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Helicity

Helicity is a concept in physics denoting the projection of a particle's spin angular momentum along the direction of its linear momentum, providing a measure of the particle's "handedness" relative to its motion.^[1] For massless particles, such as photons or neutrinos, helicity is a Lorentz-invariant quantity that can take discrete values of ±s (in units of ħ), where s is the particle's spin quantum number (e.g., +1 or -1 for photons, +1/2 or -1/2 for neutrinos), corresponding to right-handed or left-handed polarization, and remains fixed in all reference frames.^[2] In contrast, for massive particles, helicity is frame-dependent and not conserved, distinguishing it from chirality, an intrinsic property defined by the eigenvalues (±1) of the chirality operator (γ⁵ in the Dirac equation).^[3] Beyond particle physics, helicity manifests in fluid dynamics as the volume integral of the dot product of velocity and vorticity vectors, quantifying the topological linkage and knottedness of vortex lines in a flow.^[4] This fluid helicity is conserved under the ideal Euler equations for inviscid, barotropic flows, reflecting the preservation of vortex topology in the absence of viscosity or diffusion, and plays a key role in phenomena like turbulence and vortex dynamics.^[5] In meteorology, storm-relative helicity measures the potential for rotating updrafts in supercell thunderstorms, aiding severe weather forecasting by indicating environments conducive to mesocyclone formation.^[6] In magnetohydrodynamics, magnetic helicity describes the degree to which magnetic field lines are twisted, linked, or knotted, defined as the integral of the vector potential dotted with the magnetic field over a volume.^[7] This quantity is gauge-invariant in closed or bounded domains and conserved in ideal magnetohydrodynamic (MHD) plasmas, influencing processes such as magnetic reconnection, dynamo action, and solar coronal mass ejections.^[8] Its topological nature makes it a crucial invariant for understanding the evolution of complex magnetic structures in astrophysical and laboratory plasmas.

In Particle Physics

Definition and Formalism

In particle physics, helicity is defined as the quantum number representing the projection of a particle's spin angular momentum along the direction of its linear momentum. Mathematically, it is given by \lambda = \frac{\vec{J} \cdot \vec{p}}{|\vec{p}|}, where \vec{J} is the spin angular momentum operator and \vec{p} is the three-momentum vector of the particle.^[9] This quantity measures the alignment of the particle's intrinsic spin with its direction of motion, with positive helicity indicating parallel alignment and negative helicity indicating antiparallel alignment.^[10] The helicity formalism for particles with spin was systematically introduced in the context of relativistic scattering and decays by Jacob and Wick in 1959, building on earlier concepts of polarization for massless particles like photons.^[10] For massless particles, such as photons or neutrinos, the possible helicity eigenvalues are discrete and tied to the particle's spin: for spin-1/2 particles like neutrinos, the eigenvalues are \pm \hbar/2, while for spin-1 photons, they are \pm \hbar, corresponding to the two states of circular polarization (right-handed and left-handed).^[9] The helicity operator is constructed by considering the particle's state in its rest frame, where the momentum is zero, and then applying a Lorentz boost to the laboratory frame. In the rest frame of a massive particle, the spin is quantized along an arbitrary axis, but upon boosting along the momentum direction to a frame where the particle moves, the helicity becomes \vec{S} \cdot \hat{p}, with \vec{S} = \vec{J} (total angular momentum equals spin for elementary particles). For massless particles, which lack a rest frame, the boost preserves the helicity because the Lorentz transformation aligns the momentum direction without introducing transverse components that mix spin projections; thus, helicity is a Lorentz invariant for massless particles.^[11] In contrast, for massive particles, the boost induces a Wigner rotation that can mix different helicity states, making helicity frame-dependent and not Lorentz invariant.^[12] (Note: While helicity describes this extrinsic handedness based on the lab frame, it differs from chirality, an intrinsic property related to behavior under Lorentz transformations, as detailed elsewhere.) To illustrate the operator explicitly for spin-1/2 particles, the helicity operator in the Dirac representation is \hat{h} = \frac{\vec{\Sigma} \cdot \vec{p}}{2|\vec{p}|}, where \vec{\Sigma} are the spin matrices, and its eigenvalues \pm 1/2 (in units of \hbar) label the states. For photons, the helicity operator acts on the polarization vectors, yielding \pm 1 for the transverse modes, ensuring consistency with Maxwell's equations in vacuum.^[9]

