Exploring the parameter space

In this section, a study of the dependence of the solutions of the dispersion relation with respect to various physical parameters will be performed. In addition, those results will be compared with the analytical approximation shown in the previous section and the range of validity of the latter will be checked. To keep this analysis as general as possible, dimensionless parameters will be used here, while actual physical parameters will be employed later in Section 6.3.3, in which an application to solar prominence threads will be given.

Here, for simplicity, the collision frequency is chosen to have the same value in both internal and external plasmas, i.e., νni,0 =νni,ex ≡νni. In addition, the focus is put on the kink mode;

hence,m = 1.

The exploration of the parameter space is started with the study of the dependence of the solutions of the dispersion relation with respect to the shear flow velocity. Thus, ∆U is taken as a free variable. Then, the dispersion relation is solved using the following parameters:

ρi,0/ρi,ex = 2, ρn,0/ρn,ex = 2, ρn,0/ρi,0 = 1, cA,0 = 1 and kza = 0.1. The top and the bottom panels of Figure 6.5 display the normalized real and imaginary parts of the frequency, respec-tively, as functions of the normalized shear flow velocity, ∆U/c_A,0 for three different degrees of coupling. The normalization parameter ωk is known as the kink frequency, which corresponds to the frequency of the kink wave in the TT limit for a fully ionized plasma (see, e.g., Ryutov and Ryutova [1976],Spruit [1981]), and is given by

ωk =kz

sρi,0c²_A,0+ρi,exc²_A,ex ρi,0+ρi,ex

. (6.53)

Panels a), b) and c) correspond to the cases of weak coupling (ν_ni/ω_k = 0.1), intermediate coupling (νni/ωk = 1), and strong coupling (νni/ωk = 10), respectively. The red symbols

Figure 6.5: Upper panels: ωR/ωk as a function of the normalized shear flow velocity, ∆U/cA,0, for kza = 0.1, m = 1, and three different collision frequencies (a) νni/ωk = 0.1, b) νni/ωk = 1 and c) νni/ωk = 10). Lower panels: ωI/ωk as a function of ∆U/cA,0 for the same set of parameters as above. The symbols represent the solutions of the full dispersion relation, i.e., Equation (6.49); the blue solid lines correspond to the analytical approximation given by Equation (6.52), and the blue dashed lines show the unstable branch of the neutral fluid when there is no coupling (Equation (6.15)).

represent the solutions obtained numerically from the complete dispersion relation, Equation (6.49). This results are compared with the analytical solutions in the strongly coupled limit, which are represented by the blue solid lines, and in the uncoupled case, displayed as blue dashed lines. In addition, the classical shear flow velocity threshold for the KHI in a fully ionized plasma, given by Equation 6.25, is represented by the vertical dotted lines.

By inspecting the top panels of Figure6.5, it can be seen that the real part of the frequency has a very similar behavior in the three studied cases. Initially, when the shear flow velocity is zero, there are two solutions with nonzero ωR. These solutions are associated with the ionized fluid and correspond to the usual kink magnetohydrodynamic waves found in fully ionized tubes (Edwin and Roberts [1983]): the solution with ωR > 0 is the forward-propagating kink wave, while the one with ωR < 0 is the backward-propagating wave. A third solution with ωR > 0 emerges when the shear flow velocity increases from zero. It is associated with the neutral component of the plasma in the sense that it only appears in the presence of neutrals.

However, it must be noted that such simple associations between solutions and fluids cannot be made when the coupling is high and ions and neutrals behave as a single fluid. As the flow velocity continues to increase, the three solutions converge for a critical flow velocity that depends on the collision frequency. The stronger the ion-neutral coupling, the lower the critical flow. From that point on, the real part of the frequency is proportional to ∆U and is well

described by the real part of Equation (6.15) or, equivalently, Equation (6.52).

As it can be seen in the lower panel of Figure 6.5, the degree of coupling has a much more remarkable effect on the damping and growth rates of the perturbations than on their oscillation frequencies. The shaded zone in those panels highlights the region with ωI > 0, which corresponds to the area where the solutions are unstable and exponentially grow with time. Any of the three panels reveals that for low shear flow velocities there is only one unstable solution, corresponding to that originally associated with the neutral component of the plasma.

