Parametric Couplings between Kinetic Alfvén and Dispersive Ion-acousticWaves in the Solar Corona

G. Brodin∗, L. Stenflo∗ and P. K. Shukla∗
∗Dept. of Physics, Umeå university, SE-901 87 Umeå, Sweden
Abstract. The resonant interaction between kinetic Alfvén and ion-acoustic waves is considered using the Hall-MHD theory.
The results of previous authors are generalized to cover both finite Larmor radius as well as the ideal MHD results. It is found
that the three wave coupling is strongest when the wavelength is comparable to the ion-sound gyroradius. Applications of our
work to weak turbulence theories as well as to the heating of the solar corona are pointed out.
Keywords: Kinetic Alfvén waves, ion-sound waves, Hall-MHD equations
PACS: 52.35.Mw
The nonlinear interaction of magnetohydrodynamic (MHD) waves has been considered by numerous authors (see for example,
Sagdeev and Galeev, 1969; Petviashvili and Pokhotelov, 1992; Shukla and Stenflo, 1999). The applications involve fusion plasmas
(Hasegawa and Uberoi, 1982), space physics (Petviashvili and Pokhotelov, 1992; Shukla and Stenflo, 1999;Wu and Chao, 2004)
as well as solar physics (Shukla et al., 1999; Voitenko and Goossens, 2000 and 2002; Shukla and Stenflo, 2005; Chandran 2005)
and astrophysics (Ng and Bhattacharjee, 1996; Goldreich and Sridhar, 1997). The classic work on three wave interaction of ideal
MHD waves (Sagdeev and Galeev, 1969) was later generalized to account for arbitrary (but still ideal) MHD wave modes and
directions of propagations (Brodin and Stenflo, 1988). The ideal MHD processes were soon suggested to have applications for
the heating of fusion plasmas (Lashmore-Davies and Ong, 1974). Hasegawa and Chen (1976a) showed, however, that processes
involving kinetic Alfvén waves were more efficient for that purpose. The latter waves can be described by the Hall-MHD theory,
and general three wave coupling coefficients for the Hall-MHD plasmas were thus deduced by Brodin and Stenflo (1990).
Applications for the parametric decay instability of magneto-acoustic waves into two kinetic Alfvén waves, to the heating of the
solar corona, were considered by Voitenko and Goossens (2002). The Joule electron heating caused by high-frequency dispersive
Alfvén waves in the solar corona was also analysed by Shukla et al. (1999).
Much of the previous work describing parametric instabilities involving kinetic Alfvén waves (KAWs) has adopted a kinetic
theory (Hasegawa and Chen, 1976; Voitenko 1998) or multi-fluid models (Erokhin, Moiseev and Mukhin, 1978; Voitenko and
Goossens, 2002). In the present paper, we will however demonstrate that the essential characteristics of the three-wave decay
interaction involving the KAWs can be more simply described within a unified formalism of the Hall-MHD theory. An important
result of that formalism is that the decay of kinetic Alfvén waves is dominated by the excitation of modes with short perpendicular
wavelengths, of the order of the ion-sound gyroradius, that must be described by the Hall-MHD theory. We shall show that this
specific example has general significance, and that the ideal MHD typically is unable to deal with the nonlinear evolution of the
MHD waves, even if the initial conditions are within the range of the ideal MHD.

where r2 is the density perturbation of wave 2, v1,3 is the magnitude of the velocity of waves 1 and 3, respectively, q is the angle
between k1⊥ and k3⊥ (or the angle between v1 and v3 when k1,3⊥ →0), and vg j is the group velocity of wave j. The first two
terms in (11) dominate for k2j
⊥c2s
≪w2
ci, and agree with the ideal MHD coupling coefficient of Brodin and Stenflo (1988) in the
low-beta limit considered here. The third term, which dominates for large perpendicular wavenumbers, agrees with the coupling
coefficient of Hasegawa and Chen (1976), which was derived using a kinetic approach. As a specific example, we let wave 3
be a pump wave. To demonstrate the importance of the second term in (11), we assume that all waves have large perpendicular
wavenumbers, such that k2

ci ∼ 1. Furthermore, to facilitate an order of magnitude estimate of (11) we let k1⊥ and k3⊥ be
approximately perpendicular to each other. In this case, the magnitude of the third part of CAmA can be estimated as

which is much larger than the first two parts of CAmA accounted for by the ideal MHD, and which do not exceed unity. As
a consequence, the growth rate GHM into short perpendicular wavelength modes (with k2

