Modern Numerical Schemes for Solving Magnetohydrodynamic Equations

In this paper several modern shock-capturing schemes for solving hydrodynamic and magnetohydrodynamic equations are reviewed. This review covers a wide range of explicit and implicit schemes as well as those in which adaptive mesh refinement methods are adopted. As these numerical schemes are based on Riemann solvers which use Godunov-type techniques, they are well suited for strong shocks and other discontinuities without oscillations in the flow variables. Some other numerical issues as grid generation, divergence cleaning, and an application of MHD schemes to several problems in coronal and interplanetary physics are discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Nonlinear Evolution of Axisymmetric Twisted Flux Tubes in the Solar Tachocline

We study the periodicity of twisting motions in sunspot penumbral filaments, which were recently discovered from space (Hinode) and ground-based (SST) observations. A sunspot was well observed for 97 minutes by Hinode/SOT in the G-band (4305 Å) on 12 November 2006. By the use of the time?–?space gradient applied to intensity space?–?time plots, twisting structures can be identified in the penumbral filaments. Consistent with previous findings, we find that the twisting is oriented from the solar limb to disk center. Some of them show a periodicity. The typical period is about ≈?four minutes, and the twisting velocity is roughly 6 km s^?1. However, the penumbral filaments do not always show periodic twisting motions during the time interval of the observations. Such behavior seems to start and stop randomly with various penumbral filaments displaying periodic twisting during different intervals. The maximum number of periodic twists is 20 in our observations. Studying this periodicity can help us to understand the physical nature of the twisting motions. The present results enable us to determine observational constraints on the twisting mechanism.     

A Consistent one-Dimensional Model for the Turbulent Tachocline

The first consistent model for the turbulent tachocline is presented, with the turbulent diffusivity computed within the model instead of being specified arbitrarily. For the origin of the 3D turbulence a new mechanism is proposed. Owing to the strongly stable stratification, the mean radial shear is stable, while the horizontal shear is expected to drive predominantly horizontal, quasi-2D motions in thin slabs. Here I suggest that a major source of 3D overturning turbulent motions in the tachocline is the secondary shear instability due to the strong, random vertical shear arising between the uncorrelated horizontal flows in neighboring slabs. A formula for the vertical diffusivity due to this turbulence, Equation (9), is derived and applied in a simplified 1D model of the tachocline. It is found that Maxwell stresses due to an oscillatory poloidal magnetic field of a few hundred gauss are able to confine the tachocline to a thickness less than 5 Mm. The integral scale of the 3D overturning turbulence is the buoyancy scale, on the order of 10 km, and its velocity amplitude is a few m s^–1, yielding a vertical turbulent diffusivity on the order of 10⁸ cm² s^–1.     

关于日冕中稳恒态磁流体力学流动的一个定理

本文报告了在日冕或其他天体物理环境中，沿磁力线流动的稳恒态磁流体力学流动的一个定理和一个重要关系。它们是利用宏观动能密度对磁能密度的比例导出的。     

Generation of Intermediate Drift Bursts by Magnetohydrodynamic Waves in the Solar Corona

The plasma mechanism of radio emission generation in an inhomogeneous medium is investigated. In the model under study, the electron beam with loss-cone distribution generates upper-hybrid waves that, in turn, are transformed into radio emission. It is shown that the influence of the plasma density inhomogeneity limits the plasma waves’ intensity considerably due to variation in their wave vector. The results are used to interpret the intermediate drift (IMD) bursts. A model is proposed in which these bursts are reflections of propagating small-scale (with amplitudes of about 1% and sizes of hundreds of kilometers) magnetohydrodynamic (MHD) disturbances of magnetic tubes. It is shown that this model allows us to explain the spectral parameters of the bursts in question. At present, the lack of precise and independent data about the magnetic field does not allow us to decide definitively between the existing models (whistler or MHD waves) of the IMD bursts; nevertheless, if the proposed model is correct, it can be used to determine the characteristics of the coronal MHD waves.     

