International Journal of Geomagnetism and Aeronomy
Published by the American Geophysical Union
Vol. 1, No. 1, April 1998

Electric field induced in the magnetosphere by a sudden impulse

Yu. P. Mal'tsev

Polar Geophysical Institute, Apatity, Murmansk Region, Russia
The Model and Basic Equations


The induction field generated in the magnetosphere by enhancement of the solar wind dynamic pressure is calculated using the image dipole model. The field is shown to be westward directed and to cause drift of the magnetospheric plasma toward the Earth. To lowest order, the electric field is related to the varying magnetic field by El = -(r/2c) \partialBz0/\partialt, where r is the cylindrical distance, El is the eastern component of the electric field, and Bz0 is the magnetic disturbance at the Earth.


The induction electric fields are an intermediate event between the quasi-stationary electrostatic and wave fields. In contrast to the electrostatic fields, the induction fields arise owing to the time variations in the magnetic field, i.e., the Maxwell equation
The induction fields have larger spatial and temporal scales (their spatial sizes exceed the sizes of the magnetosphere, and their duration is several minutes and more) than the wave fields. While the plasma inertia is critical in the wave fields, it is not important for the induction fields.

Estimates performed by a number of authors show that the induction fields are often commensurate with the fields of quasi-stationary magnetospheric convection.

Mal'tsev et al. [1975] determined the field associated with a change in the system of field-aligned currents. At first the magnetic field was calculated, and then the electric field was found from (1) and from the condition of high electric conductivity
The induction field gives rise to two convective vortices whose intensity increases along the field line from the Earth.

From analysis of the experimental data, Sergeev [1984] showed that the induction field in the plasma sheet during the initial phase of a substorm is close in magnitude and opposite in sign to the electrostatic field, and as a result the total field in the plasma sheet turns out to be at least an order of magnitude weaker than the field in the tail lobes.

Pudovkin et al. [1991] studied the induction field resulting from a change in the plasma sheet geometry during the expansion of the substorm. It was shown that the magnitude of the induction field at the reconnection point is proportional to the velocity of motion of the auroral bulge toward the pole (under the condition that the reconnection point does not displace).

Kozlovsky et al. [1994] considered the consequences of enhancement of the solar wind dynamic pressure. The image magnetospheric model was used. The pressure enhancement was imitated by motion of an imaginary dipole toward the Earth. It was assumed that the electric field in the magnetosphere can be obtained through Lorentz transformations
where V2 is the velocity of arrival of the imaginary dipole and B2 is the magnetic field produced by this dipole. The field (3) satisfies equation (1); however, it contradicts condition (2). Thus the problem of the electric field induced by enhanced compression of the magnetosphere by the solar wind still remains unresolved.

The goal of this paper is to calculate the induction electric field in the magnetosphere during sudden increases of solar wind dynamic pressure (sudden impulse (SI) or storm sudden commencement (SSC)).

The Model and Basic Equations

A sudden change in the solar wind pressure gives rise to two current systems. One of them is an alternating part of currents on the magnetopause. The second system is the polarization (inertia) currents at the front of the magnetoacoustic wave propagating from the daytime magnetopause to the magnetospheric tail. Like Kozlovsky et al. [1994], we take into account only the effect of currents on the magnetopause. The travel time of the magnetoacoustic wave from the dayside to the tail is ~100 s. For the characteristic times exceeding this magnitude, the wave fields can be neglected.

We use the image dipole, vacuum model of the magnetosphere
The field of each dipole can be expressed as
where yk is the scalar potential, Mk is the moment of the dipole oriented antiparallel to the z axis, rk is the dipole coordinate, and k = 1 and k = 2 correspond to the Earth and imaginary dipoles. We take
where V2 is the velocity of the imaginary dipole. The general solution of (1) is given by
where Es is the solenoid part of the field satisfying (1) and condition
and Ep is the potential part equal to

The solenoid part is equal to
It is easy to make sure that expression (2) is valid by substituting (4)-(8) into (10) and also into (1). Note that the right-hand parts of (12) and (3) turned out to be the same.

By substituting (9), (11), and (12) into formula (2), we obtain the equation for the potential

Equation (13) can be solved by integrating along the field line. It is necessary to add the boundary condition. Ignoring the field-aligned and ionospheric currents and assuming that the Earth is sufficiently small, we take
We assume that V2 and r20 have only the x component equal to V2x and x2, respectively. The solution of (13), in view of (14), is given by
where integration is performed along the magnetic field line beginning from the center of the Earth dipole, and ds is the element of the field line length.

