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Electric fields and currents in the earth's polar caps
0032-0633/85 $3.08 f0.00 Pergamon press Ltd.
Pkmd. Space Sd., Vol. 33, No. 4, pp. 407-414,1985 Printed in Great Britain.
ELECTRIC FIELDS AND CURRENTS IN THE EARTH’S POLAR CAPS M.I.PUDOVRIN,
S. A. ZAlTSEVA, T. A. BAZHJWOVA and V. G. ANDREZRN Institute of Physics, Leningrad State University, Leningrad, 198904, U.S.S.R.
Abstract-A model for solar wind flow around the magnetopause incorporating a stagnation line at the frontside magnetopause is used to derive a formula for the electric field intensity and polar cap potential drop. These relationships are compared to experimental data from polar orbiting satellites. The relation between solar wind parameters and aurora1 arc velocity is also studied.
hydrod~~c aspects of the problem on the solar wind flow around the magnetosphere. An attempt to take into account the hydrodynamic aspects of the problem was made by Pudovkin (1982), Pudovkin et al. (1981, 1982a,b), and Pudovkin and Zaitseva(l983) on the base of the model of the solar wind flow around the magnetosphere with a stagnation line (Pudovkin and Semenov, 1977 ; Semenov and Pudovkin, 1978). In that model the solar wind parameters within the magnetosheath as well as the length of the stagnation line are calculated in a self-consistent manner. According to the model, the length of the stagnation (or reconnection) line approximately equals
I. ~ODU~ON
It is usua.lly supposed that the electric fields in the magnetosphere and in the polar ionosphere appear as a result of the solar wind and the Earth’s magnetic field line reconnection. The models based on that assumption (Stern, 1973; Gonzalez and Mozer, 1974; Yeh, 1976; Kan and Lee, 1979) describe the E-field relative variations in the polar ionosphere silently well. However, the absolute values of the calculated potential drop across the polar caps prove to be 3-6 times more than the experimentally observed ones. Besides, a more detailed analysis has shown that the
-in the cold plasma approximation Zaitseva, 1983
(Pudovkin
and (1)
-in the hot plasma approximation 1982a,b) coefficient of proportion~ity between the values of AcbeXp and A&eor is not constant and depends on the conditions in the solar wind (Gonzalez and Mozer, 1974). In particular, for all the models referred to above the polar cap potential drop exhibits some kind of saturation, that is the values of A&, are proportional < to A#theor for small values of the latter (A#I~~*,,~ A&, N 100 kv), and do not depend on them for A.dftheor > Ac$,, (Reiff et al., 1981; Wygant et al., 1983; Doyle and Burke, 1983). In order to reconcile the differences between the theoretical predictions and the experimental results, the authors limit the values of A#,,,,, at the magnetopause by including some additional assumptions which do not result from the models. The disagreement between the theoretical models and experimental data may be caused, as was pointed out by Gonzalez and Mozer (1974), by the neglect ofthe
(Pudovkin
et al.,
where I),,, is the diameter of the magnetosphere, Mu,, is the Alfvenic Mach number, 0:is parameter of the nonpotentiality (see Fig. l), /&, = B,(4np,,k$)‘I2 and B = dm; the symbols “SW”,and “0” and “m” mark the values of the magnetic field and of the plasma parameters within the undisturbed solar wind, at the bow shock, and at the magnetopa~e in the vicinity of the subsolar point respectiveiy. As was shown by Gonzalez and Mozer (1974), the electric field intensity at the magnetopause is described by the formulae : I$, =&~&3~
s~(e~-~)sin~ (2)
E, = i v,,,&,, sin (6, - 9) cos rp
/
where E,, E, are the electric field components along the corresponding axes of the GSM co-ordinate system;
407
i%.
I.
&JDOVKiN
et
ai.
distance between magnetic field lines in the ionosphere decreases with respect to that at the magnetopause in proportion to (~~~~~)1’2, where 3, is the normal component of the magnetic field at the magnetopause, and Bj is the magnetic field intensity at the polar cap; Bllvfx 0,1(47~p,,,,V&,)*‘~ (Pudovkin et al., 1981; Semenov and Pudovkin, 1978). Then for the field components iu the ionosphere we have : 13B = A pBSW V1i2K’i4 SW SW sin(~~~-~)sin
E, = A$swV&;1~4 O,, is the angle between the Z-axis and the B,, vector, and cp is the angle between the Z-axis and the reconnection line; dependence of the angle cp on the magnetosheath’s magnetic field orientation is assumed in the form proposed by Yeh (1976). As is seen from (2), for all the IMF orientations, the E,-component is directed from dawn to dusk, and is hence responsible for the DP-2 field. The &-component being projected onto the ionosphere is directed along the noon-midnight meridian and corresponds to the DP-4 field (or for the Svaalgard-Mansurov effect) ; the sign of the E, field depends on the sign of the Y-component of the magnetosheath’s magnetic field. Taking into account formulae (I) and (2), one obtains the fo~o~ng expressions for the polar cap dawndusk potential drop :
-in
x sin (8,, - cp)sin rplsin 91 the cold plasma approximation
I t cb)
-in
x sin (t?,, - q) sin it /sin 91 the hot plasma app~o~mat~on
(3)
i I
One can see from (3) that if the model under consideration is correct, the polar cap potential drop is proportional to the squared value of the IMF intensity and does not depend on the solar wind velocity. 2 THE POLAR CAP EK,ECTRIC FIEm
1NTENSl-N
In order to compare quan~ta~vely the ~tensity of polar cap geomagnetic disturbances with the magnetopause electric field this field has to be projected onto the ionosphere. For that, one has to take into account that the
cp
(a)
sin (@,,-- cp)cos rp
(b)
(4)
where A,, A, are constant coefficients. The product sin (BsW-(p)sin 9 = fe(@,) is approximately equal to sin28,,/2 in the cold plasma model (Pudov~n and Zaitseva, 1983); and to sin48,,/2 in the hot plasma model (Pudovkin et al., 1982a). Thus the value of the angular function in (4) may be written as 0 f,(B,,) w sin” z, 2 where 2 < k 6 4. Let us see whether formulae (4) agree with the experiment. For this purpose we shall examine va~a~ons of the ~~espond~g components of the geomagnetic field at the polar cap magnetic observatories, In this study we shah assume the Hall currents to be responsible for the observed variations of the geomagnetic field; besides, we shah take into account that these variations are caused mainly by the vacations of the electric field intensity while the conductivity of the polar ionosphere is changing insignificantly at the day hours of the summer months (Vanyan and Osipova, 1975). In Fig. 2 hourly-mean values of the dawn-dusk component of the magnetic disturbances (SY’) at the Alert station at the day time (10.00-l 5.00 M.L.T.) for the period June-July 1968 are shown in dependence on the function F, = B,,V$ sin2 @,,/2 which is proportional to the E,%eld in the ionosphere (we neglect a weak dependence of E, on n). As the level of reference, the values of the geomagnetic field components on a quiet winter day were taken. The crosses in the figure correspond to the cases when the solar wind magnetic field is significantly northward (l&,1 < 50”). One can see that most of the points are scattered near some straight line, which confirms the propo~ion~ity between expe~ment~ and theoretical values of iW. It was found that the dependence of SY‘ on the functions Fl and F, = B,,I&,, sin2 &j2 [see (2) and (4)] is approximateIy the same; the correlation coefficient equals r = 0.77 in both cases (the points with If&,1< 50” being excluded). Therefore, in order to find the optima fun~on describing the deveiopment of the magnetic disturbances, additional study is needed.
