Flux Tube Dynamics in the Presence of Mass Flows

Abstract

Mass flows observed throughout the solar atmosphere exhibit many patterns. They are observed in a wide temperature range of 3 × 10⁴–10⁷ K, and can have a steady, unsteady, or explosive character. Their amplitudes vary from a few tenths of km s⁻¹ at the photosphere up to hundreds of km s⁻¹ in the transition region and corona. Presence of mass flows drastically changes the dynamics of magnetic structures, and most importantly, plays a crucial role in the processes of the energy production, transfer, and release. Depending on the geometry and intensity of the flow, the system of magnetic flux tubes exhibits a number of unusual phenomena that are directly observed. In this chapter we consider mainly two effects associated with presence of mass flows. One is the instability of Negative-Energy waves ( ) and other lies in the range of velocities beyond the instability threshold, namely the effect of mass flows on the energy transfer by the phase mixed Alfvén waves.

Appendix: Equation for Alfvén Waves in the Presence of Parallel Mass Flows

In this section we derive the general equation for Alfvén waves in the presence of mass flows which can be used for solving problems other than those considered in the previous sections.

In particular, this equation will allow one to study the influence of mass flows on the resonance absorption of Alfvén waves and to explore the development of different kinds of hydrodynamic instabilities .

Consider for simplicity a magnetic field having only a z-component, B₀(0, 0, B(r)). The background mass flow with the velocity u(r) is assumed to be directed along the magnetic field. All the plasma parameters are monotonic functions of radius. The perturbation of the velocity v(0, v_φ(r, z, t), 0) and the magnetic field b(0, b_φ(r, z, t), 0) are axisymmetric, and we will drop the index φ further. The MHD-equations for a given geometry which include the viscous losses have the form

$$\displaystyle \begin{aligned} \rho \left( \frac {\partial v} {\partial t} + u \frac{\partial v} {\partial z} \right) = \frac{B } {4\pi} \frac{\partial b}{\partial z} + \frac{1} {r^2} \frac {\partial} {\partial r}\eta r^3 \frac{\partial} {\partial r}\frac{v } { r } \end{aligned} $$

(5.139)

$$\displaystyle \begin{aligned} \frac{\partial b}{\partial t} + u \frac{\partial b} {\partial z} = B \frac{\partial v}{\partial z}+ \frac{\partial} {\partial r} \frac {1} { r}\nu_m \frac{\partial} {\partial r}r b \end{aligned} $$

(5.140)

All notations are standard; η is ion viscosity (ω_i τ_i  ≫ 1) and ν_m is magnetic diffusivity . Since the dissipation is weak, we have neglected in the dissipative terms the derivatives ∂∕∂z.

Taking the derivative (∂∕∂t + u∂∕∂z) of (5.139) and using (5.140) we have

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \rho \left(\frac {\partial} {\partial t} + u \frac {\partial} {\partial z}\right)^2 v=\frac{B}{4\pi}\frac{\partial} {\partial z}\left[ B\frac{\partial v}{\partial z}+\frac {\partial}{\partial r}\frac{1}{ r}\nu_m\frac {\partial}{\partial r}r b \right] \\ + \left(\frac {\partial} {\partial t} + \frac {\partial} {\partial z} \right) \frac{1} {r^2}\frac {\partial}{\partial r}\eta r^3\frac{\partial}{\partial r}\frac{v}{r} \end{array} \end{aligned} $$

(5.141)

At weak dissipation, one can use in the second term in right-hand side of (5.141), the following expression for ∂b∕∂z:

$$\displaystyle \begin{aligned} \frac {B}{4\pi}\frac{\partial b} {\partial z}\simeq \rho \left(\frac{\partial v}{\partial t}+ u \frac {\partial v} {\partial z}\right), \end{aligned} $$

(5.142)

Now we can write a single equation for v:

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \rho \left(\frac{\partial}{\partial t} + u \frac {\partial}{\partial z}\right)^{2}v &\displaystyle =&\displaystyle \frac{B^2 } { 4\pi} \frac {\partial^2 v} {\partial z^2} +\frac {B} { 4\pi} \frac {\partial} {\partial r}\frac {1} { r}\nu_m \frac {\partial}{\partial r}r \frac{4\pi\rho }{ B} \left(\frac{\partial v}{\partial t}+ u \frac {\partial v}{\partial z}\right)\\ &\displaystyle &\displaystyle + \left(\frac{\partial} {\partial t}+ u \frac {\partial} {\partial z}\right) \frac{1}{ r^2} \frac {\partial}{\partial r} \eta r^3 \frac {\partial}{\partial r}\frac {v }{ r} \end{array} \end{aligned} $$

