Coronal Field Research Articles

A Data-driven Magnetohydrodynamic Simulation of the 2011 February 15 Coronal Mass Ejection from Active Region NOAA 11158

We present a boundary data-driven magnetohydrodynamic (MHD) simulation of the 2011 February 15 coronal mass ejection (CME) event of Active Region NOAA 11158. The simulation is driven at the lower boundary with an electric field derived from the normal magnetic field and the vertical electric current measured from the Solar Dynamics Observatory (SDO) Helioseismic Magnetic Imager vector magnetograms. The simulation shows the buildup of a pre-eruption coronal magnetic field that is close to the nonlinear force-free field extrapolation, and it subsequently develops multiple eruptions. The sheared/twisted field lines of the pre-eruption magnetic field show qualitative agreement with the brightening loops in the SDO Atmospheric Imaging Assembly (AIA) hot passband images. We find that the eruption is initiated by the tether-cutting reconnection in a highly sheared field above the central polarity inversion line and a magnetic flux rope with dipped field lines forms during the eruption. The modeled erupting magnetic field evolves to develop a complex structure containing two distinct flux ropes and produces an outgoing double-shell feature consistent with the Solar Terrestrial Relations Observatory B/Extreme UltraViolet Imager observation of the CME. The foot points of the erupting field lines are found to correspond well with the dimming regions seen in the SDO/AIA observation of the event. These agreements suggest that the derived electric field is a promising way to drive MHD simulations to establish the realistic pre-eruption coronal field based on the observed vertical electric current and model its subsequent dynamic eruption.

Simulating the Formation and Eruption of Flux Rope by Magneto-friction Model Driven by Time-dependent Electric Fields

Aiming to capture the formation and eruption of flux ropes (FRs) in the source active regions (ARs), we simulate the coronal magnetic field evolution of the AR 11429 employing the time-dependent magneto-friction model (TMF). The initial field is driven by electric fields that are derived from time-sequence photospheric vector magnetic field observations by invoking ad hoc assumptions. The simulated magnetic structure evolves from potential to twisted fields over the course of two days, followed by rise motion in the later evolution, depicting the formation of an FR and its slow eruption later. The magnetic configuration resembles an inverse S-sigmoidal structure, composed of the potential field enveloping the inverse J-shaped fields that are sheared past one another and a low-lying twisted field along the major polarity inversion line. To compare with observations, proxy emission maps based on averaged current density along the field lines are generated from the simulated field. These emission maps exhibit a remarkable one-to-one correspondence with the spatial characteristics in coronal extreme ultraviolet images, especially the filament trace supported by the twisted magnetic field in the southwest subregion. Further, the topological analysis of the simulated field reveals the cospatial flare ribbons with the quasi-separatrix layers, which is consistent with the standard flare models; therefore, the extent of the twist and orientation of the erupting FR is indicated to be the real scenario in this case. The TMF model simulates the coronal field evolution, correctly capturing the formation of the FR in the observed timescale and the twisted field generated from these simulations serves as the initial condition for the full MHD simulations.

Jets, Accretion and Spin in Supermassive Black Holes

The theoretical model suggests that relativistic jets of active galactic nuclei (AGNs) rely on the black hole spin and/or accretion. We study the relationship between jet, accretion, and spin using supermassive black hole samples with reliable spin of black holes. Our results are as follows: (1) There is a weak correlation between radio luminosity and the spin of the black hole for our sample, which may imply that the jet of the supermassive black hole in our sample depends on the other physical parameters besides black hole spins, such as accretion disk luminosity. (2) The jet power of a supermassive black hole can be explained by the hybrid model with magnetic field of corona. (3) There is a significant correlation between radio-loudness and black hole spin for our sample. These sources with high radio-loudness tend to have high black hole spins. These results provide observational evidence that the black hole spin may explain the bimodal phenomena of radio-loud and radio-quiet AGNs.

