4. Pulsating and explosive regimes
The time profiles of the mm-wave emission presented in Sect. 2 reflect temporal evolution of plasma heating and particle acceleration at a flare site because the radio emission at mm-wavelengths is most probably due to bremsstrahlung and/or gyrosynchrotron processes. Deviations of plasma pressure in a magnetic loop and particle acceleration rate are determined, in turn, by given in Eq. (16). We will show that different regimes of the energy release can be realized in a single current loop depending on the values of the parameter which characterizes the rate of energy release, parameter which determines the damping of MHD-oscillations, and the initial conditions.
Eq. (16) describes the pulsating and explosive energy release, but in general case the solution can be obtain only numerically. Here we consider analytically the particular case , . This case can be realized in the dense chromospheric layers where the time scale of Joule heating driven by ion-atom collisions is much less than the Alfvénic transit time . So, we can omit the term in Eq. (16) and find the solution in the form
The unknown functions and describe the slow variation of the pulse amplitude in time and gas pressure gradient averaged over the pulsation period. Substituting Eq. (24) into the non-linear equation (16) and of the averaging over the fast time we find the following set of non-linear equations for both and :
The pulsation magnitude and the function of vary slowly, with typical time scale . determines the dimensionless function (see Eq. (9)) averaged over the pulsation period:
It is quite easy to convince ourself of this fact when substituting Eq. (24) into Eq. (14) and averaging.
An important peculiarity of Eq. (26) is that the derivative is always positive. It means that at the gas pressure in the periphery of flux tube grows faster than near the axis. This effect is due to the r -dependence of the Ampère force and Joule heating (see Eqs. (7) and (14)). The gas pressure gradient becomes positive and high until it starts accelerating the plasma tongue penetrating into the current channel due to the flute instability. We will show that this process is responsible for the explosive energy release.
Let us consider the special case when the velocity of the tongue is small, , where . In this case , and from Eqs. (25) and (26) we obtain
Physically the energy release can be divided into two classes.
(i) When the initial gas pressure in the loop decreases to the periphery but is high enough, , the pulsation amplitude first grows with typical time scale and then decreases for the time . This case corresponds to the time profiles of the mm-wave emission in Fig. 2a. For the solution of Eq. (28) gives pulsations with slowly damped magnitudes as in Fig. 2b. Both cases can be realized in a flare loop when the gas pressure inside the flux tube is high, .
(ii) In the opposite case when the gas pressure increases from the tube axis towards periphery, , the gas pressure and flux of accelerated particles increase explosively. This is described by the last term on the right of Eq. (28). The characteristic time of the explosive process is
When the damping time is less compared with the explosive phase as given by (29), (low pressure plasma), the pulsations occur before the explosive phase (Fig. 4a). For high pressure in the loop, , pulsations either superimpose upon the explosive time profile (Fig. 4c) or follow the explosive phase (Fig. 4b).
© European Southern Observatory (ESO) 2000
Online publication: June 5, 2000