The addition of a third ion to the plasma increases the complexity of the problem to be analyzed, which is reflected in a much longer and more complicated formula for the dispersion relation due to the presence of a great number of additional terms associated to collisions. Hence, it will not be shown here. Nonetheless, the abridged version in which the elastic collisions between the different ions are neglected can be used to analyze some of the general properties. This simpler collisionless version is given by
ω2 ± Z1n1(ω±∓ Ω2) (ω±∓ Ω3) + Z2n2(ω±∓ Ω1) (ω±∓ Ω3) + Z3n3(ω±∓ Ω1) (ω±∓ Ω2) ±Bxk2x eµ0 (ω±∓ Ω1) (ω±∓ Ω2) (ω±∓ Ω3) = 0. (3.28) The dispersion relations for waves in a three-ion plasma are fourth-degree polynomials in ω. Thus, for the case of waves generated by an impulsive driver, an additional oscillation mode appears for each polarization with respect to the system with only two ions. By exploring analytically the limit of small wavenumbers, as in Section 3.3.1, it can be checked that the new mode is related to the cyclotron frequencies. Each polarization still has only two Alfv´enic modes, with ω ≈ ±ωA, where the Alfv´en speed is computed using the sum of the densities of
the three ions. The frequencies of the remaining modes are ω± = ±eΩ1 and ω± = ±eΩ2, where
Ω1 and eΩ2 are the solutions to
Z1n1(ω − Ω2) (ω − Ω3) + Z2n2(ω − Ω1) (ω − Ω3) + Z3n3(ω − Ω1) (ω − Ω2) = 0, (3.29)
and are given by e Ω1 = Z1n1(Ω2+ Ω3) + Z2n2(Ω1+ Ω3) + Z3n3(Ω1+ Ω2) 2ne −2n1 e Z1n1(Ω2+ Ω3) + Z2n2(Ω1 + Ω3) + Z3n3(Ω1+ Ω2) 2 −4ne(Z1n1Ω2Ω3+ Z2n2Ω1Ω3+ Z3n3Ω1Ω2) 1/2 (3.30) and e Ω2 = Z1n1(Ω2+ Ω3) + Z2n2(Ω1+ Ω3) + Z3n3(Ω1+ Ω2) 2ne + 1 2ne Z1n1(Ω2+ Ω3) + Z2n2(Ω1 + Ω3) + Z3n3(Ω1 + Ω2) 2 −4ne(Z1n1Ω2Ω3+ Z2n2Ω1Ω3+ Z3n3Ω1Ω2) 1/2 . (3.31)
In contrast, no additional solution appears in the case of a periodic driver, although a third resonance is present when ω±= ±Ω3.
As already mentioned in the previous section, the abundances of protons and doubly ionized helium in the solar corona and the solar wind are much larger than the abundances of other ions. Hence, the addition of a third ion would hardly modify the results from the two-ion model when applied to those two environments. However, the presence of a third ion can have a significant effect in upper chromosphere. The model F of Fontenla et al.  predicts that at a height of ∼ 2016 km over the top of the photosphere the number densities are np ≈ 7 × 1016 m−3,
nHe ii ≈ 6 × 1015 m−3, and nHe iii ≈ 1015 m−3. Therefore, at that height the contribution
of the three ions should be considered, although protons are still the dominant species. The temperature at that height is T ≈ 2 × 104 K and the magnetic field is B
x ≈ 35 G. This
set of parameters leads to the following collision and cyclotron frequencies: νpHeii ≈ 2000 Hz,
νHeiip ≈ 5840 Hz, νpHeiii ≈ 1260 Hz, νHeiiip ≈ 22100 Hz, νHeiiHeiii ≈ 540 Hz, νHeiiiHeii ≈ 3250 Hz,
Ωp = 335268 rad s−1, ΩHeii = 83817.1 rad s−1, and ΩHeiii = 167634 rad s−1. The Alfv´en speed
is cA ≈ 244 km s−1.
