A unified view of solar wind-magnetosphere coupling functions

A UNIFIED

VIEW OF SOLAR WIND-MAGNETOSPHERE COUPLING FUNCTIONS u’. I). C;ONZAI,EZ Institute de Pesquisas

Espaciais.

INPE. S.J. Campos-SP.

Brasil

Abstract-It is shown that all widely used coupling functions about the solar wind magnetosphere interaction can bc derived as particular casts of general expressions for the electric field and the energy transfer at the magnctopause due to large scale reconnection. Although the solar wind electric field gets transmitted along the reconnection line (e.g. Gonzale7 and Morer. 1974, J. ,qeop/~~~:v.R~.I. 79. 4186). it is also shown that the net ell‘ect of this transmission in the magnetosphcrc seems to be associated only to the transverse cotnponcnt (daun to dusk direction) of this licld with respect to the gcomagnelic field at the magnetopausc. Furthermore. the most general expression for power transfer from the solar wind to the magnetosphere via a MHD proccas is shown to lead to sm~ilar limits for the coupling function as those obtained from the gcncral expression for the reconnection power w hen the particular values for the power Iam index introduced bvi Vasvliunas c’t (I/. (1983. Pltrrwr. S~xrw .%i. 30. 159). about the dcpcndence of the power tran~fcr on the i interplanetary AIf& Mach number, arc uacd.

I.

The tnost gcncral expression for the reconnection electric ficld (Gonzalc/ and Mozer. 1974; Sonnerup, 1974) is. in Gaussian units :

INTKODLCTION

Scvcral expressions for the coupling function about the solar wind&magnctospherc interaction have been reported in the literature. The reader is referred to Table I for some of the most widely cited ones. Some of these expressions have units of electric field, some of energy rate (power) and some have simpler units. like the classical B, function. With an interest in looking at these various functions from a unified point of view . pcncral expressions fol- the clcctric field and power transfer at the magnetopausc due to the reconnection process arc cxamincd in order to see if the former arc particular limits of such pcncral expressions. Another aim of the present \vork is to identify the conditions li>r which a very general cxprcssioti for power transfer from the solat wind to the magnctosphcrc. inkoiving a MHD process (Vasqliwas c’t rrl.. 19X2), lcads to similar limits for the coupling function ;IS those that can be obtained from the general cvpression for the reconnection power.

E(S, 0) = :, VB, F(S. II).

(1)

whcrc C’ is the solar wind speed ; B,, the transvcrsc component of the interplanetary mugnctic field : B, = (B f + BS) ’ ’ in solar-magnctospheric coordinates ; and F(S. 0) is a function that describes the projection of the magnctosheath clcctric field to the reconnection line. given by :

M’ith S = IB,;+/B,,I > I ; 0 the angle bctwccn the gcomagnetic field B,, and the mugnetoshcath magnetic field B,,, at the m~tgn~‘topausc and [I the angle bctuoen B,, and the reconnection line. Note that F(S. 0) = 0 for Scox 0 > I (Gonzaler and Mozcr, 1974). Gonzalc/ and Mn;rer (1974) also presented ;Lmodificd expression for E(S. 0) when gcomctrical effeectr (introduced by the shape of the daysidc mupnctopausc on the curvature of the reconnection line) were taken into account and rcprcscntcd by the correction function i(O) = (sin 0:2) ‘. Thus. with this modification

Table la gives the most commonly used functions for clcctric field tranafcr from the solar wind to the magnetosphere. It will be shown below that they correspond to difrercnt approximations of more general cxprcssions given by Gonzalez and Mozcr (1974), Sonnerup (I 974). Kan and Lee (I 979) and Gonzalez and Gonzalez (1981) for the clcctric field transfer at the magnctopausc via magnetic rcconncction.

