Flux Accretion and Coronal Mass Ejection Dynamics

Abstract

Coronal mass ejections (CMEs) are the primary drivers of severe space weather disturbances in the heliosphere. Models of CME dynamics have been proposed that do not fully include the effects of magnetic reconnection on the forces driving the ejection. Both observations and numerical modeling, however, suggest that reconnection likely plays a major role in most, if not all, fast CMEs. Here, we theoretically investigate the accretion of magnetic flux onto a rising ejection by reconnection involving the ejection’s background field. This reconnection alters the magnetic structure of the ejection and its environment, thereby modifying the forces acting upon the ejection, generically increasing its upward acceleration. The modified forces, in turn, can more strongly drive the reconnection. This feedback process acts, effectively, as an instability, which we refer to as a reconnective instability. Our analysis implies that CME models that neglect the effects of reconnection cannot accurately describe observed CME dynamics. Our ultimate aim is to understand changes in CME acceleration in terms of observable properties of magnetic reconnection, such as the amount of reconnected flux. This flux can be estimated from observations of flare ribbons and photospheric magnetic fields.

Appendices

Appendix A: Evolution of Magnetic Helicity

Magnetic reconnection not only alters the Lorentz forces acting on an erupting flux rope, it also changes the rope’s magnetic helicity (e.g., Berger et al. 1998). To consider the effect of reconnection on the ejection’s helicity, we idealize the erupting flux system as a toroidal rope of purely azimuthal flux, $\Phi_{\mathrm{T}}$, formed by the ejection’s core, surrounded by purely poloidal flux, $\Phi_{\mathrm{P}}$. The mutual helicity of the two flux systems, $|H_{\mathrm {mut}}| = |2\Phi_{\mathrm{P}} \Phi_{\mathrm{T}}|$, quantifies linkages between the two flux systems. If the poloidal or azimuthal flux systems contained internal, “self” linkages, then the total helicity would be the sum of these self helicities and the mutual helicity. Because magnetic helicity is an ideal MHD invariant, no ideal process can lead to the increase of poloidal flux wrapping around the azimuthal core field. (Ideal processes could, however, cause poloidal flux to bunch up, leading to regions with stronger $B_{\mathrm{P}}$.)

When reconnection occurs in the wake of the outward-moving toroidal rope, however, it adds flux that wraps around the rope. Helicity is approximately globally conserved in fast reconnection (e.g., Berger 1999), and this increase in the helicity of the erupting flux system is offset by an opposing change in mutual helicity between the erupting system and surrounding fields. Figure 10 illustrates key aspects of this evolution. For simplicity, both flux tubes shown are assumed to have no internal twist; relaxing this assumption would not substantively alter results of the analysis. Essentially, some of the mutual helicity between the rising, toroidal system (A in the figure) and the initially overlying, external flux system (B in the figure) is transformed into self helicity of the erupting system (toroid plus reconnected flux that encircles it) (Berger et al., 1998).

Figure 10

Left: Flux systems A and B, each assumed untwisted, have a mutual helicity that is greater than zero, due to their positive crossing number. Middle: System A is erupting upward through B. Right: Flux from B has reconnected with itself, and thereby formed two flux systems, one that is closed and linked to system A and one that is anchored at the photosphere. The mutual helicity between A and the flux from system B that is anchored at the photosphere has changed sign, since the sign of their crossing has changed. Globally, helicity is conserved here, though it has been transferred between mutual and self helicities.

This transfer of helicity between the erupting and background flux systems may be seen by considering the sign of the crossing between the external flux and toroidal flux, which changes when the rising flux passes through the external field by driving the latter to reconnect with itself. Initially (left panel), their mutual helicity is $H_{i} = +\Phi_{A} \Phi_{B} (b_{+} + b_{-})/\pi$, and in the final state (right panel), their mutual helicity is $H_{f} = -\Phi_{A} \Phi _{B}(a_{+} + a_{-})/\pi$. (Demoulin, Pariat, and Berger (2006) review the sign convention for angles: the mutual helicity is defined from footpoint angles of the overlying loop; angles swing from the footpoint with the same sign as the apex loop’s footpoint toward the opposite-sign footpoint; and counterclockwise is positive.) The difference in mutual helicities is

$$ \Delta H_{\mathrm {mut}} = H_{f} - H_{i} = - \Phi_{A} \Phi_{B} (a_{+} + a _{-} + b_{+} + b_{-})/\pi= -2 \Phi_{A} \Phi_{B}. $$

