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On the ring current energy injection rate
Planer. Space Sci., Vol. 31. No. 8, pp. 901 91 I, 1983 Printed in Great Britam
c
ON THE RING CURRENT
ENERGY
INJECTION
00324633/83 $3.04 + 0.00 1983 Pergamon Press Ltd.
RATE
L. C. LEE, G. CORRICKand S.-I. AKASOFU Geophysical
Institute,
University
of Alaska,
Fairbanks,
AK 99701, U.S.A.
(Received 2 February 1983) Abstract-Assuming that the formation of the ring current belt is a direct consequence of an enhanced crosstail electric field and hence of an enhanced convection, we calculate the total ring current kinetic energy (KR) and the ring current energy injection rate (U,) as a function of the cross-tail electric field (Ecr); the cross-tail electric field is assumed to have a step function-like increase. The loss of ring current particles due to recombination and charge-exchange is assumed to be distributed over the whole ring current region. It is found that: (1) the steady-state ring current energy K, is approximately linearly proportional to E,,; (2) the characteristic time t, for K, to reach the saturation level is 3-4 h ; (3) the injection rate U, is proportional to EC, where b N 1.33-1.52 ; and (4) the characteristic time rp for U, to reach the peak value is l-2 h and the peak U, value is 50% higher than the steady-state value. Smce /? is now determined specifically for an enhanced convection, an observational determination of the relationship between E,, (or 4cr) and U, is essential to a better understanding of ring current formation processes. If the observed b is greater than 1.5, additional processes (e.g. an injection of heavy ions from the ionosphere to the plasma sheet and subsequently to the ring current region) may be required. INTRODUCTION
time of the injection rate is about l-2 h and the peak injection rate can be 50% more than the steady-state injection rate.
One of the major features of geomagnetic storms is the injection of energetic particles into the ring current belt region and the total energy consumption or dissipation rate U, in the inner magnetosphere during geomagnetic storms consists mostly of the ring current energy injection rate U, (e.g., Akasofu, 1981). Thus, in order to understand the relationship between the total dissipation rate U, and the solar wind energy input rate to the magnetosphere, it is important to formulate and calculate the ring current energy injection rate U, as a function of the cross-tail potential drop, which in turn can be related to the solar wind parameters for a given open topology of the magnetosphere (e.g., Sonnerup, 1974 ; Hill, 1975 ; Kan and Lee, 1979 ; Reiff et al., 1981). Recently, Lee et al. (1982) have calculated the steadystate ring current energy injection rate U, as a function of the cross-tail potential drop under the simplified assumption that the loss of the ring current particles occurs only at the point nearest to Earth along their trajectories. The loss of the charged ring current particles is due to recombination and charge-exchange (e.g., Tinsley and Akasofu, 1982). The purpose of the present study is to calculate time-dependent, as well as the steady-state, injection rates for the case in which the loss of the ring current particles is distributed over the whole ring current region. It is found that the functional dependence of the injection rate on the cross-tail potential drop can be quite different from that obtained in the previous paper, depending on the radial dependence of the mean lifetime of the ring current particles. It is further found that the characteristic rise
FORMULATION In the present study, we consider the injection of particles from the magnetotail into the ring current belt region as an initial value problem. If the cross-tail potential is maintained constant after the sudden increase, the injection rates and total energy deposited in the ring current belt region will eventually reach steady-state values. In the following, we briefly describe (a) the magnetic and electric field models, (b) the boundary conditions, (c) the mean lifetime model and (d) the injection of the ring current particles in our calculation. (a) Magnetic and electricfield models The magnetic and electric field models used in the present calculation are the same as in the previous paper (Lee et al., 1982). The magnetic field is assumed to be a dipole field and the field in the equatorial plane is given by B(r) = &(rolr)3
(1)
where r is the radial distance from the center of the Earth and B, is the magnetic field at some reference distance ro. The electric field in our model is assumed to consist of a uniform cross-tail electric field Ec, and the corotation electric field. The electric potential in the equatorial plane of the magnetosphere is given by (e.g., 901
902
Roederer,
L. C.
1970)
LEE et al.
x = -b. In our calculation, b = 15 Re and c = 20 Re.
e$(r,P)
C2r sin p
= ?-
(b) Boundary conditions The magnetospheric boundary is approximated by a parabola in the equatorial plane with vertex at (a,O), focus at (0,O) and passing through the points (-b,c) and (-b, - c) in the solar-magnetospheric coordinates as shown in Fig. 1. The contour of the magnetospheric boundary can be written as
The injection boundary
a = 10 Re,
(2)
where C, = 91.5 keV Re, C, = eE,,, eis the magnitude of an electron charge, Re is the Earth’s radius and fi is the azimuthal angle with respect to the x-axis (/l = 90” at dusk) in the solar-magnetospheric coordinates. The first term in (2) is due to the corotational electric field and the second term is due to the uniform cross-tail electric field. The cross-tail potential difference $cT is approx. 40 kV for CZ = 1 keV/Re. We assume that the cross-tail electric field E,, is increased from low values to high values at time t = 0.
x+(a+b)(y/C)*
we choose
= a.
(3)
is assumed to be in the plane
(c) Mean lifetime model of the ring current particles The mean lifetime of the ring current particles is assumed to be a function of the radial distance r and is given by 7(r)
=
7,&/rJ.
(4)
In our calculation, the parameters are chosen as zA = 0.54 h, rA = 6 Re and u = l-4, which are close to the observed values (Tinsley and Akasofu, 1982). (d) Znjection of particles The particles are injected from the plane x = -b = - 15 Re. The particle flux density in the injection plane is given by F,(r) = no&
= G&-/B(r)
(5)
where V’, is the initial particle convection velocity, r = (b2 + yg)“’ and y, is the initial particle position. In our calculation, we have divided the plane into many regions, each with a width Ay. In each region, packets of particles are continuously injected into the inner magnetosphere, each occurring with time At apart.
SUN
FIG.
