Classical MHD Research Articles

Strong solutions to the 3D full compressible magnetohydrodynamic flows

In this paper, we deal with the strong solutions to the full compressible magnetohydrodynamic flows in a domain Ω⊆R3. We prove the local existence and unique of strong solution under three different types of boundary conditions provided that the initial data satisfies a natural compatibility condition. The initial density of such a strong solution is allowed to contain vacuum states. Compared to the classical compressible MHD, the key issue in the limit passage is to obtain the a priori estimates of the absolute temperature, some new estimate ideas are introduced to estimate these results.

ANOMALOUS MAGNETIC REGIONS ON THE SUN

We studied the anomalous magnetic regions observed near the minima of solar cycles 24 and 25. The peculiarity of these areas was the deviation of their configuration from Hale's law of magnetic polarity and Joy's law about the inclination of the axes of bipolar groups to the latitudinal direction. Therefore, they belong to the class of so-called anti-Hale active regions. We paid special attention to the flare activity of anti-Hale regions, as this is important for forecasting space weather and magnetic storms in the Earth's atmosphere. The detected anomalies of the surface magnetism of the active regions studied by us may indicate the influence of the mechanisms of the deep small-scale dynamo on their evolution. In this regard we analyzed the possible mechanisms of the formation of anti-Hale magnetic regions. In particular, such mechanisms can be the mechanisms of a small-scale magnetic dynamo. In connection with this an urgent problem today is the search for observed evidence of the existence of the theoretically proposed by Brandenburg A. et al. (2012) of a new physical entity – a small-scale magnetic field hidden in the solar depths, excited by two qualitatively different mechanisms of a small-scale dynamo (SSD). The first mechanism is the SSD of macroscopic MHD (SSD1), while the second is the diffusion SSD of classical MHD (SSD2). However, the small contributions of these sources are very difficult to distinguish observationally. To solve this complication, Sokoloff, Khlystova and Abramenko (2015) proposed a test for separating the contributions of two sources based on a statistical probabilistic model. Such an important feature of the differences between of the two SSD is the behavior of the percentage of anti-Hail groups of sunspots (in relation to the total number of spots) in the minima of solar cycles. According to statistical studies of long series of observations Sokoloff, Khlystova and Abramenko (2015) found that the percentage of anti-Haile groups of spots increases during minima of the solar cycles, suggesting in favor of SSD2. We believe that the detected magnetic anomalies of the studied regions may be caused by the influence of a SSD2 in the depths of the convective zone of the Sun, since this source gives the most noticeable contribution to the surface magnetism near cycle minima.

An efficient Hermite-Galerkin spectral scheme for three-dimensional incompressible Hall-magnetohydrodynamic system on infinite domain

The Hall-magnetohydrodynamic (Hall-MHD) system plays a significant role in the fluid description of an astrophysical plasma that features fast magnetic reconnection. Compared to the classical MHD system, the appearance of Hall term brings much more challenges in the field of numerical treatment. The aim of this paper is to construct an efficient spectral scheme for incompressible Hall-MHD system on three-dimensional infinite domain R3. Combining with a novel (HN+2,HN)-Hermite-Galerkin spectral scheme for spatial approximation, a new non-zero constant function approach for the nonlinear terms, a second-order projection method for the momentum equations, and BDF2 scheme for the decoupled system, here we establish, for the first time, a fully decoupled, completely linearized, and unconditionally energy stable scheme with directly spatial approximation in R3 and second-order temporal accuracy for incompressible Hall-MHD system. Numerical results are presented to illustrate the features of the proposed scheme, including convergence/stability tests and simulations of O- and X-points appearing in fast magnetic reconnection within the Hall-MHD regime.

High-frequency magnetohydrodynamics

We consider the case that the displacement current is not neglected in the classical MHD equations, as it is usually done. This amounts to cast them in the relativistic framework of a finite speed of light. We show some consequences in describing magnetic reconnection phenomena and for hydromagnetic waves. In the first case, the equation for the magnetic induction is changed from (formally) parabolic to (formally) hyperbolic, in the second case both, the perturbed magnetic field and the particle velocity, obey to a certain third-order in time partial differential equation, rather than to the classical wave equation. We stress the role of two typically small but nonzero parameters, the magnetic diffusivity, η (corresponding to large values of the Lundquist number), and ε:=c−2.

