Non-axisymmetric Field Research Articles

Interannual Magneto–Coriolis modes and their sensitivity on the magnetic field within the Earth’s core

Linear modes for which the Coriolis acceleration is almost entirely in balance with the Lorentz force are called Magneto–Coriolis (MC) modes. These MC modes are thought to exist in Earth’s liquid outer core and could therefore contribute to the variations observed in Earth’s magnetic field. The background state on which these waves ride is assumed here to be static and defined by a prescribed magnetic field and zero flow. We introduce a new computational tool to efficiently compute solutions to the related eigenvalue problem, and study the effect of a range of both axisymmetric and non-axisymmetric background magnetic fields on the MC modes. We focus on a hierarchy of conditions that sequentially partition the numerous computed modes into those which are: (i) in principle observable, (ii) those which match a proxy for interannual geomagnetic signal over 1999–2023, and (iii) those which align with core-flows based on recent geomagnetic data. We found that the background field plays a crucial role in determining the structure of the modes. In particular, we found no examples of axisymmetric background fields that support modes consistent with recent geomagnetic changes, but that some non-axisymmetric background fields do support geomagnetically consistent modes.

Emergence of lobed wakes during the sedimentation of spheres in viscoelastic fluids

The motion of rigid particles in complex fluids is ubiquitous in natural and industrial processes. The most popular toy model for understanding the physics of such systems is the settling of a solid sphere in a viscoelastic fluid. There is general agreement that an elastic wake develops downstream of the sphere, causing the breakage of fore-and-aft symmetry, while the flow remains axisymmetric, independent of fluid viscoelasticity and flow conditions. Using a continuum mechanics model, we reveal that axisymmetry holds only for weak viscoelastic flows. Beyond a critical value of the settling velocity, steady, non-axisymmetric disturbances develop peripherally of the rear pole of the sphere, giving rise to a four-lobed fingering instability. The transition from axisymmetric to non-axisymmetric flow fields is characterized by a regular bifurcation and depends solely on the interplay between shear and extensional properties of the viscoelastic fluid under different flow regimes. At higher settling velocities, each lobe tip is split into two new lobes, resembling fractal fingering in interfacial flows. For the first time, we capture an elastic fingering instability under steady-state conditions, and provide the missing information for understanding and predicting such instabilities in the response of viscoelastic fluids and soft media.

On the vortex dynamical contribution to the emission of infrasound from tornadoes.

Previous field experiments support the claim that a tornado can radiate discernible infrasound between 0.5 and 10 Hz. The physical mechanisms of tornado sound generation are still not fully understood, although several potential mechanisms have been proposed. In this paper, the theory of vortex sound is applied to the sound radiation from two numerical tornado simulations based on large eddy simulations. There are two different vortex-related mechanisms in distinct frequency regimes. It is found that rotation of a non-axisymmetric vorticity field produces low-frequency infrasound less than 2.0 Hz. High-frequency tornado infrasound can be attributed to more complex vortex dynamics such as vortex merging.

Investigation of intrinsic error fields in MAST-U device

In magnetic fusion devices, undesired non-axisymmetric magnetic field perturbations, typically called error fields (EF), have been observed to have a detrimental effect on plasma stability and confinement and can lead to brute plasma terminations, i.e. plasma disruption events. The main strategies that can be adopted to minimize the effect of EFs on the plasma dynamics consist in a careful alignment of the coils, when assembling the fusion device, and, most commonly, in the use of EF correction coils which counteract the non-axisymmetric fields by prescribing properly designed correction currents. In this work, an assessment of the n=1 EF source in MAST-U is presented. When constructing the MAST-U device, an optimization of poloidal and divertor coil positions has been adopted to reduce the n=1 EF source. This optimization consisted in the application of coil shifts and tilts, of the order of mms and mrads, respectively. To investigate the presence of a residual n=1 EF dedicated studies have been performed during the first MAST-U campaign. The compass scan method was employed to identify the n=1 EF, relying on the detection of the locked mode (LM) onset, which proved to be a challenging task. Therefore, a methodology based on Transfer Functions (TFs) among each coil and the n=1 radial magnetic field has been developed which allows the detection of LM formation. Such method is described here, complemented with the experimental results achieved, which suggest that the intrinsic n=1 EF source on MAST-U is relatively small with respect to MAST. Indeed, the empirical correction currents for n=1 EF minimization are smaller, about 0.2 kA, than the ones used in MAST, 1 kA range. This proves that the optimal coil alignment for n=1 EF minimization has been a successful strategy in MAST-U.

