Yu. P. Maltsev and B. V. Rezhenov
Polar Geophysical Institute, Apatity, Russia
The major manifestation of a magnetic storm is the global decrease (depression) of the geomagnetic field. The quantitative measure of this perturbation is the Dst index, which is determined as the H component disturbance at low-latitude observatories averaged over longitudes. The temporal behavior of the Dst index corrected for the solar wind dynamic pressure is commonly described by the following equation [Feldstein, 1992; Gonzalez et al., 1994]
where Q is the so-called coupling function, t is the decay time of the electric currents responsible for the storm-time geomagnetic field depression. The problem of contribution of different solar wind parameters to the function Q has not been finally solved yet. The majority of the investigators relate it mainly to the interplanetary electric field Ey = -VBz [Burton et al., 1975; Feldstein et al., 1984; Grafe, 1988; Murayama, 1982; O'Brien and McPherron, 2000; Pisarsky et al., 1989; Pudovkin et al., 1985, 1988; Valdivia et al., 1996] where V is the solar wind velocity, Bz is the IMF vertical component in the GSM coordinate system. In particular, O'Brien and McPherron [ obtained the following empirical expression
Here Ec = 0.49 mV m -1, Eyr = -VBs is the "refined" electric field component directed duskward, Bs is the IMF southward component. Some investigators report that the coupling function Q depends not only on Ey but also on the proton density [Murayama, 1982] and velocity [Pisarsky et al., 1989]. Another expression for Q was suggested by Perreault and Akasofu 
Here a is a constant, B = (Bx2 + By2 + Bz2)1/2 is the IMF modulus, q = arctan (By/Bz), By is the dawn-to-dusk IMF component. The parameter e is commonly called the Akasofu parameter.
Expressions (2) and (3) have some similarity -- they both show that the magnetosphere behaves as a half-wave rectifier: the IMF southward component affects it considerably stronger than the northward component does. The main difference is that (3) presumes dependence on two other IMF components (on Bx and By, with a smaller weight, though), whereas these components are absent in (2).
For decades an opinion existed that substorms contribute to the coupling function Q. However, Iyemori and Rao  have shown convincingly that substorms rather weaken slightly the storm time depression than enhance it.
The characteristic relaxation time t of the magnetospheric currents responsible for the geomagnetic depression is of the order of 10 hr and varies in the course of a storm [Feldstein, 1992; Gonzalez et al., 1994]. It is not quite clear what factors affect this quantity. For instance, Valdivia et al.  assume t = 12.5/(1 - 0.0012 Dst0) hr whereas O'Brien and McPherron [ found t = 2.40 exp[9.74/(4.69+ Eyr) ] hr.
In this paper we examine the influence of various parameters on the coupling function Q and characteristic time t. Studying storm activity's relation to the solar wind parameters is complicated by the intercorrelation of the parameters themselves. One purpose of this paper is the analysis of the relation between different solar wind parameters. In order to exclude the effect of the intercorrelation of the parameters we shall try to examine the response of the Dst0 to each parameter separately, that is, under the other parameters being kept constant.
We used the OMNI database, which includes hourly values of all IMF components, velocity, and concentration of the solar wind protons as well as hourly Dst indices for 28 years, from 1963 to 1990. A total of 112,000 hourly data were used. The Dst index was corrected according to the following equation [Maltsev and Rezhenov, 2002]
where p = m n V2 is the solar wind proton dynamic pressure expressed in nPa, m is the proton mass, V is the solar wind velocity, and n is the proton concentration. The derivative d Dst0/dt was replaced by the ratio DDst0/ Dt where Dt = 1 hr and DDst0 is the difference between the values of Dst0 for two successive hours.
Figures 1-3 were drawn for all values of the solar wind density. The dependence of DDst0 on the density is shown in Figure 4 for 400 < V < 500 km s -1 under the northward (left panel) and southward (right panel) orientation of the IMF in several ranges of Dst0. One can see that there is no pronounced dependence of the Dst0 on the proton density.
