Evolution Of Helicity Research Articles

Energy and helicity evolution in a flux emergence simulation

Aims. The main aim of this work is to study the evolution of the recently introduced relative helicity of the magnetic polarity inversion line (PIL) in a magnetohydrodynamics simulation. Methods. The simulation used is a typical flux emergence simulation in which there is additionally an oblique, pre-existing magnetic field. The interaction of the emerging and ambient fields produces intense coronal activity, with four jets standing out. The 3D magnetic field allows us to compute various energies and helicities, and to study their evolution during the simulation, especially around the identified jets. We examine the evolution of all quantities in three different regions: in the whole volume, in three separate subvolumes of the whole volume, and in a 2D region around the PIL on the photosphere. Results. We find that the helicities are in general more responsive to the jets, followed by the free energy. The eruptivity index, the ratio of the current-carrying helicity to the relative helicity, does not show the typical behaviour it has in other cases, as its variations do not follow the production of the jets. By considering the subvolumes we find that the magnetic field gets more potential and less helical with height. The PIL relative helicity confirms the recent results it showed in observed active regions, exhibiting stronger variations during the jets compared to the standard relative helicity. Moreover, the current-carrying helicity around the PIL has a similar behaviour to the PIL relative helicity, and so this quantity could be equally useful in solar eruptivity studies.

Assessment of the near-Sun magnetic field of the 10 March 2022 coronal mass ejection observed by Solar Orbiter

Aims. We estimate the near-Sun axial magnetic field of a coronal mass ejection (CME) on 10 March 2022. Solar Orbiter’s in situ measurements, 7.8 degrees east of the Sun-Earth line at 0.43 AU, provided a unique vantage point, along with the WIND measurements at 0.99 AU. We determine a single power-law index from near-Sun to L1, including in situ measurements from both vantage points. Methods. We tracked the temporal evolution of the instantaneous relative magnetic helicity of the source active region (AR), NOAA AR 12962. By estimating the helicity budget of the pre-and post-eruption AR, we estimated the helicity transported to the CME. Assuming a Lundquist flux-rope model and geometrical parameters obtained through the graduated cylindrical shell (GCS) CME forward modelling, we determined the CME axial magnetic field at a GCS-fitted height. Assuming a power-law variation of the axial magnetic field with heliocentric distance, we extrapolated the estimated near-Sun axial magnetic field to in situ measurements at 0.43 AU and 0.99 AU. Results. The net helicity difference between the post-and pre-eruption AR is ( − 7.1 ± 1.2)×1041 Mx2, which is assumed to be bodily transported to the CME. The estimated CME axial magnetic field at a near-Sun heliocentric distance of 0.03 AU is 2067 ± 405 nT. From 0.03 AU to L1, a single power-law falloff, including both vantage points at 0.43 AU and 0.99 AU, gives an index −1.23 ± 0.18. Conclusions. We observed a significant decrease in the pre-eruptive AR helicity budget. Extending previous studies on inner-heliospheric intervals from 0.3 AU to ∼1 AU, referring to estimates from 0.03 AU to measurements at ∼1 AU. Our findings indicate a less steep decline in the magnetic field strength with distance compared to previous studies, but they align with studies that include near-Sun in situ magnetic field measurements, such as from Parker Solar Probe.

Electric field effects during disruptions

Tokamak disruptions are associated with breaking magnetic surfaces, which makes magnetic field lines chaotic in large regions of the plasma. The enforcement of quasi-neutrality in a region of chaotic field lines requires an electric potential that has both short and long correlation distances across the magnetic field lines. The short correlation distances produce a Bohm-like diffusion coefficient ∼Te/eB and the long correlation distances aT produce a large scale flow ∼Te/eBaT. This cross-field diffusion and flow are important for sweeping impurities into the core of a disrupting tokamak. The analysis separates the electric field in a plasma into the sum of a divergence-free, E→B, and a curl-free, E→q, part, a Helmholtz decomposition. The divergence-free part of E→ determines the evolution of the magnetic field. The curl-free part enforces quasi-neutrality, E→q=−∇→Φq. Magnetic helicity evolution gives the required boundary condition for a unique Helmholtz decomposition and an unfortunate constraint on steady-state tokamak maintenance.