MESSENGER Observations of Magnetohydrodynamic Waves in the Solar Corona from Faraday Rotation

During the declining phase of the longest solar minimum in a century, the arrival of the MESSENGER spacecraft at superior conjunction allowed the measurement of magnetohydrodynamic (MHD) waves in the solar corona with its 8 GHz radio frequency signal. MHD waves crossing the line of sight were measured via Faraday rotation fluctuations (FRFs) in the plane of polarization (PP) of MESSENGER’s signal. FRFs in previous observations of the solar corona (at greater offset distances) consisted of a turbulent spectrum that decreased in power with increasing frequency and distance from the Sun. Occasionally a spectral line, a distinct peak in the power spectral density spectrum around 4 to 8 mHz, was also observed in these early data sets at offset distances of about 5 to 10 solar radii. The MESSENGER FRF data set shows a spectral line at an offset distance between 1.55 to 1.85 solar radii with a frequency of 0.6±0.2 mHz. Other possible spectral lines may be at 1.2, 1.7, and 4.5 mHz; MHD waves with these same frequencies have been observed in X-ray data traveling along closed coronal loops at lower offset distances. An initial analysis of the MESSENGER spectral line(s) shows behavior similar to turbulent spectra: decreasing power with increasing frequency and distance from the Sun. Here we detail the steps taken to process the MESSENGER change in PP data set for the MHD wave investigation.     

磁流体力学流动的特性与日珥中的一些物理过程

应用并改进了由Chandrasekhar提出的一个变换,以改变磁流体力学方程组的形式．将动量守恒方程变成一个物理变量的二次方程．研究了磁流体力学流动演化的非线性性质,特别注意突变过程．本文导出的理论结果表明,磁流体力学流动能被分成一些分支,虽然在分支之间的跳跃（突变）可有大一些和小一些之分．进行了关于太阳日珥的数值试验;它们进一步直接表明理论结果的正确性．而且,虽然三个守恒定律所涉及的数量之间可以相差多个数量级,但它们在跳跃中所起的作用均不可被低估．     

A Computationally Efficient,Time-Dependent Model of the Solar Wind for Use as a Surrogate to Three-Dimensional Numerical Magnetohydrodynamic Simulations

Solar radiation variability spans a wide range in time, ranging from seconds to decadal and longer. The nearly 40 years of measurements of solar irradiance from space established that the total solar irradiance varies by \(\approx 0.1\%\) in phase with the Sun’s magnetic cycle. Specific intervals of the solar spectrum, e.g., ultraviolet (UV), vary by orders of magnitude more. These variations can affect the Earth’s climate in a complex non-linear way. Specifically, some of the processes of interaction between solar UV radiation and the Earth’s atmosphere involve threshold processes and do not require a detailed reconstruction of the solar spectrum. For this reason a spectral UV index based on the (FUV-MUV) color has been recently introduced. This color is calculated using SORCE SOLSTICE integrated fluxes in the FUV and MUV bands. We present in this work the reconstructions of the solar (FUV-MUV) color and Ca ii K and Mg ii indices, from 1749–2015, using a semi-empirical approach based on the reconstruction of the area coverage of different solar magnetic features, i.e., sunspot, faculae and network. We remark that our results are in noteworthy agreement with latest solar UV proxy reconstructions that exploit more sophisticated techniques requiring historical full-disk observations. This makes us confident that our technique can represent an alternative approach which can complement classical solar reconstruction efforts. Moreover, this technique, based on broad-band observations, can be utilized to estimate the activity on Sun-like stars, that cannot be resolved spatially, hosting extra-solar planetary systems.