Relation to Chirality

In the Dirac theory, chirality is defined as the eigenvalue of the operator \gamma^5 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3, where the eigenvalues \pm 1 correspond to right- and left-handed chiral states, respectively; this represents an intrinsic property of the fermion field that is independent of the observer's reference frame.^[13] Unlike helicity, which depends on the particle's momentum direction relative to its spin, chirality is a Lorentz-invariant quantity tied to the projection of the spinor onto chiral basis states.^[13] For massless particles, helicity and chirality coincide, such that a left-handed chiral state has negative helicity (spin antiparallel to momentum), and a right-handed chiral state has positive helicity; this equivalence arises because the massless Dirac equation conserves both quantities, and the chiral projectors align with the helicity eigenvectors.^[3] In contrast, for massive particles, helicity and chirality differ because mass mixes the chiral components, allowing a left-chiral massive neutrino, for example, to exhibit positive helicity in its rest frame or when boosted slowly, where the spin projection does not strictly oppose the momentum direction. The vector-axial vector (V-A) structure of weak interactions, as formulated in the universal Fermi theory, couples exclusively to left-handed chiral currents (and right-handed antichiral currents), which for massless fermions aligns with negative helicity but leads to helicity suppression in processes involving massive particles.^[14] This suppression manifests in the decay rates, where the amplitude for producing a right-handed helicity lepton is reduced by factors involving the mass ratio; for instance, in charged pion decays, the branching ratio \pi^+ \to e^+ \nu_e relative to \pi^+ \to \mu^+ \nu_\mu is suppressed by approximately (m_e / m_\mu)^2 \approx 2.3 \times 10^{-5}, with the observed ratio approximately $1.23 \times 10^{-4} after phase space modulation.^[15]^[16] Experimental confirmation of these chiral preferences and the resulting parity violation came from the 1957 observation of asymmetric electron emission in the beta decay of polarized cobalt-60 nuclei, where more electrons were emitted opposite the nuclear spin direction, demonstrating that weak interactions distinguish left from right. This V-A coupling to chiral states underpins the Standard Model's electroweak sector, where only left-chiral fermions participate in charged current interactions, with implications for phenomena like neutrino oscillations in massive cases.^[14]