E, (S. 0) = :T I’B, F(S, 0) (sin o/2) Gonzalez and Gonzalez nctospheric paramctcrs 621

’

(la)

( I98 I ) argued that some magcould be better related either

(p V ‘)

(pi”)’

Gonzalez and Mozer (1974) Doyle and Burke (1983) Bythrow and Potemra (1983)

Kan and Lee ( 1979) Gonzalez and Gonzalez ( 198 I ) Reiffcr f/l. (1981) Wygant (‘I Ol. (1983) Byfhrow and Potrmra (1983) Doyle and Burke (1983)

Wygnnt clt N/. (1983) Doyle and Burke (1983)

IVB, sin (0, 2)

VB, sin’ (0 2)

L’L1,sin’ (0 2)

“VU, sin’ (0 2)

’ ’ VBf sinJ (0,‘2)

(,)V’)“VB,

Doyle and Burke (1983)

V&

(1986) (1989)

et Cl/.

and Akasofu

Vasyliunas ct ul. (I 982) Bargdtze rf ~1. (1986) Gonzalez et al. (1989)

Vasyliunas (‘t 01. (1982) Gonzdle7 rl trl. (1989)

Gon~dlez

Murayama

Perreault

B;V. B’V

Holzer and Slavin (I 982) Baker et ~1. (1981)

Murayama and Hakamada (1975) Crooker ef u/. (1977) Baker c’f al. (1983) Baker e/ ~11.(1981) Holzer and Slavin (1982) B,V’,

BV’

Arnoldy (1971) Tsurutani and Meng (1972)

(c) Simple expressions

INTERACTION

&

MAGNETOSPHERE

(1978)

FOR THE SOLAR WIND

(b) Power related

FUNCTIONS

(O/2)

LJSLD COUPLING

c = VLiB’sin”

COMMONLY

Rostokcr (‘I ui. (1972) Burton c/ trl. (I 975) Holrcr and Slavin (1979)

Electric ticld related

1. Mos-I

vB,

(a)

TABLE

2 2 k

5

r;

r

A unified view of solar wind-magnetosphere to the transverse (dawn-dusk) or to the parallel component of the reconnection electric field with respect to B,;. These components are :

629

coupling functions

quantities. This electric field (with a typical value of 0.5 mV mm ‘) could be a source of particle acceleration at the magnetopause in the case that it does not get screened out.

4,(X 0) = ;I’R,(l-SCOSO) x (s-coso)~(l

fS’-2Scos0),

(3)

3. ELECTRIC

POWER

RELATED

COCPLING

FUNCTIONS

x (1 -scoso)/(l

+S’-2Scos1)).

(4)

The above expressions are general and allow different possible regimes tif S > 1. For the limit that S-, 1, namely IB,;I = jBul at the magnctopause. these expressions reduce to : 1

E( I, 0) = ~ VB, sin (o/2) C

(5a)

(5b)

E,,(l,0)

= h VB, sin’(f1/2)

E; (1,O) = k VR, sin (O/2) cos (O/2).

(5c)

(5d)

Expressions (5a), (5b) and (5~) can be identified with those given in Table la and were largely compared with several magnetospheric quantities, especially the polar cap potential (e.g. Reiff et (I/.. 1981 ; Wygant ct al., 1983: Doyle and Burke, 1983). The simple expression C’B,, extensively used as a canonical coupling function (e.g. Rostokcr c’t al., 1972; Burton c’t rd., 1975: Holzcr and Slavin, 1979). can bc thought of as a particular case of expression (5b) or (5a). when B, is not considered in the B-, component. Note that k’B, = (V x B( is the solar wind electric field. which is transmitted to the magnctosphcrc only if the reconnection process occurs at the 1Tl~~gll~tOpilllSC and along a “reconnection line” that is formed in this procc\~ In the absence of rcconneclion this clcctric ficld produces a flow only around the magnctosphcric “obstacle” (c.g. Gonzalez and Mozcr. 1974). The expression I/b, sin’ (012) of Table la. called the “intcrmcdiutc” coupling function by Wygant ct id. (I 983) in studies of the polar cap potential, is not associated to any of the clcctric ticld functions given above. However. it will bc idcntificd below with a particular case of an energy transfer function. Expression (5d). which refers to the component of the reconnection electric tield parallel to the geomagnetic field, has not been yet compared with magnetospheric

A general expression for the rate of energy transfer from the solar wind to the magnetosphere involving only a MHD flow (e.g. Vasyliunas et al.. 1982; Kan and Akasofu, 1982 ; Lee and Akasofu, 1984) is :

P=

const.pV’Lf,,f’(MA,O),

(6)

where p is the solar wind density, V is the solar wind speed, M, = (47~~)“ZV/B, is the solar wind Alfv&n Mach number (based on &), L,,. = (M,,/47ipV2)’ ’ is the Chapman--Ferraro scale length and MI, is the Earth’s magnetic dipole moment. Assuming a power law dependence of,f’(M:,O) on MA (Vasyliunas et cd., 1982) : ,f(Mi,

II) = M,;21G(0),

(7)

where G(O) is an unspecified function of II only and ZY the power law index. equation (6) can be written as :

P = c0nst.p’

3mzV7 ’ “BIG

for which the following particular cases, discussed Vasyliunas et al. (I 982), are of interest : forv = 1, for c( = l/2,

P = const. (p V’)

’ ’ VB, G(O)

P = const. (PV’)‘:“VR,G(~).