The curves shown in Figures 1 and 2 allow us to find approximation formulas for Q and t. First we averaged DDst0 in bins with a size of 20 nT in Dst0, 3 nT in Bz IMF, and 100 km s -1 in V. Then we found by the least square method the following formulae:
Our formulae (5) and (6) do not differ strongly from expression (2) and the formula for t by O'Brien and McPherron [, who suggested that the coupling function Q depends on the duskward interplanetary electric field only. We added the linear term for the solar wind velocity into the expression for Q. The dependence on V was found by Pisarsky et al.  but with the coefficients strongly different from ours in expression (5). Our Figure 4 does not confirm the dependence of Q on the solar wind proton density obtained by Murayama . While calculating the characteristic time t we looked over many kinds of dependencies including t (Dst0), suggested by Valdivia et al. . The dependence t( Eyr ) provides the best fit. One can see this fact directly in Figure 2.
As one can see from Figures 5 and 6, the Akasofu parameter e is less related to DDst0 than the electric field Eyr. Analyzing a number of storm events, Feldstein  and Murayama  have found earlier that the behavior of DDst0/ Dt is better described by the coupling function Q dependent on Eyr rather than on e. A similar result was obtained by Wu and Lundstedt  with the help of the neural network technique. The good relation between Q and e found by Perreault and Akasofu , Akasofu [1981, 1996], and Gonzalez et al.  can be explained by the strong statistical correlation between e and Eyr.
Formulas (1) and (5) are important not only for a prediction of Dst0 but also for distinguishing between the two existing theories of magnetic storms. The traditional theory regards the geomagnetic storm time depression as the ring-current effect. The ground magnetic disturbance produced by the ring current is proportional to the total energy content of magnetically trapped particles [Dessler and Parker, 1959; Sckopke, 1966]. The parameter e satisfies this concept perfectly because it is proportional to the solar wind magnetic energy flow. However, till now there have been no theoretical studies explaining quantitatively the observed dependence of Q on e (or on Eyr ) under the ring current concept. Moreover, by 2000 the Dessler-Parker-Sckopke relationship had been experimentally tested only in a few case studies. The first statistical study was performed by Greenspan and Hamilton [, who analyzed 40 storms and did not find any correlation between the total particle energy content in the dayside and Dst index.
A new theory of magnetic storms suggested by Maltsev  relates the storm-time depression to an increase of the magnetic flux in the magnetotail lobes. The magnetotail flux grows during storms due to the magnetic flux transport from the dayside to the tail as a result of the terrestrial magnetic field reconnection with the IMF southward component. The rate of this process does not depend on the solar wind energy flux. It depends on the duskward interplanetary electric field [Maltsev et al., 1996]. On the basis of this theory, Arykov and Maltsev  calculated the coupling function Q(Eyr ), which is in quantitative agreement with the empirical expression obtained by Burton et al. . The latter does not differ much from (2) and (5). Maltsev and Ostapenko [ processed a large database containing 20-year measurements of 11 satellites and found that the observed variation of the magnetotail magnetic flux is capable of contributing as much as 80% to Dst. Alexeev et al. [1996, 2001] also report on a large contribution of the magnetotail current to the storm-time depression.
We analyzed the hourly Dst and solar wind parameters for the 28-year period (112,000 data) and found that the solar wind coupling function Q in equation (1) can be presented as a linear combination of the duskward electric field Eyr (this term dominates) and velocity. The dependence on Bx and By IMF as well as on the solar wind density is negligible. The dependence on the Akasofu parameter e appeared to be weak if one examines it under nearly constant Eyr, whereas the dependence on Eyr remains strong and almost invariable when e is restricted by a narrow range. Thus, one can conclude that the storm activity is controlled by the magnetic flux transport from the dayside to the magnetotail rather than by the solar wind magnetic energy flow.
The value of Dst itself reveals a significant statistical correlation with the solar wind velocity and all IMF components. The relation of Dst to Bx, By, and e seems to be caused by intercorrelation of the majority of the solar wind parameters.
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