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Trapping of whistler-mode waves in ducts with tapered ends
Trapping of whistler-mode waves in ducts with tapered ends H. J. STRANGEWAYS Department of Physics, University
of Sheffield, Sheffield S3 7RH, U.K.
(Received in,final form 25 February
198 I)
Abstract---The trapping of whistler-mode waves in field-aligned electron density enhancements (ducts) with tapered bases is investigated by ray-tracing for ducts terminating above 1600 and 900 km corresponding to summer day and summer night conditions respectively. This complements previous ray-tracing calculations made for ducts extending down to 300 km (STRANGEWAYS,1981).The position and size of whistler exit-points is estimated for the model ducts employed. These are found to vary from 15 to 90 km in size and are located equatorwards of the foot of the field line of the duct. Two trapping modes are generally operative, corresponding to trapping through the base region and side of a duct and can result in two separated exitpoints for a single duct. The variation of the trapping properties of a duct with the altitude above which it terminates is investigated in detail for both summer day and winter night conditions for altitudes of termination between 200 and 1900 km. The altitude ofduct te~ination has a very marked effect on the proportion of waves trapped by each of the trapping modes. It also influences the range of initial latitude from which rays can become trapped in the duct, the effect being greatest for the winter night model. I. INTRODUCTKON
The altitude at which field aligned enhancements of electron density (ducts) terminate has been investigated both experimentally and theoretically. THOMSON and DOWDEN (1977) deduced from simultaneous reception of whistlers on the ground and aboard the satellite ISIS II that whistlers typically exit from ducts at an altitude of about 1800 km but that ducts may at times extend down to the ISIS II satellite altitude of 1400 km. CERISIEK (1974) reports observations of ducted and partly ducted signals received aboard the satellite FR-l from the 16.8 kHz FUB transmitter. He found from simultaneous observations aboard the satellite of VLF waves and electron density that the low altitude end of ducts was generally situated above the altitude of the satellite (750 km), but that on occasions ducts extend down to the topside ionosphere. JAMES(1972) also deduced that ducting generally occurs above 750 km altitude from observations of the wave normal directions of upgoing waves at the FR- I satellite altitude. Observations of whistler exitpoints using VLF direction-finding receivers also imply that whistlers escape from ducts at higher altitudes (above 1000 km) at least some of the time (RYCROFTet ul., 1975; MATTHEWSet al., 1979). If VLF waves escape through the side of ducts, satellite observations of a general departure of whistlers from field-aligned propagation at 750 km do not in fact imply that ducts terminate at or above this altitude; the waves may have left through the duct side at higher altitude. BFRYHARIIT and PARK (1977) have predicted the altitude of termination of ducts by numerical simu-
lation of the ionosphere and protonosphere. They found that a duct situated at J!. = 4 will extend down to 300 km altitude at night during winter and equinoxes. They also predict that the duct would terminate above 1800 km during the day and in summer above 1000 km at all local times. BERNHARDTand PARK (1977) point out that they ignore the effect of electric fields in their calculations and also the precipitation of low energy charged particles into the ionosphere, which might increase the density enhancement at low altitudes. In a previous paper (STRANGEWAYS,19811, hereafter referred to as paper 1, the exit-point of mid-latitude whistlers was investigated by ray-tracing in ducts extending down to 300 km in both summer day (SD) and winter night (WN) magnetospheric models. However, in the light of the calculations Of BERNHARIX and PARK (1977), it is also desirable to investigaie trapping in ducts with higher base altitudes. For ray-tracing through the region of the duct base to be possible (using the Haselgrove equations) a duct end was modelled in which the density enhancement was gradually reduced to zero over a range of altitude. This was chosen to be 100-200 km which is a larger distance than the effective duct width at the altitude range of the tapering region. In this tapering region the duct enhancement falls from its maximum value at and above the altitude of the top of the tapering region (h,) to zero enhancement at the bottom of the tapering region (altitude hb). A quartic relation was used to express the variation ofenhancement with altitude, viz.
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Here A’ is the reduced enhancement at altitude P in the tapering off region and A is the full enhancement at the same L-value above this region. The quartic expression was used to represent the tapering region to ensure that the partial derivative of the duct enhancement with respect to altitude was zero at both the bottom and the top of the tapering region ; a requirement both for ray-tracing continuity and for the duct end to be physically realistic. The modification required to the Haselgrove derivatives of the electron density in the ray-tracing program is given by STRANGEWAYS(1978). The enhancement above the tapering region at any given t-value was determined as in paper 1 for a Gaussian cross-section duct and as described in Section 4 for a quartic cross-section duct.
2. DUCT WITH TAPERED END AT 1800 km IN THE SUMMER DAY MODEL
Ray-tracing calculations were performed for a tapered Gaussian duct in the SD magnetospheric model (ALEXANIIER,1971; DENBYet al; 1980). The duct parameters were taken to be the same as used in paper
48 0 49 7
I 49 8
I 499
1 so that a direct comparison could be made with raytracing calculations in a similar but untapered duct. These parameters were a 15% enhancement, a 215 km effective width in the equatorial plane and a position centred on L = 2.68 (invariant latitude at 300 km altitude, A300 km = 51.31). The duct enhancement was modelled to taper off below the altitude of 1800 km, reaching zero enhancement at 1600 km, as BERNHARIX and PARK (1977) have predicted that ducts would terminate about this altitude under summer day conditions. It is assumed that there would not be a significant difference in duct tapering, between the duct L-value used (L = 2.68) and that employed by BERNHARIITand PARK in their calculations (L = 4.0). Two kiloHertz rays were started with initially vertically pointing wave normals at 300 km altitude over a fairly wide range of latitude either side of the duct position and those that were found to become trapped in the duct were traced (in the magnetic meridian only) until they reached 300 km altitude in the conjugate hemisphere. Figure 1 shows the final latitude and wave normal of the rays vs. initial latitude. A smooth curve is drawn through the points where the
I 50 0 Initial
I 50 I
I 50 2
latitude, degrees
Fig. I. Final latitude and wave normal angle of 2 kHz rays trapped in a 1.5x, 2 15 km Gaussian duct in the summer day model. The duct enhancement tapers to zero between IX00 and 1600 km. A curved line is drawn through points where the variation is sufficiently slow for interpolation to be possible. Horizontal dotted lines show the extent of the ionospheric transmission cone. Final latitude and wave normal values outside the plotted range are indicated by arrows with the actual value printed alongside.
