An integrable localized approximation for interaction of two nearly anti-parallel sheets of the generalized vorticity in 2D ideal electron-magnetohydrodynamic flows
The formalism of frozen-in vortex lines for two-dimensional (2D) flows in ideal incompressible electron magnetohydrodynamics (EMHD) is formulated. A localized approximation for nonlinear dynamics of two close sheets of the generalized vorticity is suggested and its integrability by the hodograph method is demonstrated.
Compressible ideal magnetohydrodynamics (MHD) is formulated in terms of the time evolution of potential vorticity and magnetic flux per unit mass using a compact Lie bracket notation. It is demonstrated that this simplifies analytic solution in at least one very important situation relevant to magnetic fusion experiments. Potentially important implications for analytic and numerical modelling of both laboratory and astrophysical plasmas are also discussed.
Parker's formulation of isotopological plasma relaxation process in magnetohydrodynamics (MHD) is extended to Hall MHD. The torsion coefficient alpha in the Hall MHD Beltrami condition turns out now to be proportional to the "potential vorticity." The Hall MHD Beltrami condition becomes equivalent to the "potential vorticity" conservation equation in two-dimensional (2D) hydrodynamics if the Hall MHD Lagrange multiplier beta is taken to be proportional to the "potential vorticity" as well. The winding pattern of the magnetic field lines in Hall MHD then appears to evolve in the same way as "po…
We investigate using direct numerical simulations with grids up to 1536^3 points, the rate at which small scales develop in a decaying three-dimensional MHD flow both for deterministic and random initial conditions. Parallel current and vorticity sheets form at the same spatial locations, and further destabilize and fold or roll-up after an initial exponential phase. At high Reynolds numbers, a self-similar evolution of the current and vorticity maxima is found, in which they grow as a cubic power of time; the flow then reaches a finite dissipation rate independent of Reynolds number.
The nonlinear dynamics of axisymmetric, as well as helical, frozen-in vortex structures is investigated by the Hamiltonian method in the framework of ideal incompressible electron magnetohydrodynamics. For description of current-sheet formation from a smooth initial magnetic field, local and nonlocal nonlinear approximations are introduced and partially analyzed that are generalizations of the previously known exactly solvable local model neglecting electron inertia. Finally, estimations are made that predict finite-time singularity formation for a class of hydrodynamic models intermediate bet…
We propose two sets of initial conditions for magnetohydrodynamics (MHD) in which both the velocity and the magnetic fields have spatial symmetries that are preserved by the dynamical equations as the system evolves. When implemented numerically they allow for substantial savings in CPU time and memory storage requirements for a given resolved scale separation. Basic properties of these Taylor-Green flows generalized to MHD are given, and the ideal non-dissipative case is studied up to the equivalent of 2048^3 grid points for one of these flows. The temporal evolution of the logarithmic decrem…
Inertial effects in nonlinear magnetic reconnection are studied within the context of 2D electron magnetohydrodynamics (EMHD) with resistive and viscous dissipation. Families of nonlinear solutions for relevant current sheet parameters are predicted and confirmed numerically in all regimes of interest. Electron inertia becomes important for current sheet thicknesses $δ$ below the inertial length $d_{e}$. In this case, in the absence of electron viscosity, the sheet thickness experiences a nonlinear collapse. Viscosity regularizes solutions at small scales. Transition from resistive to viscous …
Topological implications of the total generalized electron-flow magnetic helicity He in electron magnetohydrodynamics(EMHD) are explored. The invariance of He is shown to imply the invariance of the sum of the linkage of the magnetic field lines, the linkage of electron-flow vorticity field lines and the mutual linkage among these two sets of field lines. This result appears to support a change in the magnetic field topology and hence pave the way for magnetic reconnection in EMHD via a change in the concomitant electron-flow vorticity topology.
