Magnetic Helicity Conservation Research Articles

Magnetic helicity, weak solutions and relaxation of ideal MHD

AbstractWe revisit the issue of conservation of magnetic helicity and the Woltjer‐Taylor relaxation theory in magnetohydrodynamics (MHD) in the context of weak solutions. We introduce a relaxed system for the ideal MHD system, which decouples the effects of hydrodynamic turbulence such as the appearance of a Reynolds stress term from the magnetic helicity conservation in a manner consistent with observations in plasma turbulence. As by‐products we answer two open questions in the field: We show the sharpness of the L3 integrability condition for magnetic helicity conservation and provide turbulent bounded solutions for ideal MHD dissipating energy and cross helicity but with (arbitrary) constant magnetic helicity.

Parameter study of decaying magnetohydrodynamic turbulence.

It is well known that helical magnetohydrodynamic (MHD) turbulence exhibits an inverse transfer of magnetic energy from small to large scales, which is related to the approximate conservation of magnetic helicity. Recently, several numerical investigations noticed the existence of an inverse energy transfer also in nonhelical MHD flows. We run a set of fully resolved direct numerical simulations and perform a wide parameter study of the inverse energy transfer and the decaying laws of helical and nonhelical MHD. Our numerical results show only a small inverse transfer of energy that grows as with increasing Prandtl number (Pm). This latter feature may have interesting consequences for cosmic magnetic field evolution. Additionally, we find that the decaying laws E∼t^{-p} are independent of the scale separation and depend solely on Pm and Re. In the helical case we measure a dependence of the form p_{b}≈0.6+14/Re. We also make a comparison between our results and previous literature and discuss the possible reason for the observed disagreements.

Conserving Local Magnetic Helicity in Numerical Simulations

Magnetic helicity is robustly conserved in systems with very large magnetic Reynolds numbers, including most systems of astrophysical interest, and unlike kinetic and magnetic energy, it is not dissipated at small scales. This plays a major role in suppressing the kinematic large-scale dynamo and may also be responsible for driving the large-scale dynamo through the magnetic helicity flux. Numerical simulations of astrophysical systems typically lack sufficient resolution to enforce global magnetic helicity over several dynamical times. In these simulations, magnetic helicity is lost either through numerical errors or through the action of an unrealistically large resistivity. Errors in the internal distribution of magnetic helicity are equally important and typically larger. Here, we propose an algorithm for enforcing strict local conservation of magnetic helicity in the Coulomb gauge in numerical simulations, so that their evolution more closely approximates that of real systems.

Cascade conditions in electron magneto-hydrodynamic turbulence

The basic features of electron whistler waves can be described by electron magneto-hydrodynamics. In terms of this model, we discuss criteria and relevant parameter ranges for dual and forward cascades in electron whistler turbulence. It is an advantage to use the set of eigenfunctions for the “curl” operator in the analysis. It is demonstrated that three wavemodes with same helicity support a dual cascade as know also from spatially two dimensional turbulence, while modes with mixed helicities include the classical cascade of energy from long to shorter wavelengths. The analysis is based on the conservation of energy and magnetic helicity during the wave interactions. For stationary conditions in large closed systems, a detailed balance can be expected. It is, however, possible for large-scale self-organized structures to develop as transient phenomena. These can be found when solving initial value problem with a strong imbalance of positive and negative helicities. For plasmas in nature or laboratory, similar conditions can occur when the system has boundary conditions with preference for emission of oscillations with one sign of helicity.

Rigorous results on conserved and dissipated quantities in ideal MHD turbulence

We review recent mathematical results on the theory of ideal MHD turbulence. On the one hand, we explain a mathematical version of Taylor's conjecture on magnetic helicity conservation, both for simply and multiply connected domains. On the other hand, we describe how to prove the existence of weak solutions conserving magnetic helicity but dissipating cross helicity and energy in 3D Ideal MHD. Such solutions are bounded. In fact, we show that as soon as we are below the critical integrability for magnetic helicity conservation, there are weak solutions which do not preserve even magnetic helicity. These mathematical theorems rely on understanding the mathematical relaxation of MHD which is used as a model of the macroscopic behaviour of solutions of various nonlinear partial differential equations. Thus, on the one hand, we present results on the existence of weak solutions consistent with what is expected from experiments and numerical simulations, on the other hand, we show that below certain thresholds, there exist pathological solutions which should be excluded from physical grounds. It is still an outstanding open problem to find suitable admissibility conditions that are flexible enough to allow the existence of weak solutions but rigid enough to rule out physically unrealistic behaviour.