(47)

The difference in mutual helicities from the changed crossing sign has been added to the self helicity of the erupting system: the flux that reconnected to encircle the rising flux contributes to the erupting system’s self helicity. The linking number of fluxes in the erupting system is $+1$, so the helicity within that system is $+2 \Phi_{A} \Phi _{B}$.

More detailed studies of helicity evolution due to reconnection in an eruption with reconnection have been undertaken (e.g., Priest and Longcope 2017).

Appendix B: Large-$r$ Scaling of Hoop Force

From Equation 23, the hoop force $F_{\mathrm{hoop}}$ on a length of torus $r \, \Delta\phi$ scales as $R^{2} \, \Delta \phi(B_{\mathrm{mid}}^{2}/8)$ for $r \gg R$. The force per unit length therefore scales as

$$ \frac{F_{\mathrm{hoop}}}{r \, \Delta\phi} \sim\frac{R^{2} B_{\mathrm{mid}} ^{2}}{r}. $$

(48)

Shafranov (1966) derived an expression for the hoop force per unit length in terms of the total azimuthal electric current, $I$, finding that the force per unit length scaled as $I^{2}/r$. In our model, with the poloidal field in Equation 12, the electric current along the torus’s axis is

$$\begin{aligned} I =& \frac{c}{4\pi} \oint \boldsymbol {B} \cdot \mathrm{d}\boldsymbol {L} = \frac{c}{4 \pi} \int_{-\pi}^{\pi} \frac{B_{\mathrm{mid}} \, r}{r + R \cos\theta} R \, \mathrm{d}\theta \end{aligned}$$

(49)

$$\begin{aligned} =& \frac{c r R B_{\mathrm{mid}}}{4\pi} \biggl[ \frac{2}{\sqrt{r ^{2} - R^{2}}} \tan^{-1} \biggl( \frac{\sqrt{r - R}}{\sqrt{r + R}} \tan(\theta/2) \biggr) \bigg\vert _{-\pi}^{\pi} \biggr] \end{aligned}$$

(50)

$$\begin{aligned} =& \frac{c r R B_{\mathrm{mid}}}{4\pi} \frac{2 \pi}{\sqrt{r^{2} - R ^{2}}} = \frac{c r R B_{\mathrm{mid}}}{2 \sqrt{r^{2} - R^{2}}}. \end{aligned}$$

(51)

Therefore, in our model, $I^{2}/r$ varies with $r$ as

$$ I^{2}/r \sim\frac{r R^{2} B_{\mathrm{mid}}^{2}}{(r^{2} - R^{2})} = \frac{R ^{2} B_{\mathrm{mid}}^{2}}{r (1 - (R/r)^{2})} . $$

(52)

In the $r \gg R$ limit, then,

$$ I^{2}/r \sim\frac{R^{2} B_{\mathrm{mid}}^{2}}{r} . $$

(53)

This matches the scaling of our force per unit length, from Equation 48. Figure 6 shows a comparison of $1/r$ (dotted line) with the radial dependence of the term in square brackets in the expression for $F_{\mathrm{hoop}}$ in Equation 22 (solid line).

Appendix C: Pre- to Post-Dipolarization Change in Hoop Force

Another aspect of the change in hoop force is more subtle: the total change in force on the proto-ejection by the addition of reconnected flux $\Delta\Phi$ is likely larger than the value given in Equation 32. To see why, we must consider how the net force exerted by the reconnected field on the ejection changes as a result of the dipolarization.

