Scaling Laws for Dynamic Solar Loops.

The scaling laws which relate the peak temperature T M and volumetric heating rate E H to the pressure P and length L for static coronal loops were established over 40 years ago; they have proved to be of immense value in a wide range of studies. Here we extend these scaling laws to dynamic loops, where enthalpy flux becomes important to the energy balance, and study impulsive heating/filling characterized by upward enthalpy flows. We show that for collision-dominated thermal conduction, the functional dependencies of the scaling laws are the same as for the static case, when the radiative losses scale as T -1/2, but with a different constant of proportionality that depends on the Mach number M of the flow. The dependence on the Mach number is such that the scaling laws for low to moderate Mach number flows are almost indistinguishable from the static case. When thermal conduction is limited by turbulent processes, however, the much weaker dependence of the scattering mean free path (and hence thermal conduction coefficient) on temperature leads to a limiting Mach number for return enthalpy fluxes driven by thermal conduction between the the corona and chromosphere.

Reduction of Thermal Conductive Flux by Non-local Effects in the Presence of Turbulent Scattering.

The heat flux in a plasma is determined by the degree of anisotropy in the particle distribution function, which is in turn driven by gradients in the ambient density and temperature. When the mean free path at the thermal speed is substantially smaller than the scale length associated with the temperature variation, the heat flux simply depends on the local value of the temperature gradient. However, when the temperature scale length and mean free path are comparable, heat conduction becomes substantially non-local in character: the magnitude of the heat flux now depends on the overall temperature profile and is generally smaller than the locally determined value. In the presence of angular scattering associated with turbulence, the mean free path (and its velocity dependence) can be significantly smaller than its collisional value; this makes the expression for the heat flux more local in character, but also results in a heat flux that is lower than that obtained through a purely collisional analysis. Therefore, whether or not turbulence is present, the heat flux is generally smaller than the value obtained from a local collisional analysis. We here present an analytic expression for the conductive heat flux in terms of a convolution of the local heat flux with a non-local kernel function that incorporates both Coulomb collisions and turbulent scattering. We comment on the need to include both non-local and turbulent scattering effects in the modeling of quasi-static active region loops and in the conductive cooling of post-flare loops.

Identification of Multiple Hard X-Ray Sources in Solar Flares: A Bayesian Analysis of the 2002 February 20 Event.

The hard X-ray emission in a solar flare is typically characterized by a number of discrete sources, each with its own spectral, temporal, and spatial variability. Establishing the relationship among these sources is critical to determining the role of each in the energy release and transport processes that occur within the flare. In this paper we present a novel method to identify and characterize each source of hard X-ray emission. The method permits a quantitative determination of the most likely number of subsources present, and of the relative probabilities that the hard X-ray emission in a given subregion of the flare is represented by a complicated multiple source structure or by a simpler single source. We apply the method to a well-studied flare on 2002 February 20 in order to assess competing claims as to the number of chromospheric footpoint sources present, and hence to the complexity of the underlying magnetic geometry/topology. Contrary to previous claims of the need for multiple sources to account for the chromospheric hard X-ray emission at different locations and times, we find that a simple two-footpoint-plus-coronal-source model is the most probable explanation for the data. We also find that one of the footpoint sources moves quite rapidly throughout the event, a factor that presumably complicated previous analyses. The inferred velocity of the footpoint corresponds to a very high induced electric field, compatible with the fields in thin reconnecting current sheets.

Energy Deposition by Energetic Electrons in a Diffusive Collisional Transport Model.

A considerable fraction of the energy in a solar flare is released as suprathermal electrons; such electrons play a major role in energy deposition in the ambient atmosphere, and hence the atmospheric response to flare heating. Historically, the transport of these particles has been approximated through a deterministic approach in which first-order secular energy loss to electrons in the ambient target is treated as the dominant effect, with second-order diffusive terms (in both energy and angle) being generally either treated as a small correction or neglected. However, it has recently been pointed out that while neglect of diffusion in energy may indeed be negligible, diffusion in angle is of the same order as deterministic scattering and hence must be included. Here we therefore investigate the effect of angular scattering on the energy deposition profile in the flaring atmosphere. A relatively simple compact expression for the spatial distribution of energy deposition into the ambient plasma is presented and compared with the corresponding deterministic result. For unidirectional injection there is a significant shift in heating from the lower corona to the upper corona; this shift is much smaller for isotropic injection. We also compare the heating profiles due to return current ohmic heating in the diffusional and deterministic models.

Heating and Cooling of Coronal Loops with Turbulent Suppression of Parallel Heat Conduction.

Using the "enthalpy-based thermal evolution of loops" (EBTEL) model, we investigate the hydrodynamics of the plasma in a flaring coronal loop in which heat conduction is limited by turbulent scattering of the electrons that transport the thermal heat flux. The EBTEL equations are solved analytically in each of the two (conduction-dominated and radiation-dominated) cooling phases. Comparison of the results with typical observed cooling times in solar flares shows that the turbulent mean free path λT lies in a range corresponding to a regime in which classical (collision-dominated) conduction plays at most a limited role. We also consider the magnitude and duration of the heat input that is necessary to account for the enhanced values of temperature and density at the beginning of the cooling phase and for the observed cooling times. We find through numerical modeling that in order to produce a peak temperature ≃1.5 × 107 K and a 200 s cooling time consistent with observations, the flare-heating profile must extend over a significant period of time; in particular, its lingering role must be taken into consideration in any description of the cooling phase. Comparison with observationally inferred values of post-flare loop temperatures, densities, and cooling times thus leads to useful constraints on both the magnitude and duration of the magnetic energy release in the loop, as well as on the value of the turbulent mean free path λT .