The results of the study of waves generated by an impulsive driver are shown in Figure 3.6. Once more the solutions to the collisionless dispersion relation are not plotted because there are no appreciable differences in the real part of the frequency (left panel) with the case in which collisions are included and because the imaginary part is equal to zero if friction is neglected. Again, only the solutions with ωR > 0 are displayed.
Figure 3.6: Solutions to the dispersion relations of a three-ion plasma with np = 7 × 1016 m−3,
nHeii = 6 × 1015 m−3, nHeiii = 1015 m−3, Bx = 35 G, νpHeii ≈ 2000 Hz, νHeiip ≈ 5840 Hz,
νpHeiii ≈ 1260 Hz, νHeiiip ≈ 22100 Hz, νHeiiHeiii ≈ 540 Hz, and νHeiiiHeii ≈ 3250 Hz. Left:
normalized real part of the frequency as a function of the normalized wavenumber. Right: absolute value of the normalized damping as a function of kxcA/Ωp. Red dashed lines represent
the L modes and the red solid line represents the R mode. The black thin line corresponds to the solution of ideal MHD.
In the limit of small wavenumbers, it can be seen that two of the solutions of the multi-fluid model coincide with the Alfv´en frequency provided by the single-fluid description, while the
Figure 3.7: Solutions of the dispersion relations for waves generated by a periodic driver in a three-ion plasma with the same parameters as those used in Figure 3.6.
other two are given by the values eΩ1 and eΩ2. When kxcA/Ωp increases, the Alfv´enic L mode
(represented by the dashed line) turns into an ion cyclotron mode and its frequency tends to the lower cyclotron frequency. This is the same behavior as that found in the two-ion description). The remaining L modes tend to the limiting values ΩHe iii and Ωp, and they conserve their
order: the mode associated with eΩ2 (which is larger than eΩ1) tends to the upper cyclotron
frequency. Finally, the Alfv´enic R mode becomes the whistler wave and its frequency is always higher than the Alfv´en frequency.
The right panel of Figure 3.6 shows that the R mode is again the less affected by collisions and its normalized damping is |ωI|/Ωp < 10−5 for very small and very large wavenumbers, with
a maximum of |ωI|/Ωp ≈ 10−3 around kxcA/Ωp = 1. For small wavenumbers, the solutions
with stronger damping are those related to eΩ1 and eΩ2: the damping of the former (represented
by the dotted-dashed line) decreases with the wavenumber until it reaches a minimum around kxcA/Ωp = 0.5, then increases again and becomes constant for very large wavenumbers, with
|ωI|/Ωp ≈ 0.75. The damping of the mode associated with eΩ2 decreases very fast in the region
around kxcA/Ωp = 1 and then stabilizes in |ωI|/Ωp ≈ 0.025. Finally, the damping of the
Alfv´enic L mode increases with the wavenumber until it reaches the value |ωI|/Ωp ≈ 0.06 when
kxcA/Ωp ≫ 1.
The analysis of waves generated by a periodic driver is illustrated by Figure 3.7. As be- fore, only the solutions with kR > 0 are plotted. It can be noted that the collisionless L
mode, represented by the black dotted-dashed line, exhibits the expected three resonances: the wavenumber tends to infinity at the three cyclotron frequencies. There are also three cutoff re- gions, instead of the two cut-offs that exist in the two-ion case. The wavenumber is equal to zero in the following intervals: ω ∈ΩHeii, fΩ1
, ω ∈ΩHeiii, fΩ2
or ω > Ωp, where fΩ1 ≈ 0.3Ωp and
Ω2 ≈ 0.52Ωp. The solutions corresponding to the case with νst 6= 0 show that the singularities
are substituted by extrema of the normalized wavenumber, where the highest peak corresponds to the most abundant species, i.e., protons. Again, the momentum transfer removes the cutoff
regions. Regarding the R mode, the same behavior explained for the case of two-ion plasmas is found here: there are no resonances, the normalized wavenumber increases with the frequency and the spatial damping is inefficient in the whole frequency range.
Therefore, the overall results obtained in the three-ion model appear as natural extensions to the results of the two-ion case. Hence, the generalization to plasmas with a larger number of ions seems straightforward.