1. THE &~A(~ETOSPHER~C
The parmeterS
uSed are : a =
BOUNDARY
10 Re, b =
AND THE INJECTION SURFACE IN THE EQUATORIAL PLANE.
15 Re and c = 20 Re. Positions times with At apart are also plotted.
of the injected particles
at
Va~OuS
903
On the ring current energy injection rate Some which apart packet
of the particle trajectories are shown in Fig. 1, in the positions of particles at various time with At are plotted. The number of particles in each at t = 0 is given by N(t = 0) = n,(E,,/B(r))L,Ay
At
(6)
where L, = 10 Re is the plasma sheet thickness in the initial injection plane. After injection, the number of particles in each packet at time t + At is given by
N(t +At) = N(t) where z(r) is the mean lifetime in (4) and r is the particle position at time t after injection. The number of particles in each packet is very large. For n, = 0.5 cme3, L, = 10 Re, Ay = l/16 Re, EC, = 2 kV/Re, At = 0.05 h, we have N(t = 0) rr 4 x 10z4 for each packet. In our calculation, the particles in the injection plane are assumed to have the same magnetic moment pO, which can be expressed in terms of the particle energy W,, defined as w,, = POROO
(8)
where Boo = B(r = 15 Re) N 12~. W,, can be considered as the kinetic energy of a particle with p. at a radial distance of r = 15 Re. The contribution of electrons to the ring current energy is known to be small and will be neglected in the present calculation. RESULTS
The particles are continuously injected from the initial plane x = - 6, starting at t = 0, based on the field and mean lifetime models described in the last section. The total kinetic energy K,, the ring current energy injection rate U, and the loss rate Lean be calculated as a function of time t for various values of the cross-tail potential drop. The total kinetic energy K, in the ring current region can be written as RR@) = c dl sss
n,(r, r)%(r, t) dr
(9)
where the volume integral is over the bounded ring current region as shown in Fig. 1, n,(r, t) is the particle number density and W,(r, t) is the particle kinetic energy of the ccth species. The ring current energy K, is related to the energy injection rate U, and the loss rate L as =, ___ at
= u,-L
= u,+u,,-L
(10)
where u,
= u,+u,,
(11)
u,= -1
ss
norw,V; dA = U,-
U,
(12)
OL U EM =
qanaVi, - E dr
(13)
a zsss
L=C
(14) (n, WJd k 1 sss qdlis the particle charge, V, is the particle velocity, E is the electric field, t, is the mean lifetime of the ccth species, and dA is the area element on the surface of the ring current region. The injection rate U, in (11) consists of two parts : (i) the first part U, represents the total energy flux flowing through the surface area of the ring current region, and (ii) the second part U,, represents the work done by the electromagnetic field within the ring current region. The surface energy flux U, in (12) can also be written as the sum of two parts: (i) the incoming part, U,, contributed through the initial injection plane, and (ii) through the parabolic the outgoing part, -U,, boundary shown in Fig. 1. It is seen from (10) that in steady state the ring current energy injection rate equals the loss rate, i.e., U, = L. In the present calculation, we consider only the contribution from ions, since the ions are known to carry most of the ring current. Curve A in Fig. 2a shows the ring current energy K, as a function of time t after injection at t = 0. The parameters used in the calculation are : EC, = 3 kV/Re, r = 2h,r, = 6Re,cc = 2, Woo = 2keV,n, = 0.5cm-3, and L, = 10 Re. The characteristic time t, for the ring current energy K, to reach the saturation value is approx. 4 h. The ring current energy K, can be related to the Dst index by the formula KR(t) = ADst(t)
(15)
where A = 4 x 10 l3 J/y (e.g., Akasofu, 1981). The saturation ring current energy shown in Fig. 2a is - 5 x lOi J, which corresponds to Dst N 125~. Curve B in Fig. 2a is the ring current energy K, as a function of time t for the case in which the initial (t = 0) distribution is given by the steady-state distribution produced by a cross-tail electric field E,, = 1 keV/Re and Ecr = 3 kV/Re is applied for t > 0. As shown in the figure, the characteristic time t, is reduced to - 3 h. The corresponding energy injection rates U,, U,,, US and the loss rate L are shown in Figs. 2b, 2c, 2d and 2e, respectively. As shown in Fig. 2b, the peak value for the ring current energy injection rate U, is - 7.6 x 10” W, while the steady-state rate is only 5.5 x 10” W. The time t, for the injection rate to reach the peak value is - 2 h in case (A) and is - 1 h in case (B). Note that as shown in Fig. 2b and 2e the steady-state injection rate U, equals the steady-state loss rate L.
904
t (hrsf RINGC~RK~~TENERG~K~ASAFUN~TI~NOFTHETIME~APTERINIECTION:~ASE (A):No PARTICLES INSIDETHERINGCURRENTREGIONATt =OANDCASE(B):THERINGCURRENTREGIONATt=OISFILLEDUPBYTHE STEADY-STATEPARTICLEDISTRIBUTlONPRODL'CEDBYTHEPRESENCBOFAI7ROSS-TAILFIELD~r~ = f kV/Kf.
Flc.2a.T~~
A cross-tail
electric field E,,
= 3 keV/Re is applied at t > Oin both cases. Al1 other parameters text.
As shown in Fig. 2c, U,, reaches a peak value -8.3 x 10” W and has a steady-state value -7 x 10” W. The input rate UEM is due to the work done by the electromagnetic field in the ring current region and is supplied mostly by the solar wind energy input to the magnetosphere, since the power supplied by the
0
0
t 2
Earth’s corotation electric field can be shown to be very small, as pointed out by Vasyliunas (1982, private communication). In this connection, he pointed out also that it is improper to divide the injection rate U, into two parts U,, and U,, as in our previous paper. As shown in Fig. 2d, the surface injection rate Us has an
, 4
/ 6
, 8
10
t (hrs) FIG.~~.THERING
are given in the
CURRENTENERGYINJBCTIONRATE
U, ASAEUNCTIONOFTIME~.
