1 Introduction

Acceleration of protons up to several tens of MeVs and beyond may take place at different phases of solar eruptions, from the flare pulse in the low corona through the passage of the interplanetary shock beyond the Earths orbit. Some protons interact at the Sun, producing secondary $\gamma$-rays and neutrons, others escape into the interplanetary medium to produce a solar energetic particle (SEP) event at 1 AU. The current two-class paradigm of solar particle events suggests that impulsive SEP events originate in impulsive solar flares and gradual events are linked to the coronal/interplanetary shocks (Cane et al. 1986; Cliver 1996; Reames 1999 and references therein). Impulsive flares may produce ³He-rich SEP events, which typically are not strong in terms of the total number of accelerated protons. Some impulsive flares, however, can produce a large number of interacting protons and can be observed in the $\gamma$-ray band. In such events the ratio of the numbers of interplanetary-to-interacting protons is typically small ( 0.1), whereas gradual $\gamma$-ray flares show larger ( 1) values of this ratio (Kocharov & Kovaltsov 1986; Hua & Lingenfelter 1987; Ramaty et al. 1993).

Intensive production of accelerated protons implies the potential importance of self-generated waves, i.e., waves generated by the flux of protons leaking from the accelerator (see, e.g., Ng et al. 1999 in application to gradual SEP events). In application to impulsive $\gamma$-ray flares, the role of self-generated waves was emphasized by Bespalov et al. (1987, 1991). The transport conditions of protons in flaring loops determine the time scale for their precipitation. The conditions are, therefore, crucial for the flare scenario. Strong interaction with hydromagnetic waves is known to make the trapping of protons inside the loop efficient. Bespalov et al. (1987, 1991) studied the proton transport under the assumption that the protons stream away from a source located at the top of the loop, and generate Alfvén waves. They showed that for a sufficiently strong source, the waves would become so intense that the particles would essentially be locked on the waves and convect with them at Alfvén speed towards the footpoints of the loop. Electrons, being coupled to whistler and ion-cyclotron waves, would travel much faster than protons, having a bulk speed of 9 $\, V_{\mathrm{A}}$ (e.g., Vainio 2000). This would lead, consistent with observations of impulsive flares, to a delay in the $\gamma$-ray emission relative to the X-ray emission, if electrons and protons were accelerated simultaneously near the top of the flaring loop (Bespalov et al. 1987). Since the bulk speed of electrons depends on energy, however, the ultra-relativistic electrons responsible for the $\gamma$-ray continuum behave very similarly to protons. Note that the delay of the peak time of the $\gamma$-ray emission relative to the X-ray peak could also be interpreted in terms of particle transport in flaring loops with weak external turbulence and large variation of the cross-sectional area of the flux tube (Hulot et al. 1989, 1992).

One aspect of the turbulent trapping model missing from the original work of Bespalov et al. (1987, 1991) is that it could lead to interesting effects, if the flux tube is asymmetric with respect to the source position, and if transient processes are incorporated in the model. Scattering off self-generated waves at open magnetic field lines may also affect the interplanetary-to-interacting proton ratio. Vainio et al. (2000) obtained reasonable ratios for gradual flares in a model of coronal shock acceleration by arranging the turbulence around the shock in a stationary manner to mimic self-generated waves. In the case of impulsive flares, the role of self-generated waves in forming the interplanetary-to-interacting proton ratio has not yet been studied. Applying the asymmetric, strongly turbulent transport model to an open flux tube with the solar surface on one end, and a free-escape boundary, i.e., the interplanetary medium, at the other end, we could anticipate a small value of the interplanetary-to-interacting proton ratio, if the source is close to the Sun.

The purpose of this paper is to study and extend the transport model of Bespalov et al. (1987, 1991), using a time-dependent numerical method. To simplify the model at this stage, we perform the study inside a single magnetic flux tube with constant values of the magnitude of the magnetic field and plasma density, and zero bulk speed of the thermal plasma. The study includes as separate cases a symmetric loop (Case A), an asymmetric loop (Case B), and an open field line (Case C). The potentially important effects of non-constant plasma parameters (see, e.g., Kocharov et al. 1999) is left for the future. We note here, however, that interpretation of magnetogram data (McClymont & Mikic 1994) suggest that coronal loops showing almost constant cross section are possible due to twists in the magnetic flux tubes. Thus, the constant-field approximation should not be too far from reality to be applied in this study.