Nonlinear waves in a partially ionized two-fluid plasma

Here, a partially ionized two-fluid plasma is considered as a simpler, toy model that can help us to understand the numerical results given in the previous section. One of the fluids is composed of ions and electrons, and the other one is composed of neutrals. For the sake of simplicity, Hall’s term and Ohm’s diffusion are neglected from the induction equation. Therefore, the equations that describe the dynamics of this plasma are a simplified version of those used in the previous simulations, namely

∂ρ_i

∂t +∇ ·(ρ_iV_i) = 0, (5.2)

∂ρ_n

∂t +∇ ·(ρnV_n) = 0, (5.3)

∂(ρ_iV_i)

∂t +∇ ·(ρiV_iV_i) =−∇Pie+ ∇ ×B

µ0 ×B+αin(V_n−V_i), (5.4)

∂(ρnV_n)

∂t +∇ ·(ρnV_nV_n) =−∇Pn+αin(V_i−V_n) (5.5)

and ∂B

∂t =∇ ×(V_i×B), (5.6)

where Pn is the pressure of neutrals, Pie is the sum of the pressures of ions and electrons and αin is the ion-neutral friction coefficient.

To study the properties of non-linear perturbations, a perturbative expansion is performed.

Thus, each variable,f, in the previous system of equations is rewritten as follows:

f =f⁽⁰⁾+ǫf⁽¹⁾+ǫ²f⁽²⁾+. . . , (5.7) where ǫ is a dimensionless parameter proportional to the velocity amplitude of Alfv´en waves, the superscript “(0)” refers to the background values and the superscripts “(1)” and “(2)” cor-respond to the first-order and second-order perturbations, respectively. Since a static uniform background is considered, V_i⁽⁰⁾ =Vn⁽⁰⁾ = 0 and the remaining background values are constant.

Then, the terms in Equations (5.2)-(5.6) can be gathered according to their powers ofǫ, and separated systems of equations can be obtained for each order of the perturbative expansion.

If the initial perturbations are chosen to be transverse to the direction of the background magnetic field (assumed here to be in the x-direction) and let to propagate along that same direction, the first-order (or linear) system leads to the equation for Alfv´en waves,

∂³

After solving this equation, the first-order perturbation of magnetic field can be found through the equation

The solutions of Equation (5.8) in the form of normal or Fourier modes have been analyzed in the past by, e.g., Piddington [1956], Kulsrud and Pearce [1969], Pudritz [1990], Martin et al. [1997] or Kamaya and Nishi [1998], and more recently by Kumar and Roberts [2003], Zaqarashvili et al.[2011a],Mouschovias et al.[2011] orSoler et al. [2013b]. At first order, there is no coupling between the perpendicular and longitudinal components of the perturbations, which means that there is no coupling between Alfv´en and sound waves. In contrast, a coupling appears at the second-order, as shown by the following equations, which are related to the velocities in the longitudinal direction:

ρ⁽⁰⁾_i ∂V_i,x⁽²⁾

. Thus, the second-order perturbation of the velocity of ions is related to the first-order perturbation of the magnetic field and, in turn, produces a fluctuation in the rest of the variables, namely Vn,x⁽²⁾, ρ⁽²⁾_i , and ρ⁽²⁾n . It must be noted that the second-order equations corresponding to the perpendicular components have the same form as those of first-order and hence, they describe the same behavior as Equations (5.8) and (5.9).

The sound speeds of the ionized and of the the neutral fluids are defined ascie = q

γP_ie⁽⁰⁾/ρ⁽⁰⁾_i and cS,n =

q

γPn⁽⁰⁾/ρ⁽⁰⁾n , respectively. In the fully ionized single-fluid case, the second-order perturbations of pressure and density are related by the expression P_ie⁽²⁾ = c²_ieρ⁽²⁾_i (see, e.g, Hollweg [1971],Rankin et al.[1994]). When multi-fluid plasmas are considered, that relation is not accurate because of the heat transfer terms in the evolution equation of pressure (see Equa-tion (2.54)). Nevertheless, for the purposes of this analytical study, it can be taken as a good approximation. Thus, assuming that Pn⁽²⁾ ≈c²_S,nρ⁽²⁾n and combining Equations (5.10)-(5.13), it is possible to obtain the following equation that describes the second-order perturbations of the density of ions (a similar equation can be cast for neutrals and for the x-component of the velocities of ions and neutrals):

An interesting limiting case of the previous equation can be found if νni is assumed to tend to infinity, which corresponds to a strong coupling between the two fluids. The following expression is obtained:

where the relationνin/νni =χ has been used. The integration with respect to time leads to ∂²

where an integration constant has been taken equal to zero. This is the 1-dimensional inho-mogeneous wave equation, with the right-hand representing a driving term. Using the initial conditions ρ⁽²⁾_i (x, t= 0) = 0 and _∂t^∂ρ⁽²⁾_i (x, t = 0) = 0, respectively, the solution to this equation

The only speed that explicitly appears in Equation (5.17) is the effective sound speed, ec_S,

However, since the driving wave is assumed to be Alfv´enic, the evolution ofB_⊥² depends on the Alfv´en speed. Hence, the evolution ofρ⁽²⁾_i (x, t) depends on both sound and Alfv´en speeds.

