Leonardo Di G. Sigalotti

The Effects of Stratification on Oscillating Coronal Loops

Recent observations by the Solar and Heliospheric Observatory (SOHO) and the Transition Region and Coronal Explorer (TRACE) have confirmed previous theoretical predictions that coronal loops may oscillate. These oscillations and their damping are of fundamental importance, because they can provide diagnostics of the coronal plasma. In the present paper, we perform numerical hydrodynamic calculations of a one-dimensional loop model to investigate the effects of stratification on damping of longitudinal waves in the hot coronal loops observed by the Solar Ultraviolet Measurements of Emitted Radiation (SUMER) on board the SOHO satellite. In particular, we study the dissipation by thermal conduction and by compressive viscosity of standing slow magnetosonic disturbances in loops of semicircular shape. For the parameter regime that characterizes the SUMER hot loops, we find that stratification results in a ~10%-20% reduction of the wave-damping time compared to the nonstratified loop models because of increased dissipation by compressive viscosity due to gravity. We show that temperature oscillations are more strongly dissipated by thermal conduction, while density and velocity waves are mostly damped by compressive viscosity. However, the decay time of the oscillations is always governed by the thermal conduction timescale. The scalings of the decay time with wave period, temperature, and loop length all point toward higher dissipation rates in the stratified, hotter loops because of the increased effects of thermal conduction and compressive viscosity.

SPH simulations of time-dependent Poiseuille flow at low Reynolds numbers

A working Smoothed Particle Hydrodynamics (SPH) formalism for solving the equations of motion of a viscous fluid is presented. The method is based on a standard symmetrized SPH expression for the viscous forces, which involves only first-order derivatives of the kernel through a direct evaluation of the viscous stress tensor. Therefore, the interpolation can be performed using low-order kernels of compact support without compromising the accuracy and stability of the results. In principle, the scheme is suitable for treating compressible fluids with arbitrary shear and bulk viscosities. Here, we demonstrate that when it is combined with the pressure-gradient correction proposed by Morris et al., the method is also suitable for solving the Navier-Stokes equations for incompressible flows without any further assumptions. Simulations using the method show close agreement with the analytic series solutions for plane Poiseuille and Hagen-Poiseuille flows at very low Reynolds numbers. At least for these specific tests, the results obtained are essentially independent of employing either a cubic or a quintic spline kernel.

A shock-capturing SPH scheme based on adaptive kernel estimation

Here we report a method that converts standard smoothed particle hydrodynamics (SPH) into a working shock-capturing scheme without relying on solutions to the Riemann problem. Unlike existing adaptive SPH simulations, the present scheme is based on an adaptive kernel estimation of the density, which combines intrinsic features of both the kernel and nearest neighbor approaches in a way that the amount of smoothing required in low-density regions is effectively controlled. Symmetrized SPH representations of the gas dynamic equations along with the usual kernel summation for the density are used to guarantee variational consistency. Implementation of the adaptive kernel estimation involves a very simple procedure and allows for a unique scheme that handles strong shocks and rarefactions the same way. Since it represents a general improvement of the integral interpolation on scattered data, it is also applicable to other fluid-dynamic models. When the method is applied to supersonic compressible flows with sharp discontinuities, as in the classical one-dimensional shock-tube problem and its variants, the accuracy of the results is comparable, and in most cases superior, to that obtained from high quality Godunov-type methods and SPH formulations based on Riemann solutions. The extension of the method to two- and three-space dimensions is straightforward. In particular, for the two-dimensional cylindrical Nohs shock implosion and Sedov point explosion problems the present scheme produces much better results than those obtained with conventional SPH codes.

Catastrophic Cooling of Impulsively Heated Coronal Loops

The physical mechanisms that cause the heating of the solar corona are still far from being completely understood. However, recent highly resolved observations with the current solar missions have shown clear evidence of frequent and very localized heating events near the chromosphere that may be responsible for the observable high temperatures of the coronal plasma. In this paper, we perform one-dimensional hydrodynamic simulations of the evolution of a hypothetical loop model undergoing impulsive heating through the release of localized Gaussian energy pulses near the loops footpoints. We find that when a discrete number of randomly spaced pulses is released, loops of length L = 5 and 10 Mm heat up and stay at coronal temperatures for the whole duration of the impulsive heating stage, provided that the elapsed time between successive heat injections is 175 and 215 s, respectively. The precise value of the critical elapsed time is sensitive to the loop length. In particular, we find that for increased loop lengths of 20 and 30 Mm, the critical elapsed time rises to about 240 and 263 s, respectively. For elapsed times longer than these critical values, coronal temperatures can no longer be maintained at the loop apex in spite of continued impulsive heating. As a result, the loop apex undergoes runaway cooling well below the initial state, reaching chromospheric temperatures (~104 K) and leading to the typical hot-cool temperature profile characteristic of a cool condensation. For a large number of pulses (up to ~1000) having a fully random spatiotemporal distribution, the variation of the temperature along the loop is highly sensitive to the spatial distribution of the heating. As long as the heating concentrates more and more at the loops footpoints, the temperature variation is seen to make a transition from that of a uniformly heated loop to a flat, isothermal profile along the loop length. Concentration of the heating at the footpoints also results in a more frequent appearance of rapid and significant depressions of the apex temperature during the loop evolution, most of them ranging from ~1.5 × 106 to ~104 K and lasting from about 3 to 10 minutes. This behavior bears a tight relation with the strong variability of coronal loops inferred from SOHO observations in active regions of the solar atmosphere.

