Charged-particle orbits near a magnetic null point

An approximate analytical expression is obtained for the orbits of a charged particle moving in a cusp magnetic field. The particle orbits pass close to or through a region of zero magnetic field before being reflected in regions where the magnetic field is strong. Comparison with numerically evaluated orbits shows that the analytical formula is surprisingly good and captures all the main features of the particle motion. A map describing the long-time behaviour of such orbits is obtained. The motion of charged particles in spatially varying magnetic fields has received a great amount of attention because of its relevance to plasma fusion devices, particle accelerators and astrophysics. Even in the simplest cases, the motion is complicated and is now known to be an example of chaos. One simplifying assumption, which is good when the ratio of the Larmor radius to a scale length describing the spatial variation of the magnetic field, ε, is small, is that the so-called adiabatic invariant μ is a constant. This immediately leads to an explanation of charged-particle containment in the Van Allen radiation belts and in magnetic mirror fusion devices. For larger values of ε, it has been found that the adiabatic invariant undergoes jumps ∆μ where μ changes rapidly in just a few Larmor periods in special regions of symmetry, but otherwise μ is to all intents and purposes constant. The jumps are such that ∆μ£ exp(®1}ε). For a specific calculation of ∆μ for a wide range of magnetic field configurations, see for example Cohen et al. (1978). In this case, the long-time behaviour of particles can be understood in terms of a map relating the values (μ n θ n ) before a jump to the values (μ n+" , θ n+" ) after a jump. Here θ is an angle specifying the Larmor phase of the particle. It is found that, to a reasonable approximation (terms of order exp(®2}ε) being neglected), that one can write μ n+" ̄μ n ­∆μ cos θ n , where, of course, ∆μ is a function of μ n . In many applications, it is sufficient to restrict attention to changes in μ that are small, so that one may linearize the value of μ n about a chosen mean. Then the above equation reduces to δμ n+" ̄ δμ n ­K cos θ n , (1) and since the original equations of motion were Hamiltonian, the equation for θ variation can be obtained by insisting that the Jacobian is unity. This gives θ n+" ̄ θ n ­δμ n+" . (2) o Present address : Walaik University, Thailand. 256 K. Jaroensutasinee and G. Rowlands In the above, K is a constant whose value is determined by the field configuration and energy of the particle. The above map (δμ, θ) is the Chirikov map, and is used to study the long-time behaviour of nearly adiabatic particles in spatially varying magnetic fields. For sufficiently small values of ∆μ, it is found that the particle motion is such that μ changes periodically about a constant value (superadiabatic). For larger values, the motion can become chaotic ; and for sufficiently large values, the motion of the charged particle can be understood in terms of a diffusion in momentum space with diffusion coefficient proportional to exp(®1}ε). Numerous examples of this type of behaviour have now been studied in detail, and are described in the book by Lichtenberg and Lieberman (1983). It must be stressed that the direct numerical solution of the particle-orbit equations becomes prohibitively expensive in machine time because one has to follow the particle around its Larmor orbit, whereas it is the motion of the guiding centre that is really needed. Adiabatic and weakly non-adiabatic theory overcome this problem by essentially introducing a suitable averaging procedure to remove the fast motion associated with motion about the Larmor orbit. However, the whole theory is totally inadequate if, during its motion, a particle can move in a region where the field strength is small or even zero. An example of such a field is the two-dimensional cusp described by the vector potential A ̄ xyk, where k is a unit vector along the z axis. For such a field, B ̄ (x,®y, 0), and the motion of a charged particle in this field is governed by the reduced Hamiltonian (Jaroensutasinee and Rowlands, 1994) H ̄ " # [xd #­yd #­(Q®xy)#], (3) where Q is a constant proportional to the z component of the momentum and xd 3 dx}dt. An immediate consequence of the constancy of H (which in the following we normalize to " # ) is that the particle motion is confined to regions between the curves y ̄ (Q31)}x. Thus, for Q" 1, the particle is excluded from the origin, the position of the zero of the magnetic field. For Q( 1, adiabatic theory applies, and the value of the jump ∆μ was given some time ago by Howard (1971). A typical orbit is shown in Fig. 1(d). For Q! 1, the origin is no longer excluded, and particle orbits may pass through or close to the zero-magnetic-field region. Some typical orbits are shown in Fig. 1. A subset of these orbits (Figs 1a, b) are such that they remain close to the x axis, and it is for this type of orbit that we now develop a novel analytical approach. The case Q ̄ 0 is developed in detail, although the method is applicable to all Q! 1. The exact equations of motion are simply d# x dt# ̄®xy#, d# y dt# ̄®x# y, whilst the adiabatic invariant μ (the ratio of the perpendicular kinetic energy to the magnitude of the magnetic field) is given by μ ̄ 1 (x#­y#)$/# [(xd x­yd y)#­x# y# (x#­y#)]. (4) Charged-particle orbits near a magnetic null point 257