ARE TORNADO-LIKE MAGNETIC STRUCTURES ABLE TO SUPPORT SOLAR PROMINENCE PLASMA?

In a situation of pure rotation there is no plasma flow along the poloidal field (${{\boldsymbol{v}}}_{p}=0$), and the inertial term (5) reduces to the centripetal component $-\rho \;r\;{{\rm{\Omega }}}^{2}\;{{\boldsymbol{e}}}_{r}^{}$. In this case, from Equation (9), we see that the azimuthal component of the Lorentz force is zero, i.e., ${{rB}}_{\varphi }$ is constant along each poloidal field line. On the other hand, using the induction equation one can easily see that in this case Ω is constant along each given field line, in agreement with Ferraro's isorotation law (Ferraro 1937). The magnitudes of ${{rB}}_{\varphi }$ and Ω are then determined by the boundary conditions of the problem. Given the constancy of ${{rB}}_{\varphi }$, we see from the longitudinal equilibrium Equation (7) that the Lorentz force has no longitudinal component, and so the equation reduces to

$$\begin{eqnarray}&&{{\boldsymbol{F}}}_{{nm}}^{}\cdot {{\boldsymbol{e}}}_{\parallel }^{}=0.\end{eqnarray} \tag{ 10 }$$

The Lorentz force is thus purely poloidal and perpendicular to ${{\boldsymbol{B}}}_{p}$. Calling θ the angle between the poloidal field and the horizontal direction, Equation (10) may be written

$$\begin{eqnarray}&&r\;{{\rm{\Omega }}}^{2}\;\mathrm{cos}\theta -\displaystyle \frac{1}{\rho }\;\displaystyle \frac{\partial p}{\partial s}-\;g\;\mathrm{sin}\theta =0,\end{eqnarray} \tag{ 11 }$$

implying a balance of the non-magnetic forces. Note, in particular, that the centrifugal term could help support the plasma against gravity if the field lines are sufficiently close to horizontal (i.e., $\mathrm{sin}\theta \gt 0$ sufficiently small).

Is the foregoing purely rotating stationary equilibrium a realistic possibility for the observed apparent barb tornadoes? The latest observations (Su et al. 2014) reveal a rotational velocity of 5 km s⁻¹ at the edges of the structure with r = 2'' = 1.5 Mm, so Ω ≈ 3 × 10⁻³ rad s⁻¹. Wedemeyer et al. (2013) found average values of barb widths of r ≈ 2'' in agreement with Su et al. With these values we can compare the centrifugal acceleration to the gravitational acceleration,

$$\begin{eqnarray}&&\displaystyle \frac{r\;{{\rm{\Omega }}}^{2}}{g}\approx 0.06=\mathrm{tan}(3\buildrel{\circ}\over{.} 4).\end{eqnarray} \tag{ 12 }$$

With the values found by Orozco Suárez et al. (2012) this ratio is even smaller. The only way to have centrifugal support of the plasma is then for the field lines to be almost horizontal, which contradicts the tornado picture. Can a pressure gradient help support the plasma in a non-horizontal field? Hot, coronal plasma can be supported by a pressure gradient against gravity across coronal distances even in vertical field lines. However, for cool prominence plasma the pressure scale-height is too small to balance gravity in barbs as tall as those observed. Even if the field lines close to the rotation axis were filled with hot coronal plasma, the magnetic structure would have to turn almost horizontal (say, 3°, as given in Equation (12)) in the region holding cool prominence plasma. We conclude that the centrifugal force associated with the rotation cannot support the cool barb plasma against gravity in a helical field structure.

Another way to illustrate this conclusion is to estimate the rotational speed necessary to have purely centrifugal support, namely ${v}_{\varphi }=\sqrt{{rg}\mathrm{tan}\theta }.$ Assuming a magnetic field inclination of 45°, say, at the edges of the observed barb tornadoes, the rotation velocity would have to be 20 km s⁻¹ if we use the data of Su et al. (2014) and 45 km s⁻¹ when using those of Orozco Suárez et al. (2012), much larger than the measured speeds.