We define $t_{i}$ to be an instant after this flux has reconnected but before it has fully dipolarized, and $t_{f}$ to be after full dipolarization. At $t_{i}$, the field strength on the inner side of the proto-ejection is weaker than assumed in deriving Equation 20, as shown in the left magnetic field configuration in Figure 11. Consequently, the pre-dipolarization, inner-surface component of the hoop force, $F_{\mathrm {inner,pre}}$, is weaker than $F_{\mathrm{inner}}$ derived above. We assume that flux arching over the proto-ejection’s leading edge is unchanged by the reconnection, so always acts on the proto-ejection with the same inward pressure force $F_{\mathrm{outer}}$. This implies that the instantaneous hoop force acting on the proto-ejection due to just the annulus of reconnected flux is strongest after complete dipolarization, as depicted in the right panel of Figure 11. This force is equal to the value given in Equation 32, which can also be understood as the net pressure force due to only the annulus of reconnected flux $\Delta\Phi$ that wraps around the proto-ejection,

$$ \Delta F_{\mathrm{hoop}} = F_{{\mathrm {inner}}, \Delta\Phi} + F_{{\mathrm {outer}}, \Delta\Phi}, $$

(54)

where the $\Delta\Phi$ subscripts indicate that these forces are due only to the reconnected flux, and $F_{\mathrm{outer}, \Delta\Phi}$ is negative.

Figure 11

Left: Prior to complete dipolarization of post-reconnection flux, the flux density behind the ejection, $B_{\mathrm {pre}}$, is less than the inner-edge flux density $B_{\mathrm{in}}$ assumed in deriving Equation 20. The distance from the reconnection site to the trailing edge of the proto-ejection is assumed to be ${\sim}\, r/2$. The inward pressure force on the proto-ejection’s outer surface due to the reconnected flux, $F_{{\mathrm {outer}},\Delta \Phi}$, assumed to be unchanging, is depicted as a blue vector. The pre-dipolarization outward pressure force on its inner surface due to the reconnected flux, $F_{{\mathrm {inner,pre}},\Delta\Phi}$, is depicted as a red vector. For the configuration shown, at time $t_{i}$, the net pressure force from the annulus of reconnected flux is inward (negative). Right: After complete dipolarization at $t_{f}$, the outward pressure force from the reconnected flux (denoted $F_{{\mathrm {inner}},\Delta\Phi}$) is larger, and the net (outward) pressure force from the annulus of reconnected flux is strongest, and equal to $\Delta F_{\mathrm{hoop}}$ from Equation 32. The total change in the magnetic pressure force on the proto-ejection is the net pressure force at $t_{f}$ minus the net pressure force at $t_{i}$, $\Delta F_{fi}$ and this difference can exceed the value for $\Delta F_{\mathrm{hoop}}$.

This instantaneous force, however, is not the same as the change in pressure force acting on the proto-ejection due to the accretion of reconnected flux. The change in pressure force on the ejection from $t_{i}$ to $t_{f}$ is the difference between the net forces at $t_{f}$ and $t_{i}$,

$$\begin{aligned} \Delta F_{fi} =& (F_{\mathrm{inner}, \Delta\Phi} + F_{\mathrm{outer}, \Delta\Phi}) - (F_{\mathrm{inner, pre}, \Delta\Phi} + F_{\mathrm{outer},\Delta\Phi}) \end{aligned}$$

(55)

$$\begin{aligned} =& (F_{{\mathrm{inner}}, \Delta\Phi} - F_{{\mathrm{inner,pre}}, \Delta\Phi}), \end{aligned}$$

(56)

where the subscripts denote the outward and inward forces (subscripted inner and outer, respectively) on the proto-ejection from the reconnected flux $\Delta\Phi$. It can be seen that if $|F_{\mathrm{inner,pre}, \Delta\Phi}| < |F_{\mathrm{outer}, \Delta\Phi}|$, then $\Delta F_{fi} > \Delta F_{\mathrm{hoop}}$ from Equation 32. We remark that, much like the magnetic tension force from a flux tube $\Delta\Phi$ that is inward prior to tether-cutting reconnection, the net force due to inward and outward magnetic pressures from a flux tube $\Delta\Phi$ might also be inward in the pre-dipolarization state (i.e., $(F_{\mathrm{inner,pre}, \Delta\Phi} + F_{\mathrm{outer}, \Delta\Phi}) < 0$ in Equation 55).