The backreaction of stellar wobbling on accretion discs of massive protostars

black In recent years, it has been demonstrated that massive stars see their infant circumstellar medium shaped into a large irradiated, gravitationally unstable accretion disc during their early formation phase. Such discs constitute the gas reservoir from which nascent high-mass stars gain a substantial fraction of their mass by episodic accretion of dense gaseous circumstellar clumps, simultaneously undergoing accretion-driven bursts and producing close-orbit spectroscopic companions of the young high-mass stellar object. We aim to evaluate the effects of stellar motion caused by the disc non-axisymmetric gravitational field on the disc evolution and its spatial morphology. In particular, we analyse the disc's propensity to gravitational instability and fragmentation and the disc's appearance in synthetic millimetre band images pertinent to the ALMA facility. We employed three-dimensional radiation-hydrodynamical simulations of the surroundings of a young massive star in the non-inertial spherical coordinate system, adopting the highest spatial resolution to date and including the indirect star-disc gravitational potential caused by the asymmetries in the circumstellar disc. The resulting disc configurations were post-processed with the radiation transfer tool RADMC-3D and CASA software to obtain synthetic images of the disc. We confirm that the early evolution of the accretion disc is notably different when stellar wobbling is taken into account. The redistribution of angular momentum in the system makes the disc smaller and rounder, reduces the number of circumstellar gaseous clumps formed via disc gravitational fragmentation, and prevents the ejection of gaseous clumps from the disc. The synthetic predictive images at millimetre wavelengths of the accretion disc that includes stellar wobbling are in better agreement with the observations of the surroundings of massive young stellar objects, namely black AFGL 4176 mm1, G17.64+0.16, and G353.273 than our simulations of numerical hydrodynamics that omit this physical mechanism. Our work confirms that stellar wobbling is an essential ingredient to account for in numerical simulations of accretion discs of massive protostars.

A non-axisymmetric temperature field reconstruction method based on the compressive sensing background oriented Schlieren method

Measurement of the combustion temperature field is an extremely important issue in industrial production. Temperature is one of the key parameters in combustion studies. With the temperature field distribution of the combustion field obtained, heat transfer, heat convection, and heat radiation can be calculated directly and efficiently. Traditional background oriented Schlieren (BOS) is an effective method for non-axisymmetric temperature field measurements, but it requires simultaneous Schlieren imaging at multiple angles for tomographic reconstruction, which will greatly limit its application. In this paper, the compressive sensing algorithm is introduced into the temperature field reconstruction, which establishes the system of equations between the deflection angle and the refractive index gradient. Then, the reconstruction of the non-axisymmetric temperature field is realized by solving the underdetermined system of equations by the method of solving the sparse solution through the compressive sensing. First, light offsets across the non-axisymmetric temperature field are measured by the under-angled BOS system and image processing method. Second, the spatial refractive index field is reconstructed by the compressive sensing BOS method proposed in this paper. Finally, the spatial temperature field is obtained. The experimental results show that by comparing the iRadon reconstruction algorithm and the compressive sensing reconstruction algorithm, the temperature field reconstructed by the compressive sensing under the condition of the under-angled sampling of projection data had a higher accuracy than that reconstructed by the tomographic reconstruction algorithm under the same condition. The average error of the temperature field was reduced from 34.6 to 29.7 K under the same measurement conditions; however, the accuracy is better maintained by using the compressive sensing algorithm under the condition of undersampling projection.

A theory for the emission of infrasound from Tornadoes

Tornadoes have been shown to radiate infrasound to great distances, however convincing fundamental sound mechanisms are still absent. After using vortex sound theory to study sound generated by two numerical tornadoes, we found that there is a significant low-frequency signal between 0.1 Hz and 1 Hz. The sound is closely related to rotation of the non-axisymmetric vorticity field and its frequency depends on the rotational frequency. The non-axisymmetric vorticity field is represented by a Kirchhoff vortex-like flow in baseline tornado model and by multiple-vortex flow in eddy injection tornado model. Interestingly, there also exist high-frequency components in the later model which are hypothesized to originate from vortex merging process. Field detection data of tornado infrasound provides some support for the low-frequency sound.

