Energetics of the magnetosphere during the magnetic storm

Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 429 – 446

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Energetics of the magnetosphere during the magnetic storm Y.I. Feldsteina , L.A. Dremukhinaa , A.E. Levitina , U. Mallb;∗ , I.I. Alexeevc , V.V. Kalegaevc a IZMIRAN,

Troitsk, Moscow Region, 142190, Russia fur Aeronomie, 37191 Katlenburg-Lindau, D-37191, Germany c Institute of Nuclear Physics, Moscow State University, 119899 Moscow, Russia

b Max-Planck-Institute

Received 11 May 2001; received in revised form 14 November 2002; accepted 9 December 2002

Abstract The most detailed studies of the energetic budget of the magnetosphere during the magnetic storms were done on the basis of the paraboloid model using the November 23–27, 1986 and May 6 –8, 1988 magnetic storms. Calculations have shown that the energy injected in the course of the magnetic storms into the inner magnetosphere and ionosphere of both hemispheres amounts to ∼ 0:9–2.2% of the solar wind kinetic energy on the magnetospheric cross section. The total energy injected into the magnetosphere from a distance of 60RE in the tail down to the ionosphere is ∼ 4:0–7.5% of the solar wind kinetic energy during the main phase of the two magnetic storms. The injected energy into the tail ETL is 1.03–1.18 of the total energy input into the inner magnetosphere and ionosphere of both hemispheres at the main phase of the two storms. The decay parameter for the energy stored in the magnetospheric tail is ∼5 h. The total energy dissipated in the ionosphere of both hemispheres, in the inner magnetosphere and in the tail during the two storms, is 1:85 × 1017 and 3:24 × 1017 J, respectively. The total energy input into the magnetosphere is calculated to be 1:77 × 1017 and 3:16 × 1017 J. The discrepancies of 0:08 × 1017 and 0:10 × 1017 J amount to 4.3% and 3.1% of the total energy input and characterize the accuracy of the magnetospheric energy budget calculation. In the magnetotail the balance between the injected and dissipated energy of ∼1:09 × 1017 J for one storm and ∼1:7 × 1017 J for the other is preserved as well. We conclude that one-half of the energy which is injected into the magnetosphere from the solar wind during the storms enters the magnetotail and dissipates there. The coupling parameter PA is widely considered to be a measure of the energy dissipation in the inner magnetosphere. The dissipation energy in the inner magnetosphere UT = UJ + UA + UDR is deBned as the sum of the contributions of the Joule dissipation UJ , the energy of auroral particle precipitation UA , and the energy injection into the ring current UDR . In this paper, we Bnd that PA in the two storms investigated is substantially diCerent from UT . The energy injected into the ring current region at the main phase of the storm amounts to ∼ 1016 J. It is nearly 3 or 4 times smaller than the energy input into the magnetosphere via the Beld-aligned currents or the energy dissipated in the ionosphere by Joule dissipation. The energy injected into the magnetosphere is transferred mainly into processes diCerent from the ring current generation. During the development of intensive auroral electrojets, the energy dissipation in the magnetotail and the increase in the energy of the tail current system occur simultaneously. The energy dissipation in the inner magnetosphere and ionosphere US occurs not only at the expense of energy previously stored in the magnetotail, but rather at the expense of energy that is injected into the near-Earth tail. This energy transfer from the solar wind into the magnetotail and the energy dissipation in the ionosphere increased simultaneously. Thus, during disturbances in the magnetosphere, simultaneously loading–unloading and directly driven processes occur. Loading- and unloading processes manifest themselves both in the storage of the solar wind

∗ Corresponding author. E-mail address: [email protected] (U. Mall).

c 2003 Elsevier Science Ltd. All rights reserved. 1364-6826/03/$ - see front matter
doi:10.1016/S1364-6826(02)00339-5

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energy in the magnetotail and the ring current, and subsequent dissipation. The directly driven processes become manifest in the direct dissipation of the energy which enters into the ionosphere through large-scale Beld-aligned current systems. c 2003 Elsevier Science Ltd. All rights reserved.  Keywords: Magnetosphere; Magnetic storm

1. Introduction Every large-scale physical phenomena in the Earth’s environment is driven either by an energy injection or through energy dissipation (Stern, 1984; Weiss et al., 1992; Baker et al., 1997). An examination of the magnetospheric energy budget reveals a great deal about the involved processes. The energy of large-scale current systems is stored in magnetic Belds and is dissipated in the magnetosphere through various processes: (1) through Joule heating in the upper atmosphere by ionospheric currents, (2) through charged particles precipitation, (3) through ejection of plasmoids via the remote part of the magnetotail into the solar wind, (4) through charge–exchange interaction of the ring current ions with neutral particles of the exosphere and (5) through energetic ion ejection across the dayside magnetopause during their convective drift around the Earth. Using the paraboloid model of the magnetospheric magnetic Beld, we are able to calculate the intensity of the magnetic Belds for the individual contributions of the diCerent current systems. This circumstance is substantial for studying such dynamic processes as magnetic storms. Magnetic Belds of diCerent current systems change with diCerent characteristic times; this is taken into account in the model by its input parameters, which have clear physical and geometrical meaning and are determined by using low- and high-altitude satellite data measured in the magnetosphere and in the solar wind. The paraboloid model used for the investigation of the diCerent sources of the magnetospheric magnetic Beld is described in Alexeev et al. (1996), Kalegaev et al. (1998), Dremukhina et al. (1999) and Alexeev and Feldstein (2001). The model allows one to study the energy budget variations during a magnetic storm. Until now, only the energy budget of individual components, such as the energy injection into the ring current, the Joule dissipation in the ionospheric current, and the energy Jux connected with the precipitation of auroral particles has been studied in detail (Lu et al., 1998). The energy of the Beld-aligned currents and the magnetotail current systems was only roughly estimated. Stern (1984) and Baker et al. (1997) determined the averaged values of injection and dissipation for typical geomagnetic situations. The power delivered to the magnetosphere and the energy stored in the magnetosphere, which is necessary for the generation and maintenance of large-scale current systems, can be calculated using the paraboloid model relations presented by Alexeev (1997). Kalegaev et al. (2001) used

these relations in order to determine the separate parts of the magnetosphere energetic budget. First, we will analyse the data of the November, 1986 storm in detail, and then we will present the main results for May 1988 storm. 2. Energetics of the solar wind and the magnetosphere during a magnetic storm 2.1. Magnetic storm on November 23–27, 1986 The magnetic storms on November 23–27, 1986 were selected to study the magnetic Beld changes and energy storage in the magnetic Belds of the diCerent magnetospheric current systems. During this storm, detailed satellite measurements and ground-based observations of the magnetic Beld, plasma and energetic particles in the magnetosphere and on the Earth’s surface were made. These data allow one to calculate the input parameters for the paraboloid model. The solar wind plasma parameters and the Interplanetary Magnetic Field (IMF) data are available as well (King, 1989). The top panel of Fig. 1 shows variations of the solar wind pressure on the magnetopause PSW . During this magnetic storm there were several time intervals of pressure increases. The other panels of Fig. 1 show input parameters of the paraboloid model: R1 —the geocentric distance up to the subsolar point of the dayside magnetopause, R2 —the geocentric distance up to the current sheet inner boundary in the magnetospheric tail at the midnight, 0 —the magnetic Jux in the tail lobe, and DR—the magnetic Beld on the Earth’s surface, generated by the ions of the ring current. The method was used to calculate the input parameters of the paraboloid model was described extensively in Dremukhina et al. (1999). The course of a magnetic storm is usually described by the planetary magnetic activity indexes AE and Dst. The AE index characterizes the intensity of the auroral electrojets at ionospheric altitudes and is sensitive to the appearance of magnetospheric disturbances, such as storms and substorms. The Dst index is determined by current systems on the magnetopause DCF, by the ring current in the inner magnetosphere DR, and the tail current in the tail plasma sheet DT. It is traditionally assumed that the Beld of the ring current makes the main contribution to the Dst-variations. Based on the Dst-variation, a development of a magnetic storm is divided into three phases: the initial phase, the main phase and the recovery phase. The initial phase characterizes the