The solar wind heavy ion composition in the ascending phases of the solar cycles 23 and 25

The approximately 11-year solar cycle has been shown to impact the heavy ion composition of the solar wind, even when accounting for streams of differing speeds; however, the heavy ion composition observed between the same specific phases of a past solar cycle and the current cycle has rarely, if ever, been compared. Here, we compare the heavy ion composition of the solar wind, as measured in situ during the solar cycle 23 and 25 ascending phases. We examine the mean iron and oxygen charge state composition and the O7+/O6+ ratio in multiple ranges of associated bulk wind speeds. Then, we compare the iron and oxygen charge state composition and relative abundance of iron to oxygen in the traditionally defined fast and slow solar wind. Finally, to determine the impact of individual ion contributions on the solar wind iron abundance, we examine individual ratios of iron and oxygen ions. Although the charge state composition remained broadly similar between these two ascending phases, both the O7+/O6+ ratio and iron fractionation in fast-speed streams were higher in the solar cycle 25 ascending phase than they were during the solar cycle 23 ascending phase, suggesting that equatorial coronal hole fields more frequently reconnected with helmet streamers or active regions in the latter of the two ascending phases; however, more work will need to be done to connect these observations back to their coronal origins. The individual ion ratios used in this work provided a spectrum to analyze the aggregate elemental abundances, and this work, as a whole, is an important step in determining how conditions in the corona may vary between solar cycles between the same phases.

Total Solar Eclipse White-light Images as a Benchmark for Potential Field Source Surface Coronal Magnetic Field Models: An In-depth Analysis over a Solar Cycle

Potential field source surface (PFSS) models are widely used to simulate coronal magnetic fields. PFSS models use the observed photospheric magnetic field as the inner boundary condition and assume a perfectly radial field beyond a “source surface” (R ss). At present, total solar eclipse (TSE) white-light images are the only data that delineate the coronal magnetic field from the photosphere out to several solar radii (R ⊙). We utilize a complete solar cycle span of these images between 2008 and 2020 as a benchmark to assess the reliability of PFSS models. For a quantitative assessment, we apply the Rolling Hough Transform to the eclipse data and corresponding PFFS models to measure the difference, Δθ, between the data and model magnetic field lines throughout the corona. We find that the average Δθ, 〈Δθ〉, can be minimized for a given choice of R ss depending on the phase within a solar cycle. In particular, R ss ≈ 1.3 R ⊙ is found to be optimal for solar maximum, while R ss ≈ 3 R ⊙ yields a better match at solar minimum. Regardless, large (〈Δθ〉 > 10°) discrepancies between TSE data and PFSS-generated coronal field lines remain regardless of the choice of source surface. However, implementation of solar-cycle-dependent R ss optimal values does yield more reliable PFSS-generated coronal field lines for use in models and for tracing in situ measurements back to their sources at the Sun.

Spectral cleaving in solar type II radio bursts: Observations and interpretation

Context. Shock waves in the solar corona are associated with solar flares and coronal mass ejections (CMEs). Type II solar bursts are radio signatures of shock waves in the solar corona. They are driven by solar flares or CMEs. Despite extensive studies, the intricate spectral patterns observed in type II solar bursts occasionally pose new challenges for the theory of electron acceleration in shocks. Aims. We study a newly identified feature in type II solar bursts called spectral cleaving. This feature is characterized by the actual branching of a type II radio emission lane in radio spectral data. Methods. We analyzed the type II burst exhibiting spectral cleaving in high-fidelity dynamic spectra obtained using the URAN-2 radio telescope (8.25–33 MHz; Poltava region, Ukraine) on 2011 February 14. The high-resolution spectrograms were examined to ascertain its spectral morphology. Results. Our research represents the first recognition of spectral cleaving as a peculiarity of type II bursts that is yet to be classified. This effect occurs due to the shift (or migration) of radio source(s) along a shock front, which in turn is caused by changes in the magnetic field orientation ahead of the propagating shock front. Conclusions. The spectral cleaving observed in solar type II bursts reveals a distinct phenomenon that indicates complex interactions between shock waves and magnetic fields in the solar corona. This discovery enhances our understanding of the mechanisms behind solar radio emissions and emphasizes the need for further observational studies to verify these findings.

