VAF VvrnVP 2880

where symbol ( )- denotes the averaging over the particle pitch-angle - = np/p, and d(-) is the pitch-angle diffusion coefficient, determined by expression p2D(-) = -2D±± - 2-(1 - -2)-12D±// + (1 - -2)D////. (2.8.81) The solution of Eq. 2.8.79-2.8.80 has the form

where

If the particles are highly scattered then

Substituting Eq. 2.8.81 into Eq. 2.8.78 Katz and Yacobi (1997) obtain the transport equation dN- - VraaVrAN + (V^p + -\r^p2Dpr (V^N

p2 dp dp where

Kai = K//nanA + {«icd)^ Drp = Dpr = ^«/' D(P) = «/' «/ = . (2 8 87)

According to Katz and Yacobi (1997), Eq. 2.8.86 may be re-written in the form of the continuity equation in the coordinate space and in the space of the absolute values of the momentum (Fedorov et al., 1992; Dolginov and Katz, 1994):

- Vr J a + -T T" P2 JP = 0, (2.8.88) dt p2 dp where dN

is the vector of the particles flow in the coordinate space, and

is the particles flow in the space of absolute values of the particles momentum.

2.9. CR diffusion in the momentum space

Consider the collision integral in the kinetic Eq. 2.7.1 written in the spherical coordinates in the momentum space (pz = pcose, p± = psine):

(S/) = —--— sin BDqq^— + _^±sm00p d/N p2sinede eede psinede ep dp

where Dee, Dep, Dpe, Dpp are determined by Eq. 2.7.2, and Dl, D2, D3 are determined by Eq. 2.6.13-2.6.24 with respective substitution for the variables. The terms of the collision integral including the components Dee of the tensor Da% describe an elastic scattering of charged particles on turbulent pulsations of the magnetic field and the rest terms include the energy interchange between turbulent pulsations and charged particles resulting in the acceleration of the latter. Note the following property. If we compare the value of various terms in the collision integral, we obtain that the term with Dee is the main term. The effective frequency of scattering of charged particles in the collisions with the magnetic field inhomogeneities v = p~2Dee is far higher than the frequency of inelastic collisions, which is represented by the coefficients Dpe and Dpp. Otherwise the particles are very quickly got into a chaotic state owing to scattering on turbulent pulsations of a magnetic field and their further diffusion in momentum space is described by an isotropic distribution function. According to this we shall search for the solution of Eq. 2.7.1 in the form (Ryutov, 1969):

where the second term is far less than the first, and

2n 0

means averaging over e. By averaging the kinetic Eq. 2.7.1 over 0 angle we obtain d/A 1 d 2

ddlo

The equation for a correction Sf to a distribution function has the following form:

Was this article helpful?

0 0

Post a comment