Solar Tachocline Confinement by the Nonaxisymmetric Modes of a Dynamo Magnetic Field

We recently presented the first 3D numerical simulation of the solar interior for which tachocline confinement was achieved by a dynamo-generated magnetic field. In this follow-up study, we analyze the degree of confinement as the magnetic field strength changes (controlled by varying the magnetic Prandtl number) in a coupled radiative zone (RZ) and convection zone (CZ) system. We broadly find three solution regimes, corresponding to weak, medium, and strong dynamo magnetic field strengths. In the weak-field regime, the large-scale magnetic field is mostly axisymmetric with regular, periodic polarity reversals (reminiscent of the observed solar cycle) but fails to create a confined tachocline. In the strong-field regime, the large-scale field is mostly nonaxisymmetric with irregular, quasi-periodic polarity reversals and creates a confined tachocline. In the medium-field regime, the large-scale field resembles a strong-field dynamo for extended intervals but intermittently weakens to allow temporary epochs of strong differential rotation. In all regimes, the amplitude of poloidal field strength in the RZ is very well explained by skin-depth arguments, wherein the oscillating field that gives rise to the skin depth (in the medium- and strong-field cases) is a nonaxisymmetric field structure at the base of the CZ that rotates with respect to the RZ. These simulations suggest a new picture of solar tachocline confinement by the dynamo, in which nonaxisymmetric, very long-lived (effectively permanent) field structures rotating with respect to the RZ play the primary role, instead of the regularly reversing axisymmetric field associated with the 22 yr cycle.

Design of passive and structural conductors for tokamaks using thin-wall eddy current modeling

A new three-dimensional electromagnetic modeling tool (ThinCurr) has been developed using the existing PSI-Tet finite-element code in support of conducting structure design work for both the SPARC and DIII-D tokamaks. Within this framework a 3D conducting structure model was created for both the SPARC and DIII-D tokamaks in the thin-wall limit. This model includes accurate details of the vacuum vessel and other conducting structural elements with realistic material resistivities. This model was leveraged to support the design of a passive runaway electron mitigation coil (REMC), studying the effect of various design parameters, including coil resistivity, current quench duration, and plasma vertical position, on the effectiveness of the coil. The REMC is a non-axisymmetric coil designed to passively drive large non-axisymmetric fields during the plasma disruption thereby destroying flux surfaces and deconfining RE seed populations. These studies indicate that current designs should apply substantial 3D fields at the plasma surface during future plasma current disruptions as well as highlight the importance of having the REMC conductors away from the machine midplane in order to ensure they are robust to off-normal disruption scenarios.

Studies on EU-DEMO 3D coils requirements and conceptual design for error field correction and plasma control

This work presents the initial studies on the requirements for the design of 3D coils on EU-DEMO. The use of 3D coils in present machines includes two main purposes: the first one is the Error Field Correction (EFC). The EFC aims at minimizing the intrinsic non-axisymmetric fields in the machine. Fulfilling the requirements is evaluated with the minimization of residual error fields, both in vacuum and with plasma response. The second purpose relates to the application of 3D fields to control the plasma. The first of these control functions includes the manipulation of the locked neoclassical tearing modes (NTMs) phase, and their positioning in front of the electron cyclotron antennae, for optimum stabilization. A second function is the application of 3D perturbation of the plasma edge to suppress ELMs. As the fastest required time constant of the applied 3D fields is comparable or longer with respect to that of the vacuum vessel, ex-vessel coils can be considered. Another constraint considered is that the 3D fields applied do not generate 3D power exhaust effects, such as divertor lobes, that jeopardize the protection of the machine divertor/wall. This constriant is met in the finding of this work, in the case considered of the ELMs suppression. The aim of this study is to assess if one set of coils could be designed that fulfills all these purposes/functions. The results show that, by using ITER tolerances for the TF, CS and PF coils, all these functions on EFC (using the overlap criterion assumption, which includes the plasma response), NTM phase control and ELMs suppression can be met with the proposed design of one row of ex vessel coils. An additional activity has started to study if the non-resonant perturbations generated by this set of coils could lead to detrimental effects via neoclassical toroidal viscosity (NTV) torque. Preliminary considerations on the coil technology, are also presented, with no showstopper found in this preliminary phase.