Global Effect of New Active Regions on Coronal Holes and Their Wind Streams

Solar wind prediction algorithms and simulations of coronal events often employ photospheric field maps that are assembled over a 27 day solar rotation. This has stimulated efforts to update and better synchronize the maps by applying flux transport and including observations of the back side of the Sun. Here, using potential-field source-surface extrapolations, we address the question of how the emergence of a large active region (AR) on the Sun’s farside affects the coronal field and configuration of coronal holes on the Earth-facing side. We find that, if the new AR is located ∼135°–180° in longitude from Earth, the effect on the coronal field and solar wind near the central meridian will be almost negligible. This is because, when sunspot activity is relatively low, the outermost AR loops will become connected to the nearby polar fields; when sunspot activity is high, the newly emerged flux will connect to neighboring ARs. However, large ARs that emerge near the solar limb may sometimes have a significant effect on the field near the central meridian. In particular, a coronal hole having opposite polarity to that of the nearest sector of the AR may partially close down, resulting in slower wind; conversely, if the coronal hole has the same polarity as the facing AR sector, it will tend to increase in areal size, resulting in faster wind. In most cases, the main effect of a new AR will be to redistribute open flux between itself and neighboring coronal holes (including the polar holes) through interchange reconnection.

On the northward shift of the heliospheric current sheet at the end of solar cycle 24

ABSTRACT Since solar cycle 16, the heliospheric current sheet (HCS) has been found to be shifted southward during the late declining to minimum phase. However, this trend is broken at the end of solar cycle 24. In this paper, we analyse the shift of the HCS by using information obtained from coronal model and in situ data provided by the near-Earth OMNI data base and the Parker Solar Probe (PSP). Coronal potential field source surface modelling results show that the northward shift is established at the beginning of 2018 and remains stable for about 2 yr. Interplanetary magnetic field data obtained from and within 1 au also support the northward shift, as the southern polarity T appears more frequently than the northern polarity A between 2018 and 2020. Both model results and in situ observation obtained by PSP imply that the HCS shift is established in the corona, and then propagates into the heliosphere. The quadrupole term still has a significant influence on the formation of the HCS shift.

Stability of the coronal magnetic field around large confined and eruptive solar flares

Context. The coronal magnetic field, which overlies the current-carrying field of solar active regions, straps the magnetic configuration below. The characteristics of this overlying field are crucial in determining if a flare will be eruptive and accompanied by a coronal mass ejection (CME), or if it will remain confined without a CME. Aims. In order to improve our understanding of the pre-requisites of eruptive solar flares, we study and compare different measures that characterize the eruptive potential of solar active regions – the critical height of the torus instability (TI) as a local measure and the helicity ratio as a global measure – with the structural properties of the underlying magnetic field, namely the altitude of the center of the current-carrying magnetic structure. Methods. Using time series of 3D optimization-based nonlinear force-free magnetic field models of ten different active regions (ARs) around the time of large solar flares, we determined the altitudes of the current-weighted centers of the non-potential model structures. Based on the potential magnetic field, we inspected the decay index, n, in multiple vertical planes oriented alongside or perpendicular to the flare-relevant polarity inversion line, and estimated the critical height (hcrit) of TI using different thresholds of n. The critical heights were interpreted with respect to the altitudes of the current-weighted centers of the associated non-potential structures, as well as the eruptive character of the associated flares, and the eruptive potential of the host AR, as characterized by the helicity ratio. Results. Our most important findings are that (i) hcrit is more segregated in terms of the flare type than the helicity ratio, and (ii) coronal field configurations with a higher eruptive potential (in terms of the helicity ratio) also appear to be more prone to TI. Furthermore, we find no pronounced differences in the altitudes of the non-potential structures prior to confined and eruptive flares. An aspect that requires further investigation is that, generally, the modeled non-potential structures do not really reside in a torus-instable regime, so the applicability of the chosen nonlinear force-free modeling approach when targeting the structural properties of the coronal magnetic field is unclear.

