Von Kármán-Howarth relationship for helical magnetohydrodynamic flows.

513 The presence of helicity in correlation characteristics of turbulent flows is one of the observed manifestations of mirror-symmetry violation. Helicity is defined as the scalar product of the flow velocity and vorticity. Along with energy, helicity is a second-order invariant of the motion of an ideal fluid. This phenomenon is observed in both laboratory and natural geo- and astrophysical flows. Helical characteristics of turbulence related to the angular-moment accumulation in smallscale motions are among the main sources of magnetic field generation in rotating objects. Enhanced helicity is observed in many coherent vortex structures over a broad range of scales. This was devoted to the local structure of turbulent helicity. It is shown that the helicity dissipation can be determined from the results of simultaneous measurements of the velocity and vorticity components using the exact relations obtained previously [1] for the helical component of the third-order two-point correlation tensor. Among immediate consequences of these relations are (i) the power-law helicity spectrum with an exponent of ‐5/3, which has been obtained previously only qualitatively [2] and is now observed both under natural conditions and in numerical simulations [3‐5] and (ii) the more hypothetical helicity spectrum with an exponent of ‐4/3 that is typical “exclusively” of a helical cascade [2]. Very few measurements of helicity in turbulent flows are known, since such measurements are related to fairly complicated problems encountered in the measurement of multicomponent velocity-field gradients. The known experiments were carried out for turbulent flows behind a wire grid and in a jet [6, 7]. The existence of a helicity cascade, obeying a ‐5/ 3 power law, along with an energy cascade was confirmed by recent observations [8] of the atmospheric boundary layer at altitudes ~ 50 km [8] using an acoustic circulometer at the Obukhov Institute of Atmospheric Physics (Moscow) capable of measuring velocity and helicity simultaneously. The measured values of helicity were close to theoretical estimates. Without making hypotheses concerning the helicity cascade nature, let us consider correlation characteristics of homogeneous helical turbulence. The pair velocity correlation 〈 v i ( x ) v j ( x + r ) 〉 is described by the following function:

Large eddy simulation (LES) of transient magnetohydrodynamic (MHD) turbulent flow under a single-ruler electromagnetic brake (EMBr) in a laboratory-scale, continuous-casting mold is presented. The influence of different electrically-conductive boundary conditions on the MHD flow and electromagnetic field was studied, considering two different wall boundary conditions: insulating and conducting. Both the transient and time-averaged horizontal velocities predicted by the LES model agree well with the measurements of the ultrasound Doppler velocimetry (UDV) probes. Q-criterion was used to visualize the characteristics of the three-dimensional turbulent eddy structure in the mold. The turbulent flow can be suppressed by both configurations of the experiment’s wall (electrically-insulated and conducting walls). The shedding of small-scale vortices due to the Kelvin–Helmholtz instability from the shear at the jet boundary was observed. For the electrically-insulated walls, the flow was more unstable and changed with low-frequency oscillations. However, the time interval of the changeover was flexible. For the electrically-conducting walls, the low-frequency oscillations of the jets were well suppressed; a stable double-roll flow pattern was generated. Electrically-conducting walls can dramatically increase the induced current density and electromagnetic force; hence they contribute to stabilizing the MHD turbulent flow.

Magnetohydrodynamic (MHD) turbulent flows are found in the solar wind, the magnetosheath and the magnetotail plasma sheet. In this paper, we review both observational and theoretical evidence for turbulent flow in the magnetotail. MHD simulations of the global magnetosphere for southward interplanetary magnetic field (IMF) exhibit nested vortices in the earthward outflow from magnetic reconnection that are consistent with turbulence. Similar simulations for northward IMF also exhibit enhanced vorticity consistent with turbulence. These result from Kelvin-Helmholtz (KH) instabilities. However, the turbulent flows association with reconnection fill much of the magnetotail while the turbulent flows associated with the KH instability are limited to a smaller region near the magnetopause. Analyzing turbulent flows in the magnetotail is difficult because of the limited extent of the tail and because the flows there are usually sub-magnetosonic. Observational analysis of turbulent flows in the magnetotail usually assume that the Taylor frozen-in-flow hypothesis is valid and compare power spectral density vs. frequency with spectral indices derived for fluid turbulence by Kolmogorov in 1941. Global simulations carried out for actual magnetospheric substorms in the tail enable the results of the simulations to be compared directly with observed power spectra. The agreement between the two techniques provides confidence that the plasma sheet plasma is actually turbulent. The MHD results also allow us to calculate the power vs. wave number; results that also support the idea that the tail is turbulent.

Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations in more than one space dimension must confront the challenge of controlling errors in the discrete divergence of the magnetic field. One approach that has been shown successful in stabilizing MHD calculations are constrained-transport (CT) schemes. CT schemes can be viewed as predictor-corrector methods for updating the magnetic field, where a magnetic field value is first predicted by a method that does not exactly preserve the divergence-free condition on the magnetic field, followed by a correction step that aims to control these divergence errors. In Helzel, Rossmanith, and Taetz [J. Comput. Phys., 230 (2011), pp. 3803--3829] the authors presented an unstaggered CT method for the MHD equations on three-dimensional Cartesian grids. In this approach an evolution equation for the magnetic potential is solved during each time step and a divergence-free update of the magnetic field is computed by taking the curl of the magnetic potential. The evolution equation for the vector potential is only weakly hyperbolic, which requires special numerical treatment. A key step in this method is the use of dimensional splitting in order to overcome these difficulties. In this work we generalize the method of Helzel, Rossmanith, and Taetz [J. Comput. Phys., 230 (2011), pp. 3803--3829] in three important ways: (1) we remove the need for operator splitting by switching to an appropriate method of lines discretization and coupling this with a nonconservative finite volume method for the magnetic vector potential equation; (2) we increase the spatial and temporal order of accuracy of the entire method to third order; and (3) we develop the method so that it is applicable on both Cartesian and logically rectangular mapped grids. The method-of-lines approach that is used in this work is based on a third-order accurate finite volume discretization in space coupled to a third-order strong stability preserving Runge--Kutta time-stepping method. The evolution equation for the magnetic vector potential is solved using a nonconservative finite volume method based on the approach of Castro et al. [Math. Comput., 79 (2010), pp. 1427--1472]. The curl of the magnetic potential is computed via a third-order accurate discrete operator that is derived from appropriate application of the divergence theorem and subsequent numerical quadrature on element faces. Special artificial resistivity limiters are used to control unphysical oscillations in the magnetic potential and field components across shocks. Several test computations are shown that confirm third-order accuracy for smooth test problems and high resolution for test problems with shock waves.

Progress in lattice Boltzmann methods for magnetohydrodynamic flows relevant to fusion applications

A large eddy simulation of magnetohydrodynamic (MHD) turbulent channel flow is carried out to investigate the dynamo mechanism. It is shown that the streamwise component of the mean magnetic field is generated and sustained due to the effect of the turbulent electromotive force. The Reynolds-averaged turbulence model for MHD flows is assessed; it is suggested that the cross-helicity dynamo effect contributes to the turbulent electromotive force; that is, the electromotive force parallel to the mean vorticity is generated due to the turbulent cross helicity. To verify the importance of the cross-helicity dynamo, the transport equation for the turbulent electromotive force is evaluated; it is confirmed that the term involving the cross helicity and the mean vorticity is the main production term for the turbulent electromotive force. The transport equations for the turbulent kinetic and magnetic energies are also examined to discuss the dynamo mechanism from the viewpoint of the energy transfer.