Polarized dipole scattering amplitudes meet the valence quark model

The recently revised small-x helicity evolution [], resumming the double-logarithmic factor, αsln2(1/x), allows for the study of helicity distributions of quarks and gluons at small Bjorken x, corresponding to high center-of-mass energy. In this work, we calculate the moderate-x initial conditions in the regime, αsln2(1/x)∼1, for the small-x helicity evolution using a light-front valence quark model of the proton, which provides additional physical information about the target. The perturbative emission and absorption of a gluon by the valence quarks are also included. The results, given in Eqs. (35), provide a new set of initial conditions with a significantly reduced number of free parameters than conventional models []. Consequently, the predictive power of small-x helicity evolution is expected to improve once the initial conditions from this work are incorporated. Published by the American Physical Society 2024

Helicity evolution at small x: quark to gluon and gluon to quark transition operators

We include the quark to gluon and gluon to quark shock-wave transition operators into the small Bjorken-x evolution equations for helicity in the flavor-singlet channel derived earlier in [1–3]. While such transitions do not affect the large-Nc version of the evolution equations for helicity, the large-Nc & Nf equations are affected. (Nc and Nf are the numbers of quark colors and flavors, respectively.) We derive the corresponding corrected large-Nc & Nf equations for the polarized dipole amplitudes contributing to the flavor-singlet quark and gluon helicity distributions in the double-logarithmic approximation (DLA), resumming powers of αs ln2(1/x) with αs the strong coupling constant. We solve these equations iteratively and extract the polarized splitting functions up to four loops. We show that our splitting functions agree with the fixed-order perturbative calculations up to and including the existing three-loops results [4–7]. Similar to the large-Nc helicity evolution in the shock-wave approach [8], our large-Nc & Nf small-x splitting functions agree with those obtained in the infrared evolution equations framework from [9, 10] up to three loops, but appear to slightly disagree at four loops.

Magnetic helicity and energy budgets of jet events from an emerging solar active region

Using photospheric vector magnetograms obtained by the Helioseismic and Magnetic Imager onboard the Solar Dynamics Observatory and a magnetic connectivity-based method, we computed the magnetic helicity and free magnetic energy budgets of a simple bipolar solar active region (AR) during its magnetic flux-emergence phase, which lasted ∼47 h. The AR did not produce any coronal mass ejections (CMEs) or flares with an X-ray class above C1.0, but it was the site of 60 jet events during its flux-emergence phase. The helicity and free energy budgets of the AR were below established eruption-related thresholds throughout the interval we studied. However, in addition to their slowly varying evolution, each of the time profiles of the helicity and free energy budgets showed discrete localized peaks, with eight pairs of them occurring at times of jets emanating from the AR. These jets featured larger base areas and longer durations than the other jets of the AR. We estimated, for the first time, the helicity and free magnetic energy changes associated with these eight jets, which were in the ranges of 0.5 − 7.1 × 1040 Mx2 and 1.1 − 6.9 × 1029 erg, respectively. Although these values are one to two orders of magnitude smaller than those usually associated with CMEs, the relevant percentage changes were significant and ranged from 13% to 76% for the normalized helicity and from 9% to 57% for the normalized free magnetic energy. Our study indicates that jets may occasionally have a significant imprint in the evolution of helicity and free magnetic energy budgets of emerging ARs.

Magnetic field and kinetic helicity evolution in simulations of interacting disk galaxies

Context. There are indications that the magnetic field evolution in galaxies is influenced by tidal interactions and mergers between galaxies. Aims. We carried out a parameter study of interacting disk galaxies with impact parameters ranging from central collisions to weakly interacting scenarios. The orientations of the disks were also varied. In particular, we investigated how magnetic field amplification depends on these parameters. Methods. We used magnetohydrodynamics for gas disks in combination with live dark-matter halos in adaptive mesh refinement simulations. The disks were initialized using a setup for isolated disks in hydrostatic equilibrium. Since we focused on the impact of tidal forces on magnetic field evolution, adiabatic physics was applied. Small-scale filtering of the velocity and magnetic field allowed us to estimate the turbulent electromotive force (EMF) and kinetic helicity. Results. Time series of the average magnetic field in central and outer disk regions show pronounced peaks during close encounters and mergers. This agrees with observed magnetic fields at different interaction stages. The central field strength exceeds 10 μG (corresponding to an amplification factor of 2–3) for small impact parameters. As the disks are increasingly disrupted and turbulence is produced by tidal forces, the small-scale EMF reaches a significant fraction of the total EMF. The small-scale kinetic helicity is initially antisymmetric across the disk plane. Though its evolution is sensitive to both the impact parameter and inclinations of the rotation axes with respect to the relative motion of the disks, antisymmetry is generally broken through interactions and the merger remnant loses most of the initial helicity. The EMF and the magnetic field also decay rapidly after coalescence. Conclusions. The strong amplification during close encounters of the interacting galaxies is mostly driven by helical flows and a mean-field dynamo. The small-scale dynamo contributes significantly in post-interaction phases. However, the amplification of the magnetic field cannot be sustained.