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2000

10

AE, nT

PSW, nPa 1500 5

1000 0 12

500

R 1 , RE

10

0 0

8 6 12

Dst, nT

-40

R 2 , RE

10

-80 8 6

-120

4

Ukin/1012, W

40

10

5

20

Flux, 10 8 Wb 0 0

DR, nT 0 -50

300

Uem/1010, W -100 0

12 23

24

12 24

24

12 25

24

12 26

24

12 27.11.86

200

UT

100

Fig. 1. The solar wind dynamic pressure (PSW ) and the paraboloid model parameters: geocentric distance up to the subsolar point of the magnetopause on the dayside (R1 ), geocentric distance up to the current sheet inner boundary in the magnetospheric tail at the midnight (R2 ), the magnetic Jux in the tail lobe ( 0 ), the magnetic Beld on the Earth’s surface, generated by the ring current ions (DR) during November 23–27, 1986, magnetic storm

magnetic Beld state before the main phase begins. During the main phase, theDst-value sharply decreases (the ring current develops). During the recovery phase, the Dst values return to pre-storm levels (the ring current dissipates). Usually, the data of four magnetic observatories located at diCerent longitudes are used for Dst-variation calculation. In this paper, the data of seven magnetic observatories located at the diCerent longitudes were used for a more accurate elimination of the longitudinal asymmetry of the low-latitudinal magnetic Beld variations (Dremukhina et al., 1999). Based on an examination of the Dst-variation in Fig. 2, we classify the storm into the following two phases: the interval

0 0

12

0

12

0

12 UT

0

12

0

12

Fig. 2. Variations of geophysical parameters and the solar wind energetics during the magnetic storm: the auroral electrojet index (AE) and the geomagnetic Beld decreasing at low latitudes kin and the electromagnetic (Dst-index). The kinetic energy Jux Usw emag energy Jux Usw of the solar wind on the magnetosphere cross section, taking into account changes in the magnetopause geometric sizes at the subsolar point during November 23–27, 1986 magnetic storm.

from 1500 UT on November 24 until 1800 UT on November 25, referred to as the storm main phase (with a minimum of Dst = −129 nT), and the interval from 1900 UT on November 25 until 2300 UT on November 27, referred to as the recovery phase. We consider the interval from 1200 UT on November 23 until 1400 UT on November 24 as magnetic disturbances before the main phase of the storm (substorms).

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AE indices are usually calculated using magnetometer data of an observatory chain located at auroral latitudes (Kamei et al., 1991). During magnetic storms, auroral electrojets shift equatorwards. Such a shift leads to an underestimation of the electrojet intensities and consequently, to a decrease in the AE index. During magnetic storms, it is, therefore, necessary to also use data from subauroral observatories (Feldstein et al., 1994,1997). For the storm under investigation with a moderate value of Dst of about −100 nT, it is enough to use additional data from the observatories at latitudes close to 60◦ : Lerwik, Archangelsk, Sitka. The AE index presented in Fig. 2 is calculated using data from these observatories, but the contribution of these observatories is very small, and does not change the standard AE index. 2.2. Energetics of the solar wind We proceed by making a quantitative estimation of the energy transferred by the plasma Jux and the solar wind magnetic Beld onto a magnetosphere with a crosssection S (m2 ). If n denotes the density of solar wind particles (particles=cm3 ), and V denotes the speed of the solar wind (km/s), one obtains for the rate of kinetic energy transfer kin USW (W):
3 V kin = 8:35 × 10−7 n S: USW 100 The rate of transfer of electromagnetic energy in (W) onto the surface S is given by
 V emag = 0−1 VB2 S = 7:9 × 10−8 USW B2 S; 100 where the total magnetic IMF Beld B is given in (nT) and 0 = 4 × 10−7 H=m is the magnetic permeability of free space. The cross section along the dawn-dusk meridian of a magnetopause, with a shape of a paraboloid, is given by emag kin S =×(1:5R1 )2 . USW and USW were calculated, taking into account the size changes of the paraboloid magnetopause in time intervals of 1 h. The results of this calculation are presented in Fig. 2 (right panels). The kinetic energy of the solar wind is greater by one order of magnitude than the electromagnetic energy. The electromagnetic and kinetic power are highest in the interval before the storm main phase. During the main phase, the mean power of electromagnetic energy kin decreases to 50%; during the recovery phase, USW decreases emag to 45%, and USW decreases to 20% compared to the values that were before the main phase. 2.3. Energetics of the magnetosphere in traditional approximation Akasofu (1981) devised a quantitative estimate of the total magnetospheric energy consumption rate. This quantity, UT , consists of three major terms: UT = Uj + UA + UDR :

The term UJ is the hemispherically averaged Joule heating rate, and UA is the hemispherically auroral particle energy precipitation rate. Akasofu (1981) suggested that the Brst two terms had the following relationship to the AE index: Uj = 2 × 108 AE; UA = 1 × 108 AE with AE given in nanoteslas. The dissipation power is given in W. The model IZMEM (Levitin et al., 1984; Feldstein and Levitin, 1986) is used to calculate the part of UJ that is determined by the quasi-steady-state convection controlled by viscous friction and by the IMF vector component Ujcon (Faermark et al., 1985). The term Ujcon is the sum of inBnitesimal Joule dissipation qi = p E 2 , integrated over the entire high-latitude area, where, p is the altitude integrated Pedersen conductivity of the ionosphere, E is the electric Beld intensity, according to IZMEM model. Every inBnitesimal Joule dissipation takes into account the corresponding value of the IMF Bz and By components (Feldstein et al., 1986). During the magnetic storm, the value UJcon is increased by Joule dissipation due to unsteady-state substorm current systems, denoted by Ujsub . We assume with Baumjohann and Kamide (1984) that Ujsub = 0:32 × 109 AE; where Ujsub is in Watts. We obtain an upper limit of the total real Joule dissipation because Ujsub already includes the contribution from the convection component of Joule dissipation. The energy release due to auroral particle precipitation in both hemispheres can be determined from the relation 
 AE UA = 1:75 × + 1:6 × 1010 ; 100 where UA is in W, proposed by Spiro et al. (1982). The values of Ujcon ; Ujsub ; UA are presented in Fig. 3. The ring current energy injection rate UDR is given as a sum of two terms: the Brst term UDR1 describes the energy storage rate in the ring current, the second one UDR2 characterizes the energy dissipation from the ring current which is caused by diCerent processes (Liemohn et al., 1999; Ebihara and Ejiri, 2000)
 dDR DR UDR = −0:74 × 1010 + dt  = UDR1 + UDR2 : UDR is given in Watts with DR given in nanoteslas; dt and  are in hours and  is the ring current decay time scale. The sign of dDR=dt deBnes whether energy is stored (negative) or dissipated (positive). The ring current magnetic Beld on the Earth’s surface DR is determined using AMPTE/CCE ion measurements in the magnetosphere and is shown in Fig. 1. The sharp gradients arise from the fact that the satellite spent only a fraction of its orbit in the ring current.