(5.143)

For the boundary value problem with

$$\displaystyle \begin{aligned} v \mid_{z=0} = v_0 e^{i\omega t}\end{aligned} $$

(5.144)

and without the dissipation we have from (5.143) (see (5.82)):

$$\displaystyle \begin{aligned} v = v_0 \ \cos\frac {\omega z}{v_A (r)}\end{aligned} $$

(5.145)

As discussed in Section 8, the velocity profile becomes spiky with height, and its scale (5.86) at the heights z ≫ v_A∕ω becomes much less than R:

$$\displaystyle \begin{aligned} \varDelta r \ll R \end{aligned} $$

(5.146)

This condition allows us to simplify (5.141). Namely, in the dissipative terms we can neglect the radial derivatives of all the functions except v. Note that the background plasma parameters B(r), ρ(r), and u(r) change at the scale R, and from to condition (5.146) we have

$$\displaystyle \begin{aligned} \frac {db} { dr}, \ \frac{d\rho} { dr}, \ \frac{du} { dr} \ll \frac{\partial v} {\partial r} \end{aligned} $$

(5.147)

Furthermore,

$$\displaystyle \begin{aligned} \frac{\partial^2} {\partial r^2} \gg \frac{1 } {r} \frac {\partial } {\partial r}, \ \frac {1 }{ r^2} \end{aligned} $$

(5.148)

Under these conditions (5.143) becomes:

$$\displaystyle \begin{aligned} \left(\frac{\partial} {\partial t}+ u \frac {\partial } {\partial z}\right)^2 v = v_A ^2 (r) \frac{\partial^ 2 v} {\partial z^2} + \nu_* \left(\frac{\partial } {\partial t} + u \frac {\partial } {\partial z}\right) \frac {\partial^2v}{\partial r^2} \end{aligned} $$

(5.149)

where ν_∗ = η∕ρ + ν_m.

One can see that in the absence of dissipation, different radii do not communicate with each other as is clearly seen from (5.149): The Alfvén waves along each cylindrical surface propagate with the speed determined only by the surface radius r, i.e. at a speed v_A(r).

The dissipative term links the perturbations at the field lines separated by the distance not exceeding a generalized skin-depth (5.89), which means that considering the wave propagation and dissipation at a certain magnetic surface, we need to know the coefficients of (5.149) only in the vicinity of this surface. We denote the radius of some arbitrary surface by r₀, and for

$$\displaystyle \begin{aligned} r = r_0 + x , \end{aligned} $$

(5.150)

use the approximations

$$\displaystyle \begin{aligned} v_A (r)\cong v_A (r_0) +\frac{d v_A (r)}{d x}\mid_{r_0} \end{aligned} $$

(5.151)

$$\displaystyle \begin{aligned} u(r) \cong u(r_0) +\frac{d u(r)}{d x}\mid_{r_0} \end{aligned} $$

(5.152)

$$\displaystyle \begin{aligned} \nu_*\cong \nu_*(r_0) \end{aligned} $$

(5.153)

We drop the index “0” below, and will keep in mind that these quantities have a local meaning at a certain radius r₀.

For perturbations ${\sim } \ \exp (-i\omega t)$, (5.149) reduces to

$$\displaystyle \begin{aligned}{}[u^2 (r) - v_A^2 (r)] \frac{\partial^2 v}{\partial z^2}- 2i\omega u(r) \frac{\partial v}{\partial z} - \omega^2 v = \nu_* \frac{\partial^2} {\partial r^2} \left(-i\omega v + u(r)\frac{\partial v}{\partial z}\right) \end{aligned} $$

(5.154)

It is convenient to represent the velocity perturbations in a form

$$\displaystyle \begin{aligned} v = w(r,z,t)~\mbox{exp}\left (i\frac{\omega}{v_{A0} + u_0}z \right ) \end{aligned} $$

(5.155)

where v_A0 = v_A(r₀) and u₀ = u(r₀). Simple algebra gives the following equation for w (the analogue of parabolic equation in diffraction theory):