(8) by (9) (IO)

Equations (9) and (10) arc two of the expressions given in Table 1b, when G(0) = sin’ (0i2). Expression (10) has been discussed in the recent literature (c.g. Bargatze cut al., 19X6 ; Gonzalez clt trl.. 1989) in conncction with the gcomagnctic AI!_and D,, indices. If one assumes in equation (6) that the ChapmanFerraro length L, ,~remains constant at the magnetopause. equation (8) becomes :

P = const. 0’ ~’ I” Thus, for the samc particular one obtains : forz = I. forr/ = 1.2,

L”Rf”G(CI).

(1 I)

values of CItaken above

P = const. i/B:G(I))

(12)

P = const. (pV’) ’ ’ VR, G(0). (13)

When G(0) = sin’ (0/‘2) and L, t = L,, = constant. equation (12) reduces to the widely used coupling function i: (Perreault and Akasofu, 1978) if B-, is replaced by the IMF amplitude B. This latter substitution can be done if one defines the Aifvkn Mach

W. D.

630

number M, in terms of B instead of B,, When G(0) = 1,equation (13) reduces to the pressure dependent coupling function studied by Gonzalez et al. (1989). initially suggested by Murayama (1986). The approach given above. initially presented by Vasyliunas (~1al. (1982), dots not allow the understanding ofthe limits G(0) = sin’ (11,‘2)and G(O) = I, which have only been empirically used. This is because such an approach refers to a general MHD solar wind-magnetosphere interaction which dots not necessarily restrict itself only to the reconnection process, while such limits for G(O) result from this process (c.g. Gonzalez and Gonzalez, 198 1). The reconnection power at the mugnetopausc was originally defined by Gonzalez and Mozer (1974) in terms of the reconnection potential and the reconnection current, the latter being given by (c/27r)L~Bcj-B,I. where L is the radius of the dayside magnctopause, equivalent to L,, The reconnection potential is obtained from the reconnection electric field. expression (I), times the length of the rcconnection line. This length was approximated by Gonzalez and Mozer (1974) as 2L. Thus, because equation (2) can also be written (Gonzalez and Gonzalez, 1981) as :

F(S,O) = B,(l the expression

-ScosO)!‘/B,;-B,I

for the reconnection

P(S,fl) =I

VL’B,B,(l

Y(S,II) = sin’U(l-ScosU)j(l

(14)

with = (1 -ScosI))(S-cos(I)‘, (I +s~-2scos0),

(18)

For the limit when S + I one gets for P, P,, and P, : P(l,U) p,,(l,Q

-

VL’B,B,

sin’ (U/2)

_ VL’B, L& sin” (fI;‘2)

(l4a) (15a)

and P,(l,O)

_ VL’B, &, sin’(0/2cos’(0/2)

(17a)

when

L = L,, = constant and because, typically, BI& h B” (Gonzalez and Gonzalez, 1981) expression (15a) can be associated to the coupling function s (Perreault and Akasofu, 1981). Gonzalez and Mozer (1974) used B, = qB,. with y being an amplification factor of B,- across the magnctosheath. If this factor does not vary much and using L’ = (15a) reduces to Lf,. = const. (/IV’)- “j, equation another expression listed in Table 1b. namely : (pv’)-

’ ’ VB; sin’ (O/2).

(18)

In the limit of S + 1, namely B, = B,,, and using a zero order pressure balance at the magnetopausc, one can approximate B,, to B,; = const. (pV2) “?. With this approximation the other two expressions of Table 1b can also bc obtained from equation (15a). When IL* = Lf,. = const. (pV’) ’ ‘, one gets: sin’ (O/2).