Y.I. Feldstein et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 429 – 446 20

UJcon, 1010 W

15 10 5 0 60

sub

Uj , 1010 W

40

20

0 30

UA, 1010 W

20 10 0 20

UDR, 1010 W

10 0 -10

UDR 2, 1010 W

10

433

To calculate the UDR values presented in Fig. 3, we used DRmod and  = 7:7 h, as in Burton et al. (1975). Using a  value that is smaller than 1 h leads to an unjustiBed increase in UDR (Zwickl et al., 1987). For comparison reasons one may check the table of  values used in the literature, as presented in Feldstein (1992). The right panel of Fig. 3 shows the variations of the injection rate to the ring current UDR = UDR1 + UDR2 and the dissipation from the ring current UDR2 , as well as the power of integrated dissipation in the inner magnetosphere and in the ionosphere US = UJcon + UJsub + UA + UDR2 : It is necessary to keep in mind that the dissipation function US diCers from the traditionally accepted function UT by the value of the energy stored in the ring current during the magnetic storm main phase. This energy is taken from the solar wind, but is then dissipated from the ring current with a long time delay mainly during the recovery phase of the magnetic storm. It is, therefore, necessary to calculate the energy input from the solar wind and the ionosphere to the ring current region UDR as a sum of UDR1 and UDR2 during the main phase. The energy dissipated from the ring current is determined alone by the term UDR2 . 2.4. Additional components of the energetic budget in the magnetosphere

5

0

US, 1010 W

100

50

0 0

12 23

24

12 24

24

12 25

24

12 26

24

12 27.11.86

UT

Fig. 3. Variation of the traditional dissipative constituents of the magnetosphere energy budget: Joule losses due to the convective movements in the ionosphere (Ujcon ), Joule losses during substorms (Ujsub ), the energy losses during the energetic particle Juxes precipitation (UA ). Injection to the ring current UDR = UDR1 + UDR2 ; the energy dissipation from the ring current UDR2 = −0:74 × 1010 (DRm )=; total dissipation of the energy in the inner magnetosphere and in the ionosphere US = Ujcon + Ujsub + UA + UDR2 .

To obtain smoother UDR variations, we used Dst to calculate the magnetic Beld DR values from relation DRmod = Dst − DCF − DT: The values of DCF and DT were determined through the paraboloid model by Dremukhina et al. (1999). The energy dissipation from the ring current depends substantially on the decay parameter value , which is generally not a constant during a magnetic storm.

2.4.1. Energy of @eld-aligned currents Another way to deliver the energy from the solar wind to the magnetosphere is through the generation of Beld-aligned currents located at the ionosphere altitudes in Region 1, as Brst discussed by Iijima and Potemra (1982). We take into account only the Beld-aligned current of Region 1 for our calculation, because the generator of the large-scale current system is located in this region. The currents in Region 2 are secondary. The currents in Region 1 are generated under the interaction of the solar wind plasma with the Earth’s magnetosphere and are directed by the magnetic Beld lines from the magnetosphere periphery to the polar cap boundary. The energy delivered to the magnetosphere by the Beld-aligned currents is calculated as the product of the integral Beld-aligned current intensity I and the potential diCerence N’ across the polar cap UFAC = I N’ 103 ; where UFAC is in W, I is in A, N’ is in kV. The integral Beld-aligned current I = j × S is calculated as a product of the current density and the area occupied by this current. The Beld-aligned current density j , which is controlled by the parameter of the solar wind, is given by the relation of Iijima and Potemra (1982)  1=2  1=2 j = 0:0328 n · V · BT · sin + 1:4; 2 wherej is in
A=m; n is in per cm3 ; V is in km/s; BT = Bz2 + By2 is in nT, where Bz and By denote the IMF

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magnetic components, the angle  measured between the positive Z-axis and the IMF vector in the Y –Z plane is given by  = arctg By =Bz for Bz ¿ 0;  = 180◦ − arctg |By |=|Bz | for Bz ¡ 0. If the size of the Region 1 Beld-aligned current along the meridian is N$ ∼ 1:5◦ (Potemra, private communication, 1997), then the power transferred by this current has to be calculated from the relation  
1=2  9 1=2 UFAC = 4:2 × 10 0:0328 n · V · BT · sin + 1:4 2 ×sin m N’; where m is the polar cap boundary angle radius. N’ along the dawn-dusk meridian is determined by the simple relation N’ = 25–12 Bz for Bz ¡ 0, where N’ is in kV, Bz is in nT (Dremukhina et al., 1990). For Bz ¿ 0, the potential diCerence across the polar cap has to be calculated from the relation (ReiC and Luhmann, 1986) N’ = 6:7 + 0:047VBT sin3

 ; 2

 where N’ is in kV, V is in km/s, BT = Bz2 + By2 is in nT. Fig. 4 shows the energy delivered to the magnetosphere by the Beld-aligned currents UFAC during the November 23–27, 1986 magnetic storm. One sees that the power delivered to the magnetosphere from the solar wind by FAC undergoes considerable Juctuation. In the beginning and at the end of the magnetic disturbances, before the storm main phase, UFAC is about 1010 W, while it reaches a value of 25 × 1010 W during the substorm maximum. Two distinct peaks, 46 × 1010 and 38:5 × 1010 W, characterize the main phase. The power decreases to 9 × 1010 W at 1000 UT on November 25. During the recovery phase, the UFAC curve Juctuates with a sharp peak of 24 × 1010 W at 0600 UT on November 27. The UFAC variation shows the same features as the variation of the dissipative part of the energy budget 50

UFAC, 1010 W 40

at the ionospheric altitudes. However, the relative magnitudes of the extrema are not kept. UFAC show the greatest similarity with UJcon but in this case, the UFAC extremum is about Bve times greater than the UJ extremum during the recovery phase. 2.4.2. Energy of Chapman–Ferraro current system The energy of dipole geomagnetic Beld Bd and the magnetic Beld Bsd associated with the Chapman–Ferraro current at the magnetopause can be written as 

1 2 E1 = Bd2 dv + Bsd dv + 2 Bd Bsd dv ; 20 Vm Vm Vm where the integration extends over the whole volume of the magnetosphere Vm (see Alexeev, 1997). The Brst term may be calculated as the diCerence between the energy of a dipole in the inBnite volume Ed and the dipole energy outside the magnetopause E0 . To Bnd Ed the integrations are performed over all distances from the centre of the Earth to inBnity. We obtain Ed =

4 2 3 B0 RE = 2:36 × 1018 ; 0

where Ed is in J and B0 is the dipole Beld at the equator. Usually, to calculate the dipole magnetic Beld energy, the integration is performed from 1RE to inBnity. This leads to an energy value of 8 × 1017 J which is three times smaller. To calculate E0 and the other two terms of E1 , it is convenient to use Greens’ theorem (Maguire and Carovillano, 1966) to replace the volume to a surface integral. Representing the magnetopause as a paraboloid and using the fact that the geomagnetic Beld exists only inside the magnetosphere (the magnetic Beld component normal to the magnetopause is equal to zero), Alexeev (1997) has obtained a relation for the magnetic energy associated with the magnetic Beld of the Chapman–Ferarro current
3 RE 1 E m = Ed : 3 R1 Fig. 5 shows that Em values are in the range of (5 –18)×1014 J during the storm; this energy is taken from the solar wind. If the solar wind pressure on the magnetosphere increases, the size of the magnetosphere becomes smaller but the energy of the Beld associated with the Chapman–Ferraro current Em increases.