905
On the ring current energy injection rate
1°c
t (hrs) FIG. 2C.
u,,
AS A FUNCTION
initial value - 1.3 x 10’ ’ W and reaches an asymptotic value - - 1.5 x 10” W due to the loss through the parabolic boundary shown in Fig. 1. Figures 3a, 3b and 3cshow, respectively, the steadystate ring current energy K,, the injection rate Us and U,, as a function of the cross-tail electric field ECT for W,, = 1,2 and 3 keV. The other parameters used are : n, = 0.5 cm-j, L, = 10 Re, T,, = 2 h, rA = 6 Re, and
0.5
1
OF
t.
a = 2. Note that the steady-state loss rate L is equal to the injection rate U, in Fig. 3b. The ring current kinetic energy K, is approximately linearly proportional to the cross-tail electric field. The best fit is K, = 1.77 x 1Or5 E2T4 J for W,, = 2 keV, where E,, is in units of kV/Re. On the other hand, we have U, = L = 1.24x 10” E,$’ W and U,, = 1.58 x 10” E1.36 W for W,, = 2 keV. The injection rate U, is near&qua1
1
I
1
0.40.32
0.2-
-0.3-0.4 -o.50
I
I
I
I
2
4
6
8
t (hrs) FIG. 2d. Tm
SURFACE
INJECTION
RATE
U,
“S TIME t.
10
L.
906
c.
LEE
et al.
t (hrs) FIG.~~.THE
L AS AETJNCTIONOFTIME~.
LOSSRATE
to UEMIsince Us is relatively small. Note that the ring current energy K, is proportional to (L,n,). Therefore, K, in Fig. 3a can be written in the general form, for W,, = 2 keV, as K, = 3.54 x 10’4L,noE~~4 J
(16)
where L, is measured in units of Re and n, in units of cme3.
I
161
The total ring current energy K, and the ring current energy injection rate also depend on the mean lifetime of the ring current particles in (4). Figures 4a and 4b show, respectively, KR and U, as a function ofthecrosstail electric fietd E,, for rA = 6 Re, a = 2 and tA = 0.5, 1,2,4 h. All other parameters used are the same as those in Fig. 3. As shown in Fig. 4a, the ring current energy K, increases with increasing lifetime Z~ On the other hand,
I
I
I
E&V/R,) FIG.~~.THERINGCURRENTENERGY
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ANDY keV.
All other parameters are given in the text.
EC.FOR W,, = 1,2
901
On the ring current energy injection rate
0 r) a4-
_;:;:j_:::_
I 1
OO
FIG.
3b.
THE INJECTION
I 4
I I 2 3 E,,(kV/R,) RATE u,
the injection rates U, (= L) is a decreasing function of zA for ~~ 2 1 h and is an increasing function of rA 6 1 h as shown in Fig. 5b. This result can be understood as follows. As the mean lifetime of the ring current particles decreases, the ring current energy loss rate L (hence the injection rate U,) will increase. However, as shown in Fig. 1, the charged particles gain energy when drifting toward the dawndusk meridian (x = 0) plane
AS A FUNCTION
5
E,,.
and lose energy when drifting away from the x = 0 plane. Therefore, as the mean lifetime or zA decreases further (zA < 1 h),most ring current particles will be lost before they reach the dawn-dusk meridian plane and obtain maximum energy, resulting in the decrease of the total loss rate L. Figures 5a and 5b show KR and UR as a function of the cross-tail electric field E,, for various functional
9-
FIG. 3c. UI,,
OF
i
AS A FUNCTION
OF &,.
908
L. C.
LEE
et al.
FIG.~~.THESTEADY-STATERINGCURRENTENERGY K, ASAI%JNCTIONOFTHECROSS-TAILELECTRICFIELD&FOR zA = 0.5,1,2 AND 4 h.
The mean lifetime of ring current particles is given by I
forms of z(r) in (4). The mean lifetimes are specified as : rA = 6 Re, zA = 2 h and c( = 1, 2, 3 and 4. All other parameters are the same as in Fig. 4. As shown in Fig. 5a, K, is not very sensitive to c(. However, Fig. 5b shows that as for a larger value of CL,the injection rate UR increases faster as E,, increases. Let the injection rate
16
I
FIG. 4b. THESTEADY-STATEINJECTION
I
= zA(r/rA)a,where a = 2, rA = 6 Re.
UR be of the following form U, = AE& W
(17)
where EC, is in units of kV/Re. The best fits for the curves shown in Fig. 5b are: (i) A = 1.24 x 10” and /I = 1.33fora = l,(ii)A = 1.24x 10r’andB = 1.37for
I
U,ASAFUNCTIONOF&FOR
I
ZA =
0.5,1,2
AND
4h.
909
On the ring current energy injection rate
16 i
FIG.%.THESTEADY-sTATERINGCURRENTENERGY
~RASAFUNCTIONOFcROSS-TAILELECTR~CFIELD&~ 1,2,3~~~4. The mean lifetime is given by 7 = ra(r/rA)“, where zA = 2 h and rA = 6 Re.
FOR
a=
CC= 2, (iii) A = 1.19 x 1O1’ and p = 1.43 for tt = 3, and(iv)A~l.lO~lO~~and~=l.52for~=4. Tt should be pointed out that the ring current energy K,, the injection rate U,, U,, and the loss rate L are all proportional to (L,no). Therefore, (17) can he written in the general form as U,, = A’L,n,E&
W
FIG.S~.THESTEADY-STATS
(18)
ISJFCTION
RATE U,
where A’ = 0.2A, It is interesting those obtained in The injection rate can be written as
L, is in units of Re and n, in cmF3. to compare the present results with our previous paper (Lee et al., 1982). [I, as given in (20) of the earlier paper
U, = 1.14 x 10” (E~~+O.42E~~) N 1.63 x 10” ‘!P.CT
ASA FUNCTION OF &FOR
w
~1= 1,2,3 ANDY.
