5.3 Shock formation and heating

5.3.1 Slow shock

One of the important features of the Petschek-type reconnection is the formation of a pair of slow MHD shocks extending from a diffusion region. The role of the slow shocks was first considered in Cargill and Priest (1982). A pair of the slow shocks guide a reconnection jet, and the angle between these shocks is very narrow, of the order of MA. In the adiabatic case, the temperature of coronal plasma increases across a slow shock up to (Vršnak and Skender, 2005Jump To The Next Citation Point)

T ∼ T ∕ β, (28 ) slow shock corona
where
( )( )( ) β ≡ pgas∕pmag = -2nkT--∼ 0.01 ----n----- --T--- --B--- . (29 ) B2 ∕8π 1010 cm −3 106 K 100 G
Here, we assume that the total pressure is balanced between the pre-shock region (2 ∼ B ∕(8π ) because of low β plasma) and post-shock region (∼ pgas because of high β), and the gas density is roughly constant across the shock within a factor of 2 – 3. Equations (28View Equation) and (29View Equation) suggest that coronal plasma with T ∼ 1 MK could be heated up to 100 MK. However, in reality, the thermal conduction works to reduce the temperature, so the value mentioned above is somewhat overestimated, just presenting the upper limit of the temperature enhanced via slow shock heating. Moreover, when the thermal conduction is efficient, an adiabatic slow shock tends to be split into a conduction front and an isothermal slow shock across which only the density and pressure increase but the temperature does not change (Forbes et al., 1989Jump To The Next Citation Point; Yokoyama and Shibata, 1997Jump To The Next Citation Point).
View Image

Figure 44: Top panels show temperature (left) and pressure (right) distributions in the Petschek-type reconnection for the cases without and with conduction. The curves show magnetic field lines and the arrows show velocity vectors. Note that the adiabatic slow shocks dissociate into the conduction fronts and the isothermal slow shocks in the case with conduction (from Yokoyama and Shibata, 1997Jump To The Next Citation Point). Bottom panel shows the time evolution of the temperature and density distributions in the reconnection with thermal conduction and chromospheric evaporation as a model of solar flares. The curves show magnetic field lines and the arrows show velocity vectors. In this case, not only the conduction fronts and isothermal slow shocks but also dense chromospheric evaporation flow are clearly seen (from Yokoyama and Shibata, 1998Jump To The Next Citation Point).

Yokoyama and Shibata (1997Jump To The Next Citation Point) first carried out a self-consistent MHD simulation of magnetic reconnection that includes thermal conduction (see the top panels in Figure 44View Image). They confirmed that an adiabatic slow shock is split into a conduction front and an isothermal slow shock, as predicted by Forbes and Malherbe (1986). Yokoyama and Shibata (1997) further explained the structure of a cusp-shaped flare observed by Yohkoh (Tsuneta et al., 1992a), where the evaporation of chromospheric plasma heated by the conduction front is reproduced (Yokoyama and Shibata, 1998Jump To The Next Citation Point, see the bottom panels in Figure 44View Image). Typically, the conduction time is estimated to be

--3nkBT--- (----n-----)(--T---)− 5∕2( --L----)2 tcond ∼ κ0T 7∕2∕L2 ∼ 10 1010 cm −3 107 K 109 cm s, (30 )
where −6 κ0 ≃ 10 (CGS) is the heat conductivity coefficient due to the coulomb collision (Spitzer, 1962) and kB is the Boltzmann constant. Here, we note that thermal flux saturation could occur at the conduction front in the presence of strong temperature gradients (e.g., Manheimer and Klein, 1975, for application see, e.g., Somov and Titov, 1985; Vršnak, 1989).

On the other hand, the Alfvén transit time is

( n )1∕2( B )−1( L ) tA ∼ L ∕vA ∼ 10 ---------- ----- ------- s. (31 ) 1010 cm −3 50 G 109 cm
For T > 107 K, the conduction time is shorter than the Alfvén transit time, so the effect of thermal conduction should be taken into account in considering the evolution of a flare.

The radiative cooling time of plasma is written as

( ) ( ) trad ≃ -3kT---≃ 5 × 103 s --T--- 3∕2 --n---- −1. (32 ) nQ (T ) 106 K 109 cm
Here,
−22(--T---)−1∕2 7 Q (T) ∼ 10 106 K cgs for T < 10 K (33 )
−23( --T--)1∕2 7 Q (T ) ∼ 3 × 10 107 K cgs for T > 10 K (34 )
is the radiative loss function for optically thin plasma (Raymond et al., 1976). In a typical SXR loop, the temperature is about 107 K and electron density is about n = 1010 cm–3, so the radiative cooling time becomes of the order 104 s, which is much longer than both the conduction time and the Alfvén transit time. Hence, the radiative cooling could be neglected at least in the very early phase of a flare.