A high-resolution numerical investigation of unsteady wake vortices for coaxial rotors in hover

High-resolution numerical simulations for wake vortical flows have long been a challenge in rotor aerodynamics. A novel spectrum-optimized sixth-order Weighted Essentially Non-Oscillatory (WENO) scheme is proposed to discretize inviscid fluxes on moving overset grids, and the Improved Delayed Detached Eddy Simulation (IDDES) is employed to resolve turbulent vortices. The integration of these methods facilitates a comprehensive numerical investigation into the unsteady vortical flows over coaxial rotors in hover. The results highlight the substantial improvement in numerical resolution, in terms of both spatial structure and temporal evolution of unsteady multiscale wake vortices. Coaxial rotors in hover manifest three primary scales of wake vortex structures: (A) the helical evolution of primary blade tip vortices and the periodic occurrence of strong Blade-Vortex Interactions (BVI); (B) the continuous shedding of small-scale horseshoe-shaped vortices from the trailing edges of rotor blades, forming the vortex sheets; (C) the emergence of small-scale secondary vortex braids induced by interactions between rotor tip vortices and the vortex sheets. These vortex structures and their interactions cause high-frequency oscillations in rotor disk loads and induce unsteady perturbations in the local flow field. Interactions among these primary vortices, coupled with the generation of secondary vortices, result in the dissipation, distortion, and breakup of the rotor tip vortices, ultimately forming a vortex soup. Notably, a substantial quantity of seemingly weak small-scale secondary vortex braids significantly contribute to energy dissipation during the evolution of wake vortices for coaxial rotors in hover.

Magnetic helicity evolution during active region emergence and subsequent flare productivity

Aims. Solar active regions (ARs), which are formed by flux emergence, serve as the primary sources of solar eruptions. However, the specific physical mechanism that governs the emergence process and its relationship with flare productivity remains to be thoroughly understood. Methods. We examined 136 emerging ARs, focusing on the evolution of their magnetic helicity and magnetic energy during the emergence phase. Based on the relation between helicity accumulation and magnetic flux evolution, we categorized the samples and investigated their flare productivity. Results. The emerging ARs we studied can be categorized into three types, Type-I, Type-II, and Type-III, and they account for 52.2%, 25%, and 22.8% of the total number in our sample, respectively. Type-I ARs exhibit a synchronous increase in both the magnetic flux and magnetic helicity, while the magnetic helicity in Type-II ARs displays a lag in increasing behind the magnetic flux. Type-III ARs show obvious helicity injections of opposite signs. Significantly, 90% of the flare-productive ARs (flare index ≥ 6) were identified as Type-I ARs, suggesting that this type of AR has a higher potential to become flare productive. In contrast, Type-II and Type-III ARs exhibited a low and moderate likelihood of becoming active, respectively. Our statistical analysis also revealed that Type-I ARs accumulate more magnetic helicity and energy, far beyond what is found in Type-II and Type-III ARs. Moreover, we observed that flare-productive ARs consistently accumulate a significant amount of helicity and energy during their emergence phase. Conclusions. These findings provide valuable insight into the flux emergence phenomena, offering promising possibilities for early-stage predictions of solar eruptions.

Magnetic Helicity Signs and Flaring Propensity: Comparing the Force-free Parameter with the Helicity Signs of Hα Filaments and X-Ray Sigmoids

Sigmoids produce strong eruptive events. Earlier studies have shown that the ICME axial magnetic field Bz can be predicted with some credibility by observing the corresponding filament or the polarity inversion line in the region of eruption and deriving the magnetic field direction from that. Sigmoids are coronal structures often associated with filaments in the sigmoidal region. In this study, we compare filament chirality with sigmoid handedness to observe their correlation. Second, we perform nonlinear force-free approximations of the coronal magnetic connectivity using photospheric vector magnetograms underneath sigmoids to obtain a weighted-average value of the force-free parameter and to correlate it with filament chirality and the observed coronal sigmoid handedness. Importantly, we find that the sigmoids and their filament counterparts do not always have the same helicity signs. Production of eruptive events by regions that do not have the same signs of helicities is ∼3.5 times higher than when they do. A case study of magnetic energy/ helicity evolution in NOAA AR 12473 is also presented.