W (19)
910
L. c. LEE et al.
for& = IORe,n, = 0.~~-3.Theinj~ctionratein(19) is the maximum injection rate considered in Lee et al. (1982) and is a factor of 1.4 higher than those shown in Fig. 5b.
SUMMARY
AND DISCUSSION
(a) Summary Although many of the parameters used in this paper are not accurately known at the present time, it is hoped that we have adequately covered their possible ranges and the results represent a fairly realistic situation. Based on a simple dipole magnetic field model, we have calculated the ring current energy K,, the injection rate U,, UEMand the loss rate L as a function of the crosstail electric field EC, and the mean lifetime z of the ring current particles. The main results are summarized as follows :
(1) The steady-state ring current energy K,, as given in (16), is approximately linearly proportional to Ecr. The characteristic time t, for I<, to reach the saturation level is typically 3-4 h. (2) The steady-state ring current energy injection rate U, as given in (18) is proportional to E& where p 2: 1.33-1.52. These values of b may be compared with that obtained in the previous paper (b = 1.44). (3) The characteristic time for UR to reach the peak value is t, 2: 1-2 h. The peak U, value is typically 50% more than the steady-state value. (4) The contribution of the surface term Us to the injection rate U, is small compared to U,,, the contribution from the electromagnetic field in the ring current region. (b) Discussions Burton et al. (1975) reported UR cc EC, based on a correlation study of the solar wind parameters and the ring current energy injection rate. On the other hand, a study of the energy coupling between the solar wind and the magnetosphere by Perreault and Akasofu (1978) and Akasofu (1981) implies that UR = 0.7~ cc E&, where E is the solar wind-magnetosphere coupling function used by Perreault and Akasofu ; see Vasyliunas et al. (1982) and Lee et al. (1982). Reiff et al. (1981) provided a correlation study of the polar cap potential drop and the solar wind parameters, but a direct correlation study between the polar cap (or crosstail potential drop &-r) and the ring current energy injection (U,) has not been made yet. We suggest that such a direct correlation study would provide valuable information to our understanding of the ring current injection processes.
The injection rate in (18) can also be written as
where LY is the width of the plasma sheet in units of Re, d, is the cross-tail potential drop in units of kV, A’: 0.22-0.25 and /I = 1.33-1.52. We have from (20) U, cc q$., provided that the parameters L,,, L, and no are not sensitive to the solar wind parameters or (PCT.If the future correlation study of U, and c#+-=indicates that @is in the range of the values (1.3-1.5) obtained here, the formation of the ring current belt can be simply understood as a direct consequence of an enhanced cross-tail electric field as discussed in this paper. If the observational study shows otherwise, possible explanations of the discrepancy are :(i) the parameters L,, L, and n, in (20) cannot be assumed as constants during geomagnetic storms ; they may vary as a function of&r or the solar wind parameters. In fact, the width of L, of the plasma sheet may decrease during geomagnetic storms, it is sensitive to the solar wind pressure (cf. Akasofu, 1977, p. 193) and such a change tends to increase p. Both no and L, are known to fluctuate considerably during geomagnetic storms, but their general trends are not clear. If L, tends to decrease and n,, tends to increase during geomagnetic storms, we would expect that the change in L, decreases the value of p while the change in n, increases 8. (ii) Other physical processes may also play an important role in the ring current particle injection. It has been shown that energetic ions (H+ and O+) of ionospheric origin are at times the major ions in the ring current belt (e.g. Peterson et at., 1981). This suggests that these ions supplied from the ionosphere are fed to the plasma sheet and subsequently accelerated as they are injected into the ring current. Such a process tends to increase fi from our theoretical values (1.33-1.52). Acknowledgement-We would like to thank Professor V. M. Vasyliunas for his helpfui comments on our work. This work was supported in part by the Atmospheric Sciences Section of the National Science Foundation under grants ATM 8216605 and ATM 81-15321 to the University of Alaska. REFERENCES
Akasofu, S-I. (1977) Physics ojhlagnetospheric Snrbstorms.D. Reidel, Dordrecht. Akasofu, S.-I. (1981) Energy coupling between the solar wind and the magnetosphere. Space Sci. Rev. 28,121. Burton, R. K.,McPherron, R. L. and Russell, C. T. (1975) An empirical re!ationship between interplanetary conditions and Dst. J. geophys. Res. 80,4204. Hill, T. W. (1975) Magnetic merging in a collisionless plasma. J. geophys. Rex 80,4689. Kan,~.I. R. and Lee, L. C. (1979) Energy coupling function and solar wind-magnetosphere dynamo. Geophys. Res. Lett. 6, 577. Lee, L. C., Kan, J. R. and Akasofu, S-1. (1982) Ring current
On the ring current energy injection rate and solar wind-magnetosphere energy coupling. Planet. Space Sci. 30,627. Perreault, P. and Akasofu, S.-I. (1978) A study ofgeomagnetic storms. Geophys. J. R. astr. Sac. 54,547. Peterson, W. K.,Sharp, R. D., Shelley, E. G. and Johnson, R. G. (1981) Energetic ion composition of the plasma sheet. f. geoph ys. Res. 86,76 1. Reiff, P. H., Spiro, R. W. and Hill, T. W. (1981) Dependence of polar cap potential drop on interplanetary parameters. J. geophys. Rex 86,7639.
energy injection
rate
911
Roederer, J. G. (1970) Dynamics O~Geo~ag~etica~~yTrapped Radiation. Springer, Berlin, Heidelberg, New York. Sonnerup, B. U. 0. (1974) Magnetopause reconnection rate. J. geophys. Res. 19. 1546. Tinsley, B. A. and Akasofu, S.-I. (1982) A note on lifetime of the ring current particles. Planet. Space Sci. 30,733. Vasyliunas, V. M., Kan, J. R., Siscoe, G. L. and Akasofu, S-i. (1982) Scaling relations governing magnetospheric energy transfer. Planet. Space Sci. 30, 359.
c
ON THE RING CURRENT
ENERGY
INJECTION
00324633/83 $3.04 + 0.00 1983 Pergamon Press Ltd.