Applications in Particle Interactions

In particle physics, the helicity formalism plays a crucial role in computing scattering amplitudes, particularly for high-energy processes involving spin-1/2 and spin-1 particles. Introduced by Jacob and Wick in their seminal 1959 paper, this method expands the S-matrix elements in a basis of helicity states, where the amplitude for a process is expressed as f_{\lambda_c \lambda_d; \lambda_a \lambda_b}, with \lambda_i denoting the helicity of each particle. This approach is especially efficient for massless particles, as their helicity is Lorentz-invariant and conserved, simplifying the evaluation of invariants and reducing the number of independent amplitudes through parity and angular momentum constraints. The formalism has been widely adopted in quantum field theory calculations, enabling compact expressions for cross-sections in collider experiments.^[10] In quantum electrodynamics (QED), helicity conservation for massless fermions manifests prominently in tree-level processes, such as electron-photon scattering (Compton scattering). For massless electrons, the interaction vertex preserves helicity because the QED Lagrangian couples the photon to the vector current, which does not flip the fermion's chirality, and helicity aligns with chirality in the massless limit. Consequently, amplitudes involving helicity flips vanish at leading order, leading to selection rules like the absence of right-handed electron to left-handed electron transitions in e^- \gamma \to e^- \gamma. This conservation holds perturbatively to all orders in the massless QED theory, though mass effects introduce small violations proportional to m_e / E. Weak interactions exhibit helicity suppression in decays involving massive particles due to the purely left-handed V-A current structure in the Standard Model. For beta decay (n \to p e^- \bar{\nu}_e), the differential rate includes a helicity mismatch factor that suppresses contributions from wrong-helicity states, scaling as (m_e / E_e)^2 for the electron, where E_e is its energy; this reduces the overall rate compared to the massless case. The integrated decay rate is approximated by \Gamma \propto G_F^2 |M|^2 (1 - \alpha [Z](/page/Z) / [R](/page/R)), where G_F is the Fermi constant, |M|^2 the squared matrix element incorporating axial and vector couplings, and (1 - \alpha [Z](/page/Z) / [R](/page/R)) accounts for finite nuclear size corrections with R the nuclear radius, [Z](/page/Z) the atomic number, and \alpha the fine-structure constant; phase space integration further modulates the suppression for massive leptons. This effect is evident in the branching ratio disparity between \pi^- \to e^- \bar{\nu}_e (heavily suppressed) and \pi^- \to \mu^- \bar{\nu}_\mu.^[17] Neutrino oscillations provide another key application, where tiny masses induce mixing between helicity states. In the Standard Model extended with neutrino masses, flavor eigenstates |\nu_\ell\rangle (for \ell = e, \mu, \tau) are superpositions of mass eigenstates |\nu_i\rangle with masses m_i, and the latter are not pure helicity eigenstates; the left-handed weak interaction projects primarily onto negative-helicity components, but mass terms allow admixture of positive-helicity states suppressed by m_i / E. Propagation in vacuum or matter causes coherent evolution of these components, leading to flavor oscillations with probability P(\nu_\ell \to \nu_{\ell'}) = \sin^2(2\theta) \sin^2\left( \frac{\Delta m^2 L}{4E} \right), where the helicity mixing enhances sensitivity to \Delta m^2 and mixing angles. This framework explains observed oscillations in solar, atmospheric, and accelerator experiments. Experimental verification of helicity in electroweak processes is highlighted by measurements of W boson helicity in top quark decays (t \to W b) at the LHC. The Standard Model predicts dominantly left-handed (f_- \approx 0.31) and longitudinal (f_0 \approx 0.69) W bosons due to the V-A coupling and Goldstone boson equivalence, with right-handed f_+ suppressed below 0.005. An ATLAS analysis of 139 fb^{-1} of data from pp collisions at \sqrt{s} = 13 TeV, using dilepton events and angular distributions of decay products, yields f_0 = 0.684 \pm 0.015 and f_- = 0.318 \pm 0.008, consistent with theory and CMS results, constraining anomalous couplings. These results probe the top-W-b vertex and test for new physics.^[18]

In Hydrodynamics

Definition and Mathematical Expression

In hydrodynamics, kinetic helicity serves as an integral measure quantifying the linkage and mutual twisting of vortex lines within a fluid flow. It is formally defined as

H = \int_V \mathbf{v} \cdot (\nabla \times \mathbf{v}) \, dV,

where \mathbf{v} is the velocity field and the integration is over a volume V containing the flow. This quantity captures the overall "handedness" of the vorticity alignment with the flow direction. For solutions of the ideal incompressible Euler equations, kinetic helicity is conserved, alongside the total kinetic energy, linear momentum, and angular momentum. In the context of vortex filaments, where vorticity is concentrated along thin, localized structures, helicity admits a link representation expressing the topological entanglement:

H = \frac{1}{4\pi} \iint \frac{\mathbf{v}_1 \cdot \left( \mathbf{v}_2 \times (\mathbf{r}_1 - \mathbf{r}_2) \right)}{|\mathbf{r}_1 - \mathbf{r}_2|^3} \, dV_1 \, dV_2,

with \mathbf{v}_1, \mathbf{v}_2 denoting the velocity fields at positions \mathbf{r}_1, \mathbf{r}_2.^[19] The dimensions of helicity are \mathrm{m}^4 / \mathrm{s}^2, reflecting the product of velocity and vorticity integrated over volume. Its sign indicates the dominant handedness of twists in the flow: positive values correspond to right-handed configurations, while negative values denote left-handed ones. The concept of hydrodynamic helicity was introduced in the context of turbulent flows by Levich in the 1980s, building on foundational work in knot theory applied to vortex lines by Moffatt in 1969.^[20]