The Chemodynamics of the Stellar Populations in M31 from APOGEE Integrated-light Spectroscopy

We present an analysis of nearly 1000 near-infrared, integrated-light spectra from APOGEE in the inner ∼7 kpc of M31. We utilize full-spectrum fitting with A-LIST simple stellar population spectral templates that represent a population of stars with the same age, [M/H], and [α/M]. With this, we determine the mean kinematics, metallicities, α abundances, and ages of the stellar populations of M31's bar, bulge, and inner disk (∼4–7 kpc). We find a nonaxisymmetric velocity field in M31 resulting from the presence of a bar. The bulge of M31 is less metal-rich (mean [M/H] = dex) than the disk, features minima in metallicity on either side of the bar ([M/H] ∼ −0.2), and is enhanced in α abundance (mean [α/M] = ). The disk of M31 within ∼7 kpc is enhanced in both metallicity ([M/H] = ) and α abundance ([α/M] = ). Both of these structural components are uniformly old at ≃12 Gyr. We find the mean metallicity increases with distance from the center of M31, with the steepest gradient along the disk major axis (0.043 ± 0.021 dex kpc−1). This gradient is the result of changing light contributions from the bulge and disk. The chemodynamics of stellar populations encodes information about a galaxy’s chemical enrichment, star formation history, and merger history, allowing us to discuss new constraints on M31's formation. Our results provide a stepping stone between our understanding of the Milky Way and other external galaxies.

Non-axisymmetric design and flow field analysis of boundary layer ingesting fans

The inlet distortion problem exists in a boundary layer ingesting propulsion system, and the distortion position is almost stable, which seriously affects the fan performance and stability margin (SM). To effectively improve the distortion resistance of the fan, three non-axisymmetric stator (NAS) schemes are designed according to the parameter distribution law of the distorted flow field in front of the stator under the design conditions in this study, which are based on the solidity matching method of the optimization of the stator inlet geometric angle, chord length, and pitch. The impact of the NAS on the fan performance and the distorted flow field parameters under distorted conditions are numerically studied. The results indicate that NAS can effectively enhance the SM of the fan. Among the three NAS designs, the third one demonstrates a remarkable increase of 74.91% in SM compared to the original fan. The NAS design can promote the fluid re-distribution in the distortion zone, make the aerodynamic parameters more uniform, and thus improve the flow state in the distortion zone. Among the three NAS schemes that are adopted in this research, the NAS design based on the pitch optimization method has the best effect. This design can effectively improve the stator flow field structure, thus significantly improving the SM and the aerodynamic performance of the fan under distortion conditions.

A Multi-Spark Electrohydraulic Shock Wave Generator with Adjustable Pressure Field Distribution and Beam Steering Capability.

All clinical shock wave lithotripters produce an axisymmetric acoustic field without accounting for the anatomic features of the kidney or respiratory motion of the patient. This work presents a steerable and adjustable focusing electrohydraulic (SAFE) shock wave generator design with variable beam size and shape. 90 electrohydraulic transducers are mounted concentrically on a spherical basin with adjustable connection to individual transducers. Each transducer consists of 45 3D-printed titanium microelectrodes embedded in epoxy with a tip diameter of 0.3 mm. All the transducers are arranged in 5 concentric rings and sub-divided into 6 sectors. By changing the connections of individual transducers, the focused pressure field produced by the transducer array can be either axisymmetric with a -6 dB focal width of 14.8 mm in diameter, or non-axisymmetric with a long axis of 22.7 mm and a short axis of 15.1 mm. The elongated beam produces a peak positive pressure of 33.7±4.1 MPa and comminution efficiency of 42.2±3.5%, compared to 36.2±0.7 MPa and 28.6±6.1% for axisymmetric beam after 150 pulses at 20 kV. We have demonstrated that the SAFE shock wave generator can produce an elongated non-axisymmetric pressure field with higher stone comminution efficiency. The SAFE shock wave generator may provide a flexible and versatile design to achieve accurate, stable, and safe lithotripsy for kidney stone treatment.

A non-axisymmetric temperature field reconstruction method based on the interferometric fringe schlieren method

The typical non-axisymmetric temperature field distribution in the tail nozzle outlet area of the combustion chamber of an aircraft engine can reflect the operating state of the engine. As an effective non-contact method for measuring a non-axisymmetric temperature field, the traditional background-oriented schlieren (BOS) method has the problem that the extraction accuracy of light deflection is not high. In this paper, interference fringes are introduced into the schlieren measurement, and the highly sensitive fringe offset replaces the traditional background speckle offset for measurement, which realizes accurate reconstruction of the non-axisymmetric temperature field. Firstly, through the interferometric fringe schlieren (IFS) method comprising multiple sets of double-slit interferometric systems and imaging sensors, the offset of the interference fringe after passing through the non-axisymmetric temperature field is measured by image processing. The spatial refractive index field is then reconstructed by the filtered back-projection algorithm under the sector model proposed in this paper and the spatial temperature field is obtained. The experimental results show that the temperature field reconstruction error is smaller for the IFS method than the BOS method, the measurement area is accurate from 0.5 mm for the BOS method to 0.2 mm for the IFS method and the temperature error range is reduced from 50 K for the BOS method to 25 K for the IFS method.

