A numerical study of polar ionospheric currents

A numerical study of polar ionospheric currents

Planet. Space Sci. 1973, Vol. 21, pp. 1287 to 1300. Pergamon Press. Printed in Northern Ireland A NUMERICAL STUDY OF POLAR IONOSPHERIC CURRENTS HIROS...

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Planet. Space Sci. 1973, Vol. 21, pp. 1287 to 1300. Pergamon Press. Printed in Northern Ireland

A NUMERICAL STUDY OF POLAR IONOSPHERIC CURRENTS HIROSHI MAEDA* Space Science Division, NASA-Ames Research Center, Moffett Field, California 94035, U.S.A. and

KOICHIRO MAEKAWA Department of Earth Sciences, Kyoto Kyoiku University, Kyoto 612, Japan

(Received in final form 2 February 1973) Abstract--Numerical calculations for the electric current in the polar ionosphere have been made by assuming some realistic distributions of the electric field and conductivity. Two dynamo actions are taken into account; one of which is induced by ionospheric winds and the other by the solar wind. For the solar wind dynamo action, it is found that the secondary polarization field caused by non-uniform distribution of ionospheric conductivity is much larger than the primary field induced by the solar wind, suggesting its important effect on charged particles in the magnetosphere, and that the irrotational current having a source and sink is of the same order of magnitude as the solenoidal current closing its circuit in the ionosphere. It is also found that the solar wind is, in general, more effective than the ionospheric winds in producing polar current systems such as DP 1 and 2, but in some cases the ionospheric winds have a significant effect on the current distribution. 1. INTRODUCTION Geomagnetic variations in the polar regions have long been studied extensively, and it seems that they are roughly divided into two types, D P 1 and D P 2 (see a review by Obayashi and Nishida cl) and by Nishida and Kokubun(2)). The D P 1 type variations are usually observed on disturbed days and they are associated with the auroral electrojet, whereas the D P 2 type variations are observed on relatively quiet days and they are not associated with the electrojet. Anyway, the immediate cause of these variations would be an electric current in the polar ionosphere. Concerning the driving force for the electric current, we may take at least two different causes; ionospheric winds and the solar wind. The ionospheric wind dynamo action has been postulated in order to explain the daily geomagnetic variations on quiet days (Sq), when the ionosphere is in a quiescent state. On disturbed days, the ionosphere in the polar regions is influenced by the precipitation of charged particles, resulting in an anomalous enhancement of conductivity, so that the electric current would be much enhanced and distorted from that on quiet days. The possibility that the geomagnetic variations on disturbed days are interpreted in terms of ionospheric winds by taking into account an enhanced conductivity has been considered by a number of authors. (3-m However, it has been pointed out by Maeda, (~3) Cole, (14) Swift, (aS) Bostr~m (le) and others that the disturbed geomagnetic variations cannot be produced by ionospheric winds alone, even if an appropriate enhancement of conductivity is taken into consideration. The importance of electric field and current of external origin was emphasized by Birkeland (17) and Alfv6n. clang) Several different mechanisms for such a field and current were proposed/2°-2s) and associated current systems were calculated. (29-34) * Now at the Geophysical Institute, Kyoto University, Kyoto 606, Japan. 1287

1288

HIROSHI MAEDA and KOICHIRO MAEKAWA

In recent years observations of electric lie[d and current have been carried out cxtc:~ sively. Field-aligned currents are deduced from observations of magnetic field and'o~ ~ particle flux (Zmuda e t al. laS-a:~ and others). Electric fields are also observed by severai direct and indirect methods (e.g. a review by MaynardlaS)); i.e. the double probe technique on space vehicles, the Barium release experiments developed by the group at the Max Planck Institute, drift measurements by whistler dispersion or by incoherent scatter ra~l,-u at the polar region. These observations show strong evidences for the existence of ti~e electric field and current of external (solar wind) origin. Even if the solar wind plays an important role in producing current systems in the polar ionosphere, the contribution from ionospheric winds might not be negligible, The purpose of this paper is to calculate current systems generated by these two causes assuming some realistic models, and to make clear the relative importance of contributions from the tao causes in producing current systems in the polar ionosphere. 2. BASIC EQUATIONS W e consider, for simplicity, a two-dimensional spherical current sheet with the height-

integrated layer conductivity [K]. If the electric current is induced by ionospheric winds, then the current density is given by J -- [K](Ez, -~- Es)

