In the case in which the regular field is strong enough the approximation Ap(r) = 0 is not valid, and one should include a spiral character of particle motion at the distances of the order of the correlation radius of a stochastic field.
2.4. Kinetic equation for CR propagation including fluctuations of plasma velocity
Fluctuations of plasma velocity u(r,t) = uG(r)+ ui(r,t); (u(r,t)) = uG(r); (u^r,t)) = 0 (2.4.1)
were taken into account in (Dorman and Katz, 1972a) besides the fluctuations of a magnetic field. In this case, one should take into account the action of the induced electric field
c on a particle. Because of the non-stationary character of the processes on the Sun, as well as a development of turbulence immediately in the interplanetary space, a widespread spectrum of turbulent pulsations (Alfven, magneto-sonic waves, etc.) is generated in the solar wind plasma parallel with stochastic inhomogeneities frozen in it. The stochastic electromagnetic fields of these pulsations sufficiently affect the motion of charged particles. The distribution function f (r, p, t) of non-interacting charged particles moving in the magnetic and electric fields which are determined by the Eq. 2.2.1 and Eq. 2.4.1 satisfy the collisionless kinetic equation f + v f + F f = o, (2.4.2b)
dt dr dp where v = c2p/E is the velocity, E = c(p2 + m2c21 is the total energy of a particle with the momentum p and rest mass m. The term F is the force acting on a particle:
Let us present F in the form of the sum of regular
and a stochastic
component. If the magnetic field is completely frozen in plasma, the regular component of the electric field Eo has a form
and a stochastic component will be
According to Eq. 2.4.3 - 2.4.7 we write the kinetic Eq. 2.4.2b in the form
or d is the operator related to the regular component Fo of the force F and L = — . As
9p for the case of fluctuations of a magnetic field alone, the distribution function f (r,p, t) varies irregularly in space and time following the variations of a stochastic field, so that the actual meaning is proper to the distribution function < f (r, p, t )> which is averaged over a statistical ensemble corresponding to a stochastic field. To deduce the equation which the function < f (r, p, t )> satisfies, let us use the method of characteristic functional considered in Section 2.2. Then in a general case the correlation tensor of electromagnetic field (in contrast to Eq.
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