Poloidal Field Research Articles

Measurements of Sunspot Group Tilt Angles Based on SOHO/MDI and SDO/HMI Magnetograms

The tilt angle of sunspot groups plays a crucial role in solar dynamo models for the generation of the poloidal field, yet the statistical properties of the tilt angle are not fully comprehended. This study employs magnetograms from the Solar and Heliospheric Observatory/Michelson Doppler Imager and Solar Dynamics Observatory/Helioseismic and Magnetic Imager to measure the tilt angles of 11,373 sunspot groups over the period from 2008 to 2023. This comprehensive analysis examines the relationship between the tilt angle and latitude of the sunspot groups, as well as the correlation between the tilt angle and solar cycle strength. The methodology involves calculating tilt angles within the ±45° central meridian distance, comparing mean-based and median-based measurements, and applying specific angular separation criteria. The findings reveal that during solar cycle 24, the tilt angles increase by approximately 4° for every 10° increase in latitude, in line with Joy’s law. A significant anticorrelation is observed between the latitude-normalized tilt angle (γ/∣L∣) and solar cycle strength. The research also uncovers a substantial hemispheric asymmetry in tilt angle parameters, with the southern hemisphere (m Joy: 0.23 ± 0.092 ∼ 0.24 ± 0.074, γ = 8.°14 ± 0.°43 ∼ 9.°04 ± 0.°486) consistently showing larger tilt angles than the northern hemisphere (m Joy: 0.47 ± 0.096 ∼ 0.51 ± 0.062, γ = 6.°14 ± 0.°304 ∼ 6.°64 ± 0.°334).

Investigation of the poloidal magnetic flux at the PF-3 plasma focus within the framework of the program of laboratory simulation of astrophysical jets

Astrophysical jets are collimated plasma outflows observed in diverse astrophysical settings covering seven decades of spatial scale and twenty decades of power, which, nevertheless, share many common features. This similarity over wide range of scales indicates a common core of physics underlying this phenomenon, leading to considerable interest in observational, theoretical and numerical studies. Laboratory astrophysics experiments for simulating astrophysical jets are premised on this common core of physics responsible for multi-scale similarity of jets remaining valid down to laboratory spatial scales of millimeters. Jets formed after the disassembly of the non-cylindrical z-pinch formed in a plasma focus installation have recently been subjects of observational studies. They offer an important complementarity to the main lines of investigations in two respects. Firstly, the multi-faceted role of gravity, radiation, nuclear reactions and related astrophysics is eliminated retaining only a rapid implosion of a compact plasma object in a magnetohydrodynamic environment as a common feature. Secondly, observations can be made using techniques of laboratory plasma diagnostics. In this paper, we report preliminary results regarding presence of poloidal magnetic flux associated with the jets lasting long after the pinch disassembly. This is significant in the context of uncertainty regarding the origin of poloidal magnetic field postulated in several MHD models of astrophysical jet phenomena. Evidence indicating presence of a radial component of electric field suggests existence of plasma rotation as well. These results suggest that more refined experiments can provide insights into the astrophysical jetting phenomena not available from observational astronomy techniques.

A simple and fast method to calculate the coil currents and the external poloidal flux and magnetic field for fixed boundary equilibria

A simple and fast method to calculate the coil currents and the external poloidal flux and magnetic field consistent with a given internal magnetohydrodynamic (MHD) equilibrium and coil geometry is presented. The constrained optimization technique employed allows to fix the exact position of the X-points. The external poloidal flux and field are calculated using Green’s functions expressed in terms of elliptic integrals that are calculated using a fast and accurate routine. The calculation can be limited to the region between the plasma and the device wall. An example for a double null configuration is presented.

Startup experiment with the newly installed lower tungsten divertor of KSTAR

Plasma startup experiments after the KSTAR upgrade of a lower divertor from carbon to tungsten have been studied during the KSTAR 2023 plasma campaign. In-vessel eddy currents, especially those induced on the divertor supporting structure in the toroidal direction, are significantly altered by the upgrade of the lower divertor configuration. These changes could affect the poloidal field configuration of the plasma startup phase, which is essential for the reliable breakdown and burn-through of deuterium neutral gas and the stable rise of the plasma current. Here, we present the development process of a new startup scenario that considers these upgrades. The vacuum magnetic field design code was utilized to estimate the changes in the poloidal field due to the new divertor configuration. Modifications to the startup scenario were prepared to compensate for the different effects of eddy currents, and experimental validation and optimization were performed during the initial phase of the KSTAR 2023 campaign. The final version of the startup scenario and plasma startup data are presented, and the reliability of the new startup scenario was also confirmed throughout the remainder of the KSTAR campaign.

