLY OSCIUATIONS IN DAILY AVERAGE VALUE OF N,,,F2 ANDhJ-7. Solar
drift = 14.6 m/set, lunar drift = 1.46 m/set.
1080
M. F. K. AEWR-ROBBand E. DUNFORD (Whola
magnetic equator can be explained satisfactorily by the superposition of a small lunar drift on a large solar drift. (2) The calculated lunar tides at low latitudes show better agreement with observations from data spread over all solar hours than from data confined to any one particular solar hour. (3) The abrupt change of phase with latitude predicted from the average (and mid-day) lunar tidal variations results mostly from the fact that the n-peak has, on the average, a steep latitudinal gradient.
Lunatlon)
work has been largely done by one of us (M.F.K.A.-R.). We shouldliketothank Professor P. C. Kendall and Dr. D. W. Windle for their interest and encouragement. The numerical calculations of this paper were carried out on the University of Sheffield ICL Computer. Acknowledgements-This
FIO, 7. TnE
LATITUDINAL
VARIATIONS
P2
CURVES)
PHASE
(BOlTOM
LUNAR
SEMI-DIURNAL ATION
AVERAQE
AND
t2
OSCILLATIONS VALUE
OF
OF THE (TOP
IN
N,,,F2
AhiFLITUDE
CURVES)
THE AND
WHOLE
OF THE LUN-
h,F2.
Solar drift = 14.6 m/set, lunar drift = 1.46 m/set. h,F2 occurs at 10 hr in low latitudes
and at about 9 hr in moderate latitudes. Thus there is a time lag of about 2 lunar hr between the phases of N,F2 and h,,,F2 at moderate latitudes and about 6 hr in the magnetic equatorial zone. These results are in good agreement with Matsushita’s (1967) deductions and the experimental results of Rush et al. (1968). The following explanation of the above described results is based on the diffusion-drift theory. At low latitudes, an upwards drift decreases N,F2 by transporting the ionization along the geomagnetic field lines to settle at the latitudinal crests of the while a downward drift equatorial anomaly, increases N,,,F2 by transporting the ionization back to settle at the equator and very low latitudes. At moderate latitudes, an upwards drift increases N,F2 while a downward drift decreases N,F2. This is expected since the loss coefficient is greater at lower heights. 6. CONCLUSIONS The
following conclusions may be drawn from the calculations of this paper. (1) The main features of the observed lunar tidal variations of the F2-region within 20’ of the
REFERENCES Abur-Robb, M. F. K. (1970). Ph.D. Thesis, Department of Applied Mathematics and Computing Science, University of Sheffield. Abur-Robb M. F. K. and Dunford, E. (1969). Proc. 3rd Int. Symp. Equatorial Aeron., p. 238. Abur-Robb, M. F. K. and Windle, D. W. (1969). Planet. Space Sci. 17, 97.
Anderson, D. N., Matsushita, S. and Tarpley, J. (1973). J. atmos. terr. Phys. 35,753.
Balsley, B. B. (1969). J. geophys. Res. 74,1213. Baxter, R. G. and Kendall, P. C. (1968). Proc. Roy. Sot. A304,171. Brown, R. A. (1956). J. atmos. terr. Phys. 9, 144. Chapman, S. and Bartels, J. (1940). Geomagnetism. Oxford University Press, London. Dunford, E. (1970). J. atmos. terr. Phys. 32,421. Dunford, E. and Lawden, M. D. (1969). Planet. Space Sci. 17, 923. Gliddon, J. E. C. and Kendall, P. C. (1964). J. atmos. terr. Phys. 26,721.
Kendall, P. C. (1962). J. atmos. terr. Phys. 24,805. Lyon, A. J. (1963). J. geophys. Res. 68,253l. Maeda, H. (1955). J. Geomagn. Geoelect. 7,75. Martyn, D. R. (1947). Proc. Roy. Sot. A190,273. Matsushita, S. (1967). EncyclopediaofPhysics (Handbuch der Physik), 49/111/2,p. 547. Springer, New York. Rastogi, R. G. (1961). J. atmos. ferr. Phys. 22,290. Rastogi, R. G. (1963). Proc. Indim Acad. Sci. 58, 38. Rastogi, R. G. (1965). Z. Geophys. 31,27. Rastogi, R. G. (1968). J. atmos. terr. Phys. 30,497. Rastogi, R. G. and Alurkar, S. K. (1964). J. atmos. terr. Phys. 26,641.
Rastogi, R. G. and Alurkar, S. K. (1966). Proc. Indian Acad. Sci. 63,75.
Rastogi, R. G. and Alurkar, S. K. (1970). Pageoph. 80, 288.
Rush, C. M. and Venkateswaran, S. V. (1968). J. atmos. terr. Phys. 30,633.
Tarpley, J. D. (1970). Planet. Space Sci. 18, 1075. Zagulyaeva, V. A. and Fatkullin, M. N. (1969). J. atmos. terr. Phys. 31, 1.
Scl. Vol. 23, PP. 1071 to 1080.
Permmon
A THEORETICAL VARIATIONS Department
Press. 1975.