How much weaker would the pre-dipolarization magnetic field, $B_{\mathrm{pre}}$, have to be for the pre-dipolarization force to be inward? As a first step to addressing this question, we use Equation 20 to crudely approximate the outward pressure force prior to complete dipolarization, $F_{\mathrm{inner,pre},\Delta\Phi}$, to be

$$ F_{\mathrm{inner,pre}, \Delta\Phi} \simeq(B_{\mathrm{pre}}/B_{\mathrm{in}})^{2} F_{\mathrm{inner}, \Delta\Phi}. $$

(57)

We expect that the behavior of $F_{\mathrm{inner}, \Delta\Phi}$ follows $F_{\mathrm{inner}}$, and similarly that $F_{\mathrm{outer}, \Delta\Phi}$ follows $F_{\mathrm{outer}}$. It is helpful to refer again to Figure 9, which plots the ratio of $|F_{\mathrm{outer}}/F _{\mathrm{inner}}|$ as a function of major radius $r$, in units of minor radius $R$. For $(r/R)$ near unity, $|F_{\mathrm{outer}}| \ll |F_{\mathrm{inner}}|$, but $|F_{\mathrm{outer}}|$ is more comparable to $|F_{\mathrm{inner}}|$ at larger $(r/R)$. For the ratio $(r/R) = 3$ assumed above, $|F_{\mathrm{outer}}/F_{\mathrm{inner}}|$ is near 0.6, implying that a ratio of $(B_{\mathrm {pre}}/B_{\mathrm{inner}})$ of 0.7, for instance, would yield an initially inward force:

$$ F_{\mathrm{inner,pre}, \Delta\Phi} + F_{\mathrm{outer}, \Delta\Phi} \simeq0.49 F_{\mathrm{inner}, \Delta\Phi} - 0.6 F_{\mathrm{inner}, \Delta\Phi} < 0. $$

(58)

A flux density of 0.7 times the dipolarized value would occur if the average thickness of the shaded region in Figure 11 were more than about $(0.7)^{-1} \simeq1.4$ times the final thickness, $\Delta R$.

It might be suggested that having an $F_{\mathrm{inner,pre}, \Delta \Phi}$ weaker than $F_{\mathrm{outer}, \Delta\Phi}$ is somehow unphysical, and that $(F_{\mathrm{inner}, \Delta\Phi} + F_{\mathrm{outer}, \Delta\Phi})$ must for some reason always be outward for an accelerating CME. But our analysis does not consider all forces on an ejection, just forces from a single flux tube $\Delta\Phi$, and forces from individual tubes may oppose the ejection. As noted above, a similar analysis is implicit in the tether-cutting model, which supposes that many individual tethers oppose an eruption’s upward motion, but nonetheless that eruption can have a net outward force.

How much larger is the change in hoop force $\Delta F_{fi}$ than the estimate in Equation 32? It is problematic to estimate the diminished pressure force on the inner surface of the ejection prior to complete dipolarization without assuming a model of the field structure in reconnection outflow, which is beyond the scope of this analysis.

Appendix D: Tension Force for Torus

We revisit the change in tension estimated in Equation 8 above to investigate the effect, if any, of the axial curvature that gives rise to the hoop force, since this could also affect the change in tension from $B_{\mathrm{P}}$. To get the total (sum of upward and downward) tension, we now use Equation 12 in Equation 5 and integrate over $[-\pi,\pi]$,

$$\begin{aligned} \Delta F_{\mathrm{tension}} =& \frac{1}{4\pi} \int \mathrm{d}V \, \hat{z} \cdot(\boldsymbol {B} \cdot \boldsymbol {\nabla } ) \boldsymbol {B} \end{aligned}$$

(59)

$$\begin{aligned} =& \frac{1}{4\pi} \int R \, \mathrm{d}\theta\, \mathrm{d}A \, B_{\mathrm{P}}(\theta) \frac{ \hat{z} }{R} \cdot\frac{\partial(B_{\mathrm{P}}(\theta) \hat{b})}{\partial \theta} \end{aligned}$$

(60)

$$\begin{aligned} =& \frac{\Delta\Phi}{4\pi} \int\, \mathrm{d}\theta\, \hat{z} \cdot\frac{ \partial( B_{\mathrm{P}}(\theta) \hat{b})}{\partial\theta} \end{aligned}$$