Three-dimensional rapid visualization of flame temperature field via compression and noise reduction of light field imaging

Light field (LF) imaging, as an emerging temperature measurement technique, can simultaneously capture the intensity and direction information of radiation in the combustion flow field and has recently attracted extensive attention. However, the dense sampling of the LF introduces a large amount of redundant information and makes the plenoptic data susceptible to noise pollution, which can significantly affect the performance of 3D temperature field reconstruction. To address this problem, a novel approach, i.e., light field compression and noise reduction (LFCNR), is proposed to extract the main features of the projection matrix and separate the noise-related and signal-related subspaces of the plenoptic data, thus improving the reconstruction accuracy and efficiency. The performance of the proposed LFCNR method and the effect of the hyper-parameters on reconstructed quality are investigated thoroughly. It has been observed that the proposed LFCNR-based method for 3D temperature tomography imaging has higher accuracy, more efficiency, and better anti-noise ability compared to the well-established method. LFCNR can improve the reconstructed temperature accuracy by approximately 20%, and the calculation time is shortened to 10%. Priori smoothing is introduced into the optimization objective to constrain the fluctuation of the retrieval temperature, called LFCNR-PS, which can further improve the reconstructed accuracy of axisymmetric and non-axisymmetric fields about 1.79% and 1.23%. Furthermore, experiments of the LFCNR-based method are carried out to reconstruct the 3D temperature field of ethylene diffusion flames. The reconstructed temperature slices demonstrate that the flame topology can be clearly and rapidly identified at different depths. The efficiency and robustness of the proposed method can be applied to the harsh environment, which will be beneficial to the reliable operation of thermal equipment, the study of turbulent chemical reaction mechanisms, fuel utilization, and the optimization design of the engine combustion chamber.

Origin of eclipsing time variations: Contributions of different modes of the dynamo-generated magnetic field

Context. The possibility to detect circumbinary planets and to study stellar magnetic fields through eclipsing time variations (ETVs) in binary stars has sparked an increase of interest in this area of research. Aims. We revisit the connection between stellar magnetic fields and the gravitational quadrupole moment Qxx and compare different dynamo-generated ETV models with our simulations. Methods. We present magnetohydrodynamical simulations of solar mass stars with rotation periods of 8.3, 1.2, and 0.8 days and perform a detailed analysis of the magnetic and quadrupole moment using spherical harmonic decomposition. Results. The extrema of Qxx are associated with changes in the magnetic field structure. This is evident in the simulation with a rotation period of 1.2 days. Its magnetic field has a more complex behavior than in the other models, as the large-scale nonaxisymmetric field dominates throughout the simulation and the axisymmetric component is predominantly hemispheric. This triggers variations in the density field that follow the magnetic field asymmetry with respect to the equator, affecting the zz component of the inertia tensor, and thus modulating Qxx. The magnetic fields of the two other runs are less variable in time and more symmetric with respect to the equator, such that the variations in the density are weaker, and therefore only small variations in Qxx are seen. Conclusions. If interpreted via the classical Applegate mechanism (tidal locking), the quadrupole moment variations obtained in the current simulations are about two orders of magnitude below those deduced from observations of post-common-envelope binaries. However, if no tidal locking is assumed, our results are compatible with the observed ETVs.