MHD theory) is larger than the growth rate GIM due to the ordinary ideal MHD modes by a factor GHM/GIM ∼ wci/w. Thus,
the increased coupling strength into short wavelength modes with perpendicular wavelengths of the order of the ion-sound
gyroradius (cs/wci) affects the parametric decay processes significantly. This is very important as the wave cascade processes
(Goldreich and Sridhar, 1997) of weak turbulence theories are based on the resonant three-wave coupling mechanism. Moreover,
while the general features of such processes lead to a broadening of the frequency spectrum, and energy transfer towards lower
frequencies, we note that the energy transfer will mainly occur in the direction of higher coupling strength, i.e. into modes
with short perpendicular wavelengths. Thus, even for an initial turbulent spectrum well within the range of the ideal MHD, wave
cascade processes will eventually lead to the excitation of short perpendicular wavelengths and the necessity to use the Hall-MHD
rather than the ideal MHD.
Assuming that wave 3 is a pump wave with magnetic field magnitude B3, and using the estimate B3 ∼ v3B0/cA, we find the

It should, however, be pointed out that the present decay channel for a KAW into an ion-acoustic wave and another KAW is
not unique. Other decay channels that have been investigated for KAWs can compete with it (e.g. Voitenko and Goossens 2000;
Onishchenko et. al. 2004). These processes can spread out the KAW spectrum and thus prevent the parametric decay into ionacoustic
waves. To find out the relative importance of the decay into ion-acoustic waves as compared to the above mentioned
processes, we should therefore compare our estimate (13) with the growth rates GAA of Voitenko and Goossens (2000), and GJGR
of Onishchenko et al. (2004). We then use the estimates GAA ∼ 0.2w3k2
3riB3/k3zB0, where ri = (Ti/mi)1/2 /wci is the ion Larmor

, where D is a factor of order unity (Onishchenko et al., 2004). Onishchenko et al. (2004)
showed that GJGR is smaller than GAA if B3/B0 is smaller than a factor of the order k3zri. A comparison between GHM of the
present paper and GAA reveals that GHM/GAA ∼ 5wcik3z/w3k2
3ri. Although the estimates above are very crude, they show that the
process we consider in the present paper can be even more important than those of previous papers for a significant range of
parameters.
Our coupling coefficient (11) includes both the ideal MHD results of Brodin and Stenflo (1988), and the effects due to the
kinetic approach of Hasegawa and Chen (1976a), in a unified formalism. As can be seen from (11), the wave coupling is
strongest for perpendicular wavelengths of the order of the ion-sound gyroradius. As has been argued above, this has important
consequences for several processes, such as for the parametric decay instabilities and wave cascades in weak turbulence theories.
Moreover, the formalism presented above is relevant for plasma particle energization in the solar corona by kinetic Alfvén
waves. In the solar corona, a kinetic Alfvén pump wave can be excited by a linear transformation of an Alfvén surface wave
in the neighbourhood of the resonance region (Hasegawa and Chen, 1976b). The mode converted kinetic Alfvén wave can then
further decay into a daughter kinetic Alfvén wave and a dispersive ion sound wave, as described here. The nonlinearly excited
kinetic Alfvén waves can attain large amplitudes and small perpendicular wavelengths (Hasegawa and Chen, 1976b), and they
could therefore be our most efficient agents for energization of ions and electrons by kinetic Alfvén wave phase mixing and Joule
heating (Hasegawa and Uberoi, 1982; Shukla et al., 1994; Cramer, 2001), as well as for turbulent heating and particle-KAW
interactions.
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