Helicity observations of active regions during the exchange period of Solar Cycle 24 and 25

ABSTRACT Using vector magnetic field data from the Helioseismic and Magnetic Imager instrument onboard the Solar Dynamics Observatory, we study the signs of helicity (magnetic twist αav, z-component of current helicity Hc) and tilt angle of 85 sample active regions (ARs) that appeared on the central solar disc (within ±45° from disc centre) between December 2018 and November 2020. This time range spans the exchange period of Solar Cycle 24 and 25. The main findings are as follows: (1) As a whole, 62 per cent of sample ARs follow the helicity hemispherical sign rule, and our observational results do not show any hemispherical rule sign reversal at the end or beginning of a Solar Cycle. (2) Interestingly, there is no significant statistical relationship between helicity and tilt angle, as in contrast with the general idea on the conservation of magnetic helicity in the solar convection zone inferred by the relationship between the photospheric helicity and tilt angle. It is also found that the hemispherical tendency of helicity and tilt angle is more obvious for ARs at high latitudes, and the hemispherical preference is more obvious for ARs with magnetic twist and writhe of the opposite signs than for ARs with the same signs.

The magnetic helicity density patterns from non-axisymmetric solar dynamo

We study the helicity density patterns which can result from the emerging bipolar regions. Using the relevant dynamo model and the magnetic helicity conservation law we find that the helicity density patterns around the bipolar regions depend on the configuration of the ambient large-scale magnetic field, and in general they show a quadrupole distribution. The position of this pattern relative to the equator can depend on the tilt of the bipolar region. We compute the time–latitude diagrams of the helicity density evolution. The longitudinally averaged effect of the bipolar regions shows two bands of sign for the density distributions in each hemisphere. Similar helicity density patterns are provided by the helicity density flux from the emerging bipolar regions subjected to surface differential rotation.

Estimates of Current Helicity and Tilt of Solar Active Regions and Joy’s Law

The tilt angle, current helicity and twist of solar magnetic fields can be observed in solar active regions. We carried out estimates of these parameters by two ways. Firstly, we consider the model of turbulent convective cells (super-granules) which have a loop floating structure towards the surface of the Sun. Their helical properties are attained during the rising process in the rotating stratified convective zone. The other estimate is obtained from a simple mean-field dynamo model that accounts magnetic helicity conservation. The both values are shown to be capable to give important contributions to the observable tilt, helicity and twist.

The origin and effect of hemispheric helicity imbalance in solar dynamo

In this paper we study the effects of the net magnetic helicity density on the hemispheric symmetry of the dynamo generated large-scale magnetic field. Our study employs the axisymmetric dynamo model which takes into account the nonlinear effect of magnetic helicity conservation. We find that, on the surface, the net magnetic helicity follows the evolution of the parity of the large-scale magnetic field. Random fluctuations of the $\unicode[STIX]{x1D6FC}$ -effect and the helicity fluxes can invert the causal relationship, i.e. the net magnetic helicity or the imbalance of magnetic helicity fluxes can drive the magnetic parity breaking. We also found that evolution of the net magnetic helicity of the small-scale fields follows the evolution of the net magnetic helicity of the large-scale fields with some time lag. We interpret this as an effect of the difference of the magnetic helicity fluxes out of the Sun from the large and small scales.

