Stability and allied problems in magnetohydrodynamics

Abstract

We note that the expressions (1.20)�(1.27) are real for all real values of the angular momentum MMM. In the numerical computations, we fixed V0V_0V0? and MMM, then obtained the values of (r,z)(r, z)(r,z) from the given constraints. These values were substituted into the corresponding expressions for RRR and ZiZ_iZi? to determine the set of points (R,Zi)(R, Z_i)(R,Zi?) that define the guiding center. Computations were carried out for V0=1V_0 = 1V0?=1 and M=?1,0,1M = -1, 0, 1M=?1,0,1. In Figure 11, we have plotted the allowed regions for the guiding center and its trajectory for case (i). We find that the guiding center lies near a line of force and, in most cases, does not cross the boundaries of the allowed region. Figure 12, corresponding to case (ii), depicts the so-called mirror reflection encountered by the particle in strong field regions because the velocity of the particle tangential to a field line tends to zero. Figure 13 traces the path of the guiding center for case (iii). The guiding center remains within the allowed region, indicating that the particle抯 motion is of a hybrid type and stays in the region of interest for all ?M??1|M| \leq 1?M??1. We also solved the actual non-linear set of equations governing the particle抯 motion for several values of MMM using the Runge朘utta method, to compare these results with those obtained from the guiding center approach. Figure 14 shows the particle抯 orbit in the (r,z)(r, z)(r,z) plane for M=0M = 0M=0 with initial conditions: r=1,?=0,z=0r = 1, \theta = 0, z = 0r=1,?=0,z=0 and velocity components r?=0,??=0.3900,z?=0\dot{r} = 0, \dot{\theta} = 0.3900, \dot{z} = 0r?=0,??=0.3900,z?=0. We observe that the particle does not escape from the region of interest defined by the corresponding St鴕mer boundaries. This holds true for all cases considered by Srivastava and Bhat. Furthermore, for V0=1V_0 = 1V0?=1, the present results are in qualitative agreement with those obtained by numerical integration.