Printed
in Northern
Ireland
EVALUATION OF THE LUNAR TIDAL IN THE IONOSPHERIC F2-LAYER M. P. K. ABUR-ROBB of Mathematics, Kuwait University,
Kuwait, Arabia
and E. DUNFORD
S.R.C., Appleton Laboratory, Ditton Park, Slough, England (Received infiM1 form 13 Jamuny 1975) Abstract-Time-varying solutions of the full continuity equation for electrons in the n-region are obtained. The effects of production, loss, diffusion and electrodynamic ‘E x B’ drift are taken into account. The ‘E X B’ drift term consists of a solar and a lunar component. The solar component of drift is assumed diurnal with 14.6 m/xc maximum upward speed at mid-day. The lunar component is assumed sinusoidal with period of half lunar day and amplitude one tenth of the solar drift; the phase is assumed to remain constant in lunar time, in accordance with Chapman’s phase law. The results show that the lunar variations in the F2-region are markedly dependent on solar time and latitude. It is also shown that the average semi-diurnal lunar variations in N,,,M and h,F2 at any particular hmar time are almost oppositz in phase to each other (i.e. out of phase by 6 hr) in thk magnetic eauatorial zone. and out of chase bv 2 hr at moderate latitudes. The phase of &.I?? is 10 hr at 1:~ latitides and 9 hrat moderate’ latitudks. The phase of 6N,F2 is 4 hr it low latit&es and 11 lunar hr at moderate latitudes. The results also show that the phase of the lunar semi-monthly oscillations in N,F2 undergoes a rapid shift of about 5 lunar hr in going from 8 to 12” and the so called phase reversal occurs at about 10” lat at which the amplitude of NJ3 becomes extremely small. These and other results are in good agreement with observations. Thus it is shown that the main features of the observed lunar tidal variations of the F2-region within 20” of the magnetic equator can
be explained satisfactorily by the superposition of a small lunar drift on a large solar drift. 1. INTRODUCTION
studies by Martyn (1947) which established the existence of a semi-diurnal lunar tide in the F2-region above Huancayo, led to the determination of lunar variations at many other stations. Detailed experimental studies of the amplitude and phase of the predominant semi-diurnal component of the lunar tide in the F2-region indicate a remarkable dependence on magnetic latitude. The maximum in lunar variations in N,F2. averaged over the whole lunation, occurs at about 10 lunar hr in middle latitudes and around 3-5 hr in low latitudes. In other words, there is a 6-hr difference (i.e. completely opposite phase) between middle and low latitudes. The maximum of the lunar tide in h,F2 occurs around 6-7 lunar hr in middle latitudes and around 8-10 hr in low latitudes. Thus, there is a time lag of about 3 lunar hr between the phases of h,F2 and N,F2 in middle latitudes and about 6 hr in the magnetic equatorial zone (see review article by Matsushita, 1967). Recent published data were collected and studied The
pioneering
by Rastogi (1961,1963, 1968), Rastogi and Alurkar (1964, 1966), Rush and Venkateswaran (1968) and Zagulyaeve and Fatkullin (1969). These studies have shown that the lunar tidal variations in the F2-region vary with both latitude and solar time. Rastogi (1961) has shown that the amplitude of the predominant semi-monthly component of the critical frequency fvF2, when plotted against magnetic latitude near solar noon, has peaks at the equator and at about f20’ magnetic lat.. and the phase, which is otherwise (almost) inde pendent of latitude, has reversal at about fl0’ lat. Rastogi’s analysis was, however, restricted to the mid-day values off,FZ. Rush et al. (1968) have shown that the maximum tidal amplitude in fOF2 is not reached at local noon at all the low latitude stations they investigated. They have also shown that the average lunar semi-diurnal variation in f,F2 undergoes an abrupt
phase-shift from about 4 to 10 lunar hr as we proceed away from the magnetic equatorial zone to higher latitudes and that the phase-shift occurs
1071
1072
M. F. K. ABUR-RO~B and E. DLJNNXD
roughly at about f 10” lat. These deductions about the phase-shift are in essential agreement with Rastogi’s (1961) results. These observed features of the lunar tidal variations in the FZregion are unexplained, although several authors have put forward simplified models which are valid at the magnetic equator (Martyn, 1947 and Maeda, 1955) and at midlatitudes (Gliddon and Kendall, 1964). Dunford and Lawden (1969) have shown that the variations present over the range of latitudes within 25’ of the magnetic equator can be explained in terms of the mechanism first suggested by Martyn (1947). They found that changes in the n-layer equatorial anomaly brought about by the superposition of a small (lunar) electrostatic field on a large (solar) field could lead to changes in the FZlayer distributions similar to those observed. However, the calculations of Dunford and Lawden were based on numerical solutions of the equilibrium continuity equation and their results are therefore only applicable to the quasi-stationary conditions around mid-day. A complete understanding of the lunar tidal phenomena in the F2-layer requires solution of the full time varying continuity equation, a problem which has been studied so far by Abur-Robb and Dunford (1969), Abur-Robb (1970) and Anderson et al. (1973). These authors have either restricted their discussion to the magnetic equatorial zone (Abur-Robb and Dunford, 1969 and Abur-Robb, 1970) or to the region within 32’ of the magnetic equator (Anderson et al., 1973). All these authors have shown that the amplitudes and phases of the lunar oscillations in N,F2 are consistent with observational results. It is the purpose of the present work to present a fairly comprehensive study of the luni-solar variations in both N,,,F2 and h,F2 which result from the complex interaction between solar and lunar effects. The discussion is based on numerical solutions of the full time varying continuity equation for the F2-layer ionization. Terms representing production by photoionization, loss by recombination, diffusion and electrodynamic ‘E x B’ drift are included. A static model atmosphere is adopted and the loss coefficient is assumed to vary linearly with the concentration of molecular nitrogen. It is also assumed that the drift velocity due to the solar component of the electric field is diurnal with a maximum upward drift of 14.6 m/set at mid-day. If the electric field is assumed to drive E-region currents, the magnetic field observed at the ground would be -100 y at the magnetic equator. The lunar component of drift is assumed to be sinusoidal
with a period of half lunar day and amplitude one tenth of the solar drift amplitude. This fraction does not seem unreasonable on the evidence of magnetograms; the magnetic field observed at the ground would be -20 y at the magnetic equator. The theory of the present work is described in Section 2, parameters in Section 3 and numerical procedure in Section 4. The numerical results are presented and discussed in Section 5 ; these are compared with experimental results. The discussion is divided into two parts, for convenience. Part (A) is devoted to discussing the results for the region close to the magnetic equator and part (B) those for the range of latitudes up to 20” north and south of the equator. Conclusions are drawn in Section 6. 2. THEORY The continuity equation which governs the electron number-density, N, in the FZ-region is given by: aN/at = q - KN - div (NV). (1)
Here q is the production rate given by the Chapman function, K the linear loss coefficient and V is the velocity. The term div (NV) can be expressed in the form div (NV) = -DglN
- gzN - 9sN,
(2)
where D is the coefficient of ambipolar diffusion and .Qr, 4 and 9s denote linear differential operators representing the effects of diffusion, transport by ‘E x B’ drift and neutral air transport respectively (see Baxter and Kendall, 1968 and Abur-Robb and Windle, 1969). We shall neglect any contribution due to the motion of the neutral air so that 9sN = 0. Approximating the geomagnetic field to a dipole, 9r as derived by Kendall (1962) and Lyon (1963) is given by: 2H2g1 = sin21 + 2(15,u4 + 10,~~ - 1)H/(rA4) + 3H sin Z[l + H(3 + 5p2)/(r A2)1d+ 2H2d2, (3) where I is geomagnetic dip angle, H the scale height, r the geocentric distance. and if 0 is the geomagnetic colatitude, /J = cos 8, A = d/(1 + 3~3, Z = arctan (2 cot 0) and d=sinIalar
+cosIa/ra&
The equation of a typical field line is r = a sin2 0, where the parameter a is the value of r at the magnetic equator for a given field line.