Generation of chiral asymmetry via helical magnetic fields

It is well known that helical magnetic fields undergo a so-called inverse cascade by which their correlation length grows due to the conservation of magnetic helicity in classical ideal magnetohydrodynamics (MHD). At high energies above approximately $10$ MeV, however, classical MHD is necessarily extended to chiral MHD and then the conserved quantity is $\langle\mathcal{H}\rangle + 2 \langle\mu_5\rangle / \lambda$ with $\langle\mathcal{H}\rangle$ being the mean magnetic helicity and $\langle\mu_5\rangle$ being the mean chiral chemical potential of charged fermions. Here, $\lambda$ is a (phenomenological) chiral feedback parameter. In this paper, we study the evolution of the chiral MHD system with the initial condition of nonzero $\langle\mathcal{H}\rangle$ and vanishing $\mu_5$. We present analytic derivations for the time evolution of $\langle\mathcal{H}\rangle$ and $\langle\mu_5\rangle$ that we compare to a series of laminar and turbulent three-dimensional direct numerical simulations. We find that the late-time evolution of $\langle\mathcal{H}\rangle$ depends on the magnetic and kinetic Reynolds numbers ${\rm Re}_{_\mathrm{M}}$ and ${\rm Re}_{_\mathrm{K}}$. For a high ${\rm Re}_{_\mathrm{M}}$ and ${\rm Re}_{_\mathrm{K}}$ where turbulence occurs, $\langle\mathcal{H}\rangle$ eventually evolves in the same way as in classical ideal MHD where the inverse correlation length of the helical magnetic field scales with time $t$ as $k_\mathrm{p} \propto t^{-2/3}$. For a low Reynolds numbers where the velocity field is negligible, the scaling is changed to $k_\mathrm{p} \propto t^{-1/2}\mathrm{ln}\left(t/t_\mathrm{log}\right)$. After being rapidly generated, $\langle\mu_5\rangle$ always decays together with $k_\mathrm{p}$, i.e. $\langle\mu_5\rangle \approx k_\mathrm{p}$, with a time evolution that depends on whether the system is in the limit of low or high Reynolds numbers.

The mean tilt of sunspot bipolar regions: theory, simulations and comparison with observations

ABSTRACT A theory of the mean tilt of sunspot bipolar regions (the angle between a line connecting the leading and following sunspots and the solar equator) is developed. A mechanism of formation of the mean tilt is related to the effect of the Coriolis force on meso-scale motions of super-granular convection and large-scale meridional circulation. The balance between the Coriolis force and the Lorentz force (the magnetic tension) determines an additional contribution caused by the large-scale magnetic field to the mean tilt of the sunspot bipolar regions at low latitudes. The latitudinal dependence of the solar differential rotation affects the mean tilt, which can explain deviations from Joy’s law for the sunspot bipolar regions at high latitudes. The theoretical results obtained and the results from numerical simulations based on the non-linear mean-field dynamo theory, which takes into account conservation of the total magnetic helicity and the budget equation for the evolution of the Wolf number density, are in agreement with observational data of the mean tilt of sunspot bipolar regions over individual solar cycles 15–24.

Application of Synoptic Magnetograms to Global Solar Activity Forecast

Abstract Synoptic magnetograms provide us with knowledge about the evolution of magnetic fields on the solar surface and present important information for forecasting future solar activity. In this work, poloidal and toroidal magnetic field components derived from synoptic magnetograms are assimilated, using the Ensemble Kalman Filter method, into a mean-field dynamo model based on Parker’s migratory dynamo theory complemented by magnetic helicity conservation. It was found that the predicted toroidal field is in good agreement with observations for almost the entire following solar cycle. However, poloidal field predictions agree with observations only for the first 2–3 yr of the predicted cycle. The results indicate that the upcoming Solar Maximum of Cycle 25 (SC25) is expected to be weaker than the current Cycle 24. The model results show that a deep extended solar activity minimum is expected during 2019–2021, and that the next solar maximum will occur in 2024–2025. The sunspot number at the maximum will be about 50 with an error estimate of 15%–30%. The maximum will likely have a double peak or show extended periods (for 2–2.5 yr) of high activity. According to the hemispheric prediction results, SC25 will start in 2020 in the southern hemisphere, and will have a maximum in 2024 with a sunspot number of about 28. In the northern hemisphere the cycle will be delayed for about 1 yr (with an error of ±0.5 yr), and reach a maximum in 2025 with a sunspot number of about 23.

Determining the Transport of Magnetic Helicity and Free Energy in the Sun’s Atmosphere

Abstract The most important factors determining solar coronal activity are believed to be the availability of magnetic free energy and the constraint of magnetic helicity conservation. Direct measurements of the helicity and magnetic free energy in the coronal volume are difficult, but their values may be estimated from measurements of the helicity and free energy transport rates through the photosphere. We examine these transport rates for a topologically open system such as the corona, in which the magnetic fields have a nonzero normal component at the boundaries, and derive a new formula for the helicity transport rate at the boundaries. In addition, we derive new expressions for helicity transport due to flux emergence/submergence versus photospheric horizontal motions. The key feature of our formulas is that they are manifestly gauge invariant. Our results are somewhat counterintuitive in that only the lamellar electric field produced by the surface potential transports helicity across boundaries, and the solenoidal electric field produced by a surface stream function does not contribute to the helicity transport. We discuss the physical interpretation of this result. Furthermore, we derive an expression for the free energy transport rate and show that a necessary condition for free energy transport across a boundary is the presence of a closed magnetic field at the surface, indicating that there are current systems within the volume. We discuss the implications of these results for using photospheric vector magnetic and velocity field measurements to derive the solar coronal helicity and magnetic free energy, which can then be used to constrain and drive models for coronal activity.