Orbital angular momentum at small x revisited

We revisit the problem of the small Bjorken-x asymptotics of the quark and gluon orbital angular momentum (OAM) distributions in the proton utilizing the revised small-x helicity evolution derived recently in [1]. We relate the quark and gluon OAM distributions at small x to the polarized dipole amplitudes and their (first) impact-parameter moments. To obtain the OAM distributions, we derive novel small-x evolution equations for the impact-parameter moments of the polarized dipole amplitudes in the double-logarithmic approximation (summing powers of αs ln2(1/x) with αs the strong coupling constant). We solve these evolution equations numerically and extract the leading large-Nc, small-x asymptotics of the quark and gluon OAM distributions, which we determine to beLq+q¯xQ2~LGxQ2~∆ΣxQ2~∆GxQ2~123.66αsNc2π,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {L}_{q+\\overline{q}}\\left(x,{Q}^2\\right)\\sim {L}_G\\left(x,{Q}^2\\right)\\sim \\Delta \\Sigma \\left(x,{Q}^2\\right)\\sim \\Delta G\\left(x,{Q}^2\\right)\\sim {\\left(\\frac{1}{2}\\right)}^{3.66\\sqrt{\\frac{\\alpha_s{N}_c}{2\\uppi}}}, $$\\end{document}in agreement with [2] within the precision of our numerical evaluation. (Here Nc is the number of quark colors.) We also investigate the ratios of the quark and gluon OAM distributions to their helicity distribution counterparts in the small-x region.

Helicity: A Possible Indicator of Negative Feedback Initiation of Tropical Cyclone–Ocean Interaction

AbstractThe development of large‐scale vortex dynamo during tropical cyclogenesis through the mutual intensification of primary and secondary circulation in terms of helical evolution is well studied. However, the influence of atmospheric helicity on ocean surface heat and moisture fluxes associated with tropical cyclone (TC) evolution is yet to be understood. At its development stage, the heat and moisture flux from the ocean surface increases with the increase in intensity of the TC. This time, TC‐Ocean interaction works in a positive feedback mechanism. However, after a particular stage, the very intense wind of the TC causes strong turbulent mixing in the underlying ocean. Hence, the upper ocean layer temperature starts decreasing, leading to a decrease in heat and moisture flux exchange to the TC. We have analyzed the emergence of vortex helical structure in TC and the role of helicity in modulating surface heat and moisture fluxes. The result shows that helical development is associated with the axisymetrization of the vertical velocity and diabatic heating at the middle and upper troposphere. There is no definite TC intensity at which the initiation of negative feedback of TC‐upper ocean interaction occurs. However, the increase in helicity above 200 × 10−2 ms−2 reversed the TC‐ocean interaction trend. Therefore, TC helicity evolution and upper ocean interaction are essential to understand TC evolution.

Helical and non-helical large-scale dynamos in thin accretion discs

ABSTRACT The dynamics of accreting and outgoing flows around compact objects depends crucially on the strengths and configurations of the magnetic fields therein, especially of the large-scale fields that remain coherent beyond turbulence scales. Possible origins of these large-scale magnetic fields include flux advection and disc dynamo actions. However, most numerical simulations have to adopt an initially strong large-scale field rather than allow them to be self-consistently advected or amplified, due to limited computational resources. The situation can be partially cured by using sub-grid models where dynamo actions only reachable at high resolutions are mimicked by artificial terms in low-resolution simulations. In this work, I couple thin-disc models with local shearing-box simulation results to facilitate more realistic sub-grid dynamo implementations. For helical dynamos, detailed spatial profiles of dynamo drivers inferred from local simulations are used, and the non-linear quenching and saturation is constrained by magnetic helicity evolution. In the inner disc region, saturated fields have dipole configurations and the plasma β reaches ≃0.1 to 100, with correlation lengths ≃h in the vertical direction and ≃10 h in the radial direction, where h is the disc scale height. The dynamo cycle period is ≃40 orbital time scale, compatible with previous global simulations. Additionally, I explore two dynamo mechanisms which do not require a net kinetic helicity and have only been studied in shearing-box set-ups. I show that such dynamos are possible in thin accretion discs, but produce field configurations that are incompatible with previous results. I discuss implications for future general-relativistic magnetohydrodynamics simulations.