The putative center in NGC 1052

Context: Many active galaxies harbor powerful relativistic jets, however, the detailed mechanisms of their formation and acceleration remain poorly understood. Aims: To investigate the area of jet acceleration and collimation with the highest available angular resolution, we study the innermost region of the bipolar jet in the nearby low-ionization nuclear emission-line region (LINER) galaxy NGC\,1052. Methods: We combined observations of NGC\,1052 taken with VLBA, GMVA, and EHT over one week in the spring of 2017. Our study is focused on the size and continuum spectrum of the innermost region containing the central engine and the footpoints of both jets. We employed a synchrotron-self absorption model to fit the continuum radio spectrum and we combined the size measurements from close to the central engine out to $ to study the jet collimation. Results: For the first time, NGC\,1052 was detected with the EHT, providing a size of the central region in-between both jet bases of $43\ perpendicular to the jet axes, corresponding to just around $250\,R_ S $ (Schwarzschild radii). This size estimate supports previous studies of the jets expansion profile which suggest two breaks of the profile at around $3 10^3\,R_ S $ and $1 10^4\,R_ S $ distances to the core. Furthermore, we estimated the magnetic field to be 1.25\,Gauss at a distance of $22\ from the central engine by fitting a synchrotron-self absorption spectrum to the innermost emission feature, which shows a spectral turn-over at $ 130\,$GHz. Assuming a purely poloidal magnetic field, this implies an upper limit on the magnetic field strength at the event horizon of $2.6 10^4\,$Gauss, which is consistent with previous measurements. Conclusions: The complex, low-brightness, double-sided jet structure in NGC\,1052 makes it a challenge to detect the source at millimeter (mm) wavelengths. However, our first EHT observations have demonstrated that detection is possible up to at least 230\,GHz. This study offers a glimpse through the dense surrounding torus and into the innermost central region, where the jets are formed. This has enabled us to finally resolve this region and provide improved constraints on its expansion and magnetic field strength.

Real-time equilibrium reconstruction by multi-task learning neural network based on HL-3 tokamak

Abstract A neural network model, EFITNN, has been developed capable of real-time magnetic equilibrium reconstruction based on HL-3 tokamak magnetic measurement signals. The model processes inputs from 68 channels of magnetic measurement data gathered from 1159 HL-3 experimental discharges, including plasmas current, loop voltage, and the poloidal magnetic fields measured by probes. The outputs of the model feature eight key plasmas parameters, alongside high-resolution ( 129 × 129 ) reconstructions of the toroidal current density J P and poloidal magnetic flux distributions Ψ rz . Moreover, the network’s architecture employs a multi-task learning structure, which enables the sharing of weights and mutual correction among different outputs, and leads to increase the model’s accuracy by up to 32%. The performance of EFITNN demonstrates remarkable consistency with the offline EFIT, achieving average R 2 = 0.941, 0.997 and 0.959 for eight plasmas parameters, Ψ rz and J P , respectively. The model’s robust generalization capabilities are particularly evident in its successful predictions of quasi-snowflake (QSF) divertor configurations and its adept handling of data from shot numbers or plasmas current intervals not previously encountered during training. Compared to numerical methods, EFITNN significantly enhances computational efficiency with average computation time ranging from 0.08 ms to 0.45 ms, indicating its potential utility in real-time isoflux control and plasmas profile management.

Shear-driven magnetic buoyancy in the solar tachocline: Dependence of the mean electromotive force on diffusivity and latitude

Abstract The details of the dynamo process that is responsible for driving the solar magnetic activity cycle are still not fully understood. In particular, whilst differential rotation provides a plausible mechanism for the regeneration of the toroidal (azimuthal) component of the large-scale magnetic field, there is ongoing debate regarding the process that is responsible for regenerating the Sun’s large-scale poloidal field. Our aim is to demonstrate that magnetic buoyancy, in the presence of rotation, is capable of producing the necessary regenerative effect. Building upon our previous work, we carry out numerical simulations of a local Cartesian model of the tachocline, consisting of a rotating, fully compressible, electrically conducting fluid with a forced shear flow. An initially weak, vertical magnetic field is sheared into a strong, horizontal magnetic layer that becomes subject to magnetic buoyancy instability. By increasing the Prandtl number we lessen the back reaction of the Lorentz force onto the shear flow, maintaining stronger shear and a more intense magnetic layer. This in turn leads to a more vigorous instability and a much stronger mean electromotive force, which has the potential to significantly influence the evolution of the mean magnetic field. These results are only weakly dependent upon the inclination of the rotation vector, i.e. the latitude of the local Cartesian model. Although further work is needed to confirm this, these results suggest that magnetic buoyancy in the tachocline is a viable poloidal field regeneration mechanism for the solar dynamo.

