Time of flight calculations for high latitude geomagnetic pulsations

phnr spcla ScL. Vd. 27. pp. 679-669. --Ltd.

1979.printrdinNorthn~

TIME OF FLIGHT CALCULATIONS FOR HIGH LATITUDE GEOMAGNETIC PULSATIONS M.RWARNER

andD.ORR

Department of Physics, University of York, Heslington, York YO15DD, U.K. (Receiued 16 June 1978)

Abduct-High latitude geomagnetic field lines dier significantly from a dipole geometry. Time of tlight calculations using the Mead-Fairfield (1975) model of the geomagnetic field are presented for different tilt angles and K, conditions. Typical standing wave periods of geomagnetic pulsations are estimated for three different magnetospheric cold plasma regions, corresponding to waves guided in (i) the plasmatrough, (ii) the extended plasmasphere and (iii) regions of enhanced proton density (detached plasma) within the plasmatrough. Pc4/5 pulsation studies at high latitudes are briefly reviewed and some new results from Tromso are given. Many of the observations reveal hydromagnetic waves whose location and period are consistent with ducting in a region of enhanced plasma density within the plasmatrough. INTRODUCIlON

In this paper we are concerned with pulsations observed at latitudes greater than 60”, with periods between 50 and 1000s. Pc5 pulsations form an important subset of these and it is to them that most attention will be given. We are primarily interested in the way in which the period of these pulsations varies with latitude. This variation has been studied experimentally many times, and some of these results are summarized in Fig. 1. The various methods which were used to obtain these results are given in Table 1. The studies shown here all indicate that the period increases approximately exponentially with latitude. However, the details depend quite strongly upon the method used to produce them, the year in which the observations were made and the general conditions used to select data for inclusion in the study. The main purpose of this paper is to give a quantitative explanation for this latitude dependence of period. At the same time we hope to identify the region(s) of the magnetosphere which is (are) most closely associated with these pulsations. We will also briefly investigate the dependence of period upon other less important parameters-local time, K, index and dipole tilt angle. THE TIME OF

FLIGHT

local standing waves can be approximated by the symmetric toroidal wave mode (Radoslci and Carovillano, 1966) or by the guided poloidal wave mode (Radoski, 1967). This approximation has been used successfully by Orr and Matthew (1971) to give a quantitative explanation of the latitude dependence of Pc3/4 pulsation period at moderate latitudes (C 60”). If one considers a magnetic field that is more simple than the dipole field, then wave modes that are more sophisticated than those mentioned above may be developed. Conversely, if one considers a more complex magnetic field then only very simple wave modes can be realistically considered. Now at high latitudes the dipole approximation to the Earth’s magnetic field becomes progressively worse as the field is distorted by the action of the solar wind. Thus at the high latitudes we are interested in in this paper one may use a poor approximation to the Earth’s magnetic field (i.e. the dipole) and a sophisticated wave mode, or a more accurate and hence complicated field model and only a simple wave mode. We have chosen the latter course and represent the wave mode as an Alfven wave standing upon a single field line. The period of such an oscillation is then given by the time of flight approximation, i.e. Period = 2 c $,

APPROXIMA’ITON

of the authors listed in Table 1 have indicated that the dependence of period upon latitude may be explained by considering the pulsations as the manifestation of some kind of hydromagnetic standing oscillation of the local magnetic field lines. This explanation is the one explored in this paper. For a pure dipole field the period of

J VA

Many

679

where tion is The amined dipole tained

V.., is the Alfven velocity, and the integraperformed along the field line. validity of this approximation may be exby considering time of flight results for a field and comparing them with results obusing more realistic wave modes. The most

680

M. R. Wm

and D. Oaa

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B. l* a

+

b

4

c

n

d

4

l-

I I

a

t

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1

f

60

12

69 GEOMAGNETIC

13

LATITUDE

FIG. la. EWERIMENTWY DETERMINED VAlU4TION OF

MICROPULSATION

PERIOD

I,

18

WITH

GEOh4AGNJZllC

LATITUDE.

(a-d) 01’ (1963), (e) Hirasawa (1970), (f) Samson et al. (1971). See Table 1 for details.