A second unstable branch (originally associated with ions) appears for higher flow velocities above the classical super-Alfv´enic threshold. By comparing the three panels, it is possible to conclude that ion-neutral collisions reduce to a great extent the growth rate of the instability that appears for sub-Alfv´enic flows but are not able to completely suppress it (Watson et al.

[2004],Soler et al.[2012b]). In addition, the analytical approximation shows that, in its range of applicability, the growth rate is directly proportional to the square of the shear flow velocity and inversely proportional to the ion-neutral collision frequency. As expected, the approximation agrees well with the numerical results for small shear flow velocities. For weak coupling, the approximation is reasonably good for flow velocities up to 40% of the internal Alfv´en speed.

When the collision frequency is increased, the range of agreement between the numerical results and the approximation is greatly extended even to super-Alfv´enic speeds.

Now, to investigate in more detail the effect of ion-neutral collisions on the instability for slow flows, the shear flow velocity is fixed to the value ∆U/cA,0 = 1 and the solutions of Equation (6.49) are computed as functions of the collision frequency, νni. The chosen flow velocity is below the classical threshold for the KHI in fully ionized plasmas (Chandrasekhar [1961]), so that in principle only the neutral component is unstable in this configuration. The results of this study are displayed in Figure 6.6, where the solutions originally associated with

Figure 6.6: a) ωR/ωk and b) ωI/ωk for the kink mode (m=1) as a function of νni/ωk, with

∆U/cA,0 = 1 and kza= 0.1. The red diamonds are the solutions originally associated with the ions when there is no coupling, while the blue crosses are the solutions for neutrals. The solid line is the analytical approximation given by Equation (6.52). In b) the shaded area denotes the region of instability.

ions are shown as red diamonds and those originally associated with neutrals are plotted with blue crosses. The right panel shows that there is always one unstable solution for any value of νni, although its growth rate decreases when the collision frequency increases. The growth rate is reduced because neutrals feel indirectly, through the collisions with ions, the stabilizing effect of the magnetic field. In addition, as discussed before, the analytical approximation for the growth rate, agrees well with the numerical results for high values of the collision frequencies, as is consistent with the assumptions behind the approximation. Regarding the real part of the solutions, it can be checked that the absolute value of the frequency of the modes associated with ions decreases until it reaches a plateau for νni/ωk > 1 (Soler et al. [2013c]), while the frequency of the solutions associated with the neutral stays constant all over the range. The reason of the decrease in frequency of the former is that, due to the strong collisional coupling, the two species oscillate together as a single fluid. Hence, the inertia of neutrals is added to that of ions and the Alfv´en frequency depends on the total density of the plasma (Kumar and Roberts [2003], Soler et al. [2013b]).

For the sake of completeness, the next step is to study the dependence of the solutions to Equation (6.49) on the azimuthal wavenumber. As shown by Equation (6.50), the results in the TT limit are independent of the value of m for m 6= 0; this fact implies that in the range of applicability of that approximation there will not be substantial variations in the behavior of the different modes. Hence, to observe some dissimilarities, it is necessary to choose parameters beyond the TT case. Figure 6.7 shows the results corresponding to a case with the same densities and magnetic field as in the previous analyses but with a higher value of the dimensionless longitudinal wavenumber, namely,kza= 2. Five different modes are represented, which correspond to the following azimuthal wavenumbers: m = 0,m = 1, m= 2, m= 10 and m = 100. The left panel shows that the oscillation frequency decreases when m increases. In

Figure 6.7: Solutions of Equation (6.49) as functions of the normalized shear flow for a nor-malized wavenumber of kza= 2. The gray area on the right panel shows the region where the solutions are unstable. The modes with m = 0, m = 1, m = 2, m = 10 and m = 100 are represented by the blue solid lines, red diamonds, black dotted lines, black dashed lines and green triangles, respectively.

contrast, the right panel reveals that the modes with larger growth rates are those with higher m. However, the dependence of the solutions on the azimuthal wavenumber is weak, specially when modes with large m are compared with each other.