We restrict ourselves to calculating the integral in (15) in the first approximation when inequalities
are fulfilled. Then expression (15) can be rewritten as
is the element of the dipole field line length, re is the distance to the most remote point of the field line in the equatorial plane, and q is the polar angle;
is the magnetic field of the Earth dipole. By calculating the field components B1 and B2 from (5) and (6), substituting them into (18), and taking into account (17), (19), and (20), we obtain the potential in the equatorial plane
where B20 = B2z    (r = 0). Substituting (21) into (11) and (9) and taking into account (12), (5), (6), and (17), we obtain the electric field in the equatorial plane (in the cylindrical coordinates)

El = (V2x/c)Bz0 1.5r/x2 where r is the cylindrical distance, and l is the longitude counted to the east.

The field El can be written in a different manner. By differentiating (5) with respect to time at the point r = 0 and taking into account (6) and (8), we obtain
Substitution of (23) into (22) yields


Let us estimate the induction field for the typical parameters of sudden impulses. Assuming that Bz0 grows by 10 nT during 100 s, we obtain from (24) that at the distance r = 7RE the electric field El ~ 1.5 10-7 CGSE = 4.5 mV m-1 is induced. This field exceeds somewhat the field of the steady state magnetospheric convection which typically amounts to ~1 mV m-1.

In contrast to the electrostatic field, the induction field is not transmitted to the ionosphere. The condition of freezing-in (2) is not violated in this case. The magnetic field lines remain as if fixed at the level of the ionosphere and experience maximum displacements in the equatorial plane of the magnetosphere.

As compression of the magnetosphere by the solar wind increases, the induction electric field, according to (24), is westward directed and causes the plasma drift toward the Earth both in daytime and at night. This is the difference between our result and the conclusions of Kozlovsky et al. [1994], who state that the induction field is directed from evening to morning and forces the plasma to move from the Sun in all longitudinal sectors. We have noted above that Kozlovsky et al. [1994] use a wrong expression for the induction field (3) contradicting the condition of freezing-in (2).

The trapped particles are energized during compression of the magnetosphere by the solar wind. This occurs because of the betatron acceleration. The induction electric field enhances this effect by several fold because in the reference system associated with the displacing magnetic field tube the magnetic field is enhanced to a larger degree than in the magnetosphere as a whole. Variations in the magnetic field in the tube can be written as
where dBz/dt is the substantial (Lagrangian) derivative; \partialBz/\partialt is the local (Euler) derivative; and Vr = cEl/Bz is the velocity of radial drift in the induction electric field. Under condition (16), we obtain from (4)-(6) \partialBz/\partialr = -3Bz/r. In view of (24), we infer from (25)
Thus taking into account the induction electric field, the betatron acceleration in the magnetosphere during SI or SSC is enhanced by a factor of 2.5.

Knott et al. [1985] and Laakso and Schmidt [1989] reported the electric field measurements by the geosynchronous satellite r = 6.6RE immediately after a SSC. Knott et al. [1985] studied two SSC events during each of which the electric amplitude burst reached ~5 mV m-1. During the first event, the magnetic field increased by 12 nT. These magnitudes are consistent with (24). As far as the electric field direction is concerned, it contained a radial (from the Earth) component somewhat exceeding the azimuthal one, in addition to the westward component. The measurement was performed at the pre--noon sector (~1100 LT). In the second case, the measurement was carried out almost at noon, and the electric field proved to be directed eastward, which agrees with formula (24) because the magnetic field impulse was negative. The magnitude of the magnetic impulse of the second event was not given by Knott et al. [1985].

Laakso and Schmidt [1989] investigated 27 cases of SSC. Figure 3 of Laakso and Schmidt [1989] showed vectors E for the moments when DBz reached 50% of the maximum value. On average, there was a westward directed azimuthal component of the order of 1 mV m-1. The average values of \partialBz/\partialt were not given. It follows from (24) that they should be \partialBz0/ \partialt ~0.025 nT s-1.

In addition to the azimuthal component of the electric field, Figure 3 of Laakso and Schmidt [1989] shows a radial component of 1-2 mV m-1. In the morning sector it is directed away from the Earth, and in the evening sector it is directed toward the Earth. Probably the appearance of the radial component is caused by the polarization currents at the front of the magnetoacoustic wave propagating from the daytime magnetopause to the magnetospheric tail. Propagation of such a wave with the velocity of ~1000 km s-1 was detected by several satellites [Nopper et al., 1982] for one SI event.