30 20 10 0 0

12

24

12

24

12

24

12

24

12

UT Fig. 4. The variation of the power UFAC delivered from the solar wind to the magnetosphere through the Beld-aligned currents Region 1 during the magnetic storm.

2.4.3. Energetics of the magnetospheric tail current system The magnetic energy of the magnetospheric tail current ETL is determined by the relation Rk 1 1 2 BTL dv = 0 i dl; ETL = 20 Vm 2 R2

Y.I. Feldstein et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 429 – 446

435

20 Em , 1014 J

15 10 5 0 250

ETL, 1014 J

200 150 100 50 0 EDR, 1014 J

30

20

10

0 0

12

24

12

24

12

24

12

24

12

UT

Fig. 5. Variations of the total energy of the magnetosphere magnetic Beld generated by large-scale current systems calculated using the paraboloid model of the magnetospheric magnetic Beld: variations of the energy of the Chapman–Ferraro currents on the magnetopause Em , variations of the energy of the magnetotail current system closed on the magnetopause ETL during November 23–27, 1986 magnetic storm. For comparison, variations of the ring current energy determined using DRM values on the Earth’s surface are presented.

where 0 is the magnetic Jux in the tail lobe, and i is the linear current density in the central part of the tail cross section from the dawnside to the duskside. The integration is performed from the inner boundary of the current sheet R2 to the geocentric distance Rk that is given in the Earth’s radii. The interaction of the magnetic tail Beld with the dipole Beld and the Chapman–Ferraro current Beld leads to two additional terms for the energy calculation Edtl and Etlsd . The energy of the interaction of Bd with BTL is determined by the integral Edtl =

1 0

Vm

Bd BTL dv:

Using Green’s formula and the fact that the magnetic potential of the dipole has a singularity at some point inside Vm , one can obtain Edtl = −

4 B0 R3E BTO ; 0

where BTO is the tail current sheet magnetic Beld in the Earth’s centre. For the paraboloid model we take BTO ≈ 10 nT and Edtl ≈ −Em . The energy of the interaction

of BTL with Bsd is calculated as 1 Etlsd = Bsd BTL dv: 0 V m In the paraboloid model, Etlsd equals zero, due to the condition that the normal component of the magnetic Beld on the magnetopause is equal to zero. The value ETL is a function of the tail current i which decreases with increasing tailward distance. The dependence of the magnetic Beld intensity (and, therefore, also the current intensity in the tail current sheet) on geocentric distance was studied by many investigators. The following functional forms are discussed in the literature: B ∼ 1=X 0:3 in Behannon (1968), B ∼ 1=R0:77 in Mihalov and Sonett (1968), B∼1=X 0:58 in Sonett et al. (1971), B ∼ 1=X 0:53 in Slavin et al. (1985). Newer investigations show a faster decrease in the magnetic Beld intensity with increasing distance in the tail: B ∼ 1=R1:2 in Nakai et al. (1991), and B ∼ 1=X 1:25 (or B ∼ 1=R1:46 ) in FairBeld and Jones (1996). In summary, depending on the author, the tail current density decreases with increasing geocentric distance in the range between 1=R0:5 and 1=R1:5 . Tsyganenko (2000) showed that changes in the magnetic Beld intensity in the magnetotail during weakly disturbed conditions at the geocentric distances

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−10RE ¿ r ¿ − 60RE in the tail comply with these relations. Assuming that i falls oC with distance R from the Earth as 1=R, we obtain after integration the following expression for ETL :
  2Rk =R1 + 1 2R2 1 ; + 1 ln  ETL = 0 i0 R1 2 R1 2R2 =R1 + 1 where i0 = 2bt0 =0 is the linear current density on the inner boundary of the tail current sheet, bt0 is the tail magnetic Beld at this boundary connected with 0 by the relation  R1 2 0 bt0 = 2R2 + R1 R21 and for ETL , we have   2Rk =R1 + 1 2R2 2 02 : + 1 ln  ETL = 0 R1 R1 2R2 =R1 + 1 The faster the current density falls oC with increasing distance from the Earth, the less energy is stored in the tail. According to the √ paraboloid model, the tail radius grows proportional to R. Since the magnetic Jux in the tail is conserved, the magnetic Beld intensity in the tail lobe is decreasing with 1=R. The calculations of ETL presented below have used current densities which fall oC with 1=R. To calculate the magnetic energy of the tail Beld, we chose Rk =60RE . The ETL variations in the magnetospheric tail are presented in Fig. 5. If one uses a i ∼ 1=R1:5 instead of a i ∼ 1=R dependence, then the maximal values of ETL on November 24, 1986, 1900 UT, and November 25, 1986, 1800 UT, are smaller by a factor of 1.6. For the case where the current density dependence is described by a relation which is between 1=R and 1=R1:5 , then the ETL values presented in Fig. 5 change by ∼30%. Our calculations include the magnetic Beld energy up to a tail distance of 60RE . At these geocentric distances the magnetospheric tail is modeled by a cylinder with a radius determined from pressure balance between the solar wind thermal energy density and the tail lobe magnetic Beld energy density. By taking average solar wind parameters, we Bnd B∞ ∼ 9:2 nT. FairBeld and Jones (1996) found at a distance of ∼ 60RE a tail radius of ∼ 30RE , which gives a magnetic Jux of ∞ ∼ 5:2 × 108 Wb. Using this value of ∞ as an input parameter of the paraboloid model, together with the other parameters, R1 and R2 for the characteristic main phase storm time, one determines a magnetospheric tail radius of ∼ 33RE . The discrepancy with the experimental data found by FairBeld and Jones (1996) is equal to 3RE , i.e., ∼ 10%. During a storm, the tail current energy ETL is greater by one order of magnitude than the energy of the interaction of tail currents with the dipole Beld Edtl . We can, therefore, assume that ETL determines the total magnetospheric tail contribution to the energetics during a storm. With the beginning of the main storm phase, ETL increases from 8:0 × 1014 J at 0800 UT on November 24 to

2:1 × 1016 J at 1900 UT on November 24. This increase is not steady. On Fig. 5, ETL shows, superimposed on the total energy increases, smaller energy drops. The dissipation processes begin to dominate around 2000 UT on November 24. Looking at the ETL decreases, one notices that the energy dissipation does occur monotonously, but with diCerent speeds as well. During the interval up to 1100 UT on November 25, energy injection occurred simultaneously with energy dissipation in the magnetospheric tail. The next sharp ETL increase occurs during the Bnal interval of the main phase. The energy stored in the tail lobe has reached a maximum of 2:4 × 1016 J around 1900 UT on November 25 and then starts to decrease rapidly and monotonously. This ETL decrease starts at the beginning of the storm recovery phase and coincides with a similar decrease in the Dst-variation. But while Dst continues to decrease during the storm recovery phase, ETL is roughly constant at a rather high level ∼ (40 ÷ 60) × 1014 J. Calculations with the paraboloid model show that the energy stored at geocentric distances tailward from 100RE to 200RE is 0:35 ETL (60RE ), where ETL (60RE ) is the energy in the magnetotail from the inner boundary of the tail current sheet to 60RE tailward. During the storm main phase, ETL (60RE ) peaks twice: 2:1 × 1016 and 2:4 × 1016 J. By taking the mean value of these two values, we calculate that between 100RE and 200RE the stored energy is ETL ∼ 7:7 × 1015 J. This value is in very good agreement with the results from Nakamura et al. (1997). They

1.6

1.4

1.2

ln (ETL1/ETLi)

436

1.0

0.8

0.6

0.4

0.2

1

2

3

4

5

6

7

dT Fig. 6. Temporal variation of the magnetotail energy during two intervals with rapid energy dissipation, which are used for the calculation of the TL parameter (from 2300 UT on November 23 until 0400 on November 24, and from 1900 UT on November 25 until 0200 UT on November 26, 1986).