Electric fields and currents in the Earth’s polar caps
409
.
FIG. 3. THE SAME As IN FIG. 2, BUT IN Dm?NDENcE B#MF); (b) /B,j(IMF), WHENIS,1 < 0.57. FIG. 2. THE
DEPENDENCE
OF GEOMAGNEXC
DISTURBANCE
VECTOR (ALFXT OBS.,30.00-15.00 M.L.T.;JuN&JuLY 1968) PROJECTBDONTOTHBPERPIINDICXJLAR TOTHBEARTH-SIJNLINE 6Y' ON THE FUNCTION F, = B,,Vj,f sin’ i&/2.
The cases when l6’,,,,l < 50” are pointed by crosses. In this co~eci~on let us see whether the inclination of the regression lines for the two functions under consideration is constant or if it changes in its dependence on the solar wind velocity. The regression coeflicients for functions I;, and F, for two intervals of V,, (V,, < 450 km s-* and V,, > 450 km s-l) are sho~inTable 1. Onecansee from theTable, that ofthe two functions, only F, is linearly related to 6 Y’. For both functions the factorf,(8,,J was taken above in the form sin2@,,,,/2 which is correct for the cold plasma only, and in a general casef, = sink 6,,/2. The value of the exponent k may be found from the experiment. For this purpose, we expand E, in the vicinity of the point (IS,, = 0; e,, = E/Z ; y,, = V,& into Taylor series ; asbefore, we neglect the dependence of Ei, on II.Then having restricted ourselves by the first terms of the series we have :
The equality (6) shows that the dawn-dusk polar cap electric field is determined by the variations of B, and lB,IIMF (Stern, 19’73; Gonzalez and Mozer, 1974; Burton et QZ., 1975; Mishin, 1978). The relative “sensitivity” of the E,-field to the variations of B, and lByj IMF is :
and according to the mdhel, 2 < k < 4, i.e. 1 < (&i&J < 2. Let us check the expression (6) using the data on the geomagnetic field variations. In Fig. 3 the intensity of the magnetic disturbances 6 Y’is plotted vs the values of B, (Fig. 3a) and lB,( IMF (for IB,i < OSy, (Fig. 3b). One can see that 6Y’ really depends on &I, which seems to confirm the model. Correlation coefficients between SY’ and B,, jB,i and corresponding regression coefficients (the latter in arbitrary units) are: r, = -0.7, a, = 20.6; rv = 0.6, ay = 14.1. Hence (E&J = @,/a,) E 1.5, then k sz 3 and fDRSsin3 0,,,,/2. Inserting this value of fa into (3) we have A+, z B&,ps-wli=sin4 @,,/2, the last expression being very close to the Akasofu-Perreault index E. In Fig. 4, the hourly-man values of the magnetic
TABLE 1. THE COEPFICIENTS CHARACTBRIZED INCLINATION
THE REGRESSION LINE
Thefunction Theinterval of K, $450kms-f >450kms-’
ON:(a)
Icl= B,,V$’ 1.40 1.34
sin’ 6,,/;!
F, = B,,f&
sin’ OS,,,/2
0.69 0.34
410
M. L PUDOVKIN et
FIG. 4. THE D=~ENCE OF ~EOMA~NEBC DISTURBANCE VECTOR (MOULD BAY; 11.00-13.00 M.L.T.; JUNE--JULY1968) PROJECTED ONT~THE&RTH-SUNLINE SX'ON(B,V~~~).
variations 6x’ (the component along the Barth-Sun line) at the Mould Bay station are plotted vs B,YiLz for the interval 22.00-24.00 UT. ([email protected] M.L.T.) in June-July 1968 ; as the zero level the values of the geomagnetic field components at a quiet summer day are chosen. All the points in the figure correspond to IQml> SO”.One can see that the relation between 6X’ and (B,V~~z)is close to the linear one; the correlation coefficient equals I = -0.9. Thus, the& component of the electric field in the polar cap is described by the model under discussion also sufficiently well.
dependence of the polar cap potential drop on solar wind parameters was thoroughly studied by Reiff et aZ. (1981), Wygant et al. (1983), and by Doyle and Burke (1983). Having considered various reconnection models, the authors have concluded that all those models describe the behaviour of the polar cap potential drop more or iess successfmly when the interplanetary electric field is relatively weak (E,, < 0.5 mV m-l). For stronger solar wind electric fields, the polar cap potential drop is almost constant (A+ % 120-130 kv). This effect is considered an indication of a strong saturation of the reconnection process at the magnetopause. On the other hand, as we have seen above, the geomagnetic disturbance intensity in the polar cap is proportional to the reconnection electric field within a wide diapason of the latter (up to Es, = 2 mV m- ‘). Taking into account that the potential drop equals the electric field intensity multiplied by the The
al.