(l)

where E D is the dynamo electric field, and E s is a polarization field produced by a nonuniform distribution of conductivity. In this case, the current is solenoidal, so that its distribution may be obtained by solving equation div J = 0, if we ignore a current flow between the northern and southern hemispheres along the geomagnetic field lines. On the other hand, if the current is induced by an externally applied electric field E o, which is induced by the solar wind, then the current density may similarly be given by a : : [K](E o ~- Es)

(2)

where E s is again a polarization field produced by the non-uniform distribution of conductivity. In this case, the current has two components; one is the irrotational current having a source and sink, and the other is the solenoidal current closing its circuit in the current sheet. Even though the current layer includes such a source and sink, we may obtain the current distribution by solving equation div J == 0 in regions except for the source and sink. (In general, however, we have to solve equation div J = j~, wherej, is the vertical component of field-aligned currents at the source and sink.) In the most general case where the current is induced by the above two causes, the current density will be given by J := [K](E o q - E D @ Es)

(3)

and we may find the current distribution by solving div J = 0 in regions except for the source and sink in the same way as above. 3. MODELS

In order to solve Equations (1), (2) and (3), we need to know the distribution of the height-integrated layer conductivity [K], the external electric field E o, and the dynamo electric field E D.

A NUMERICAL

STUDY

OF POLAR

IONOSPHERIC

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3.1 D&tribution of[K] A model by Fejer ~39) is used for the latitudinal distribution of [K] at equinoctial noon. The variation of [K] is taken into account in the form [ K ] = [K0l 1--' ~ k . c o s ' z

(4)

n=l

where Z is the Sun's zenith angle. The coefficients kn are estimated as to agree with the noon and midnight value of [K], where the night values are deduced from rocket observations of the ionosphere. The distribution of [K] is shown in Fig. 1, where Kxx, K~ and Kx~

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FIG. 1. LATITUDINAL DISTRIBUTIONS OF THE LAYER CONDUCTIVITIES (K~, K~v and Kxu) AT NOON AND MIDNIGHT, WHERE (1), (2) AND (3) SHOW MODIFICATIONS IN HIGH LATITUDES.

are the components of conductivity tensor [K]. The symbols (1), (2) and (3) indicate modifications of K's in the polar region (called K1, K2 and K3 models, respectively). 3.2 Distribution olEo We suppose two types (EA and EB) of charge separation by which the polarization field E o is produced (Table 1). The EA field is produced by a pair of positive (at P1) and negative (at P2) charges, and it may correspond to a DP 2 field. The EB field is produced by another pair of positive (at P1) and negative (at/'2) charges, and it may correspond to a DP 1 field. The positions of these point charges in the ionosphere are as follows: Type of E o EA EB

Point

Charge

P1 P2 P1 P2

+q --q +q --q

Colatitude 21-25 ° 21"25° 18.75° 18-75°

Longitude (or LT) 86"25° ( ~ 266"25° ( ~ 33.75 ° ( ~ 326"25° ( ~

06 hr) 18 hr) 02 hr) 22 hr)

1290

H1ROSHI MAEDA and KOICHIRO MAEKAWA [ ABI,E 1, CASES O F CALCULATION

Electric field Distribution applied from of external conductivity ~zh, io@6

Solar-wind dynamo (Eo -t- E s,~

Ionosphericwind dynamo (ED i- E,,,)

Generalized dynamo (Eo i E:~ ~Es)