This paper is concerned with a physical understanding of the main features of interstellar plasma turbulence. Our observational knowledge of this turbulence is provided by radio-wave propagation observations, generically referred to as interstellar scintillations. Distinctive features of the observations are the nearly omnipresent anisotropy of scattering, revealed by elliptical rather than circular scattering disks, drastic differences in the magnitude of scattering between closely spaced lines of sight through the interstellar medium, evidence from Faraday rotation observations that the interstellar vector magnetic field changes markedly on small spatial scales, and the existence of a power-law spectrum of density irregularities over a wide range of spatial scales. This power-law density spectrum strongly suggests the existence of similar spatial power spectra for the other magnetohydrodynamic (MHD) variables such as flow velocity and magnetic field. In this paper, it is pointed out that the aforementioned features arise or may naturally be explained by an approximate theory of magnetohydrodynamic turbulence, two-dimensional magnetohydrodynamics. In this theory, the plasma turbulence is described by two scalar functions (a velocity stream function and one component of the magnetic vector potential) that are coupled by nonlinear partial differential equations. These equations are physically transparent, possess some relevant analytic results, and are easily solved numerically. Arguments for the relevance of this reduced plasma description are presented. Although obviously an incomplete description of the interstellar plasma, these equations provide plausible explanations for the observational features described above. Anisotropy of scattering arises as an obvious consequence of the conditions for validity of the two-dimensional MHD description, i.e., that spatial gradients along a large-scale magnetic field are much smaller than those perpendicular to the field. The equations of two-dimensional MHD predict the formation of intense electrical current and vorticity sheets from broad classes of initial conditions. It is highly plausible that these sheets are the loci of elevated turbulence, which could explain the variations of radio-wave scattering and provide a physical explanation for intermittency in interstellar turbulence. The strong current sheets would also produce localized reversals in the turbulent component of the interstellar magnetic field. Finally, the equations of two-dimensional MHD produce spectral flattening of initial conditions with steep spatial power spectra. The calculations presented here, as well as elsewhere in the literature, indicate that the equilibrium magnetic field and velocity spectra may be power laws with indices close to those observed for the density fluctuations. In summary, the equations of two-dimensional magnetohydrodynamics are proposed as a simplified theoretical tool for use in understanding interstellar plasma turbulence.

Context. For a better understanding of the dynamics of the solar corona, it is important to analyse the evolution of the helicity of the magnetic field. Since the helicity cannot be directly determined by observations, we have recently proposed a method to calculate the relative magnetic helicity in a finite volume for a given magnetic field, which however required the flux to be balanced separately on all the sides of the considered volume. Aims. We developed a scheme to obtain the vector potential in a volume without the above restriction at the boundary. We studied the dissipation and escape of relative magnetic helicity from an active region. Methods. In order to allow finite magnetic fluxes through the boundaries, a Coulomb gauge was constructed that allows for global magnetic flux balance. The property of sinusoidal function was used to obtain the vector potentials at the 12 edges of the considered rectangular volume extending above an active region. We tested and verified our method in a theoretical fore-free magnetic field model. Results. We applied the new method to the former calculation data and found a difference of less than 1.2%. We also applied our method to the magnetic field above active region NOAA 11429 obtained by a new photospheric-data-driven magnetohydrodynamics (MHD) model code GOEMHD3. We analysed the magnetic helicity evolution in the solar corona using our new method. We find that the normalized magnetic helicity (H∕Φ2) is equal to −0.038 when fast magnetic reconnection is triggered. This value is comparable to the previous value (−0.029) in the MHD simulations when magnetic reconnection happened and the observed normalized magnetic helicity (−0.036) from the eruption of newly emerging active regions. We find that only 8% of the accumulated magnetic helicity is dissipated after it is injected through the bottom boundary. This is in accordance with the Woltjer conjecture. Only 2% of the magnetic helicity injected from the bottom boundary escapes through the corona. This is consistent with the observation of magnetic clouds, which could take magnetic helicity into the interplanetary space. In the case considered here, several halo coronal mass ejections (CMEs) and two X-class solar flares originate from this active region.