Y.I. Feldstein et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 429 – 446

437

1400

AE, nT

1200 1000 800 600 400 200 0 0

-50

Dst, nT

-100

-150

-200 6000

Ukin, 1010 W

5000 4000 3000 2000 1000 800

Um , 1010 W

600

400

200

0

12

0

12

0

12

UT Fig. 7. The same that is shown in Fig. 2 for the May 6 –8, 1988 magnetic storm.

calculated for an average of 13 magnetic storms the diCerence in the energy between the main phase maximum of the magnetic storm (Dst ∼ −100 nT) and the quiet time. The diCerence was ∼5 × 1015 J in the distant tail between −100RE and −200RE . As a Brst approximation one assumes that the ETL decrease during the interval from 1900 UT on November 25 until 0500 UT on November 26 is caused mainly by dissipation processes with hardly any energy injection occurring

in the tail. We use this interval for determining the rate of the energy dissipation from the magnetospheric tail in the same fashion as one Bnds the decay parameter DR for the ring current. The magnetic disturbances before the storm main phase (substorms) are also accompanied by a signiBcant storage of energy in the magnetospheric tail from 1200 to 2300 UT on November 23, and then by a rapid dissipation of energy. It is clearly seen that the energy storage proceeds non-monotonically. After reaching a maximum,

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Y.I. Feldstein et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 429 – 446

the energy ETL dissipates rather rapidly and monotonically, which allows one to determine the parameter TL for this interval as well. The bottom panel of Fig. 5 shows the energy stored in the ring current calculated by using the values of DRmod = Dst − DCF − DT for comparison with ETL . It is seen that the energy stored in the magnetospheric tail is several times greater than the ring current energy. In addition, the energy stored in the magnetotail dissipates faster than the ring current energy. By having two intervals where we can assume that there is the only energy dissipation occurring, we can determine the decay parameter for the energy dissipation in the tail TL :

24 Em, 1014 J 20 16 12 8 E TL, 1014 J

300

200

ETL = ETL1 exp{(t − t1 )=TL }: The value of ln ETL1 =ETL as a function of Nt = (t − t1 ) is shown in Fig. 6. Using the least-squares method, one obtains the relation ETL1 ln = 0:20(t − t1 ) + 0:013; ETL TL = t − t1

for ln

ETL1 = 1; ETL

TL = 4:94 h: One should keep in mind that we have assumed zero injections to occur during the time intervals under investigation, so that the true value for TL will be somewhat smaller. To obtain a more accurate TL determination, one would clearly have to take such injections into account. 2.5. Magnetic storm on May 6–8, 1988 Based on an examination of the Dst and AE indexes in Fig. 7 (left side), the interval from 0600 UT until 1400 UT on May 6, 1988 should be referred to as the storm main phase with a minimum Dst value of Dst = −175 nT. We deBne the recovery phase to occur between 1500 UT on May 6 and 2300 UT on May 8, 1988. The end of main phase of the storm determined from the AE index (1800 UT) occurs 4 h later than if one uses the minimum of the Dst to deBne the end of the main phase. For every UT hour the kinetic and electromagnetic energy emag kin rates USW and USW were calculated, which account for the cross section changes of the magnetosphere. The results are presented in Fig. 7 (right panels). As for the November 1986 magnetic storm, the kinetic energy of the solar wind is greater by one order of magnitude than the electromagnetic energy. Both energy types show a tendency to have their maximum during the beginning of the main storm phase. kin During the recovery phase, USW decreases approximately emag three times, while USW decreases about one order of magnitude during the same time interval. All energetic budget components were calculated for every UT hour: Joule heating, particles precipitation, energy injection, energy dissipation in the ring current and

100

0 35 EDR, 1014 J 30 25 20 15 12

24

12

24

12

UT

Fig. 8. The same that is shown in Fig. 5 for the May 6 –8, 1988 magnetic storm.

the Beld-aligned current power. As an example, we show in Fig. 8 the energy stored in the magnetic Beld of the Chapman–Ferraro current Em , in the tail Beld ETL , and the Beld associated with the ring current EDR . The energy Em changes from 7:4×1014 to 22×1014 J during the storm. The energy ETL reaches a value of 2:8×1016 J towards the end of the main phase and then decreases to 8×1014 J in the recovery phase. It increases again to a value of 1:2 × 1016 J in the substorm occurring at 2000 UT on May 7, 1988. The bottom panel of Fig. 8 shows the energy EDR as the total ion energy of the ring current obtained from the AMPTE/CCE spacecraft and interpolated between measurements, as it is described in Dremukhina et al. (1999). The spacecraft intersected the ring current region in the afternoon (1300 LT) and dusk (1900 LT) sectors. Obviously, there is a strong LT-asymmetry of the ion density distribution in the ring current that causes the variations of EDR , which were measured by the satellite: EDR decreases from 32 × 1014 to 24 × 1014 J and is then followed by a steep increase to 31×1014 J. The other components of the energetic budget for the May 1988 magnetic storm will be discussed in the next section.

Y.I. Feldstein et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 429 – 446

3. Discussion

The dependence now reads

This paper has outlined methods to calculate the energy injection and dissipation in diCerent regions of the magnetosphere during moderate magnetic storms and presented the results obtained with these methods for two speciBc magnetic storms. In addition to the traditional methods for calculating the energy content of the magnetosphere, we have made use of the paraboloid model. We want to compare the PA parameter advanced by Perreault and Akasofu (1978) with the total magnetospheric consumption rate. The parameter PA represents the Poynting Jux from the solar wind into the magnetosphere over a magnetospheric collecting area l20

∗ = VB2 sin4

PA = V × B2 sin4

 × l20 : 2

Empirically, Perreault and Akasofu (1978) assumed that l0 is a constant and found that PA can roughly be equated to the rate of energy dissipation in the magnetosphere by taking l0 to be 7RE . The PA index can be viewed as an indicator of the solar wind–magnetosphere coupling, as well as an indicator of the rate of injected energy into the magnetosphere (Gonzalez et al., 1990) It is, therefore, natural to compare PA with the total energy consumption rate UT or the total energy dissipation rate US in the inner magnetosphere and the ionosphere. The left panel of Fig. 9 (Fig. 9a) shows PA and US during the November 1986 magnetic storm. It is seen, that in general, the quantities shown follow each other closely, but during some intervals of the storm, there is a substantial apparent deviation. During the largest part of the main phase, PA essentially exceeds US . Close to the maximum of the main phase, PA ∼ US . During the recovery phase, the energy dissipation US exceeds PA . We calculate that the dissipation US during the main phase in one hemisphere equals 50% of PA for the Brst storm and 18% of PA for the second storm. As outlined above, we emphasize that US matches the energy dissipation in the magnetosphere more accurately than UT . The term UDR1 , describing the energy storage rate in the ring current, contributes additionally to UT during the storm main phase. In the literature, UT is usually used for describing the energy consumption or dissipation rate in the magnetosphere. However, the above obtained discrepancy between PA and US is not the result of calculating the energy consumption in the magnetosphere by two diCerent methods. The right panel of Fig. 9 (Fig. 9b) presents UT instead of US for the May 1988 storm. It is seen that the discrepancy also exists between PA and UT . It is possible that the reason for this discrepancy arises from choosing l0 equal to 7RE . Monreal-MacMahon and Gonzalez (1997) took into account the size variation of the magnetosphere which results from changes in the solar wind pressure by replacing the Akasofu constant l0 with the Chapman–Ferraro distance lCF .