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Trapping of whistler-mode waves variation with initial latitude is sufficiently slow for interpolation between them. Horizontal dashed lines on the final wave normal angle graph indicate the extent of the ionospheric transmission cone which is taken to be + 0.9” to the vertical at 2 kHz as in paper 1. It can be seen that the variation of both final latitude and wave normal with initial latitude is quasisinusoidal, each cycle corresponding to a different number of ray path excursions to both sides of the duct centre. This number is indicated on the final latitude curve. The form of the variation is different from that resulting from trapping in a duct extending down to 300 km in the same magnetospheric model, for which (see Fig. 1 in paper 1) the minima in the final latitude curve are much broader than the maxima. This difference arises because many rays are trapped thrqugh the base region of the tapered duct whereas for the duct which extended down to 300 km all the rays were trapped through &he duct side. Trapping through the duct base region is facilitated when this is situated between 1600 and 1800 km, as rays entering through the base region then avoid propagating in the duct at lower altitudes, where the refractive index decreases sufficiently along a field-aligned path for untrapping generally to occur. The range of initial latitudes trapped in the tapered duct is a little larger than for the duct which extended down to 300 km. This increase is at the high latitude limit of trapping (from 50.21 to 50.29), at which trapping occurs through the duct base region. The low latitude limit of trapping is the same for both ducts as rays from this latitude are incident on the duct well above 1800 km. The variation of final latitude and wave normal angle is also similar for both ducts at the low latitude end of the range of trapping for the same reason. Rays started at higher initial latitude, however, become trapped through the duct base region and generally propagate inside the duct along its entire length, leaving through the duct base region in the conjugate hemisphere. The final latitude of these ray paths is determined by the wave normal angle of the ray when it leaves the base region. This was found to vary between - 14” and + 12” to the geomagnetic field direction at 1600 km altitude. This large cone of wave normals produced a range of final latitude for the ray paths of about 4”. The variation of final latitude and final wave normal angle with initial latitude are very similar as was found for trapping in the duct extending down to 300 km in paper 1. This is because the unducted parts of the ray paths in the conjugate hemisphere have a well defined locus of initial positions on leaving the duct, corresponding to its side and the base of the tapering region. Although rays leave the duct base region with a fairly wide cone of
wave normal angles, the nature of the ducted propagation (SMITH et al., 1960) results in the final wave normal angle and the position at the base of the tapering region where the ray leaves being systematically related. If rays leave through the centre of the duct base, corresponding to the position of maximum density enhancement, they will have the largest {positive or negative) wave normal angle to the geomagnetic field. If however, a ray leaves the duct near its edge, it will have a nearly field-aligned wave normal direction. Thus for any number of ray path excursions about the duct centre there will be a cyclic variation of the related parameters, the wave normal angle at the duct base and the distance from the duct centre at which the ray leaves the duct. For a smaller number of guide wavelengths in the duct there is a larger range of wave normal angles at the duct base resulting in a wider range of final latitudes as can be seen in Fig. 1. The good correlation between the final wave normal angle and latitude at 300 km altitude results in the range of latitudes for which rays have final wave normal angles inside the transmission cone being significantly smaller than the range of final latitudes for all the ducted paths. This range was found to be 0.83” (49.53-50.36”) which represents a meridional extent of about 90 km. This is a little larger than the exit-point resulting from trapping in the 300 km base altitude duct which was 75 km rO.68” (49&50.28”)]. A comparison of Fig. 1 with fig. 1 in paper 1 shows that fewer rays have wave normal angles inside the transmission cone when the altitude above which the duct terminates is increased. This suggests that ducts extending down to 300 km might be more effective than tapered ducts in producing ground-observed l-hop whistlers. However, more ray-tracing calculations for a range of wave frequencies in different ducts at various L-values should be performed before any general conclusion is reached.
3. DUCT WITHTAPEREDENDAT
1OOOkmINSDAND
WN
MODELS
For night-time
conditions
in summer BEKNHARDT that ducts will terminate above 1000 km. Ray-tracing calculations were therefore performed to investigate ducting for a duct terminating about this altitude. Since no summer night magnetospheric model was readily available, calcuIations were performed in both the SD and WN models. This also has the advantage that results obtained for these two extremes should indicate the total amount of variation that might be expected for any time of day or year for this duct termination altitude. It should also be pointed out that the and PARK (1977) predict
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symmetry of the ray-tracing model employed does not permit seasonal difference between the two hemispheres to be modelled. A Gaussian duct was incorporated in each magnetospheric model which had the same position (centred on L = 2.68) and parameters (15% enhancement, 215 km effective equatorial width) used for the previous ray-tracing calculations. The duct enhancement was modelled to taper off to zero between 1000 and 900 km altitude. Two kiloHertz rays were started at small intervals of initial latitude at 300 km altitude with vertically pointing wave normals as in Section 2. Rays, found to be trapped in the duct, were ray-traced to 300 km altitude in the conjugate hemisphere. Plots of final wave normal angle and final latitude vs. initial latitude are shown in Figs 2 and 3 for ducting in the SD model and in Fig. 4 for the WN model. There were found to be two initial latitude ranges of trapping for this duct in the SD model. This results from there being separate initial latitude ranges corresponding to trapping through the duct side and through the duct base region. The lower latitude range of trapping, which is also the largest in extent, corresponds to trapping through the duct side and ray-
tracing results for this are shown in Fig. 2. Comparison of Fig. 2 with fig 1 ofpaper 1 shows that the variation of final latitude with initial latitude is very similar in both these figures. (There are broad low latitude troughs and narrow high latitude peaks.) This arises because trapping occurs through the duct side only in both cases. The form of the variation of final latitude and wave normal for the initial latitude range corresponding to trapping through the duct base region (Fig. 3) is similar to the form of the variation in Fig. 1 for trapping in the duct with tapered end between 1600 and 1800 km. (There is a quasisinusoidal variation of final latitude and final wave normal angle.) This is because trapping through the duct base occurs for all the ray paths in Fig. 3 and for the great majority of those in Fig. 1. Trapping through either the side or base region of the duct results in a significant number of rays with wave normals inside the transmission cone in the conjugate hemisphere (taken to be kO.9” and shown as horizontal dashed lines in Figs 2 and 3). The exitpoint was determined by examining the range of final latitude of rays which had final wave normal angles inside this cone. For trapping through the duct side,
”
------0
b c
Y
0
s
497 Inltlal
latitude,
degrees
Fig. 2. Final latitude and wave normal angle of 2 kHz rays trapped Gaussian
through the side of a 15x, 215 km duct in the summer day model. The duct enhancement tapers off to zero between 1000 and 900 km. Horizontal dotted lines show the extent of the ionospheric transmission cone.
