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Register a new userCoronal Loop Scaling Laws for Various Forms of Parallel Heat Conduction
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The solar atmosphere is dominated by loops of magnetic flux which connect the multi-million-degree corona to the much cooler chromosphere. The temperature and density structure of quasi-static loops is determined by the continuous flow of energy from the hot corona to the lower solar atmosphere. Loop scaling laws provide relationships between global properties of the loop (such as peak temperature, pressure, and length); they follow from the physical variable dependencies of various terms in the energy equation, and hence the form of the loop scaling law provides insight into the key physics that controls the loop structure. Traditionally, scaling laws have been derived under the assumption of collision-dominated thermal conduction. Here we examine the impact of different regimes of thermal conduction -- collision-dominated, turbulence-dominated, and free-streaming -- on the form of the scaling laws relating the loop temperature and heating rate to its pressure and half-length. We show that the scaling laws for turbulence-dominated conduction are fundamentally different than those for collision-dominated and free-streaming conduction, inasmuch as the form of the scaling laws now depend primarily on conditions at the low-temperature, rather than high-temperature, part of the loop. We also establish regimes in temperature and density space in which each of the applicable scaling laws prevail.
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To understand the nonlinear dynamics of the Parker scenario for coronal heating, long-time high-resolution simulations of the dynamics of a coronal loop in cartesian geometry are carried out. A loop is modeled as a box extended along the direction of the strong magnetic field $B_0$ in which the system is embedded. At the top and bottom plates, which represent the photosphere, velocity fields mimicking photospheric motions are imposed. We show that the nonlinear dynamics is described by different regimes of MHD anisotropic turbulence, with spectra characterized by intertial range power laws whose indexes range from Kolmogorov-like values ($sim 5/3$) up to $sim 3$. We briefly describe the bearing for coronal heating rates.
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 {it 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 corona and chromosphere.
Long-time high-resolution simulations of the dynamics of a coronal loop in cartesian geometry are carried out, within the framework of reduced magnetohydrodynamics (RMHD), to understand coronal heating driven by motion of field lines anchored in the photosphere. We unambiguously identify MHD anisotropic turbulence as the physical mechanism responsible for the transport of energy from the large scales, where energy is injected by photospheric motions, to the small scales, where it is dissipated. As the loop parameters vary different regimes of turbulence develop: strong turbulence is found for weak axial magnetic fields and long loops, leading to Kolmogorov-like spectra in the perpendicular direction, while weaker and weaker regimes (steeper spectral slopes of total energy) are found for strong axial magnetic fields and short loops. As a consequence we predict that the scaling of the heating rate with axial magnetic field intensity $B_0$, which depends on the spectral index of total energy for given loop parameters, must vary from $B_0^{3/2}$ for weak fields to $B_0^{2}$ for strong fields at a given aspect ratio. The predicted heating rate is within the lower range of observed active region and quiet Sun coronal energy losses.
Shibata & Yokoyama (1999, 2002) proposed a method of estimating the coronal magnetic field strengths ($B$) and magnetic loop lengths ($L$) of solar and stellar flares, on the basis of magnetohydrodynamic simulations of the magnetic reconnection model. Using the scaling law provided by Shibata & Yokoyama (1999, 2002), $B$ and $L$ are obtained as functions of the emission measure ($EM=n^2L^3$) and temperature ($T$) at the flare peak. Here, $n$ is the coronal electron density of the flares. This scaling law enables the estimation of $B$ and $L$ for unresolved stellar flares from the observable physical quantities $EM$ and $T$, which is helpful for studying stellar surface activities. To apply this scaling law to stellar flares, we discuss its validity for spatially resolved solar flares. $EM$ and $T$ were calculated from GOES soft X-ray flux data, and $B$ and $L$ are theoretically estimated using the scaling law. For the same flare events, $B$ and $L$ were also observationally estimated with images taken by Solar Dynamics Observatory (SDO)/ Helioseismic and Magnetic Imager (HMI) Magnetogram and Atmospheric Imaging Assembly (AIA) 94{AA} pass band. As expected, a positive correlation was found between the theoretically and observationally estimated values. We interpret this result as indirect evidence that flares are caused by magnetic reconnection. Moreover, this analysis makes us confident in the validity of applying this scaling law to stellar flares as well as solar flares.
Coronal loops reveal crucial information about the nature of both coronal magnetic fields and coronal heating. The shape of the corresponding flux tube cross section and how it varies with position are especially important properties. They are a direct indication of the expansion of the field and of the cross-field spatial distribution of the heating. We have studied 20 loops using high spatial resolution observations from the first flight of the Hi-C rocket experiment, measuring the intensity and width as a function of position along the loop axis. We find that intensity and width tend to either be uncorrelated or to have a direct dependence, such that they increase or decrease together. This implies that the flux tube cross sections are approximately circular under the assumptions that the tubes have non-negligible twist and that the plasma emissivity is approximately uniform along the magnetic field. The shape need not be a perfect circle and the emissivity need not be uniform within the cross section, but sub-resolution patches of emission must be distributed quasi-uniformly within an envelope that has an aspect ratio of order unity. This raises questions about the suggestion that flux tubes expand with height, but primarily in the line-of-sight direction so that the corresponding (relatively noticeable) loops appear to have roughly uniform width, a long-standing puzzle. It also casts doubt on the idea that most loops correspond to simple warped sheets, although we leave open the possibility of more complex manifold structures.
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