Particle Acceleration in Collapsing Magnetic Traps with a Braking Plasma Jet

Abstract

Collapsing magnetic traps (CMTs) are one proposed mechanism for generating non-thermal particle populations in solar flares. CMTs occur if an initially stretched magnetic field structure relaxes rapidly into a lower-energy configuration, which is believed to happen as a by-product of magnetic reconnection. A similar mechanism for energising particles has also been found to operate in the Earth’s magnetotail. One particular feature proposed to be of importance for particle acceleration in the magnetotail is that of a braking plasma jet, i.e. a localised region of strong flow encountering stronger magnetic field which causes the jet to slow down and stop. Such a feature has not been included in previously proposed analytical models of CMTs for solar flares. In this work we incorporate a braking plasma jet into a well studied CMT model for the first time. We present results of test particle calculations in this new CMT model. We observe and characterise new types of particle behaviour caused by the magnetic structure of the jet braking region, which allows electrons to be trapped both in the braking jet region and the loop legs. We compare and contrast the behaviour of particle orbits for various parameter regimes of the underlying trap by examining particle trajectories, energy gains and the frequency with which different types of particle orbit are found for each parameter regime.

Appendix A: Description of Transformation

Here we discuss how the transformation defining the analytical fields is obtained. For a detailed discussion of the theory behind the transformation method see Giuliani, Neukirch, and Wood (2005). We start with an initial stretched configuration similar to that in Giuliani, Neukirch, and Wood (2005) to which a front with a sharp increase in magnetic field strength is added. Behind the front the field lines are closer to their equilibrium configuration, similar to what is suggested in Artemyev (2014). The $y$-component of the transformation used in this paper, restricted to $x = 0~\mbox{Mm}$ is shown in Figure 12 for multiple values of $t$.

Figure 12

Transformation with $x = 0~\mbox{Mm}$ (in centre of CMT). Regions for which $y_{\infty}< y$ correspond to stretching, while regions with $y_{\infty}> y$ correspond to compression. For a fixed time, the front is located in the region where $y_{\infty}$ increases rapidly. The dashed line is a visual aid, set at $y_{\infty}= y$.

Vertical stretching corresponds to $y_{\infty}< y$, whereas for compression $y_{\infty}> y$. Figure 12 shows that when $t = 0~\mbox{s}$ and $y < 60~\mbox{Mm}$, $y_{\infty}< y$, which corresponds to a vertical stretching. As the front passes a given $y$ the value of $y_{\infty}$ increases rapidly, indicating compression caused by the front. The transformation shown in Figure 12 is given by

$$ y_{\infty}= S + y\frac{1 + \tanh\phi}{2}, $$

(13)

where the first term, $S$, defines the shape of the field before the front has passed. The second term describes the shape of the front, where $\phi= 0$ is the location of the front. The function $\phi$ depends on position and time. It satisfies $\phi< 0$ before the front passes a particular location and $\phi> 0$ after the front has passed.

One choice for the first term of Equation (13) that produces the desired vertical stretching is

$$\begin{aligned} S =& s\log \biggl(1 + \frac{y}{s} \biggr) \biggl(1 - \frac{1 - \tanh(y - y')}{2}\cdot\frac{\tanh(x + x') - \tanh(x - x')}{2} \biggr) \\ & {}+ y \biggl(\frac{1 - \tanh(y - y')}{2}\cdot\frac{\tanh(x + x') - \tanh(x - x')}{2} \biggr). \end{aligned}$$

(14)

This choice imposes a stretching of the form $s\log (1 + y/s )$ for points outside the box given by $0 \leq y \leq y'$ and $-x' \leq x \leq x'$, and leaves the interior untransformed (i.e. in its final configuration). We currently do not confine the box in the $x$ direction by setting $x' = 100$ (outside of the region we investigate in this paper).

To choose a functional form for $\phi$ we consider the front position determined by

$$ \phi= y - v_{\phi}\sigma\tanh (t/\sigma )- y_{0}. $$

(15)

The location of the front is given by $\phi= 0$. For small values of $t$ the location of the front is $y = y_{0} + v_{\phi}\sigma\tanh (t/\sigma )\simeq y_{0} + v_{\phi}t$, which means that the front propagates with a constant speed. For larger values of $t$, $\tanh (t/\sigma )$ approaches a constant, meaning the front slows down and stops. Calculating

$$ \lim_{t \to\infty} \bigl(y_{0} + v_{\phi}\sigma \tanh (t/\sigma ) \bigr)= y_{0} + v_{\phi}\sigma $$

(16)

shows that the parameter $\sigma$ controls how deeply the front penetrates into the equilibrium loops at the bottom of the trap because it determines the final location of the front.

It is possible to modify the steepness of the front (which in turn affects the strength of the electric and magnetic fields at the front) by increasing the gradient of $\phi$. We achieve this by multiplying the expression given in Equation (15) by the factor $\alpha ( (y + 1 ) (s_{o} x^{2} + 1 ) )^{\beta}$. The factor $(y+1)^{\beta}$ causes the transformation to be steeper for larger values of $y$. Multiplication of $\phi$ by the factor $(y+1)^{\beta}$ changes the shape of the loops, so the factor $(s_{o} x^{2} + 1 )^{\beta}$ is added to correct for this effect. The functions $J$ and $T$ (given in Equation (17)) are added to further modify the shape of the CMT:

$$ J = d \mathrm{e}^{-x^{2} y /w}, \qquad T = k \tan \biggl( \frac{\pi}{2} \frac{x^{2}}{w_{2}} \biggr)\tanh y. $$

(17)

The function $J$ produces an indentation in the jet braking region. The indentation takes the form of an exponential with its depth and width determined by the parameters $d$ and $w$. The function $T$ modifies the shape of the trap for large values of $|x|$ to maintain the shape of the loops.

Artemyev (2014) suggests that as the braking jet propagates the front may become steeper. In the transformation described above the front becomes shallower as the jet propagates towards the solar surface. The reason for this is that as the braking jet approaches the lower loops it is travelling slow enough that the magnetic flux that piled up previously spreads out. The spreading out of the magnetic flux causes the front to become less steep and eventually disappear. Nevertheless, closer to the reconnection region the outflow is faster and the magnetic field not as strong so we expect to see a steepening front. To incorporate this steepening into our model $\phi$ and $T$ are multiplied by the factor $1 + \frac{1}{2} [\chi\sin (\pi y/y_{0} )- 1 ] [1 - \tanh(\zeta(t - t')) ]$. This corresponds to multiplying $\phi$ and $T$ by $\chi\sin (\pi y/y_{0} )$ when $t \ll t'$, producing a steepening of the transformation near $y = y_{0}$. The result is the transformation presented in Equations (3)–(7).