RATE
L. C. LEE, G. CORRICKand S.-I. AKASOFU Geophysical
Institute,
University
of Alaska,
Fairbanks,
AK 99701, U.S.A.
(Received 2 February 1983) Abstract-Assuming that the formation of the ring current belt is a direct consequence of an enhanced crosstail electric field and hence of an enhanced convection, we calculate the total ring current kinetic energy (KR) and the ring current energy injection rate (U,) as a function of the cross-tail electric field (Ecr); the cross-tail electric field is assumed to have a step function-like increase. The loss of ring current particles due to recombination and charge-exchange is assumed to be distributed over the whole ring current region. It is found that: (1) the steady-state ring current energy K, is approximately linearly proportional to E,,; (2) the characteristic time t, for K, to reach the saturation level is 3-4 h ; (3) the injection rate U, is proportional to EC, where b N 1.33-1.52 ; and (4) the characteristic time rp for U, to reach the peak value is l-2 h and the peak U, value is 50% higher than the steady-state value. Smce /? is now determined specifically for an enhanced convection, an observational determination of the relationship between E,, (or 4cr) and U, is essential to a better understanding of ring current formation processes. If the observed b is greater than 1.5, additional processes (e.g. an injection of heavy ions from the ionosphere to the plasma sheet and subsequently to the ring current region) may be required. INTRODUCTION
time of the injection rate is about l-2 h and the peak injection rate can be 50% more than the steady-state injection rate.
One of the major features of geomagnetic storms is the injection of energetic particles into the ring current belt region and the total energy consumption or dissipation rate U, in the inner magnetosphere during geomagnetic storms consists mostly of the ring current energy injection rate U, (e.g., Akasofu, 1981). Thus, in order to understand the relationship between the total dissipation rate U, and the solar wind energy input rate to the magnetosphere, it is important to formulate and calculate the ring current energy injection rate U, as a function of the cross-tail potential drop, which in turn can be related to the solar wind parameters for a given open topology of the magnetosphere (e.g., Sonnerup, 1974 ; Hill, 1975 ; Kan and Lee, 1979 ; Reiff et al., 1981). Recently, Lee et al. (1982) have calculated the steadystate ring current energy injection rate U, as a function of the cross-tail potential drop under the simplified assumption that the loss of the ring current particles occurs only at the point nearest to Earth along their trajectories. The loss of the charged ring current particles is due to recombination and charge-exchange (e.g., Tinsley and Akasofu, 1982). The purpose of the present study is to calculate time-dependent, as well as the steady-state, injection rates for the case in which the loss of the ring current particles is distributed over the whole ring current region. It is found that the functional dependence of the injection rate on the cross-tail potential drop can be quite different from that obtained in the previous paper, depending on the radial dependence of the mean lifetime of the ring current particles. It is further found that the characteristic rise
FORMULATION In the present study, we consider the injection of particles from the magnetotail into the ring current belt region as an initial value problem. If the cross-tail potential is maintained constant after the sudden increase, the injection rates and total energy deposited in the ring current belt region will eventually reach steady-state values. In the following, we briefly describe (a) the magnetic and electric field models, (b) the boundary conditions, (c) the mean lifetime model and (d) the injection of the ring current particles in our calculation. (a) Magnetic and electricfield models The magnetic and electric field models used in the present calculation are the same as in the previous paper (Lee et al., 1982). The magnetic field is assumed to be a dipole field and the field in the equatorial plane is given by B(r) = &(rolr)3
(1)
where r is the radial distance from the center of the Earth and B, is the magnetic field at some reference distance ro. The electric field in our model is assumed to consist of a uniform cross-tail electric field Ec, and the corotation electric field. The electric potential in the equatorial plane of the magnetosphere is given by (e.g., 901
902
Roederer,
L. C.
1970)
LEE et al.
x = -b. In our calculation, b = 15 Re and c = 20 Re.
e$(r,P)
C2r sin p
= ?-
(b) Boundary conditions The magnetospheric boundary is approximated by a parabola in the equatorial plane with vertex at (a,O), focus at (0,O) and passing through the points (-b,c) and (-b, - c) in the solar-magnetospheric coordinates as shown in Fig. 1. The contour of the magnetospheric boundary can be written as
The injection boundary
a = 10 Re,
(2)
where C, = 91.5 keV Re, C, = eE,,, eis the magnitude of an electron charge, Re is the Earth’s radius and fi is the azimuthal angle with respect to the x-axis (/l = 90” at dusk) in the solar-magnetospheric coordinates. The first term in (2) is due to the corotational electric field and the second term is due to the uniform cross-tail electric field. The cross-tail potential difference $cT is approx. 40 kV for CZ = 1 keV/Re. We assume that the cross-tail electric field E,, is increased from low values to high values at time t = 0.
x+(a+b)(y/C)*
we choose
= a.
(3)
is assumed to be in the plane
(c) Mean lifetime model of the ring current particles The mean lifetime of the ring current particles is assumed to be a function of the radial distance r and is given by 7(r)
=
7,&/rJ.
(4)
In our calculation, the parameters are chosen as zA = 0.54 h, rA = 6 Re and u = l-4, which are close to the observed values (Tinsley and Akasofu, 1982). (d) Znjection of particles The particles are injected from the plane x = -b = - 15 Re. The particle flux density in the injection plane is given by F,(r) = no&
= G&-/B(r)
(5)
where V’, is the initial particle convection velocity, r = (b2 + yg)“’ and y, is the initial particle position. In our calculation, we have divided the plane into many regions, each with a width Ay. In each region, packets of particles are continuously injected into the inner magnetosphere, each occurring with time At apart.
SUN
FIG.