Optical magnetic helicity with binormal electromagnetic vortex filament flows in MHD

Pressure drops and velocity distributions of liquid metal MHD flow in pipes are closely related to the geometry of the cross-section. Liquid metal flows in two types of electrically conducting circular pipes under a uniform magnetic field applied transversely to the flow are investigated numerically using large eddy simulation with a coherent structure model developed in the OpenFOAM environment. One pipe has a typical circular cross-section (N-pipe), whereas the other has four V-shaped strips at the wall parallel and perpendicular to the magnetic field (F-pipe). The increasing transverse magnetic field causes a turbulent MHD flow evolves into a single-sided turbulent flow and a laminar flow in the N-pipe. The V-shaped strips on the Hartmann wall generate a low-velocity belt along the magnetic field direction in the center of the pipe and weak high-velocity jets near the V-shaped strips. Furthermore, the inflection points in the velocity profile improve instability and turbulence. The root mean square of the velocity fluctuation, in particular, the transverse components increase with the Hartmann number increasing along the magnetic field direction in the F-pipe. The V-shaped strips delay the MHD turbulent-laminar transition. However, the skin friction coefficient is slightly higher in the F-pipe than in the N-pipe, with the same dimensionless parameters. The research results can be applied to the turbulence promoter in the MHD pipe flow. Key words: Magnetohydrodynamic turbulent flow, large eddy simulation, pipe, V-shaped strips. Tables 5, Figs 9, Refs 23.

An investigation of the decay laws of energy and of higher moments of the Elsässer fields z±=v±b in the self-similar regime of magnetohydrodynamic (MHD) turbulence is presented, using phenomenological models as well as two-dimensional numerical simulations with periodic boundary conditions and up to 20482 grid points. The results are compared with the generalization of the parameter-free model derived by Galtier et al. [Phys. Rev. Lett.79, 2807 (1997)], which takes into account the slowing down of the dynamics due to the propagation of Alfvén waves. The new model developed here allows for a study in terms of one parameter governing the wavenumber dependence of the energy spectrum at scales of the order of (and larger than) the integral scale of the flow. The one-dimensional compressible case is also dealt with in two of its simplest configurations. Computations are performed for a standard Laplacian diffusion as well as with a hyperdiffusive algorithm. The results are sensitive to the amount of correlation between the velocity and the magnetic field, but rather insensitive to all other parameters such as the initial ratio of kinetic to magnetic energy or the presence or absence of a uniform component of the magnetic field. In all cases, the decay is significantly slower than for neutral fluids in a way that favours for MHD flows the phenomenology of Iroshnikov [Soviet Astron.7, 566 (1963)] and Kraichnan [Phys. Fluids8, 1385 (1965)] as opposed to that of Kolmogorov [Dokl. Akad. Nauk. SSSR31, 538 (1941)]. The temporal evolution of q-moments of the generalized vorticities 〈[mid ]ω±[mid ]q〉 =〈[mid ]ω±j[mid ]q〉 up to order q=10 is also given, and is compared with the prediction of the model. Less agreement obtains as q grows – a fact probably due to intermittency and the development of coherent structures in the form of eddies, and of vorticity and current sheets.

