By Bowie W.
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Additional resources for Present Status of Geodesy and Some of the Problems of This Branch of Geophysics (1920)(en)(11s)
Example text
These equations can be generalized to the case in which the system shows helical symmetry. Omitting the calculations, we write the result [26] [see Eq. £. ~ ~ + ~ . £. £. 33) Here ~ is a function of the two coordinates r and e, e = cp- az, a = 21r /L. f3 = 1 + a 2r 2, and Lis the pitch of the helix. § 2. Axisymmetric Flow across an Azimuthal Magnetic Field An important case of MHD flow is the flow across an azimuthal magnetic field with velocity v = v 11 and field H = HfP. The equations describing flow of this type can be found from Eqs.
37) must be solved together for p and ~; the functions U(0 and B(O and the equations of the boundary surfaces must be given. If the field Hr,o is known we can write the current density j in terms of H r,o by means of . _ (rH In the first approximationp we can replace p in Eq. 115) by p0 (z). Equating the flow velocity v (z) to the critical velocity c s. we find the equation of the surface at which the critical velocity is reached: v 2 (z) (l+y) ---(2--y) B 2 r 2 p0 (z)=(y-l)U. 116) Since v(z) is an increasing function, while p 0(z) is a decreasing function, the critical surface is located at progressively larger values of z as the radius increases (Fig. lOa). B. In the other limiting case, (3 « 1, a cold plasma, in the zeroth approximation in the parameter (3, the density is p 0(r, z) = (U - v 2/2) jr2B 2• We now write c~ as c~= U- v2 /2 + ('\'- 2)W (p).