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Goodrich & Scudder 1984
Goodrich, C.C. and Scudder, J.D. (1984). The adiabatic energy change of plasma electrons and the frame dependence of the cross-shock potential at collisionless magnetosonic shock waves. Journal of Geophysical Research 89: doi: 10.1029/JA089iA08p06654. issn: 0148-0227.

In collisionless magnetosonic shock waves, ions are commonly thought to be decelerated by dc electrostatic cross-shock electric field along the shock normal nˆ. In a frame where ions are normally incident to the shock the change in the potential energy [qϕN> in the quasi-perpendicular geommetry is of the order of the change of the energy of normal ion flow: [qϕN>≈[1/2mi(ViNnˆ)2>, which is approximately 200-500 eV at the earth's bow shock. We show that the electron energy gain, typically 1/10 this number, is consistent with such a large potential jump in this geometry. Key facts are the different paths taken by electrons an ions through the shock wave and the frame dependence of the potential jump in the geometry. In the normal incidence frame, electrons lose energy by doing work against the solar wind motional electric field EMN, which partially offsets the energy gain from the cross-shock electrostatic potential energy [eϕ*N>. In the de Hoffman-Teller frame the motional electric field vanishes; the elctrons gain the full electrostatic potential energy jump e*HT> of that frame, which is not, however, equal to the electrostatic potential energy jump e*N> of that frame, which is not, however, equal to the electrostatic potential energy jump e*N> in the normal incidence frame. We estimate the ratio of these potential jumps to be [ϕ*HT>/[ϕ*N>≈&ggr;/(&ggr;-1)[kTe> {[-1/2mi(ViNnˆ>}-1, where &ggr; is the effective polytrope index for electrons. By observation this ratio is ~1/10 at the earth's bow shock. When viewed in the de Hoffman-Teller frame, corresponding changes in the ion kinematics occur. Since the e*HT> is an order of magnitude smaller than the ion energy, the ions are significantly affected by the electrostatic force. They are instead primarily retarded in this frame by the magnetic force. Since this latter force is proportional to the component of B out of the coplanarity plane, infinitesimally thin shock models may not be realistic for the study of the ion and electron dynamics in the de Hoffman-Teller frame.

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Journal of Geophysical Research
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