1073
The E&layer lunar tidal variations The electrodynamic drift term may be derived in the following manner. We assume that the electric field is derived from a potential R and that its component in the direction of the magnetic field is zero. Furthermore, we assume that the vertical electrodynamic drift at the magnetic equator is independent of height. It may then be shown that the potential can be written in the form h2 = -gBeRs/a2,
e/A)li + 2g sin8 &,
(5)
where ii and i are unit vectors upwards and normal to the geomagnetic field B in the meridian plane I, and in the east-west direction, respectively, and f = dg/dv. Using this expression for V, we may derive the electrodynamic drift operator which is given by: --g2 = 2g a/aae + f[sin” O(cos Z a/ CJr- sin Z a/r
f, =
(4)
where g is a function of longitude (time) Q only, Be the equatorial magnitude of the magnetic field at the Earth’s surface and R is the Earth’s radius. The drift velocity is then V, =f(sin8
to the F2-region along the highly conducting geomagnetic field lines (Martyn, 1947). Dynamo theory (Tarpley, 1970) and direct observations (Balsley, 1969) show that the dominant electric field is diurnal, being eastwards during the day and westwards at night. We assume therefore a sinusoidal variation with a maximum at mid-day. The solar contribution to f is thus taken to be
ae)
+ W - r3Ka A’)l/A. (6)
w, sin (p::>g,
=
-
w, cos
Q,
(11)
where 9, is the local solar time measured in radians from sunrise. If W, = 14.6 m/see, the electric field giving rise to such drifts would also produce E-region currents whose mid-day magnetic field would be observed at the ground to be with a magnitude of about 100 y at the magnetic equator. Such a value is fairly representative for stations in the magnetic equatorial zone. The dynamo action of the Moon’s gravitational field produces an electric field which is an order of magnitude smaller than the solar generated field and has yet not been observed directly. We may, however, estimate this field from observations of the magnetic field at the ground (Dunford, 1970) using the dynamo theory as a guide. Figure 1 AP’fER
CHAPMAN 6 BARTELS
=,,, ,
THEORETICAL
The system of coordinates (r, 8, t) is changed to the system (x, a’, q’) moving with the electrodynamic drift, by means of the transformations X2 = (e-Pl2H - e-o12H)/(e-r,12= - ea/=) (7) a’ = a/(1 + 2gT/a)
(8)
ql) = q = t/r,
(9)
where x is positive in the northern hemisphere and negative in the southern hemisphere. Thus x = rt 1 at the level of maximum production r = r. and x = 0 at the magnetic equator r = a. The symbol 7 refers to the number of seconds in a radian. In exactly similar manner to that used by AburRobb and Windle (1969), Equation (2) is transformed to an equation of the form ~~ atflap, = I;,a2vjax2 + Fzavjax + F,V + F,,
(IO)
where v is proportional to N and F. to F4 are complicated functions of position and time. Further details may be obtained from Abur-Robb and Windle (1969) and references cited therein. It remains to define the form of the function f inthe expressions for drift velocity and electrostatic potential. We assume that the electric field has its origins in the dynamo layer of the E-region and that it is communicated without serious attenuation
LUNAR
TIME
FIG. 1. CONTOURSOF THE LUNAR VARIATIONS
IN THE HORIZONTAL MAGNETIC INTENSITY H AT THE MAGNETIC EQUATOR ABOVE HUANCAYO (AFIZR CHAPMAN AND BARTELS; SEE RASTOGI, 1965), AS COMPARED WITH THOSE CALCULATED FOR A SINUSOIDAL VARIATIiN OF ELECTRICFIELD USINGTHEFOR~KJLA dH= H,crE.
Ho = 2Oy, u = 41~0s (I+ Y + n-)1and E = -sin 21, where 2 is the local lunar time (0 --t 2~) and v is the local lunar age (0 at new Moon, n at full Moon). The theoretical contours are shifted in time by 1 lunar hr to get better agreement with observations. The shaded areas correspond to solar night-time.