We consider now a more general scenario allowing for flows in the poloidal direction. Now the simple situation of constant Ω and ${{rB}}_{\varphi }$ along the field lines no longer applies. Checking for instance the azimuthal equation of motion (9), we see that a change of specific axial angular momentum of the plasma elements associated with the poloidal motion must be associated with a non-zero toroidal component of the Lorentz force, $({{\boldsymbol{B}}}_{p}/r)\cdot {\rm{\nabla }}(r\;{B}_{\varphi })\;{{\boldsymbol{e}}}_{\varphi }^{}$. Hence, in general ${{rB}}_{\varphi }$ can no longer be constant along field lines. In this situation, the Lorentz force also has a non-zero projection along poloidal field lines (Equation (7)), ${F}_{M\;\mathrm{long}}^{}$,

$$\begin{eqnarray}&&{F}_{M\;\mathrm{long}}^{}\stackrel{\mathrm{def}}{=}-\displaystyle \frac{1}{2{\mu }_{0}^{}{r}^{2}}\displaystyle \frac{\partial {({{rB}}_{\varphi })}^{2}}{\partial s}.\end{eqnarray} \tag{ 13 }$$

In spite of the added complication of this general scenario, there is a set of conserved quantities along the field lines. One can obtain them by following the general procedure used in the theory of astrophysical jets (Mestel 1961; Lovelace et al. 1986). The induction equation requires that ${\rm{\nabla }}\times ({{\boldsymbol{v}}}_{p}\times {{\boldsymbol{B}}}_{p})=0$, so the poloidal velocity must be parallel to the poloidal field lines, for otherwise a singularity would arise in the toroidal electric field at the z-axis. So, we can write

$$\begin{eqnarray}&&{{\boldsymbol{v}}}_{p}=\kappa (r,z){{\boldsymbol{B}}}_{p}.\end{eqnarray} \tag{ 14 }$$

Integrating the MHD equations along field lines and simplifying the resulting expressions, a set of conserved quantities results, namely,

$$\begin{eqnarray}&&{\mu }_{0}^{}\rho \kappa =K,\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{\rm{\Omega }}-\displaystyle \frac{{{KB}}_{\varphi }}{{\mu }_{0}^{}\rho \;r}=W,\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{\rm{\Omega }}{r}^{2}-\displaystyle \frac{{{rB}}_{\varphi }}{K}={\rm{\Lambda }},\end{eqnarray} \tag{ 17 }$$

where K, W, and Λ are all constant along each poloidal field line. Those relations allow us to find an explicit expression for ${{rB}}_{\varphi }$ along the poloidal field. One can write it in terms of the poloidal Alfvén Mach number

$$\begin{eqnarray}&&{M}_{{{\rm{A}}}_{p}}^{2}\stackrel{\mathrm{def}}{=}\displaystyle \frac{{v}_{p}^{2}}{{B}_{p}^{2}/{\mu }_{0}^{}\rho }=\displaystyle \frac{{K}^{2}}{{\mu }_{0}^{}\rho },\end{eqnarray} \tag{ 18 }$$

and of the Alfvén radius, r_A, defined by

$$\begin{eqnarray}&&{r}_{{\rm{A}}}^{2}\;\stackrel{\mathrm{def}}{=}\;\displaystyle \frac{{\rm{\Lambda }}}{W},\end{eqnarray} \tag{ 19 }$$

as follows:

$$\begin{eqnarray}&&{{rB}}_{\varphi }={KW}\displaystyle \frac{{r}^{2}-{r}_{{\rm{A}}}^{2}}{1-{M}_{{{\rm{A}}}_{p}}^{2}}.\end{eqnarray} \tag{ 20 }$$

Expression (20) becomes singular when ${M}_{{{\rm{A}}}_{p}}^{2}\to 1$. In the classical wind solutions, the flow speed increases from sub-Alfvénic to super-Alfvénic at a given point, and, to avoid the singularity, this transition must happen precisely at the Alfvén radius, $r={r}_{{\rm{A}}}$. In such models, this requirement serves as an internal boundary condition to pick up the desired trans-Alfvénic solution instead of the a completely sub-Alfvénic or completely super-Alfvénic solution. In our case, however, it is unlikely that the flows reach Alfvénic values within the tornado; rather, as shown below, we expect ${M}_{{{\rm{A}}}_{p}}^{2}$ to remain below 1 throughout.

