2 The model

Our model consist of a (possibly curved) magnetic field with a constant magnitude along the line of force considered. Either one (Case C) or both (Cases A and B) ends of the flux tube are tied to the solar surface, where we assume that the density of the plasma is so large that all particles reaching this region interact, producing secondary emissions. No mirroring of particles at the footpoints of the flux tube is taken into account. Trapping of protons by mirroring at footpoints leads to the escape time of $2\rho _{0}(L/v)$ (e.g., Hulot et al. 1989), where $\rho _{0}$ is the footpoint mirror ratio. If this time is small relative to the trapping time by self-generated waves, L/(2V_A) (Bespalov et al. 1987, 1991), one may regard neglecting the footpoint mirroring reasonable (cf. Kocharov et al. 1999). Thus, this approximation is justified at $\rho _{0}\ll v/(4V_{\mathrm {A}})$, but for very large values of , trapping by footpoint mirroring should become important. The basic scale length of the (1-D) system is denoted by L. In the case of a closed magnetic field, L is the length of the loop and in the case of an open field, L is the height at which the particles are assumed to freely escape into the interplanetary medium.

The plasma parameters are fixed to representative values for flaring loops (Bespalov et al. 1987): Alfvén speed V_A=10⁸ cm s^-1, plasma electron (and proton) density n₀=10¹¹ cm^-3, and the length of the loop L=10⁹ cm. In Case C, we choose the length of the trapping region as $L=5\times 10^{9}$ cm being less than or of the order of the scale height of the magnetic field in the lower corona. The magnetic field in such a flux tube (with electron-proton composition) is $B\approx 145$ Gauss, and the ion skin length, $V_{\mathrm{A}}/\Omega _{\mathrm{p}}\approx 72$ cm, where $\Omega _{\mathrm{p}}\approx 1.4\times 10^{6}$ s^-1 is the proton gyro-frequency. The plasma temperature is taken to be T=10⁷ K for both electrons and protons. This gives a value of plasma beta of $\beta =16\, \pi \, n_{0}k_{\mathrm{B}}T/B^{2}\approx 0.33$ for the thermal component.

The Alfvén waves responsible for the collisionless scattering of the accelerated protons are assumed to propagate along the mean magnetic field. We, therefore, assume that the flux tube acts as a wave guide in case it is curved. This is valid, if the considered flux tube has a higher plasma density than its surroundings (Mazur & Stepanov 1984). For simplicity, no absorption of the waves by the thermal plasma nor any wave-wave interactions are thought to occur. Sunward-propagating waves are absorbed and anti-sunward propagating waves are emitted at the footpoints of the flux tube. The emitted wave flux at the footpoints has to be, of course, given as a boundary condition.

We consider energetic protons of momentum p=p₀ emitted from a point source located inside the flux tube at x=x₀. Protons are followed under the guiding-center approximation along the magnetic field as they undergo wave-particle interactions with Alfvén waves of wavenumber that is assumed to be fixed and given by $k=m\Omega _{\mathrm{p}}/p_{0}$. This approximation to the full resonance condition, $k=m\Omega _{\mathrm{p}}/p\mu$ ($p\mu$ is the parallel momentum in the wave frame), allows us to follow a single wavenumber instead of the full spectrum of them, and makes the simulations much simpler and faster. Particles hitting the ends of the flux tube are assumed to be absorbed (by stopping at the solar surface or by escape to the interplanetary medium). The wave-particle interactions are modeled as pitch-angle scattering that is elastic in each wave frame and occurs at the scattering rates (Skilling 1975)

$$\nu _{\pm }(x,p,t)=\frac{\pi }{4}\, \Omega \, \frac{U_{\pm }(x,t)}{U_{B}},$$ (1)

where $U_{B}=B^{2}/8\pi$, $\Omega =\Omega _{\mathrm{p}}/\gamma$ is the relativistic gyro-frequency of the proton, and $U_{\pm }(x,t)$ is the total (kinetic+magnetic) energy density (per logarithmic bandwidth in wavenumber) of waves propagating parallel (+) or anti-parallel (-) to the field line. The scatterings lead to pitch-angle diffusion with isotropic diffusion coefficients $D_{\mu \mu }^{\pm }=\frac{1}{2}(1-\mu ^{2})\nu _{\pm }$.

Energetic protons interact with the waves self-consistently in the sense of conserving the total energy (as measured in the plasma frame) of waves and particles at a microscopic level. Let $p_{\pm}$ and $\mu _{\pm}$ denote the momentum and pitch-angle cosine as measured in the wave frame indicated by the subscript. A particle scattering in pitch-angle cosine by an amount of $\Delta \mu _{\pm }$ (in the wave frame) suffers an energy loss of $-\Delta E=\mp V_{\mathrm{A}}p_{\pm }\Delta \mu _{\pm}$ in the plasma frame. We use isotropic scattering, so $\langle \Delta \mu _{\pm}\rangle /\Delta t=\partial D_{\mu \mu }/\partial \mu =-\mu _{\pm }\nu _{\pm }$. This leads to a growth of the waves at the rate

 

$$\Gamma ^{\pm } \pm \frac{1}{U_{\pm }}\int \! \! \mathrm{d}^{3}p_{\pm }\, \nu _{\pm }V_{\mathrm{A}}p_{\pm }\mu _{\pm }\, f_{\pm }$$ =

$$\pm \Omega _{\mathrm{p}}\frac{\pi S_{\pm }}{2n_{0}V_{\mathrm{A}}}$$ = (2)

where $f_{\pm }(\mu _{\pm },p_{\pm },x,t)$ is the distribution function of the accelerated particles and $S_{\pm }=\int \mathrm{d}^{3}p_{\pm }\, v_{\pm }\mu _{\pm }\, f_{\pm }$ gives their flux. A more detailed description of the numerical method is given in the Appendix.