Case SA 1

Case 1 1

Case GA 1

Case SA 2

Case 1 2

Case GA 2

Case SB 1

Case 1 1

Case GB 1

Case SB 2

Case 1 2

Case GB 2

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3.3 Distribution o r E D Since we do not yet know what kind of winds exists in the polar ionosphere, we use a wind system obtained by Maeda, t4°~ and calculate the dynamo electric field E D. 3.4 Models Characteristics of the models adopted in this paper are summarized in Table 1, where the case of ED = 0 is called the solar-wind dynamo, the case of Eo = 0 is called the ionospheric-wind dynamo, the case of ED =~ 0 and E o ~ 0 is called the generalized dynamo. <41~ The symbols EA and EB show the type of the external electric field Eo, and/£1, Ka and K z show the model of conductivity distribution in the polar region, where the regions of enhanced conductivity are hatched. 4. METHOD OF SOLUTION If we take the most general case as shown by (3), the equation to be solved is given by div {[KI(E o + ED + Es)} = 0

(5)

where [K], Eo and ED are known, and Es is unknown. By using spherical polar coordinates (0, 4) where 0 is the colatitude and 2 is the east longitude, Equation (5) is reduced to the form (see Maeda and Murata ~4~)) 02S O~S 3S OS ~ - - ~ + fl~-~ + ~ ' - ~ -k b--O-~= e

(6)

02S O~S 00---2+ A - - ~ - k

(7)

or

OS OS B-o~-k- C--~ = D

A NUMERICAL STUDY OF POLAR IONOSPHERIC CURRENTS

1291

where 0¢ =

K~

sin 0

fl = K~/sin 0 7 = 0~/00 -- OKxJ02 6 = OKx~/O0 + 0K~J(sin 0 02)

e = OFjO0 + OF./O~ a = radius of the current sheet S = electrostatic potential of E s F~ = a sin O.[K~x(Eox + E ~ ) -+- Kx~(Eo~ + ED~)]. F u = a{--Kxu(Eox + EDx) @ K~(Eou + E ~ ) } A

=

fl/~,B =

7/~,C

=

~/~,0

=

~/~.

and K~, Kyy, K~, = components of [K]. Since the coefficients A, B, C and D are given by numerical values, the solution is obtained by a method of numerical integration. The boundary conditions are such that S = 0 at the pole OS O0

--

0 at the Equator

and s ( a = o) = s ( ~ = 2~).

Note that the Eo field becomes infinite at the source and sink, and therefore the integration has been done in such a manner that these singular points are taken as a center of the mesh of calculation. The mesh is taken to be 2.5 ° for 0, and 7.5 ° for 2, and solutions have, in most cases, been converged within limits of error for several minutes by computer. This seems to be due to the reason that our basic Equation (7) is, fortunately, of elliptic type. 5. RESULTS AND DISCUSSION The distributions of the electrostatic potential S of the E o field for Cases A and B are shown in Fig. 2, where the charge q is taken as 0.2 Coulomb. The corresponding distributions of the potential S of the E s field for Cases SA-1 and SB-1 are shown in Fig. 3. Since the E o field is the applied field and the E s field is the resultant field, it is to be noted that the latter is much (about ten times) larger than the former, though the patterns of the distribution do not differ significantly from each other. This surprising result may be due to the fact that the Hall current caused by the applied field is, in general, larger than the Pedersen current, and therefore the resultant field is mainly associated with the Hall current. For this reason, the current distributions are more influenced by the E s field than the Eo field. This result seems to be very important in that the E s field could have an important effect on charged particles in the magnetosphere, and therefore the E o field itself may be significantly altered. F o r comparison, the distribution of the dynamo (applied) field ED and the resultant field E s for Case I-1 is shown in Fig. 4. It is found that the applied field and the resultant

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field have a l m o s t the same o r d e r o f m a g n i t u d e , so that the current is d e t e r m i n e d by b o t h o f these fields. I n general, we have two kinds o f i o n o s p h e r i c currents; one is the Pedersen current J v a n d the o t h e r is the Hall c u r r e n t J ~ . I n the case o f the i o n o s p h e r i c - w i n d d y n a m o , the c u r r e n t density J ( = J p + J•) is s o l e n o i d a l everywhere, so that its d i s t r i b u t i o n can be expressed in terms o f a c u r r e n t function or s t r e a m line. I n the case o f the solar w i n d d y n a m o , however, the resultant c u r r e n t J consists o f i r r o t a t i o n a l Ji,r a n d s o l e n o i d a l J~ol parts, where