I

I

61

64 GEOMAGNETIC

I

67

I

1

70

13

-I 70

LATITUDE

FIG. lb. As ABOVE. (a) Oguti (1963), (b) Voelker (1968), (c and e) .?$nn and Rostoker (1972), (d) Obertz and Raspopov

simple guided mode is the symmetric toroidal mode and Radoski (1966) has shown that the time of flight approximation corresponds exactly with the results obtained from this mode if the plasma density varies as rm6,i.e. the Alfv6n velocity is radially independent. In the magnetosphere the density will generally vary less rapidly than this and for an r4

variation the time of flight approximation leads to results that are about 16% less than those given by the symmetric toroidal mode and for an rC3 variation this increases to about 21%. From these figures it appears that the approximation is not a particularly close one, however it must be remembered that the symmetric toroidal mode is itself

calculations on lligb latltlKIe

681

geanagnetic pulsations

TAEZ~1 Year of Reference

(d) (4

i

01’, 1%3 OI’, 1963 Ol’, 1963 Ol’, 1963 Hirasawa, 1970 Samson er al., 1971

1957-1958 1957-1958 1951-1952 1951-1952 1958 1969

Afternoon, large amplitude. Pg Morning, large amplitude. Pg Aftemecn, large amplitude. Pg Morning, large amplitude. Pg More than 3 waves, >lOy. Pc5 0000-0700 L.G.T., 3 days data. Pc4/5

111

oguti, 1963 Voelker, 1968 Samson and Rostoker, 1972 Obertz and Raspopov, 1968 Samson and Rostoker, 1972

I.G.Y. 1966 1969-1970

Data from I.G.Y., sunspot max. Pc5 Single event, 7, January afternoon. 0730-1730 L.G.T., summer, P&5

i ii iv

1958

Morning, Jan-April, Pc5

iv

1969-1970

1730-0730 L.G.T., summer, Pc4/5

iv

Fig. lb. ;:; (4

Method used

Selection criteria, etc.

ObSt%WtiOll

i

i i ..i.

Methods used

(i) (ii) (iii) (iv)

Period determined from average at individual stations. Observation of single latitude dependent event. Latitude determined from position of polarization reversal. Latitude determined from position of amplitude maximum.

only an approximate description of the system. The two major factors that this mode ignores are: the effect of azimuthal asymmetry, i.e. localization in longitude and the effect of coupling between the guided and isotropic modes. Radoski (1972) has studied the effects of asymmetry, in the absence of coupling, for the toroidal mode. He shows that the period decreases with increasing asymmetry. These results indicate that the decrease in period for a mildly asymmetric wave mode is of the same order as the ditference between the symmetric mode and the time of tlight approximation. This means that the time of flight approximation can give quite a good description of the period of toroidal oscillations with a small, but significant, degree of azimuthal asymmetry; indeed this description is at least as good as that given by the symmetric toroidal mode. The effect of coupling the isotropic and guided modes has been considered by Chen and Hasegawa (1974a,b) and Southwood (1974) These studies show that coupling effects are important for the polarization characteristics of the system, but do not greatly atlect the resonant period. We therefore believe that the time of flight approximation is both useful and reasonably accurate, but in doing so we make the following assumptions about the pulsations we are considering: the pulsations represent the first harmonic of localized,

standing, MHD oscillations and there is a small degree of azimuthal asymmetry. We also assume that the approximation remains good even when the guided and isotropic modes are coupled. In order to use the time of 8ight approximation we must be able to define the Alfven velocity (VA) accurately at all points along the field line. This means that we must use accurate models of both the magnetic field and the plasma density. These models are described in the following sections. TEE PLASMA DFNSI’IY

MODEL

The density of cold plasma in the magnetosphere was modelled using the results from the OGO 5 mass spectrometer as presented in published papers-see Chappell (1972) for a review. Note that these results are for 1968 and early 1969. The distribution of plasma along individual field lines at distances greater than 2& can be simply represented as aR_“. A value of n = 3 corresponds to diffusive equilibrium and II = 4 to a collisionless distribution. Both of these are typical of the plasma distribution within the magnetosphere. Or-r and Matthew (1971) have shown that the period depends only weakly upon the value of n chosen. The equatorial distribution across field lines was modelled for four different regions of the magnetosphere: the plasmatrough, the dusk plasmasphere, the extended quiet time plasmasphere and

M. R. Wm

682

andD.0~~

TAEKE2 Typical density at L=4

Equatorial density variation

Extended plasmasphere Dusk plasmasphere Plasmatrough 0800 L.T.

