where X is the free path length. The numerical coefficient a in this case is equal to 1/2; a more systematic treatment can show that its value is somewhat different. An acceleration of charged particles, therefore, occurs in a shear flow of a collisionless plasma, and the rate of energy increase is proportional to the square of the curl of the hydrodynamic velocity of a plasma (in the analyzed case (rot u)2 ~ (du/dy)2).

Fig. 4.19.1. The idealized picture of the two-dimensional shear flow of collisionless plasma. Horizontal rows show velocities of plasma u(y); B'(x,y), B(x,y),and A(x,y) are particle scattering inhomogeneities frozen in plasma and moved in x direction with velocity u(y). According to Berezhko (l98l).

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