The previous sections have enumerated several reasons why the study of the Sun, and its at- mosphere in particular, is of great interest. Waves are ubiquitous in such environment and are

deeply related to fundamental processes that take place in the Sun. The general characteris- tics of such processes are known in most of the cases but more specific details are yet to be determined and there are still open questions. The answers to those issues cannot be reached without a better understanding about the generation and propagation of waves. Hence, the goal of this Thesis is to make a step forward by investigating some of the effects that collisions between different species have on the properties of waves and instabilities. The main focus is put on partially ionized plasmas of the solar atmosphere and on the low-frequency Alfv´en modes and the high-frequency ion-cyclotron and whistler modes, in addition to the Kelvin-Helmholtz instability. The following paragraphs describe how the research performed in this Thesis is organized:

• Chapter 2 presents the equations that will be used throughout this Thesis and explains the reasons why that set of equations has been chosen instead of other alternatives. There are two main descriptions that can be used in the investigation of plasmas, namely kinetic and fluid theories. The results presented in this Thesis are based in the latter, the fluid approach. Within such description, there are several approximations that can be taken, depending on the degree of accuracy that may be needed to explain the properties and evolution of a given fluid. Moreover, even when they may be composed of various distinct species, plasmas can be analyzed from a single-fluid or a multi-fluid perspective. This work is based on a multi-fluid approach where each species is described by the so-called five-moment approximation, with the inclusion of the corresponding momentum and heat transfer terms due to elastic collisions. The interaction with electromagnetic fields is taken into account by a generalized Ohm’s law.

• The non-linearity of the model chosen for this research allows the study of perturbations with both small and large amplitudes. In this Thesis, such studies are performed by a combination of analytical methods and numerical simulations. The investigation of small-amplitude perturbations starts in Chapter 3, where a general dispersion relation for transverse waves in static homogeneous multi-component plasmas is derived and the details of the numerical code MolMHD are explained. Then, Chapter 3 focus on fully ionized plasmas and the effects that Coulomb collisions have in the propagation and damping of low- and high-frequency Alfv´enic waves and on ion cyclotron resonances. In this chapter, the multi-fluid model is applied to three regions of the solar atmosphere: an upper layer of the chromosphere, the lower corona and the solar wind at a distance of 1 AU.

• Chapter4continues with the investigation of small-amplitude perturbations, but applied to the more general case of partially ionized plasmas. In addition, the effects of resistivity or magnetic diffusivity, which were previously ignored, are taken into account. Following a similar procedure to that presented in the preceding chapter, the multi-fluid approach is used to analyze the properties of waves at several heights of the chromosphere and in a quiescent prominence. A comparison of the damping caused by the different collisional interactions (Coulomb and ion-neutral elastic collisions, and resistivity) and their depen- dence on frequency, wavenumber and polarization of the perturbation is performed. In addition, the great influence of Hall’s current in weakly ionized plasmas is studied. Fur- thermore, a non-linear effect is briefly analyzed: the dissipation of kinetic energy of the

initial perturbation and its transformation into internal energy of the plasma by means of the collisional friction.

• Chapter 5 is devoted to the investigation of large-amplitude perturbations. In the first place, numerical simulations of standing waves excited by an impulsive driver in a plasma with prominence conditions are performed. In contrast with the small-amplitude case, it can be seen that noticeable fluctuations in the pressure and density of the plasma are generated. This fact is explained in terms of the ponderomotive force created by non-linear Alfv´en waves, and comparisons between the fully and partially ionized cases are performed. Then, the heating due to ion-neutral collisions caused by a Gaussian perturbation is studied as a function of its width.

• And last but not least, in Chapter6, a simplified version of the multi-fluid model is applied to the investigation of the Kelvin-Helmholtz instability in partially ionized magnetic flux tubes. A two-fluid approach is used to study the properties of waves in cylindrical promi- nence threads made of protons and neutral hydrogen when they are subject to shear flows. The influence of the collisional coupling on the onset of the instability is analyzed and comparisons between the growth times of the instability and typical lifetimes of threads are performed.

Finally, the last pages of this Thesis present a brief summary of the obtained results, out- lining the main conclusions of the research described in the previous chapters, and a discussion of possible lines of future work.