 2 lCF ; 2

where lCF =




B02 4nV 2

439

1=6 :

Inspection of the data presented in Fig. 1 reveals that during the storm under study, the magnetopause subsolar point is located at geocentric distances greater than 7RE . Using values greater than 7RE increases the discrepancy either between ∗ and UT or between ∗ and US during the main phase of the magnetic storm. We, therefore, scale the Monreal-MacMahon and Gonzalez parameter lCF by a factor of 0:7R1 . Such a normalisation allows one to take into account both the magnetosphere size variations and, in addition, brings the value of l0 back to Akasofu’s 7RE during normal magnetospheric situations. Figs. 9a and b show the calculated values of (0:7R1 ) = VB2 sin4 2 (0:7Ri )2 (broken line). The diCerence between (0:7R1 ) and US (or UT ) has decreased during most of the main phase of the November 1986 magnetic storm and the diCerence is also smaller for the main phase of the May 1988 magnetic storm. During the Bnal interval of the main phase of the Brst storm, US , (UT ) is close to  using Akasofu’s value of l0 . During the recovery phase, R1 ∼10RE and the value PA diCer little from (0:7R1 ) and agree rather well with UT and US . The rate of energy dissipation in the inner magnetosphere and ionosphere US equals 68% of (0:7R1 ) in one hemisphere during the main phase for the Brst storm, and 31% of (0:7R1 ) for the second storm. The energy injection and dissipation occur simultaneously in both hemispheres. This circumstance is taken into account for calculating US and UT , both of which are presented in Fig. 10a and b. In this case US is calculated from the relation US ∗ = 2(Ujsub + Ujcon ) + UA + UDR2 ; UT is computed using the relation UT∗ = 2(Ujsub + Ujcon ) + UA + UDR . Including both hemispheres US ∗ comes close to  during the main phase of the November 1986 storm, but exceeds the value of  during the recovery phase. For the magnetic storm of May 1988, UT∗ is always substantially smaller than  for the main phase and slightly exceeds  during the recovery phase. A thorough investigation of the dependence of the -parameter on other combinations of parameters of the interplanetary medium, including energy injection in diCerent regions of the magnetosphere, is necessary. The well-known notion about the -parameter as an integral Jux of the solar wind electromagnetic energy into the magnetosphere has to be speciBed and thoroughly reviewed. As a result of such investigations, new quantitative relations between injection and dissipation of the energy and the parameters of the interplanetary medium need to be obtained for magneto-quiet and magnetic storm time intervals. The total rate of the energy dissipation in the magnetosphere of the two hemispheres as determined by US ∗ =UA +2(Ujcon +Ujsub )+UDR2 needs to be compared with the energy injected to the magnetosphere. As US ∗ is consumed in the inner magnetosphere and in the ionosphere, the energy input into this region must be determined. The energy input through

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Y.I. Feldstein et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 429 – 446 200

EPSPA, EPS0.7R1, US, 1010 W

500

EPSPA , EPS0.7R1, UT , 1010 W

400

150

300

100 200

50 100

0

0 12

12

0

(a)

12

0

0

12

12

0

(b)

UT

0

12

12

UT

Fig. 9. Variations for November 1986 magnetic storm of the PA parameter (solid line) and the rate of energy dissipating in the inner magnetosphere and in the ionosphere US (Fig. 9a, left panel). The broken line is the variation of the PA (0:7R1 ) parameter, taking into account the PA normalised to the magnetosphere size changing according to the geocentric distance up to the subsolar point. On the right panel (Fig. 9b), there are similar data for May 1988 magnetic storm, but UT = Ujcon + Ujsub + UA + UDR1 + UDR2 has been used instead of US .

200

EPSPA, EPS0.7R1, US*, 1010 W

500

EPSPA, EPS0.7R1, UT * 1010 W

400 150 300 100 200 50 100

0 12

(a)

0

12

0

12

0

12

UT

0

(b)

12

0

12

0

12

UT

Fig. 10. The same that is on Fig. 9 but in two hemispheres for US ∗ during November 1986 magnetic storm (Fig. 10a, left panel) and for UT during the May 1988 magnetic storm (Fig. 10b, right panel).

Beld-aligned currents and injection into the ring current is Uin = 2UFAC + UDR . Figs. 11a and b show the total energy injection rate into the two hemispheres (solid line), and the total rate of energy dissipation in two hemispheres for the November 1986 (left panel) and May 1988 storms (right panel, dashed line). The components of the total injection and dissipation rates have been calculated independently. One has to admit that the fact that injection and dissipation rates show the same trend during the magnetic storm is surprising. At the beginning and at the end of the storm, their absolute values are even in good agreement. During the

storm main phase and during the substorms in the recovery phase, the dissipation is greater than the injection. This may indicate the existence of an additional energy source to the considered region: the magnetotail. The rate of the energy change in the magnetotail dETL =dt = UTL1 can be obtained from the energy conservation equation as a diCerence of the energy injected into the magnetotail UTL and the rate of the energy dissipation in the tail UTL2 : UTL1 = UTL − UTL2 . During intervals with UTL1 ¿ 0, energy is stored in the magnetotail. If UTL1 ¡ 0, dissipation of the previously stored energy takes place. The

Y.I. Feldstein et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 429 – 446

24-27.11.1986

441

6-8.05.1988

200

UIN, US*, 1010 W

U IN , U S* , 10 10 W 150

150 US *

100

100

50

50 UIN

0 (a)

0 12

0

12

0

12

0

12

UT

12

0

12

(b)

0

12

UT

Fig. 11. The total energy injection rate into the two hemispheres Uin for the case where the energy injection into the inner magnetosphere is due to Beld-aligned currents and the injection into the ring current is Uin = 2UFAC + UDR (solid line). The total rate of the energy dissipation US = 2(Ujcon + Ujsub ) + UA + UDR2 (broken line) for November 1986 magnetic storm is shown on the left panel (Fig. 11a). On the right panel (Fig. 11b), the same quantities are shown for the May 1988 magnetic storm.

term UTL2 describes the rate of the energy dissipation in the magnetotail and can be calculated from the relation UTL2 = ETL =TL , where TL is a decay parameter that characterizes the rate of the energy dissipation from the magnetotail. As was described earlier, TL is approx. 5 h. The rates of the energy injection into the magnetotail are given by dETL ETL ; + UTL = UTL1 + UTL2 = dt TL where UTL is calculated for intervals during the magnetic storm when the energy of the magnetospheric current system was sharply increased. Fig. 12 (solid line) shows the sharp changes in the energy injection rate to the tail UTL during active storm intervals for the November 1986 magnetic storm. During two adjacent hours, UTL changed by two orders of magnitude. On November 25 at 1700 UT, UTL reached a value UTL = 4:76 × 1012 W that is greater by an order of magnitude than the maximum rate of the energy injection into the inner magnetosphere. The energy dissipation rate UTL2 is shown in Fig. 12 by a dashed line, and US ∗ by a thick line. Table 1 and Table 2 list all the relevant parts of the energy budget for the diCerent phases of the two considered storms. In each cell of the table—from top to bottom—we present the maximum energy Jux (W), the hourly averaged energy Jux (W), and the total energy (J) for the diCerent phases of the magnetic storms. The length of the diCerent phases are: 27 h for the interval before the main phase, 28 h during the main phase and 54 h during the recovery phase for the Brst storm and 9 hours for the main phase and 58 h for recovery phase for second storm, respectively.