Small x physics beyond eikonal approximation: an effective Hamiltonian approach

Understanding the spin structure of hadrons in the small x regime is an important direction to unravel the spin puzzle in hadronic physics. To include spin degrees of freedom in the small x regime requires going beyond the usual eikonal approximation in high energy QCD. We developed an effective Hamiltonian approach to study spin related observables in the small x regime using the shockwave formalism. The small-x effective Hamiltonian incorporates both quark and gluon propagators in the background fields and the background field induced interaction vertices up to next-to-eikonal order. A novel feature of sub-eikonal interactions is the background gluon field induced gluon radiation inside the shockwave. Its relation to chromo-electrically polarized Wilson line correlator is established both in small x helicity evolution and in longitudinal double-spin asymmetry for gluon production.

Temporal Behaviors of Magnetic Helicity Injections by Self and Mutual Sunspot Rotations

Magnetic helicity is a physical parameter used to quantify the complexity of magnetic fields, providing an indication of the energy state in the coronal magnetic structure. We investigate the temporal evolution of magnetic helicity and its relationship to the occurrence of a variety of flares in the solar active region NOAA 12297, which was well observed using the Solar Dynamics Observatory/Helioseismic and Magnetic Imager in 2015 March. The active region produced many M-class flares and an X-class flare in two distinctive areas, both of which had a similar magnetic evolution, i.e., the opposite polarity of an emerging flux developed beside a preexisting sunspot, but exhibited flares with different magnitudes and frequencies. We derived the spatiotemporal evolution of the magnetic helicity injections and evaluated how spinning and braiding helicity injections evolved with time in the two areas. In one area, we observed a remarkable evolution, in which a negative spinning helicity injection in the preexisting sunspot increased in a positive helicity system, followed by the occurrence of the X-class flare. The negative helicity injection was clearly caused by the flux emergence that developed along the outer edge of the preexisting sunspot. The other area showed positive braiding helicity injections, with spinning helicity injections fluctuating concurrently with flux emergence, changing their signs several times, i.e., variable energy, and helicity input. The observed temporal behaviors of the helicity injections may explain different types of flare occurrences in the regions.

Changes of Magnetic Energy and Helicity in Solar Active Regions from Major Flares

Magnetic free energy powers solar flares and coronal mass ejections, and the buildup of magnetic helicity might play a role in the development of unstable structures that subsequently erupt. To better understand the roles of energy and helicity in large flares and eruptions, we have characterized the evolution of magnetic energy and helicity associated with 21 X-class flares from 2010 to 2017. Our sample includes both confined and eruptive events, with 6 and 15 in each category, respectively. Using the Helioseismic and Magnetic Imager vector magnetic field observations from several hours before to several hours after each event, we employ (a) the Differential Affine Velocity Estimator for Vector Magnetograms to determine the photospheric fluxes of energy and helicity, and (b) nonlinear force-free field extrapolations to estimate the coronal content of energy and helicity in source-region fields. Using superposed epoch analysis, we find, on average, the following: (1) decreases in both magnetic energy and helicity, in both photospheric fluxes and coronal content, that persist for a few hours after eruptions, but no clear changes, notably in relative helicity, for confined events; (2) significant increases in the twist of photospheric fields in eruptive events, with twist uncertainties too large in confined events to constrain twist changes (and lower overall twist in confined events); and (3) on longer timescales (event time +12 hr), replenishment of free magnetic energy and helicity content to near preevent levels for eruptive events. For eruptive events, magnetic helicity and free energy in coronal models clearly decrease after flares, with the amounts of decrease proportional to each region’s pre-flare content.