Conservation and Invariants

In ideal, incompressible, and inviscid fluid flows governed by the Euler equations, hydrodynamic helicity is conserved, with its time derivative satisfying \frac{dH}{dt} = 0. This invariance holds for barotropic fluids under conservative body forces and bounded domains where the normal component of vorticity vanishes on the boundary.^[21] The conservation arises from Kelvin's circulation theorem, which states that the circulation around any material loop remains constant, implying that vortex lines are advected with the fluid without reconnection, thereby preserving the topological linkage of vorticity flux tubes.^[21] Additionally, the proof relies on the vector calculus identity \nabla \cdot (\mathbf{v} \times \boldsymbol{\omega}) = 0 in the context of the helicity evolution equation, where \boldsymbol{\omega} = \nabla \times \mathbf{v}, leading to vanishing volume integrals of the transport terms after integration by parts, assuming suitable boundary conditions.^[22] In viscous flows described by the Navier-Stokes equations, helicity is no longer conserved due to diffusive effects that allow vortex line reconnection and dissipation. The time evolution becomes \frac{dH}{dt} = -2\nu \int_V \boldsymbol{\omega} \cdot (\nabla \times \boldsymbol{\omega}) \, dV, where \nu is the kinematic viscosity.^[23] This dissipation term links helicity decay to the curl of vorticity, \nabla \times \boldsymbol{\omega}, which quantifies vortex stretching and tilting; in regions of high vorticity gradients, it drives helicity transfer, contrasting with enstrophy dissipation \frac{dZ}{dt} = -\nu \int_V |\nabla \times \boldsymbol{\omega}|^2 \, dV, where enstrophy Z = \frac{1}{2} \int_V |\boldsymbol{\omega}|^2 \, dV measures total vorticity intensity.^[23] In turbulent regimes, helicity participates in cascade processes that redistribute energy and enstrophy across scales. In three-dimensional hydrodynamic turbulence, helicity undergoes a forward cascade, transferring from large to small scales where it is ultimately dissipated by viscosity, consistent with the direct energy cascade predicted by Kolmogorov phenomenology.^[24] By contrast, in three-dimensional magnetohydrodynamic (MHD) turbulence, magnetic helicity exhibits an inverse cascade, accumulating at larger scales and influencing dynamo action, while kinetic helicity may follow a forward path.^[24] For comparison, MHD also conserves cross-helicity H_c = \int_V \mathbf{v} \cdot \mathbf{B} \, dV in ideal limits, where \mathbf{B} is the magnetic field, quantifying the correlation between velocity and magnetic fluctuations and affecting imbalanced turbulence spectra.^[25] A canonical numerical example illustrating helicity conservation is the Arnold-Beltrami-Childress (ABC) flow, a steady-state solution of the Euler equations from the 1960s featuring exact helical eigenmodes of the curl operator. In this flow, the velocity field satisfies \mathbf{v} = \nabla \times \mathbf{v} (up to a scalar multiple), yielding uniform helicity density and serving as a benchmark for studying chaotic advection and invariant preservation in bounded domains.^[21]