Suppression of non-axisymmetric field-induced α-particle loss channels in a quasi-axisymmetric stellarator

In future fusion reactors, the confinement of α-particles is a crucial issue. The perfect omnigenity may be difficult to achieve in the quasi-isodynamic and quasi-symmetric stellarators when a multi-objective optimization is considered. Non-axisymmetric field can result in collisionless particles’ transport via localized trapping by ripples. Specific loss channels have been revealed to essentially exist in quasi-axisymmetric stellarators [Yang et al., Europhys. Lett. 129, 35001 (2020)] and W7-X [J. M. Faustin et al., Nucl. Fusion 56, 092006 (2016)]. It indicates a drastic loss of collisionless ions through these channels. This paper is devoted to investigate the effects of axisymmetry-breaking magnetic fields on collisionless α-particle transport in the CFQS (Chinese First Quasi-axisymmetric Stellarator) -like reactor configuration. A semi-analytic representation of radial and poloidal drifts in Boozer coordinates is given, by which we found an effective route to mitigate α-particle losses, i.e., adjusting the location of the quasi-axisymmetric radial position. Such a route enables the enhancement of the poloidal drift and decrease of radial drift in peripheral regions of the identified loss channels. The particles launched inside the quasi-axisymmetric radial surface can be well confined because localized particles that may fall in loss channels can transit into blocked particles near the quasi-axisymmetric surface, escaping from loss channels, which is beneficial for the improvement of the particle confinement. Moreover, this paper may provide a set of proxy functions for suppression of energetic particle losses to optimize stellarator configurations.

Near-infrared measurement of temperature fields formed by mixed convection from a small heating sphere in water

This study proposed a method for reconstructing three-dimensional (3D) non-axisymmetric temperature fields in water formed by mixed convection surrounding a small heating sphere. The proposed method is based on the temperature dependence of the absorption coefficient of water at a wavelength of 1150 nm in the near-infrared (NIR) region. The absorbance images of a 10-mm-thick water cell containing a 1-mm diameter steel sphere heated via electromagnetic induction were acquired using a two-orthogonal-direction NIR imaging system. The 3D reconstruction was implemented by applying a non-axisymmetric inverse Abel transform to the absorbance images. In addition, particle tracking velocimetry (PTV) was applied to the same images to visualize and quantify the flow field. The results indicated that the temperature distribution within the plume was distorted compared to an axisymmetric one, and the plume shape, length, and angle were dependent on the forced flow and heating power. Further, employing a thermal balance model with temperature distributions, the heat production rates were determined and discussed based on the plume characteristics. The PTV results revealed a mixed convection field inconsistent with the plume shape, which was also verified by comparison with numerical simulations.

Nonaxisymmetry and flow transition in evaporating water drops

Understanding the internal flow in evaporating sessile drops is of paramount importance in a myriad of applications such as ink-jet printing, surface patterning, and medical diagnostics. Marangoni flow driven by a gradient in surface tension is an essential internal flow mechanism, whose characteristics in evaporating water drops remain elusive in the literature. Here, by employing infrared thermography and particle image velocimetry, we show that the manifestation of Marangoni flow as a convective cell at the liquid–vapor interface results in a nonaxisymmetric internal flow field. Eventually, during evaporation, the flow transitions to a buoyancy-dominated regime, where an axisymmetric toroidal flow is observed. This transition marks a reversal in the flow along with an order of magnitude decrease in velocity. We corroborate this experimentally observed transition using previously reported analytical and scaling frameworks. Finally, we present hitherto unreported features correlating the three aspects of evaporating water drops, viz., contact line dynamics, thermal field, and internal flow field, which are generally investigated independently.

Controlling the size of non-axisymmetric magnetic footprints using resonant magnetic perturbations

The structure of the non-axisymmetric heat load distribution at the divertor plates is determined not only by the toroidal but also from the poloidal spectrum of non-axisymmetric field perturbations. Whether they are intrinsic, like error fields, or they are applied through 3D coils, the non-axisymmetric fields produce complex 3D edge magnetic topologies (footprints) that alter the properties of the heat and particle flux distributions on the divertor target plates. In this manuscript, a study of the impact of applied 3D field poloidal spectrum on the footprint size and structure is done for the DIII-D tokamak using the resistive MHD code M3D-C1 coupled with the field line tracing code TRIP3D. To resolve the impact of the poloidal spectrum of the magnetic perturbation, the relative phase of the two rows of in-vessel 3D coils used to produce both a n = 2 and a n = 3 perturbation is varied, where n is the toroidal harmonic of the magnetic perturbation. This shows that the largest footprint is predicted when the relative phase of the two rows is close to zero, which is also where the resonant coupling with the plasma is maximized. These results suggest that it will be challenging to decouple the footprint size from the requisite resonant coupling for RMP–ELM control. The correlation between the measured heat load and particle flux distributions at the outer divertor plates in DIII-D and the magnetic measurements is in good agreement with the predicted dependence of the magnetic footprint size on the amplitude of the resonant component of the plasma response.