Following Equation (7) and the results of Section 2.1, to support the cool plasma along field lines the magnetic force ${F}_{M\;\mathrm{long}}^{}$ of Equation (13) should be larger than the centrifugal force, ${f}_{{\rm{C}}}=\rho \;{{\rm{\Omega }}}^{2}\;r$. From Equations (15) and (17), the ratio of these forces is

$$\begin{eqnarray}&&|\displaystyle \frac{{F}_{M\;\mathrm{long}}^{}}{{f}_{{\rm{C}}}}|=\left|\displaystyle \frac{{T}_{B}}{{T}_{v}}\right|\left(\displaystyle \frac{r}{{L}_{{\rm{\Omega }}}}\right),\end{eqnarray} \tag{ 21 }$$

in terms of the ratios (T_B and T_v) of azimuthal to poloidal components for the magnetic fields and plasma velocities:

$$\begin{eqnarray}&&{T}_{B}\stackrel{\mathrm{def}}{=}\displaystyle \frac{{B}_{\varphi }}{{B}_{p}}\quad {\rm{and}}\quad {T}_{v}\stackrel{\mathrm{def}}{=}\displaystyle \frac{{v}_{\varphi }}{{v}_{p}},\end{eqnarray} \tag{ 22 }$$

and the length ${L}_{{\rm{\Omega }}}$, defined as

$$\begin{eqnarray}&&{L}_{{\rm{\Omega }}}={\left|\displaystyle \frac{\partial \mathrm{ln}({r}^{2}{\rm{\Omega }})}{\partial s}\right|}^{-1}.\end{eqnarray} \tag{ 23 }$$

The characteristic length (23) can be assumed to be of order the radial size of the system, i.e., ${L}_{{\rm{\Omega }}}\sim O(r)$. Hence, from Equation (21), it is necessary for the value of the parameter $\left|{T}_{B}/{T}_{v}\right|$ to be large in order to have a barb-tornado dominated by the magnetic force. We see that a high level of magnetic twist, and a comparatively large poloidal velocity (compared to the rotational velocity) are necessary to provide magnetic support.

We can derive a more exact estimate of the possibilities of magnetic support of tornado plasma against gravity using expression (13) for the longitudinal magnetic force combined with expression (20). First of all, we can use the definitions of W and K to write their product in the form

$$\begin{eqnarray}&&K\;W={M}_{{{\rm{A}}}_{p}}^{2}\displaystyle \frac{{B}_{p}}{r}\left({T}_{v}-{T}_{B}\right)\end{eqnarray} \tag{ 24 }$$

keeping in mind, nevertheless, that K, W (and r_A) are constant along each field line. We can now derive Equation (20) with respect to the arc-length and, after a little algebra, obtain the general expression

$$\begin{eqnarray}&&{F}_{M\;\mathrm{long}}^{}\;=\;\displaystyle \frac{2}{r}\displaystyle \frac{{B}_{\varphi }\;{B}_{p}}{{\mu }_{0}}\displaystyle \frac{{M}_{{{\rm{A}}}_{p}}^{2}}{1-{M}_{{{\rm{A}}}_{p}}^{2}}\mathrm{cos}\theta \left[\;\left(1+\displaystyle \frac{\alpha }{2}\right)\;{T}_{B}-{T}_{v}\right],\end{eqnarray} \tag{ 25 }$$

where we have used $\mathrm{cos}\theta =\partial r/\partial s$, and called α the scale of variation of ρ along poloidal field lines in terms of the cylindrical radius, as follows:

$$\begin{eqnarray}&&\alpha \quad \stackrel{\mathrm{def}}{=}\;\displaystyle \frac{r}{\mathrm{cos}\theta }\displaystyle \frac{\partial \mathrm{log}\rho }{\partial s}.\end{eqnarray} \tag{ 26 }$$

