Single-fluid Magnetohydrodynamics Research Articles

Damping of MHD turbulence in a partially ionized medium

ABSTRACT The coupling state between ions and neutrals in the interstellar medium plays a key role in the dynamics of magnetohydrodynamic (MHD) turbulence, but is challenging to study numerically. In this work, we investigate the damping of MHD turbulence in a partially ionized medium using 3D two-fluid (ions + neutrals) simulations generated with the athenak code. Specifically, we examine the velocity, density, and magnetic field statistics of the two-fluid MHD turbulence in different regimes of neutral-ion coupling. Our results demonstrate that when ions and neutrals are strongly coupled, the velocity statistics resemble those of single-fluid MHD turbulence. Both the velocity structures and kinetic energy spectra of ions and neutrals are similar, while their density structures can be significantly different. With an excess of small-scale sharp density fluctuations in ions, the density spectrum in ions is shallower than that of neutrals. When ions and neutrals are weakly coupled, the turbulence in ions is more severely damped due to the ion-neutral collisional friction than that in neutrals, resulting in a steep kinetic energy spectrum and density spectrum in ions compared to the Kolmogorov spectrum. We also find that the magnetic energy spectrum basically follows the shape of the kinetic energy spectrum of ions, irrespective of the coupling regime. In addition, we find large density fluctuations in ions and neutrals and thus spatially inhomogeneous ionization fractions. As a result, the neutral-ion decoupling and damping of MHD turbulence take place over a range of length-scales.

Partially ionized two-fluid shocks with collisional and radiative ionization and recombination – multilevel hydrogen model

ABSTRACT Explosive phenomena are known to trigger a wealth of shocks in warm plasma environments, including the solar chromosphere and molecular clouds where the medium consists of both ionized and neutral species. Partial ionization is critical in determining the behaviour of shocks, since the ions and neutrals locally decouple, allowing for substructure to exist within the shock. Accurately modelling partially ionized shocks requires careful treatment of the ionized and neutral species, and their interactions. Here we study a partially ionized switch-off slow-mode shock using a multilevel hydrogen model with both collisional and radiative ionization and recombination rates that are implemented into the two-fluid (PIP) code, and study physical parameters that are typical of the solar chromosphere. The multilevel hydrogen model differs significantly from magnetohydrodynamic (MHD) solutions due to the macroscopic thermal energy loss during collisional ionization. In particular, the plasma temperature both post-shock and within the finite-width is significantly cooler that the post-shock MHD temperature. Furthermore, in the mid to lower chromosphere, shocks feature far greater compression than their single-fluid MHD analogues. The decreased temperature and increased compression reveal the importance of non-equilibrium ionized in the thermal evolution of shocks in partially ionized media. Since partially ionized shocks are not accurately described by the Rankine-Hugoniot shock jump conditions, it may be incorrect to use these to infer properties of lower atmospheric shocks.

The Impact of Multifluid Effects in the Solar Chromosphere on the Ponderomotive Force under SE and NEQ Ionization Conditions

The ponderomotive force has been suggested to be the main mechanism to produce the so-called first ionization potential (FIP) effect—the enrichment of low-FIP elements observed in the outer solar atmosphere, in the solar wind, and in solar energetic events. It is well known that the ionization of these elements occurs within the chromosphere. Therefore, this phenomenon is intimately tied to the plasma state in the chromosphere and the corona. For this study, we combine IRIS observations, a single-fluid 2.5D radiative magnetohydrodynamics (MHD) model of the solar atmosphere, including ion–neutral interaction effects and nonequilibrium (NEQ) ionization effects, and a novel multifluid multispecies numerical model (based on the Ebysus code). Nonthermal velocities of Si iv measured from IRIS spectra can provide an upper limit for the strength of any high-frequency Alfvén waves. With the single-fluid model, we investigate the possible impact of NEQ ionization within the region where the FIP may occur, as well as the plasma properties in those regions. These models suggest that regions with strongly enhanced network and type II spicules are possible sites of large ponderomotive forces. We use the plasma properties of the single-fluid MHD model and the IRIS observations to initialize our multifluid models to investigate the multifluid effects on the ponderomotive force associated with Alfvén waves. Our multifluid analysis reveals that collisions and NEQ ionization effects dramatically impact the behavior of the ponderomotive force in the chromosphere, and existing theories may need to be revisited.