From Equation (5.14) it is also possible to recover the differential equation that describes the second-order perturbations of density in a fully ionized plasma. If the collision frequencies are taken equal to zero (meaning that neutrals are decoupled and do not interact with ions), it is possible to rewrite Equation (5.14) as

∂²

an equation that has already been derived by Hollweg [1971], Tikhonchuk et al. [1995] or Terradas and Ofman[2004]. It can be seen that Equations (5.16) and (5.20) represent the same type of behavior, with differences appearing in the velocity of propagation of waves and the amplitude of the driving term. These are two effects caused by the ion-neutral interaction.

If the initial perturbation applied to the equilibrium state is given by

V_y⁽¹⁾(x, t) =Vy,0cos(kxx), (5.21) and the strongly coupled limit is applied to Equations (5.8) and (5.9), the first-order perturba-tion of the magnetic field is

B⊥(x, t) = −B0

e cA

Vy,0sin(ecAkxt) sin(kxx), (5.22) with ecA the Alfv´en speed modified by the inclusion of the inertia of neutrals, i.e., ecA = B0/p

µ0ρi,0(1 +χ). Then, the solution to Equation (5.16) is

ρ⁽²⁾_i (x, t) = B₀²V_y,0² [ec²_A−¯c²_S + ¯c²_Scos(2ec_Ak_xt)−ec²_Acos(2¯c_Sk_xt)] cos(2k_xx)

8ec²_Ac¯²_S(ec²_A−¯c²_S)µ₀(1 +χ) . (5.23) The resulting perturbation is the combination of two standing modes with frequencies 2ecAkx

and 2ecSkx, respectively, and whose wavenumber is twice the wavenumber of the original per-turbation. The solution for the fully ionized case is recovered by substituting ec_S with c_ie, ec_A with cA and taking χ= 0.

If the sound speed is much lower than the Alfv´en speed, as it occurs in the simulations performed in this work, Equation (5.23) can be approximated as

ρ⁽²⁾_i (x, t)≈ B₀²V_y,0²

which shows that the perturbation is dominated by the oscillation mode associated with the weighted sound speed.

Then, the relative variation of density, which in Figures 5.1-5.3is represented as ∆ρ/ρ0, can be computed as the ratio between the second-order perturbation and the background density.

Hence,

∆ρi

ρi,0 ≡ ρ⁽²⁾_i (x, t)

ρi,0 ≈ V_y,0²

8ec²_S [1−cos(2ec_Sk_xt)] cos(2k_xx), (5.25) An interesting conclusion can be extracted from the previous equation: since the relative varia-tion of density is proporvaria-tional toV_y,0² /ec²_S for partially ionized plasmas while it is proportional to V_y,0² /c²_ie for fully ionized fluids andcie >ecS, the relative variation of density is larger when the effect of partial ionization is taken into account. This is an important result caused by partial ionization.

A comparison between the fully ionized case and the partially ionized case with strong coupling is shown in Figure5.4. Numerical simulations with the same total mass, withn_p =n_H (i.e.,χ= 1) for the partially ionized plasma, and the same amplitude of the initial perturbation in velocity have been performed. The collision frequency for the partially ionized case is much larger than the oscillation frequency. It can be seen that the amplitude of the relative variation of density is larger when the plasma is partially ionized. Furthermore, the fully ionized plasma oscillates with a slightly higher frequency than the partially ionized one, as it would be expected since c_ie > ec_S. The numerical results are in almost perfect agreement with the approximate analytical expressions. The small differences that appear in the partially ionized case are caused by the collisional friction between ions and neutrals. This friction causes the damping of the amplitude of the oscillation and a slight modification of its frequency. For simplicity, damping has not been taken into account in the analytic approximations given above.

Figure 5.4: Relative variation of density caused by the ponderomotive force due to an Alfv´en wave withkx =π/10⁴ m⁻¹ in a fully ionized plasma (left) and a partially ionized plasma with a strong coupling between the two fluids (right). The solid lines represent the analytical solution given by Equation (5.25) and the red symbols represent the results of the numerical simulations.