450 ions cm-3

_ R-3

188 ions cm+

R+

4.4 ions cm-3

R-4

1400 L.T.

30 ions cm+

R-3

400 ions cm+ 100 ions cm-3

R-4 R-4

Detached plasma Upper lit Lower lit

detached plasma regions within the plasmatrough. The normal daytime and night-time plasmasphere was not included as it does not extend significantly into the latitude range of interest in this paper. Each of these regions has its own identifying characteristics; a summary of these is given in Table 2. The plasmatrough This is a low density region and it covers the entire latitude range that we are interested in. It is characterized by an increase in density during the day due to filling from the ionosphere. The degree of filling is greatest for quiet magnetic conditions. The density generally returns to a low level around late afternoon. The dusk plasmasphere This is a medium density region which generally displays a smooth R-“ distribution. Around dusk the plasmasphere can extend to fairly high latitudes for times of low magnetic activity, as far as an L-value of 8 or more for EC,< 1‘. The extended plasmasphere During long periods of abnormally low magnetic activity the dayside plasmasphere may extend to very high latitudes. The density of this extended plasmasphere is somewhat higher than that of the normal dayside plasmasphere. The dayside of the magnetosphere does not react quickly to changes in magnetic activity and so the extended plasmasphere may survive for several hours after an increase in magnetic activity. Detached plasma regions These have been detected at all latitudes and local times within the dayside plasmatrough (Chap-

Effect of changing magnetic activity Only occurs at times of very low magnetic activity. Extends to higher L-values for lower magnetic activity. Increase in magnetic activity reduces effect of dayside 6Uing. (Note: density values are for low I$.) Most often seen for moderately disturbed conditions.

pell, 1974). They are regions of limited extent, both radially and azimuthally, and have densities which are above those normally associated with the plasmatrough. They are detected most often at times of moderate magnetic activity, K, = 3, 4, 5. The density of detached plasma regions can only be modelled within fairly wide limits and falls between that of the plasmatrough and plasmasphere. This is shown schematically in Fig. 2. The characteristics of each of these regions together with their respective time of 5ght results should be useful in identifying which of them is involved in producing long period pulsations. THE MAGNEITC

FIELD MODEZL

The magnetic field model serves two purposes, firstly the field lines define the integration path, and secondly the magnetic intensity determines the Alfven velocity. The model we chose to describe the magnetic field was that presented by Mead and Fairfleld (1975). This is an empirical model obtained by a least squares fit to a large set of satellite measurements of the field. It describes the. geomagnetic field accurately to a distance of 17&. Four sets of coe5cients are available corresponding to four I& ranges. The model also incorporates the tilt angle of the dipole axis, this is defined as the angle between the Earth-Sun line and the magnetic equator; it is positive for northern hemisphere summer. The model contains built in dawn-dusk and North-South symmetries. Thus, for example, an 0800 h field line will be identical to a 1600 h field line, and a field line at co-latitude 30” for a tilt angle of +15” will be identical to a field line at co-latitude 150” for a tilt angle of -15”. Because of this symmetry we need consider only one hemisphere.

Calculationson high latitudegeomagneticpulsations

I

683

,

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1

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I

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2

3

4

5

6

1

8

9

10

L-VALUE FIG. Curves

2. A

SCHEMATIC

DIAGRAM

SHOWING

DETACHED

PLASMA

REGIONS

WlTHlN

THE PLASMATROUGH.

(a) and (b) show the upper and lower limits for detached plasma, curve (c) shows the maximum density in the plasmatroughdue to dayside filling. (Chappell, 1974).