Helicity evolution during the life cycle of tropical cyclones formed over the north Indian Ocean

Tropical cyclones (TCs) are violent multiscale convections occurring in the atmosphere. The destructive impact of the TCs is commensurate to the helicity associated with their evolution. The evolution of helicity of three TCs viz. Fani, Luban, and Ockhi formed over the north Indian ocean have been analyzed in this study. The analysis of kinetic energy density of primary (EP), secondary (ES) circulation and total helicity has shown that TCs showed helical features when the secondary overturning circulation knotted with primary tangential circulation in a moist convective situation. This condition can be considered a starting of the self-sustained helical feedback process. At this time, the core region became a rotation-dominated region that suppressed strain-dominated surroundings. The Okubo-Weiss parameter demonstrates the similar qualitative behavior of deep convection as total helicity. The local maximas in helicity time series are related with the changes in tropical cyclones' stages (intensification/dissipation). Therefore, consideration of helicity analysis is essential to analyze the TC intensification and dissipation.

Magnetic Helicity Evolution and Eruptive Activity in NOAA Active Region 11158

Coronal mass ejections are among the Sun’s most energetic activity events yet the physical mechanisms that lead to their occurrence are not yet fully understood. They can drive major space weather impacts at Earth, so knowing why and when these ejections will occur is required for accurate space weather forecasts. In this study we use a 4 day time series of a quantity known as the helicity ratio, ∣H J ∣/∣H V ∣ (helicity of the current-carrying part of the active region field to the total relative magnetic helicity within the volume), which has been computed from nonlinear force-free field extrapolations of NOAA active region 11158. We compare the evolution of ∣H J ∣/∣H V ∣ with the activity produced in the corona of the active region and show this ratio can be used to indicate when the active region is prone to eruption. This occurs when ∣H J ∣/∣H V ∣ exceeds a value of 0.1, as suggested by previous studies. We find the helicity ratio variations to be more pronounced during times of strong flux emergence, collision and reconnection between fields of different bipoles, shearing motions, and reconfiguration of the corona through failed and successful eruptions. When flux emergence, collision, and shearing motions have lessened, the changes in helicity ratio are somewhat subtle despite the occurrence of significant eruptive activity during this time.

Tracking of magnetic helicity evolution in the inner heliosphere

Context. Magnetic helicity is one of the invariants in ideal magnetohydrodynamics, and its spectral evolution has a substantial amount of information to reveal the mechanism that are behind turbulence in space and astrophysical plasmas. Aims. The goal of our study is to observationally characterize the magnetic helicity evolution in the inner heliosphere by resolving the helicity transport in a scale-wise fashion in the spectral domain. Methods. The evolution of the magnetic helicity spectrum in the inner heliosphere was tracked using a radial alignment event achieved by Parker Solar Probe at a distance of 0.17 astronomical units (AU) from the Sun and BepiColombo at 0.58 AU with a delay of about 3.5 days. Results. The reduced magnetic helicity resolved in the frequency domain shows three main features: (1) a coherent major peak of a highly helical component at the lowest frequency at about 5 × 10−4 Hz, (2) a damping of helicity oscillation at the intermediate frequencies from 10−3 to 10−2 Hz when observed at 0.58 AU, and (3) a coherent nonhelical component in the ion-kinetic range at frequencies of about 0.1 − 1 Hz. Conclusions. Though limited in the frequency range, the main message from this work is that the solar wind develops into turbulence by convecting large-scale helicity components on the one hand and creating and annihilating helical wave components on the other hand. Excitation of waves can overwrite the helicity profile in the inner heliosphere. By comparing this with the typical helicity spectra at a distance of 1 AU (that is, a randomly oscillating helicity sign in the intermediate frequency range up to about 1 Hz), the helicity evolution reaches a nearly asymptotic state at the Venus orbit (about 0.7 AU) and beyond.

Single-Logarithmic Corrections to Small-$x$ Helicity Evolution

The small-xx quark helicity evolution equations at double-logarithmic order, with the kernel \sim\alpha_s\ln^2(1/x)∼αsln2(1/x), have been derived previously. In this work, we derive the single-logarithmic corrections to the equations, to order \alpha_s\ln(1/x)αsln(1/x) of the evolution kernel. The new equations include the effects of the running coupling and the unpolarized small-xx evolution, both of which are parametrically significant at single-logarithmic order. The large-N_cNc and large-N_c\ N_fNcNf approximations to the equation are computed. (Here, N_cNc and N_fNf are the numbers of quark colors and flavors, respectively.) Their solutions will provide more precise estimates of the quark helicity distribution at small xx, contributing to the resolution of the proton spin puzzle.