1074
M. F. K. ABIJR-ROBBand E. DUNF~RD
(left) shows contours of the lunar daily variations of the horizontal component of the magnetic field at Huancayo (Rastogi, 1965). We may interpret the variations as being the result of a predominantly semi-diurnal electric field modulated by a conductivity which is small during solar night-time and rises to a peak value at mid-day. The phase of this semi-diurnal electric fleld remains constantin lunartime in accordance with Chapman’s phase law (Chapman and Bartels, 1940). In an attempt to reproduce these features, we have constructed the following model. We assume for simplicity that the Moon makes 29 rotations round the Earth in a period of 30 solar days, as compared with 28.984 rotations in reality. The lunar contribution to f we will then assume to be of the form fi = w, sin (WV + 6,) 3 = -w,
wg,
cos (WV + 6,),
(12)
where w = 29115, and we give W, a value one tenth of the corresponding solar value, namely 1.46 m/set. The phase 6, was assigned a fixed value in accordance with Chapman’s phase law and for simplicity the value chosen is zero. To check the reality of these assumptions, the magnetic field variations caused by such an electric field have been calculated. The conductivity is assumed to vary as 4~0s x (where x is the solar zenith angle) during the day and very small at night. The calculated variations are illustrated in Fig. 1 (right). We see that the agreement with Rastogi’s experimental results is satisfactory during the sunlit hours, but there is insufBcient evidence at night, when the variations are extremely small. We shall assume that the electric field has the same form during day and night. In this case, the functions f and g take the forms
f = W, sin cp + W, sin (29~115) g = - W,cos Q - (15/29)W,cos
(29&5).
(13)
It is clear that the solar drift has a period of 24 solar hr whereas the lunar drift has a period of 15 solar days, i.e. half a lunar month; one solar day representing a lunar phase of 08 lunar hr, reckoning 24 lunar hr in a lunar month. The lunar drift has also a period of half a lunar day when measured with respect to local lunar time. It can be shown that the lunar drift maximizes at 9hr local lunar time. Thus in our simple model, the lunar drift is controlled by the lunar time rather than by the lunar age, i.e. by the position of the Moon irrespective of the position of the Sun.
3. PARAMETERS OF THE PROBLEM
In solving Equation (10) the following parameters are used: K,, = 3.8 x lO-a/sec, D, = 6.1 x 106m2/ set, H=SOkm, 6=0 and I=1.75, where the s&ix zero applies to the level of maximum production at the base of the FZlayer (180 km above the ground). The loss coefficient is denoted by K, the diffusion rate by D, the scale height by H, the solar declination angle by 6 and A is the vertical gradient of In K. The values for Hand 6 correspond to sunspot minimum and equinoctial conditions, respectively. With these fixed parameters, two sets of results are obtained using the following combinations. (1) W, = 14.6 m/set, W, = 0 (2) W, = 14.6 m/set, W, = 1.46 m/set. Here W, and W, denote the amplitude and lunar drifts, respectively.
of the solar
4. NUMERICAL PROCEDURE
Equation (10) is integrated numerically on a digital computer for a series of fixed values of a’ (> r0 = 6374 + 180 km) using the Crank-Nicolson technique. On the boundaries r = r,,, we take v =
-F,/F,
at
x -
fl.
i.e. we assume photochemical equilibrium at and below the base of the F2-layer (r < r,,). The integration is performed over the two hemispheres covering the entire region -1 < x < 1 and started at q = 0 (sunrise) with u = 0 and o = 2. The integration is continued stepwise in q until the quasi-steady state is reached (normally at Q = 6~). The resulting u’s at q = 6~ are then taken as starting values and the integration is recommenced at Q = 0 with o = 29/15 and continued stepwise in q until q = 32n (i.e. for 16 Earth rotations). The steplengths used are 0.05 in x and 1.5’ (6 min) in 8. The results for 30~ < q < 32n (full Moon) were found to be identical to those for 0 < q < 2n (new Moon) and the numerical integration is regarded as satisfactory. This procedure is used to obtain the results with W, # 0 whereas the solar tidal variations with W, = 0 are obtained by terminating the integration at v = 6n, i.e. after three Earth rotations. The values of N,,,F2 and h,F2 are obtained as functions of local solar time at fixed lunar ages by interpolating the computer results. The difference between the results for W, # 0 and those for W, = 0 gives the lunar tidal variations.
1075
The 2%layer lunar tidal variations 5(A). RESULTS (FOR THE MAGNETIC EQUATORIAL ZONE) AND DISCUSSION
was found to be larger than the other harmonics by orders of magnitude. In Table 1 are shown the amplitude (P2) and phase (t2) of the calculated lunar semi-monthly oscillations, as compared with observations at the solar hours 0, 3, . . . ,21. The phases are given in terms of local lunar time measured in lunar hr. The results presented here are for the case WJ W, = 0.1, but those marked (*) are obtained with W,/ W, = 0.2. The last line in Table 1 gives the daily average values; the phases being given in terms of local lunar age measured in lunar hr from new Moon. No precise comparison with observations is attempted here because of the difference between the various determinations of the observed lunar The two sets of experimental results variations.
The diurnal variations As a typical example of the F2-layer behaviour when both solar and lunar tides are taken into account, the theoretical values of the maximum height h,F2 + 6h,F2 at the magnetic equator are plotted in Fig. 2 against local solar time for eight phases of the Moon. Figure 2 indicates the presence of strong lunar tides. The lunar semi-monthly oscillations The lunar tidal variations in N,F2 and h,F2 at a fixed solar time are analysed into different harmonics. The predominant component, being semi-monthly,
a
J-4
42
0
d‘3
x
I 0
I
I
I
24
0
12
LHT
I 24
HEIGHT
OF
THE
I
12
DAILV
WT
VARIATIONS
DF
THE
8 RlASES
Fro. 2. THE CALCULATED
I 0 F2 PEAK
DF
THE
I 24
I
12
LMT
AT THE
I 0
I
12
MAGNETIC
LMT
EQUATOR
I 24
FOR
MOON
DIURNAL VARIATIONS OF MAXIMUM HEIGHT
h,J?2 AT
THE
MAGNETIC
EQUATOR
POR 8 PHASES OF THE MOON.
drift = 14*6m/sec, lunar drift = l-I.6 m/set.