The Sun’s Non-Potential Corona over Solar Cycle 24

The global magnetic field in the solar corona is known to contain free magnetic energy and magnetic helicity above that of a current-free (potential) state. But the strength of this non-potentiality and its evolution over the solar cycle remain uncertain. Here we model the corona over Solar Cycle 24 using a simplified magneto-frictional model that retains the magnetohydrodynamic induction equation but assumes relaxation towards force-free equilibrium, driven by solar surface motions and flux emergence. The model is relatively conservative compared to some others in the literature, with free energy approximately 20 – 25% of the potential field energy. We find that unsigned helicity is about a factor 10 higher at Maximum than Minimum, while free magnetic energy shows an even greater increase. The cycle averages of these two quantities are linearly correlated, extending a result found previously for active regions. Also, we propose a practical measure of eruptivity for these simulations, and show that this increases concurrently with the sunspot number, in accordance with observed coronal mass ejection rates. Whilst shearing by surface motions generates 50%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$50\\%$\\end{document} or more of the free energy and helicity in the corona, we show that active regions must emerge with their own internal helicity otherwise the eruptivity is substantially reduced and follows the wrong pattern over time.

Bi-parametric MRI of the Diaphragm Using Dynamic and Static Images: The Initial Experience.

With recent technological advances, magnetic resonance imaging (MRI) has offered new sequences that can evaluate the real-time motion of anatomic structures. This studyaims to evaluate the interobserver agreement in the diagnosis of diaphragmatic dysfunctions using bi-parametric MRI, in which dynamic sequences for diaphragm movement and static sequences for soft tissue resolution are used together to provide a visualization of the diaphragm. Twenty-nine cases that underwent a bi-parametric magnetic resonance examination which includes coronal T2 single-shot turbo spin echo and the coronal SENSE single-shot balanced turbo field echo real-time sequences were retrospectively evaluated. The images of the patients were assessed by two independent observers. Cohen's kappa coefficient was calculated to evaluate the interobserver agreement. The mean age of the patients was 44.86 ± 17.57, ranging from 18 to 80 years. The kappa value was calculated as 0.889, indicating a strong agreement between the interobservers. Our experience suggests that bi-parametric MRI is a promising tool in the evaluation of diaphragmatic abnormalities.

Exploring self-consistent 2.5D flare simulations with MPI-AMRVAC

Context. Multidimensional solar flare simulations have not yet included a detailed analysis of the lower atmospheric responses, such as downflowing chromospheric compressions and chromospheric evaporation processes. Aims. We present an analysis of multidimensional flare simulations, including an analysis of chromospheric upflows and downflows that provides important groundwork for comparing 1D and multidimensional models. Methods. We followed the evolution of a magnetohydrodynamic standard solar flare model that includes electron beams and in which localized anomalous resistivity initiates magnetic reconnection. We varied the background magnetic field strength to produce simulations that cover a large span of observationally reported solar flare strengths. Chromospheric energy fluxes and energy density maps were used to analyze the transport of energy from the corona to the lower atmosphere, and the resultant evolution of the flare. Quantities traced along 1D field lines allowed for detailed comparisons with 1D evaporation models. Results. The flares produced by varying the background coronal field strength between 20 G and 65 G have GOES classifications between B1.5 and M2.3. All produce a lobster claw reconnection outflow and a fast shock in the tail of this flow with a similar maximum Alfvén Mach number of ∼10. The impact of the reconnection outflow on the lower atmosphere and the heat conduction are the key agents driving the chromospheric evaporation and “downflowing chromospheric compressions”. The peak electron beam heating flux in the lower atmospheres varies between 1.4 × 109 and 4.7 × 1010 erg cm−2 s−1 across the simulations. The downflowing chromospheric compressions have kinetic energy signatures that reach the photosphere, but at subsonic speeds they would not generate sunquakes. The weakest flare generates a relatively dense flare loop system, despite having a negative net mass flux, through the top of the chromosphere, that is to say, more mass is supplied downward than is evaporated upward. The stronger flares all produce positive mass fluxes. Plasmoids form in the current sheets of the stronger flares due to tearing, and in all experiments the loop tops contain turbulent eddies that ring via a magnetic tuning fork process. Conclusions. The presented flares have chromospheric evaporation driven by thermal conduction and the impact and rebound of the reconnection outflow, in contrast to most 1D models where this process is driven by the beam electrons. Several multidimensional phenomena are critical in determining plasma behavior but are not generally considered in 1D flare simulations. They include loop-top turbulence, reconnection outflow jets, heat diffusion, compressive heating from the multidimensional expansion of the flux tubes due to changing pressures, and the interactions of upward and downward flows from the evaporation meeting the material squeezed downward from the loop tops.