If the wavenumber of the perturbation increases, the frequency of the Alfv´en wave increases

as well and departs from the limit whereω/(2π)≪ν_ni, which means that the coupling between the two fluids is not as strong as for smaller wavenumbers. Hence, it would be expected that Equation (5.25) becomes inaccurate at larger wavenumbers. Moreover, it has been shown in Chapters 3 and 4 that Hall’s term should be taken into account in the large wavenumber range. However, such term has been neglected here in the derivation of the equations for the second-order perturbations.

The three-fluid simulations represented in Figures 5.2 and 5.3 show that, under the chosen physical parameters, the friction due to ion-neutral collisions is more efficient in attenuating the Alfv´enic waves than the acoustic modes. For instance, it can be checked that in Figure5.3 the first-order Alfv´en wave has almost disappeared aftert= 0.5 s, but the second-order perturbation in thex-component of the velocity lasts for a longer time. In a two-fluid plasma, the oscillation frequency and damping rate of the remaining second-order wave may be obtained from Equation (5.14) in the following way. Since the driving wave, i.e., the first-order Alfv´en wave, vanishes due to collisions, after a given time the term on the right-hand side of Equation (5.14) becomes equal to zero. Then, the remaining oscillations are governed by the homogeneous version of the differential equation, with the initial conditions given by the wave previously induced by the driver. After the primary Alfv´en wave is completely damped, the second-order perturbation of the density of ions can be expressed as

ρ⁽²⁾_i ∼exp [i(−ωt+κx)], (5.26) where, in this case, the wavenumber is twice the wavenumber of the original driving wave, i.e., κ= 2kx. This procedure leads to the following dispersion relation,

ω⁴+i(ν_ni+ν_in)ω³−κ² c²_S,n+c²_ie

ω²−iκ² ν_inc²_S,n+ν_nic²_ie

ω+c²_iec²_S,nκ⁴ = 0, (5.27) which depends on the sound speeds but not on the Alfv´en speed. This is the same dispersion relation that would be obtained for linear acoustic waves in a two-species fluid in which only the collisional interaction between ions and neutrals is taken into account and the influence of magnetic fields is neglected (see, e.g., Vranjes and Poedts [2010]). It coincides with Equation (9) from Vranjes and Poedts [2010] if the factors proportional to the electron-neutral collision frequency of that formula are neglected, and it can also be recovered from Equation (47) ofSoler et al. [2013a], where magnetoacoustic waves in partially ionized plasmas have been studied, if the Alfv´en speed is set equal to zero.

It must be noted that for a certain range of collision frequencies, the driving wave may last more than the acoustic wave and, strictly, the dispersion relation, Equation (5.27) should not be applicable because the driver is still working. This is a consequence of the damping due to ion-neutral collisions being most efficient when the oscillation frequency is similar to the collision frequency (Zaqarashvili et al. [2011a],Soler et al. [2013b]). Since ecS ≪ecA, the acoustic modes are more damped than the Alfv´enic ones at low collision frequencies and the opposite would occur at high frequencies. Nevertheless, as shown by Equations (5.23) and (5.24), if the Alfv´en speed is much larger than the sound speed, the second-order oscillation is dominated by the acoustic mode. Hence, the results from Equation (5.27) are still good approximations at any range of collision frequencies. This statement can be checked by inspecting Figure 5.5, which illustrates a study of the dependence of the oscillation frequency, ωR, and the damping rate,ωI, of the second-order acoustic wave on the ion-neutral collision frequency (only the modes with ωR ≥ 0 are displayed). According to the dispersion relation, at low values of νin/ωS (where

ω_S = κec_S is a normalization parameter), the Alfv´en wave induces four acoustic modes with ωR=±κcS,n and ωR=±κcie, respectively. At higher collision frequencies, two of those modes tend toωR=±κecS, while the other two become evanescent, i.e., they have ωR= 0 and do not oscillate. In addition, the damping of the oscillatory modes is larger at the intermediate range of collision frequencies. The results from the simulation show a general good agreement with the predictions from Equation (5.27). In addition, the behavior represented by this figure is analog to that shown in Figures 4(c)-(d) of Soler et al. [2013a] for the modes denoted as slow, acoustic and modified slow.

Figure 5.5: Oscillation frequency (left) and damping rate (right) of the second-order acoustic mode generated by the Alfv´en wave in a two-fluid plasma as functions of the collision frequency.

The normalization constant is ωS =κecS. The lines correspond to the solutions from Equation (5.27) while the symbols represented the results from the simulations. At low νin/ωS, the blue diamonds and the solid lines represent the mode associated with neutrals, and the red stars and dashed lines represents the mode associated with ions. At large collision frequencies, the green dotted lines correspond to an evanescent mode.