length of the reconnection line, the only way for the two results mentioned above to be consistent is to suppose that the length of the reconnection line decreases with the increase of the Em value. In this connection, let us return to formula (1). One can see that according to the models with a stagnation line, the length of the reconnection line increases with the increase of the solar wind magnetic field ; and it decreases with the increase of the solar wind velocity (L, N V;w’)and also decreases with the increase of the solar wind plasma density (I,, - ~s;y”‘~).The values of Mru,, and of 8, which are in the denominators of the expressions for L, also increase with the increase of&,. Thus, one may expect the value of L, to decrease with the increase of the solar wind electric field, and this has to limit the value of A& at the magnetopause and in the polar cap. Let us see whether the predictions of the model agree with the experimental data. For this purpose we use the data on the polar cap potential drop obtained onboard the satellites A&C, AE-D, S3-2 and S3-3 and discussed by ReiKet al. (1981), Wygant et al. (1983) and by Doyle and Burke (1983). As the parameters of the satellite trajectories across the polar cap were quite different for all three sets of data, we shall consider them separately. In this analysis we have to take into account that formulae (3) were obtained for a steady-state flow of the solar wind around the magnetosphere. It is important to note that it requires several hours for the polar cap potential drop to respond to the rapid change of the solar wind electric field (Wygant et al., 1983). For that reason, we shall use for this analysis only those passes which meet the following conditions : 1.5y-when B, changes from minus to plus, 2y -in all the other cases, I IAnI < 10cme3,
?&I 6
where AE, (An) is the difference between the values of 3, (or ra)at the hour of the polar cap crossing and at the preceding hour. Concerning the data considered by Reiff et al. (1981), only 26 passes of 32 meet those requirements. In Fig. 5a the values of A&,P obtained during those passes are plotted vs the values A@calculated for the cold plasma model [expression 3(a)]. One can see in the Egure that : (1) The agreement between the experimental and calculated values of Ad is sufficiently close (the coefficient of correlation between them equals T = 0.90). (2) The relation between the values of A4,, and A4ea1cis linear for the entire range of the variation of
411
Electric fields and currents in the Earth’s polar caps A
“exp, Kv 120
90
1
4
calo and no saturation effect in the polar cap potential drop takes place. (3) The coefficient of the proportionality between A&l and A@calcequals a, = 0.8 ; thus, the models under consideration predict the absolute values of Ad
.
.
(a)
. .
.
.
. .
i
.
. :
.
.
..
.
l
.
. t=0!77
. 0, 0
I 30
A~~*~(xv)'34+CL6?~pl=*~ I I I 90 60 fz4A%&,
.w
FIG. 5. EXPERIMENTAL DATA ON THE POLAR CAP POTENTIAL VSTHECALCULATED A+ VALUES. DRop&'q
(a) Data by Reset al. (198l);(b)Data by Wygant etal. (1983); (c) Data by Doyle and Burke (1983).
TABLE
2. CULTS
well. (4) The “residual” value of A4exp corresponding to Male = 0 equals A&!b x 33 kV and may be explained according to Reiff et al. (1981) by the quasi-viscous interaction of the solar wind with the magnetosphere. An analogous comparison of the experimental and calculated data for the passes studied by Wygant et al. (1983)is presented in Fig. 5b. In addition to the criterion used above we have omitted here the passes whose duration AT significantly differs from the mean value, that is the passes with AT 2 50 min and with AT < 10 min. One can see that the correlation between the values of A4exp and A$ca,c is significantly worse in this case (r = 0.70) than in the previous one. At the same time, the saturation effect is not observable in this case either, and all the other parameters of the least square fitting are approximately the same as earlier (see Table 2). The third set of data (after Doyle and Burke, 1983) is presented in Fig. 5c (with the same criterion of the data selection). The coefficient of correlation between the values of A4expand A4ca,cis here higher than in Fig. 5b (r = 0.77), and the saturation of the polar cap potential drop is clearly absent in this case. An analogous analysis was carried out for the hot plasma model [formula 3(b)], and the results are given in Table 2. Wygant et al. (1983) have summarized the results of their analysis of various reconnection models in a table. Having complemented their table with the results of our analysis for the cold and the hot plasma models with the stagnation line, we reproduce this table in Table 3. One can see that those two models taking into account the contraction of the stagnation line with the increase of the solar wind electric field, permit us to obtain a better agreement with the experimental data.
OF LEAST SQUARE FITS OF POLAR CAP PO TFINTIAL DROP REcQNNECTIoNPO'IENTiALMODEL
To HOURLY
AVERAGES
Data
r
Cold plasma a,
A&b
I
Hot plasma a,
A&$
Reiffet al. (1981) Wygant et al. (1983) Doyle and Burke (1983)
0.90 0.70
0.79 0.92
33 24
0.90 0.68
1.16 1.30
32 34
0.77
0.67
34
0.76
0.94
34
Altogether
0.77
0.8
34
0.74
1.1
33
OF
M. I.
412 TABLE
PUDoVKIN
A#~~~wI~~A~
3. THE ~~~ONOFT~O~~~~T~~LDROP
Model
e$ al.
Model electric field EY
&i
Corr. coeE
~~~~~A~~=,~
I&, = 30R,
0.47
0.15
1. Burton et al., 1975
V,,& 0
2. Akasofu
E = V,J3zWsin4%
?
0.61
?
i&B,, sin4 +
RI
0.56
0.18
9 V B sin2 z SWSW 2
D,
0.54
0.17
RI!