Solar
TABLE1.
~EFFICIENTS
AND PHASES OF THE TEIBGRBTICAL AND OBSRRVBD LUNAR SEMI-MONTHLY OF THE F2-LAYER AT THE MAGNETIC EQUATOR
Experimental fa2, Rastogi Local solar time
(1968)
MHz
Theoretical
Rush et al. (1968) 2F,I Mean (%)
h,,,FZ, km
J’s
ts
PI
ts
5.9 :.;*
0.10 0.04 0.11
11.6 0.0 5.8
3.39 5.64 1.90
0.22 0.22 0.13
1.5 6.5 1.8
7.46 11.3 6.19
2.36 0.33 0.82
0.258 0.056 0.037
0.9 0.8 1.2
10.9 11.3 6.83
1;
8.4 7.9
0.33 0.17
2.7 3.7
4.05 a.35
0,33 0.18
3.3
4.28 8.35
4.64 3.91
0.205 0.098
4.4 4.3
4.42 2.51
12 15 18
88:;
aTo 6.12
0;7 0.23
8;10 5.41
5*29
0.40* 0.194 0.244
4*2*
75 5.6
co 5.2
f.?
oY4 0.26
;..i
8.62* 4.07 4.61
21
7.5
0.17
8.0
4.53
0.16
9.1
4.27
4.21
0.167
9.1
Mean
6.27
0.11
3.51
-
-
-
3.125
0.11
10.5
0 :
10
results
100 N,,,F2/qTse
2FsI Mean (%)
Mean foF2
OSCIUATIONS
Fsl Mean
Mean
P,
t8
(%I
Fsl Mean
Mean
PS
t,
(%)
9.9 10.3 10.2 9.2 7.2 7.4. 7.9
0.74 1.47 I.57 1.32 1.74 3.08’
2.25 4.50 5.20 4.50 7.00 12.4*
3.97
306 305 331 341 403 403* 389 383 351
66
1’0.“9
1.96 2.73
3.52
355
2.5
7.5
0.71
The theoretical results are for the case WC/W, = 0.1, but those marked (*) are for the case W,/W, variations (last line) are given in terms of local lunar age measured in lunar hr from new Moon.
7’;’
= 0.2.
.
1.41
Phases of the mean
1076
M. F. K.
ABLJR-ROBB
given in Table 1 give an illustration of this variability; in these two cases the same methods were used but the data were from different periods. However, the main features of the observed variations are fairly well reproduced theoretically here; the phases are in fair agreement and the amplitudes are in better agreement with the results of Rush et al. than with Rastogi’s results. We note here that, in spite of the noticablediscrepancies between theory and experiment at particular solar hours, the observed daily average variation in N,,,F2 is predicted almost exactly (see last line in Table 1). Table 1 shows that the amplitude of the lunar semi-monthly variations in N,F2 ranges from 251 per cent at 09:Ofl hr to 11.3 per cent at 06:OO hr solar time. The amplitude at Huancayo during the period 1942-1944 was found by Martyn (1947) to vary from 2 per cent at 07:OO hr to 14.6 per cent at 05:OO hr; this is in fair agreement with our theoretical predictions. Table 1 also shows the amplitude of the lunar tide in h,F2 ranging from 2~25kmat00:OOhrtoasmuchas10kmat21:OOhr. The determination of the lunar variations in N,,,F2 and h,F2 near solar noon have received the attention of many workers. We mention here Martyn’s analysis for Huancayo; he obtained an 8.6 per cent change from the noon mean value of N,,,F2 and a phase of 4 lunar hr, together with 8 km and 7 lunar hr for the amplitude and phase of the semi-monthly variation in h,F2. It can be seen from Table 1 that, for the case W,/WS = 0.2, the daytime theoretical results are in close agreement with observations. During the night, however, the calculated variations in this case were much larger than those observed. It should be noted that the value of N,,,F2 or h+,,F2 averaged over all hours of a lunar day is found to depend on the lunar age. The daily average value, when plotted against lunar age, exhibits a pure sinusoidal wave having two peaks of the same magnitude within one lunar month. The amplitudes and phases are given in Table 1 (last line) as 352 per cent and 10.5 lunar hr for NmF2 and 2.5 km and 7.5 hr for h,F2. The results for N,,,F2 are in close agreement with the experimental results of Rastogi (1968) who obtained 351 per cent and 10 hr, respectively, for the amplitude and phase of the lunar semi-monthly variation at Huancayo. The lunar daily variations The lunar tidal variations in N,,,F2 and h,F2 at the magnetic equator can also be analysed as
and E. Dunroan functions of local lunar time at fixed lunar ages. It is found that each curve shows two waves of unequal amplitude indicating the presence of luni-solar diurnal and lunar semi-diurnal oscillations of comparable magnitude. The average whole lunation curve is almost a pure sinusoidal wave having two peaks of the same magnitude within one lunar day. The diurnal component seems to have been largely cancelled out in the averaging over the whole lunation. The coefficients and phases of the calculated semi-diurnal lunar tides in N,F2 and h,F2 when averaged over the whole lunation are given in Table 2. Experimental results are also given for N,F2 at Ibadan (after Brown, 1956), for N,F2 and h,F2 at Huancayo (after Martyn, 1947) and for N,F2 at Huancayo (after Rastogi, 1968). In their analysis, these authors used the annual mean values and the data were taken during periods of low solar activity. The theoretical mean value of N,F2 is found to be O.O3125rq,e and the experimental mean value of f0F2 is given by 6.154 MHz (after Martyn, 1947) a.ndby6.27MHz(after Rastogi,1968). It can be seen from Table 2 that the calculated semi-diurnal oscillations agree fairly well with observations at Huancayo. The theoretical oscillations in N,,,F2 is 3.1 per cent with a phase of 4 lunar hr which compares well with Martyn’s results for Huancayo, viz. 3.25 per cent with phase 4.33 hr. The theoretical results also compare well with those of Rastogi who obtained 3.03 per cent and 4 hr for the amplitude and phase of the lunar tide at Huancayo. The calculated amplitude of h,F2 is 4.6km with a phase of 1Ohr which compares well with Martyn’s results, 5.2 km and 8.43 hr. It is worth pointing out that the discrepancy between the observed results for Ibadan and Huancayo may be due to the fact that these stations lie in different ionospheric zones. 2.