An incompletely understood consequence of SI and SSC is the train of Alfvén oscillations in the range of periods of Pc 4-5 with the amplitude (in the magnetosphere) commensurate with the intensity of the primary magnetoacoustic impulse [Knott et al., 1985; Laakso and Schmidt, 1989; Nopper et al., 1982; Tverskaya and Khorosheva, 1982]. In a uniform medium, the Alfvén and magnetoacoustic waves propagate independently. Different mechanisms for transformation of the magnetoacoustic wave into an Alfvén wave have been suggested, such as changes in the ionospheric conductivity [Safargaleev and Mal'tsev, 1987] or irregularities of the magnetospheric plasma [Kozlovsky et al., 1994; Lysak and Lee, 1992].

SIs and SSCs are also accompanied by enhancement of particle injections. On the whole, the injected flux in the auroral region can be enhanced by 2-3 fold [Craven et al., 1986].

Measurements of fluxes of trapped particles by several geosynchronous satellites during one of the SSC events showed that energization of particles agrees with the theory of betatron acceleration to an order of magnitude [Wilken et al., 1986], though it should be taken into account that the plasma motion toward the Earth during an SSC leads to additional temporal variations in the fluxes because of the presence of a radial gradient in the particle pressure. In addition to acceleration, a change in the pitch angle distribution was observed; it acquired the shape of a "butterfly." Wilken et al. [1986] suppose that this is caused by splitting of the drift shells for particles with different pitch angles. Note that splitting is accompanied by separation of electrical charges, which can give a certain contribution to generation of the Alfvén wave observed after SI or SSC.


Changes in the currents on the magnetopause during enhancement of the solar wind dynamic pressure induce in the magnetosphere a westward directed electric field with an amplitude of up to several millivolts per meter consistent with magnetospheric plasma motion toward the Earth. The substantial (in the system of coordinates of the frozen-in plasma) increase in the magnetic field proves to be 2.5 times as high as the local increase (in the magnetospheric coordinate system).


Craven, J. D., et al., Global auroral responses to magnetospheric compressions by shocks in solar wind: Two case studies, in Solar Wind-Magnetosphere Coupling, edited by Y. Kamide and J. A. Slavin, pp. 367-380, TERRAPUB, Tokyo, 1986.

Knott, K., A. Pedersen, and U. Wedeken, GEOS 2 electric field observations during a sudden commencement and subsequent substorms, J. Geophys. Res., 90, (A2), 1283-1288, 1985.

Kozlovsky, A. E., V. V. Safargaleev, and W. B. Lyatskiy, The transformation of magnetoacoustic waves into Alfvén waves inside the magnetosphere, Ann. Geophys., 12 (10/11), 1022-1026, 1994.

Laakso, H., and R. Schmidt, Pc 4-5 pulsations in the electric field at geostationary orbit (GEOS 2) triggered by sudden storm commencements, J. Geophys. Res., 94 (A6), 6626-6632, 1989.

Lysak, R. L., and D.-H. Lee, Response of the dipole magnetosphere to pressure pulses, Geophys. Res. Lett., 19 (9), 937-940, 1992.

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Nopper, R. W., Jr., W. J. Hughes, C. G. MacLennan, and R. L. McPherron, Impulse-excited pulsations during the July 29, 1977, event, J. Geophys. Res., 87 (A8), 5911-5916, 1982.

Pudovkin, M. I., V. S. Semyenov, G. V. Starkov, and T. A. Kornilova, On separation of the potential and vortex parts of the magnetotail electric fields, Planet. Space Sci., 39 (4), 563-568, 1991.

Safargaleev, V. V., and Yu. P. Mal'tsev, Generation of the preliminary impulse and long-period pulsations during SI, Geomagn. Aeron., 27 (2), 247-252, 1987.

Sergeev, V. A., Regularities in development of intense isolated substorms from observations in the plasma sheet of the magnetosphere, Magnitosfer. Issled., 5, 71-82, 1984.

Tverskaya, L. V., and O. V. Khorosheva, On the nature of the source of packets of decaying long-period pulsations, Geomagn. Aeron., 22 (5), 824-830, 1982.

Wilken, B., et al., Magnetospheric configuration and energetic particle effects associated with a SSC: A case study of the CDAW 6 event on March 22, 1979, J. Geophys. Res., 91 (A2), 1459-1473, 1986.

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