Breakdown of classical electron transport in the corona

4.3 Breakdown of classical electron transport in the corona

As was shown by Dorelli and Scudder (2003 ), the deceleration of suprathermal electrons in the coronal polarization electric field may even allow electron heat to flow radially outward against the local coronal temperature gradient, in contrast to the LTE relation (29

), in which heat is constrained to flow down the local temperature gradient. The reason being, that the Sun’s gravitational field and the electric polarization field in the transition region cause a trapping of thermal electrons, but cannot prevent runaway of suprathermals, which can thus carry heat radially outward below the temperature maximum. This is illustrated in Figure 9

, which shows the normalised value, $θ = qe∕qsat$ , with the saturation heat flux, being $q = ρ V3 sat e e$ , versus the kappa index. For $κ ≈ 10$ , the heat flux direction reverses.

Figure 9:

Electron heat flux in transition region as calculated for a kappa VDF at the chromospheric lower boundary. The horizontal dash-dot line (left frame) and dashed line (right) are the Spitzer and Härm (1953

) predictions drawn for reference. Left: The squares show the predicted normalised heat flux from the kappa-function model equation. The vertical dashed line shows the value of $κ$ for which the electron heat flux vector changes sign. Positive values of $θ$ correspond to heat flowing up the temperature gradient (antiSunward) for strong tails ( $κ < 10$ ). Negative values of $θ = q∕qsat$ correspond to heat flowing down the temperature (Sunward) gradient (after Dorelli and Scudder, 1999

). Right: Similar plot, but for a full numerical solution obtained by solving the kinetic problem according to Fokker–Planck diffusion. Note that here non-classical conduction occurs only for strong suprathermal tails, $κ < 5$ , assumed to prevail at the lower boundary (after Landi and Pantellini, 2001

).

Dorelli and Scudder (1999 ) modelled this effect, while describing the zeroth-order local electron VDF by a kappa function with exponent $κ$ , and retaining only the linear terms in an expansion in terms of the pitch angle variable $μ = cos ϑ$ . This linearization badly fails in the distant solar wind, as was shown by Dum et al. (1980 ), and was also criticised for its application to the lower corona by Landi and Pantellini (2001 ), both authors arguing that higher-order polynomials must be retained to model the collisional energy exchange between the thermal core electrons and suprathermal halo electrons. Also some time ago, Anderson (1994 ) criticised the exospheric velocity filtration model and argued that a collisional treatment of this effect is needed.

According to Dorelli and Scudder (1999 ), if one considers in the electron VDF power-law suprathermal tails (kappa function) and then accounts for the associated velocity filtration effect, one gets a corrected energy balance for a steady-state transition region (for a general review of the classical approach see the book of Mariska , 1992 about the TR), reading:

where $K1$ and $K2$ are positive constants depending on the $κ$ -exponent, and $Fe$ denotes the combined external force related to the gravitational and electric potential. Here $z$ is the altitude, and $R$ and $H$ are the radiative loss function, respectively local heating function. If the net force is attractive ( $F < 0 e$ ), then the last term represents a heating source adding to $H$ . In conclusion, non-local electron heat flow coupled to the force $Fe$ through velocity filtration must be taken into account in the internal energy balance in the TR.

This result is a kinetic consequence of heat flow in a weakly collisional and a non-uniform medium such as the corona, and will likely remain valid in any realistic future model. Landi and Pantellini (2001 ) have already demonstrated this with their more refined kinetic model of electron heat conduction. In the solar corona the collisional mean free path for a thermal particle (electrons or protons) is small, of the order of $10− 2$ to $10 −3$ times the typical scale height, $h$ , of macroscopic fluid quantities like density or temperature. Despite this relative smallness of $λc∕h$ , the coronal plasma cannot be described satisfactorily by theories supposing that the local VDFs are close to Maxwellians (see our previous discussion again).

It was shown in particular that if the electron VDFs at the base of the corona have sufficiently strong suprathermal power-law tails, the heat flux may indeed flow upwards, i.e., in the direction of increasing temperature. Using kappa functions as prototypes for non-thermal VDFs at the base, they found that heat conduction can only be adequately described by the classical law provided that $κ > 5$ . This value is much smaller than the one found by Dorelli and Scudder (1999 ). The results from both groups are illustrated and compared in Figure 9 . Landi and Pantellini (2001 ) further showed that, unless extremely strong electron tails are assumed near the base of the corona ( $κ < 4$ ), a local heating mechanism (most likely by waves) is needed to sustain the steep temperature gradient between the base of the corona and the location of its temperature maximum.