TABLE AND
COEFFICIENTS
OBSERVED WHOLE
AND
SEMI-DIURNAL LUNATION
PHASES LUNAR
AT THE w,/w,
PI
=
OF TIiJ3 THEORETICAL VARIATIONS
MAGNETIC
FOR
THE
EQUATOR.
0.1
tl, lunar hi-
Reference
Theoretical N,,,F2 hmF2 Huancayo N,m
h,F2 N,,,F2 Ibadan N,F2 &J2
3-l % 4.6km
4 10
Present work
3.25% 5.2 km 3.03 %
4.33 8.43 4.00
Martyn 1(1974) Rastogi (1968)
3.83 % 7.7 km
3.80 8.10
Brown (1956)
The F2-layer lunar tidal variations Contours of lunar tidal variations In Figs. 3 and 4 are shown contours of constant values of 6N,,,F2 and 6h,F2 on the coordinates of local lunar time and local lunar age. The contours of the lunar variations are taken to be free from any solar tidal component. However, their values are governed by the position of both the Sun and the 10F2 AmER
00
Nm F2
(EX~~IWENTAL)
(THEORY)
RASTOQI AN0 ALURKAR
_ 06
1.2
24
..._-z.
LUNAR
00
06
1.2
12
24
TIME
FIG. 3. tiNTOuRS OF CONSTANT VALUES OF THE LUNAR VARIATIONS IN f,F2 (EXPERIMENTAL RESULTS AFTER RASTOGIAND ALLJRKAR; SEE RASTOGI, 1965)~~~ CALCULA~D N,,,F2 AT THE MAGNETIC EQUATOR As FUNCTIONSOPLOCALLUNARTIMEANDLUNARAGE.
Solar drift = 14.6 m/w, lunar drift = 1.46 m/set.
Lunar Time,L.hr.
FIG. 4. CoNTow
1077
Moon relative to the Earth. The lunar age Y is indirectly representing the local solar time t and is related to the local lunar time I by the equation: t = 1 + v. The solar night-time period is indicated by the shaded portion of the diagrams. Figure 3 shows contours of the calculated lunar tidal variations in N,,,F2 as compared with contours of foF2 quoted by Rastogi (1965). The numbers on the theoretical curves are normalized values, five units representing approximately a 15.2 per cent change from the mean value of N,,,F2. The experimental value of 0.6 MHz for the lunar variation in foF2 gives a variation of 18 per cent in NmF2 compared with a variation of 14.6 per cent obtained by Martyn (1947). It can be seen from Fig. 3 that during daylight hours, the observed variations are quite well reproduced; the amplitude and phase agreeing well. During the night however, the theoretical variations are far too large. Figure 4 shows contours of the lunar variations at Huancayo (after Rastogi, 1970) obtained for a period of moderate solar activity. The section on the 1.h.s. indicates that large changes in NmF2 do exist during the night-time and that the maximum amplitude tends to occur near both solar noon and mid-night; these features are in good agreement with the theoretical results. Figure 4 (right) shows the observed variations in h,F2 with a maximum amplitude of 10 km which is almost identical to the calculated maximum amplitude (see Table 1). Again, a feature which is common in both observed
LunarTime,L.hr.
OF THE OBSERVED LUNAR VARL~TIONS AT HUANCAYO OBTAINED FOR A PERIOD OF MODERATE SGLARACTIVITY(AFTER RASTOGIAND ALURKM,~~~~).
M. F. K. ABUR-ROBBand E. DUNPORD
1078
5(B). RESULTS (FOR THE LATITUDES UP TO &20°) AND DISCUSSION
and theoretical results is that the maximum amplitude tends to occur near solar noon and mid-night. We note here that the theoretical phases are arbitrary and if the correct ‘E X B’ drift phase is chosen, a better agreement with observations may be obtained.