0.57
0.15
0.77
0.8
et al.,
1973
when B, i 0 when B, > 0
P~I~BYAV~~~~OFMO~~
3. Mod&xi Akasofu-Perreault (Wygant et nl., 1982) 4. Kan and Lee, 1979
5. Gonzalez and Mom, 1974
V,,B,,(B,,-
14 cos
sin 6,,,,) %
(B&,-28 cos %,,+196) 6. Pudovkin and Zaitseva, 1983
% YSW BSWsin’= 2
Dm2.85B,,sin~ M-@+aJm asw
4. THE
ELJZCTRIC FIELD INl’ENSlTY IN THE
NIGHT SECTOR OF THE MAGNETOSPHERE
Let us consider responsible
now what processes may be for the generation of the large-scale electric
field in the plasma sheet and in the magnetic tail during tbemagnetosphericdisturbances. Using the data on the delay of the magnetic disturbances in the aurora1 zone @E-indices) with respect to the solar wind electric field variations Pudovkin et al. (1968,1982b), Pudovkin and Shnchtina (1983) have calculated the intensity of the electric field in the magnetot~l and found that the electric field associated with the magnetic field reconnection at the day side magnetopause decreases noticeably while penetrating into the night magnetosphere. But the electric field intensity and solar wind parameters were averaged in those studies for relatively large intervals (7&100 h), so their results are rather rough. In this paper, the velocity of homogeneous aurora1 arc movement was taken as a measure of the electric field intensity (the component tangential to the arc) in the night ma~etosphere. In spite of the insurgent theoretical ground of the method, we shall assume that the aurora1 arc velocity during the initial substorm phase is determined by the
velocity of the magnetospheric plasma convection (Davis, 1971; Kelley el at., 1971; Meng C.-I., 1980). The data on the movement of the homogeneous aurora1 arcs in the midnight sector (00.0043.00 L.T.) during the initial substorm phase (that is during the l-h interval preceding aurora1 break-ups) were obtained at the observatories Loparskaya, Dixon, Tixi, Wrangel, Shmidt. In order to enhance the accuracy of calculations, the arcs observed no farther than 60” from the zenith and existed for the periods longer than 5 min were selected for the analysis. The arc altitude was not measured and was accepted to be 110 km. The hourlymean values of the solar wind parameters were taken Erom the King (1977) catalogue. Among various mechanisms which could generate electric fields in the magnetosphere, is (beside the above-mentioned process of the magnetic field line reconnection) the mechanism of the quasi-viscous interaction of the solar wind with the magnetosphere (Reiff et al., 1981; Mishin, 1978; Axford and Hines, 1961.; Troshichev, 1982). In thisconnection, the auroral arc velocity V,, was compared with a set of solar wind characteristics : (I) fi = V, ; for th6 solar wind velocity V,is assumed to determine the electric Geld intensity E in case of the quasi-viscous interaction of the solar wind with thegeomagneticfield;(2) fi = B,,, sin3 8,,,,/2n~Q,11/2
Electric fields and currents in the Earth’s polar caps
413
.
.
IZ.O.82 1
.
.
.
4 Iz;
FIG.~.THE AURORALARCVELOCITYATTHEINITIALSUBSTORMPHASEINDEPENDENCEONDIFFERENTSOLAR\NIND cHAKAcTERIsTIcs.
(a) fi =
V,,;
(b) fi = B,“,sin3 l&J2 n,-,‘/’ ; (c)f3 = E,,&
- the value which is proportional to the potential drop at the magnetopause in the reconnection model (Pudovkin and Zaitseva, 1983); (3) j’s = IQ&,, sin’ t&,/Z-the Y-component of the magnetopause electric field in the reconnection model (Stern, 1973 ; Gonzalez and Mozer, 1974; Yeh, 1976; Semenov and Pudovkin, 1978 ; Pudovkin and Zaitseva, 1983) ; and (4) f4 = V,,,,Bi’”(By is the southward component of the IMF)electric field intensity in the magnetosphere in the model by Burton et al. (1975). The obtained results are shown in Fig. 6 a-d respectively. From Fig. 6a one can see that the auroral arc velocity does not correlate well with the solar wind velocity; so the quasi-viscous interaction of the solar wind with the geomagnetic field is not the primary process for generation of the electric field in the magnetosphere’s tail. Concerning the penetration of electric fields into the magnetosphere as a consequence of magnetopause field line reconnection, one might come to the conclusion that this electric field determines aurora1 arc velocity (see Fig. 6b,c). But some pecularities of the observed connection of V,,, with the functions f2andf, cause us to be careful with this conclusion.
sin’ 0,,,,/2; (d) f4 =
VJS~“‘.
As can be seen from Fig. 6 b,c the arc velocity correlates better with the magnetopause electric fieldf, (r-s = 0.82) than with f2 (r2 = 0.66) which is proportional to the potential drop produced by this field. If the primary process responsible for electric field generation in the tail were the field line reconnection, the results obtained would suggest that the solar wind flows from the reconnection line along the magnetopause not in a fan-like manner but rather along a relatively narrow stripe. This result seems to be strange or unexpected at least. Because ofthis result, the study of the V,,, relation to the reconnection electric field was continued in the following way. It was shown (Stern, 1973 ; Gonzalez and Mozer, 1974 ; Yeh, 1976) that the magnitude of this field had to depend on the absolute value ofthe Y-component ofthe IMF. The experimental data confirm this (see Fig. 3). At the same time, the auroral arc velocity does not depend on lByl IMF : correlation coefficients between V,,, and lB,,l or IV>,,\ equal 0.1 and 0.07 respectively. So, if the reconnection electric field penetrates into the magnetosphere’s tail, it significantly decreases or is distorted. Then how can the observed correlation between
M. I.
414
FTJDOVKINet al.
aurora1 arc velocity and the value off, be explained? In order to clear up this question let us see Fig. 6d. From the data presented here there is obviously a close correlation between V,,, and the functionf, = V,sW which corresponds to the Y-component of the solar wind electric field. In their turn the values f3 and f4 correlate between themselves (rs,& = 0.95); this fact may explain the observed connection between V,,, and I&B, sin’ 8,,/2. At the same time, the product V,,J~” does not include any dependence ofE, on /I?,,]IMF, and this is observed
in the reality. 5. CONCLUSION
The data presented above show that the saturation of the polar cap potential drop observed for relatively intense solar wind electric fields (Reiff et al., 1981; Wygant et al., 1983 ; Doyle and Burke, 1983) is not to be understood as a result of a non-linear response of the magnetosphere to the electric field generated by the solar wind at the magnetopause. Indeed, one can see in Figs. 2,4 and 5 that both the electric field intensity and the potential drop at the polar cap are proportional to
the values of E, and A4 at the magnetopause. At the same time, one has to keep in mind that the value of Ac$being determined by the product of E, by the length of the reconnection line L,,, is not changing generally in direct proportion to E,, and this is most probably the cause of the observed saturation of Ati,, with respect to the values of A$ calculated in framework of the models neglecting the variation of the length of the reconnection line. On the whole, the analysis carried out above shows the magnetic field line reconnection to be one of the basic mechanism of generating the electric fields at the dayside magnetopause and in the day sector of the polar caps. On the other hand, the electric fields in the magnetotail plasma sheet seem to be (judging by the regularities of the aurora1 arc motion) weakly related to the reconnection field and must be generated by some other mechanism(s).