The details and limitations of this model are described in the paper above and also in Fairfield and Mead (1975). RESUJXS Time of flight periods were computed for the models described in the previous sections. Both the magnetic field model and the plasma density model contain a number of parameters that can be varied for a given latitude. The effect of each of these parameters upon the calculated period is described below. In general, those which alter the magnetic field will tend to have a larger effect at higher latitudes, whereas those which affect the plasma distribution will produce similar changes at all latitudes. (a) Magnetic field

model used

Figure 3 shows the variation of period with geomagnetic latitude for the dipole field model and for the Mead and Fairfield model for various conditions. For all four profiles shown, the plasma distribution assumed was the same so that they indicate only the effect of changing the magnetic field. This figure is presented primarily to show the inadequacies of the dipole field as a basis for calculating pulsation periods at high latitudes. For example, at a latitude of 70” the dipole model gives a period of 300 s for this particular plasma distribution, whilst the Mead and Fairfield model gives results which vary from 220 s to periods in excess of 1000 s. Thus it is clear that, at high latitudes,

account must be taken of the non-dipolar nature of the field lines. (b) Variation with latitude Figures 3 and 4 both show the variation of period with latitude. Clearly the precise nature of this variation depends upon the various conditions imposed, however it can be seen that all the profiles show a smooth increase of period with latitude. We will see below that the rate of increase is greatest for local times around midnight and for high EC, values. A more detailed discussion of the dependence of period upon latitude is given below when experimental results are considered. (c) Variation between different plasma regions

In Fig. 4, time of flight profiles are shown for three dzerent regions in the magnetosphere-the plasmatrough, the extended plasmasphere and detached plasma. The upper axis indicates the equatorial crossing point of the field lines; they are considered to be “open” if this is greater than 17&. The relative positions of the profiles simply reflect their relative densities since the time of flight period is proportional to the square root of the equatorial plasma density. The lowest curve is thus for the plasmatrough, and the upper curve is for the extended plasmasphere. The two intermediate curves give the upper and lower limits for detached plasma. The variation of period with local time is also diEerent for the diRerent regions as indicated below.

684 11

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PC5 TIME=

120(

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PC4

z P

PC3

I

I

I

I

I

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I

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61

64

81

70

73

18

GEOMAGNETIC

hC+.3. m

VARIATION

OF MICROPUUATION

PERIOD

FHGHT

LATITUDE

WIW

LATITUDE

CAlBJLAlFDm(oMTHEllMEOF

APPROXIMATTON.

Results are shown for the Mead and Fairfield model for three sets of parameters and also for the dipole field.The plasma distriiution assumedwas identicalfor all four pro&s.

EQUATORIAL 4.00

464

CROSSING

531

641

1.61

POINT ¶.17

(R,) 11.53 Open

field lines

PC 5

PC 3

58

81

64

Sl

GEOMAGNETIC

h0.4.

LATlTUDE--PERIOD

PROFILes

MAspHEREANDDmACfiEDpLAshu

FOR

THE

DAYSIDR

RRG~ONs

70

73

76

19

LATITUDE

PLASMATROUGH, FOR THE ~&AD

Time=O8OO,K,b2,tilt=o”.

RXlXNDRDO~TlMRPLAS AND

FMRFIELDMODEL.

685

Csleulations on high latitude geomagnetic pulsations

gree of magnetic activity. If conditions have been relatively quiet the dawn plasmatrough will generally contain more plasma and so the period wiIl start at a higher level, but still reach approximately the same final level. If conditions are relatively disturbed during the day then the effects of dayside tilling will be reduced and the linal afternoon period will be lower. The results shown here correspond to the maximum rate of idling, i.e. quiet conditions after relatively disturbed conditions. In the late afternoon the plasmatrough generally loses plasma to the magnetopause and so the period will return to its original low level as indicated by the dotted section of the graph. The exact speed and time of plasmatrough emptying appears to be highly variable. The dusk bulge plasmasphere is, of course, localized in local time and so it can only support pulsations for a limited time. The period in this region simply follows that of the extended plasmasphere, though at a lower level. The variation of plasma density with local time within detached plasma has not been studied in detail, however the results shown here assume that dayside filling operates at the same rate as for the plasmatrough but since the background density is much higher for detached plasma, the relative effect wiIl be reduced. This leads to the variation shown in Fig. 5 which has a minimum around