For the estimate that follows, it is best to write Equation (25) in terms of an observed quantity, namely ${v}_{\varphi }$, and of the ratio ${T}_{B}/{T}_{v}$, which turns out to be the essential dimensionless variable in the resulting expression. Further, to calibrate the possibilities of magnetic support, we normalize (25) with respect to gravity along the poloidal field, $\rho \;g\;\mathrm{sin}\theta$. The result is

$$\begin{eqnarray}\displaystyle \frac{{F}_{M\;\mathrm{long}}^{}}{\rho \;g\;\mathrm{sin}\theta } & = & \displaystyle \frac{2}{1-{M}_{{{\rm{A}}}_{p}}^{2}}\;\left(\displaystyle \frac{{v}_{\varphi }^{2}}{g\;r}\right)\\ & & \cdot \left(\displaystyle \frac{{T}_{B}}{{T}_{v}}\right)\left[\left(1+\displaystyle \frac{\alpha }{2}\right)\;\displaystyle \frac{{T}_{B}}{{T}_{v}}-1\right]\mathrm{cot}\theta .\end{eqnarray} \tag{ 27 }$$

For the applications to prominence barbs we envisage cases with ${M}_{{{\rm{A}}}_{p}}^{2}\lt 1$. The term $1\;-\;{M}_{{{\rm{A}}}_{p}}^{2}$ in Equation (27) therefore favors magnetic support. In fact, in many cases ${M}_{{{\rm{A}}}_{p}}^{2}\ll 1$, and so for simplicity this is the case we consider. Assuming the value of the ratio ${v}_{\varphi }^{2}/(g\;r)$ to be given from observations expression (27) is basically a quadratic polynomial in ${T}_{B}/{T}_{v}$ with parameters α and θ. In Figure 2, we show (27) for θ = 60° (upper panel) and θ = 20° (lower panel), for different relevant values of α. To draw the figure we have used ${v}_{\varphi }^{2}/(g\;r)=0.06$ (i.e., ${v}_{\varphi }=5$ km s⁻¹ and $r=1.5$ Mm), which is the value that led us in Section 2.1, Equation (12), to conclude that there is no possible support for the plasma in the purely rotating case. For magnetic support, the relevant stretches of the curves are those near the dashed horizontal line at ordinate = 1. For ease, we have marked the cut of each curve with that horizontal line with a large black dot and a thin dotted vertical line. The following guidelines have been used in the choice of values for α: in a solar tornado α is probably negative, i.e., the density decreases as one goes outward along the tornado field lines. Also, $| \alpha |$ should not be larger than of order unity, since the lengthscale of variation of ρ should be not too different from r itself. Finally, in the figure we restrict ourselves to concave-upward parabolas (i.e., $\alpha \gt -2$), since those are the most favorable cases for magnetic support.

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We see that it is not difficult to find values of ${T}_{B}/{T}_{v}$ that yield magnetic support of the plasma against gravity. In all cases shown, though, the basic variable $| {T}_{B}/{T}_{v}|$ must be above 1: for θ = 60°, for instance, the values marked in the figure range from $-6.3$ to $-3.3$, on the negative side and from 4.3 to 7.3 on the positive side. Note that the actual ranges are $[-14.2,-3.3]$ and $[4.3,\infty ]$. For θ = 20°, the absolute values are smaller. We expect T_v to be perhaps 1 or 2, reflecting the fact that the outward motion of the tornado is possibly as fast as the rotating motion or a little less so. Thus, for magnetic support one would need a substantial level of magnetic twist, possibly $| {B}_{\varphi }/{B}_{p}|$ from a few to several units. If the poloidal flow becomes less important, then the amount of magnetic twist would increase to unrealistic values.