Trapping ,4r
(14 Id
of whistler-mode
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waves
t (15.12)
E 0
i
so 01 cici3 3
5035
I 504 Imtlal
I 50 45 latitude
,
I 50 5
0
I 5055
degrees
Fig. 3. Final latitude and wave normal angle of 2 kHz rays trapped through the base region of a 15x, 215 km Gaussian duct in the SD model. The duct enhancement tapers off to zero between 1000 and 900 km. Horizontal dotted lines show the extent of the ionospheric transmission cone.
the extent of the exit-point was found to be 0.76” (49.48 -50.24”) but there was also one ray which had a final latitude well outside this range (50.70”). This final latitude was atypical because it resulted from a ray being deflected by passing through the duct between 960 and 940 km altitude after it had escaped from the duct at 3000 km altitude. The exit-point extent for trapping through the duct base region was much smaller, 0.13” (50&t-50.57”), and was located at a slightly higher latitude than that for trapping through the duct side. This is the same result as for the corresponding initial latitude ranges (entry points) for these two trapping modes. The extent of the exit-point is about 85 km for trapping through the duct side and about 15 km for trapping through the duct base. Figure 4 shows the variation of final latitude and wave normal angle with initial latitude for trapping in the duct with tapered end between 1000 and 900 km in the WN model. There is only one range of trapping which corresponds to trapping through the duct base region and the form of the variation of final latitude with initial latitude is similar to that for trapping
through the duct base region in the SD model (Fig. 3). This can be explained by the fact that the altitude of the refractive index minimum along geomagnetic field lines, corresponding to the duct position, is at a lower altitude (1500 km) in the WN than in the SD model. Thus the altitude of the duct base (1000 km) is nearer to the altitude of the refractive index minimum. For trapping through the duct base, this reduces the distance of propagation in a region where the decreasing refractive index along the direction of geomagnetic field lines, corresponding to the duct position, promotes untrapping. The range of initial latitudes for which rays can become trapped in the duct (0.21”) is found to be comparable in extent to that for trapping through the duct base region in the SD model (0.3 1”). The estimated size of the exit-point (determined as previously) was found to be 0.43” (50.6651.09”) which is of the same order of magnitude as the exit-point sizes for trapping in the SD model through the duct base region (0.13”) or the duct side (0.76”). The range of initial latitudes for which rays are trapped in the duct lies nearer to the latitude of the field line of the duct at
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STRANGEWAYS
0 <>
4968
491
49 8
499 Inhal
latitude,
50 0
50
I
5G2
503
degrees
Fig. 4. Final latitude and wave normal angle of 2 kHz rays trapped
in a 15%, 215 km Gaussian
duct in the WN
model. The duct enhancement tapers to zero between 1000 and 900 km. Horizontal dotted lines show the extent of the ionospheric
300 km than for the SD model. In the WN model, the large mid-latitude topside electron density gradient results in equatorwards refraction of upgoing waves. Thus, rays become field-aligned, a favourable condition for trapping, at a lower L-value in the WN than in the SD model if started at the same initial latitude. Consequently, for the same duct position in both models, rays will become trapped in the duct when started at a higher initial latitude in the WN than in the SD model. For summer night conditions, the trapping properties of the duct can be considered to lie between those for summer day and winter night. Trapping will then occur through the duct base region as it does for both the SD and WN models. Some trapping may also take place through the duct side (as for the SD model) but this is unlikely to be the dominant trapping mechanism. The exit-point extent, if the average of that determined for the SD and WN models, would be about 0.5” of latitude and would lie towards the middle of the latitude range 49.5-51.1”. The different results obtained for the initial latitude range of trapping and position and size of the resulting exit-point when ducts terminate at different altitudes show that the altitude of duct termination has a
transmission
cone.
significant effect on the trapping properties of a duct. This is investigated further in the next section by performing ray-tracing calculations in both the SD and WN models for a complete range of duct termination altitudes. 4.
THE VARIATION OF DUCT TRAPPING WITH ALTITUDE OF TERMINATION
The variation of trapping characteristics of a duct with altitude of termination from 200 to 1900 km, was investigated for a duct centred on 1, = 2.5 (A3,,0r_, = 49.67”) in both the SD and WN models. A quartic rather than Gaussian cross-section was used for the duct. The enhancement, A, at any distance x from the duct centre (mapped into the equatorial plane) was given by A=6[1+~($2)]
for
O
where 6 is the maximum enhancement at the duct centre and g,, the duct width either side of the duct centre in the equatorial plane. This duct cross-section is similar to that employed by WALKER (1971). The advantage of the quartic cross-section is that it results
Trapping
of whistler-mode
in a duct with better defined edges (at x = kcr, when mapped into the equatorial plane). When examining a ducted ray path, it can then be more clearly discerned whether trapping (or untrapping) occurred through the duct side or base region. A value c,, = 150 km was chosen corresponding to an equatorial duct width of 300 km. This is a little larger than that of the Gaussian duct employed in Section 3 but still comparable with the satellite observations of duct width made by ANGERAMI(1970). A maximum electron density enhancement at the duct centre of 15% was used as for the previous ray-tracing calculations. The duct enhancement was reduced to zero in 100 km of altitude using the (longitudinal) quartic taper described in Section I. The range of initial latitude which could become trapped in the duct was investigated for ducts which terminated at altitudes between 200 and 1900 km altitude in 100 km steps, Where the variation in the range with altitude could not be easily interpreted with this resolution, smaller altitude steps were employed. In each case the range of trapping was determined as in Sections 2 and 3, by starting 2 kHz rays at small latitude intervals over a latitude range from which it was considered they might become trapped in the duct. The results of the ray-tracing calculations are presented in Figs 5 and 6 for the SD and WN models respectively. The range of trapping which exists at higher latitude than the duct in the WN model (see
1011
waves
paper 1) is not included, since for this range trapping occurs at such a high altitude that the variation of the duct termination height in the range 200-1900 km will have a negligible effect. The initial latitude range of trapping for any altitude of duct termination is indicated by the hatched region, different hatching being used for trapping through the base region and side of the duct as indicated in the figure captions. Figure 5 shows that the low latitude limit of trapping in the SD model is constant with altitude of termination in the range 200-1900 km. This is because trapping at this latitude limit occurs through the duct side above 2000 km in the SD model. There is no variation of high latitude limit of trapping through the duct side in the SD model when the duct terminates below 850 km, but when the duct exists above 440 km there is an additional trapping range corresponding to rays entering the duct through the base region. This is a similar result to the two latitude ranges of trapping found for the 90&1000 km tapered end Gaussian duct in Section 3. It is evident, however, that for the quartic duct the two trapping ranges merge for an altitude of duct termination of 870 km, whereas for a Gaussian duct this merging occurs for a duct termination somewhere between 900 and 1600 km. This difference may be due to the narrower effective equatorial width of the Gaussian duct (215 km) than the quartic duct (300 km). The initial latitude range corresponding to trapping
Extentof duct at 00
km altitude
50 Inlttollatitude at 300km
cltltude,
degrees
Fig. 5. Variation with altitude of duct termination of the initial latitude range at 300 km from which rays can become trapped in a 15x, 300 km quartic duct centred on L = 2.5 in the SD model. 4 hatching indicates trapping takes place through the duct side. Q hatching indicates trapping occurs through the duct base region.
