4 Discussion and conclusions

We have modeled the transport of energetic protons in impulsive flares through self-generated waves using the Monte Carlo method. Our model uses a scattering law that is proportional to the energy densities of single-k Alfvén waves propagating in both directions along the mean field. The energy lost (gained) by the particles in the scatterings is given to (taken from) the waves. The full quasi-linear resonance condition would involve a spectrum of waves even in case of mono-energetic particle distribution. The artificially sharpened resonance condition $\alpha_0 kp = m\Omega_\mathrm{p}$, where $\alpha_0$ is a numerical constant of the order unity, is, however, a commonly used approximation in diffusive particle transport and acceleration theories involving self-generated waves (e.g., Skilling 1975; Bell 1978; Lee 1983). It introduces an inaccuracy to the scattering rates, but for a power-law spectrum of particles it can be tuned by choosing $\alpha_0$ properly to yield correct growth rates for waves (Skilling 1975). Our simulations make use of a still simpler form, $k\propto p_{0}^{-1}$, which is a reasonable choice, if the particles approximately conserve their energies during the propagation. Note that this assumption is made implicitly in the analytical calculations of Bespalov et al. (1987, 1991). Because the turbulent trap is expanding along the magnetic field at a constant rate, however, there exists adiabatic deceleration of the trapped particles at a rate $\dot{p}/p=-1/(3t)$. This estimate is the lower limit, since it assumes that the particles spend a negligible fraction of time scattering off the turbulent walls of the trap and that stochastic acceleration inside the trap can be neglected. However, if the particle is trapped at t=t₀, it has an average final momentum of $\langle p\rangle =p_{0}(2V_{\mathrm{A}}t_{0}/L)^{1/3}$, when the trap opens (symmetric case). Thus, significant deceleration could be expected, if the characteristic trapping time t₀ is small. The simulations reveal that precipitating particles have average momenta $\ln \langle p\rangle /p_{0}>-1$ (see Fig. 5), so the approximative resonance condition seems acceptable for scattering estimates. Note, however, that the change in energy during trapping can be substantial in the context of $\gamma$-ray production, up to a factor of $\sim$5 in the non-relativistic case. For a typical integral particle spectrum, $N(>\!E)\propto E^{-2}$, this means a factor of $\sim$25 fewer particles capable of $\gamma$-ray production, and may suppress the delayed peak of emission completely. Note, however, that the problem of adiabatic deceleration is most severe in the case of strong, impulsive injections, as indicated by Fig. 6, where the average momentum is plotted for a constant number of injected particles varying the duration of the injection; for prolonged injections with , the particles conserve their energies more effectively, because most of them are injected in a trap that has a large initial dimension.

Figure 5: Average momentum of precipitating particles in Case A as a function of time for $Q=2\times 10^{14}$ cm-2s-1. The curves are as in Fig. 1.

Figure 6: Average momentum of precipitating particles in Case A as a function of time for $Q\Delta t=6.67\times 10^{13}$ 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).

Another simplification in our model is the assumption of constant magnetic field and plasma density inside the loop. A constant magnetic field is often seen as near-uniform cross sections of the flare loops (McClymont & Mikic 1994). Non-constant plasma parameters, however, affect wave propagation, particle transport, and wave growth in several ways: (i) wave-frequency conservation changes the wavenumber of a wave propagating at a spatially varying phase speed; (ii) resonant wavenumber changes with position in a non-constant magnetic field; (iii) particles suffer mirroring and adiabatic energy changes in a non-constant magnetic field. Because of all these effects, the wave growth is also affected. Our next step in developing the code is to replace the single wave fields and mono-energetic injections with with wavenumber and particle-energy spectra to estimate the effects of non-constant plasma parameters on wave growth. Once this is done, the resonance conditions can also be modeled more realistically.

Wave damping by plasma ions should be much less important than the growth or decay due to the energetic particles, because we are considering non-compressive Alfvén waves having only gyro-resonant interactions. We note, however, that if one drops the assumption of a wave-guiding loop, even initially parallel-propagating right-handed (left-handed) Alfvén waves convert to oblique fast-magnetosonic (Alfvén) waves and may suffer Landau damping (Wentzel 1976). From other processes, at least wave-wave interactions, as described by weak-turbulence theory, between Alfvén waves ( $A_{\angle }^{\pm }\rightleftharpoons A_{\perp }+A_{\angle }'^{\pm }$) and between Alfvén waves and sound waves ( $A_{\parallel }^{\pm }\rightleftharpoons S^{\pm }+A_{\parallel }^{\mp }$) have potential importance (see, e.g., Skilling 1975). The former cascade operates in the perpendicular wavenumber component (and involves a population of nearly perpendicular waves, $A_{\perp }$, and oblique waves, $A_{\angle }^{\pm }$) and the latter one in the parallel component. The kinetic equations for these processes, easily adaptable to a Monte Carlo simulation, can be obtained using quantum-mechanical formalism (Melrose 1980). Both cascades lead to an effective damping of the resonant waves. The parallel cascade, however, also helps spreading the wave power to wavenumbers, where the resonant particles are unable to generate waves on their own. In addition, it provides waves propagating in the stable direction, so it may also lead to efficient stochastic acceleration of resonant particles. It is left for future simulations to address the effects of the wave-wave interactions to the flare scenario.

We have neglected particle drifts due to field curvature in our simulations, which is justified because of the long time scales (hours) associated with the curvature drift (Ramaty & Mandzhavidze 1994). We have also neglected Coulomb losses and nuclear interactions during the propagation in our model. This is justified, since the trapping times we consider are relatively short ( 10 s): a Coulomb-loss estimate (e.g., Hulot et al. 1989) gives a momentum-loss rate of $(\dot{p}/p)_{\mathrm{c}}=-3\times 10^{-3}$ s^-1 at E=30 MeV and n₀=10¹¹ cm^-3.

In conclusion, the impulsive flare scenario in light of our simulations is the following: promptly (  s) after the start of the proton acceleration process near the loop top, the footpoints will brighten in $\gamma$-rays because the the turbulent trap will not be developed at that time. For a high level of ambient turbulence or large loop length, the footpoints may show rather different emission levels if the source is positioned somewhat off the loop top. After the first brightening, the turbulent particle trap develops and emission from the loop legs will stay at a level determined by the convective flux of particles. After one Alfvénic propagation time from the source, the closer loop leg will brighten in $\gamma$-rays once again, provided that the number of high-energy particles after the adiabatic losses in the trap is large enough, $p\, \mathrm{d}N/(\mathrm{d}A\, \mathrm{d}p)\gg (4/\pi )(n_{0}V_{\mathrm{A}}/\Omega _{\mathrm{p}})\sim 10^{13}\, n_{11}^{1/2}$ cm^-2 with n₁₁=n₀/(10¹¹ cm^-3). The footpoint farther away from the source is unlikely to show the delayed peak in $\gamma$-ray emission in any case. We note that this feature is observationally testable, especially with the images of solar $\gamma$-ray emission anticipated from the HESSI mission. We also found that the small interplanetary-to-interacting proton ratios observed in impulsive solar flares do not necessarily imply closed field line topology, but can result also from the turbulent trapping on open magnetic field lines. In the next stage of the modeling, we intend to take better account of the resonance conditions and consider non-constant background plasma parameters. The effects of wave-wave interactions will also be studied.