The lunar semi-monthly variations In Table 3 the phases of the lunar ‘E X B’ drift, 6N,F2 and 6h,,,F2 are given at fixed solar times for the magnetic latitudes 48,. . . , 20°. The phase measured in lunar hr represents the local lunar time when the maximum positive variation occurs. Clearly, the lunar drift (and so the electric field) is controlled by the local lunar time, i.e. by the position of the Moon irrespective of the position of the Sun. It can be seen that the response of the F2-layer to small perturbations in the electric field is not immediate, and there is a phase Jag whose magnitude depends on solar time. Since the height of the F2-peak is controlled more by the solar drift than by the lunar drift, the magnitude of the phase lag depends on the current height of the peak. The phase of 6N,,,F2 is seen to vary more rapidly with solar time than does the phase of 6h,F2. However, during the 6-hr period centred at solar mid-day, the phase of 6NmF2 remains almost constant. During the same period the drift and 6N,,,F2 are out of phase at low latitudes and are in phase at moderate latitudes with a maximum delay of about 3 lunar hr at 15 hr solar time. Figure 5 shows the amplitudes and phases of the semi-monthly oscillations in N,F2 and h,F2 plotted against magnetic latitude at solar mid-day. This figure shows that the amplitude of N,F2 increases slowly in going from the equator to higher latitudes and reaches a maximum value at about 5’, then it decreases rapidly and reaches a minimum value at about 10”. Beyond 10” lat., the amplitude has a single maximum developed at about 14’. Otherwise constant, the phase of N,,,F2 undergoes a rapid shift of about 5 lunar
Physical interpretations According to observations, the phase of the lunar variations in N,,,F2 averaged over the whole lunation is about 6 hr earlier than the phase of h,F2. This feature is in good agreement with the theoretical results (see Table 2). To explain this phenomenon we note that when 6h,F2 is positive, the F2-layer is raised above the daily mean value of h,F2, and when it is negative the n-layer is lowered below this level. In the former case the ionization is transported upwards from the region near the magnetic equator and gravitates along the geomagnetic field lines to settle at the crests of the equatorial anomaly; this leads to the negative values of 6N,F2. In the latter case the ionization is transported back to settle at the magnetic equator and very low latitudes giving rise to positive values of dN,,,F2. It seems that the assumed form of the ‘E x B drift provides an explanation as to why the maximum positive variation in h,F2 occurs at 10 lunar hr. The lunar drift has been shown to maximize at 9 hr local lunar time. Thus the response of h,F2 to the drift is not immediate; the dissipative processes cause a delay of about one hour. It should be noticed that while changes in the ‘E x B’ drift are felt in h,F2 with a delay of an hour, N,,,F2 at the magnetic equator responds almost immediately to any changes in h,F2.
TABLZ3.
CALCULATRD DENSITY
PHASES OF THE SEMI-MONTHLY
N&Z
AND MAXIMUM
HRIGHT
m
OSCILLATIONS w,/w,
Solar
time
Lunar drift
0”
Nm 0.9 0.8 1.2 4.3 4.4 4.4 ;:;’
IN THE LUNAR
6 MAGNRTIC
POR
=
4”
12O
80
Nm
hm
Nm
hm
9.9
o-9
9.9
O-4
0.8
9.9 10.3 10.2
0.9
10.3 10.2 9-2 7.2 7.9 10.9 9.6
O-8 1.2
10.3 10.0 *i: ’
0.8 l-2 11.4 11.6
4.3 :I: 4.7 6.1
P-8
9.2 8.3 7.7 . I;.:
44
6.2 7-6 11.2
DRIFT,
MAXIMUM
ELECTRON
AT FIXED SOLAR HOURS.
o-1
hm
1.2
‘E X B’
LATITUDES
Nm
109.: .
11.9
10.8
10.6 0.5
20”
16”
h, 9.9 9’:; 9.4 10.3 11-s 11.5 10.0
N,
h,
N,
h,
9.9 8.9
0.4 0.8
9.2 8.5
lA.3
95 9.6
10.0 1.2
;:;
11.4 11.9 1-l 0.5
10.1 10.2 10.0 9.8
11.2 o-1 0.4 0.4
10.0 10.0 9.8 9.5
0.4 0.8
The n-layer
lunar tidal variations
1079
the individual curves for 6N,,,F2 and 6h,,J2 at fixed magnetic latitudes. The amplitudes and phases of the predominant semi-monthly components plotted against magnetic latitude are shown in Fig. 6. The resulting variations are similar to those obtained at solar mid-day. Here we have smaller amplitudes and the phase shift occurs at about 13-14’lat as compared with 10” for mid-day 6N+,,F2. The lunar daily average variations
i
0'
0
I
t
4 6 hlognetic
I
I
I
12 16 20 Latitude , Degs.
24’
FIG.
5. THE LATITUDINAL VARIATION OF TI-IBAMPLITUDE (BOITOMCURVES)ANDPHASE f2 (TOP CURvBS)OPTHE LUNARSEMI-MONTHLYOSCILLATIONSIN MAXIMUMELFCTRON DBNSITY N,,,F2 AND MAXIMUM HEIGHT hz2 AT SOLAR MID-DAY.
P2
Solar drift =
These are the lunar variations in NmF2 and h,F2 obtained for the whole hmation by averaging over all lunar ages at fixed magnetic latitudes. Figure 7 shows the amplitudes and phases of the predominant semi-diurnal oscillations plotted against magntic latitude. It can be seen that the latitude of phase reversal is 9’ and the amplitude of SN,,,F2 achieves maximum values at the equator and 14” lat. The amplitude of Sh,F2 develops a single maximum at about 9’ which coincides with the latitude of phase reversal. The top curves show that the maximum of lunar daily average variations in N,,,F2 occurs at about 11 lunar hr in moderate latitudes and at 4 hr in low latitudes. In other words, there is a 5-hr difference between low and moderate latitudes. The maximum of the lunar variations in
146 m/set, lunar drift = 146 m/set.
hours in going from 8 to 12’ and the so-called phase reversal occurs at about 10” lat. These theoretical details are in fair agreement with Rastogi’s (1961) experimental results. The same Fig. 5 shows that the amplitude of 6h,F2 is a minimum at the equator and increases by 3 kmlin going from the equator to 6’ lat. Beyond 6” it decreases with increasing latitude. Figure 5 (top curves) show that 8N,,,F2 and Sh,F2 are out of phase by 3 hr at the equator and by 1 hr at moderate latitudes. The one hour phase difference at moderate latitudes can be interpreted as being the time constant for the ionization to re spond to changes in h,F2. There is also a phase lag of about 1 hr between h,F2 and the electric field. Thus near solar mid-day, the maximum height responds to the applied electric field with a delay of about 1 hr whilst the maximum concentration responds to the same field with a delay of about 2 lunar hr at moderate latitudes. It must be pointed out here that N,,,F2 responds to the electric field via h,F2. The solar dairy average variations The average semi-monthly variations for the whole solar day may be obtained by averaging
(Doily-Average)
0'
0
I
I
I
I
4
I
20 hiognk
24
IZitude YDegs.