500

UTL, UTL2, US* 1010 W

450 400 350 300 250 200 150 100 50 0 -50 0

12 23

24

12 24

24

12 25

24

12 26

24

12 27.11.86

Fig. 12. The rate of energy injection into the magnetotail UTL = dETL =dt + ETL =TL (solid line) during the sharp increase of energy in the magnetotail during 1200 –2300 UT on November 23, during 0900 –1900 UT on November 24, and during 1100 –1800 UT on November 25, 1986, and the rate of the energy dissipation from the magnetotail ETL =TL (broken line), the rate of dissipation into the inner magnetosphere and the two ionospheres US ∗ (thick line).

The numbers in Tables 1 and 2 show the following: (1) The kinetic and electromagnetic energy of the solar wind sharply decreases from the interval before the main

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Y.I. Feldstein et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 429 – 446

Table 1 Maxima values of power (Peak), medium values of power (Um ) and total energy (Esum ) of various components of the magnetospheric energy budget in the interval of November 23–27, 1986 magnetic storm: 1—prior to main phase onset; 2—main phase; 3—recovery phase 1

2

3

Ukin SW

Peak/1012 W Um =1012 W Esum =1017 J

46.27 25.10 24.40

37.80 20.01 20.17

17.33 11.49 22.34

Uem SW

Peak/1010 W Um =1010 W Esum =1015 J

311.64 168.62 163.90

316.51 111.83 112.73

74.96 32.65 63.48

PA

Peak/1010 W Um =1010 W Esum =1015 J

179.09 98.96 99.75

73.39 13.63 26.50

(0:7R1 )

Peak/1010 W Um= =1010 W Esum =1015 J

140.42 72.59 73.17

53.15 12.35 24.01

UJcon

Peak/1010 W Usum =1010 W Esum =1015 J

29.35 10.62 10.32

66.78 33.53 33.80

33.19 11.04 21.46

UA

Peak/1010 W Um =1010 W Esum =1015 J

13.99 5.93 5.76

30.67 14.66 14.78

15.52 6.38 12.40

UDR = UDR1 + UDR2

Peak/1010 W Um =1010 W Esum =1015 J

22.37 4.92 4.78

21.52 10.17 10.25

15.73 5.87 11.40

UFAC

Peak/1010 W Um =1010 W Esum =1015 J

24.62 12.05 11.72

45.98 28.72 28.95

23.90 9.90 19.24

UDR2

Peak/1010 W Um =1010 W Esum =1015 J

9.22 4.38 4.26

11.16 8.73 8.80

9.79 6.65 12.92

Uinj = UFAC + UDR

Peak/1010 W Um =1010 W Esum =1015 J

37.95 16.98 16.50

55.60 38.90 39.21

38.76 15.76 30.64

Peak/1010 W Um =1010 W

49.58 20.95

108.30 56.92

58.62 24.06

Esum =1015 J

20.36

57.37

46.78

UTL = UTL1 + UTL2

Peak/1010 W Um =1010 W Esum =1015 J

195.50 38.85 37.76

476.09 82.22 82.87

317.21 25.30 49.18

UTL2

Peak/1010 W Um =1010 W Esum =1015 J

85.99 36.10 35.09

128.72 63.20 63.70

132.72 35.18 68.39

+

UJsub

Udis = UA + UJcon +UJdis + UDR2

Y.I. Feldstein et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 429 – 446 Table 2 Maxima of values of power (Peak), mean values of power (Um ) and the total energy (Esum ) of various components of the magnetospheric energy budget in the interval of the magnetic storm on May 6 –8, 1988: 1—the main phase; 2—the recovery phase 1

2

Ukin SW

Peak/1012 W Um =1012 W Esum =1017 J

59.32 47.23 15.30

58.93 24.24 50.61

Uem SW

Peak/1010 W Um =1010 W Esum =1015 J

589.72 345.38 111.90

286.86 78.97 164.89

Peak/1010

W Um =1010 W Esum =1015 J

448.16 320.68 103.90

358.37 29.16 60.89

(0:7R1 )

Peak/1010 W Um= =1010 W Esum =1015 J

252.09 188.23 60.99

180.65 18.98 39.64

UJcon

Peak/1010 W Usum =1010 W Esum =1015 J

58.33 38.15 12.36

47.50 10.79 22.52

UA

Peak/1010 W Um =1010 W Esum =1015 J

24.94 17.07 5.53

21.43 6.70 13.99

UDR = UDR1 + UDR2

Peak/1010 W Um =1010 W Esum =1015 J

33.21 17.89 5.80

25.31 3.61 7.54

UFAC

Peak/1010 W Um =1010 W Esum =1015 J

49.88 36.86 11.94

43.99 7.33 15.31

UDR2

Peak/1010 W Um =1010 W Esum =1015 J

15.25 11.67 3.78

15.11 5.06 10.56

Uinj = UFAC + UDR

Peak/1010 W Um =1010 W Esum =1015 J

81.44 54.75 17.73

69.30 10.94 22.85

Peak/1010 W Um =1010 W

94.19 66.89

83.41 22.55

Esum =1015 J

21.67

47.08

W Um =1010 W Esum =1015 J

535.46 93.76 30.38

448.48 37.55 78.40

Peak/1010 W Um =1010 W Esum =1015 J

91.66 56.56 18.33

137.05 39.40 82.26

PA

+

UJsub

Udis = UA + UJcon UJsub + UDR2

UTL = UTL1 + UTL2

UTL2

Peak/1010

(2)

phase to the recovery phase, reaching an intermediate value during the main phase. The electromagnetic en-

(3)

(4)

(5)

(6)

443

ergy is less than the kinetic energy by one order of magnitude. Epsilon (0:7R1 ) is the corrected value of PA , taking into account the change in the size of the magnetosphere. It makes up ∼ 60% of the solar wind electromagnetic energy falling on the magnetosphere cross section during the main phase of the storm, and ∼40% during the recovery phase. The energy injected Ein = EDR + 2EFAC into the inner magnetosphere and ionosphere of both hemispheres makes up ∼0:9–2:2% of the solar wind kinetic energy during the magnetic storm. The injected energy into the tail ETL is greater than the total energy injected to the inner magnetosphere and ionosphere of both hemispheres by ∼ 10% during the storm main phase. The total energy injected into the magnetosphere (from the ionosphere altitudes up to 60RE tailward) amounts to ∼ 4:0–7.5% of the solar wind kinetic energy falling on the magnetospheric cross section during the main phase, ∼ 2:3– 4.4% during the recovery phase, and ∼2:7% before the main phase commencement. The energy transferred by Beld-aligned currents from their region of generation to the ionosphere is as big as the Joule heating produced by ionispheric currents.