H.J.
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1900 -
E 1
1500-
,
f-l
of duct at 300 km altitude I
Extent
, 200 48 5
_i
I 490
Inhal
495
latitude
I 505
500
at 300 km althde,
degrees
Fig. 6. As Fig. 5 but for the WN model. through the duct base region increases fairly rapidly with increasing altitude of termination in the range 440-900 km, but remains fairly constant for ducts terminating at higher altitude. This can be explained by the fact that the refractive index minimum along the L = 2.5 field line is at about 1850 km. Rays entering the duct base region well below this altitude will have to propagate up to it in the region where the decreasing refractive index along the field line promotes untrapping. The longer this distance is, that is the lower the altitude of the duct termination below 1850 km, the less likely rays will be able to propagate in the duct to higher altitude. Thus the initial latitude range resulting in successful duct trapping will increase fairly rapidly with altitude of duct termination for altitudes well below that of the refractive index minimum. For higher termination altitudes, the increase in initial latitude range of trapping (which partially results from the increased duct width at its base) is less rapid. Also evident in Fig. 5 are two very narrow latitude ranges about O.ll(3.2” wide. These result from trapping through the duct side of rays which have completely passed through the duct at lower altitude and become incident on the duct again at higher altitude. When the duct extends down to 300 km such ray paths generally start inside the duct, near its edge. Figure 6 shows the variation of the initial latitude range of trapping with altitude of termination for the WN model. It is evident that the lowest and highest initial latitudes from which rays can become trapped in the duct are more dependent on the altitude at which the duct terminates than for the SD model. The large
topside mid-latitude electron density gradient results in refraction of the ray paths to lower L-values, which increases with the altitude of the ray position. Thus, for a higher termination altitude, rays must start at 300 km altitude at a higher latitude to enter the base region. It can also be seen from Fig. 6 that rays are trapped through the duct side from a much smaller initial latitude range than occurs for the SD model. The greater refraction of the rays that occurs in the WN model generally results in more rays travelling in nearly field-aligned directions at the altitude of duct termination. This results in fewer rays becoming incident on the duct side, except when the duct terminates at low altitude (~800 km). The lower altitude of the refractive index minimum along the L = 2.5 field line than in the SD model, also favours trapping through the duct base region. The increase of the initial latitude range corresponding to trapping through the duct base region with increasing altitude of termination (in the range 400-800 km) is also much more rapid in the WN than in the SD model. It is clear from both Figs 5 and 6 that the altitude of termination of ducts has a significant effect on their trapping properties. Apart from altering the extent and limits of the initial latitude range of trapping, changes in termination altitude significantly influence the percentage of wave energy trapped through the duct side or base region. For the quartic cross-section duct considered in this section, no trapping occurs through the duct base region in either the SD or WN models when the duct terminates below 400 km. Conversely,
Trapping
of whistler-mode
trapping through the duct side does not occur for the quartic duct in the WN model when the duct terminates above 800 km. For night-time during winter and equinox, BERNHARDT and PARK’S (1977) prediction of ducts extending down to 300 km altitude would suggest that trapping will only occur through the duct side. However, for summer months at all local times [when BERNHARDTand PARK (1977) predict that ducts will terminate above 1000 km] and for summer and winter day conditions (when it is predicted they will terminate above 1800 km), trapping through the duct base region is likely to be the dominant mechanism. It is hoped that VLF direction-finding observations of whistler exit-points, interpreted in the light of raytracing calculations in ducts with tapered ends, may yield evidence on the altitude of duct termination, its diurnal and seasonal variation and the relative importance of the two trapping modes. It must be recognised, however, that in the real magnetosphere, the plasma distribution is not symmetric about the equatorial
waves
plane as in the ray-tracing models employed in this paper. The seasonal difference in altitude of duct termination predicted by BERNHARDTand PARK (1977) can also result in ducts terminating at different altitudes in the northern and southern hemispheres, particularly for day-time conditions. Thus, in order that more realistic results can be obtained, it is important to investigate whistler ducting employing a ray-tracing model with a plasma distribution which is asymmetric about the equatorial plane and which can also incorporate model ducts with different altitudes of termination in the northern and southern hemispheres. This will be the subject of future work. Acknowkdgements--I
am grateful for the facilities made available to me at the Physics Departments at Southampton, Leicester and Sheffield Universities. I would like to thank Dr M. J. RYCRO~~ for helpful discussions and Professor T. R. KAISER, Dr K. BULLOUGH and Dr A. J. SMITH for comments on the draft manuscript. This work was partially funded by the Science Research Council.
REFERENCES ANGERAMIJ. J. BERNHARDTP. A. and PARK C. G. CERISIERJ. C. DENLIY M.. BULL~IJGH K., ALEXANDER P. D. and RYCRIXT M. J. JAM= H. G. MATTHEWSJ. P.. SMITH A. J. and SMITH I. D. SMITH R. L., HELLIWELL R. A. and YABROFFI. W. STRANGEWAYSH. J. STRANGEWAYSH. J. and RYCROFT M. J. THOMWN R. J. and JIOWDEN R. L. WALKER A. D. M.
1970 1977 1974
J. geophys. Res. 75,6115. J. geophys. Res. 82, 5222. J. atmos. terr. Phys. 36, 1443.
1980 1972 1979 1960 1981 1980 1977 1971
J. atmos. terr. Phvs. 42. 51. Annls Geophys. 28, 3Oi. Planet. Soace Sci. 21. 1391. J. geophis. Res. 65, il5. J. atmos. terr. Phys. 43, 231. J. atmos. terr. Phys. 42,983. J. atmos. terr. Phys. 39, 879. Proc. R. Sot. A321,69.
1971 1975 1978
Ph.D. thesis, University of Southampton. ESRO special pub]. No. 107, p. 225. Ph.D. thesis, University of Southampton.
Reference is also made to the following unpublished material ALEXANDERP. D. RYCROFT M. J., JARVIS M. J. and STRANGEWAYSH. J. STRANGEWAYSH. J.