Physical Interpretations and Examples

Hydrodynamic helicity serves as a topological measure of the "knottedness" or mutual linkage of vortex lines within a fluid flow, where configurations with intertwined or knotted vortex filaments exhibit non-zero helicity, while unlinked ones possess zero. This quantity captures the degree to which velocity and vorticity align in a correlated manner, reflecting the flow's intrinsic handedness or chirality. In ideal, inviscid flows, the conservation of helicity underpins the persistence of complex vortex structures by resisting their diffusive decay, as briefly referenced in prior discussions of invariants. For instance, smoke rings—toroidal vortex loops—maintain their form over extended distances due to self-helicity, which quantifies the internal twisting and linkage within the vortex core, preventing rapid dissipation in air.^[26] In atmospheric dynamics, helicity influences cyclone and thunderstorm formation by promoting rotational organization in updrafts. Positive storm-relative helicity, calculated as the integral of wind shear relative to storm motion, correlates with supercell development, where values greater than 300 m²/s² in the 0–3 km layer signal environments favorable for persistent mesocyclones and severe weather events like tornadoes. These conditions arise from veering wind profiles that inject helicity into the storm, sustaining the low-level rotation essential for supercell longevity.^[27] Laboratory studies provide direct visualizations of helicity through controlled flows exhibiting helical instabilities. In Taylor-Couette setups, where fluid resides between counter-rotating concentric cylinders, azimuthal perturbations evolve into helical waves and vortex pairs at moderate Reynolds numbers, manifesting non-zero helicity in the resulting three-dimensional structures. These dynamics are quantified using particle image velocimetry (PIV), a non-intrusive technique that maps velocity fields across planes to compute local and integrated helicity, revealing how helical modes transport momentum axially.^[28] Astrophysical applications highlight helicity's role in large-scale transport processes. In accretion disks surrounding compact objects, helical turbulence—often driven by the helical variant of the magnetorotational instability—facilitates outward angular momentum transfer, enabling inward mass accretion despite hydrodynamic stability.^[29] This mechanism generates correlated velocity-vorticity alignments that amplify effective viscosities by factors of 10–100, as seen in simulations of protoplanetary and quasar disks.^[30] In forced turbulence, selective injection of helicity at large scales induces a sign reversal across the inertial range, breaking chiral symmetry and favoring one-handed vortex structures over mirror images. This asymmetry alters cascade statistics, with positive (or negative) helicity injection promoting imbalanced energy transfer and enhanced small-scale dissipation, as observed in direct numerical simulations of isotropic turbulence.

In Magnetohydrodynamics

Magnetic Helicity Definition

In plasma physics, magnetic helicity serves as a topological measure quantifying the degree of twisting, linking, and overall knottedness of magnetic field lines within a volume. It provides insight into the structural complexity of magnetic fields in magnetohydrodynamic (MHD) systems, distinguishing configurations that cannot be continuously deformed into one another without breaking field lines. The mathematical formulation of magnetic helicity H_m is given by the volume integral

H_m = \int_V \mathbf{A} \cdot \mathbf{B} \, dV,

where \mathbf{B} is the magnetic field and \mathbf{A} is the vector potential satisfying \mathbf{B} = \nabla \times \mathbf{A}, typically in the Coulomb gauge \nabla \cdot \mathbf{A} = 0. This quantity has units of weber squared (Wb²), reflecting the product of magnetic flux linkages.^[31] The sign of H_m indicates the handedness of the field lines: positive for right-handed twisting and negative for left-handed. This concept was first introduced by Woltjer in 1958 to analyze the conservation properties of force-free magnetic fields in astrophysical plasmas, such as those in stellar atmospheres. Moffatt extended the interpretation in 1969, linking magnetic helicity explicitly to the topological invariants of field line configurations, analogous to the knottedness in vortex lines. In closed volumes or domains with periodic boundary conditions, H_m is gauge-invariant, as surface terms in the gauge transformation vanish. For open domains, a gauge-invariant relative magnetic helicity is defined by subtracting the helicity of a reference potential field, often using the Weyl gauge where \mathbf{A} = 0 on the boundary. Magnetic helicity can be decomposed into contributions from self-helicity (internal twisting within individual flux tubes), linking helicity (interconnections between distinct flux tubes), and twist helicity (writhe of the flux tube axis), particularly when considering mutual helicity between multiple flux tubes. This decomposition highlights how H_m captures both local and global topological features of the magnetic structure. The formulation bears a formal similarity to kinetic helicity in hydrodynamics, \int \mathbf{v} \cdot (\nabla \times \mathbf{v}) \, dV, both serving as measures of rotational complexity in their respective fields.^[31]