НЕЗВИЧАЙНА СОНЯЧНА АКТИВНА ОБЛАСТЬ NOAA 13088/13102

Background. Under certain conditions, deep fluctuating magnetic fields lead to violations of Hale’s and Joy’s laws of observed magnetism on the surface of the Sun. These magnetic fluctuations can be excited by two qualitatively different mechanisms of a small-scale dynamo. The first mechanism is a macroscopic MHD dynamo, while the second mechanism is a classical MHD diffusion dynamo. An important difference between the two mechanisms is the percentage of observed anti-Hale sunspot groups (relative to the total number of sunspots) in solar cycle minima. In the case of the first mechanism, the percentage of anti-Hale groups does not depend on the phase of the cycle, while the specified percentage associated with the second mechanism should reach its maximum value in solar minima. To separate the minor contributions of the two named sources of magnetic fluctuations, the researchers proposed a theoretical test based on statistical analysis of observational data over long periods of time (Sokoloff, & Khlystova, 2010). According to the proposed test, the percentage of anti-Hale groups of spots increases during the minima of the cycles, which indicates the favor of the diffusion dynamo. In order to confirm the dominant contribution of the diffusion dynamo to the surface magnetism, this work investigates a specific anomalous active region near the solar minimum. Methods. Macroscopic and classical MHD, which study the behavior of electromagnetic and hydrodynamic fields in turbulent plasma. Analysis of data from observations of the surface magnetism of the Sun. Results. We investigated the evolution of the NOAA 13088/13102 active region observed on August 24, 2022, shortly after the cycle 25 minimum. For the analysis, data from observations using instruments installed on board space observatories were used. A feature was revealed, which consists in the deviation of the surface magnetic configuration of this active region from Hale’s law of the magnetic polarity of groups of spots and Joy’s law of the inclination of the axes of bipolar groups to the latitudinal direction. In addition, it was established that the active region of NOAA 13088/13102 is characterized by rather high flare activity. Conclusions. We believe that the magnetic anomalies of the active region of NOAA 13088/13102 that we found were caused by the influence of magnetic fluctuations excited by the mechanism of the deep small-scale diffusion dynamo, since it is this source that gives the most noticeable contribution to the surface magnetism near the cycle minima. The detection and study of unusual anti-Hale’s AOs with increased eruptive activity, similar to NOAA 13088/13102, may find application in predicting periods of dangerous manifestations of space weather and in forecasting the dynamics of solar cycles.

A Posteriori Subcell Finite Volume Limiter for General P_NP_M Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics

In this work, we consider the general family of the so called ADER P_NP_M schemes for the numerical solution of hyperbolic partial differential equations with arbitrary high order of accuracy in space and time. The family of one-step P_NP_M schemes was introduced in Dumbser (J Comput Phys 227:8209–8253, 2008) and represents a unified framework for classical high order Finite Volume (FV) schemes (N=0), the usual Discontinuous Galerkin (DG) methods (N=M), as well as a new class of intermediate hybrid schemes for which a reconstruction operator of degree M is applied over piecewise polynomial data of degree N with M>N. In all cases with M ge N > 0 the P_NP_M schemes are linear in the sense of Godunov (Math. USSR Sbornik 47:271–306, 1959), thus when considering phenomena characterized by discontinuities, spurious oscillations may appear and even destroy the simulation. Therefore, in this paper we present a new simple, robust and accurate a posteriori subcell finite volume limiting strategy that is valid for the entire class of P_NP_M schemes. The subcell FV limiter is activated only where it is needed, i.e. in the neighborhood of shocks or other discontinuities, and is able to maintain the resolution of the underlying high order P_NP_M schemes, due to the use of a rather fine subgrid of 2N+1 subcells per space dimension. The paper contains a wide set of test cases for different hyperbolic PDE systems, solved on adaptive Cartesian meshes that show the capabilities of the proposed method both on smooth and discontinuous problems, as well as the broad range of its applicability. The tests range from compressible gasdynamics over classical MHD to relativistic magnetohydrodynamics.