Nonlinear Coupling of Alfvén and Slow Magnetoacoustic Waves in Partially Ionized Solar Plasmas: The Effect of Thermal Misbalance

Solar chromosphere and photosphere, as well as solar atmospheric structures, such as prominences and spicules, are made of partially ionized plasmas. Observations have reported the presence of damped or amplified oscillations in these solar plasmas, which have been interpreted in terms of magnetohydrodynamic (MHD) waves. Slow magnetoacoustic waves could be responsible for these oscillations. The present study investigates the temporal behavior of the field-aligned motions that represent slow magnetoacoustic waves excited in a partially ionized prominence plasma by the ponderomotive force. Starting from single-fluid MHD equations, including radiative losses, a heating mechanism and ambipolar diffusion, and using a regular perturbation method, first- and second-order partial differential equations have been derived. By numerically solving second-order equations describing field-aligned motions, the temporal behavior of the longitudinal velocity perturbations is obtained. The damping or amplification of these perturbations can be explained in terms of heating–cooling misbalance, the damping effect due to ambipolar diffusion and the variation of the first adiabatic exponent with temperature and ionization degree.

Entropy-stable Gauss collocation methods for ideal magneto-hydrodynamics

In this paper, we present an entropy-stable Gauss collocation discontinuous Galerkin (DG) method on 3D curvilinear meshes for the GLM-MHD equations: the single-fluid magneto-hydrodynamics (MHD) equations with a generalized Lagrange multiplier (GLM) divergence cleaning mechanism. For the continuous entropy analysis to hold and to ensure Galilean invariance in the divergence cleaning technique, the GLM-MHD system requires the use of non-conservative terms.Traditionally, entropy-stable DG discretizations have used a collocated nodal variant of the DG method, also known as the discontinuous Galerkin spectral element method (DGSEM) on Legendre-Gauss-Lobatto (LGL) points. Recently, Chan et al. [1, “Efficient Entropy Stable Gauss Collocation Methods”. SIAM (2019)] presented an entropy-stable DGSEM scheme that uses Legendre-Gauss points (instead of LGL points) for conservation laws. Our main contribution is to extend the discretization technique of Chan et al. to the non-conservative GLM-MHD system.We provide a numerical verification of the entropy behavior and convergence properties of our novel scheme on 3D curvilinear meshes. Moreover, we test the robustness and accuracy of our scheme with a magneto-hydrodynamic Kelvin-Helmholtz instability problem. The numerical experiments suggest that the entropy-stable DGSEM on Gauss points for the GLM-MHD system is more accurate than the LGL counterpart.

Optimum set for completely supporting existing toroidal field coils toward the investigation of two-fluid physics in toroidal plasmas

Beyond single-fluid magnetohydrodynamics, multi-scale plasmas have been widely employed in recent studies on toroidal plasmas. To comparatively investigate two-fluid plasmas in a single machine, we devised an efficient method for designing an optimal set for comprehensively supporting existing toroidal field coils (TFCs) linked to a torus chamber. Electromagnetic forces acting on the TFCs and the strain in the TFCs were precisely calculated by using a three-dimensional (3D) finite element analysis method. Based on the obtained results, various additional support components to be attached to specific weak places of the TFCs were determined. Experimental studies revealed that the strain observed in the TFCs was reduced to considerably less than its yield strain by using the additional support components. The value of the strain was consistent with those of 3D simulations. The aid of additional support components to existing TFCs allows modern comparative studies such as those on two-fluid plasmas in the same machine, in which tokamak edges as well as reversed-field pinches with low densities and distinct safety factors are considered. The results can provide insight into the two-fluid plasma mechanism, thus aiding future fusion plasma development.