(d) Variation with local time This variation will be controlled by two features-the changing shape of the field lines and the changing plasma distribution. The combined effect of both of these is shown in Fig. 5 for a latitude of 68”. This figure is not intended to show the variation with local time on one particular day since both the K,, value and tilt angle are fixed and on any one day the magnetic activity will generally not remain at a constant level and the tilt angle of the dipole axis will certainly change with local time as it rotates about the spin axis of the Earth. Thus for a particular station on a particular day a figure such as this must be modified to take into account the variation in K, and tilt for that situation. Returning to Fig. 5 we see that the local time variation is quite different for the diiTerent regions of the magnetosphere. In the extended plasmasphere the density does not vary appreciably with local time, so that the variation is dominated by the changing magnetic field. This produces a U-shaped variation with a minimum around midday. The degree of variation will, of course, be greater at higher latitudes. In the plasmatrough the dominant effect is due to dayside lilling from the ionosphere. This will cause the period to increase from a minimum around dawn to a maximum value in the afternoon. The level of plasmatrough filling is affected by the de-

6.14

EQUATOR 7.61

IAL

CROSSING

POINT

(R,)

7.19

1.61

6.74

9

1200

1600

2200

2400

I2

600-

PC

5

PC 3

20

A

0400

6000

0600 LOCAL

FIG.

5. ?-HE! VARIATION OF TROUGH,

(B) DUSK

PERIOD

WlTH

SPHERE,

GEOMAGNETIC LOCAL.

TIME

(C) EXTENDED Latitude

=

FOR

TIME VARIOUS

PLAsMAspHERE,

68”, K, 3 2, tilt = 0”.

PLASMA

lWXONS-(A)

(D) DETACHED

PLASMA.

PLASMA-

686

M.RWARNBR~~~D.ORR

midday and a tendency for the periods in the afternoon to be longer than the corresponding morning period. (e) k, value The level of magnetic activity can a&ct the results in three ways. Firstly it controls the position and density of the various plasma regions, for example the extended plasmasphere only exists for very quiet conditions, and the rate of dayside tilling is reduced for disturbed conditions. Secondly the level of magnetic activity will affect the demarcation latitude between closed and open field lines. As the magnetic activity increases more field lines will be swept back into the tail and so be unable to support the type of oscillations we are considering. For example, at 1200 h and a tilt of O’, the 80” field line is closed for &, C l- but is open together with the 77” field line for & 3 3”. Increasing magnetic activity therefore reduces the latitude range for stable oscillations. Changing magnetic activity also afIect.s the calculated periods by changing the shape of the field lines. Increasing I&, tends to increase the period, the effect itself increasing at higher latitudes and at local times closer to midnight. At a latitude of 70” the effect around midday is generally of the order of 20% for a change in Ic, of O-3, and around midnight can be as large as a factor of two for a similar change in K,,.

(f) Tilt angle The effect of changing the tilt angle upon the period is quite complex and depends upon all the other parameters involved, however some broad generalizations can be made. The main effect is to increase the local time variation for increasingly positive tilt angles, and to decrease this variation for negative tilt angles. This happens in such a way that the period around dawn and dusk remains approximately constant while the midnight and midday periods move further apart or closer together. Changing the tilt angle can also be responsible for altering the positions of conjugate points by as much as *4” of latitude and *18” of longitude at high latitudes and certain local times. Tilt angle also affects the closure and opening of field lines. COMPARISON

