c where the first part is caused by the electron decrement/increment and the second is due to Landau damping of the thermal ions.

Overheating of turbulent regions in the current sheet. Turbulent plasma heating by anomalous current dissipation is another cause of the unstable state of current sheets (Pustil'nik, 1980). This phenomenon is due to the rapid heating of the plasma over an extremely short time

It leads to changes of the current velocity/thermal velocity ratio over this time from u/Vje > 1 (necessary for plasma wave generation) to u/Vje < 1 with very rapid plasma wave dissipation by Landau absorption of the plasma waves by the thermal electrons (the time is of the order of Q-/). This heating stops the local plasma turbulence and leads to a transition to a normal, high-conductive state. During the next stage, this region will be cooled by the thermal front of collision-less hot electrons, and will return into its initial turbulent state (the anomalous thermo-conductivity will restore the turbulent state after short cooling times T— = £((/// VTe ). Subsequent anomalous heating in the turbulent state will repeat these local transitions in a pulsation regime, and create numerous normal and turbulent anomalous regions in the current sheet.

Splitting of current sheet at regions of discontinuous conductivity. According to Pustil'nik (1997), the processes of rapid plasma wave generation in a turbulent regime lead to the rapid increase of the anomalous resistance and create a jump in the conductivity a at the boundary between the normal and anomalous stage of magnetic field structure is caused by the slower diffusion process (then plasma instability), and at the first stage electric current is conserved, this will lead to a jump in the electric field at the boundary of the current sheet. This discontinuity in the electric field leads to a rapid redistribution of the current, with a decrease of the current density in the turbulent region to below ucr, simultaneously, there is a rapid increase of the current density in the external region up to values exceeding ucr near the boundary. In the inner region, with u < ucr , plasma turbulence will disappear, and this local layer turns into the normal state. In the external region, we have the opposite result, with plasma turbulence and the generation of anomalous resistance for short times. New jumps of the conductivity arise at the new boundary between the normal and abnormal plasma, which will lead to a new splitting of the current sheet boundary. The resulting final state is a dynamic equilibrium of the current sheet, which contains numerous compact and short-lived turbulent and normal regions.

4.18.12. Unsteady state of turbulent current sheet and percolation

As it was demonstrated above, the standard steady state of a current sheet is unstable, and it must be disrupted into numerous local, short-lived, small-scale domains of "normal" and "abnormal" plasma (see Fig. 4.18.5). These domains form numerous virtual bond clusters from the "normal" and "turbulent" elements. Therefore, current propagation through a flare current sheet should be similar to percolation through a stochastic network of "good" and "bad" resistors with a constant source of electric current. This process has been studied both in experiments: in superconductive ceramic samples (Vedernikov et al., 1994) and in conductive graphite paper with random holes (Levinshtein et al., 1976; Last and

Thouless, 1971), in numerical simulations (Kirkpatrik, 1973), and in theoretical models (Render, 1983).

However, this important phenomenon was not taken into account in models of energy release and particle acceleration. There are some additional effects of current percolation in our situation - positive and negative feedback between elements, due to the dynamic redistribution of currents and thermo-conductive fluxes:

(i) Current conservation leads to a redistribution of the currents in the sheet as a result of the permanent stochastic rebuilding of the network resistors. This will change the current density in the elements of the network and will lead to the generation of induced turbulence, with switch-on of plasma turbulence in some neighborhood and switch-off in the initial area. This will lead to constant transitions of the resistors in the network from the "bad" state to the "good" state and back.

(ii) Thermo-conductive flux from the heated turbulent elements will escape into the surrounding cold plasma, heat this plasma, and thereby change the threshold current value ucr ^ Vje .

This very complex pattern, with intricate feedback between current propagation and the plasma turbulence state in local regions can be described using elementary transition probabilities in a stochastic resistor network, with the properties of the resistors dependent on the local current value, cross connections between resistors, and some delay effects. The best approximation to this process is percolation through a fractal network characterized by some cluster factuality, fractal dimension, and threshold of the percolation as infinite cluster disruption. Some general conclusions can be drawn from the first principles of percolation theory (Feder, M1988; see also Mogilevsky, M2004): 1) A fundamental property of a hold dependence of the global network conductivity on the density of bad elements, and, hence, on the current:

2) Another general property of a percolation process is the universal power-law dependence of the statistical properties in a global system (number versus amplitude, for example) on the characteristics of the domains (scaling):

Here x is a parameter of the domain (size, amplitude, energy, etc.). The exponents a in Eq. 4.18.32 and k in Eq. 4.18.33 are determined by the fractal dimension of the clusters and the global dimensions of the system n, and are

a= \ k = \ (4.18.34) I 0.40 for n = 3 I 2.5 for n = 3

Pustil'nik (1997) compared the general conclusions of the percolation approach with real flare observations in the solar atmosphere and flare stars, and obtained a remarkable correspondence: for all flare stars (UV Ceti-type red dwarfs; Gershberg and Shakhovskaya, 1983; Gershberg, 1989) and for various manifestations of solar flares in Ha storms (Kurochka, 1987; Aschwanden et al., 1995; Merceier and Trottet, 1997) and hard X-ray solar bursts (Crosby et al., 1992), there is the same statistical dependence of the flare frequency and energy with the value of P = 1.7^1.8, similar to that expected from percolation theories. The role of the percolation process and fractal formation in the formation of the frequency-energy spectrum was successfully considered by Wentzel et al. (1992) in a percolation model of active region formation from the convective zone, and by Vlahos et al. (1995) in a fractal model for the structure of magnetic elements over active regions.

4.18.13. Acceleration of particles in a fragmented turbulent current sheet

The fundamental property of solar flares is the acceleration of charged particles up to very high energies of several GeV/nucleon with energy spectrum of a power law with the slope y from 2 up to 7, but the mean value {y)~ 2 + 3 (Dorman, M1957,

M1963a,b, M1978; Dorman and Miroshnichenko, M1968; Duggal, 1979; Dorman and Venkatesan, 1993; Stoker, 1994; Miroshnichenko, M2001). Pustil'nik (1997) considered two types of models for the particle acceleration during solar flares:

(i) a turbulent boiler with a "particle-plasmon" energy exchange (Kaplan and Tsytovich, M1973),

(ii) direct run-away of the particles in the DC electric field of a current sheet (Spicer, 1982).

The first model (i) is able to explain naturally the power-law energy spectrum as a consequence of turbulent diffusion in momentum space, but requires extreme assumptions about the turbulent energy. The second model (ii) can accelerate particles up to the maximum energies, but has difficulties in explaining the power-law dependence of the energy spectrum (the typical spectrum for run-away particles is an exponential). This approach does not take into account the fact that the motion of rapid particles in turbulent plasma is not a direct run-away, but rather a space diffusion, caused by effective elastic particle-plasmon scattering. Taking into account the cluster structure of a turbulent current sheet with numerous bad resistors (which play the role of plasma double layers and act as compact line accelerators in

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