Particle acceleration and fast magnetic reconnection

Mathematics demonstrates that the exponential separation of neighboring magnetic field lines, which naturally increases during an ideal evolution in three dimensions, leads to an exponentially increasing connection-breaking nonideal magnetic field. On a time scale that depends only logarithmically on the magnitude of the nonideal terms, a fast magnetic reconnection will generally occur, which has a rate determined by Alfvénic, not resistive, physics. The traditional assumption that the reconnecting flux must be dissipated by an electric field is false. In three dimensions, an ideal evolution can spatially mix the magnetic flux. Flux mixing conserves magnetic helicity, which limits the energy that can be transferred from the magnetic field to the plasma. The magnetic evolution is quasi-ideal during a fast magnetic reconnection, and the energy loss is given by the dot product of the magnetic field line velocity u→⊥ with the j→×B→ Lorentz force. Energy loss occurs through Alfvén waves and two other effects, which are also present in an ideal evolution. One is an effective parallel electric field E||, which can accelerate particles despite the particle acceleration due to the true parallel electric field E|| being negligible, and a coefficient νK, which gives a rate for exponentiation of the kinetic energy of particle motion along the magnetic field.

Stability of a force‐free Hall equilibrium and release of magnetic energy

Conservation of magnetic helicity by the Hall drift does not prevent Hall instability of helical fields. This conclusion follows from stability analysis of a force‐free spatially periodic Hall equilibrium. The growth rates of the instability scale as σ ∝ B3/4η1/4 with the field strength B and magnetic diffusivity η and can be large compared to the rate of resistive decay of the background field. The instability deviates the magnetic field from the force‐free configuration. The unstable eigenmodes include a fine spatial structure that evolves into current sheets at the nonlinear stage of the instability. The instability catalyzes the resistive release of magnetic energy. The energy is released in a sequence of spikes, every spike emits several percent of the total energy. A numerically defined scaling for the energy released in a single spike permits an extrapolation to astrophysically relevant values of the Hall number. The instability can be relevant to magnetic energy release in a neutron star crust and, possibly, in stellar coronae.

Proof of Taylor’s Conjecture on Magnetic Helicity Conservation

We prove Taylor's conjecture which says that in 3D MHD, magnetic helicity is conserved in the ideal limit in bounded, simply connected, perfectly conducting domains. When the domain is multiply connected, magnetic helicity depends on the vector potential of the magnetic field. In that setting we show that magnetic helicity is conserved for a large and natural class of vector potentials but not in general for all vector potentials. As an analogue of Taylor's conjecture in 2D, we show that mean square magnetic potential is conserved in the ideal limit, even in multiply connected domains.

On magnetohydrodynamic gauge field theory

Clebsch potential gauge field theory for magnetohydrodynamics is developed based in part on the theory of Calkin (1963 Can. J. Phys. 41 2241–51). It is shown how the polarization vector in Calkin’s approach naturally arises from the Lagrange multiplier constraint equation for Faraday’s equation for the magnetic induction , or alternatively from the magnetic vector potential form of Faraday’s equation. Gauss’s equation, (divergence of is zero) is incorporated in the variational principle by means of a Lagrange multiplier constraint. Noether’s theorem coupled with the gauge symmetries is used to derive the conservation laws for (a) magnetic helicity, (b) cross helicity, (c) fluid helicity for non-magnetized fluids, and (d) a class of conservation laws associated with curl and divergence equations which applies to Faraday’s equation and Gauss’s equation. The magnetic helicity conservation law is due to a gauge symmetry in MHD and not due to a fluid relabelling symmetry. The analysis is carried out for the general case of a non-barotropic gas in which the gas pressure and internal energy density depend on both the entropy S and the gas density ρ. The cross helicity and fluid helicity conservation laws in the non-barotropic case are nonlocal conservation laws that reduce to local conservation laws for the case of a barotropic gas. The connections between gauge symmetries, Clebsch potentials and Casimirs are developed. It is shown that the gauge symmetry functionals in the work of Henyey (1982 Phys. Rev. A 26 480–3) satisfy the Casimir determining equations.

