3 Results

3.1 Case A: Symmetric loop

In the first set of simulations, we study a loop which is spatially symmetric about the point source of energetic protons. We inject $p_{0}=0.25\, mc$(corresponding to an energy of about 30 MeV) protons isotropically at a rate $Q=\epsilon \, n_{0}V_{\mathrm{A}}=\epsilon \times 10^{19}$ cm^-2 s^-1, where $\epsilon =2\times 10^{-5}$ is chosen to give a large flux resulting in a rapid wave growth, but still keeping the energetic-proton pressure at least an order of magnitude below the thermal proton pressure even if all particles are trapped by Alfvén waves inside $\vert x\vert<V_{\rm A}t$. We vary the total amount of injected particles by varying the duration of the injection between $\Delta t=1/15$ s and 10/15 s, resulting in a total number of injected particles between $Q\, \Delta t=1.33\times 10^{13}$ cm^-2 and $1.33\times 10^{14}$ cm^-2. The background wave flux emitted from the footpoints is fixed by assuming that the wave mode emitted from each footpoint has an energy density of $2U_{\pm }(\mp L/2,t)=2U_{0}=10^{-4}\times U_{B}$. We assume that this emission of waves is steady; thus the initial condition for both wave modes is also $U_{\pm }(x,0)=U_{0}$. The adopted value for 2U₀ corresponds to an rms velocity amplitude of 10 kms^-1 per logarithmic bandwidth at $f=V_{\mathrm{A}}k/2\pi \sim 3$ kHz. Assuming that this velocity amplitude holds for all frequencies from f₀=V_A/L=0.1 Hz up to $f_{\mathrm{p}}=\Omega _{\mathrm{p}}/2\pi \approx 220$ kHz, the total rms amplitude is 38 kms^-1, which agrees very well with the typical observed non-thermal velocity in a quiescent active region of 45 km s^-1 (Antonucci & Dodero 1995). The non-thermal velocities in flaring plasmas are larger, consistent with wave growth. We do not consider any emission of waves from the acceleration site; we assume that the waves are absorbed by the particles during the acceleration. However, some external forcing of the Alfvénic turbulence is modeled by keeping the wave-energy densities above a minimum level of $U_{\pm }\geq U_{\mathrm{min}}=5\times10^{-7}\times U_{B}$ everywhere inside the flux tube.

The results of the simulations are presented in Fig. 1 in form of flux of protons precipitating at the footpoints of the loop. One can see a peak in the precipitating flux at $t\approx 5$ s corresponding to the travel time of Alfvén waves from the center of the loop to the footpoints, as predicted by the theory of Bespalov et al. (1987, 1991). What is not predicted by the steady-state theory is the rather intense precursor peak immediately after the particle release, that corresponds to the first phase, when the waves have not yet grown enough to suppress the diffusive particle transport. The number of particles in the precursor seems to be independent of the number of injected particles, if this number exceeds a threshold level. This can be understood as follows. We consider an initial turbulence level, where the diffusion length, $\kappa /V_{\mathrm{A}}$ (where $\kappa =\frac{1}{3}v^{2}/\nu$ is the spatial diffusion coefficient), is larger than the distance from the source to the footpoints. Until the wave-energy density has grown to a level, denoted by U^*, that suppresses diffusive transport, the particles propagate quickly (relative to the time scale of wave transport) towards the footpoints adjusting the value of particle flux to $\vert S_{\pm }\vert\approx \frac{1}{2}Q$ (directed away from the source). The wave-energy density obeys

$$\ln \frac{U_{\pm }(x,t)}{U_{0}} \int _{0}^{t}\! \! \! \mathrm{d}t'\, \Gamma _{\pm }(x\mp V_{\mathrm{A}}t',t')$$ =

$$\pm \Omega _{\mathrm{p}}\frac{\pi }{2n_{0}V_{\rm A}}\int _{0}^{t}\! \! \! \mathrm{d}t'\, S_{\pm }(x\mp V_{\mathrm{A}}t',t')$$ =

$$\approx \pm \Omega _{\mathrm{p}}\frac{\pi }{4n_{0}V_{\mathrm{A}}}Qt$$ (3)

where the approximation on the last line is valid until the time t^* when the waves have reached the level U^*. This can be turned around to give the number of precursor particles per unit area, dN_prec/dA=Qt^*, as

$$\frac{\mathrm{d}N_{\mathrm{prec}}}{\mathrm{d}A}\sim \frac{4}... ...\mathrm{A}}}{\Omega _{\mathrm{p}}}\ln \frac{U^{*}}{U_{0}}\cdot$$ (4)

Figure 1: Flux of precipitating particles for four values of the total number of injected protons. The rate of injection is kept constant, $Q=2\times 10^{-14}$ cm-2s-1, and its duration, $\Delta t$, is varied, as indicated with their product in the legend.