How Small-scale Jetlike Solar Events from Miniature Flux Rope Eruptions Might Produce the Solar Wind

We consider small-scale jetlike events that might make the solar wind, as has been suggested in recent studies. We show that the events referred to as “coronal jets” and as “jetlets” both fall on a power-law distribution that also includes large-scale eruptions and spicule-sized features; all of the jetlike events could contribute to the solar wind. Based on imaging and magnetic field data, it is plausible that many or most of these events might form by the same mechanism: Magnetic flux cancelation produces small-scale flux ropes, often containing a cool-material minifilament. This minifilament/flux rope erupts and reconnects with adjacent open coronal field, along which “plasma jets” flow and contribute to the solar wind. The erupting flux ropes can contain twist that is transferred to the open field, and these become Alfvénic pulses that form magnetic switchbacks, providing an intrinsic connection between switchbacks and the production of the solar wind.

Data-driven Simulations of Magnetic Field Evolution in Active Region 11429: Magneto-frictional Method Using PENCIL CODE

Coronal magnetic fields evolve quasi-statically over long timescales and dynamically over short timescales. As of now there exist no regular measurements of coronal magnetic fields, and therefore generating the coronal magnetic field evolution using observations of the magnetic field at the photosphere is a fundamental requirement to understanding the origin of transient phenomena from solar active regions (ARs). Using the magneto-friction (MF) approach, we aim to simulate the coronal field evolution in the solar AR 11429. The MF method is implemented in the open source Pencil Code along with a driver module to drive the initial field with different boundary conditions prescribed from observed vector magnetic fields at the photosphere. In order to work with vector potential and the observations, we prescribe three types of bottom boundary drivers with varying free-magnetic energy. The MF simulation reproduces the magnetic structure, which better matches the sigmoidal morphology exhibited by Atmospheric Imaging Assembly (AIA) images at the pre-eruptive time. We found that the already sheared field further driven by the sheared magnetic field will maintain and further build the highly sheared coronal magnetic configuration, as seen in AR 11429. Data-driven MF simulation is a viable tool to generate the coronal magnetic field evolution, capturing the formation of the twisted flux rope and its eruption.

Analysis of force-free fields in the solar corona at flare sites

Force-free magnetic fields can be used to compute the magnetic structure above active regions where the field strength is large. Using vector magnetograms for two active regions that gave rise to solar flares, the force-free fields are obtained for times before and after the flare. The current density in the upper atmosphere obtained from nonlinear force-free fields is analyzed using Singular Value Decomposition (SVD) methods to determine how it is distributed around the times of flare occurrence. Regions of localized and relatively intense currents are sought at the different scales since these are usually linked to magnetic reconnection. It is found that the probability distribution functions (PDF) for small scales exhibit long tails and they are compared for times before and after the flare. Previous to the flare they seem to be slightly wider but not significant enough to draw a definite conclusion. This is likely due to break down of the applicability of the Force-Free fields after the flare. The effect of having long-tailed PDFs is equivalent to the one found in temperature fluctuations about the flare occurrence (1) which could point to a correlation between electrical currents and temperature. It is also obtained that the energy content in the middle scales of the currents is larger before than after the flare which would point to a magnetic dissipation at those scales during the flare.