Acknowledgements-The authors are greatly indebted to an anonymous referee for assistance in revising the paper and for the data from S3-3 satellite, and to J. H. King for the interplanetary medium data. REFERENCES
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Pkmd. Space Sd., Vol. 33, No. 4, pp. 407-414,1985 Printed in Great Britain.
ELECTRIC FIELDS AND CURRENTS IN THE EARTH’S POLAR CAPS M.I.PUDOVRIN,
S. A. ZAlTSEVA, T. A. BAZHJWOVA and V. G. ANDREZRN Institute of Physics, Leningrad State University, Leningrad, 198904, U.S.S.R.
Abstract-A model for solar wind flow around the magnetopause incorporating a stagnation line at the frontside magnetopause is used to derive a formula for the electric field intensity and polar cap potential drop. These relationships are compared to experimental data from polar orbiting satellites. The relation between solar wind parameters and aurora1 arc velocity is also studied.
hydrod~~c aspects of the problem on the solar wind flow around the magnetosphere. An attempt to take into account the hydrodynamic aspects of the problem was made by Pudovkin (1982), Pudovkin et al. (1981, 1982a,b), and Pudovkin and Zaitseva(l983) on the base of the model of the solar wind flow around the magnetosphere with a stagnation line (Pudovkin and Semenov, 1977 ; Semenov and Pudovkin, 1978). In that model the solar wind parameters within the magnetosheath as well as the length of the stagnation line are calculated in a self-consistent manner. According to the model, the length of the stagnation (or reconnection) line approximately equals
I. ~ODU~ON
It is usua.lly supposed that the electric fields in the magnetosphere and in the polar ionosphere appear as a result of the solar wind and the Earth’s magnetic field line reconnection. The models based on that assumption (Stern, 1973; Gonzalez and Mozer, 1974; Yeh, 1976; Kan and Lee, 1979) describe the E-field relative variations in the polar ionosphere silently well. However, the absolute values of the calculated potential drop across the polar caps prove to be 3-6 times more than the experimentally observed ones. Besides, a more detailed analysis has shown that the
-in the cold plasma approximation Zaitseva, 1983
(Pudovkin
and (1)
-in the hot plasma approximation 1982a,b) coefficient of proportion~ity between the values of AcbeXp and A&eor is not constant and depends on the conditions in the solar wind (Gonzalez and Mozer, 1974). In particular, for all the models referred to above the polar cap potential drop exhibits some kind of saturation, that is the values of A&, are proportional < to A#theor for small values of the latter (A#I~~*,,~ A&, N 100 kv), and do not depend on them for A.dftheor > Ac$,, (Reiff et al., 1981; Wygant et al., 1983; Doyle and Burke, 1983). In order to reconcile the differences between the theoretical predictions and the experimental results, the authors limit the values of A#,,,,, at the magnetopause by including some additional assumptions which do not result from the models. The disagreement between the theoretical models and experimental data may be caused, as was pointed out by Gonzalez and Mozer (1974), by the neglect ofthe
(Pudovkin
et al.,
where I),,, is the diameter of the magnetosphere, Mu,, is the Alfvenic Mach number, 0:is parameter of the nonpotentiality (see Fig. l), /&, = B,(4np,,k$)‘I2 and B = dm; the symbols “SW”,and “0” and “m” mark the values of the magnetic field and of the plasma parameters within the undisturbed solar wind, at the bow shock, and at the magnetopa~e in the vicinity of the subsolar point respectiveiy. As was shown by Gonzalez and Mozer (1974), the electric field intensity at the magnetopause is described by the formulae : I$, =&~&3~
s~(e~-~)sin~ (2)
E, = i v,,,&,, sin (6, - 9) cos rp
/
where E,, E, are the electric field components along the corresponding axes of the GSM co-ordinate system;
407
i%.
I.
&JDOVKiN
et
ai.
distance between magnetic field lines in the ionosphere decreases with respect to that at the magnetopause in proportion to (~~~~~)1’2, where 3, is the normal component of the magnetic field at the magnetopause, and Bj is the magnetic field intensity at the polar cap; Bllvfx 0,1(47~p,,,,V&,)*‘~ (Pudovkin et al., 1981; Semenov and Pudovkin, 1978). Then for the field components iu the ionosphere we have : 13B = A pBSW V1i2K’i4 SW SW sin(~~~-~)sin
E, = A$swV&;1~4 O,, is the angle between the Z-axis and the B,, vector, and cp is the angle between the Z-axis and the reconnection line; dependence of the angle cp on the magnetosheath’s magnetic field orientation is assumed in the form proposed by Yeh (1976). As is seen from (2), for all the IMF orientations, the E,-component is directed from dawn to dusk, and is hence responsible for the DP-2 field. The &-component being projected onto the ionosphere is directed along the noon-midnight meridian and corresponds to the DP-4 field (or for the Svaalgard-Mansurov effect) ; the sign of the E, field depends on the sign of the Y-component of the magnetosheath’s magnetic field. Taking into account formulae (I) and (2), one obtains the fo~o~ng expressions for the polar cap dawndusk potential drop :
-in
x sin (8,, - cp)sin rplsin 91 the cold plasma approximation
I t cb)
-in
x sin (t?,, - q) sin it /sin 91 the hot plasma app~o~mat~on
(3)
i I
One can see from (3) that if the model under consideration is correct, the polar cap potential drop is proportional to the squared value of the IMF intensity and does not depend on the solar wind velocity. 2 THE POLAR CAP EK,ECTRIC FIEm
1NTENSl-N
In order to compare quan~ta~vely the ~tensity of polar cap geomagnetic disturbances with the magnetopause electric field this field has to be projected onto the ionosphere. For that, one has to take into account that the
cp
(a)
sin (@,,-- cp)cos rp
(b)
(4)
where A,, A, are constant coefficients. The product sin (BsW-(p)sin 9 = fe(@,) is approximately equal to sin28,,/2 in the cold plasma model (Pudov~n and Zaitseva, 1983); and to sin48,,/2 in the hot plasma model (Pudovkin et al., 1982a). Thus the value of the angular function in (4) may be written as 0 f,(B,,) w sin” z, 2 where 2 < k 6 4. Let us see whether formulae (4) agree with the experiment. For this purpose we shall examine va~a~ons of the ~~espond~g components of the geomagnetic field at the polar cap magnetic observatories, In this study we shah assume the Hall currents to be responsible for the observed variations of the geomagnetic field; besides, we shah take into account that these variations are caused mainly by the vacations of the electric field intensity while the conductivity of the polar ionosphere is changing insignificantly at the day hours of the summer months (Vanyan and Osipova, 1975). In Fig. 2 hourly-mean values of the dawn-dusk component of the magnetic disturbances (SY’) at the Alert station at the day time (10.00-l 5.00 M.L.T.) for the period June-July 1968 are shown in dependence on the function F, = B,,V$ sin2 @,,/2 which is proportional to the E,%eld in the ionosphere (we neglect a weak dependence of E, on n). As the level of reference, the values of the geomagnetic field components on a quiet winter day were taken. The crosses in the figure correspond to the cases when the solar wind magnetic field is significantly northward (l&,1 < 50”). One can see that most of the points are scattered near some straight line, which confirms the propo~ion~ity between expe~ment~ and theoretical values of iW. It was found that the dependence of SY‘ on the functions Fl and F, = B,,I&,, sin2 &j2 [see (2) and (4)] is approximateIy the same; the correlation coefficient equals r = 0.77 in both cases (the points with If&,1< 50” being excluded). Therefore, in order to find the optima fun~on describing the deveiopment of the magnetic disturbances, additional study is needed.
