Multi-scale solar magnetic activity and the resulting generation (Axford and McKenzie, 1997) and dissipation of Alfvén waves may play an important role in coronal heating. The popular cascading approach assumes a gradual evolution of the wave energy to the small dissipative scale of the order of by local non-linear interactions among MHD waves. Voitenko and Goossens (2005a,b) investigated an alternative non-local transfer of Alfvén wave energy from the large MHD length scales directly into the small dissipation-range scales, which are naturally associated with kinetic Alfvén waves (KAWs). As discussed in Subsection 5.4, these waves have very short wavelengths across the magnetic field, irrespectively of the frequency which their dispersion relation (44) contains.

A non-linear mechanism for the excitation of KAWs is via the resonant decay of a pump low-frequency Alfvén wave: AW KAW1 + KAW2. The decay is still efficient for the amplitudes expected for coronal Alfvén waves, which thus will, through the non-linearly driven KAWs, suffer significant dissipation. Therefore, the cross-scale non-linear coupling of Alfvén waves can provide a mechanism for the replenishment of the energy in the dissipation domain, and consequently may lead to heating of the corona and fast solar wind. The maximal non-linear growth rate, , of the KAWs may reach the order of the lower-hybrid frequency, and is given by an expression that only involves the relative pump-wave amplitude but has a sensitive parametric dependence on the plasma via a dimensionless function , such that

The function ranges between about zero and unity. Therefore, the decay can be very fast by MHD standards. A detailed quantitative discussion is provided in the original references. Given the existence of low-frequency KAW or other small-scale turbulence in the corona, Voitenko and
Goossens (2004) studied the possible cross-field heating of coronal ions by these waves or turbulence. They
showed that test ions moving in the electromagnetic field of a KAW may get locally detached from
cyclotron motion in the demagnetizing phase of the wave field, and thereby undergo strong cross-field
heating. In particular, heavy O^{5+} ions were found to enhance their perpendicular energy by up to two
orders of magnitude. The required small perpendicular wavelength, being of the order of , may
for example be produced by phase mixing of MHD Alfvén waves, or could result from the above discussed
decay process.

Continuing the discussion of kinetic wave dissipation, we mention that besides linear wave damping also non-linear mechanisms have recently been considered. Voitenko and Goossens (2002a) studied the non-linear excitation of kinetic Alfvén waves (KAWs) by fast magnetoacoustic waves in the solar atmosphere. Since these waves have very small wavelengths in the direction perpendicular to the background magnetic field, they are very efficient in exchanging energy with kinetic plasma waves or MHD waves. It was shown that the non-linear supply of energy by the finite-amplitude fast-mode pump wave to the small-scale KAWs can be much faster than dissipation mechanisms such as viscous damping and Landau damping. Transient heating events observed by Yohkoh and SOHO may thus be produced by KAWs which are excited by parametric decay of fast waves originating from reconnection sites.

Turbulence simulations generally show that the MHD cascade process transfers fluctuation energy from long to short wavelengths in directions predominantly perpendicular to the background magnetic field. If the resulting fluctuations are kinetic Alfvén waves, then the Gary and Borovsky (2004) results indicate that (in the low-beta solar corona) electron Landau damping is the most likely consequence. Computer simulations of KAWs clearly show electron heating, and therefore MHD turbulence in the corona may not easily result in heating or acceleration of the ions. On the other hand, several different hybrid simulations (Liewer et al., 2001; Ofman et al., 2002; Xie et al., 2004) already demonstrated that Alfvén-cyclotron waves at parallel propagation can provide strong perpendicular heating of the protons and heavy ions in association with the frequency sweeping mechanism. The differences between the turbulent cascade model and the frequency sweeping scenario deserve further discussion and detailed evaluation which is beyond the scope of this review. For further reading we therefore refer to the recent comprehensive articles by Cranmer and van Ballegooijen (2003, 2005), who presented a model of magnetohydrodynamic Alfvénic turbulence in the extended solar corona, which contains collisionless dissipation and anisotropic particle heating, and discussed the global properties of Alfvén waves in the solar atmosphere and fast solar wind.

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