Gauge Invariance and Topology

Magnetic helicity H_m = \int_V \mathbf{A} \cdot \mathbf{B} \, dV, where \mathbf{B} = \nabla \times \mathbf{A}, exhibits gauge invariance under the transformation \mathbf{A}' = \mathbf{A} + \nabla \chi for a scalar function \chi, provided the volume V is closed or satisfies appropriate boundary conditions such as tangential electric fields or perfectly conducting walls. Under this gauge shift, the change in helicity is \Delta H_m = \int_V \nabla \chi \cdot \mathbf{B} \, dV = \oint_S \chi \mathbf{B} \cdot d\mathbf{S}, which vanishes for closed volumes or suitable boundary conditions (e.g., \mathbf{B} normal to the surface or \chi=0 on the boundary), since \nabla \cdot \mathbf{B} = 0. This invariance holds in ideal magnetohydrodynamics (MHD) where magnetic fields are frozen into the plasma, ensuring H_m serves as a robust topological invariant.^[32] Topologically, magnetic helicity quantifies the knottedness and linkage of magnetic field lines, with H_m / (2\pi) corresponding to the average linking number between pairs of field lines, as captured by the Gauss linking integral. For a collection of flux tubes, this integral weights the mutual linkages by the product of enclosed fluxes, providing a measure of the overall entanglement invariant under continuous deformations in ideal MHD. The topological structure of helicity decomposes into linking, twist, and writhe components via White's theorem (1969), which states that the self-linking number Lk = Tw + Wr for a magnetic ribbon or flux tube, where Tw measures internal twisting along the axis, and Wr quantifies the writhing or coiling of the axis itself. The magnetic helicity is then H_m = \Phi^2 Lk for a flux tube of flux \Phi (in normalized units).^[33] This decomposition highlights how helicity partitions the total linkage into local (twist) and global (linking and writhe) contributions, preserving the invariant under smooth changes. In the context of plasma relaxation, Taylor's theory (1974) posits that turbulent reconnection in a low-resistivity plasma drives the system to a minimum-energy state that maximizes the absolute value of conserved magnetic helicity, resulting in force-free fields where \mathbf{J} = \mu \mathbf{B} for some scalar \mu. This relaxed state, often termed the Taylor state, emerges because helicity constrains the topology while energy dissipates, leading to configurations with uniform \mu in simply connected volumes. Computing magnetic helicity numerically in finite volumes introduces challenges due to gauge dependence and boundary effects, where surface terms can dominate if the computational domain does not fully enclose the flux or if the solenoidal condition \nabla \cdot \mathbf{B} = 0 is imperfectly enforced. Finite-volume methods must carefully handle these artifacts, often requiring relative helicity formulations with reference fields to ensure gauge invariance and mitigate spurious contributions from open boundaries.^[34]

Applications in Plasma Physics

In resistive magnetohydrodynamics (MHD), the evolution of magnetic helicity H_m is governed by the equation