An overview of some recent progress on magnetohydrodynamic stability and current sheet formation in a line-tied system is given. Key results on the linear stability of the ideal internal kink mode and resistive tearing mode are summarized. For nonlinear problems, a counterexample to the recent demonstration of current sheet formation by Low \emph{et al}. [B. C. Low and Å. M. Janse, Astrophys. J. \textbf{696}, 821 (2009)] is presented, and the governing equations for quasi-static evolution of a boundary driven, line-tied magnetic field are derived. Some open questions and possible strategies to…
\large{\bf Abstract-} Unsteady Hall Magnetohydrodynamics (MHD) near a hyperbolic magnetic neutral line is investigated. An exact analytical solution describing a self-similar evolution is given. This solution shows a negligible impact on the current-sheet formation process near the hyperbolic magnetic neutral line at small times by the Hall effect but, subsequently, a quenching by the Hall effect of the finite-time singularity exhibited in ideal MHD and, hence a prevention of the current density blow-up at large times. The asymptotic result given by this time-dependent solution is in full quan…
Three dimensional electron-magnetohydrodynamic (EMHD) simulations of electron current sheet dynamics in a background of stationary and unmagnetized ions and the subsequent generation of electromagnetic fluctuations are carried out. The physical parameters and initial magnetic configuration in the simulations are chosen to be similar to those in the \textsc{Vineta}.II magnetic reconnection experiment. Consistent with the experimental results, our 3D EMHD simulations show the formation of an elongated electron scale current sheet together with the excitation of electromagnetic fluctuations withi…
The Okubo [2]-Weiss [3] criterion is recast by using the 2D hydrodynamic Beltrami condition (Shivamoggi et al.[13]) that approximates the slow flow-variation ansatz imposed in its derivation. This turns out to provide an interesting interpretation of the Okubo-Weiss criterion very logically in terms of the topological properties of the underlying vorticity manifold. These developments are then extended to 2D quasi-geostrophic flows (via the potential divorticity framework), magnetohydrodynamic flows and electron magnetohydrodynamic flows (via the generalized magnetic flux framework) and the Ok…
We develop a framework for studying the statistical properties of current sheets in numerical simulations of 3D magnetohydrodynamic (MHD) turbulence. We describe an algorithm that identifies current sheets in a simulation snapshot and then determines their geometrical properties (including length, width, and thickness) and intensities (peak current density and total energy dissipation rate). We then apply this procedure to simulations of reduced MHD turbulence and perform a statistical analysis on the obtained population of current sheets. We evaluate the role of reconnection by separately stu…
Newcomb's Lagrangian for ideal magnetohydrodynamics (MHD) in Lagrangian labeling is discretized using discrete exterior calculus. Variational integrators for ideal MHD are derived thereafter. Besides being symplectic and momentum-preserving, the schemes inherit built-in advection equations from Newcomb's formulation, and therefore avoid solving them and the accompanying error and dissipation. We implement the method in 2D and show that numerical reconnection does not take place when singular current sheets are present. We then apply it to studying the dynamics of the ideal coalescence instabil…
We investigate the origins of $k^{-2}$ spectrum in a decaying Taylor-Green magnetohydrodynamic flow with zero large scale magnetic flux that was reported in Lee et al. (2010). A possible candidate for this scaling exponent has been the weak turbulence phenomenology. From our numerical simulations, we observe that current sheets in the magnetic Taylor-Green flow are formed in regions of magnetic discontinuities. Based on this observation and by studying the influence of the current sheets on the energy spectrum, using a filtering technique, we argue that the discontinuities are responsible for …
The scaling properties of three-dimensional magnetohydrodynamic turbulence are obtained from direct numerical simulations of decaying turbulence using $512^3$ modes. The results indicate that the turbulence does not follow the Iroshnikov-Kraichnan phenomenology.In the case of hyperresistivity, the structure functions exhibit a clear scaling range yielding absolute values of the scaling exponents $ζ_p$. The scaling exponents agree with a modified She-Leveque model $ζ_p=p/9 + 1 - (1/3)^{p/3}$, corresponding to Kolmogorov scaling but sheet-like geometry of the dissipative structures.