MHD simulations of astrophysical and laboratory jets under different magnetic field configurations

This paper presents the results of MHD simulations of astrophysical and laboratory supersonic jets under a superposition of poloidal (Br, Bz) and toroidal (Bϕ) magnetic fields. It is shown that the escaping matter is quickly collimated by the magnetic field. A shock wave of an elongated shape is formed, which moves from the target to the boundary of the chamber, leaving behind a stable flow. A periodic shock wave structure is observed inside the main conical expanding shock wave. It is shown that the toroidal component of the magnetic field remains in the region throughout the entire calculation and plays a role in the collimation of the flow. The poloidal magnetic field decreases in the region of the jet cone, but remains in the simulation region throughout the calculation and also participates in flow collimation. Thus, both components Bz and Bϕ take part in the collimation of the flow by the magnetic field. The width of the jet and the opening angle of the cone Ɵ depend on the magnitude of the magnetic field induction. As the field increases, the jet becomes narrower and the cone angle decreases. Initially, we do not specify the rotation of the jet Ω. However, due to the presence of the Bϕ field, the substance acquires angular velocity and twists along the z axis. The simulation results are in agreement with laboratory jets arising in the experiment at the Neodymium laser installation, and with the previously obtained results of MHD modeling of jet formation separately, in poloidal or toroidal magnetic field.

Optimization of DC Energy Storage in Tokamak Poloidal Coils

Tokamaks are a very promising option to exploit nuclear fusion as a programmable and safe energy source. A very critical issue for the practical use of tokamaks consists of the power flow required to initiate and sustain the fusion process, in particular in the poloidal field coils. This flow can be managed by introducing a DC energy storage based on supercapacitors. Because such storage may be the most expensive and largest part of the poloidal power supply system, an excessive size would cancel its potential advantages. This paper presents innovative strategies to optimize the DC storage in poloidal power supply systems. The proposed solution involves the sharing of the DC storage between different coil circuits. The study is supported by novel analytical formulas and by a circuital model developed for this application. The obtained results show that this method and the related algorithms can noticeably reduce the overall size of the storage and the power exchange with the grid, providing a practical contribution toward the feasibility and the effectiveness of nuclear fusion systems.

Development of a high current density, high temperature superconducting cable for pulsed magnets

Abstract A low AC loss Rare Earth Barium Copper Oxide (REBCO) cable, based on the VIPER cable technology has been developed by Commonwealth Fusion Systems for use in high field, REBCO based tokamaks. The new cable is composed of partitioned and transposed copper ‘petals’ shaped to fit together in a circular pattern with each petal containing a REBCO tape stack and insulated from each other to reduce AC losses. A stainless steel jacket adds mechanical robustness—also serving as a vessel for solder impregnation—while a tube runs through the middle for cooling purposes. Additionally, fiber optic sensors are placed under the tape stacks for quench detection. To qualify this design, a series of experiments were conducted as part of the SPARC tokamak Central Solenoid Model Coil program—to retire the risks associated with full scale, fast ramping, high flux HTS Central Solenoid (CS) and Poloidal Field (PF) coils for tokamak fusion power plants and net energy demonstrators. These risk study and risk reduction experiments include (1) AC loss measurement and model validation in the range of ~5 T/s, (2) an IxB electromagnetic loading of over 850 kN/m at the cable level and up to 300 kN/m at the stack level, (3) a transverse compression resilience of over 350 MPa, (4) manufacturability at tokamak relevant speeds and scales, (5) cable to cable joint performance, (6) fiber optic based quench detection speed, accuracy, and feasibility, and (7) overall winding pack integration and magnet assembly. The result is a cable technology, now referred to as PIT VIPER, with AC losses that measure fifteen times lower (at ~5 T/s) than its predecessor technology; a 2% or lower degradation of critical current (Ic) at high IxB electromagnetic loads; no detectable Ic degradation up to 570 MPa of transverse compression on the cable unit cell; end to end magnet manufacturing, consistently producing Ic values within 7% of the model prediction; cable to cable joint resistances at 20 K on the order of ~15 nΩ; and fast, functional quench detection capabilities that do not involve voltage taps. This cable technology will be tested comprehensively in a Central Solenoid Model Coil to prove its readiness for compact, high field tokamak operation.