1079
of Sheffield, Sheffield S3 7RH, U.K.
(Received in,final form 25 February
198 I)
Abstract---The trapping of whistler-mode waves in field-aligned electron density enhancements (ducts) with tapered bases is investigated by ray-tracing for ducts terminating above 1600 and 900 km corresponding to summer day and summer night conditions respectively. This complements previous ray-tracing calculations made for ducts extending down to 300 km (STRANGEWAYS,1981).The position and size of whistler exit-points is estimated for the model ducts employed. These are found to vary from 15 to 90 km in size and are located equatorwards of the foot of the field line of the duct. Two trapping modes are generally operative, corresponding to trapping through the base region and side of a duct and can result in two separated exitpoints for a single duct. The variation of the trapping properties of a duct with the altitude above which it terminates is investigated in detail for both summer day and winter night conditions for altitudes of termination between 200 and 1900 km. The altitude ofduct te~ination has a very marked effect on the proportion of waves trapped by each of the trapping modes. It also influences the range of initial latitude from which rays can become trapped in the duct, the effect being greatest for the winter night model. I. INTRODUCTKON
The altitude at which field aligned enhancements of electron density (ducts) terminate has been investigated both experimentally and theoretically. THOMSON and DOWDEN (1977) deduced from simultaneous reception of whistlers on the ground and aboard the satellite ISIS II that whistlers typically exit from ducts at an altitude of about 1800 km but that ducts may at times extend down to the ISIS II satellite altitude of 1400 km. CERISIEK (1974) reports observations of ducted and partly ducted signals received aboard the satellite FR-l from the 16.8 kHz FUB transmitter. He found from simultaneous observations aboard the satellite of VLF waves and electron density that the low altitude end of ducts was generally situated above the altitude of the satellite (750 km), but that on occasions ducts extend down to the topside ionosphere. JAMES(1972) also deduced that ducting generally occurs above 750 km altitude from observations of the wave normal directions of upgoing waves at the FR- I satellite altitude. Observations of whistler exitpoints using VLF direction-finding receivers also imply that whistlers escape from ducts at higher altitudes (above 1000 km) at least some of the time (RYCROFTet ul., 1975; MATTHEWSet al., 1979). If VLF waves escape through the side of ducts, satellite observations of a general departure of whistlers from field-aligned propagation at 750 km do not in fact imply that ducts terminate at or above this altitude; the waves may have left through the duct side at higher altitude. BFRYHARIIT and PARK (1977) have predicted the altitude of termination of ducts by numerical simu-
lation of the ionosphere and protonosphere. They found that a duct situated at J!. = 4 will extend down to 300 km altitude at night during winter and equinoxes. They also predict that the duct would terminate above 1800 km during the day and in summer above 1000 km at all local times. BERNHARDTand PARK (1977) point out that they ignore the effect of electric fields in their calculations and also the precipitation of low energy charged particles into the ionosphere, which might increase the density enhancement at low altitudes. In a previous paper (STRANGEWAYS,19811, hereafter referred to as paper 1, the exit-point of mid-latitude whistlers was investigated by ray-tracing in ducts extending down to 300 km in both summer day (SD) and winter night (WN) magnetospheric models. However, in the light of the calculations Of BERNHARIX and PARK (1977), it is also desirable to investigaie trapping in ducts with higher base altitudes. For ray-tracing through the region of the duct base to be possible (using the Haselgrove equations) a duct end was modelled in which the density enhancement was gradually reduced to zero over a range of altitude. This was chosen to be 100-200 km which is a larger distance than the effective duct width at the altitude range of the tapering region. In this tapering region the duct enhancement falls from its maximum value at and above the altitude of the top of the tapering region (h,) to zero enhancement at the bottom of the tapering region (altitude hb). A quartic relation was used to express the variation ofenhancement with altitude, viz.
H. J.
107’
STRANGEWAYS
Here A’ is the reduced enhancement at altitude P in the tapering off region and A is the full enhancement at the same L-value above this region. The quartic expression was used to represent the tapering region to ensure that the partial derivative of the duct enhancement with respect to altitude was zero at both the bottom and the top of the tapering region ; a requirement both for ray-tracing continuity and for the duct end to be physically realistic. The modification required to the Haselgrove derivatives of the electron density in the ray-tracing program is given by STRANGEWAYS(1978). The enhancement above the tapering region at any given t-value was determined as in paper 1 for a Gaussian cross-section duct and as described in Section 4 for a quartic cross-section duct.
2. DUCT WITH TAPERED END AT 1800 km IN THE SUMMER DAY MODEL
Ray-tracing calculations were performed for a tapered Gaussian duct in the SD magnetospheric model (ALEXANIIER,1971; DENBYet al; 1980). The duct parameters were taken to be the same as used in paper
48 0 49 7
I 49 8
I 499
1 so that a direct comparison could be made with raytracing calculations in a similar but untapered duct. These parameters were a 15% enhancement, a 215 km effective width in the equatorial plane and a position centred on L = 2.68 (invariant latitude at 300 km altitude, A300 km = 51.31). The duct enhancement was modelled to taper off below the altitude of 1800 km, reaching zero enhancement at 1600 km, as BERNHARIX and PARK (1977) have predicted that ducts would terminate about this altitude under summer day conditions. It is assumed that there would not be a significant difference in duct tapering, between the duct L-value used (L = 2.68) and that employed by BERNHARIITand PARK in their calculations (L = 4.0). Two kiloHertz rays were started with initially vertically pointing wave normals at 300 km altitude over a fairly wide range of latitude either side of the duct position and those that were found to become trapped in the duct were traced (in the magnetic meridian only) until they reached 300 km altitude in the conjugate hemisphere. Figure 1 shows the final latitude and wave normal of the rays vs. initial latitude. A smooth curve is drawn through the points where the
I 50 0 Initial
I 50 I
I 50 2
latitude, degrees
Fig. I. Final latitude and wave normal angle of 2 kHz rays trapped in a 1.5x, 2 15 km Gaussian duct in the summer day model. The duct enhancement tapers to zero between IX00 and 1600 km. A curved line is drawn through points where the variation is sufficiently slow for interpolation to be possible. Horizontal dotted lines show the extent of the ionospheric transmission cone. Final latitude and wave normal values outside the plotted range are indicated by arrows with the actual value printed alongside.