It is not necessary to know the value of U^* exactly since it appears in the logarithm, and the numerical factor can be determined from simulations. However, a lower-limit estimate is given by equating the diffusion length to the distance from the source to the footpoint, i.e.,

$$\frac{4}{3\pi }\frac{U_{B}}{U^{*}}\frac{v^{2}}{V_{\mathrm{A}}\Omega }\sim \frac{L}{2}$$ (5)

giving

$$U^{*}\sim \frac{8}{3\pi }\frac{\gamma v^{2}}{V_{\mathrm{A}}^... ...pi }\frac{V_{\mathrm{A}}}{L\Omega _{\mathrm{p}}}\, pv\, n_{0}.$$ (6)

Inserting the numerical values gives $U^{*}/U_{0}\sim 7$ and $\mathrm{d}N_{\mathrm{prec}}/\mathrm{d}A\sim 10^{13}$ cm^-2, which is seen to agree well with the number of precursor particles in the simulation.

Let us estimate the spectrum of the promptly-precipitating protons. If a spectrum of particles is emitted, and if energy changes during the transport are neglected, we should identify the number of precursor protons with $Qt^{*}=p\, \mathrm{d}N_{\mathrm{prec}}/(\mathrm{d}A\, \mathrm{d}p)$ in the spirit of the resonance condition $k\propto p^{-1}$. For a wave spectrum $U_{0}\propto p^{q-1}$ with q<3, $p\, \mathrm{d}N_{\mathrm{prec}}/(\mathrm{d}A\, \mathrm{d}p)$ increases with (non-relativistic) energy, although logarithmically ( $\propto \ln vp^{2-q}$). Typical turbulence models give 1<q<2. Up to the energy where the total number of injected particles (per logarithmic momentum interval), $p\, \mathrm{d}N/(\mathrm{d}A\, \mathrm{d}p)$, equals the calculated value for the precursor, Eqs. (4-6), the energy spectrum of the first peak is $\mathrm{d}N_{\mathrm{prec}}/(\mathrm{d}A\, \mathrm{d}E)\propto (\ln vp^{2-q})/(vp)$. At larger energies, only the first peak is observed with a spectrum identical to that of the source.

Using the assumption of a steady state, Bespalov et al. (1991) deduced that the rate of particle injection determines whether particle transport is governed by diffusion or convection with the waves. That is correct in the steady-state regime of continual injection. In the case of short injection, however, the total number of injected particles, $Q\Delta t$, is the only controlling factor in the relative importance of the prompt and delayed components of the precipitating flux. To confirm this, we ran a simulations of using injection rates of $Q=2\times 10^{-6}\, n_{0}V_{\mathrm{A}}$, $2\times 10^{-5}\, n_{0}V_{\mathrm{A}}$, and $1\times 10^{-4}\, n_{0}V_{\mathrm{A}}$and with $Q\Delta t=6.67\times 10^{13}$ cm^-2 in all cases (Fig. 2). The resulting flux of precipitating particles looks nearly identical for the shortest injections ( $\Delta t=1/15$ s and 1/3 s) since the diffusive time scale is large enough to regard both of these injections as impulsive. The prolonged injection ( $\Delta t=10/3$ s), however, suppresses the height of the precursor while keeping the total number of precursor particles very similar to the impulsive ones, which is in accord with diffusive transport. Further, increasing the loop length or the level of ambient turbulence inside the loop clearly suppresses the importance of the precursor predicted by the estimation and confirmed by simulations. We, thus, regard the Eqs. (4-6), up to a numerical factor of the order unity, as a valid scaling law for arranging observations of precipitating proton fluences.