Poloidal magnetic field reconstruction by laser-driven ion-beam trace probe in spherical tokamak

The poloidal magnetic field ( plays a critical role in plasma equilibrium, confinement and transport of magnetic confinement devices. Multiple diagnostic methods are needed to complement each other to obtain a more accurate profile. Recently, the laser-driven ion-beam trace probe (LITP) has been proposed as a promising tool for diagnosing and radial electric field ( ) profiles in tokamaks [Yang X Y et al 2014 Rev. Sci. Instrum. 85 11E429]. The spherical tokamak (ST) is a promising compact device with high plasma beta and naturally large elongation. However, when applying LITP to diagnosing in STs, the larger invalidates the linear reconstruction relationship for conventional tokamaks, necessitating the development of a nonlinear reconstruction principle tailored to STs. This novel approach employs an iterative reconstruction method based on Newton’s method to solve the nonlinear equation. Subsequently, a simulation model to reconstruct the profile of STs is developed and the experimental setup of LITP is designed for EXL-50, a middle-sized ST. Simulation results of the reconstruction show that the relative errors of reconstruction are mostly below 5%. Moreover, even with 5 mm measurement error on beam traces or 1 cm flux surface shape error, the average relative error of reconstruction remains below 15%, initially demonstrating the robustness of LITP in diagnosing profiles in STs.

The Role of Meridional Flow in the Generation of Solar/Stellar Magnetic Fields and Cycles

Meridional flow is crucial in generating the solar poloidal magnetic field by facilitating poleward transport of the field from decayed bipolar magnetic regions (BMRs). As the meridional circulation changes with the stellar rotation rate, the properties of stellar magnetic cycles are expected to be influenced by this flow. In this study, we explore the role of meridional flow in generating magnetic fields in the Sun and Sun-like stars using the STABLE (surface flux transport and Babcock–Leighton) dynamo model. We find that a moderate meridional flow increases the polar field by efficiently driving the trailing polarity flux toward the pole, while a strong flow tends to transport both polarities of BMRs poleward, potentially reducing the polar field. Our findings are in perfect agreement with what one can expect from the surface flux transport model. Similarly, the toroidal field initially increases with moderate flow speeds and then decreases beyond a certain value. This trend is due to the competitive effects of shearing and diffusion. Furthermore, our study highlights the impact of meridional flow on the strength and duration of stellar cycles. By including the meridional flow from a mean-field hydrodynamics model in STABLE, we show that the magnetic field strength initially increases with the stellar rotation rate and then declines in rapidly rotating stars, offering an explanation of the observed variation of stellar magnetic field with rotation rate.

Juno Observations of Large‐Scale Azimuthal Fields in Jupiter's Nightside Magnetosphere and Related Radial Currents

AbstractWe combine magnetic data from the first 46 data‐taking periapsides of the polar orbiting Juno spacecraft spanning dawn to dusk via midnight to investigate azimuthal fields and related currents in Jupiter's nightside magnetosphere. Data are binned by perpendicular radial distance ρ from the magnetic axis over 4–32 RJ along empirical poloidal model field lines spanning from tail to middle magnetosphere regions (ionospheric colatitudes θi ∼ 5°–17°), and by local time (LT). The data are well organized by these parameters. On southern tail field lines () the azimuthal field is well represented as the sum of sweepback fields falling as 1/ρ that are near‐independent of θi and LT, and a near‐constant field consistent with ∼3.5 nT pointing sunward. The combination is swept back at dawn/midnight but swept forward at dusk outside ∼5 RJ. Outer magnetosphere (θi ∼ 12°–15.5°) azimuthal fields are instead swept back near‐independent of LT, near‐continuous with the tail field in the dawn sector, but with large shear at the tail interface and across outer magnetosphere field lines in the dusk‐midnight sector. The tail region 1/ρ field is associated with a nightside inward polar axial current ∼5.8 MA located within θi ∼ 5° of the magnetic axis, while the dusk‐midnight field shear measured near the ∼30 RJ study boundary provides an inward current ∼15.4 MA, related to the near‐constant field. Azimuthal fields fall to small values across middle magnetosphere field lines (θi ∼ 15.5°–17°), associated with outward currents ∼21.2 MA per hemisphere near‐independent of LT forming the nightside equatorial current sheet, balancing these inward currents.