The adiabatic limit of a recently proposed dynamical extension of Taylor relaxation, \emph{multi-region relaxed magnetohydrodynamics} (MRxMHD) is summarized, with special attention to the appropriate definition of relative magnetic helicity. The formalism is illustrated using a simple two-region, sheared-magnetic-field model similar to the Hahm--Kulsrud--Taylor (HKT) rippled-boundary slab model. In MRxMHD a linear Grad--Shafranov equation applies, even at finite ripple amplitude. The adiabatic switching on of boundary ripple excites a shielding current sheet opposing reconnection at a resonant…
The plasmoid instability in evolving current sheets has been widely studied due to its effects on the disruption of current sheets, the formation of plasmoids, and the resultant fast magnetic reconnection. In this Letter, we study the role of the plasmoid instability in two-dimensional magnetohydrodynamic (MHD) turbulence by means of high-resolution direct numerical simulations. At sufficiently large magnetic Reynolds number ($R_m=10^6$), the combined effects of dynamic alignment and turbulent intermittency lead to a copious formation of plasmoids in a multitude of intense current sheets. The …
The interacting vorticity wave formalism for shear flow instabilities is extended here to the magnetohydrodynamic (MHD) setting, to provide a mechanistic description for the stabilising and destabilising of shear instabilities by the presence of a background magnetic field. The interpretation relies on local vorticity anomalies inducing a non-local velocity field, resulting in action-at-a-distance. It is shown here that the waves supported by the system are able to propagate vorticity via the Lorentz force, and waves may interact; existence of instability then rests upon whether the choice of …
Introducing a Clebsch-like parameterization, we have formulated a canonical Hamiltonian system on a symplectic leaf of reduced magnetohydrodynamics. An interesting structure of the equations is in that the Lorentz-force, which is a quadratic nonlinear term in the conventional formulation, appears as a linear term -ΔQ, just representing the current density (Q is a Clebsch variable, and Δ is the two-dimensional Laplacian); omitting this term reduces the system into the two-dimensional Euler vorticity equation of a neutral fluid. A heuristic estimate shows that current sheets grow exponentially (…
We present a set of three-dimensional (3D) direct numerical simulations of incompressible decaying magnetohydrodynamic turbulence in which we investigate the influence of an external uniform magnetic field B_0. A parametric study in terms of B_0 intensity is made where, in particular, we distinguish the shear- from the pseudo-Alfven waves dynamics. The initial kinetic and magnetic energies are equal with a negligible cross-correlation. Both the temporal and spectral effects of B_0 are discussed. A sub-critical balance is found between the Alfven and nonlinear times with both a global and a spe…
We present high resolution 2D and 3D simulations of magnetized decaying turbulence in relativistic resistive magneto-hydrodynamics. The simulations show dynamic formation of large scale intermittent long-lived current sheets being disrupted by the tearing instability into plasmoid chains. These current sheets are locations of enhanced magnetic field dissipation and heating of the plasma. We find magnetic energy spectra $\propto k^{-3/2}$, together with strongly pronounced dynamic alignment of Elsasser fields and of velocity and magnetic fields, for strong guide-field turbulence, whereas we ret…
Intermittency of heating in weakly collisional plasma turbulence is an active subject of research, with significant potential impact on understanding of the solar wind, solar corona and astrophysical plasmas. Recent studies suggest a role of vorticity in plasma heating. In magnetohydrodynamics small scale vorticity is generated near current sheets and this effect persists in kinetic plasma, as demonstrated here with hybrid and fully kinetic Particle-In-Cell (PIC) simulations. Furthermore, vorticity enhances local kinetic effects, with a generalized resonance condition selecting sign-dependent …
Simulations of decaying magnetohydrodynamic (MHD) turbulence are performed with a fluid and a kinetic code. The initial condition is an ensemble of long-wavelength, counter-propagating, shear-Alfvén waves, which interact and rapidly generate strong MHD turbulence. The total energy is conserved and the rate of turbulent energy decay is very similar in both codes, although the fluid code has numerical dissipation whereas the kinetic code has kinetic dissipation. The inertial range power spectrum index is similar in both the codes. The fluid code shows a perpendicular wavenumber spectral slope of…
The present paper provides comprehensive description of various regimes involved in the two-fluid model of the resistive tearing instability. These include two novel regimes of this instability, which correspond to the long-wave modes that can develop in a highly-elongated current sheet. This issue is relevant to the study of fast magnetic reconnection and magnetic turbulence in magnetohydrodynamic (MHD) objects with a large value of the Lundquist number.