Rigorous derivation and well-posedness of a quasi-homogeneous ideal MHD system

The goal of this paper is twofold. On the one hand, we introduce a quasi-homogeneous version of the classical ideal MHD system and study its well-posedness in critical Besov spaces Bp,rs(Rd), d≥2, with 1<p<+∞ and under the Lipschitz condition s>1+d∕p and r∈[1,+∞], or s=1+d∕p and r=1. A key ingredient is the reformulation of the system via the so-called Elsässer variables. On the other hand, we give a rigorous justification of quasi-homogeneous MHD models, both in the ideal and in the dissipative cases: when d=2, we will derive them from a non-homogeneous incompressible MHD system with Coriolis force, in the regime of low Rossby number and for small density variations around a constant state. Our method of proof relies on a relative entropy inequality for the primitive system, and yields precise rates of convergence, depending on the size of the initial data, on the order of the Rossby number and on the regularity of the viscosity and resistivity coefficients.

Вариационный принцип для электромагнитной гидродинамики плазмы

Within the framework of the Lagrangian approach, a mathematically correct procedure is given for obtaining the necessary conditions of locextr from the variational principle for a number of continuous media that fill a certain geometric region at each moment of time and does not leave it over time. The proposed method is based on the geometric properties of the set of diffeomorphisms of the indicated domain, which is the configuration space of a continuous medium, and on a certain embedding theorem. The efficiency of the method is demonstrated by examples of obtaining the necessary locextr conditions from variational principles for ideal gas dynamics and classical MHD plasma. The central result of the work is the construction of the variational principle, in particular, the Lagrangian and the action functional for the theory of electromagnetic hydrodynamics of plasma.

Systematic construction of upwind constrained transport schemes for MHD

The constrained transport (CT) method reflects the state of the art numerical technique for preserving the divergence-free condition of magnetic field to machine accuracy in multi-dimensional MHD simulations performed with Godunov-type, or upwind, conservative codes. The evolution of the different magnetic field components, located at zone interfaces using a staggered representation, is achieved by calculating the electric field components at cell edges, in a way that has to be consistent with the Riemann solver used for the update of cell-centered fluid quantities at interfaces. Albeit several approaches have been undertaken, the purpose of this work is, on the one hand, to compare existing methods in terms of robustness and accuracy and, on the other, to extend the upwind constrained transport (UCT) method by Londrillo & Del Zanna (2004) [22] and Del Zanna et al. (2007) [23] for the systematic construction of new averaging schemes. In particular, we propose a general formula for the upwind fluxes of the induction equation which simply involves the information available from the base Riemann solver employed for the fluid part, provided it does not require full spectral decomposition, and 1D reconstructions of velocity and magnetic field components from nearby intercell faces to cell edges. Our results are presented here in the context of second-order schemes for classical MHD, but they can be easily generalized to higher than second order schemes, either based on finite volumes or finite differences, and to other physical systems retaining the same structure of the equations, such as that of relativistic or general relativistic MHD.

IMEX and exact sequence discretization of the multi-fluid plasma model

Multi-fluid plasma models, where an electron fluid is modeled in addition to multiple ion and neutral species as well as the full set of Maxwell's equations, are useful for representing physics beyond the scope of classic MHD. This advantage presents challenges in appropriately dealing with electron dynamics and electromagnetic behavior characterized by the plasma and cyclotron frequencies and the speed of light. For physical systems, such as those near the MHD asymptotic regime, this requirement drastically increases runtimes for explicit time integration even though resolving fast dynamics may not be critical for accuracy. Implicit time integration methods, with efficient solvers, can help to step over fast time-scales that constrain stability, but do not strongly influence accuracy. As an extension, Implicit-explicit (IMEX) schemes provide an additional mechanism to choose which dynamics are evolved using an expensive implicit solve or resolved using a fast explicit solve. In this study, in addition to IMEX methods we also consider a physics compatible exact sequence spatial discretization. This combines nodal bases (H-Grad) for fluid dynamics with a set of vector bases (H-Curl and H-Div) for Maxwell's equations. This discretization allows for multi-fluid plasma modeling without violating Gauss' laws for the electric and magnetic fields. This initial study presents a discussion of the major elements of this formulation and focuses on demonstrating accuracy in the linear wave regime and in the MHD limit for both a visco-resistive and a dispersive ideal MHD problem.