Inferring the scrape-off layer heat flux width in a divertor with a low degree of axisymmetry

Plasma facing components (PFCs) in the next generation of tokamak devices will operate in challenging environments, with heat loads predicted to exceed 10 MWm−2. The magnitude of these heat loads is set by the width of the channel, the ‘scrape-off layer’ (SOL), into which heat is exhausted, and can be characterised by an e-folding length scale for the decay of heat flux across the channel. It is expected this channel will narrow as tokamaks move towards reactor relevant conditions. Understanding the processes involved in setting the SOL heat flux width is imperative to be able to predict the heat loads PFCs must handle in future devices. Measurements of the SOL width are performed on the high-field spherical tokamak, ST40, using a newly commissioned infrared thermography system. With its high on-axis toroidal magnetic field (≥1.5 T) ST40 is uniquely positioned to investigate the influence of toroidal field on the heat flux width in spherical tokamaks, whilst also extending measurements of the SOL width in spherical tokamaks to increased poloidal field (≥0.3 T). Due to the divertor on ST40 having a low degree of axisymmetry, it is necessary for a set of radial measurements of the heat flux to be taken across the divertor, made possible using an automated toolchain that fully incorporates its 3D geometry. These radial profiles are combined with the magnetic geometry of the plasma to infer the width of the SOL, with both Eich and double exponential profiles of heat flux observed. A reduction in the heat flux is observed toroidally across part of the divertor, along with increased heat loads observed locally around the edges of the tiles. Future work in characterising the impact of tile misalignment and uncertainties in the reconstructed divertor magnetic geometry is required in order to further understand the observed heat flux patterns, as are additional investigations into the role potentially being played by an inhomogeneous sheath electric field.

Validation of prediction capability of operating space for plasma initiation in MAST-U

Abstract DYON is a plasma initiation modelling code that solves the differential equation system of the full circuit equations (plasma current, active coil currents and eddy currents in full passive structures) and 0D global energy and particle balance equations[1]. In order to test the capability of the full electromagnetic plasma initiation model to predict individual discharges in experiments and thus the operating space in the device, a dedicated experimental database was built in MAST-U by scanning the prefilled gas pressure p0 and the induced loop voltage Vloop . In the experimental operating space of p0 and Vloop the lower and the upper limits of p0 are determined by the plasma breakdown failure and the plasma burn-through failure, respectively. The lower limit of Vloop is determined by the plasma burn-through failure. By directly reading the control room data used in each discharge (i.e. currents in the solenoid, poloidal field coils, and toroidal field coils, p0 , and gas puffing rate), the full electromagnetic DYON consistently predicted the failed breakdown, failed burn-through, and successful plasma initiation discharges in the experimental database, demonstrating its capability to predict the operating space for inductive plasma initiation. The Paschen curve calculated with the effective connection length in MAST-U indicates a much higher p0 required for plasma breakdown than the experimental data, indicating that individual field line evaluation is necessary to calculate the quantitative requirements for Townsend breakdown. The demonstration in this paper shows that the full electromagnetic DYON could be a useful simulation tool to assess the feasibility of inductive plasma initiation and to optimise operating scenarios in future devices. [1] Hyun-Tae Kim 2022 Nuclear Fusion 62 126012

Energization Result of JT-60SA Poloidal Field Coil

Energization Result of JT-60SA Poloidal Field Coil

Investigation of pedestal parameters and divertor heat fluxes in small ELM regimes in DIII-D

Abstract Divertor heat flux and its correlation with pedestal parameters within various small edge localized mode (ELM) regimes, including high beta poloidal, type-II and ELMs with negative triangularity H-modes were investigated in DIII-D. The parallel energy fluences of type-II and high beta poloidal small ELM regimes fall below the linear scaling with pedestal electron pressure for type-I ELMs put forward in Eich et al 2017 (Nucl. Mater. Energy 12 84–90). The negative triangularity of H-mode ELMs follow the Eich scaling for type-I ELMs. The parallel heat flux and total heat loads to the divertor were determined using high-time resolution infrared thermography, while pedestal parameters were obtained through self-consistent kinetic equilibrium reconstructions. Linear regressions for the type-II and high beta poloidal regimes demonstrate that an equivalent 7.5 MA small ELM scenario in ITER would fall below the ~5 MJ m − 2 leading edge melting limit for tungsten (Gunn et al 2017 Nucl. Fusion 57 046025). Utilizing fast thermography, the scrape-off layer power fall-off length for both inter-ELM and intra-ELM was determined and compared to the Eich scaling with poloidal magnetic field in Eich et al (ASDEX Upgrade Team and JET EFDA Contributors 2013 Nucl. Fusion 53 093031). Except for the high beta poloidal scenario, all the small ELM regimes during both inter- and intra-ELM periods had power fall-off lengths ( λ q ) larger then would be expected from the B pol , MP − 1 scaling associated with type-I ELMs, signifying their potential in managing heat loads and offering a solution for core–edge integration.