Two-dimensional MHD equilibria of diamagnetic bubble in gas-dynamic trap

This article presents a magnetohydrodynamic (MHD) two-dimensional numerical model of diamagnetic bubble equilibria in an axisymmetric open trap. The theoretical model consists of the Grad–Shafranov equilibrium equation and the transport equation obtained within the resistive single-fluid MHDs with isotropic pressure. Found are the numerical solutions corresponding to the diamagnetic confinement mode. In particular, the equilibria of the diamagnetic bubble in the gas-dynamic multimirror trap are calculated. We investigate the effect of magnetic field corrugation on the equilibrium; the corrugation of the vacuum field is shown to lead to a rather moderate corrugation of the bubble boundary if the period of corrugation is sufficiently small. A valuable numerical result is the distribution of the diamagnetic field, which would be useful for optimizing the position of the wall-stabilization plates.

Anti-symmetric representation of the extended magnetohydrodynamic equations

We introduce the anti-symmetric representation of the extended magnetohydrodynamic (MHD) equations. In this representation, the use of the anti-symmetric flux operator (∇·v+v·∇) results in conservation theorems with discrete analogs. Inherently robust numerical applications are achieved with little effort, and conservation to machine precision is possible with simple numerical schemes. Starting from the two-fluid equations, we construct a single-fluid MHD model based on generalized center-of-mass variables for the mass (ρ), momentum (ρv), and pressure (p). This model is shown to possess identical conservation properties to the two-fluid system, with the only restriction being the use of a single temperature. Common approximations to the Braginskii heat fluxes and to the gyroviscous stress tensor are cast into our representation for convenience. The discrete conservation properties are verified using the classic Orszag–Tang vortex problem. In addition to the favorable mass, momentum, and energy conservation properties, the time reversibility of the simulations is demonstrated.

Coherent nonlinear oscillations in magnetohydrodynamic plasma

Single fluid magnetohydrodynamic (MHD) equations have been studied through direct numerical simulations using pseudospectral methods in two as well as three spatial dimensions. At Alfvén resonance, a reversible periodic exchange of energy between kinetic and magnetic variables is observed. The oscillations are identified as nonlinear dispersionless Alfvén waves that have been predicted earlier on theoretical grounds but not observed in large scale numerical simulations. A systematic study of their occurrence for various initial conditions and a range of Alfvén velocities is carried out. An analysis based on a finite mode representation of the incompressible single fluid MHD equations in two spatial dimensions reproduces the essential features of these oscillations.

Dominant two-fluid magnetohydrodynamic instabilities in CFETR upgrade phase-I scenario in presence of perfect conducting wall

The dominant two-fluid magnetohydrodynamic (MHD) stabilities of China Fusion Engineering Test Reactor upgrade phase-I baseline equilibria in presence of perfect conducting wall have been evaluated using the initial value code NIMROD. Our MHD simulations show that for equilibrium with qaxis < 1, high-n edge localized modes are slightly stabilized by toroidal shear flow, and n = 1 mode is found to be unstable in the core region with the characteristic of sawtooth instability. For qaxis > 2 equilibrium, all modes are found to be unstable but edge localized both in single-fluid and two-fluid MHD models. Specifically for n ≤ 13 modes, the two-fluid MHD model gives a slightly higher growth rate than the single-fluid MHD model, while for n > 13 modes, this trend becomes the opposite, which means the two-fluid MHD model is needed for high-n mode analysis. In addition, n = 1–10 modes are found to be more unstable with increasing wall position and eventually their growth rates approach values in the no-wall limit.