When L = L,, = constant 0 = 180 ‘, one gets : (,,V’)’

Gonzalez and Gonralez (1981) argued that the reconnection electric field and current vectors can bc deparallel their transverse and composed in components. with respect to the geomagnetic field at the magnetopausc nose. and also derived the corresponding projection functions. Thus using those functions one can define expressions fol- the power associatcd to Lhosc transverse and parallel components. They are :

K(S.0)

+S’-22scosO).

(pV’)“‘VB,

power is : ~~SCOS~l).

GONZALEZ

and

B, --f B,.

‘I’B,.

(19) namely

(20)

For cases when the ram pressure of the solar wind, p V’, remains constant. expression (I 9) reduces to VB, sin4 (O/2). which can bc identified with the “intercoupling function (Table la) used by mediate” Wygant ct r/l. ( 1983) and by Doyle and Burke (1983). Expression (20) was initially used by Murayama (1986) and was shown by Gonzalez ct al. (1989) to correlate well with the ring current intensity, as measured by the /I,, index, during intense magnetic storms and during intervals with large solar windy ram pressure variability. Finally. cxprcssions (18), (19) and (20) can also be written in terms of the total IMF amplitude, B. instead of B,, if the Alfvi-n Mach number is based on R instead of on B,

(16)

and 4. SIMPLE

P,,(S, 0) = ; I’LLR, B,, Y(S, II), with

(17)

(‘OUPIJNG

KNCl‘IONS

Table lc shows some of the most commonly used interplanetary coupling functions among simple combinations of interplanetary parameters. They are the

A unified view of solar wind--m: Ignetosphere B,, VBj and C”B, functions, in addition to VB,. which has already been mentioned above in connection with coupling functions of the electric field type. The simplest function B,. originally tested by Arnotdy (1971) and Tsurutani and Meng (1972). can be understood as a particular case of the electric field function I/R, assuming that the variability of V is ncgtigibtc as compared with that of B,. The functions L’B, and VB’ can bc considered as particular cases of functions (1Sa). approximated by equation (I 8) and by the function i;. rcspcctivcty. when II = I80 , namely when the reconnection tine tics along the equator (c.g. Gonzalez and Mozcr, 1974). The function L”B, can also hc obtained from equation (I %I). with the approximation given in equation (20) and when the solar wind density is assumed conslant. The similar expression V’R is obtained when the Alfvi-n Mach number is defined in terms of B instead of B, classical

5. DISCIJSSION

It has been shown above that all (or at tcast most) coupling functions reported in the titcraturc for correlation studies with magnetospheric parameters can bc derived as particular cases of general expressions for the ctcctric field transfer and for the power transfer at the magnctopause due 10 large scale reconnection (Gonzalez and Mozer. 1974: Sonnerup. 1974: Gonzalez and Gonzalez, 1981). As shown in Table I, Ihe most successfully tested functions involve the angular function sin’ (O/2) for the electric field and sin’ (/j/2) for the power. Note that they do not refer to Ihe total reconnection etcctric licld and poher but to the transverse components of them as defined by expressions (3) and (15). respcctivcty. Thus. although ~hc solar uind clcctric fictd gets ~ransmilted along ~hc tilted rcconncclion tine (c.g. GonzateL and Morer. 1974). the net effccl of this transmission in the magnetosphcrc seems to bc associatcd onl! Lo the transvcrsc componcnL of this fictd (dawn to dusk direction). Expression (t7a). which refers to the power obtained I-I-om IIIC parattet conponcnts of the reconnection ctcctric lietd and current, could be compared with puramctcrs ~hs~ mcasurc the aurora] activity at the daysidc. It is in(ercsting 10 obscrvc that the most gencrat cxprcssion for ~hc power 11.ansfcr from the solat- wind 10 the magnetosphere via ;I MHD process. originally suggested by Vasyliunas c’t ol. (tYX2) and given by equations (6) and (8), suggcsls forms of the coupling function similar to those oblaincd from the reconnection power cxprcssions when ~atucs of the power law index of equation (8) are equal to 1/2 and

coupling

631

functions

to I. Those similar functions (18). (t9), (20), respectively.

are (9), (I 0), (I 3) and

.4ck,rol~,/~,r!yer,7c,~7/--This work was p;u%ally supported by the “Fundo N’acional de Desenvolvimento Cientifico e Technolhgico” of Brarll. REFERENCES

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