In an electrically conducting fluid, vortices stretch out a weak, large-scale magnetic field to form strong current sheets on their edges. Associated with these current sheets are magnetic stresses, which are subsequently released through reconnection, leading to vortex disruption, and possibly even destruction. This disruption phenomenon is investigated here in the context of two-dimensional, homogeneous, incompressible magnetohydrodynamics. We derive a simple order of magnitude estimate for the magnetic stresses --- and thus the degree of disruption --- that depends on the strength of the ba…
Parker's formulation of isotopological plasma relaxation process toward minimum magnetics energy states in magnetohydrodynamics (MHD) is extended to electron MHD (EMHD). The lower bound on magnetic energy in EMHD is determined by both the magnetic field and the electron vorticity field topologies, and is shown to be reduced further in EMHD by an amount proportional to the sum of total electron-flow kinetic energy and total electron-flow enstrophy. The EMHD Beltrami condition becomes equivalent to the potential vorticity conservation equation in two-dimensional (2D) hydrodynamics, and the torsi…
The Lagrangian mechanical consideration of the dynamics of ideal incompressible hydrodynamic, magnetohydrodynamic, and Hall magnetohydrodynamic media, which are formulated as dynamical systems in appropriate Lie groups equipped with Riemannian metrics, leads to the notion of generalized vorticities, as well as generalized coordinates, velocities, and momenta. The action of each system is conserved against the integral path variation in the direction of the generalized vorticity, and this invariance is associated with the particle relabeling symmetry. The generalized vorticities are formulated …
Numerical simulations and experimental results have shown that current sheets formation in space plasmas can be associated with enhanced vorticity. Also, in simulations the generation of such structures is associated with strong plasma heating. Here, we compare four-point measurements in the terrestrial magnetosheath turbulence from the Multiscale magnetospheric mission (MMS) of the flow vorticity and the magnetic field curlometer versus their corresponding one-point proxies PVI(V) and PVI(B) based on the Partial Variance of Increments (PVI) method. We show that the one-point proxies are suffi…
We analyze the effect of the Hall term in the magnetohydrodynamic turbulence under a strong externally supported magnetic field, seeing how this changes the energy cascade, the characteristic scales of the flow and the dynamics of global magnitudes, with particular interest in the dissipation. Numerical simulations of freely evolving three-dimensional reduced magnetohydrodynamics (RHMHD) are performed, for different values of the Hall parameter (the ratio of the ion skin depth to the macroscopic scale of the turbulence) controlling the impact of the Hall term. The Hall effect modifies the tran…
A set of reduced Hall magnetohydrodynamic (MHD) equations are used to evaluate the stability of large aspect ratio current sheets to the formation of plasmoids (secondary islands). Reconnection is driven by resistivity in this analysis, which occurs at the resistive skin depth $d_η\equiv S_L^{-1/2} \sqrt{L v_A/γ}$, where $S_L$ is the Lundquist number, $L$ the length of the current sheet, $v_A$ the Alfvén speed, and $γ$ the growth rate. Modifications to a recent resistive MHD analysis [N.\ F.\ Loureiro, A.\ A.\ Schekochihin, and S.\ C. Cowley, Phys.\ Plasmas {\bf 14}, 100703 (2007)] arise when …
To investigate the impact of current sheet motion on the reconnection process, we perform resistive magnetohydrodynamic (MHD) simulations of two closely located reconnection sites which move apart from each other as reconnection develops. This simulation develops less quickly than an otherwise equivalent single perturbation simulation but eventually exhibits a higher reconnection rate. The unobstructed outflow jets are faster and longer than the outflow jets directed towards the magnetic island that forms between the two current sheets. The X-line and flow stagnation point are located near the…
The evolution of piecewise constant distributions of a conserved quantity related to the frozen-in canonical vorticity in effectively two-dimensional incompressible ideal EMHD flows is analytically investigated by the Hamiltonian method. The study includes the case of axisymmetric flows with zero azimuthal velocity component and also the case of flows with the helical symmetry of vortex lines. For sufficiently large size of such a patch of the conserved quantity, a local approximation in the dynamics of the patch boundary is suggested, based on the possibility to represent the total energy as …
Low-frequency turbulence in magnetised plasmas is intrisically influenced by gyroscale effects across ion Larmor orbits. Here we show that fundamental vortex interactions like merging and co-advection in gyrofluid plasmas are essentially modified under the influence of gyroinduced vortex spiraling. For identical initial vorticity, the fate of co-rotating eddies is decided between accelerated merging or explosion by the asymmetry of initial density distributions. Structures in warm gyrofluid turbulence are characterised by gyrospinning enhanced filamentation into thin vorticity sheets.