\frac{\partial H_m}{\partial t} = -2 \int \mathbf{E} \cdot \mathbf{B} \, dV,

where \mathbf{E} is the electric field and the integral is over the plasma volume; this reflects dissipation primarily through the non-ideal term in the induction equation, linking helicity changes to ohmic heating and reconnection processes.^[35] In high-conductivity plasmas, such as those in astrophysical and laboratory settings, H_m is nearly conserved on dynamical timescales, except during rapid reconnection events, allowing it to serve as a topological invariant that influences long-term plasma behavior.^[36] In solar physics, the buildup of magnetic helicity in active regions through photospheric twisting motions drives the storage of free magnetic energy, ultimately triggering flares and coronal mass ejections (CMEs) when the accumulated helicity exceeds stability thresholds.^[37] Observations from SOHO/MDI magnetograms of active regions associated with halo CMEs reveal significant helicity injection via horizontal flows, with budgets on the order of $10^{42} Mx² correlating with eruptive events.^[38] STEREO and SOHO data further support the hemispheric helicity sign rule, where active regions in the northern (southern) hemisphere predominantly exhibit negative (positive) helicity—corresponding to left-handed (right-handed) twists—observed across multiple solar cycles with over 80% adherence in mature regions.^[39] During magnetic reconnection in solar plasmas, helicity is injected primarily through photospheric shearing and emergence motions, which transport twist from the convection zone into the corona, while the quantity remains largely conserved globally except at 3D null points where localized reconnection can annihilate or redistribute it.^[40] In these events, the magnetic energy release rate per unit area scales approximately as (B^2 / \mu_0) v_A in fast reconnection regimes (with v_{in} \sim 0.1 v_A), corresponding to the inflow of magnetic energy through Poynting flux in the reconnection diffusion region. The associated helicity dissipation follows from the evolution equation.^[36] This conservation property, tied to the topological nature of H_m, enables tracking of helicity transport from photosphere to interplanetary CMEs, with dissipation minimal away from nulls.^[41] In laboratory fusion devices like the reversed field pinch (RFP), plasma relaxation processes minimize magnetic energy for a fixed H_m, leading to Taylor states where the field aligns with the lowest-energy eigenmode of the Beltrami equation. Experiments at RFX-mod in the 2000s demonstrated this through pulsed operations at currents up to 1.5 MA, observing spontaneous transitions to quasi-single-helicity (QSH) configurations that reduce transport by factors of 2–3 compared to multi-helicity states, validating Taylor's theory in toroidal geometry.^[42] These QSH states exhibit sustained helicity conservation during sawtooth-like relaxations, enhancing confinement in RFP regimes. In astrophysical contexts, such as active galactic nuclei (AGN) jets, helical magnetic fields inferred from helicity transport power collimated outflows over kiloparsec scales. Very Long Baseline Interferometry (VLBI) polarimetric imaging of sources like 3C 273 and M87 reveals transverse gradients in rotation measure (RM) across jet widths, with sign reversals indicating helical field structures wound around the jet axis, consistent with helicity injection from the accretion disk. These observations, spanning frequencies from 1–22 GHz, show RM gradients of 10–100 rad m⁻² arcsec⁻¹, supporting models where conserved H_m stabilizes jets against instabilities.^[43]

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E · B dV + 2 ZV. ∂φ. ∂t. ∇ · Ap dV. − 2 Z∂V. (B · Ap)v · dS + 2 Z∂V ... helicity variation rate between these 2 instants: ∆HV/∆t = ... evolution of the ...
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Mar 5, 2010 · A question is whether magnetic helicity is still conserved in a highly conducting plasma undergoing reconnection. The essence of the Taylor ...
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[PDF] Modelling and observations of photospheric magnetic helicity
Nov 26, 2008 · Their accumulation in the solar corona leads to flares and coronal mass ejections. Since reconnections occur during the evolution of the flux.
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The Magnetic Helicity Budget of Solar Active Regions and Coronal ...
Sep 10, 2003 · We compute the magnetic helicity injected by transient photospheric horizontal flows in six solar active regions associated with halo ...Missing: buildup SOHO STEREO hemispheric rule<|separator|>
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Solar-Cycle Variation of Magnetic Helicity in Active Regions
In this paper we consider the hemispheric distribution of helicity over three solar cycles (cycles 21–23, 1983–2001) using magnetograms obtained with the two ...Missing: STEREO | Show results with:STEREO
[37]
Modelling and observations of photospheric magnetic helicity
Most of the helicity injection is found during magnetic flux emergence, whereas the effect of differential rotation is small, and the long-term evolution of ...Missing: transfer | Show results with:transfer
[38]
Magnetic reconnection with null and X-points - AIP Publishing
Dec 3, 2019 · Null and X-points are not themselves directly important to magnetic reconnection because distinguishable field lines do not approach them closely.
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Aug 21, 2000 · The reversed field pinch (RFP) is a configuration for plasma magnetic confinement. It has been traditionally viewed as dominated by a bath ...
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Probing the Innermost Regions of AGN Jets and Their Magnetic ...
Jan 19, 2022 · ... rotation measure across the jet width, indicates that the VLBI core is threaded by a helical magnetic field, in agreement with jet formation ...