Assuming that cooling via thermal conduction (κ0T7∕2∕L2) is balanced by heating via magnetic reconnection (Q erg cm–3 s–1) in an SXR loop with the size of L, the temperature in this loop is given by (Fisher and Hawley, 1990Jump To The Next Citation Point)

Tloop ∼ (QL2 ∕κ0)2∕7. (35 )
The heating rate based on magnetic reconnection is given by
2 Q ∼ B vA∕(4πL ). (36 )
This expression has been derived by Yokoyama and Shibata (1998Jump To The Next Citation Point) for the Petschek-type reconnection with a pair of slow shocks. In this case, the volumetric heating rate due to reconnection is given by the Poyinting flux entering into the reconnection region divided by the thickness of the reconnection region (bounded by a pair of the slow shock):
Q = [B2 ∕(4π)](Vin∕L )(1∕sin𝜃 ), (37 )
where Vin is the velocity of the inflow into the slow shock, L is the length of the slow shock and 𝜃 is the angle between the slow shock and the initial field line. Since the 𝜃 is approximately given by sin𝜃 ≈ Vin∕VA, we have Equation (36View Equation). Note that this is a crude estimate, though it is roughly consistent with the numerical simulation results of Yokoyama and Shibata (1998Jump To The Next Citation Point, 2001Jump To The Next Citation Point) within a factor of a few.

Combining Equation (35View Equation) and (36View Equation), we obtain

2 T ∼ ( B-vAL-)2∕7 loop 4 πκ0 B n L ∼ 4 × 107( ------)6∕7(--10----−3)−1∕7(--9----)2∕7 K. (38 ) 100 G 10 cm 10 cm
These values are measured behind the slow shock from which the conduction front and isothermal slow shock extend (Forbes et al., 1989; Yokoyama and Shibata, 1998Jump To The Next Citation Point, 2001). The temperature of 107 K is comparable to the one observed in the superhot region formed at the top of an SXR loop (Masuda, 1994; Kosugi et al., 1994; Tsuneta, 1996; Nitta and Yaji, 1997). Note that this value is smaller than the value obtained from adiabatic MHD simulations, in which the temperature becomes of the order 108 K as mentioned before.

It should be noted that the conduction could be strongly reduced due to a large difference in the magnetic field strength in inflow and outflow region, as well as due to thermal flux saturation and the flow/field geometry (for details see Vršnak et al., 2006, and references therein).

5.3.2 Chromospheric evaporation

During a flare, the chromospheric evaporation (i.e., ablation of chromospheric plasma) plays a fundamental role in creating an SXR loop via the injection of hot plasma into a loop (Hirayama, 1974). The evaporation occurs when the heat flux coming from the corona overcomes radiative cooling rate in the chromosphere. That heat flux then increases the gas pressure in the upper chromosphere significantly to produce an upflow toward the corona against gravity (called evaporation flow Antonucci et al., 1982, 1984) as well as downflow (Ichimoto and Kurokawa, 1984Jump To The Next Citation Point; Canfield et al., 1990) to the lower chromosphere. In a steady state, the heat flux of thermal conduction from the corona is balanced by the enthalpy flux of evaporation, such as

κ T 7∕2∕L2 ∼ 5pv . (39 ) 0 2 evap
Here v evap is the velocity of an evaporation flow, which is the same order of the sound velocity,
T vevap ∼ cs ∼ 500(------)1∕2 km s−1, (40 ) 107 K
since the flow is driven by gas pressure. When the heat flux reaches a deep layer in the chromosphere and is balanced by enhanced radiative cooling rate due to large gas density, evaporation ceases.

Nagai (1980) first performed a one-dimensional hydrodynamic simulation of chromospheric evaporation. Since then, similar one-dimensional hydrodynamic simulations have been performed extensively (Somov et al., 1981; Nagai and Emslie, 1984; Peres et al., 1987; MacNeice et al., 1984; Mariska et al., 1989; Fisher and Hawley, 1990; Gan et al., 1991), which qualitatively explained the blue shift of Bragg Crystal Spectrometer (BCS) lines observed by Yohkoh as well as the red shift of Hα line observed during the impulsive phase of a flare (Ichimoto and Kurokawa, 1984). Investigations into the quantitative agreement between one-dimensional models and observations are still in progress. Later, pseudo two-dimensional models have been developed, reported by several authors (Hori et al., 1997; Warren et al., 2003). Yokoyama and Shibata (1998) performed a two-dimensional MHD simulation reproducing the chromospheric evaporation driven by thermal conduction (see the bottom panels in Figure 44View Image). By combining magnetic reconnection, thermal conduction and radiative cooling, they derived a scaling law about the temperature observed in a loop filled with evaporated plasma, as shown in Equation (38View Equation). Later, Shibata and Yokoyama (1999Jump To The Next Citation Point) applied this scaling law to stellar flares (see Section 6).

5.3.3 Fast shock

A fast MHD shock is formed by the downward reconnection jet colliding with the top of an SXR loop (Forbes and Priest, 1983; Ugai, 1987; Magara et al., 1996; Aurass et al., 2002) where high-energy electrons are possibly produced (see Figure 42View Image). In the adiabatic case, the temperature just behind the fast shock becomes

2 8 ( B )2 ( ne )−1 Tfast shock ∼ mivjet ∕(6kB) ∼ 2 × 10 100-G- 1010-cm−-3 K, (41 )
where mi is the proton mass (Vršnak and Skender, 2005). This value might be overestimated because the thermal conduction is not taken into account, unless magnetic field lines are so tangled as to reduce the efficiency of thermal conduction.

A fast shock could also be formed at the bottom of an ejecting plasmoid when it moves much slowly compared to the local Alfvén velocity (Magara et al., 1997, 2000).


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