c I 50 3
1073
Trapping of whistler-mode waves variation with initial latitude is sufficiently slow for interpolation between them. Horizontal dashed lines on the final wave normal angle graph indicate the extent of the ionospheric transmission cone which is taken to be + 0.9” to the vertical at 2 kHz as in paper 1. It can be seen that the variation of both final latitude and wave normal with initial latitude is quasisinusoidal, each cycle corresponding to a different number of ray path excursions to both sides of the duct centre. This number is indicated on the final latitude curve. The form of the variation is different from that resulting from trapping in a duct extending down to 300 km in the same magnetospheric model, for which (see Fig. 1 in paper 1) the minima in the final latitude curve are much broader than the maxima. This difference arises because many rays are trapped thrqugh the base region of the tapered duct whereas for the duct which extended down to 300 km all the rays were trapped through &he duct side. Trapping through the duct base region is facilitated when this is situated between 1600 and 1800 km, as rays entering through the base region then avoid propagating in the duct at lower altitudes, where the refractive index decreases sufficiently along a field-aligned path for untrapping generally to occur. The range of initial latitudes trapped in the tapered duct is a little larger than for the duct which extended down to 300 km. This increase is at the high latitude limit of trapping (from 50.21 to 50.29), at which trapping occurs through the duct base region. The low latitude limit of trapping is the same for both ducts as rays from this latitude are incident on the duct well above 1800 km. The variation of final latitude and wave normal angle is also similar for both ducts at the low latitude end of the range of trapping for the same reason. Rays started at higher initial latitude, however, become trapped through the duct base region and generally propagate inside the duct along its entire length, leaving through the duct base region in the conjugate hemisphere. The final latitude of these ray paths is determined by the wave normal angle of the ray when it leaves the base region. This was found to vary between - 14” and + 12” to the geomagnetic field direction at 1600 km altitude. This large cone of wave normals produced a range of final latitude for the ray paths of about 4”. The variation of final latitude and final wave normal angle with initial latitude are very similar as was found for trapping in the duct extending down to 300 km in paper 1. This is because the unducted parts of the ray paths in the conjugate hemisphere have a well defined locus of initial positions on leaving the duct, corresponding to its side and the base of the tapering region. Although rays leave the duct base region with a fairly wide cone of
wave normal angles, the nature of the ducted propagation (SMITH et al., 1960) results in the final wave normal angle and the position at the base of the tapering region where the ray leaves being systematically related. If rays leave through the centre of the duct base, corresponding to the position of maximum density enhancement, they will have the largest {positive or negative) wave normal angle to the geomagnetic field. If however, a ray leaves the duct near its edge, it will have a nearly field-aligned wave normal direction. Thus for any number of ray path excursions about the duct centre there will be a cyclic variation of the related parameters, the wave normal angle at the duct base and the distance from the duct centre at which the ray leaves the duct. For a smaller number of guide wavelengths in the duct there is a larger range of wave normal angles at the duct base resulting in a wider range of final latitudes as can be seen in Fig. 1. The good correlation between the final wave normal angle and latitude at 300 km altitude results in the range of latitudes for which rays have final wave normal angles inside the transmission cone being significantly smaller than the range of final latitudes for all the ducted paths. This range was found to be 0.83” (49.53-50.36”) which represents a meridional extent of about 90 km. This is a little larger than the exit-point resulting from trapping in the 300 km base altitude duct which was 75 km rO.68” (49&50.28”)]. A comparison of Fig. 1 with fig. 1 in paper 1 shows that fewer rays have wave normal angles inside the transmission cone when the altitude above which the duct terminates is increased. This suggests that ducts extending down to 300 km might be more effective than tapered ducts in producing ground-observed l-hop whistlers. However, more ray-tracing calculations for a range of wave frequencies in different ducts at various L-values should be performed before any general conclusion is reached.
3. DUCT WITHTAPEREDENDAT
1OOOkmINSDAND
WN
MODELS
For night-time
conditions
in summer BEKNHARDT that ducts will terminate above 1000 km. Ray-tracing calculations were therefore performed to investigate ducting for a duct terminating about this altitude. Since no summer night magnetospheric model was readily available, calcuIations were performed in both the SD and WN models. This also has the advantage that results obtained for these two extremes should indicate the total amount of variation that might be expected for any time of day or year for this duct termination altitude. It should also be pointed out that the and PARK (1977) predict
H.J.
1074
STRANGEWAYS
symmetry of the ray-tracing model employed does not permit seasonal difference between the two hemispheres to be modelled. A Gaussian duct was incorporated in each magnetospheric model which had the same position (centred on L = 2.68) and parameters (15% enhancement, 215 km effective equatorial width) used for the previous ray-tracing calculations. The duct enhancement was modelled to taper off to zero between 1000 and 900 km altitude. Two kiloHertz rays were started at small intervals of initial latitude at 300 km altitude with vertically pointing wave normals as in Section 2. Rays, found to be trapped in the duct, were ray-traced to 300 km altitude in the conjugate hemisphere. Plots of final wave normal angle and final latitude vs. initial latitude are shown in Figs 2 and 3 for ducting in the SD model and in Fig. 4 for the WN model. There were found to be two initial latitude ranges of trapping for this duct in the SD model. This results from there being separate initial latitude ranges corresponding to trapping through the duct side and through the duct base region. The lower latitude range of trapping, which is also the largest in extent, corresponds to trapping through the duct side and ray-
tracing results for this are shown in Fig. 2. Comparison of Fig. 2 with fig 1 ofpaper 1 shows that the variation of final latitude with initial latitude is very similar in both these figures. (There are broad low latitude troughs and narrow high latitude peaks.) This arises because trapping occurs through the duct side only in both cases. The form of the variation of final latitude and wave normal for the initial latitude range corresponding to trapping through the duct base region (Fig. 3) is similar to the form of the variation in Fig. 1 for trapping in the duct with tapered end between 1600 and 1800 km. (There is a quasisinusoidal variation of final latitude and final wave normal angle.) This is because trapping through the duct base occurs for all the ray paths in Fig. 3 and for the great majority of those in Fig. 1. Trapping through either the side or base region of the duct results in a significant number of rays with wave normals inside the transmission cone in the conjugate hemisphere (taken to be kO.9” and shown as horizontal dashed lines in Figs 2 and 3). The exitpoint was determined by examining the range of final latitude of rays which had final wave normal angles inside this cone. For trapping through the duct side,
”
------0
b c
Y
0
s
497 Inltlal
latitude,
degrees
Fig. 2. Final latitude and wave normal angle of 2 kHz rays trapped Gaussian
through the side of a 15x, 215 km duct in the summer day model. The duct enhancement tapers off to zero between 1000 and 900 km. Horizontal dotted lines show the extent of the ionospheric transmission cone.