1. THE &~A(~ETOSPHER~C
The parmeterS
uSed are : a =
BOUNDARY
10 Re, b =
AND THE INJECTION SURFACE IN THE EQUATORIAL PLANE.
15 Re and c = 20 Re. Positions times with At apart are also plotted.
of the injected particles
at
Va~OuS
903
On the ring current energy injection rate Some which apart packet
of the particle trajectories are shown in Fig. 1, in the positions of particles at various time with At are plotted. The number of particles in each at t = 0 is given by N(t = 0) = n,(E,,/B(r))L,Ay
At
(6)
where L, = 10 Re is the plasma sheet thickness in the initial injection plane. After injection, the number of particles in each packet at time t + At is given by
N(t +At) = N(t) where z(r) is the mean lifetime in (4) and r is the particle position at time t after injection. The number of particles in each packet is very large. For n, = 0.5 cme3, L, = 10 Re, Ay = l/16 Re, EC, = 2 kV/Re, At = 0.05 h, we have N(t = 0) rr 4 x 10z4 for each packet. In our calculation, the particles in the injection plane are assumed to have the same magnetic moment pO, which can be expressed in terms of the particle energy W,, defined as w,, = POROO
(8)
where Boo = B(r = 15 Re) N 12~. W,, can be considered as the kinetic energy of a particle with p. at a radial distance of r = 15 Re. The contribution of electrons to the ring current energy is known to be small and will be neglected in the present calculation. RESULTS
The particles are continuously injected from the initial plane x = - 6, starting at t = 0, based on the field and mean lifetime models described in the last section. The total kinetic energy K,, the ring current energy injection rate U, and the loss rate Lean be calculated as a function of time t for various values of the cross-tail potential drop. The total kinetic energy K, in the ring current region can be written as RR@) = c dl sss
n,(r, r)%(r, t) dr
(9)
where the volume integral is over the bounded ring current region as shown in Fig. 1, n,(r, t) is the particle number density and W,(r, t) is the particle kinetic energy of the ccth species. The ring current energy K, is related to the energy injection rate U, and the loss rate L as =, ___ at
= u,-L
= u,+u,,-L
(10)
where u,
= u,+u,,
(11)
u,= -1
ss
norw,V; dA = U,-
U,
(12)
OL U EM =
qanaVi, - E dr
(13)
a zsss
L=C
(14) (n, WJd k 1 sss qdlis the particle charge, V, is the particle velocity, E is the electric field, t, is the mean lifetime of the ccth species, and dA is the area element on the surface of the ring current region. The injection rate U, in (11) consists of two parts : (i) the first part U, represents the total energy flux flowing through the surface area of the ring current region, and (ii) the second part U,, represents the work done by the electromagnetic field within the ring current region. The surface energy flux U, in (12) can also be written as the sum of two parts: (i) the incoming part, U,, contributed through the initial injection plane, and (ii) through the parabolic the outgoing part, -U,, boundary shown in Fig. 1. It is seen from (10) that in steady state the ring current energy injection rate equals the loss rate, i.e., U, = L. In the present calculation, we consider only the contribution from ions, since the ions are known to carry most of the ring current. Curve A in Fig. 2a shows the ring current energy K, as a function of time t after injection at t = 0. The parameters used in the calculation are : EC, = 3 kV/Re, r = 2h,r, = 6Re,cc = 2, Woo = 2keV,n, = 0.5cm-3, and L, = 10 Re. The characteristic time t, for the ring current energy K, to reach the saturation value is approx. 4 h. The ring current energy K, can be related to the Dst index by the formula KR(t) = ADst(t)
(15)
where A = 4 x 10 l3 J/y (e.g., Akasofu, 1981). The saturation ring current energy shown in Fig. 2a is - 5 x lOi J, which corresponds to Dst N 125~. Curve B in Fig. 2a is the ring current energy K, as a function of time t for the case in which the initial (t = 0) distribution is given by the steady-state distribution produced by a cross-tail electric field E,, = 1 keV/Re and Ecr = 3 kV/Re is applied for t > 0. As shown in the figure, the characteristic time t, is reduced to - 3 h. The corresponding energy injection rates U,, U,,, US and the loss rate L are shown in Figs. 2b, 2c, 2d and 2e, respectively. As shown in Fig. 2b, the peak value for the ring current energy injection rate U, is - 7.6 x 10” W, while the steady-state rate is only 5.5 x 10” W. The time t, for the injection rate to reach the peak value is - 2 h in case (A) and is - 1 h in case (B). Note that as shown in Fig. 2b and 2e the steady-state injection rate U, equals the steady-state loss rate L.
904
t (hrsf RINGC~RK~~TENERG~K~ASAFUN~TI~NOFTHETIME~APTERINIECTION:~ASE (A):No PARTICLES INSIDETHERINGCURRENTREGIONATt =OANDCASE(B):THERINGCURRENTREGIONATt=OISFILLEDUPBYTHE STEADY-STATEPARTICLEDISTRIBUTlONPRODL'CEDBYTHEPRESENCBOFAI7ROSS-TAILFIELD~r~ = f kV/Kf.
Flc.2a.T~~
A cross-tail
electric field E,,
= 3 keV/Re is applied at t > Oin both cases. Al1 other parameters text.
As shown in Fig. 2c, U,, reaches a peak value -8.3 x 10” W and has a steady-state value -7 x 10” W. The input rate UEM is due to the work done by the electromagnetic field in the ring current region and is supplied mostly by the solar wind energy input to the magnetosphere, since the power supplied by the
0
0
t 2
Earth’s corotation electric field can be shown to be very small, as pointed out by Vasyliunas (1982, private communication). In this connection, he pointed out also that it is improper to divide the injection rate U, into two parts U,, and U,, as in our previous paper. As shown in Fig. 2d, the surface injection rate Us has an
, 4
/ 6
, 8
10
t (hrs) FIG.~~.THERING
are given in the
CURRENTENERGYINJBCTIONRATE
U, ASAEUNCTIONOFTIME~.