(61)

$$\begin{aligned} =& \frac{\Delta\Phi}{4\pi} \int_{-\pi}^{\pi} \, \mathrm{d}\theta \biggl( - \frac{\partial B_{\mathrm{P}}(\theta)}{\partial\theta} \sin\theta- B _{\mathrm{P}}(\theta) \cos\theta \biggr) \end{aligned}$$

(62)

$$\begin{aligned} =& \frac{- \Delta\Phi B_{\mathrm{mid}}}{4\pi} \int_{-\pi}^{\pi} \,\mathrm{d} \theta \biggl( \frac{r R \sin^{2} \theta}{(r + R \cos\theta)^{2}} + \frac{r \cos\theta}{r + R \cos\theta} \biggr) \end{aligned}$$

(63)

$$\begin{aligned} =& \frac{- \Delta\Phi B_{\mathrm{mid}}}{4\pi} \int_{-\pi}^{\pi} \,\mathrm{d} \theta \biggl( \frac{r R + r^{2} \cos\theta}{(r + R \cos\theta)^{2}} \biggr) \end{aligned}$$

(64)

$$\begin{aligned} =& \frac{- \Delta\Phi B_{\mathrm{mid}}}{4\pi} \int_{-\pi}^{\pi}\, \mathrm{d} \theta \biggl( \frac{(r/R) + (r/R)^{2} \cos\theta}{(r/R + \cos \theta)^{2}} \biggr) \end{aligned}$$

(65)

$$\begin{aligned} =& \frac{- \Delta\Phi B_{\mathrm{mid}} \, c}{4\pi} \int_{-\pi}^{\pi} \,\mathrm{d} \theta \biggl( \frac{1 + c \cos\theta}{(c + \cos\theta)^{2}} \biggr), \end{aligned}$$

(66)

where, as before, the unit vector of the poloidal field is given by $\hat{b} = \hat{\theta}= (\cos\theta\hat{x} - \sin\theta\hat{z})$, $\mathrm{d}V = \mathrm{d}A \, R \,\mathrm{d}\theta$, $\Delta\Phi= B_{\mathrm{P}} \, \Delta A$ is the flux that reconnects, and in the last line we have defined $c = r/R$.

From Dwight (1961) (p. 106), we first evaluate

$$\begin{aligned} \int_{-\pi}^{\pi}\, \mathrm{d}\theta\frac{1}{(c + \cos\theta)^{2}} =& \frac{\sin\theta}{(1 - c^{2})(c + \cos\theta)} \bigg\vert _{-\pi}^{\pi}- \frac{c}{1 - c^{2}} \int_{-\pi}^{\pi} \,\mathrm{d}\theta\frac{1}{(c + \cos\theta)} \end{aligned}$$

(67)

$$\begin{aligned} =& \frac{c}{c^{2} - 1} \int_{-\pi}^{\pi} \,\mathrm{d}\theta\frac{1}{(c + \cos\theta)}. \end{aligned}$$

(68)

Second, and also from Dwight (1961) (p. 106), we evaluate

$$\begin{aligned} \int_{-\pi}^{\pi} \,\mathrm{d}\theta \frac{c \cos\theta}{(c + \cos\theta)^{2}} =& \frac{c^{2} \sin\theta}{(c^{2} - 1)(c + \cos\theta)} \bigg\vert _{-\pi} ^{\pi}- \frac{c}{c^{2} - 1} \int_{-\pi}^{\pi} \,\mathrm{d}\theta\frac{1}{(c + \cos\theta)} \end{aligned}$$

(69)

$$\begin{aligned} =& - \frac{c}{c^{2} - 1} \int_{-\pi}^{\pi} \,\mathrm{d}\theta\frac{1}{(c + \cos\theta)} . \end{aligned}$$

(70)

Equations 68 and 70 are equal and opposite.

This cancellation is reasonable, given that our model poloidal field is essentially circular, so the curvature is uniform around the flux tube over which the force density is integrated. It is true that, due to the higher flux density on the inner section of the flux tube, the force density is higher there. But the force density is not integrated over equal volumes for the inner and outer segments; it is over different volumes threaded by equal amounts of flux.