Electric fields and currents in the Earth’s polar caps
409
.
FIG. 3. THE SAME As IN FIG. 2, BUT IN Dm?NDENcE B#MF); (b) /B,j(IMF), WHENIS,1 < 0.57. FIG. 2. THE
DEPENDENCE
OF GEOMAGNEXC
DISTURBANCE
VECTOR (ALFXT OBS.,30.00-15.00 M.L.T.;JuN&JuLY 1968) PROJECTBDONTOTHBPERPIINDICXJLAR TOTHBEARTH-SIJNLINE 6Y' ON THE FUNCTION F, = B,,Vj,f sin’ i&/2.
The cases when l6’,,,,l < 50” are pointed by crosses. In this co~eci~on let us see whether the inclination of the regression lines for the two functions under consideration is constant or if it changes in its dependence on the solar wind velocity. The regression coeflicients for functions I;, and F, for two intervals of V,, (V,, < 450 km s-* and V,, > 450 km s-l) are sho~inTable 1. Onecansee from theTable, that ofthe two functions, only F, is linearly related to 6 Y’. For both functions the factorf,(8,,J was taken above in the form sin2@,,,,/2 which is correct for the cold plasma only, and in a general casef, = sink 6,,/2. The value of the exponent k may be found from the experiment. For this purpose, we expand E, in the vicinity of the point (IS,, = 0; e,, = E/Z ; y,, = V,& into Taylor series ; asbefore, we neglect the dependence of Ei, on II.Then having restricted ourselves by the first terms of the series we have :
The equality (6) shows that the dawn-dusk polar cap electric field is determined by the variations of B, and lB,IIMF (Stern, 19’73; Gonzalez and Mozer, 1974; Burton et QZ., 1975; Mishin, 1978). The relative “sensitivity” of the E,-field to the variations of B, and lByj IMF is :
and according to the mdhel, 2 < k < 4, i.e. 1 < (&i&J < 2. Let us check the expression (6) using the data on the geomagnetic field variations. In Fig. 3 the intensity of the magnetic disturbances 6 Y’is plotted vs the values of B, (Fig. 3a) and lB,( IMF (for IB,i < OSy, (Fig. 3b). One can see that 6Y’ really depends on &I, which seems to confirm the model. Correlation coefficients between SY’ and B,, jB,i and corresponding regression coefficients (the latter in arbitrary units) are: r, = -0.7, a, = 20.6; rv = 0.6, ay = 14.1. Hence (E&J = @,/a,) E 1.5, then k sz 3 and fDRSsin3 0,,,,/2. Inserting this value of fa into (3) we have A+, z B&,ps-wli=sin4 @,,/2, the last expression being very close to the Akasofu-Perreault index E. In Fig. 4, the hourly-man values of the magnetic
TABLE 1. THE COEPFICIENTS CHARACTBRIZED INCLINATION
THE REGRESSION LINE
Thefunction Theinterval of K, $450kms-f >450kms-’
ON:(a)
Icl= B,,V$’ 1.40 1.34
sin’ 6,,/;!
F, = B,,f&
sin’ OS,,,/2
0.69 0.34
410
M. L PUDOVKIN et
FIG. 4. THE D=~ENCE OF ~EOMA~NEBC DISTURBANCE VECTOR (MOULD BAY; 11.00-13.00 M.L.T.; JUNE--JULY1968) PROJECTED ONT~THE&RTH-SUNLINE SX'ON(B,V~~~).
variations 6x’ (the component along the Barth-Sun line) at the Mould Bay station are plotted vs B,YiLz for the interval 22.00-24.00 UT. ([email protected] M.L.T.) in June-July 1968 ; as the zero level the values of the geomagnetic field components at a quiet summer day are chosen. All the points in the figure correspond to IQml> SO”.One can see that the relation between 6X’ and (B,V~~z)is close to the linear one; the correlation coefficient equals I = -0.9. Thus, the& component of the electric field in the polar cap is described by the model under discussion also sufficiently well.
dependence of the polar cap potential drop on solar wind parameters was thoroughly studied by Reiff et aZ. (1981), Wygant et al. (1983), and by Doyle and Burke (1983). Having considered various reconnection models, the authors have concluded that all those models describe the behaviour of the polar cap potential drop more or iess successfmly when the interplanetary electric field is relatively weak (E,, < 0.5 mV m-l). For stronger solar wind electric fields, the polar cap potential drop is almost constant (A+ % 120-130 kv). This effect is considered an indication of a strong saturation of the reconnection process at the magnetopause. On the other hand, as we have seen above, the geomagnetic disturbance intensity in the polar cap is proportional to the reconnection electric field within a wide diapason of the latter (up to Es, = 2 mV m- ‘). Taking into account that the potential drop equals the electric field intensity multiplied by the The
al.