Figure 2: Precipitating flux for a constant number of injected particles, $Q\Delta t=6.67\times 10^{13}$ cm-2, and a variable duration of the injection, $\Delta t=1/15$ s (dashed curve), 1/3 s (dotted curve), and 10/3 s (solid curve).

3.2 Case B: Asymmetric loop

As became evident from the simulations of Case A, the majority of the particles in a powerful impulsive flare precipitate once the turbulent walls have reached the footpoints of the flux tube, at t=L/2V_A. It is of interest to investigate what happens if the particles are not released in the central point of the flux tube; if the trap reaches one end first, we should see a peak in the precipitating flux at this footpoint, but if the precipitation is rapid, the particle reservoir might empty before the trap opens from the other end. We have simulated this by varying the position of the source as $x_{0}=0.05\, L$, $0.1\, L$, and $0.25\, L$ (from the center of the loop). The number of injected particles was $6.67\times 10^{13}$ cm^-2, and other parameters were chosen as in Case A. The results are plotted in Fig. 3. The flux towards the footpoint closer to the source resembles the symmetric case with a prompt and a delayed peak. The flux at the other footpoint, however, behaves differently. Even a slightly asymmetric position of the source can lead to a huge difference in the precipitating flux for the delayed component; in practice, we can expect a delayed brightening in $\gamma$-rays at only one of the footpoints based on our simulations. However, the observed asymmetry in the precursor brightening provides a measure of the asymmetry of the source position, if the ambient turbulence level is high enough to yield a diffusive delay in the particle flux reaching the footpoints at different distances from the source.

Figure 3: The flux of precipitating protons in a loop with an asymmetric source position at $x_{0}=0.05\, L$, $0.1\, L$, and $0.25\, L$ (top to bottom). The curves should be multiplied by the indicated constants. Solid curves correspond to the footpoint closer to and dashed curves to the footpoint farther from the source. The total number of injected 30-MeV protons is $6.67\times 10^{13}$ cm-2.

Without a possibility to distinguish between the emission from different footpoints, the observations in the asymmetric case are very similar to the symmetric case, and the general conclusions drawn from Case A - two-component structure, importance of the total number of injected particles - are valid also for the asymmetric case.

3.3 Case C: Open flux tube

As a last study, we performed simulations in a flux tube that is connected to the solar surface from one footpoint at x=0, only. The other end at $x=L=5\times 10^{9}$ cm is assumed to leak the incident particles directly to the interplanetary medium, modeling a flux tube rapidly expanding above x=L; a large and rapid expansion would probably eject most of the particles reaching this distance, since the particle bulk speed is directed outwards, and particles diffusing backward relative to the flow at see the boundary as a strong magnetic mirror. In a more gently opening geometry, the effects of the diverging field should be taken into account explicitly in the simulation. The source is positioned at $x=0.1\, L$, and the total number of injected particles is $1.33\times 10^{14}$ cm^-2. This case differs from Case B simulations in that we assume that the background Alfvén waves have a large cross helicity: the outward propagating waves have $U_{+}(x=0,t)=U_{+}(x,t=0)=U_{0}=5\times10^{-5}\times U_{B}$ as in Cases A and B, but the inward propagating waves have only the minimum-level intensity, $U_{\mathrm{min}}=5\times10^{-7}\times U_{B}$. Other parameters were taken as in Cases A and B.

The results of the open-field simulation are shown in Fig. 4. The precipitating particle flux again shows a double peaked structure, with a precursor well described by Eqs. (4-6). The escaping flux, however, rises more slowly to a stationary level that is destroyed by the boundary effects, when the turbulent trap hits the free-escape boundary, opening it also from this end. Towards the end of the event, the escaping flux becomes dominant over the precipitating flux. The number of precipitating particles is, however, an order of magnitude larger than the number of escaping particles, which shows that energetic particles accelerated in impulsive flares can be effectively trapped near the Sun even on open field lines. This demonstrates that the small ratio of interplanetary-to-interacting protons observed in impulsive flares do not necessarily result from closed field line topology.

Figure 4: The flux of particles precipitating (solid curve) and escaping into the interplanetary medium (dashed curve) in an open magnetic flux tube. The hatched area is disturbed by a somewhat oversimplified outer boundary condition.