905
On the ring current energy injection rate
1°c
t (hrs) FIG. 2C.
u,,
AS A FUNCTION
initial value - 1.3 x 10’ ’ W and reaches an asymptotic value - - 1.5 x 10” W due to the loss through the parabolic boundary shown in Fig. 1. Figures 3a, 3b and 3cshow, respectively, the steadystate ring current energy K,, the injection rate Us and U,, as a function of the cross-tail electric field ECT for W,, = 1,2 and 3 keV. The other parameters used are : n, = 0.5 cm-j, L, = 10 Re, T,, = 2 h, rA = 6 Re, and
0.5
1
OF
t.
a = 2. Note that the steady-state loss rate L is equal to the injection rate U, in Fig. 3b. The ring current kinetic energy K, is approximately linearly proportional to the cross-tail electric field. The best fit is K, = 1.77 x 1Or5 E2T4 J for W,, = 2 keV, where E,, is in units of kV/Re. On the other hand, we have U, = L = 1.24x 10” E,$’ W and U,, = 1.58 x 10” E1.36 W for W,, = 2 keV. The injection rate U, is near&qua1
1
I
1
0.40.32
0.2-
-0.3-0.4 -o.50
I
I
I
I
2
4
6
8
t (hrs) FIG. 2d. Tm
SURFACE
INJECTION
RATE
U,
“S TIME t.
10
L.
906
c.
LEE
et al.
t (hrs) FIG.~~.THE
L AS AETJNCTIONOFTIME~.
LOSSRATE
to UEMIsince Us is relatively small. Note that the ring current energy K, is proportional to (L,n,). Therefore, K, in Fig. 3a can be written in the general form, for W,, = 2 keV, as K, = 3.54 x 10’4L,noE~~4 J
(16)
where L, is measured in units of Re and n, in units of cme3.
I
161
The total ring current energy K, and the ring current energy injection rate also depend on the mean lifetime of the ring current particles in (4). Figures 4a and 4b show, respectively, KR and U, as a function ofthecrosstail electric fietd E,, for rA = 6 Re, a = 2 and tA = 0.5, 1,2,4 h. All other parameters used are the same as those in Fig. 3. As shown in Fig. 4a, the ring current energy K, increases with increasing lifetime Z~ On the other hand,
I
I
I
E&V/R,) FIG.~~.THERINGCURRENTENERGY
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ANDY keV.
All other parameters are given in the text.
EC.FOR W,, = 1,2
901
On the ring current energy injection rate
0 r) a4-
_;:;:j_:::_
I 1
OO
FIG.
3b.
THE INJECTION
I 4
I I 2 3 E,,(kV/R,) RATE u,
the injection rates U, (= L) is a decreasing function of zA for ~~ 2 1 h and is an increasing function of rA 6 1 h as shown in Fig. 5b. This result can be understood as follows. As the mean lifetime of the ring current particles decreases, the ring current energy loss rate L (hence the injection rate U,) will increase. However, as shown in Fig. 1, the charged particles gain energy when drifting toward the dawndusk meridian (x = 0) plane
AS A FUNCTION
5
E,,.
and lose energy when drifting away from the x = 0 plane. Therefore, as the mean lifetime or zA decreases further (zA < 1 h),most ring current particles will be lost before they reach the dawn-dusk meridian plane and obtain maximum energy, resulting in the decrease of the total loss rate L. Figures 5a and 5b show KR and UR as a function of the cross-tail electric field E,, for various functional
9-
FIG. 3c. UI,,
OF
i
AS A FUNCTION
OF &,.
908
L. C.
LEE
et al.
FIG.~~.THESTEADY-STATERINGCURRENTENERGY K, ASAI%JNCTIONOFTHECROSS-TAILELECTRICFIELD&FOR zA = 0.5,1,2 AND 4 h.
The mean lifetime of ring current particles is given by I
forms of z(r) in (4). The mean lifetimes are specified as : rA = 6 Re, zA = 2 h and c( = 1, 2, 3 and 4. All other parameters are the same as in Fig. 4. As shown in Fig. 5a, K, is not very sensitive to c(. However, Fig. 5b shows that as for a larger value of CL,the injection rate UR increases faster as E,, increases. Let the injection rate
16
I
FIG. 4b. THESTEADY-STATEINJECTION
I
= zA(r/rA)a,where a = 2, rA = 6 Re.
UR be of the following form U, = AE& W
(17)
where EC, is in units of kV/Re. The best fits for the curves shown in Fig. 5b are: (i) A = 1.24 x 10” and /I = 1.33fora = l,(ii)A = 1.24x 10r’andB = 1.37for
I
U,ASAFUNCTIONOF&FOR
I
ZA =
0.5,1,2
AND
4h.
909
On the ring current energy injection rate
16 i
FIG.%.THESTEADY-sTATERINGCURRENTENERGY
~RASAFUNCTIONOFcROSS-TAILELECTR~CFIELD&~ 1,2,3~~~4. The mean lifetime is given by 7 = ra(r/rA)“, where zA = 2 h and rA = 6 Re.
FOR
a=
CC= 2, (iii) A = 1.19 x 1O1’ and p = 1.43 for tt = 3, and(iv)A~l.lO~lO~~and~=l.52for~=4. Tt should be pointed out that the ring current energy K,, the injection rate U,, U,, and the loss rate L are all proportional to (L,no). Therefore, (17) can he written in the general form as U,, = A’L,n,E&
W
FIG.S~.THESTEADY-STATS
(18)
ISJFCTION
RATE U,
where A’ = 0.2A, It is interesting those obtained in The injection rate can be written as
L, is in units of Re and n, in cmF3. to compare the present results with our previous paper (Lee et al., 1982). [I, as given in (20) of the earlier paper
U, = 1.14 x 10” (E~~+O.42E~~) N 1.63 x 10” ‘!P.CT
ASA FUNCTION OF &FOR
w
~1= 1,2,3 ANDY.
