Model, equations and derivation of the dispersion relation

Here, the equilibrium state is a partially ionized cylindrical magnetic flux tube of radius a embedded in an unbounded medium. The system is described by means of cylindrical coordi-nates, namely r,ϕ, andz,for the radial, azimuthal, and longitudinal coordinates, respectively.

A sketch of the model can be found in Figure 6.4. The subscripts “0” and “ex” denote quan-tities related to the internal and the external plasma, respectively. The densities of ions and neutrals areρi and ρn and only depend on the radial direction as

ρi(r) =

ρi,0 if r ≤a,

ρi,ex if r > a, (6.26)

ρn(r) =

ρn,0 if r ≤a,

ρ_n,ex if r > a. (6.27)

Hence, there is an abrupt jump in density between the internal and external plasmas. The

Figure 6.4: Sketch of the model

magnetic field is constant and pointing along the flux tube axis, with the same value in both media, B₀(r, ϕ, z) = (0,0, B₀)^T. In addition, there is a longitudinal mass flow with constant velocity denoted byU. The flow velocity is discontinuous at the boundary of the flux tube.

To study this system, a two-fluid theory, i.e., a simplified version of the system of equations presented in Chapter2, is used. It assumes that the plasma is composed of an ionized fluid made of ions and electrons and a second fluid made of neutral particles only. Those two fluids may interact by means of momentum-transfer collisions between ions and neutrals. In addition, only small-amplitude perturbations are studied, so the linearized version of the equations is employed. The set of two-fluid equations that describe the behavior of linear incompressible perturbations superimposed on the equilibrium state is

ρi

∂

∂t +U ∂

∂z

V_i=−∇Pie+ 1 µ0

(∇ ×B1)×B0−ρnνni(Vi−V_n), (6.28) ρn

∂

∂t +U ∂

∂z

V_n=−∇Pn−ρnνni(Vn−V_i), (6.29)

∂

where V_i and V_n are the velocity perturbations of ions and neutrals, and Pie and Pn are the pressure perturbations of the ion-electrons and neutrals fluids.

Once again, it is helpful to use the Lagrangian displacements instead of the velocities as primary variables. In the present investigation, the relation between the perturbations of velocity and the Lagrangian displacements is given by

V_i= ∂ξi

The next step in the procedure to obtain the dispersion relation is to perform a normal mode analysis. Since the equilibrium is uniform in the azimuthal and longitudinal directions, the perturbations are expressed as proportional to exp(imϕ+ikzz), where m and kz are the azimuthal and longitudinal wavenumbers, respectively. The dependence of the perturbations on the radial direction is retained and the temporal dependence is set as exp(−iωt), where ω is the angular frequency.

From the radial component of Equations (6.33), after combining it with the induction equa-tion, and (6.34), it is possible to obtain the following expressions:

∂PT frequency, and the variable PT is the sum of the thermal and magnetic pressures of the ionized fluid, and is defined as

PT =Pie+B0·B1

µ =Pie+B0B1z

µ₀ . (6.39)

Then, the azimuthal and longitudinal components of Equations (6.33), (6.34) and (6.35) can be combined with the incompressibility condition, Equation (6.36), to obtain the other two equations that are needed to solve the system, namely,

ρi

where the parameter Θ, which will be referred to as the modified frequency, is defined throughe the following relation: Now, the combination of Equations (6.37), (6.38), (6.40), and (6.41) leads to two uncoupled equations for the pressures, namely

whose solutions are combinations of modified Bessel functions of the first and second kind, Im(kzr) and Km(kzr), respectively. It is required that the solutions are regular at r = 0 and where A1 −A4 are arbitrary constants. In turn, the radial components of the Lagrangian displacements of the two fluids are related toPT and pn by

ξr,i = 1

The dispersion relation that describes the behavior of the waves in this system is found by imposing the conditions that PT, Pn, ξr,i and ξr,n are continuous at r = a, that is, at the boundary of the tube. After applying the boundary conditions, a system of algebraic equations for the constants A₁ −A₄ is obtained. The non-trivial solution to the system provides the dispersion relation, which is given by where the prime denotes the derivative of the modified Bessel function with respect to its argument.

For a given longitudinal wavenumber, Equation (6.49) yields an infinite number of oscillation modes, each one associated to a particular azimuthal wavenumber, m. Some of these modes are known with specific names due to their importance in the investigations of waves in flux tubes (see, e.g., Edwin and Roberts [1983]). For instance, the mode with m = 0, which produces expansions and contractions of the plasma tube but without displacing its axis, is called the sausage mode, and the mode with m = 1, which is the only one that causes displacements of the axis of the cylinder, is known as kink mode.

In the limit when k_za ≪ 1, which is known as the thin tube (TT) aproximation, the dispersion relation becomes much simpler and some of the previous known results about the KHI can be recovered. For any azimuthal wavenumber m 6= 0, if an asymptotic expansion of the modified Bessel functions for small arguments and only the first term in the expansion is kept, the TT dispersion relation is given by

ρi,0 Θ0(Θ0+iχ0νni,0)−ω_A,0²

+ρi,ex Θex(Θex+iχexνni,ex)−ω_A,ex²

×[ρn,0Θ0(Θ0+iνni,0) +ρn,exΘex(Θex+iνni,ex)]

+ [ρn,0Θ0νni,0+ρn,exΘexνni,ex]² = 0. (6.50) The expression above coincides with that already derived by Soler et al. [2012b] in their study of the KHI in a Cartesian interface. Hence, the geometrical effect associated with the cylindrical magnetic tube disappears when the TT approximation is considered. Furthermore, if the terms associated with the ion-neutral collisions are neglected, which means that ions are decoupled from neutrals, the dispersion relation becomes

ρi,0(Θ²₀−ω_A,0² ) +ρi,ex(Θ²_ex−ω_A,ex² )

[ρn,0Θ²₀+ρn,exΘ²_ex] = 0, (6.51) from which it is easy to recover the solutions corresponding to the classical hydrodynamic and magnetohydrodynamic KHI already shown in Section 6.2.

To obtain the solutions for the coupled case, the full dispersion relation is generally solved using numerical methods. However, it is possible to find an approximate analytical solution when the ion-neutral coupling is strong and sub-Alfv´enic flows are considered. It is known

that for sub-Alfv´enic flows, the only unstable solution in the uncoupled case is that associated with neutrals, since the magnetic field is able to stabilize ions. Thus, the way of obtaining the approximate solution is trying to find a correction to the neutrals-related solution due to ion-neutral collisions. To do so, the frequency is written as ω = ω₀ +iγ, where ω₀ is the neutrals’ unstable solution given by Equation (6.15) and γ is a small correction. The previous expression for the frequency is inserted into the TT dispersion relation and only the terms up to first order in γ and second order inU0 are kept, while the external flow velocity is taken to be zero, Uex = 0. After some algebraic manipulations, a solution for γ can be found and the the approximate solution for the frequency is given by

ω≈ kzU0ρn,0

ρn,0+ρn,ex

+i 2k²_zU₀²ρn,0ρn,ex

(ρn,0+ρn,ex)(νni,0ρn,0+νni,exρn,ex). (6.52) The previous formula shows that the approximated growth rate has a quadratic dependence on the flow velocity and is inversely proportional to the collision frequencies. Hence, the growth rate is lower in the strongly coupled case than in the uncoupled case, which means that ion-neutral collisions have a stabilizing effect on the ion-neutral fluid. The question then arises: are ion-neutral collisions able to completely stabilize neutrals for sub-Alfv´enic flows, as in the fully ionized case, or is the neutral component of the plasma always unstable?