Here I(k) is the spectrum of Alfven waves normalized to the magnetic energy of the mean field:

Wave numbers k with +(-) sign correspond to right(left) hand circularly polarized wave. The equation obtained describes the influence of waves on the mean distribution function of thermal particles, in particular, well known in plasma theory quasilinear 'plateau' formation breaking by thermal collisions (Galeev and Sagdeev, 1979). The solution of this equation should be substituted into Eq. 3.5.8. This expression, together with the Eq. 3.5.2 for the flux determines the nonlinear electric current density (the input of thermal protons is taken into account only):

Jk = ink Jdki Jv2dvMc—\Ak_k1 xez]] F=o . (3.5.10)

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Substituting this current into the Maxwell equations and ensemble averaging one can derive an equation for the Alfven wave spectrum dl (k)/ dt = -2TnlI (k) (3.5.11)

with the nonlinear Alfven wave damping rate:

Tnl = -nMVLk IdkiI(-kl))ln2dvv2 J d?£+±f(-?(k + ki)) 8 B2 -« 0 -« \k + kl|

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