## Abstract

We study wave propagation in the low-β coronal plasma using a collisionless multi-fluid model. Neglecting the electron inertia, this model allows us to take into account ion-cyclotron wave effects that are absent in the magnetohydrodynamics model. To accomplish this, we perform a Fourier plane-wave perturbation analysis. Solving numerically the dispersion relations obtained from a two- and three-fluid model, dispersion curves for representative parameters of the solar corona are presented. The results reveal the presence of resonance frequencies that might play a role in coronal heating.

## 1. Introduction

Evidence for magnetohydrodynamics (MHD) waves in the solar corona has been revealed recently by Solar and Heliospheric Observatory (SOHO) and Transition Region and Coronal Explorer (TRACE) observations. In order to interpret these observations, progress in terms of waves modelling has to be made. MHD is a large-scale theory dealing with wave frequencies well below the ion-cyclotron frequency *Ω*_{i}. This theory does not allow one to address the microphysics of the dissipation, which occurs at smaller time and length scales (Marsch *et al*. 2003). In order to describe these scales, we propose in the present study to use a multi-fluid model. Neglecting the electron inertia, this model permits the consideration of frequencies of the order of *Ω*_{i} and is often referred to as Hall-MHD, since the Hall term is retained in the generalized Ohm's law (Goossens 2003). The Hall term introduces dispersion and is important if the wavelength of interest, *λ*_{A}, is of the order of the gyro-radius *r*_{Ai} of an ion moving at the Alfvén velocity *v*_{A}, i.e. . In the weakly collisional corona *r*_{Ai} is much smaller than the collisional mean free path *λ*_{coll}, *r*_{Ai}≪*λ*_{coll}. Thus *λ*_{A}≪*λ*_{coll}, which justifies the collisionless limit we are using in the present study.

## 2. Basic equations and dispersion relation

The collisionless fluid equations for a particle species j are:(2.1)(2.2)(2.3)where *m*_{j}, *n*_{j}, *q*_{j}, **v**_{j}, *p*_{j} and *γ*_{j} are respectively the mass, density, electric charge, velocity, pressure (which for simplicity is here assumed to be isotropic) and the polytropic index of a species j. Subscript ‘j’ stands for electron e, proton p or alpha particle α (He^{2+}). The electric field * E*, the magnetic field

*and the current density*

**B***are given by Faraday's and Ampere's law (neglecting the displacement current):(2.4)Considering the quasi-neutrality, we perform a Fourier plane-wave perturbation analysis and obtain the dispersion relation using the dielectric approach (Stix 1992).*

**J**The dispersion relation is solved numerically for representative parameters of the solar corona and oblique propagation, *θ*=30°. The dispersion diagram for a two-fluid (*a*,*c*) and three-fluid (*b*,*d*) model are shown in figure 1. For comparison, cold plasma results are also presented (*a*,*b*). In the two-fluid model three modes are present, which are the extensions of the usual Slow (dotted line), Alfvén (dashed line) and Fast (solid line) MHD modes into the high-frequency domain around *ω*≈*Ω*_{p} (*Ω*_{p}=*eB*_{0}/*m*_{p}) where they are dispersive. The Alfvén mode, which in the cold plasma experiences a resonance at *ω*=*Ω*_{p}, approaches the proton-cyclotron resonance, but due to the finite pressure it couples to the acoustic branch, which enables it to emerge from this resonance frequency as the ion acoustic mode at higher frequencies. Note that the Slow mode experiences a resonance at *ω*≈*Ω*_{p} cos *θ*. This result, which has been verified numerically for different values of oblique propagation *θ*, confirms the earlier results obtained by Stringer (1963) who studied waves in a low-β dissipationless two-fluid plasma and showed both numerically and analytically that for large wavenumber the Slow mode is in a resonance regime at *ω*≈*Ω*_{p} cos *θ*. The three-fluid dispersion diagram shows the presence of five modes, with the occurrence of the phenomenon of mode coupling (or conversion). It also shows the appearance of a cutoff frequency corresponding to the Fast mode (in terms of phase velocity ordering). The resonances experienced in the cold plasma case by the two intermediate modes (dotted and dashed line) at *Ω*=*Ω*_{α} and *Ω*=*Ω*_{p} disappear in the warm plasma case due to the presence of a finite pressure term. Note in the warm plasma case the appearance of two Slow modes. The numerical results show that these modes experience resonance frequencies which, similarly to the two-fluid Slow mode, depend on the angle of propagation *θ*. Indeed, the first Slow mode (dashed-dot-dot line) experiences a resonance at *ω*≈*Ω*_{α} cos *θ* (with *Ω*_{α}=*Ω*_{p}/2), and the second Slow mode (dashed-dot line) approaches the same resonance but then couples to the first Slow mode and experiences a resonance only at *ω*≈*Ω*_{p} cos *θ*. Mode coupling effects, the appearance of cutoff and resonance frequencies, are commonly encountered effects in the multi-ions plasma models (Li & Habbal 2001; Mann *et al*. 1997).

## 3. Conclusion

Coronal waves propagating in a collisionless two- and three-fluid plasma have been studied. In the two-fluid model, we discussed the extension of the three standard MHD modes (Slow, Alfvén, Fast) to the high-frequency domain (*ω*≈*Ω*_{p}) where they are dispersive. The oblique Slow mode shows a resonance frequency at *ω*≈*Ω*_{p} cos *θ*. Five modes exist in the three-fluid model. They are subject to mode coupling or mode conversion and show the appearance of a cut-off frequency and an additional resonance frequency at *ω*≈*Ω*_{α} cos *θ*. All these effects, and particularly the possible wave absorption and dissipation near the resonance frequencies of minor heavy ions might play a role in coronal heating (Li & Habbal 2001).

## Footnotes

One contribution of 20 to a Discussion Meeting Issue ‘MHD waves and oscillations in the solar plasma’.

- © 2005 The Royal Society