Trapping ,4r
(14 Id
of whistler-mode
1075
waves
t (15.12)
E 0
i
so 01 cici3 3
5035
I 504 Imtlal
I 50 45 latitude
,
I 50 5
0
I 5055
degrees
Fig. 3. Final latitude and wave normal angle of 2 kHz rays trapped through the base region of a 15x, 215 km Gaussian duct in the SD model. The duct enhancement tapers off to zero between 1000 and 900 km. Horizontal dotted lines show the extent of the ionospheric transmission cone.
the extent of the exit-point was found to be 0.76” (49.48 -50.24”) but there was also one ray which had a final latitude well outside this range (50.70”). This final latitude was atypical because it resulted from a ray being deflected by passing through the duct between 960 and 940 km altitude after it had escaped from the duct at 3000 km altitude. The exit-point extent for trapping through the duct base region was much smaller, 0.13” (50&t-50.57”), and was located at a slightly higher latitude than that for trapping through the duct side. This is the same result as for the corresponding initial latitude ranges (entry points) for these two trapping modes. The extent of the exit-point is about 85 km for trapping through the duct side and about 15 km for trapping through the duct base. Figure 4 shows the variation of final latitude and wave normal angle with initial latitude for trapping in the duct with tapered end between 1000 and 900 km in the WN model. There is only one range of trapping which corresponds to trapping through the duct base region and the form of the variation of final latitude with initial latitude is similar to that for trapping
through the duct base region in the SD model (Fig. 3). This can be explained by the fact that the altitude of the refractive index minimum along geomagnetic field lines, corresponding to the duct position, is at a lower altitude (1500 km) in the WN than in the SD model. Thus the altitude of the duct base (1000 km) is nearer to the altitude of the refractive index minimum. For trapping through the duct base, this reduces the distance of propagation in a region where the decreasing refractive index along the direction of geomagnetic field lines, corresponding to the duct position, promotes untrapping. The range of initial latitudes for which rays can become trapped in the duct (0.21”) is found to be comparable in extent to that for trapping through the duct base region in the SD model (0.3 1”). The estimated size of the exit-point (determined as previously) was found to be 0.43” (50.6651.09”) which is of the same order of magnitude as the exit-point sizes for trapping in the SD model through the duct base region (0.13”) or the duct side (0.76”). The range of initial latitudes for which rays are trapped in the duct lies nearer to the latitude of the field line of the duct at
1076
H. J.
STRANGEWAYS
0 <>
4968
491
49 8
499 Inhal
latitude,
50 0
50
I
5G2
503
degrees
Fig. 4. Final latitude and wave normal angle of 2 kHz rays trapped
in a 15%, 215 km Gaussian
duct in the WN
model. The duct enhancement tapers to zero between 1000 and 900 km. Horizontal dotted lines show the extent of the ionospheric
300 km than for the SD model. In the WN model, the large mid-latitude topside electron density gradient results in equatorwards refraction of upgoing waves. Thus, rays become field-aligned, a favourable condition for trapping, at a lower L-value in the WN than in the SD model if started at the same initial latitude. Consequently, for the same duct position in both models, rays will become trapped in the duct when started at a higher initial latitude in the WN than in the SD model. For summer night conditions, the trapping properties of the duct can be considered to lie between those for summer day and winter night. Trapping will then occur through the duct base region as it does for both the SD and WN models. Some trapping may also take place through the duct side (as for the SD model) but this is unlikely to be the dominant trapping mechanism. The exit-point extent, if the average of that determined for the SD and WN models, would be about 0.5” of latitude and would lie towards the middle of the latitude range 49.5-51.1”. The different results obtained for the initial latitude range of trapping and position and size of the resulting exit-point when ducts terminate at different altitudes show that the altitude of duct termination has a
transmission
cone.
significant effect on the trapping properties of a duct. This is investigated further in the next section by performing ray-tracing calculations in both the SD and WN models for a complete range of duct termination altitudes. 4.
THE VARIATION OF DUCT TRAPPING WITH ALTITUDE OF TERMINATION
The variation of trapping characteristics of a duct with altitude of termination from 200 to 1900 km, was investigated for a duct centred on 1, = 2.5 (A3,,0r_, = 49.67”) in both the SD and WN models. A quartic rather than Gaussian cross-section was used for the duct. The enhancement, A, at any distance x from the duct centre (mapped into the equatorial plane) was given by A=6[1+~($2)]
for
O
where 6 is the maximum enhancement at the duct centre and g,, the duct width either side of the duct centre in the equatorial plane. This duct cross-section is similar to that employed by WALKER (1971). The advantage of the quartic cross-section is that it results
Trapping
of whistler-mode
in a duct with better defined edges (at x = kcr, when mapped into the equatorial plane). When examining a ducted ray path, it can then be more clearly discerned whether trapping (or untrapping) occurred through the duct side or base region. A value c,, = 150 km was chosen corresponding to an equatorial duct width of 300 km. This is a little larger than that of the Gaussian duct employed in Section 3 but still comparable with the satellite observations of duct width made by ANGERAMI(1970). A maximum electron density enhancement at the duct centre of 15% was used as for the previous ray-tracing calculations. The duct enhancement was reduced to zero in 100 km of altitude using the (longitudinal) quartic taper described in Section I. The range of initial latitude which could become trapped in the duct was investigated for ducts which terminated at altitudes between 200 and 1900 km altitude in 100 km steps, Where the variation in the range with altitude could not be easily interpreted with this resolution, smaller altitude steps were employed. In each case the range of trapping was determined as in Sections 2 and 3, by starting 2 kHz rays at small latitude intervals over a latitude range from which it was considered they might become trapped in the duct. The results of the ray-tracing calculations are presented in Figs 5 and 6 for the SD and WN models respectively. The range of trapping which exists at higher latitude than the duct in the WN model (see
1011
waves
paper 1) is not included, since for this range trapping occurs at such a high altitude that the variation of the duct termination height in the range 200-1900 km will have a negligible effect. The initial latitude range of trapping for any altitude of duct termination is indicated by the hatched region, different hatching being used for trapping through the base region and side of the duct as indicated in the figure captions. Figure 5 shows that the low latitude limit of trapping in the SD model is constant with altitude of termination in the range 200-1900 km. This is because trapping at this latitude limit occurs through the duct side above 2000 km in the SD model. There is no variation of high latitude limit of trapping through the duct side in the SD model when the duct terminates below 850 km, but when the duct exists above 440 km there is an additional trapping range corresponding to rays entering the duct through the base region. This is a similar result to the two latitude ranges of trapping found for the 90&1000 km tapered end Gaussian duct in Section 3. It is evident, however, that for the quartic duct the two trapping ranges merge for an altitude of duct termination of 870 km, whereas for a Gaussian duct this merging occurs for a duct termination somewhere between 900 and 1600 km. This difference may be due to the narrower effective equatorial width of the Gaussian duct (215 km) than the quartic duct (300 km). The initial latitude range corresponding to trapping
Extentof duct at 00
km altitude
50 Inlttollatitude at 300km
cltltude,
degrees
Fig. 5. Variation with altitude of duct termination of the initial latitude range at 300 km from which rays can become trapped in a 15x, 300 km quartic duct centred on L = 2.5 in the SD model. 4 hatching indicates trapping takes place through the duct side. Q hatching indicates trapping occurs through the duct base region.