FIG. 6. THE LATITUDINAL VARIATIONS OF TIIBAMPLIT~E P2 (BOlTOh CUBvBS)AND PHASE V2 (TOPCURvBs)OPTHE LUNAR SBMI-MONTHLY OSCIUATIONS IN DAILY AVERAGE VALUE OF N,,,F2 ANDhJ-7. Solar
drift = 14.6 m/set, lunar drift = 1.46 m/set.
1080
M. F. K. AEWR-ROBBand E. DUNFORD (Whola
magnetic equator can be explained satisfactorily by the superposition of a small lunar drift on a large solar drift. (2) The calculated lunar tides at low latitudes show better agreement with observations from data spread over all solar hours than from data confined to any one particular solar hour. (3) The abrupt change of phase with latitude predicted from the average (and mid-day) lunar tidal variations results mostly from the fact that the n-peak has, on the average, a steep latitudinal gradient.
Lunatlon)
work has been largely done by one of us (M.F.K.A.-R.). We shouldliketothank Professor P. C. Kendall and Dr. D. W. Windle for their interest and encouragement. The numerical calculations of this paper were carried out on the University of Sheffield ICL Computer. Acknowledgements-This
FIO, 7. TnE
LATITUDINAL
VARIATIONS
P2
CURVES)
PHASE
(BOlTOM
LUNAR
SEMI-DIURNAL ATION
AVERAQE
AND
t2
OSCILLATIONS VALUE
OF
OF THE (TOP
IN
N,,,F2
AhiFLITUDE
CURVES)
THE AND
WHOLE
OF THE LUN-
h,F2.
Solar drift = 14.6 m/set, lunar drift = 1.46 m/set. h,F2 occurs at 10 hr in low latitudes
and at about 9 hr in moderate latitudes. Thus there is a time lag of about 2 lunar hr between the phases of N,F2 and h,,,F2 at moderate latitudes and about 6 hr in the magnetic equatorial zone. These results are in good agreement with Matsushita’s (1967) deductions and the experimental results of Rush et al. (1968). The following explanation of the above described results is based on the diffusion-drift theory. At low latitudes, an upwards drift decreases N,F2 by transporting the ionization along the geomagnetic field lines to settle at the latitudinal crests of the while a downward drift equatorial anomaly, increases N,,,F2 by transporting the ionization back to settle at the equator and very low latitudes. At moderate latitudes, an upwards drift increases N,F2 while a downward drift decreases N,F2. This is expected since the loss coefficient is greater at lower heights. 6. CONCLUSIONS The
following conclusions may be drawn from the calculations of this paper. (1) The main features of the observed lunar tidal variations of the F2-region within 20’ of the
REFERENCES Abur-Robb, M. F. K. (1970). Ph.D. Thesis, Department of Applied Mathematics and Computing Science, University of Sheffield. Abur-Robb M. F. K. and Dunford, E. (1969). Proc. 3rd Int. Symp. Equatorial Aeron., p. 238. Abur-Robb, M. F. K. and Windle, D. W. (1969). Planet. Space Sci. 17, 97.
Anderson, D. N., Matsushita, S. and Tarpley, J. (1973). J. atmos. terr. Phys. 35,753.
Balsley, B. B. (1969). J. geophys. Res. 74,1213. Baxter, R. G. and Kendall, P. C. (1968). Proc. Roy. Sot. A304,171. Brown, R. A. (1956). J. atmos. terr. Phys. 9, 144. Chapman, S. and Bartels, J. (1940). Geomagnetism. Oxford University Press, London. Dunford, E. (1970). J. atmos. terr. Phys. 32,421. Dunford, E. and Lawden, M. D. (1969). Planet. Space Sci. 17, 923. Gliddon, J. E. C. and Kendall, P. C. (1964). J. atmos. terr. Phys. 26,721.
Kendall, P. C. (1962). J. atmos. terr. Phys. 24,805. Lyon, A. J. (1963). J. geophys. Res. 68,253l. Maeda, H. (1955). J. Geomagn. Geoelect. 7,75. Martyn, D. R. (1947). Proc. Roy. Sot. A190,273. Matsushita, S. (1967). EncyclopediaofPhysics (Handbuch der Physik), 49/111/2,p. 547. Springer, New York. Rastogi, R. G. (1961). J. atmos. ferr. Phys. 22,290. Rastogi, R. G. (1963). Proc. Indim Acad. Sci. 58, 38. Rastogi, R. G. (1965). Z. Geophys. 31,27. Rastogi, R. G. (1968). J. atmos. terr. Phys. 30,497. Rastogi, R. G. and Alurkar, S. K. (1964). J. atmos. terr. Phys. 26,641.
Rastogi, R. G. and Alurkar, S. K. (1966). Proc. Indian Acad. Sci. 63,75.
Rastogi, R. G. and Alurkar, S. K. (1970). Pageoph. 80, 288.
Rush, C. M. and Venkateswaran, S. V. (1968). J. atmos. terr. Phys. 30,633.
Tarpley, J. D. (1970). Planet. Space Sci. 18, 1075. Zagulyaeva, V. A. and Fatkullin, M. N. (1969). J. atmos. terr. Phys. 31, 1.