Linear response of a Hall magnetic drift wave for verification of Hall MHD algorithms

Numerical implementations of Hall magnetohydrodynamics (Hall MHD) can be challenging due to the nonlinear multidimensional nature of the Hall term. Here, a model problem is presented that couples the hydrodynamic motion of the plasma to Hall MHD evolution of the magnetic field. The Hall MHD equations are linearized about unperturbed solutions in both cylindrical and Cartesian coordinates in two dimensions. The magnetic field is assumed to lie in the ignorable direction, and the linear response about the unperturbed solution is considered. The resulting ordinary differential equation is used to numerically compute the eigenfunctions and eigenfrequencies of the mode. The resulting eigenfunctions do not make the local wave approximation but are instead global solutions that depend on the spatial dependence of the unperturbed Alfvén speed. Hall MHD simulations are then performed in the Ares multiphysics code and shown to agree with the predicted phase velocities of the wave, and the simulated solutions are shown to numerically converge to the semianalytic modes. By varying the background density of the plasma (and correspondingly, the ion inertial length), the importance of Hall physics can be varied. This allows the test problem to transition from the classical MHD limit to the extreme Hall MHD limit. This problem is a useful tool for the verification of Hall MHD routines implemented in various codes, and the robustness of a routine can be tested in regimes in which Hall physics is dominant.

Covariant and 3 + 1 Equations for Dynamo-Chiral General Relativistic Magnetohydrodynamics

The exponential amplification of initial seed magnetic fields in relativistic plasmas is a very important topic in astrophysics, from the conditions in the early Universe to the interior of neutron stars. While dynamo action in a turbulent plasma is often invoked, in the last years a novel mechanism of quantum origin has gained increasingly more attention, namely the Chiral Magnetic Effect (CME). This has been recognized in semi-metals and it is most likely at work in the quark-gluon plasma formed in heavy-ion collision experiments, where the highest magnetic fields in nature, up to B~10^18 G, are produced. This effect is expected to survive even at large hydrodynamical/MHD scales and it is based on the chiral anomaly due to an imbalance between left- and right-handed relativistic fermions in the constituent plasma. Such imbalance leads to an electric current parallel to an external magnetic field, which is precisely the same mechanism of an alpha-dynamo action in classical MHD. Here we extend the close parallelism between the chiral and the dynamo effects to relativistic plasmas and we propose a unified, fully covariant formulation of the generalized Ohm's law. Moreover, we derive for the first time the 3+1 general relativistic MHD equations for a chiral plasma both in flat and curved spacetimes, in view of numerical investigation of the CME in compact objects, especially magnetars, or of the interplay among the non-ideal magnetic effects of dynamo, the CME and reconnection.

Fast magnetic reconnection: The ideal tearing instability in classic, Hall, and relativistic plasmas.

Magnetic reconnection is believed to be the driver of many explosive phenomena in Astrophysics, from solar to gamma-ray flares in magnetars and in the Crab nebula. However, reconnection rates from classic MHD models are far too slow to explain such observations. Recently, it was realized that when a current sheet gets sufficiently thin, the reconnection rate of the tearing instability becomes “ideal”, in the sense that the current sheet destabilizes on the “macroscopic” Alfvenic timescales, regardless of the Lundquist number of the plasma. Here we present 2D compressible MHD simulations in the classical, Hall, and relativistic regimes. In particular, the onset of secondary tearing instabilities is investigated within Hall-MHD for the first time. In the frame of relativistic MHD, we summarize the main results from Del Zanna et al. [1]: the relativistic tearing instability is found to be extremely fast, with reconnection rates of the order of the inverse of the light crossing time, as required to explain the high-energy explosive phenomena.