Multi-fluid theory

2.1

Motivation for a multi-fluid theory

The most fundamental description that has been developed in the research of plasmas is the kinetic one, an extension of the kinetic theory of gases. This description is of statistical nature, i.e., it can be applied when a large number of particles is present in the system of interest. It is based on the use of a distribution function, f (x, v, t), that describes how the particles are allocated at a given time throughout the phase space, that is the space of all possible values of position, x, and velocity, v. Then, the dynamics of each species s of the plasma is obtained by solving the Boltzmann transport equation, which describes the statistical behavior of a thermodynamic system. This equation is given by

∂fs ∂t + vs· ∇fs+ F ms · ∇vs fs =  ∂fs ∂t  collisions , (2.1)

where ∇vs is the gradient in the velocity space, i.e., ∇vs ≡ (∂/∂vs,x, ∂/∂vs,y, ∂/∂vs,z)

T

. The function F is the sum of forces acting on the particles, whose mass is given by ms. The term in

the right hand side accounts for the variation in the distribution caused by collisions between particles. Macroscopic quantities such as number densities, velocities and pressures can then be obtained by integrating over velocity space the resulting distribution function. For instance, the number density and the average velocity are defined as

ns(x, t) ≡ Z fs(x, v, t) d3v (2.2) and us≡ 1 ns(x, t) Z vsfs(x, v, t) d3v, (2.3) respectively.

The kinetic approach is the most accurate but, due to its complexity, it is not the most efficient under some circumstances in which there are alternatives that yield the same results by means of much simpler computations. One of such alternatives is the MHD theory, which is based on macroscopic quantities instead of distribution functions. In magnetohydrodynamics, the plasma is treated as a continuum, which means that the length-scales of its dynamics is larger than the cyclotron radii. It is also assumed to be in thermodynamic equilibrium, i.e.,

the time-scales are much larger than the collision times between the components of the plasma and the length-scales are also larger than the mean free path of the particles. Regarding the investigation of oscillatory motions, ideal MHD is applicable when the frequencies of the waves are much lower than the electron and ion plasma frequencies, ωpe and ωpi, respectively, and the

ion gyrofrequencies. The latter are given by Equation (1.14), while the former are computed as

ωpe,i =

s

ne,iZe,i2 e2

ǫ0me,i

, (2.4)

and represent the rate at which electrons and ions react to changes in the electrostatic potential. The two descriptions mentioned in the previous paragraphs are not unrelated, but the fluid approach can be derived from the kinetic theory. The fluid equations that rule the evolution of macroscopic quantities like the density, velocity, pressure and energy can be obtained by taking velocity moments of the Boltzmann equation (see, e.g., Braginskii [1965], Bittencourt [1986],

Meier [2011], Khomenko et al.[2014]).

Since its origin with the prediction by Alfv´en [1942] about the existence of low-frequency waves driven by magnetic tension, ideal MHD has become a remarkable successful theory for understanding the general properties of plasmas. For instance, Alfv´en’s prediction was soon experimentally confirmed in a laboratory by Lundquist [1949] and corroborated later by Lehnert [1954] and Jephcott [1959]. Then, Alfv´en waves have also been found in Earth’s aurora (Chmyrev et al. [1988]), in planetary ionospheres (Berthold et al. [1960], Gurnett and Goertz [1981]), the interstellar medium (Arons and Max [1975], Balsara [1996]), and in the solar atmosphere, as already mentioned in Section 1.3.

However, ideal MHD is not accurate enough when applied to the investigation of waves with higher frequencies, which in the solar atmosphere may be driven by small-scale magnetic activity in the chromospheric network and reconnection of field lines (Axford and McKenzie

[1992],Tu and Marsch[1997]) or by cascading from low frequencies in the corona (Isenberg and Hollweg [1983], Tu [1987]). Ideal MHD resorts to simplifications such as treating the plasma as if it were fully ionized, that is ignoring the possible presence of neutral particles, neglecting Hall’s term in the induction equation and considering all ionized species together as a single fluid. Those approximations are reasonable when the frequencies of the oscillations are much lower than the collision frequencies between the different species in the plasma. In that case, collisions cause a strong coupling between all species and they can be treated as a single-fluid. On the other hand, when the oscillation frequencies are larger (or, equivalently, when the relaxation time of collisions between unlike species is much larger than the period of the waves), each species may react to perturbations in different time-scales and the coupling is weaker than before. At that range of frequencies, the use of a multi-fluid theory is more appropriate. The multi-fluid approach consists in a set of equations for each species of the plasma, with addi- tional terms that describe the interactions between them. Depending on the specific equations employed and the interactions considered, a multi-fluid model may be applicable up to frequen- cies of the order of the ion-cyclotron frequencies, but does not reach the range of electron or ion plasma frequencies, in which the kinetic theory is required. Nevertheless, the applicability range of the multi-fluid theory is enough to study the phenomena of interest for this Thesis. Hence, in the present work a multi-fluid model that takes into account the effect of partial ionization, Hall’s current and resistivity is used. The equations that compose such model are detailed in the following section.

In document High-frequency waves and instabilities in multi-fluid partially ionized solar plasmas (Page 37-45)