Multi-region relaxed magnetohydrodynamics in plasmas with slowly changing boundaries—Resonant response of a plasma slab

The adiabatic limit of a recently proposed dynamical extension of Taylor relaxation, multi-region relaxed magnetohydrodynamics (MRxMHD), is summarized, with special attention to the appropriate definition of a 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 surface. The perturbed magnetic field as a function of ripple amplitude is calculated by invoking the conservation of magnetic helicity in the two regions separated by the current sheet. At low ripple amplitude, “half islands” appear on each side of the current sheet, locking the rotational transform at the resonant value. Beyond a critical amplitude, these islands disappear and the rotational transform develops a discontinuity across the current sheet.

Coarse-grained incompressible magnetohydrodynamics: analyzing the turbulent cascades

We formulate a coarse-graining approach to the dynamics of magnetohydrodynamic (MHD) fluids at a continuum of length-scales ℓ. In this methodology, effective equations are derived for the observable velocity and magnetic fields spatially-averaged at an arbitrary scale of resolution. The microscopic equations for the ‘bare’ velocity and magnetic fields are ‘renormalized’ by coarse-graining to yield macroscopic effective equations that contain both a subscale stress and a subscale electromotive force (EMF) generated by nonlinear interaction of eliminated fields and plasma motions. Particular attention is given to the effects of these subscale terms on the balances of the quadratic invariants of ideal incompressible MHD—energy, cross-helicity and magnetic helicity. At large coarse-graining length-scales, the direct dissipation of the invariants by microscopic mechanisms (such as molecular viscosity and Spitzer resistivity) is shown to be negligible. The balance at large scales is dominated instead by the subscale nonlinear terms, which can transfer invariants across scales, and are interpreted in terms of work concepts for energy and in terms of topological flux-linkage for the two helicities. An important application of this approach is to MHD turbulence, where the coarse-graining length ℓ lies in the inertial cascade range. We show that in the case of sufficiently rough velocity and/or magnetic fields, the nonlinear inter-scale transfer need not vanish and can persist to arbitrarily small scales. Although closed expressions are not available for subscale stress and subscale EMF, we derive rigorous upper bounds on the effective dissipation they produce in terms of scaling exponents of the velocity and magnetic fields. These bounds provide exact constraints on phenomenological theories of MHD turbulence in order to allow the nonlinear cascade of energy and cross-helicity. On the other hand, we prove a very strong version of the Woltjer-Taylor conjecture on conservation of magnetic helicity. Our bounds show that forward cascade of magnetic helicity to asymptotically small scales is impossible unless 3rd-order moments of either velocity or magnetic field become infinite.

Flux-Rope Twist in Eruptive Flares and CMEs: Due to Zipper and Main-Phase Reconnection

The nature of three-dimensional reconnection when a twisted flux tube erupts during an eruptive flare or coronal mass ejection is considered. The reconnection has two phases: first of all, 3D “zipper reconnection” propagates along the initial coronal arcade, parallel to the polarity inversion line (PIL); then subsequent quasi-2D “main-phase reconnection” in the low corona around a flux rope during its eruption produces coronal loops and chromospheric ribbons that propagate away from the PIL in a direction normal to it. One scenario starts with a sheared arcade: the zipper reconnection creates a twisted flux rope of roughly one turn (2pi radians of twist), and then main-phase reconnection builds up the bulk of the erupting flux rope with a relatively uniform twist of a few turns. A second scenario starts with a pre-existing flux rope under the arcade. Here the zipper phase can create a core with many turns that depend on the ratio of the magnetic fluxes in the newly formed flare ribbons and the new flux rope. Main phase reconnection then adds a layer of roughly uniform twist to the twisted central core. Both phases and scenarios are modeled in a simple way that assumes the initial magnetic flux is fragmented along the PIL. The model uses conservation of magnetic helicity and flux, together with equipartition of magnetic helicity, to deduce the twist of the erupting flux rope in terms the geometry of the initial configuration. Interplanetary observations show some flux ropes have a fairly uniform twist, which could be produced when the zipper phase and any pre-existing flux rope possess small or moderate twist (up to one or two turns). Other interplanetary flux ropes have highly twisted cores (up to five turns), which could be produced when there is a pre-existing flux rope and an active zipper phase that creates substantial extra twist.