Fast reconnection in relativistic plasmas: the magnetohydrodynamics tearing instability revisited

Fast reconnection operating in magnetically dominated plasmas is often invoked in models for magnetar giant flares, for magnetic dissipation in pulsar winds, or to explain the gamma-ray flares observed in the Crab nebula, hence its investigation is of paramount importance in high-energy astrophysics. Here we study, by means of two dimensional numerical simulations, the linear phase and the subsequent nonlinear evolution of the tearing instability within the framework of relativistic resistive magnetohydrodynamics, as appropriate in situations where the Alfven velocity approaches the speed of light. It is found that the linear phase of the instability closely matches the analysis in classical MHD, where the growth rate scales with the Lundquist number S as S^-1/2, with the only exception of an enhanced inertial term due to the thermal and magnetic energy contributions. In addition, when thin current sheets of inverse aspect ratio scaling as S^-1/3 are considered, the so-called "ideal" tearing regime is retrieved, with modes growing independently on S and extremely fast, on only a few light crossing times of the sheet length. The overall growth of fluctuations is seen to solely depend on the value of the background Alfven velocity. In the fully nonlinear stage we observe an inverse cascade towards the fundamental mode, with Petschek-type supersonic jets propagating at the external Alfven speed from the X-point, and a fast reconnection rate at the predicted value R~(ln S)^-1.

The ideal tearing mode: theory and resistive MHD simulations

Classical MHD reconnection theories, both the stationary Sweet-Parker model and the tearing instability, are known to provide rates which are too slow to explain the observations. However, a recent analysis has shown that there exists a critical threshold on current sheet's thickness, namely a/L ∼ S-1/3, beyond which the tearing modes evolve on fast macroscopic Alfvénic timescales, provided the Lunquist number S is high enough, as invariably found in solar and astrophysical plasmas. Therefore, the classical Sweet-Parker scenario, for which the diffusive region scales as a/L ∼ S-1/2 and thus can be up to ∼ 100 times thinner than the critical value, is likely to be never realized in nature, as the current sheet itself disrupts in the elongation process. We present here two-dimensional, compressible, resistive MHD simulations, with S ranging from 105 to 107, that fully confirm the linear analysis. Moreover, we show that a secondary plasmoid instability always occurs when the same critical scaling is reached on the local, smaller scale, leading to a cascading explosive process, reminiscent of the flaring activity.

A subluminal relativistic magnetohydrodynamics scheme with ADER-WENO predictor and multidimensional Riemann solver-based corrector

The relativistic magnetohydrodynamics (RMHD) set of equations has recently seen increased use in astrophysical computations. Even so, RMHD codes remain fragile. The reconstruction can sometimes yield superluminal velocities in certain parts of the mesh. In this paper we present a reconstruction strategy that overcomes this problem by making a single conservative to primitive transformation per cell followed by higher order WENO reconstruction on a carefully chosen set of primitives that guarantee subluminal reconstruction of the flow variables. For temporal evolution via a predictor step we also present second, third and fourth order accurate ADER methods that keep the velocity subluminal during the predictor step. The RMHD system also requires the magnetic field to be evolved in a divergence-free fashion. In the treatment of classical numerical MHD the analogous issue has seen much recent progress with the advent of multidimensional Riemann solvers. By developing multidimensional Riemann solvers for RMHD, we show that similar advances extend to RMHD. As a result, the face-centered magnetic fields can be evolved much more accurately using the edge-centered electric fields in the corrector step. Those edge-centered electric fields come from a multidimensional Riemann solver for RMHD which we present in this paper. In this paper we also develop several new test problems for RMHD. We show that RMHD vortices can be designed that propagate on the computational mesh as self-preserving structures. These RMHD vortex test problems provide a means to do truly multidimensional accuracy testing for RMHD codes. Several other stringent test problems are presented. We show the importance of resolution in certain test problems. Our tests include a demonstration that RMHD vortices are stable when they interact with shocks.