On the magnetohydrodynamic limits of the ideal two-fluid plasma equations

The two-fluid plasma equations describing a magnetized plasma, originating from truncating moments of the Vlasov-Boltzmann equation, are increasingly used to describe an ion-electron plasma whose transport phenomena occur on a time scale slower and a length scale longer than those of particle collisions. A similar treatment under more stringent constraints gives the single-fluid magnetohydrodynamic (MHD) equations for low-frequency macroscopic processes. Since both stem from kinetic theory, the two-fluid plasma and MHD equations are necessarily related to each other. Such a connection is often established via ad hoc physical reasoning without a firm analytical foundation. Here, we perform a sequence of formal expansions for the dimensionless ideal two-fluid plasma equations with respect to limiting values of the speed-of-light c, the ion-to-electron mass ratio M, and the plasma skin depth dS. Several different closed systems of equations result, including separate systems for each limit applied in isolation and those resulting from limits applied in combination, which correspond to the well-known Hall-MHD and single-fluid ideal MHD equations. In particular, it is shown that while the zeroth-order description corresponding to the c→∞ limit, with M and dS fixed, is strictly charge neutral, it nonetheless uniquely determines the perturbation charge non-neutrality at the first order. Furthermore, the additional M→∞ limit is found to be not required to obtain the single-fluid MHD equations despite being essential for the Hall-MHD system. The hierarchy of systems presented demonstrates how plasmas can be appropriately modeled in situations where only one of the limits applies, which lie in the parameter space in between where the two-fluid plasma and Hall-MHD models are appropriate.

Multicomponent fluid model for two-temperature plasmas derived from kinetic theory : application to magnetic reconnection

This contribution deals with the fluid modeling of multicomponent magnetized plasmas in thermo-chemical non-equilibrium from the partially- to fully-ionized collisional regimes, aiming at the predictive simulation of magnetic reconnection in Sun chromosphere conditions. Such fluid models are required for large-scale simulations by relying on high performance computing. The fluid model is derived from a kinetic theory approach, yielding a rigorous description of the dissipative and non-equilibrium effects and a well-identified mathematical structure. We start from a general system of equations that is obtained by means of a multiscale Chapman-Enskog method, based on a non-dimensional analysis accounting for the mass disparity between the electrons and heavy particles, including the influence of the electromagnetic field and transport properties. The latter are computed by using a spectral Galerkin method based on a converged Laguerre-Sonine polynomial approximation. Then, in the limit of small Debye length with respect to the characteristic scale in the Sun chromosphere, we derive a two-temperature single-momentum multicomponent diffusion model coupled to Maxwell’s equations, which is able to describe fully- and partially-ionized plasmas, beyond the multi-fluid model of Braginskii, valid for the whole range of the Sun chromosphere conditions. The second contribution is the development and verification of an accurate and robust numerical strategy that is based on CanoP, a massively parallel code with adaptive mesh refinement capability, which is able to cope with the full spectrum of scales of the magnetic reconnection process, without additional constraint on the time steps compared to single-fluid Magnetohydrodynamics (MHD) models. The final contribution is a study of the physics of magnetic reconnection in collaboration with the heliophysics team of NASA Ames Research Center. We show that the model and methods allow us to retrieve the results of usual single-fluid MHD models in the highly collisional case at equilibrium, while achieving a more detailed physics description relevant to such applications in the weakly collisional case, where non-equilibrium effects become important.

The temporal behaviour of MHD waves in a partially ionized prominence-like plasma: Effect of heating and cooling

Context.During heating or cooling processes in prominences, the plasma microscopic parameters are modified due to the change of temperature and ionization degree. Furthermore, if waves are excited on this non-stationary plasma, the changing physical conditions of the plasma also affect wave dynamics.Aims.Our aim is to study how temporal variation of temperature and microscopic plasma parameters modify the behaviour of magnetohydrodynamic (MHD) waves excited in a prominence-like hydrogen plasma.Methods.Assuming optically thin radiation, a constant external heating, the full expression of specific internal energy, and a suitable energy equation, we have derived the profiles for the temporal variation of the background temperature. We have computed the variation of the ionization degree using a Saha equation, and have linearized the single-fluid MHD equations to study the temporal behaviour of MHD waves.Results.For all the MHD waves considered, the period and damping time become time dependent. In the case of Alfvén waves, the cut-off wavenumbers also become time dependent and the attenuation rate is completely different in a cooling or heating process. In the case of slow waves, while it is difficult to distinguish the slow wave properties in a cooling partially ionized plasma from those in an almost fully ionized plasma, the period and damping time of these waves in both plasmas are completely different when the plasma is heated. The temporal behaviour of the Alfvén and fast wave is very similar in the cooling case, but in the heating case, an important difference appears that is related with the time damping.Conclusions.Our results point out important differences in the behaviour of MHD waves when the plasma is heated or cooled, and show that a correct interpretation of the observed prominence oscillations is very important in order to put accurate constraints on the physical situation of the prominence plasma under study, that is, to perform prominence seismology.