     

On the Compatibility of a Flux Transport Dynamo with a Fast Tachocline Scenario

The compatibility of the fast-tachocline scenario with a flux-transport dynamo model is explored. We employ a flux-transport dynamo model coupled with simple feedback formulae relating the thickness of the tachocline to the amplitude of the magnetic field or to the Maxwell stress. The dynamo model is found to be robust against the nonlinearity introduced by this simplified fast-tachocline mechanism. Solar-like butterfly diagrams are found to persist and, even without any parameter fitting, the overall thickness of the tachocline is well within the range admitted by helioseismic constraints. In the most realistic case of a time- and latitude-dependent tachocline thickness linked to the value of the Maxwell stress, both the thickness and its latitudinal dependence are in excellent agreement with seismic results. In nonparametric models, cycle-related temporal variations in tachocline thickness are somewhat larger than admitted by helioseismic constraints; we find, however, that introducing a further parameter into our feedback formula readily allows further fine tuning of the thickness variations.     

Magnetohydrodynamic boundary-layer flow for a non-Newtonian fluid past a wedge

The flow of a power law fluid past a symmetrical wedge is studied in the neighbourhood of the stagnation point when an external magnetic field is applied. The nonlinear equation of motion is transformed to a similarity differential equation which is solved using the method of successive approximations. The analytical solutions found by this method yield numerical values in good agreement with tabulated calculations obtained before via numerical methods for electrically nonconducting fluids. Analytical expressions are derived for both the velocity profile and the local non-dimensional skin friction coefficient. Also the three thicknesses of displacement, momentum and kinetic energy are given in closed forms.     

Numerical Magnetohydrodynamic Experiments for Testing the Physical Mechanisms of Coronal Mass Ejections Acceleration

Analysis of observations from both space-borne (LASCO/SOHO, Skylab and Solar Maximum Mission) and ground-based (Mauna Loa Observatory) instruments show that there are two types of coronal mass ejections (CMEs), fast CMEs and slow CMEs. Fast CMEs start with a high initial speed, which remains more or less constant, while slow CMEs start with a low initial speed, but show a gradual acceleration. To explain the difference between the two types of CMEs, Low and Zhang (2002) proposed that it resulted from a difference in the initial topology of the magnetic fields associated with the underlying quiescent prominences, i.e., a normal prominence configuration will lead to a fast CME, while an inverse quiescent prominence results in a slow CME. In this paper we explore a different scenario to explain the existence of fast and slow CMEs. Postulating only an inverse topology for the quiescent prominences, we show that fast and slow CMEs result from different physical processes responsible for the destabilization of the coronal magnetic field and for the initiation and launching of the CME. We use a 2.5-D, time-dependent streamer and flux-rope magnetohydrodynamic (MHD) model (Wu and Guo, 1997) and investigate three initiation processes, viz. (1) injecting of magnetic flux into the flux-rope, thereby causing an additional Lorentz force that will destabilize the streamer and launch a CME (Wu et al., 1997, 1999); (2) draining of plasma from the flux-rope and triggering a magnetic buoyancy force that causes the flux-rope to lift and launch a CME; and (3) introducing additional heating into the flux-rope, thereby simulating an active-region flux-rope accompanied by a flare to launch a CME. We present 12 numerical tests using these three driving mechanisms either alone or in various combinations. The results show that both fast and slow CMEs can be obtained from an inverse prominence configuration subjected to one or more of these three different initiation processes.     

Magnetohydrodynamic equilibrium conditions in the post-Newtonian approximation of general relativity

A set of equations, which are magnetohydrodynamic (MHD) equilibrium conditions in the post-Newtonian approximation of general relativity (PNA of GR), is obtained. The given system generalizes the previously obtained magnetohydrodynamic equilibrium conditions of classical mechanics and the hydrodynamic equilibrium conditions in the PNA of GR.     

