By P. H Roberts
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Extra info for An introduction to magnetohydrodynamics
C) The creation of magnetic flux outside a fluid conductor. Motions of conducting fluid in a fixed volume V will generally convect, distort, rotate and dilate the area, A, in which a given flux tube intersects S, the surface of V. P and IJIt denote a typical length and velocity for V. If I] = 0, this rearrangement of field lines is the only mechanism by which motions in V can affect the field in V, the exterior of V. If I] = 0, the fluid motions cannot alter the net flux out of A as it moves; nor can they change the net flux of either sign out of S itself.
The second of these made use of (6) and Figs. lA, and is set out below (88) itself. Use this argument, with suitable modifications, to deduce (91) from (90). 1 The magnetic Reynolds number . (a) Introduction. In this chapter and the next we examine one side of the magnetohydrodynamic relationship in isolation; viz. the effect of the motion of a conducting fluid upon a magnetic field in which it lies. e. when the Alfven number, A, given by (1: 78) is large. We regard the fluid velocity u as being prescribed everywhere,t and solve the electrodynamic equations alone.
7] :~ (d) Small displacements. In later chapters of this book, we often have occasion to introduce a displacement field ~ defined by x = x o + ~(xo). (126) The value of any field, resulting from the displacement ~, may be expanded in a power series in ~ in one of two ways. Consider, for example, the density P, and temporarily denote the density at x resulting from a displacement ~ from xo by p(x: ~), so that p(x : 0) = Po(x). The eulerian expansion of p is the Taylor expansion of p(x, ~) in powers of ~: p(x,~) = po(X) + op(x) + 02P(X)+ ...