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle &\displaystyle \frac{\partial w}{\partial z}+ i\omega\frac{v^{\prime}_{A0} + u^{\prime} _0} {(v_{A0} + u_0)^2} x w \\ &\displaystyle &\displaystyle - \nu_* \left[\frac{1}{2(v_{A0 }+u_0)}\frac{\partial^2 w} {\partial^2 x} -\frac {u^{\prime} (r)} {v_{A0 }(v_{A0 } + u_0)} \frac{\partial w}{\partial x} - \frac{u^{\prime\prime} (r)}{2 v_{A0}(v_{A0} + u_0)} w \right] = 0 \end{array} \end{aligned} $$

(5.156)

Prime means the derivative over radius.

Equation (5.156) is the basic equation describing the Alfvén wave propagation in the presence of plasma flows. Magnetic field and plasma parameters have a smooth dependence on radius. This equation can be written in especially convenient form using a function Q which we define as follows:

$$\displaystyle \begin{aligned} w = Q(x,z,t)~\mbox{exp}\left (\frac{u^{\prime} (x)}{v_{A0}} x \right ), \end{aligned} $$

(5.157)

and in the variables:

$$\displaystyle \begin{aligned} \xi = x \left[\frac {\omega (v^{\prime}_{A0} + u^{\prime} _0)} {\nu_* (v_{A0 } + u_0)} \right]^{1/3} \end{aligned} $$

(5.158)

$$\displaystyle \begin{aligned} \zeta = z\frac {\omega (v^{\prime}_{A0} + u^{\prime} _0)}{(v_{A0 } + u_0)^2} \left[\frac {\omega (v^{\prime}_{A0} + u^{\prime} _0)}{\nu_* (v_{A0}+ u_0 )} \right]^{-1/3} \end{aligned} $$

(5.159)

With (5.157)–(5.159), the basic parabolic equation (5.156) takes a form:

$$\displaystyle \begin{aligned} \frac{\partial Q }{\partial\zeta } +i\xi Q -\frac{\partial^2 Q } {\partial \xi^2} + \alpha(\xi) \cdot Q = 0 \end{aligned} $$

(5.160)

with

$$\displaystyle \begin{aligned} \alpha (\xi) \sim \left[\frac {(u^{\prime} )^2(\xi)}{ v_{A0 }^2} + \frac{u^{\prime\prime} (\xi)}{v_{A0 }} \right] \end{aligned} $$

(5.161)

Equation (5.160) allows one to study the various instabilities in the presence of mass flows.

It is important to note that the asymptotic behavior of the total power flux of the wave varies much more slowly with height in the presence of mass flows than without them. Namely, the total power flux $\mathscr {P}$ in the presence of flows scales with height as $\mathscr {P}\sim z^{(-3/2)}$, while it scales as $\mathscr {P}\sim z^{-6}$ in their absence.

Thus, a regular subsonic upward and downward mass flows modify considerably the propagation of phase mixed Alfvén waves. First of all, the radial redistribution of the energy input occurs, which is determined by the fact that the extrema of the Alfvén speed and flow velocity are not spatially coincident. In this case, the strongest absorption occurs at those radii at which the gradients of the phase velocity are steepest, while those radii at which the phase velocity has extrema form an escape channel for the wave energy. Therefore, even for moderate flow velocities (compared with the magnitude of the Alfvén speed), both upward and downward flows lead to the creation of a complex mosaic of highly localized bright regions with varying heights. These effects can be considered as a reasonable explanation for understanding the complex intermittent emission such as observed in coronal bright points (Huang et al. 2012; Alexander et al. 2011; Habbal and Withbroe 1981; Habbal et al. 1990).

Note also that the strongest effect occurs in the case of downflows: if at some geometrical height the Alfvén speed and flow velocity become equal, the wave comes to extinction and gives off its energy completely. This process is independent of the heating due to phase mixing and can occur much earlier and at lower altitudes than the onset of the latter. Near this region the absorption length Δ = l ^∗− l_d along the direction of the magnetic field becomes very small. Since both the Alfvén speed v_A and flow velocity u depend on r and l, the points where v_A = u constitute a surface l = l(r) in which essentially all the wave energy is released. This effect, as mentioned earlier, should manifest itself as a complex picture of localized bright regions, as complex as the topology of the magnetic fields and plasma flows throughout the solar atmosphere.