Gravitational Collapse and Fragmentation of Molecular Cloud Cores with GADGET-2

The collapse and fragmentation of molecular cloud cores is examined numerically with unprecedentedly high spatial resolutions, using the publicly released code GADGET-2. As templates for the model clouds we use the standard isothermal test case in the variant calculated by Burkert & Bodenheimer in 1993 and the centrally condensed, Gaussian cloud advanced by Boss in 1991. A barotropic equation of state is used to mimic the nonisothermal collapse. We investigate both the sensitivity of fragmentation to thermal retardation and the level of resolution needed by smoothed particle hydrodynamics (SPH) to achieve convergence to existing Jeans-resolved, finite-difference (FD) calculations. We find that working with 0.6-1.2 million particles, acceptably good convergence is achieved for the standard test model. In contrast, convergent results for the Gaussian-cloud model are achieved using from 5 to 10 million particles. If the isothermal collapse is prolonged to unrealistically high densities, the outcome of collapse for the Gaussian cloud is a central adiabatic core surrounded by dense trailing spiral arms, which in turn may fragment in the late evolution. If, on the other hand, the barotropic equation of state is adjusted to mimic the rise of temperature predicted by radiative transfer calculations, the outcome of collapse is a protostellar binary core. At least, during the early phases of collapse leading to formation of the first protostellar core, thermal retardation not only favors fragmentation but also results in an increased number of fragments, for the Gaussian cloud.

Coronal Loop Heating by Random Energy Releases

It was suggested by Parker that the solar corona is heated by numerous small localized events called nanoflares. High-resolution satellites (the Solar and Heliospheric Observatory and Transition Region and Coronal Explorer [TRACE]) have shown a kind of very small scale activity at transition region temperatures (i.e., explosive events, microflares, blinkers, etc.). These events may serve as the building blocks of the heating mechanism(s) of the solar atmosphere. In this Letter we present the results of numerical calculations that detail the response of the coronal plasma to microscale heating pulses in a magnetic loop. The energy input pulses are at periodical and random injections, located near the footpoint where the temperature is ≈104 K. It is found that these successive energy inputs can maintain the plasma along the loop at typical coronal temperatures. This result is in good qualitative agreement with TRACE observations studied by Aschwanden et al.

Adaptive kernel estimation and SPH tensile instability

We propose an alternative method to remove the tensile instability in standard SPH simulations of a fluid. The method relies on an adaptive density kernel estimation (ADKE) algorithm, which allows the width of the kernel interpolant to vary locally in such a way that only the minimum necessary smoothing is applied to the data. By means of a linear perturbation analysis of the SPH equations for a heat-conducting, viscous, van der Waals fluid, we derive the corresponding dispersion relation. Solution of the dispersion relation in the short wavelength limit shows that the tensile instability is effectively removed for a wide range of the ADKE parameters. Application of the method to the formation of equilibrium liquid drops confirms the analytical results of the linear stability analysis. Examples of the resolving power of the method are also given for the nonlinear oscillations of an excited drop and the Sedov blast wave problem.

On El Naschie's complex time, Hawking's imaginary time and special relativity

Abstract The idea of complex time, as first proposed by El Naschie in 1995, not only provided a very important mathematical utility in clarifying the nature of nowness, but also opened a definite possibility for the instantaneous transmission of information through the theoretical prediction of massless particles travelling at velocities larger than the speed of light. Based on a very simple thought experiment, here we show that the complex nature of time arises when two independent inertial observers, in relative uniform motion, communicate via a light signal in order to compare their own proper time measurements for the same event. The observation that the time employed by the signal to go from one observer to the other is calculable, but not measurable, permits to build up a general expression for the complex time, which not only complies with the possibility of time decomposition into two dimensions, but also conciliates with the idea of a complex space. In particular, we find that El Naschie’s complex time can be interpreted as an asymptotic limit when the velocity of the moving observer equals that of light. Within this new formulation, the inverse Lorentz transformations of special relativity follow as a direct consequence of the complex time.

An adaptive SPH method for strong shocks

We propose an alternative SPH scheme to usual SPH Godunov-type methods for simulating supersonic compressible flows with sharp discontinuities. The method relies on an adaptive density kernel estimation (ADKE) algorithm, which allows the width of the kernel interpolant to vary locally in space and time so that the minimum necessary smoothing is applied in regions of low density. We have performed a von Neumann stability analysis of the SPH equations for an ideal gas and derived the corresponding dispersion relation in terms of the local width of the kernel. Solution of the dispersion relation in the short wavelength limit shows that stability is achieved for a wide range of the ADKE parameters. Application of the method to high Mach number shocks confirms the predictions of the linear analysis. Examples of the resolving power of the method are given for a set of difficult problems, involving the collision of two strong shocks, the strong shock-tube test, and the interaction of two blast waves.

GRAVITATIONAL COLLAPSE AND FRAGMENTATION OF MOLECULAR CLOUD CORES

The detected multiplicity of main-sequence and pre-main-sequence stars along with the emerging evidence for binary and multiple protostars, imply that stars may ultimately form by fragmentation of collapsing molecular cloud cores. These discoveries, coupled with recent observational knowledge of the structure of dense cloud cores and of the properties of young binary stars, provide serious constraints to the theory of star formation. Most theoretical progress in the field of star formation is largely based on numerical calculations of the early collapse and fragmentation of protostellar clouds. Although these models have been quite successful at predicting the formation of binary protostars, a direct comparison between theory and observations has not yet been established. The results of recent observations as well as of early and recent analytic and numerical models, on which the present theory of star formation is based, are reviewed here in a self-consistent manner.