Abstract
We present some results of the analytical integration of the energy rate balance equation, assuming that the input energy rate is proportional to the azimuthal interplanetary electric field, Ey, and can be described by simple rectangular or triangular functions, as approximations to the frequently observed shapes of Ey, especially during the passage of magnetic clouds. The input function is also parametrized by a reconnection-transfer efficiency factor α (which is assumed to vary between 0.1 and 1). Our aim is to solve the balance equation and derive values for the decay parameter τ compatible with the observed Dst peak values. To facilitate the analytical integration we assume a constant value for τ through the main phase of the storm. The model is tested for two isolated and well-monitored intense storms. For these storms the analytical results are compared to those obtained by the numerical integration of the balance equation, based on the interplanetary data collected by the ISEE-3 satellite, with the τ values parametrized close to those obtained by the analytical study. From the best fit between this numerical integration and the observed Dst the most appropriate values of τ are then determined. Although we specifically focus on the main phase of the storms, this numerical integration has been also extended to the recovery phase by an independent adjust. The results of the best fit for the recovery phase show that the values of τ may differ drastically from those corresponding to the main phase. The values of the decay parameter for the main phase of each event, τm, are found to be very sensitive to the adopted efficiency factor, α, decreasing as this factor increases. For the recovery phase, which is characterized by very low values of the power input, the response function becomes almost independent of the value of α and the resulting values for the decay time parameter, τr, do not vary greatly as α varies. As a consequence, the relative values of α between the main and the recovery phase, τm/τr, can be greater or smaller than one as α varies from 0.1 to 1.
Highlights
The energization of the ring current is usually described by a ®rst order dierential equation, known as the energy rate balance equation
In the Appendix we show the results of a numeric integration of Eq (3) and the resulting D t output for dierent values of s are compared to the observed storm evolution, so that the input functions and the values of s that best reproduce this evolution can be estimated
One of the assumptions is that the ring-current decay time is constant during the main phase of the storm (s sm const:), as has frequently been assumed in the literature, and the other that the input function of the equation, Q t; can be approximated by simple rectangular or triangular shapes
Summary
The energization of the ring current is usually described by a ®rst order dierential equation, known as the energy rate balance equation. Let us consider ®rst the case for which the input function, Q, has a rectangular shape, with a negative constant value during a given time interval and is equal to zero outside it. This ideal input would resemble the limiting case for a disturbing interplanetary ®eld that according to Gonzalez and Tsurutani (1987), is able to produce an intense storm, namely, a southward interplanetary magnetic ®eld Bz < ÿ10 nT acting over at least 3 h. As discussed the variable Qo is directly related to the eciency in the energy transfer to the ring current and, since this eciency is not known a priori, this parameter should be determined along with s
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