In situ spacecraft data on the solar wind show events identified as magnetic reconnection with outflows and apparent "`$X$-lines" $10^{3-4}$ times ion scales. To understand the role of turbulence at these scales, we make a case study of an inertial-range reconnection event in a magnetohydrodynamic (MHD) simulation. We observe stochastic wandering of field-lines in space, breakdown of standard magnetic flux-freezing due to Richardson dispersion, and a broadened reconnection zone containing many current sheets. The coarse-grain magnetic geometry is like large-scale reconnection in the solar wind…
We present a systematic analysis of statistical properties of turbulent current and vorticity structures at a given time using cluster analysis. The data stems from numerical simulations of decaying three-dimensional (3D) magnetohydrodynamic turbulence in the absence of an imposed uniform magnetic field; the magnetic Prandtl number is taken equal to unity, and we use a periodic box with grids of up to 1536^3 points, and with Taylor Reynolds numbers up to 1100. The initial conditions are either an X-point configuration embedded in 3D, the so-called Orszag-Tang vortex, or an Arn'old-Beltrami-Chi…
Collisionless dissipation of macroscopic energy into heat is an unsolved problem of space and astrophysical plasmas, e.g., solar wind and Earth's magnetosheath. The most viable process under consideration is the turbulent-cascade of macroscopic energy to kinetic-scales where collisionless-plasma-processes dissipate the energy. Space observations and numerical simulations show the formation of kinetic scale current sheets in turbulent plasmas. Instabilities in these CS can provide collisionless dissipation and influence the turbulence. Spatial gradients of physical quantities and non-Maxwellian…
Ideal systems like MHD and Euler flow may develop singularities in vorticity (w = curl v). Viscosity and resistivity provide dissipative regularizations of the singularities. In this paper we propose a minimal, local, conservative, nonlinear, dispersive regularization of compressible flow and ideal MHD, in analogy with the KdV regularization of the 1D kinematic wave equation. This work extends and significantly generalizes earlier work on incompressible Euler and ideal MHD. It involves a micro-scale cutoff length lambda which is a function of density, unlike in the incompressible case. In MHD,…
We examine the effect of an external guide field and current sheet thickness on the growth rates and nature of three dimensional unstable modes of an electron current sheet driven by electron shear flow. The growth rate of the fastest growing mode drops rapidly with current sheet thickness but increases slowly with the strength of the guide field. The fastest growing mode is tearing type only for thin current sheets (half thickness $\approx d_e$, where $d_e=c/ω_{pe}$ is electron inertial length) and zero guide field. For finite guide field or thicker current sheets, fastest growing mode is non…
Reconnection physics at micro-scales is investigated in an electron magnetohydrodynamics frame. A new process of collapse of the neutral current sheet is demonstrated by means of analytical and numerical solutions. It shows how at scales smaller than ion inertia length a compression of the sheet triggers an explosive evolution of current perturbation. Collapse results in the formation of a intense sub-sheet and then an X-point structure embedded into the equilibrium sheet. Hall currents associated with this structure support high reconnection rates. Nonlinear static solution at scales of the e…
Models of magnetohydrodynamic (MHD) equilibia that for computational convenience assume the existence of a system of nested magnetic flux surfaces tend to exhibit singular current sheets. These sheets are located on resonant flux surfaces that are associated with rational values of the rotational transform. We study the possibility of eliminating these singularities by suitable modifications of the plasma boundary, which we prescribe in a fixed boundary setting. We find that relatively straightforward iterative procedures can be used to eliminate weak current sheets that are generated at reson…
The cat-eyes steady state solution in the framework of hydrodynamics describing an infinite row of identical vortices is extended to the magnetohydrodynamic equilibrium equation with incompressible flow of arbitrary direction. The extended solution covers a variety of equilibria including one- and two-dimensional generalized force-free and Harris-sheet configurations which are preferable from those usually employed as initial states in reconnection studies. Although the vortex shape is not affected by the magnetic field, the flow in conjunction with the equilibrium nonlinearity has a strong im…