A NUMERICAL STUDY OF POLAR IONOSPHERIC CURRENTS

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FIG. 4. DISTRIBUTIONS OF THE DYNAMO FIELD E D (LEFT) AND THE RESULTANF FIELD E S (RIGHT) FOR CASE I - 1 .

curl Jirr ~ 0 and div J~o] = 0. Jirr is a current flowing from the source to the sink, and J~o~ is a current flowing in a closed circuit in the ionosphere. It is very difficult, in general, to separate Jirr and Jsol from J. Kern (43) has tried to separate these two parts, but his method cannot be applied for the case of non-uniform conductivity. If the electric field is derived from a potential, J ~ = J p and J~ol --~ Jii when the conductivity is uniform, but Ji~r = J P + J n and J~ol ---- Jt, + J n when the conductivity, is non-uniform. However, if we neglect the non-uniformity of conductivity, these two parts may be estimated from the following relations: Jirr,~c ~ Jirr,y ~

Jp,~, --~ K~E~. JP,v ~- K~Ey

(8~

and KxuEv Jsol, u - ~ JH,v ~ --K~.uE~,. Jsol,x ~ ' J H . ~ ~

(9)

An example calculated in such a manner is shown in Fig. 5. It is found that Jsol iS only a few times larger than Jirr as a whole. This result suggests that the total current flowing in a closed circuit in the ionosphere is of the same order of magnitude as that flowing from the source to the sink and therefore as a field-aligned current. The distributions of calculated current density J are shown in Figs. 6(a) and (b) for the K 1 and K 2 models of conductivity, and for Cases SA-1 and 2 (left),/-1 and 2 (middle), and GA-1 and 2 (right), respectively, where the arrows indicate the magnitude and direction o( the current density in logarithmic scale. It is seen that the current patterns for Cases SA-I and 2 resemble those of the DP 2 system, and are quite different from those for Cases 1-t and 2. This suggests a difficulty in interpreting the DP 2 system by ionospheric winds alone, even if the enhancement of conductivity is taken into consideration. However, the ionospheric wind dynamo seems to have an equal importance to the solar wind dynamo. if we take q = 0.1 Coulomb which gives an applied field E o of about 0.5 mV/m at the pole.

1294

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J irr(SA-I)

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10-Semtl ~ Io"

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KOICHIRO

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F I G . 5. SEPARATION OF J INTO J i r r AND,]sol FOR CASE S A - 1 , WHERE J i r r FLOWS FROM THE SOURCE TO THE SINK, AND Jsol FLOWS IN A CLOSED CIRCUIT IN THE CURRENT SHEET.

If we compare this value with that of the dynamo field ED of about 2.6 mV/m, it is found that the solar wind dynamo is several times more effective than the ionospheric wind dynamo in producing current systems in the polar ionosphere. It is, however, to be noted that the ionospheric wind dynamo may have a significant effect on the current distribution, when the applied field is relatively weak. The distributions of J for the EB type field are shown in Figs. 7(a)-(c) for the K1, K2 and K3 models of conductivity, respectively. It is seen, in general, that the current patterns for Cases S B - 1 , 2 and 3 resemble a combination o f D P 1 and DP 2 systems, and are quite different from those for Cases I-1, 2 and 3, again suggesting a difficulty for interpreting the DP system by ionospheric winds alone. It may be interesting to note from the comparison between Cases SB's and GB's that the ionospheric wind dynamo has much more influence on the current pattern in the day-side than in the night-side. In other words, the generalized dynamo in the day-side is driven mainly by ionospheric winds, whereas that in the night-side is driven mainly by the solar wind. This result may suggest that the current in the electrojet region can be explained only by the solar wind dynamo, whereas the current distribution in the remaining region is much influenced by the ionospheric wind dynamo. In Fig. 7(c) an electrojet can be seen, where the conductivity is several times enhanced (see Fig. 1). In order to compare the calculated results with observations, a greater enhancement (about several times ten) of conductivity than the assumed one might be necessary. Geomagnetic variations associated with the current distributions obtained above are illustrated in Fig. 8(a) for Cases SA-1 (left) and SA-2 (right), and in Fig. 8(b) for Cases SB-1 (left) and SB-2 (right), where only the effect of Jsol is calculated, because the effect of Jirr is approximately cancelled by the effect of field-aligned currents, cu~ and the effect of induced currents within the Earth is ignored. It is seen from Fig. 8(a) that the magnetic variations of about 50 gammas (SA-1) to 100 gammas (SA-2) are expected in the polar cap region. Since this is the same order of magnitude as the observed DP 2 field, an applied