length of the reconnection line, the only way for the two results mentioned above to be consistent is to suppose that the length of the reconnection line decreases with the increase of the Em value. In this connection, let us return to formula (1). One can see that according to the models with a stagnation line, the length of the reconnection line increases with the increase of the solar wind magnetic field ; and it decreases with the increase of the solar wind velocity (L, N V;w’)and also decreases with the increase of the solar wind plasma density (I,, - ~s;y”‘~).The values of Mru,, and of 8, which are in the denominators of the expressions for L, also increase with the increase of&,. Thus, one may expect the value of L, to decrease with the increase of the solar wind electric field, and this has to limit the value of A& at the magnetopause and in the polar cap. Let us see whether the predictions of the model agree with the experimental data. For this purpose we use the data on the polar cap potential drop obtained onboard the satellites A&C, AE-D, S3-2 and S3-3 and discussed by ReiKet al. (1981), Wygant et al. (1983) and by Doyle and Burke (1983). As the parameters of the satellite trajectories across the polar cap were quite different for all three sets of data, we shall consider them separately. In this analysis we have to take into account that formulae (3) were obtained for a steady-state flow of the solar wind around the magnetosphere. It is important to note that it requires several hours for the polar cap potential drop to respond to the rapid change of the solar wind electric field (Wygant et al., 1983). For that reason, we shall use for this analysis only those passes which meet the following conditions : 1.5y-when B, changes from minus to plus, 2y -in all the other cases, I IAnI < 10cme3,
?&I 6
where AE, (An) is the difference between the values of 3, (or ra)at the hour of the polar cap crossing and at the preceding hour. Concerning the data considered by Reiff et al. (1981), only 26 passes of 32 meet those requirements. In Fig. 5a the values of A&,P obtained during those passes are plotted vs the values A@calculated for the cold plasma model [expression 3(a)]. One can see in the Egure that : (1) The agreement between the experimental and calculated values of Ad is sufficiently close (the coefficient of correlation between them equals T = 0.90). (2) The relation between the values of A4,, and A4ea1cis linear for the entire range of the variation of
411
Electric fields and currents in the Earth’s polar caps A
“exp, Kv 120
90
1
4
calo and no saturation effect in the polar cap potential drop takes place. (3) The coefficient of the proportionality between A&l and A@calcequals a, = 0.8 ; thus, the models under consideration predict the absolute values of Ad
.
.
(a)
. .
.
.
. .
i
.
. :
.
.
..
.
l
.
. t=0!77
. 0, 0
I 30
A~~*~(xv)'34+CL6?~pl=*~ I I I 90 60 fz4A%&,
.w
FIG. 5. EXPERIMENTAL DATA ON THE POLAR CAP POTENTIAL VSTHECALCULATED A+ VALUES. DRop&'q
(a) Data by Reset al. (198l);(b)Data by Wygant etal. (1983); (c) Data by Doyle and Burke (1983).
TABLE
2. CULTS
well. (4) The “residual” value of A4exp corresponding to Male = 0 equals A&!b x 33 kV and may be explained according to Reiff et al. (1981) by the quasi-viscous interaction of the solar wind with the magnetosphere. An analogous comparison of the experimental and calculated data for the passes studied by Wygant et al. (1983)is presented in Fig. 5b. In addition to the criterion used above we have omitted here the passes whose duration AT significantly differs from the mean value, that is the passes with AT 2 50 min and with AT < 10 min. One can see that the correlation between the values of A4exp and A$ca,c is significantly worse in this case (r = 0.70) than in the previous one. At the same time, the saturation effect is not observable in this case either, and all the other parameters of the least square fitting are approximately the same as earlier (see Table 2). The third set of data (after Doyle and Burke, 1983) is presented in Fig. 5c (with the same criterion of the data selection). The coefficient of correlation between the values of A4expand A4ca,cis here higher than in Fig. 5b (r = 0.77), and the saturation of the polar cap potential drop is clearly absent in this case. An analogous analysis was carried out for the hot plasma model [formula 3(b)], and the results are given in Table 2. Wygant et al. (1983) have summarized the results of their analysis of various reconnection models in a table. Having complemented their table with the results of our analysis for the cold and the hot plasma models with the stagnation line, we reproduce this table in Table 3. One can see that those two models taking into account the contraction of the stagnation line with the increase of the solar wind electric field, permit us to obtain a better agreement with the experimental data.
OF LEAST SQUARE FITS OF POLAR CAP PO TFINTIAL DROP REcQNNECTIoNPO'IENTiALMODEL
To HOURLY
AVERAGES
Data
r
Cold plasma a,
A&b
I
Hot plasma a,
A&$
Reiffet al. (1981) Wygant et al. (1983) Doyle and Burke (1983)
0.90 0.70
0.79 0.92
33 24
0.90 0.68
1.16 1.30
32 34
0.77
0.67
34
0.76
0.94
34
Altogether
0.77
0.8
34
0.74
1.1
33
OF
M. I.
412 TABLE
PUDoVKIN
A#~~~wI~~A~
3. THE ~~~ONOFT~O~~~~T~~LDROP
Model
e$ al.
Model electric field EY
&i
Corr. coeE
~~~~~A~~=,~
I&, = 30R,
0.47
0.15
1. Burton et al., 1975
V,,& 0
2. Akasofu
E = V,J3zWsin4%
?
0.61
?
i&B,, sin4 +
RI
0.56
0.18
9 V B sin2 z SWSW 2
D,
0.54
0.17
RI!