H.J.
107x
STRANGEWAYS
1900 -
E 1
1500-
,
f-l
of duct at 300 km altitude I
Extent
, 200 48 5
_i
I 490
Inhal
495
latitude
I 505
500
at 300 km althde,
degrees
Fig. 6. As Fig. 5 but for the WN model. through the duct base region increases fairly rapidly with increasing altitude of termination in the range 440-900 km, but remains fairly constant for ducts terminating at higher altitude. This can be explained by the fact that the refractive index minimum along the L = 2.5 field line is at about 1850 km. Rays entering the duct base region well below this altitude will have to propagate up to it in the region where the decreasing refractive index along the field line promotes untrapping. The longer this distance is, that is the lower the altitude of the duct termination below 1850 km, the less likely rays will be able to propagate in the duct to higher altitude. Thus the initial latitude range resulting in successful duct trapping will increase fairly rapidly with altitude of duct termination for altitudes well below that of the refractive index minimum. For higher termination altitudes, the increase in initial latitude range of trapping (which partially results from the increased duct width at its base) is less rapid. Also evident in Fig. 5 are two very narrow latitude ranges about O.ll(3.2” wide. These result from trapping through the duct side of rays which have completely passed through the duct at lower altitude and become incident on the duct again at higher altitude. When the duct extends down to 300 km such ray paths generally start inside the duct, near its edge. Figure 6 shows the variation of the initial latitude range of trapping with altitude of termination for the WN model. It is evident that the lowest and highest initial latitudes from which rays can become trapped in the duct are more dependent on the altitude at which the duct terminates than for the SD model. The large
topside mid-latitude electron density gradient results in refraction of the ray paths to lower L-values, which increases with the altitude of the ray position. Thus, for a higher termination altitude, rays must start at 300 km altitude at a higher latitude to enter the base region. It can also be seen from Fig. 6 that rays are trapped through the duct side from a much smaller initial latitude range than occurs for the SD model. The greater refraction of the rays that occurs in the WN model generally results in more rays travelling in nearly field-aligned directions at the altitude of duct termination. This results in fewer rays becoming incident on the duct side, except when the duct terminates at low altitude (~800 km). The lower altitude of the refractive index minimum along the L = 2.5 field line than in the SD model, also favours trapping through the duct base region. The increase of the initial latitude range corresponding to trapping through the duct base region with increasing altitude of termination (in the range 400-800 km) is also much more rapid in the WN than in the SD model. It is clear from both Figs 5 and 6 that the altitude of termination of ducts has a significant effect on their trapping properties. Apart from altering the extent and limits of the initial latitude range of trapping, changes in termination altitude significantly influence the percentage of wave energy trapped through the duct side or base region. For the quartic cross-section duct considered in this section, no trapping occurs through the duct base region in either the SD or WN models when the duct terminates below 400 km. Conversely,
Trapping
of whistler-mode
trapping through the duct side does not occur for the quartic duct in the WN model when the duct terminates above 800 km. For night-time during winter and equinox, BERNHARDT and PARK’S (1977) prediction of ducts extending down to 300 km altitude would suggest that trapping will only occur through the duct side. However, for summer months at all local times [when BERNHARDTand PARK (1977) predict that ducts will terminate above 1000 km] and for summer and winter day conditions (when it is predicted they will terminate above 1800 km), trapping through the duct base region is likely to be the dominant mechanism. It is hoped that VLF direction-finding observations of whistler exit-points, interpreted in the light of raytracing calculations in ducts with tapered ends, may yield evidence on the altitude of duct termination, its diurnal and seasonal variation and the relative importance of the two trapping modes. It must be recognised, however, that in the real magnetosphere, the plasma distribution is not symmetric about the equatorial
waves
plane as in the ray-tracing models employed in this paper. The seasonal difference in altitude of duct termination predicted by BERNHARDTand PARK (1977) can also result in ducts terminating at different altitudes in the northern and southern hemispheres, particularly for day-time conditions. Thus, in order that more realistic results can be obtained, it is important to investigate whistler ducting employing a ray-tracing model with a plasma distribution which is asymmetric about the equatorial plane and which can also incorporate model ducts with different altitudes of termination in the northern and southern hemispheres. This will be the subject of future work. Acknowkdgements--I
am grateful for the facilities made available to me at the Physics Departments at Southampton, Leicester and Sheffield Universities. I would like to thank Dr M. J. RYCRO~~ for helpful discussions and Professor T. R. KAISER, Dr K. BULLOUGH and Dr A. J. SMITH for comments on the draft manuscript. This work was partially funded by the Science Research Council.
REFERENCES ANGERAMIJ. J. BERNHARDTP. A. and PARK C. G. CERISIERJ. C. DENLIY M.. BULL~IJGH K., ALEXANDER P. D. and RYCRIXT M. J. JAM= H. G. MATTHEWSJ. P.. SMITH A. J. and SMITH I. D. SMITH R. L., HELLIWELL R. A. and YABROFFI. W. STRANGEWAYSH. J. STRANGEWAYSH. J. and RYCROFT M. J. THOMWN R. J. and JIOWDEN R. L. WALKER A. D. M.
1970 1977 1974
J. geophys. Res. 75,6115. J. geophys. Res. 82, 5222. J. atmos. terr. Phys. 36, 1443.
1980 1972 1979 1960 1981 1980 1977 1971
J. atmos. terr. Phvs. 42. 51. Annls Geophys. 28, 3Oi. Planet. Soace Sci. 21. 1391. J. geophis. Res. 65, il5. J. atmos. terr. Phys. 43, 231. J. atmos. terr. Phys. 42,983. J. atmos. terr. Phys. 39, 879. Proc. R. Sot. A321,69.
1971 1975 1978
Ph.D. thesis, University of Southampton. ESRO special pub]. No. 107, p. 225. Ph.D. thesis, University of Southampton.
Reference is also made to the following unpublished material ALEXANDERP. D. RYCROFT M. J., JARVIS M. J. and STRANGEWAYSH. J. STRANGEWAYSH. J.
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