By Vassily Beskin, Gilles Henri, Francois Menard, Guy Pelletier, Jean Dalibard
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Extra info for Accretion discs, jets and high energy phenomena in astrophysics: , 29 July-23 August, 2002
22) The magnetic ﬁeld is supposedly given by B = B(x) ey , B(x) being an odd function of x. The ﬁeld lines are straight and there is a neutral sheet at x = 0, towards which the ﬂow converges. 23) J. Heyvaerts: Accretion and Ejection-Related MHD 29 This equation cannot be satisﬁed at x = 0 in the absence of resistivity. 23) for very small x. Thus, B(x) ≈ Ea V0 x (x large) B(x) ≈ Ex ηm (x small). 25) The smaller the resistivity, the thinner this resistive layer. For very large magnetic Reynolds numbers Rm , it takes the character of a boundary layer.
The latter is usually negligible under astrophysical conditions . The ﬂuid parcel is also subject to volume forces, such as gravity or Lorentz forces. Let f vol be the total volume force density. The fundamental law of mechanics applied to an inﬁnitesimal ﬂuid element reduces to the equation of motion of hydrodynamics: ρ (∂v/∂t + (v · ∇)v) = −∇P + div σ + f vol . 4): ∂(ρv)/∂t + div(ρvv + P δ − σ) = f vol . 10) The ab component of the tensor vv is v a vb . 10) is the equation of conservation for matter momentum.
12) B 2 is related to B 1 by the chain rule: B 2 = B(P2 , t2 ) − B(P2 , t1 ) + B(P2 , t1 ) − B(P1 , t1 ) + B(P1 , t1 ). 13) For inﬁnitesimal dt, this gives: B2 = ∂B dt + ((v(P1 ) · ∇) B) dt + B 1 . 14) The theorem claims that if K 1 × B 1 vanishes, so does K 2 × B 2 . 5) and the identity K 2 × B 2 = (K 1 + (K 1 · ∇) v dt)) × B 1 + rot(v × B) = (div B) v − (div v) B + (B · ∇)v − (v · ∇)B. 16) The vectors B 1 and K 1 being parallel, they can be written as B 1 = B1 t and K 1 = K1 t, where t is a common unit vector.
Accretion discs, jets and high energy phenomena in astrophysics: , 29 July-23 August, 2002 by Vassily Beskin, Gilles Henri, Francois Menard, Guy Pelletier, Jean Dalibard