Global solar photospheric and coronal magnetic field over activity cycles 21–25

The evolution of the global solar magnetic field from the beginning of cycle 21 (mid-1970s) until the currently-ascending cycle 25 is described using photospheric full-disk and synoptic magnetograms from NSO Kitt Peak Vacuum Telescope (KPVT) 512-channel and Spectromagnetograph (SPMG) and the Synoptic Optical Long-term Investigation of the Sun (SOLIS) Vector Spectro-Magnetograph (VSM) and Global Oscillations Network Group (GONG), and Stanford University’s Wilcox Solar Observatory (WSO). The evolving strength and symmetry of the global coronal field are described by potential-field source-surface models decomposed into axisymmetric and non-axisymmetric, and even- and odd-ordered magnetic multipoles. The overall weakness of the global solar magnetic field since cycle 23 splits the 50-year observing window into the stronger, simpler, more hemispherically symmetric cycles 21 and 22 and the weaker, more complex cycles 23 and 24. An anomalously large decrease in the global solar field strength occurred during cycles 23, and an anomalously weak axial/polar field resulted from that cycle, accompanied by an anomalously weak radial interplanetary magnetic field (IMF) during cycle 23 activity minimum and a weakened radial IMF overall since cycle 23. The general long-term decline in solar field strength and the development during cycle 24 of strong swings of hemispheric and polar asymmetry are analyzed in detail, including their transfer through global coronal structural changes to dominate mean in situ interplanetary field measurements for several years. Although more symmetric than cycle 24, the rise phase of cycle 25 began with the southern leading the northern hemisphere, but the north has recovered to lead this cycle’s polar field reversal. The mean polar flux (poleward of ±60°) has reversed at each pole, so far more symmetrically than the cycle 23 and 24 polar reversals.

High-altitude Spider-type Prominence above the Magnetic Null Point

Rather unique observations of a high-altitude spider-type prominence in 2023 February are presented. The prominence or corresponding filament on the disk was not visible all the time but could appear and disappear in the course of a particular day. However, it persisted during the whole half of a solar rotation, being observable from day to day starting from the east limb of the Sun to the west limb. We show that the prominence was located in sagged coronal field lines just above a coronal magnetic null point. The presence of the null point and magnetic dips above it is confirmed by calculations of the potential magnetic field. The mass of the prominence apparently was appearing due to the condensation of hot coronal plasma after several eruptions that occurred in an active-region complex where the prominence was located. The prominence material flowed down along widely spread large coronal loops as coronal rain and was sometimes swept away by subsequent eruptions.

The accretion history of the Milky Way: III. Hydrodynamical simulations of Galactic dwarf galaxies at first infall

ABSTRACT Most Milky Way dwarf galaxies are much less bound to their host than are relics of Gaia–Sausage–Enceladus and Sgr. These dwarfs are expected to have fallen into the Galactic halo less than 3 Gyr ago, and will therefore have undergone no more than one full orbit. Here, we have performed hydrodynamical simulations of this process, assuming that their progenitors are gas-rich, rotation-supported dwarfs. We follow their transformation through interactions with the hot corona and gravitational field of the galaxy. Our dedicated simulations reproduce the structural properties of three dwarf galaxies: Sculptor, Antlia II, and with somewhat a lower accuracy, Crater II. This includes reproducing their large velocity dispersions, which are caused by ram-pressure stripping and Galactic tidal shocks. Differences between dwarfs can be interpreted as due to different orbital paths, as well as to different initial conditions for their progenitor gas and stellar contents. However, we failed to suppress in a single orbit the rotational support of our Sculptor analogue if it is fully dark matter dominated. In addition, we have found that classical dwarf galaxies like Sculptor may have stellar cores sufficiently dense to survive the pericentre passage through adiabatic contraction. On the contrary, our Antlia II and Crater II analogues are tidally stripped, explaining their large sizes, extremely low surface brightnesses, and velocity dispersion. This modelling explains differences between dwarf galaxies by reproducing them as being at different stages of out-of-equilibrium stellar systems.