ITER materials irradiation within the D–T neutron environment at JET: post-irradiation radioactivity analysis following the DTE2 experimental campaign

Abstract This work presents the results following the first irradiation of ITER materials samples in a tokamak D--T plasma environment operating at significant fusion power. The materials exposed to this nuclear environment at the Joint European Torus during the DTE2 experimental campaign that took place in 2021 include representative ITER samples from various components such as poloidal field (PF) coil jacket samples, toroidal field coil radial closure plate steels, EUROFER 97 steel, W and CuCrZr materials from the divertor, Inconel-718, CuCrZr, and 316L stainless steel for blanket modules, as well as vacuum vessel forging samples.

The experimental results discussed include high-resolution gamma spectrometry measurements and analysis conducted with the post-irradiated samples, of which there were 68 in total. These samples were exposed through different experimental campaigns, including deuterium, deuterium--tritium and tritium phases. Supporting the analysis were 25 dosimetry foil-based neutron diagnostics and two 'VERDI' neutron spectrometry diagnostics. A further 12 samples for positron annihilation spectroscopy (PALS) were also irradiated. The irradiation of all these samples took place in a long-term irradiation assembly located near the JET vacuum vessel.

The post-irradiation analysis of the ITER material samples has yielded valuable insights into their material activation levels and radiation fields. Comparative assessments between experimental measurements and comprehensive neutronics simulations have demonstrated a significant level of agreement in this work, while also revealing some discrepancies in specific material instances. The data and interpretation from this work not only serve as a robust experimental foundation for enhancing the precision and predictability of neutronics simulation approaches for ITER and next-step devices but also present some opportunities for the refinement of simulation methodologies. In light of these findings, a series of recommendations have been proposed, aimed at improving confidence in nuclear predictions associated with materials that have been exposed to fusion nuclear environments and advancing understanding in this important domain.

Divertor Tokamak Test facility project: status of design and implementation

Abstract An overview is presented of the progress since 2021 in the construction and scientific programme preparation of the Divertor Tokamak Test (DTT) facility. Licensing for building construction has been granted at the end of 2021. Licensing for Cat. A radiologic source has been also granted in 2022. The construction of the toroidal field magnet system is progressing. The prototype of the 170GHz gyrotron has been produced and it is now under test on the FALCON facility. The design of the vacuum vessel, the poloidal field coils and the civil infrastructures has been completed. The shape of the first DTT divertor has been agreed with EUROfusion to test different plasma and exhaust scenarios: single null, double null, XD divertor and negative triangularity plasmas. A detailed research plan is being elaborated with the involvement of the EUROfusion laboratories.

The mean solar butterfly diagram and poloidal field generation rate at the surface of the Sun

The difference between individual solar cycles in the magnetic butterfly diagram can mostly be ascribed to the stochasticity of the emergence process. We aim to obtain the expectation value of the butterfly diagram from observations of four cycles. This allows us to further determine the generation rate of the surface radial magnetic field. We used data from Wilcox Solar Observatory to generate time-latitude diagrams of the surface radial and toroidal magnetic fields spanning cycles 21 to 24. We symmetrized them across the equator and cycle-averaged them. From the mean butterfly diagram and surface toroidal field, we then inferred the mean poloidal field generation rate at the surface of the Sun. The averaging procedure removes realization noise from individual cycles. The amount of emerging flux required to account for the evolution of the surface radial field is found to match that provided by the observed surface toroidal field and Joy's law. Cycle-averaging butterfly diagrams removes realization noise and artefacts due to imperfect scale separation and corresponds to an ensemble average that can be interpreted in the mean-field framework. The result can then be directly compared to $ dynamo models. The Babcock-Leighton alpha -effect is consistent with observations, a result that can be appreciated only if the observational data are averaged in some way.