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A NUMERICAL STUDY OF POLAR IONOSPHERIC CURRENTS

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field E o of about 1.0 mV/m may be appropriate. This value does not agree with the observed field of about 10 mV/m on quiet days. If, however, we take the resultant field E s into account, a general agreement can be seen. This result strongly suggests that the effect of electric coupling between the ionosphere and the magnetosphere should be taken into account, as has already been pointed out by Bostrt~m, a6) Fejer, t~9~Swift, t32'aa), Akasofu and Meng, 14~)Vasyliunas, 134~Wolf/46) Oguti ~47~and others. In particular, Vasyliunas has proposed an interesting scheme of self-consistent calculation. It is seen from Fig. 8(b) that bay-type variations of about 200 gammas are enhanced to about 2000 gammas, when the conductivity is enhanced from K 1 to K s. This may be due to the fact that the effective conductivity is changed from the Pedersen conductivity to the Cowling conductivity, because of the local enhancement of conductivity. In order to make a more detailed comparison with observations, the following points should be taken into account: (1) More realistic distribution of the applied field and ionospheric conductivity. (2) Self-consistent calculation including electric coupling between the ionosphere and the magnetosphere. A study along this line is in progress, and the result will be published elsewhere. 6. CONCLUSION The main results obtained in the present study are as follows: (1) In the solar wind dynamo, the secondary polarization field caused by a non-uniform distribution of ionospheric conductivity is much larger than the primary applied field, suggesting its important effect on charged particles in the magnetosphere; whereas in the ionospheric wind dynamo, both of these fields are of the same order of magnitude. (2) In the solar wind dynamo, the total irrotational current flowing from the source 1o the sink as a field-aligned current is of the same order of magnitude as the total solenoidal current closing its circuit in the ionosphere. (3) The solar wind dynamo is, in general, more effective in producing polar current systems such as DP 1 and 2. In some cases, e.g. for a weak EA type field or in the day-side region of the EB type field, the ionospheric wind dynamo has an important effect on the resultant current distribution. (4) Geomagnetic variations caused by the EA type field are very similar to the observed Sq~ (or D P 2) type variations, and those caused by the EB type field resemble the observed bay (or D P 1) type variations. Very strong electrojets as usually observed in the auroral zone cannot be seen in the present calculation. The assumed slight enhancement of conductivity is not sufficient in generating the jet. For detailed comparison with observations, however, a further study is necessary in which the following points should be taken into consideration: (1) More realistic distribution of the applied electric field and ionospheric conductivity. (2) Self-consistent calculation including electric coupling between the ionosphere and the magnetosphere. Acknowledgements--One of us (H. M) is very much obliged to Drs. R. C. Whitten and I. G. Poppoff for their hospitality during his stay at the Ames Research Center, NASA. He also expresses his appreciation to the NAS-NRC for a Senior Research Associateship. Part of the numerical calculations were made on a FACOM 230-60 at the Data Processing Center of Kyoto University. 1. 2. 3. 4.

REFERENCES T. OBAYASHIand A. NISHIDA,Space Sci. Rev. 8, 3 (1968). A. NISmDAand S. KO~:UBUN,Rev. Geophys. Space Phys. 9, 417 (1971). T. RIS:ITAKE,Rep. lonosph. Res. Japan 2, 58 (1948). N. FUKUSHIMAand T. OGUTI,Rep. lonosph. Res. Japan7, 137 (1953).

1300

HIR()SHI MAEI)A and K t ) I ( H I R ( ) M A E K A W A

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

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