The total dissipated energy in the magnetosphere during the two storms intervals (in the ionosphere of both hemispheres, in the inner magnetosphere and in the tail) equals 3:57×1017 and 2:04 × 1017 J. The total energy input into the magnetosphere equals 3:16 × 1017 and 1:76 × 1017 J for the Brst and the second storms, respectively. In the two storms considered, the dissipated energy exceeds the input energy by 0:41 × 1017 and 0:28 × 1017 J. We argue here that this energy diCerence results from including twice the energy loss caused by precipitating particles. One adds up the energy loss by precipitation EA and the energy loss in the tail ETL2 . One has to keep in mind the energy lost by particle precipitation occurs mainly from the tail. We, therefore, have to substract the values EA = 32:9 × 1015 and 19:52 × 1015 J found for the two storms from the total dissipation. Hence, the total energy dissipated in the magnetosphere is equal to 3:24 × 1017 and 1:85 × 1017 J. This value diCers by only 2.5% and 4.8% from the energy input into the magnetosphere. The injected and dissipated energies are ∼ 1:7 × 1017 J for the Brst and ∼ 1:1 × 1017 J for the second storms, so that we obtain approximately a zero energy balance in the magnetospheric tail. This may indicate that the relations and input parameters which we used in our calculations of the energy budget are correct. (7) During the storm main phase, the average rate of the energy injected into the ring current makes up ∼ (6 –10)% of PA and ∼(10 –14)% of (0:7R1 ). According to an estimation by Feldstein (1992), the rate of the energy injection to the ring current during the

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main phase of magnetic storms makes up ∼ 15% of PA , i.e., the energy injected into the magnetosphere is consumed mainly by processes diCerent from the ring current generation. According to Feldstein and Grafe (1996), the rate of the energy injection into the tail exceeds the injection into the ring current by a factor of three. These results demonstrate that the ring current plays only a secondary role in the consumption of energy injected into the magnetosphere. If one looks at the energy injected into the ring current during the whole storm of November 1986, one Bnds EDR =2:64×1016 J, i.e. ≈ 15% of PA . Kozyra et al. (1998) calculated the energy budget for the November 2– 6, 1993 magnetic storm interval and obtained 8:1 × 1015 J for the ring current energy input and 8 × 1016 J for PA . Therefore, the storm under study is not an exception and the ring current is not the main consumer of the energy during the magnetic storm. (8) According to Baker et al. (1997), the ring current injection rate during a magnetic storm is about 1011 –1012 W, the rate of ionospheric Joule heating is about 1010 –1011 W, and the rate of auroral precipitation is about 109 –1010 W. Results existing in the literature are, however, pretty contradictory. For example, Weiss et al. (1992) estimate the energy dissipation as UDR ∼ 4 × 1011 –1012 W, UJ ∼ 8 × 1010 –2 × 1012 W, UA ∼ 1011 ÷ 5 × 1011 W, but Stern (1984) gave values of UDR ∼ 1:5 × 1010 –2:0 × 1011 W, UJ ∼ 1011 W, UA ∼ 4 × 1010 –1011 W for the same processes. On the other hand, we found in this study UDR ∼1011 W. Even during the main phase UDR is not greater than UA , which is about 1:3 × 1011 W and UDR is several times less than UJ , which is about 3 × 1011 W. In the two storms investigated in this paper, UDR is smaller than UJ as long as we use for the calculation of UDR a  value longer than ≈ 5 h. Only if one assumes that  is less than 1 h, does the energy dissipation from the ring current occur much faster. In this case, the term DR= begins to make a main contribution to UDR (Zwickl et al., 1987). For the November 1986 magnetic storm, we Bnd EJ =6:56×1016 J, EA =3:3×1016 J, EDR =2:6×1016 J for the total storm energies, and hence, EJ :EA :EDR = 2:5:1:3:1; for the May 1988 storm the relations between total storm energies are EJ :EA :EDR =2:6:1:5:1. Kozyra et al. (1998) reported approx. the same values of ratios EJ :EA :EDR = 3:8:1:3:1 for the November 1993 storm, but Knipp et al. (1998) noticed that these ratios change during the storm. 4. Conclusion The investigation of the energy budget in the magnetosphere during the magnetic storms November 23–27, 1986 and May 6 –8, 1988 was done on the basis of the paraboloid model, which allows one to calculate the intensity of the

magnetic Beld and, therefore, the energy of all large-scale magnetospheric current systems as well. The paraboloid model allows us to study the variations of the energy budget of magnetic storms on a hourly basis. Approximating the dayside magnetosphere with a shape of a paraboloid of cross section S, one can calculate the kin rate of solar wind kinetic energy USw , and the rate of soemag lar wind electromagnetic energy USW impinging onto this cross section. By changing the cross section, one can take into account a size change of the magnetosphere. During the storm main phase, the mean values of the solar wind particle kin kinetic and electromagnetic power are USW ∼ 2 × 1013 W, emag 12 USW ∼ 1:1 × 10 W. The electromagnetic energy is less than kinetic one, by one order of magnitude. The energy injected during the magnetic storms into the inner magnetosphere and ionosphere of both hemispheres makes up ∼0:9–2.2% of the solar wind kinetic energy. The total energy injected into the magnetosphere (from the ionosphere altitudes up to 60RE tailward) makes up ∼4:0–7.5% of the solar wind kinetic energy falling on the magnetosphere cross section during the main phase. The energy injected into the tail ETL is greater than the total energy injected into the inner magnetosphere and ionosphere of both hemispheres by 3–18% during the storms main phases. The total dissipated energy in the magnetosphere (in the ionosphere of both hemispheres, in the inner magnetosphere and in the tail) during the storm intervals equals 1:85 × 1017 –3:24 × 1017 J with a total input into the magnetosphere energy equating 1:77 × 1017 –3:16 × 1017 J. These values are distinguished by only 4.5 –3.1%. The total injected and dissipated energy in the magnetotail is for the May 1988 storm ∼ 1:09 × 1017 J and for the November 1986 storm, ∼ 1:7 × 1017 J. Therefore, the injected and dissipated energy into the magnetotail is bigger than the energy which is injected and dissipated in the ionosphere and inner magnetosphere. In the magnetic storms under investigation, the average ratios between the energies dissipated in the diCerent regions (ionosphere, auroral region and ring current) were EJ :EA :EDR = 2:5:1:4:1. During the storm main phase, the average rate of the energy injection into the ring current accounts for ∼(6 –10)% of the PA parameter. The energy injected into the magnetosphere is consumed mainly in processes diCerent from the ring current generation. Thus, our investigation shows that the ring current is a secondary consumer of the energy injected into the magnetosphere. During the intense development of the auroral electrojets (AE-index increases sharply), the energy stored in the magnetotail and the dissipation in the tail increases simultaneously. The power dissipation in the inner magnetosphere and in the ionosphere US during the magnetic storms main phases do not occur at the expense of the energy previously stored in the magnetotail ETL . If this scenario is realized then the increase in US should be accompanied by a decrease in ETL and hence a decrease in UTL . However, during the main phase of the magnetic storm, the

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