Advanced numerical methods based on exponential propagation have been applied to magnetohydrodynamic (MHD) simulations. This recently developed numerical technique evolves the system of nonlinear equations using exponential propagation of the Jacobian matrix. The exponential of the matrix is approximated by projecting it onto the Krylov subspace using the Arnoldi algorithm. The primary advantage of the exponential propagation method is that it allows time steps exceeding the Courant-Friedrichs-Lewy (CFL) limit. Another important aspect is faster convergence of the iteration computing the Krylov subspace projection compared to solving an implicit formulation of the system with similar iterative methods. Since the time scales in the resistive MHD equations are widely separated, the exponential propagation methods are especially advantageous for computing the long term evolution of a low-beta plasma. We analyze several types of exponential propagation methods and highlight important issues in the development of such techniques. Our analysis also suggests new ways to construct schemes of this type. Implementation issues, including scalability properties of exponential propagation methods, and performance are also discussed. In the second part of this work we present numerical MHD models which are constructed using exponential propagation methods and which describe the evolution of the magnetic arcades in the solar corona. Since these numerical methods have not been used before for large evolutionary systems like resistive MHD, we first validate our approach by demonstrating application of the exponential schemes to two existing magnetohydrodynamic models. We simulate the reconnection process resulting from shearing the footpoints of two-dimensional magnetic arcades and compute the three-dimensional linear force-free states of plasma configurations. Analysis of these calculations leads us to new insights about the topology of the solutions. The final chapter of this work is dedicated to a new three-dimensional numerical model of the dynamics of coronal plasma configurations. The model is motivated by observations and laboratory experiments simulating the evolution of solar arcades. We analyze the results of numerical simulations and demonstrate that our numerical approach provides an accurate and stable way to compute the solution to the zero-resistive MHD system. Based on comparisons of the simulation results and the observational data, we offer an explanation for the observed structure of eruptive events in the corona called coronal mass ejections (CME). We argue that the diversity of the images of CMEs obtained by the observational instruments can be explained as two-dimensional projections of a unique three-dimensional plasma configuration and suggest an eruption mechanism.

The statistical properties of active and passive scalar fields transported by the same turbulent flow are investigated. Four examples of active scalar have been considered: temperature in thermal convection, magnetic potential in two-dimensional (2D) magnetohydrodynamics (MHD), vorticity in 2D Ekman turbulence and potential temperature in surface flows. In the cases of temperature and vorticity, it is found that the active scalar behaviour is akin to that of its co-evolving passive counterpart. The two other cases indicate that this similarity is in fact not generic and differences between passive and active fields can be striking: in 2D MHD, the magnetic potential performs an inverse cascade, whereas the passive scalar cascades towards the small scales; in surface flows, although both perform a direct cascade, the potential temperature and the passive scalar have different scaling laws already at the level of low-order statistical objects. These significant differences are rooted in the correlations between the active scalar input and the particle trajectories. The role of such correlations in the issue of universality in active scalar transport and the behaviour of dissipative anomalies is addressed.

We present a set of numerical simulations that model the emergence of active region magnetic flux into an initially field-free model corona. We simulate the buoyant rise of twisted magnetic flux tubes initially positioned near the base of a stable stratified model convection zone and use the results of these calculations to drive a three-dimensional magnetohydrodynamic model corona. The simulations show that time-dependent subsurface flows are an important component of the dynamic evolution and subsequent morphology of an emerging magnetic structure. During the initial stages of the flux emergence process, the overlying magnetic field differs significantly from a force-free state. However, as the runs progress and boundary flows adjust, most of the coronal field—with the exception of those structures located relatively close to the model photosphere—relaxes to a more force-free configuration. Potential field extrapolations do not adequately represent the magnetic structure when emerging active region fields are twisted. In the dynamic models, if arched flux ropes emerge with nonzero helicity, the overlying field readily forms sigmoid-shaped structures. However, the chirality of the sigmoid and other details of its structure depend on the observer's vantage point and the location within a given loop of emitting plasma. Thus, sigmoids may be an unreliable signature of the sign and magnitude of magnetic twist.

Magnetohydrodynamic (MHD) buoyancy-driven instability has much in common with gravity-driven natural convection instability in thermally stratified fluids, in that account must be taken of the temperature dependence of conductivity in the former case and of density in the latter case. Employing the Monin–Obukhov theory of mixed convection in a stratified atmosphere, temperature and velocity profiles in MHD turbulent shear flow are obtained. The problem of the boundary-layer development is solved, and the heat and momentum fluxes are found. It is shown that the buoyancy force effects drastically change the fluxes when the Stuart number exceeds unity. Anode–cathode asymmetry observed experimentally on MHD devices is explained.