2^')^ p=0 + iFfo (F')l exp( - V (f3 -F'3 )) = 0. (3.16.9)

Here I (k) is the spectrum of Alfven waves normalized to the magnetic energy of the mean field: I SB 2\ = j dkl (k) . Wave-numbers with + or - sign correspond to right or left hand circularly polarized wave. Eq. 3.16.9 describes the influence of waves on the mean distribution function of thermal particles, in particular, well known in plasma theory quasi-linear 'plateau' formation breaking by thermal collisions (Galeev and Sagdeev, 1979). The solution of this equation should be substituted into Eq. 3.16.8. This equation, together with the Eq. 3.16.2 for the flux, determines the nonlinear electric current density (the input of thermal protons is taken into account only)

Jk = ink j dk1 jv2dvMc—\Ak-k1 Xez] F=0 . (3.16.10)

0 Bo

Substituting this current into Maxwell's equations and ensemble averaging one can derive an equation for the Alfven wave spectrum dl (k)/ dt = -2TnlI (k) (3.16.11)

with the nonlinear Alfven wave damping rate:

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