W (19)
910
L. c. LEE et al.
for& = IORe,n, = 0.~~-3.Theinj~ctionratein(19) is the maximum injection rate considered in Lee et al. (1982) and is a factor of 1.4 higher than those shown in Fig. 5b.
SUMMARY
AND DISCUSSION
(a) Summary Although many of the parameters used in this paper are not accurately known at the present time, it is hoped that we have adequately covered their possible ranges and the results represent a fairly realistic situation. Based on a simple dipole magnetic field model, we have calculated the ring current energy K,, the injection rate U,, UEMand the loss rate L as a function of the crosstail electric field EC, and the mean lifetime z of the ring current particles. The main results are summarized as follows :
(1) The steady-state ring current energy K,, as given in (16), is approximately linearly proportional to Ecr. The characteristic time t, for I<, to reach the saturation level is typically 3-4 h. (2) The steady-state ring current energy injection rate U, as given in (18) is proportional to E& where p 2: 1.33-1.52. These values of b may be compared with that obtained in the previous paper (b = 1.44). (3) The characteristic time for UR to reach the peak value is t, 2: 1-2 h. The peak U, value is typically 50% more than the steady-state value. (4) The contribution of the surface term Us to the injection rate U, is small compared to U,,, the contribution from the electromagnetic field in the ring current region. (b) Discussions Burton et al. (1975) reported UR cc EC, based on a correlation study of the solar wind parameters and the ring current energy injection rate. On the other hand, a study of the energy coupling between the solar wind and the magnetosphere by Perreault and Akasofu (1978) and Akasofu (1981) implies that UR = 0.7~ cc E&, where E is the solar wind-magnetosphere coupling function used by Perreault and Akasofu ; see Vasyliunas et al. (1982) and Lee et al. (1982). Reiff et al. (1981) provided a correlation study of the polar cap potential drop and the solar wind parameters, but a direct correlation study between the polar cap (or crosstail potential drop &-r) and the ring current energy injection (U,) has not been made yet. We suggest that such a direct correlation study would provide valuable information to our understanding of the ring current injection processes.
The injection rate in (18) can also be written as
where LY is the width of the plasma sheet in units of Re, d, is the cross-tail potential drop in units of kV, A’: 0.22-0.25 and /I = 1.33-1.52. We have from (20) U, cc q$., provided that the parameters L,,, L, and no are not sensitive to the solar wind parameters or (PCT.If the future correlation study of U, and c#+-=indicates that @is in the range of the values (1.3-1.5) obtained here, the formation of the ring current belt can be simply understood as a direct consequence of an enhanced cross-tail electric field as discussed in this paper. If the observational study shows otherwise, possible explanations of the discrepancy are :(i) the parameters L,, L, and n, in (20) cannot be assumed as constants during geomagnetic storms ; they may vary as a function of&r or the solar wind parameters. In fact, the width of L, of the plasma sheet may decrease during geomagnetic storms, it is sensitive to the solar wind pressure (cf. Akasofu, 1977, p. 193) and such a change tends to increase p. Both no and L, are known to fluctuate considerably during geomagnetic storms, but their general trends are not clear. If L, tends to decrease and n,, tends to increase during geomagnetic storms, we would expect that the change in L, decreases the value of p while the change in n, increases 8. (ii) Other physical processes may also play an important role in the ring current particle injection. It has been shown that energetic ions (H+ and O+) of ionospheric origin are at times the major ions in the ring current belt (e.g. Peterson et at., 1981). This suggests that these ions supplied from the ionosphere are fed to the plasma sheet and subsequently accelerated as they are injected into the ring current. Such a process tends to increase fi from our theoretical values (1.33-1.52). Acknowledgement-We would like to thank Professor V. M. Vasyliunas for his helpfui comments on our work. This work was supported in part by the Atmospheric Sciences Section of the National Science Foundation under grants ATM 8216605 and ATM 81-15321 to the University of Alaska. REFERENCES
Akasofu, S-I. (1977) Physics ojhlagnetospheric Snrbstorms.D. Reidel, Dordrecht. Akasofu, S.-I. (1981) Energy coupling between the solar wind and the magnetosphere. Space Sci. Rev. 28,121. Burton, R. K.,McPherron, R. L. and Russell, C. T. (1975) An empirical re!ationship between interplanetary conditions and Dst. J. geophys. Res. 80,4204. Hill, T. W. (1975) Magnetic merging in a collisionless plasma. J. geophys. Rex 80,4689. Kan,~.I. R. and Lee, L. C. (1979) Energy coupling function and solar wind-magnetosphere dynamo. Geophys. Res. Lett. 6, 577. Lee, L. C., Kan, J. R. and Akasofu, S-1. (1982) Ring current
On the ring current energy injection rate and solar wind-magnetosphere energy coupling. Planet. Space Sci. 30,627. Perreault, P. and Akasofu, S.-I. (1978) A study ofgeomagnetic storms. Geophys. J. R. astr. Sac. 54,547. Peterson, W. K.,Sharp, R. D., Shelley, E. G. and Johnson, R. G. (1981) Energetic ion composition of the plasma sheet. f. geoph ys. Res. 86,76 1. Reiff, P. H., Spiro, R. W. and Hill, T. W. (1981) Dependence of polar cap potential drop on interplanetary parameters. J. geophys. Rex 86,7639.
energy injection
rate
911
Roederer, J. G. (1970) Dynamics O~Geo~ag~etica~~yTrapped Radiation. Springer, Berlin, Heidelberg, New York. Sonnerup, B. U. 0. (1974) Magnetopause reconnection rate. J. geophys. Res. 19. 1546. Tinsley, B. A. and Akasofu, S.-I. (1982) A note on lifetime of the ring current particles. Planet. Space Sci. 30,733. Vasyliunas, V. M., Kan, J. R., Siscoe, G. L. and Akasofu, S-i. (1982) Scaling relations governing magnetospheric energy transfer. Planet. Space Sci. 30, 359.