Numerical Simulation of Acoustic Wave Generated by DC Corona Discharge Based on the Shock Wave Theory

The audible noise generated by corona discharge has the N-type characteristic at the initial generation stage, and it is a typical shock wave. This shock wave usually only exists around the corona source with a tiny range, making it difficult to obtain its characteristics through experimental measurements. An electrosound-combined simulation of the corona discharge based on the shock wave theory was conducted, and the development process involving the corona discharge, shock wave, and sound wave was simulated. First, the corona was numerically simulated based on the 2D pin–plate axisymmetric hydrodynamic model. It was found that the plasma was mainly distributed near the axis of the corona field where the electric field changed violently, and the maximum value of the electric field appeared at the head of the discharge channel. Then, the plasma energy was equivalent to the explosive energy, and a plasma explosion shock model was established. It was found that the shock wave pressure had obvious positive and negative pressure zones, and the propagation velocity decays to the sound velocity gradually. Finally, the shock wave pressure derived by the explosion model was used as the acoustic source, and the acoustic wave propagation process was simulated. The simulated sound pressure waveform had the same characteristics as the relevant experimental measurement results, proving that the developed method possessed strong applicability and gave rise to a new angle for the simulation of corona-generated audible noise.

Applying the Lagrange multiplier technique to reconstruct a force-free magnetic field

Context. The magnetic field in the solar corona can be reconstructed by extrapolating the data obtained by measuring the magnetic field in the photosphere. It is widely thought that the dynamics of the solar corona is governed by the magnetic field. The magnetic field in the corona can therefore be reconstructed using the physics of force-free fields. Several models have been developed so far to reconstruct the magnetic field, but shortcomings in this regard remain. Therefore, alternative models in this respect can still be proposed. Here we apply the Lagrange multiplier technique to render an optimization model. Aims. The main aim of the present paper is to propose a method of constrained optimization using the vector Lagrange multiplier and compare the results with those of preexisting models. Methods. Our main focus is on the conceptual modification of the optimization models. Since these models are computationally efficient and easy to implement, any possible progressive step would be welcome. The Lagrange multiplier technique is a powerful mathematical tool that has been successfully applied to many areas in physics. It may serve this purpose. In the absence of nonmagnetic forces such as pressure, gravity, and dissipative forces, the coronal medium is dominated by magnetic force. Thus, the function that is considered to be minimized may include a divergence term subject to the constraint force-free term, which yields a solenoidal and force-free magnetic field by an iterative process. Results. The numerical analysis of the proposed model was conducted through an artificial magnetogram and an observational vector magnetogram obtained by SDO/HMI images. The results obtained confirm that (i) the Lagrangian to be minimized in the present model converges slightly faster toward zero, at least for initial iteration steps, (ii) the energy variation during the optimization is compatible with the variations in previous studies, (iii) the numerical results seem to be compatible with a semi-analytical solution as the test case, and (iv) the model is applied to a real magnetogram, and relevant quantities such as magnetic energy content, the current-weighted angle between the current density vector and magnetic field, and the fractional flux errors are computed. Conclusions. The methods and techniques that convert a constrained problem into an unconstrained one are the penalty method, the barrier method, the augmented Lagrangian method, and the Lagrange multiplier technique, for instance. We have employed the Lagrange multiplier technique, by which any constrained condition could be added to the Lagrangian by an appropriate Lagrange multiplier. In our case, the constraint is the force-freeness of the magnetic field and is therefore a special case. The method has the following advantages: (i) The convergence rate is slightly higher for the initial iteration steps, which may help us for time-series events, while several magnetograms must be considered and limited steps of iteration may be needed. (ii) The Lagrangian is introduced based on the Lagrange multiplier technique, which facilitates first fixing a physical compromise such as a divergence-free condition, and subsequently adding any given constraint term. (iii) The quantities obtained by the constrained optimization with the vector Lagrange multiplier model, that is, the relative magnetic energy, the ratio of the magnetic energy to its initial value, the angle between the electric current and the magnetic field, fractional flux errors, the normalized vector error, the vector correlation, and the Cauchy-Schwarz indicator, are comparable with those of the comparison models considered in this article.