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Unformatted text preview: 17. Hydromagnetic shock waves 17.1 The jump conditions A magnetized and electrically conducting plasma can carry different types of small-amplitude waves, depending on the angle between the large-scale magnetic field and their direction of propagation or on the values of the Alfv en velocity and the sound speed. We can therefore expect to also find different classes of hydromagnetic shocks. Let us again consider a steady-state plane-parallel shock in its rest frame in the absence of an external force. Let us further assume ideal MHD, i.e. the conductivity in infinitely large. Then the MHD equations read in Einstein convention (sum over common indices) x k ( V k ) = 0 (17 . 1) x k bracketleftbigg V i V k + P ik 1 4 parenleftbigg B i B k 1 2 | vector B 2 | ik parenrightbiggbracketrightbigg = 0 (17 . 2) x k bracketleftBigg V k parenleftBigg 2 V 2 + 1 P parenrightBigg + 1 4 ( vector B vector V ) vector B bracketrightBigg = 0 (17 . 3) x k B k = 0 (17 . 4) vector ( vector B vector V ) = 0 (17 . 5) Calculations show that it is possible to choose a coordinate system, the de Hoffmann-Teller frame, in which the velocity vector vector V and the magnetic field vector vector B on both sides of the shock lie in same plane. In the de Hoffmann-Teller frame the problem is therefore effectively two- dimensional. The corresponding vector components can be indexed with and bardbl , respectively, meaning perpendicular to the shock front or parallel to the shock front (in the shock plane). The integration of the MHD equation across the shock front then yields V = const (17 . 6) V V + P 1 8 parenleftBig B 2 B 2 bardbl parenrightBig = const (17 . 7) V V bardbl 1 4 B B bardbl = const (17 . 8) V parenleftBigg 2 ( V 2 + V 2 bardbl ) + 1 P parenrightBigg 1 4 B bardbl ( B V bardbl B bardbl V ) = const (17 . 9) B = const (17 . 10) 1 B V bardbl B bardbl V = const (17 . 11) The six equations 17.6 to 17.11 determine the downstream values of the six fluid variables , P , V , V bardbl , B , and B bardbl , if their upstream values are given. Using 17.6 and 17.10 we can rewrite equation 17.8 as V d bardbl V u bardbl = B 4 V parenleftBig B d bardbl B u bardbl parenrightBig (17 . 12) A discontinuity occurs in V bardbl because a current sheet exists in the shock plane on account of vector vector B negationslash = 0.= 0....
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- Fall '08