Magnetohydrodynamic modelling of interplanetary disturbances between the Sun and Earth

A time-dependent, nonplanar, two-dimensional magnetohydrodynamic computer model is used to simulate a series, separately examined, of solar flare-generated shock waves and their subsequent disturbances in interplanetary space between the Sun and the Earth's magnetosphere. The ‘canonical’ or ansatz series of shock waves include initial velocities near the Sun over the range 500 to 3500 km s^?1. The ambient solar wind, through which they propagate, is taken to be a steady-state homogeneous plasma (that is, independent of heliolongitude) with a representative set of plasma and magnetic field parameters. Complete sets of solar wind plasma and magnetic field parameters are presented and discussed. Particular attention is addressed to the MHD model's ability to address fundamental operational questions vis-à-vis the long-range forecasting of geomagnetic disturbances. These questions are: (i) will a disturbance (such as the present canonical series of solar flare shock waves) produce a magnetospheric and ionospheric disturbance, and, if so, (ii) when will it start, (iii) how severe will it be, and (iv) how long will it last? The model's output is used to compute various solar wind indices of current interest as a demonstration of the model's potential for providing ‘answers’ to these questions.     

Magnetohydrodynamic simulations of gamma-ray burst jets: Beyond the progenitor star

Achromatic breaks in afterglow light curves of gamma-ray bursts (GRBs) arise naturally if the product of the jet’s Lorentz factor $\gamma$ and opening angle ${\Theta }_j$ satisfies $\gamma {\Theta }_j?1$ at the onset of the afterglow phase, i.e., soon after the conclusion of the prompt emission. Magnetohydrodynamic (MHD) simulations of collimated GRB jets generally give $\gamma {\Theta }_j?1$, suggesting that MHD models may be inconsistent with jet breaks. We work within the collapsar paradigm and use axisymmetric relativistic MHD simulations to explore the effect of a finite stellar envelope on the structure of the jet. Our idealized models treat the jet–envelope interface as a collimating rigid wall, which opens up outside the star to mimic loss of collimation. We find that the onset of deconfinement causes a burst of acceleration accompanied by a slight increase in the opening angle. In our fiducial model with a stellar radius equal to ${10}^{4.5}$ times that of the central compact object, the jet achieves an asymptotic Lorentz factor $\gamma ～500$ far outside the star and an asymptotic opening angle ${\Theta }_j?0.04\text{rad}?2°$, giving $\gamma {\Theta }_j～20$. These values are consistent with observations of typical long-duration GRBs, and explain the occurrence of jet breaks. We provide approximate analytic solutions that describe the numerical results well.     

A Simple Model for the Solar Cycle

A simple nonlinear model which describes the 11-year solar cycle can be derived from the usual – dynamo theory in the form of a Van der Pol equation. Solar activity displays also small-scale inter-cycle persistent stochastic oscillations with a Hurst exponent of the order of H0.76±0.01. The results obtained from the Van der Pol oscillator superimposed on a fractional Brownian motion which describes the stochastic fluctuations are presented.     

Magnetohydrodynamic behaviour of a finite amplitude disturbance in the solar magnetic field

In this paper, dynamic processes in the solar atmosphere are studied numerically from a complete set of MHD equations. Dynamic evolution of the non-linear magnetic field is produced by the finite amplitude of the azimuthai magnetic field at the base of the flux tube of the solar atmosphere. It is assumed that the initial configuration of the magnetic field is a force-free and potential field, the magnetic field is disturbed at the base, the plasma is driven and a part of the magnetic energy is transformed into the kinetic energy of the plasma.The compressed flow of the plasma has the features of fast MHD waves. The computation results give quantitatively the non-linear evolution of strong magnetic fields. These results could be used in an explanation of coronal transients, surge, spray and eruptive prominence events in the solar atmosphere, as well as in a modelling of plasma behaviour in high-β structure experiments in the laboratory.     

Magnetohydrodynamic free convective effect for an incompressible viscous fluid past an infinite limiting surface

An analysis of the effect of a magnetic field on the free convective flow of an incompressible, electrically conducting and viscous fluid past an infinite vertical limiting surface, has been carried out. Also the limiting surface are unmoving, we have constant heat flux at the limiting surface, the free velocity is constant and the magnetic Reynolds number is not small. The effects of the magnetic parameter and the Grashoff number on the flow are discussed.