ALFVÉNIC TURBULENCE BEYOND THE AMBIPOLAR DIFFUSION SCALE

We investigate the nature of the Alfv\'enic turbulence cascade in two fluid MHD simulations in order to determine if turbulence is damped once the ion and neutral species become decoupled at a critical scale called the ambipolar diffusion scale (L$_{AD}$). Using mode decomposition to separate the three classical MHD modes, we study the second order structure functions of the Alfv\'en mode velocity field of both neutrals and ions in the reference frame of the local magnetic field. On scales greater than L$_{AD}$ we confirm that two fluid turbulence strongly resembles single fluid MHD turbulence. Our simulations show that the behavior of two fluid turbulence becomes more complex on scales less than L$_{AD}$. We find that Alfvenic turbulence can exist past L$_{AD}$ when the turbulence is globally super-Alfv\'enic, with the ions and neutrals forming separate cascades once decoupling has taken place. When turbulence is globally sub-Alfvenic and hence strongly anisotropic with a large separation between the parallel and perpendicular decoupling scales, turbulence is damped at L$_{AD}$. We also find that the power spectrum of the kinetic energy in the damped regime is consistent with a $k^{-4}$ scaling (in agreement with the predictions of Lazarian, Vishniac & Cho 2004).

Zero dielectric constant limit of the full Magnet-Hydro-Dynamics system

In this paper, we study the zero dielectric constant limit to the full Magnet-Hydro-Dynamics system (or more precisely the Maxwell–Navier–Stokes system). For well prepared initial data, the convergences of solutions of the full MHD system towards to the solutions of the classical 2D parabolic MHD system are justified rigorously by adapting the elaborate energy method as the dielectric constant tends to zero. Furthermore, the corresponding convergence rates are also obtained.

Observation of shocks associated with CMEs in 2007

Abstract. The interaction of CMEs with the solar wind can lead to the formation of interplanetary shocks. Ions accelerated at these shocks contribute to the solar energetic protons observed in the vicinity of the Earth. Recently a joint analysis of Venus Express (VEX) and STEREO data by Russell et al. (2009) have shown that the formation of strong shocks associated with Co-rotating Interaction Regions (CIRs) takes place between the orbits of Venus and the Earth as a result of coalescence of weaker shocks formed earlier. The present study uses VEX and Advanced Composition Explorer (ACE) data in order to analyse shocks associated with CMEs that erupted on 29 and 30 July 2007 during the solar wind conjunction period between Venus and the Earth. For these particular cases it is shown that the above scenario of shock formation proposed for CIRs also takes place for CMEs. Contradiction with shock formation resulting from MHD modelling is explained by inability of classical MHD to account for the role of wave dispersion in the formation of the shock.

If substorm onset triggers tail reconnection, what triggers substorm onset?

Despite the claim that tail reconnection triggers substorm onset, there is an abundance of cases wherein substorm onset triggers tail reconnection. In such cases, the first observable precursor to onset is a periodic rippling (beads) along an equatorward auroral arc. In this study, through an example, we show that substorms arising out of arcs of this type have the classical “inside‐out” evolution, including the triggering of tail reconnection as a possible result. We then investigate what the magnetospheric mode underlying the ripples along the arcs might be. The classical MHD ballooning invoked by some substorm theories is inconsistent with the observation, which exhibits a finite azimuthal wavelength comparable to the local ion gyroradius and the propensity of onset to occur under moderately high (1–10) rather than extremely high plasmaβ. We show that the onset is due to a modified ballooning mode, subject to corrections by the General Ohm's Law and ion heat flux. The net result is that the necessary condition for the instability remains unchanged from the classical MHD, but the growth rate of the instability is heavily attenuated or quenched in the high β and short azimuthal wavelength limits. In the actual magnetosphere, the mode has a wavelength ∼1,500 km, growth timescale ∼10 s, and critical plasma beta in the 3–13 range, all consistent with observations.