Braginskii magnetohydrodynamics for arbitrary magnetic topologies: coronal applications

We investigate single-fluid magnetohydrodynamics (MHD) with anisotropic viscosity, often referred to as Braginskii MHD, with a particular eye to solar coronal applications. First, we examine the full Braginskii viscous tensor in the single-fluid limit. We pay particular attention to how the Braginskii tensor behaves as the magnetic field strength vanishes. The solar corona contains a magnetic field with a complex and evolving topology, so the viscosity must revert to its isotropic form when the field strength is zero, e.g. at null points. We highlight that the standard form in which the Braginskii tensor is written is not suitable for inclusion in simulations as singularities in the individual terms can develop. Instead, an altered form, where the parallel and perpendicular tensors are combined, provides the required asymptotic behaviour in the weak-field limit. We implement this combined form of the tensor into the Lare3D code, which is widely used for coronal simulations. Since our main focus is the viscous heating of the solar corona, we drop the drift terms of the Braginskii tensor. In a stressed null point simulation, we discover that small-scale structures, which develop very close to the null, lead to anisotropic viscous heating at the null itself (that is, heating due to the anisotropic terms in the viscosity tensor). The null point simulation we present has a much higher resolution than many other simulations containing null points, so this excess heating is a practical concern in coronal simulations. To remedy this unwanted heating at the null point, we develop a model for the viscosity tensor that captures the most important physics of viscosity in the corona: parallel viscosity for strong fields and isotropic viscosity at null points. We derive a continuum model of viscosity where momentum transport, described by this viscosity model, has the magnetic field as its preferred orientation. When the field strength is zero, there is no preferred direction for momentum transport and viscosity reverts to the standard isotropic form. The most general viscous stress tensor of a (single-fluid) plasma satisfying these conditions is found. It is shown that the Braginskii model, without the drift terms, is a specialization of the general model. Performing the stressed null point simulation with this simplified model of viscosity reveals very similar heating profiles to those of the full Braginskii model. The new model, however, does not produce anisotropic heating at the null point, as required. Since the vast majority of coronal simulations use only isotropic viscosity, we perform the stressed null point simulation with isotropic viscosity and compare the heating profiles to those of the anisotropic models. It is shown that the fully isotropic viscosity can overestimate the viscous heating by an order of magnitude.

ON THE SPATIAL SCALES OF WAVE HEATING IN THE SOLAR CHROMOSPHERE

Dissipation of magnetohydrodynamic (MHD) wave energy has been proposed as a viable heating mechanism in the solar chromospheric plasma. Here, we use a simplified one-dimensional model of the chromosphere to theoretically investigate the physical processes and the spatial scales that are required for the efficient dissipation of Alfv\'en waves and slow magnetoacoustic waves. We consider the governing equations for a partially ionized hydrogen-helium plasma in the single-fluid MHD approximation and include realistic wave damping mechanisms that may operate in the chromosphere, namely Ohmic and ambipolar magnetic diffusion, viscosity, thermal conduction, and radiative losses. We perform an analytic local study in the limit of small amplitudes to approximately derive the lengthscales for critical damping and efficient dissipation of MHD wave energy. We find that the critical dissipation lengthscale for Alfv\'en waves depends strongly on the magnetic field strength and ranges from 10~m to 1~km for realistic field strengths. The damping of Alfv\'en waves is dominated by Ohmic diffusion for weak magnetic field and low heights in the chromosphere, and by ambipolar diffusion for strong magnetic field and medium/large heights in the chromosphere. Conversely, the damping of slow magnetoacoustic waves is less efficient, and spatial scales shorter than 10~m are required for critical damping. Thermal conduction and viscosity govern the damping of slow magnetoacoustic waves and play an equally important role at all heights. These results indicate that the spatial scales at which strong wave heating may work in the chromosphere are currently unresolved by observations.