0.57
0.15
0.77
0.8
et al.,
1973
when B, i 0 when B, > 0
P~I~BYAV~~~~OFMO~~
3. Mod&xi Akasofu-Perreault (Wygant et nl., 1982) 4. Kan and Lee, 1979
5. Gonzalez and Mom, 1974
V,,B,,(B,,-
14 cos
sin 6,,,,) %
(B&,-28 cos %,,+196) 6. Pudovkin and Zaitseva, 1983
% YSW BSWsin’= 2
Dm2.85B,,sin~ M-@+aJm asw
4. THE
ELJZCTRIC FIELD INl’ENSlTY IN THE
NIGHT SECTOR OF THE MAGNETOSPHERE
Let us consider responsible
now what processes may be for the generation of the large-scale electric
field in the plasma sheet and in the magnetic tail during tbemagnetosphericdisturbances. Using the data on the delay of the magnetic disturbances in the aurora1 zone @E-indices) with respect to the solar wind electric field variations Pudovkin et al. (1968,1982b), Pudovkin and Shnchtina (1983) have calculated the intensity of the electric field in the magnetot~l and found that the electric field associated with the magnetic field reconnection at the day side magnetopause decreases noticeably while penetrating into the night magnetosphere. But the electric field intensity and solar wind parameters were averaged in those studies for relatively large intervals (7&100 h), so their results are rather rough. In this paper, the velocity of homogeneous aurora1 arc movement was taken as a measure of the electric field intensity (the component tangential to the arc) in the night ma~etosphere. In spite of the insurgent theoretical ground of the method, we shall assume that the aurora1 arc velocity during the initial substorm phase is determined by the
velocity of the magnetospheric plasma convection (Davis, 1971; Kelley el at., 1971; Meng C.-I., 1980). The data on the movement of the homogeneous aurora1 arcs in the midnight sector (00.0043.00 L.T.) during the initial substorm phase (that is during the l-h interval preceding aurora1 break-ups) were obtained at the observatories Loparskaya, Dixon, Tixi, Wrangel, Shmidt. In order to enhance the accuracy of calculations, the arcs observed no farther than 60” from the zenith and existed for the periods longer than 5 min were selected for the analysis. The arc altitude was not measured and was accepted to be 110 km. The hourlymean values of the solar wind parameters were taken Erom the King (1977) catalogue. Among various mechanisms which could generate electric fields in the magnetosphere, is (beside the above-mentioned process of the magnetic field line reconnection) the mechanism of the quasi-viscous interaction of the solar wind with the magnetosphere (Reiff et al., 1981; Mishin, 1978; Axford and Hines, 1961.; Troshichev, 1982). In thisconnection, the auroral arc velocity V,, was compared with a set of solar wind characteristics : (I) fi = V, ; for th6 solar wind velocity V,is assumed to determine the electric Geld intensity E in case of the quasi-viscous interaction of the solar wind with thegeomagneticfield;(2) fi = B,,, sin3 8,,,,/2n~Q,11/2
Electric fields and currents in the Earth’s polar caps
413
.
.
IZ.O.82 1
.
.
.
4 Iz;
FIG.~.THE AURORALARCVELOCITYATTHEINITIALSUBSTORMPHASEINDEPENDENCEONDIFFERENTSOLAR\NIND cHAKAcTERIsTIcs.
(a) fi =
V,,;
(b) fi = B,“,sin3 l&J2 n,-,‘/’ ; (c)f3 = E,,&
- the value which is proportional to the potential drop at the magnetopause in the reconnection model (Pudovkin and Zaitseva, 1983); (3) j’s = IQ&,, sin’ t&,/Z-the Y-component of the magnetopause electric field in the reconnection model (Stern, 1973 ; Gonzalez and Mozer, 1974; Yeh, 1976; Semenov and Pudovkin, 1978 ; Pudovkin and Zaitseva, 1983) ; and (4) f4 = V,,,,Bi’”(By is the southward component of the IMF)electric field intensity in the magnetosphere in the model by Burton et al. (1975). The obtained results are shown in Fig. 6 a-d respectively. From Fig. 6a one can see that the auroral arc velocity does not correlate well with the solar wind velocity; so the quasi-viscous interaction of the solar wind with the geomagnetic field is not the primary process for generation of the electric field in the magnetosphere’s tail. Concerning the penetration of electric fields into the magnetosphere as a consequence of magnetopause field line reconnection, one might come to the conclusion that this electric field determines aurora1 arc velocity (see Fig. 6b,c). But some pecularities of the observed connection of V,,, with the functions f2andf, cause us to be careful with this conclusion.
sin’ 0,,,,/2; (d) f4 =
VJS~“‘.
As can be seen from Fig. 6 b,c the arc velocity correlates better with the magnetopause electric fieldf, (r-s = 0.82) than with f2 (r2 = 0.66) which is proportional to the potential drop produced by this field. If the primary process responsible for electric field generation in the tail were the field line reconnection, the results obtained would suggest that the solar wind flows from the reconnection line along the magnetopause not in a fan-like manner but rather along a relatively narrow stripe. This result seems to be strange or unexpected at least. Because ofthis result, the study of the V,,, relation to the reconnection electric field was continued in the following way. It was shown (Stern, 1973 ; Gonzalez and Mozer, 1974 ; Yeh, 1976) that the magnitude of this field had to depend on the absolute value ofthe Y-component ofthe IMF. The experimental data confirm this (see Fig. 3). At the same time, the auroral arc velocity does not depend on lByl IMF : correlation coefficients between V,,, and lB,,l or IV>,,\ equal 0.1 and 0.07 respectively. So, if the reconnection electric field penetrates into the magnetosphere’s tail, it significantly decreases or is distorted. Then how can the observed correlation between
M. I.
414
FTJDOVKINet al.
aurora1 arc velocity and the value off, be explained? In order to clear up this question let us see Fig. 6d. From the data presented here there is obviously a close correlation between V,,, and the functionf, = V,sW which corresponds to the Y-component of the solar wind electric field. In their turn the values f3 and f4 correlate between themselves (rs,& = 0.95); this fact may explain the observed connection between V,,, and I&B, sin’ 8,,/2. At the same time, the product V,,J~” does not include any dependence ofE, on /I?,,]IMF, and this is observed
in the reality. 5. CONCLUSION
The data presented above show that the saturation of the polar cap potential drop observed for relatively intense solar wind electric fields (Reiff et al., 1981; Wygant et al., 1983 ; Doyle and Burke, 1983) is not to be understood as a result of a non-linear response of the magnetosphere to the electric field generated by the solar wind at the magnetopause. Indeed, one can see in Figs. 2,4 and 5 that both the electric field intensity and the potential drop at the polar cap are proportional to
the values of E, and A4 at the magnetopause. At the same time, one has to keep in mind that the value of Ac$being determined by the product of E, by the length of the reconnection line L,,, is not changing generally in direct proportion to E,, and this is most probably the cause of the observed saturation of Ati,, with respect to the values of A$ calculated in framework of the models neglecting the variation of the length of the reconnection line. On the whole, the analysis carried out above shows the magnetic field line reconnection to be one of the basic mechanism of generating the electric fields at the dayside magnetopause and in the day sector of the polar caps. On the other hand, the electric fields in the magnetotail plasma sheet seem to be (judging by the regularities of the aurora1 arc motion) weakly related to the reconnection field and must be generated by some other mechanism(s).
Acknowledgements-The authors are greatly indebted to an anonymous referee for assistance in revising the paper and for the data from S3-3 satellite, and to J. H. King for the interplanetary medium data. REFERENCES
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