Wnll EX.PERlMENTAL RESULTS

The most striking feature of both the time of fhght and experimental results is the increase in period with latitude. There are various ways of

producing latitude-period pro&s experimentally as indicated in Table 1. The most meaningful and consistent of these is to determine the latitude at which a single event shows a maximum amplitude. This was the method used by Samson and Rostoker (1972) and their results represent, perhaps, the most detailed and reliable study of this kind. Kato and Saito (1964) have shown that the relative sunspot number is an important parameter in determining Pc4/5 period and it is worth noting that the results of Samson and Rostoker discussed in this paper were obtained using data from 1969 and 1970 and that the plasma density measurements used were taken in 1968. All three years had very similar sunspot numbers and so a comparison is quite straight forward. Figure 6 shows Samson and Rostoker’s results together with the appropriate time of flight results. The experimental protile shown represents a linear regression of a large number of events and the dotted lines give 95% confidence limits. The experimental data is for the northern hemisphere summer with K,, <4- at 0730-1730 L.T. The time of flight data is for a tilt of +30”, &, 32 and 1400 L.T. It is clear from this figure that the rate of change of period with latitude is approximately the same for theory and experiment. It can also be seen that the results fall below those of the plasmasphere, above those of the plasmatrough and between the limits for detached plasma. Although there are inaccuracies in both the plasma density model assumed and in the time of flight approximation itself, it is very unlikely that these will be suiIiciently large to shift these results into either the plasmatrough or the plasmasphere. The time of flight results presented here thus indicate that high latitude Pc4 and PCS pulsations represent standing waves within regions of detached plasma in the plasmatrough. This possibility has been mentioned before (Chappell, 1974; Rostoker and Samson, 1972; Obertz and Raspopov, 1968) and these results offer quantitative evidence for it. If long period pulsations are associated with detached plasma regions then we should find that they occur preferentially well outside the plasmapause. An analysis of the frequency of occurrence of pulsation events with latitude, K, and local time will indicate whether this is so. Figures 7 and 8 show the results of such an analysis. Figure 7 is for so-called giant pulsations and shows the results of 01’ (1963) superimposed upon contours of latitude and local time obtained from the Mead and

687

Calculation6 on high latitude geomagnetic pulsations 101

50

PC 5

l! ;; :: =

PC4

8 pr <

k

PC 3

1

I

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61

64

67

70

73

76

GEOMAGNETIC

FIG. 6. A

COMPARLSON

OF THE OF Fl.IGHT

-AL FROFlLES

LATITUDE

RRSULX3 FOR SIMILAR

OF !hMSGN

AND

ROSTOKER (1972) wlTHTIME

CONDITIONS

(SE

TEXT).

(a) extended plasmasphere. (b) detached plasma-upper limit, (c) detached plasma-lower limit, (d) plasmatrough. The barred region shows the experimentally observed continuum of periods seen above 75”. The experimental results fall within the limits for detached plasma. Fairfield model. The position of the plasmapause as given by Chappell et al. (1971) is also shown. Figure 8 is for Pc5 pulsations observed at Tromso (latitude 66.7”) and indicates the frequency of occurrence as a function of local time and &. The NIGHT

position of the plasmapause overhead line is also shown. Both of these figures show that long period pulsations do indeed fall well outside the plasmapause, within the plasmatrough, i.e. in the region where we find detached plasma.

SIDE

PLASMASPHERE

> 35

EVENTS

--Paint of

> 150 EVENTS

,

RE ,

Equatorial crossing field line Ieavinq earth

liar DAY SIDE Point of field line leaving given local time.

at

earth

OF GIANT PUISATIONS (pti) RECORDED AT FIVE tiW hG. 7. THE FltEQIJENCY OF MZURRENCR JNTOTHE EQUATORIAL OBSERVATORIES DURING 1951-1952 AND 1957-1958 (OL’, 1963) MAPFWD MODEL OF THE GECMAGNEI’K! FD3.D. PLANE USMO THE MEAD-F9

688

M. R. WARNERand D. Ouu

I/

0

I

3

2 AVERAGE

Fro.

8.

Quwcy

CONTOUR OF

TRohfso

IN

MAP

SHOWING

OOXRRENCE

OF

4

NIGHT

PC5

THE

3

TIME

Kp

NORMALED

EVENTS

1969-70 IN TERMSOF

TIMB

AVERAGE

INDEX.

