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**Magnetohydrodynamic shock structure without collisions.**
*(English)*
Zbl 0111.39603

### Keywords:

fluid mechanics
Full Text:
DOI

### References:

[1] | Marshall, Proc. Roy. Soc. (London) A233 pp 367– (1956) |

[2] | C. S. Morawetz, ”Magneto-hydrodynamic shock structure using friction,” Institute of Mathematical Sciences, New York University, NYO-8677 (1959). |

[3] | M. H. Rose, ”On the structure of a steady hydromagnetic shock,” Institute of Mathematical Sciences, New York University, NYO-7693 (1956). |

[4] | Shafranov, Zhur. Eksptl. i Teoret. Fiz. 32 pp 1453– (1957) |

[5] | Soviet Phys.-JETP 5 pp 1183– (1957) |

[6] | H. Grad, ”Shock heating,” Conference on Controlled Thermonuclear Reactions, Berkeley, California, 1957; TID-7536 (1957). |

[7] | C. S. Gardner, H. Goertzel, H. Grad, C. S. Morawetz, M. H. Rose, and H. Rubin,Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atoinic Energy(United Nations, New York, 1958), Vol. 31, p. 230. |

[8] | Fishman, Revs. Modern Phys. 32 pp 959– (1960) |

[9] | Colgate, Phys. Fluids 2 pp 485– (1959) |

[10] | Adlam, Phil. Mag. 3 pp 448– (1958) · Zbl 0082.20803 · doi:10.1080/14786435808244566 |

[11] | M. H. Rose, ”On plasma magnetic shocks,” Institute of Mathematical Sciences, New York University, NYO-2883 (I960). |

[12] | The units in the above equations are mks rationalized. |

[13] | If Uc is sufficiently small compared to e2,Kis small compared to 1, and we may take ū+=ū-. In this case the total charge is zero. This approximation which is frequently made will be valid unlessdEdxturns out to be very large. However, it is not necessary to make this assumption, and we do not. |

[14] | A general theory of such asymptotic expansions has been developed by C. S. Gardner (unpublished). |

[15] | J. Calkin has given a general proof (unpublished). |

[16] | Bernstein, Phys. Rev. 108 pp 549– (1957) |

[17] | Hain, Z. Naturforsch. 13 pp 11– (1958) |

[18] | We shall not therefore be able to say anything about the motion of the particles and hence about the fields forxpositive and greater than o(l). But the original problem is not correctly set if we look for a solution in the range -<x<X, where X<+, unless we prescribe on x=X the values of f+ (and even of f-) for negative values ofu. We are therefore forced to impose some conditions forxlarge. These rather weak conditions will be discussed in case D of this section. |

[19] | We recollect from lemma 1 that there are no trapped ions. |

[20] | Note that the ion in question is characterized by the condition that it crosses x=x. with u=u, and v=v, wherex,u,v are independent of . Everything else depends on . Hence, in particular,rdepends on . |

[21] | If the cutoff radiusRis sufficiently small, no ions bounce. Hence by a continuity argument one finds that by increasingRone gets first ions that bounce only a few times. For these,vremains O(1). |

[22] | In general, it is sufficient to takeHconstant,dEdxof fixed sign. |

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