Magnetic field fluctuations in MHD turbulence can be viewed as current sheets that are progressively more anisotropic at smaller scales. As suggested by Loureiro & Boldyrev (2017) and Mallet et al (2017), below a certain critical thickness $λ_c$ such current sheets become tearing-unstable. We propose that the tearing instability changes the effective alignment of the magnetic field lines in such a way as to balance the eddy turnover rate at all scales smaller than $λ_c$. As a result, turbulent fluctuations become progressively less anisotropic at smaller scales, with the alignment angle increa…
The effect of compressibility in charged particle energization by magnetohydrodynamic (MHD) fields is studiedin the context of test particle simulations. This problem is relevant to the solar wind and the solar corona due to the compressible nature of the flow in those astrophysical scenarios. We consider turbulent electromagnetic fields obtained from direct numerical simulations of the MHD equations with a strong background magnetic field. In order to explore the flow compressibilty effect over the particle dynamics we performed different numerical experiments: an incompressible case, and two…
The magnetic topology and field line random walk properties of a nanoflare-heated and magnetically confined corona are investigated in the reduced magnetohydrodynamic regime. Field lines originating from current sheets form coherent structures, called Current Sheet Connected (CSC) regions, extended around them. CSC field line random walk is strongly anisotropic, with preferential diffusion along the current sheets' in-plane length. CSC field line random walk properties remain similar to those of the entire ensemble but exhibit enhanced mean square displacements and separations due to the stron…
We investigated the dynamic evolution of a 3-dimensional (3D) flux rope eruption and magnetic reconnection process in a solar flare, by simply extending 2-dimensional (2D) resistive magnetohydrodynamic simulation model of solar flares with low $β$ plasma to 3D model. We succeeded in reproducing a current sheet and bi-directional reconnection outflows just below the flux rope during the eruption in our 3D simulations. We calculated four cases of a strongly twisted flux rope and a weakly twisted flux rope in 2D and 3D simulations. The time evolution of a weakly twisted flux rope in 3D simulation…
Numerical evidence for a finite-time singularity in ideal 3D magnetohydrodynamics (MHD) is presented. The simulations start from two interlocking magnetic flux rings with no initial velocity. The magnetic curvature force causes the flux rings to shrink until they come into contact. This produces a current sheet between them. In the ideal compressible calculations, the evidence for a singularity in a finite time $t_c$ is that the peak current density behaves like $|J|_\infty \sim 1/(t_c-t)$ for a range of sound speeds (or plasma betas). For the incompressible calculations consistency with the c…
We clarify how magnetic reconnection can be derived from magnetohydrodynamics (MHD) equations in a way that is easily understandable to university students. The essential mechanism governing the time evolution of the magnetic field is diffusion dynamics. The magnetic field is represented by two components. It is clarified that the diffusion of a component causes a generation of another component that is initially zero and, accordingly, that the magnetic force lines are reconnected. For this reconnection to occur correctly, the initial magnetic field must be directed oppositely in the two regio…
The Parker model for coronal heating is investigated through a high resolution simulation. An inertial range is resolved where fluctuating magnetic energy E_M (k_perp) \propto k_\perp^{-2.7} exceeds kinetic energy E_K (k_\perp) \propto k_\perp^{-0.6}. Increments scale as δb_\ell \simeq \ell^{-0.85} and δu_\ell \simeq \ell^{+0.2} with velocity increasing at small scales, indicating that magnetic reconnection plays a prime role in this turbulent system. We show that spectral energy transport is akin to standard magnetohydrodynamic (MHD) turbulence even for a system of reconnecting current sheets…
In a recent numerical study [Ng et al., Astrophys. J. {\bf 747}, 109, 2012], with a three-dimensional model of coronal heating using reduced magnetohydrodynamics (RMHD), we have obtained scaling results of heating rate versus Lundquist number based on a series of runs in which random photospheric motions are imposed for hundreds to thousands of \al time in order to obtain converged statistical values. The heating rate found in these simulations saturate to a level that is independent of the Lundquist number. This scaling result was also supported by an analysis with the assumption of the Sweet…