NIGHT-TIME

K,

FREL

RECORDED OF

DAY

AT AND

to 50 and 25% of the maximum normalized frequency of occurrence. The estimated plasmapause overhead position for Trotuse is also piven. (For an explanation of this technique see Orr and Webb, 1975.) Two

contours

are drawn,

that there may be a link between long period pulsations and detached plasma. We can note that they share a number of common characteristicsthey are both almost exclusively dayside phenomena, they both occur for moderately disturbed conditions (Chappell, 1974; Hirasawa, 1970) and both are localized in latitude and longitude. The time of Sight results also indicated that the period should vary with local time and that the variation will be dserent for the various regions of the magnetosphere. There have been a number of experimental studies of this and they have produced two main conclusions. Firstly that the period in the morning is shorter than that in the afternoon (Hirasawa, 1970; Saito, 1964; Ol’, 1963), and secondly that the period is a minimum around noon (Kitamura, 1963; Samson and Rostoker, 1972). Both these facts are explained by assuming that these pulsations are associated with detached plasma--see Fig. 5. The increase of period from morning to afternoon is generally of the order of 30% and this is similar to that obtained from the time of Sight calculations. The variation of period with I& and tilt angle has received little experimental attention, but Obertx and Raspopov (1968) have indicated that there is a tendency for the period to increase with increasing K,, a result which was also obtained from the time of Sight calculations.

corresponding

Chappell (1974) shows that detached plasma occurs at all local times on the dayside of the magnetosphere, while Fig. 7 indicates that long period pulsations are primarily seen in the dawn to midday sector. This apparent discrepancy can be explained by assuming that the local time distribution of long period pulsations will reflect both the characteristics of detached plasma and also the characteristics of the excitation source. We are thus assuming that these pulsations are initially excited by a source which tends to be localixed in the pre-midday sector. Such a source would be, for example, a Kelvin-Hehnholtx instability on the magnetopause which will tend to produce guided waves before midday but isotropic waves after midday. In addition to the evidence presented above there are a number of other reasons for thinking

SWY

ANDcoNcLusxoN

In this paper we have used the time of tight approximation, together with empirical models of the plasma density and magnetic field to investigate micropulsations at high latitudes. From this investigation we present the following conclusions: (i) At high latitudes, the non-dipolar nature of the magnetic field is important in determining pulsation period. Changes in the magnetic field due to local time and & index can vary the period by a factor as large as four at a latitude of 70”. (ii) The predicted period varies as the square root of the equatorial plasma density. The period will thus be different in different plasma regions. (iii) The period is largely independent of the detailed distribution of plasma along the field lines. (iv) The time of flight results indicate that, within any single magnetospheric plasma region, the period will increase smoothly with latitude. The rate of increase depends upon the conditions imposed and is greatest around midnight and for high

K,.

Calculations on bigb latitude geomagnetic pulsations

(v) The period varies with local time and the way in which it does so depends upon the plasma region which is supporting the pulsation. In the plasmasphere and in detached plasma regions, the variation is U-shaped with a minimum around noon. In the plasmatrough the period increases from dawn to dusk. (vi) The period tends to increase with increasing magnetic activity, and the rate of increase of period with latitude also increases; however these effects may in practice be masked by changes in plasma density brought about by the changing magnetic act%@ which has not been modelled. (vii) The angle between the Earth-Sun line and the magnetic equator also effects the period. This variation is smallest around dawn and dusk. A comparison of these conclusions with experimentally determined periods indicates that: (i) The rate of increase of period with latitude as determined from these time of flight results shows good agreement with the best experimental data currently available over the latitude range of interest in this paper (i.e. -6O”-78”). The time of tight results presented here thus represent a quantitative explanation for this increase of period with latitude. (ii) The comparison of time of Hight and expcrimental results indicates that the plasma density required is the same as that observed in regions of enhanced plasma density (i.e. detached plasma) within the plasmatrough. An analysis of the occurrence of long period pulsations with latitude, local time and K, index indicates that this is broadly similar to that observed for detached plasma. (iii) The variation of period with local time and rC, predicted by the time of flight results agrees with the limited experimental data available and is consistent with the belief that these pulsations occur within regions of detached plasma. We are currently involved in further experimental and theoretical work to test the link between detached plasma and long period pulsations.

689

Acknowledgements-We want to thank the observatory staff at Tromso for their geomagnetic pulsation data. This work was supportezl by the Science Research Council under grant SG/D/OOl78.

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