Closure of the single fluid magnetohydrodynamic equations in presence of electron cyclotron current drive

In the presence of electron cyclotron current drive (ECCD), the Ohm’s law of single fluid magnetohydrodynamics (MHD) is modified as E + v × B = η(J – JECCD). This paper presents a new closure relation for the EC driven current density appearing in this modified Ohm’s law. The new relation faithfully represents the nonlocal character of the EC driven current and its main origin in the Fisch-Boozer effect. The closure relation is validated on both an analytical solution of an approximated Fokker-Planck equation as well as on full bounce-averaged, quasi-linear Fokker-Planck code simulations of ECCD inside rotating magnetic islands.

Three-dimensional magnetohydrodynamic model of the interaction between the solar wind and cometary atmospheres

A magnetohydrodynamic (MHD) model of the interaction between the solar wind and an expanding gas flow issuing from a comet nucleus is presented. The processes of photoionization of the neutral component of the comet matter and the resonance charge exchange between charged and neutral particles are taken into account, together with the interplanetary magnetic field effect. The adequacy of the single-fluid MHD approximation in describing the plasma flow with different temperatures of the components is shown. The model is numerically realized on the basis of the higher-order Godunov method using an approximate solution of the arbitrary MHD discontinuity breakdown problem. The results of a numerical investigation of the plasma and magnetic field parameters in the vicinity of a comet are presented for the conditions corresponding to the flyby of several spacecraft near comet Halley in March 1986 and the expected fly-round of comet Churyumov-Gerasimenko by the Rosetta spacecraft in 2015.

On the possibility of Alfvén wave resonance in collisionless magnetic reconnection

Alfvén wave resonance and magnetic reconnection are among the potential candidates for efficient dissipation of magnetic energy in space and astrophysical plasmas. In this paper, the correspondence between Alfvén resonance and the electron-inertia driven reconnection in a sheared force-free magnetic field is discussed. By analytical scaling the linear regimes of compressible tearing instability in the two-fluid magnetohydrodynamic (MHD) model, we present parametric conditions for the possibility of Alfvén resonance existence. Meanwhile, it is argued that the slow MHD Alfvénic resonance can take place only in the “intermediate” – called Hall-MHD regime when β>μ. β is the ratio of plasma thermal pressure to the pressure in equilibrium magnetic field and μ is the electron to ion mass ratio. There is no room for such a resonant dissipation phenomenon either in the single-fluid MHD or the electron-MHD regimes.

Linear stability studies including resistive wall effects with the CASTOR/STARWALL code

CASTOR/STARWALL is a code package for linear stability studies consisting of the CASTROR_3DW code, the STARWALL code, and a hybrid version of both codes named CASTOR3D that is currently under development. The latter solves an extended eigenvalue problem consisting of the plasma part defined by the weak form of the perturbed single-fluid MagnetoHydroDynamic (MHD) equations, and a vacuum part which is derived from an energy functional. This new code can describe wall resistivity and plasma inertia simultaneously. Furthermore, besides the so far used straight field line coordinates, additionally, general flux coordinates are implemented which are more appropriate for the description of instabilities located close to the separatrix (e.g. Edge Localized Modes (ELMs)). The formulation of the eigenvalue problem in the new coordinates is not limited to axisymmetric equilibria, so that, finally, stability studies of resistive and rotating, 3D equilibria should be possible. In this paper we will sketch the theory, report on the progress of the code development, and present first results.

Closure of the single fluid magnetohydrodynamic equations in presence of electron cyclotron current drive

In the presence of electron cyclotron current drive (ECCD), the Ohm's law of single fluid magnetohydrodynamics is modified as E + v × B = η(J – JEC). This paper presents a new closure relation for the EC driven current density appearing in this modified Ohm's law. The new relation faithfully represents the nonlocal character of the EC driven current and its main origin in the Fisch-Boozer effect. The closure relation is validated on both an analytical solution of an approximated Fokker-Planck equation as well as on full bounce-averaged, quasi-linear Fokker-Planck code simulations of ECCD inside rotating magnetic islands. The new model contains the model put forward by Giruzzi et al. [Nucl. Fusion 39, 107 (1999)] in one of its limits.