Quadrupole Families Research Articles

Radial Profiles of Formation Mass Density, Bulk and Shear Moduli From Borehole Flexural and Stoneley Dispersions.

Exploration wells are liquid-filled boreholes drilled into formations with different geophysical and petrophysical properties. These boreholes support axisymmetric, flexural, and quadrupole family of guided modes that can probe radially varying formation properties at different frequencies. Radially varying formation properties are caused by drilling-induced fractures or near-wellbore stress concentrations. This work describes a novel workflow that inverts borehole flexural and Stoneley dispersions to obtain radially varying formation mass density and shear and bulk moduli away from the borehole surface. An integral equation relates fractional changes in guided mode velocities at different frequencies caused by fractional changes in radially varying mass density and shear and bulk moduli from a radially uniform reference state. A solution of this integral equation is based on extending the Backus-Gilbert (B-G) method for obtaining radial profile of a single to radial profiles of three formation properties away from the borehole surface. Inverted radial profiles from synthetic flexural and Stoneley dispersions have been validated against input formation parameters used to generate synthetic (measured) dispersions.

Parametrization, characterization, and optimization of double-bend achromat cell

We continue the study of single-particle dynamics in a more realistic cell, namely, the double-bend achromat with three families of quadrupoles. The cell is parametrized with the optical parameters including the phase advances and the horizontal beta functions at its entry and center. At zero chromaticity, we find that the landscape of the dynamic aperture in the tune plane can be captured by a simple formula constructed from the effective Hamiltonian of fourth order. Furthermore, the optimal dynamic aperture can be found by simply minimizing the third-order driving terms with equal weight in the effective Hamiltonian.

Development of a tune knob for lattice adjustment in the HLS-II storage ring

The betatron tune is an important parameter in a storage ring to enable stable operation. A tune adjustment tool with a small impact on the beam dynamics is useful for user operation and machine studies. Therefore, a tune knob is developed for the Hefei light source-II (HLS-II) storage ring. Owing to the compactness of the storage ring, a global adjustment mechanism is adopted. To reduce the impact on beam injection, only quadrupole families outside the injection section are used by the tune knob, and the $$\beta$$ functions of the injection section remain unchanged. A code is developed based on the accelerator simulation software, MAD-X, to calculate the adjustment of the quadrupole strengths. The accelerator toolbox is used to double check the accuracy of the tune knob. Online measurement of the tune knob is also performed. The result shows that the tune knob works well when the tune is adjusted in a specific range. Betatron coupling measurement is also carried out, showing an application of the tune knob on machine studies. In this paper, the development of the tune knob and its experimental results in the HLS-II storage ring are reported in detail.

Improving Touschek lifetime in ultralow-emittance lattices through systematic application of successive closed vertical dispersion bumps

In present ultralow-emittance storage ring designs the emittance coupling required for the production of vertically diffraction-limited synchrotron radiation in the hard x-ray regime is achieved and in many cases surpassed by a correction of the orbit and the linear optics alone. However, operating with a vertical emittance lower than required is disadvantageous, since it decreases Touschek lifetime and reduces brightness due to the transverse emittance increase from intrabeam scattering. In this paper we present a scheme consisting of closed vertical dispersion bumps successively excited in each arc of the storage ring by skew quadrupoles that couple horizontal dispersion into the vertical plane to a desired level and thereby raise the vertical emittance in a controlled fashion. A systematic approach to vertical dispersion bumps has been developed that suppresses dispersion and betatron coupling in the straight sections in order to maintain a small projected emittance for insertion devices. In this way, beam lifetime can be significantly increased without negatively impacting insertion device source properties and hence brightness. Using simulation results for the MAX IV 3 GeV storage ring including magnet and alignment imperfections we demonstrate that Touschek lifetime can be increased by more than a factor 2 by adjusting the vertical emittance from 1.3 pm rad (after orbit correction) to 8 pm rad (after application of dispersion bumps) using two to three independent skew quadrupole families all the while ensuring deviations from design optics are restrained to a minimum.

A new model for the (geo)magnetic power spectrum, with application to planetary dynamo radii

We propose two new analytical expressions to fit the Mauersberger–Lowes geomagnetic field spectrum at the core–mantle boundary. These can be used to estimate the radius of the outer liquid core where the geodynamo operates, or more generally the radius of the planetary dynamo regions. We show that two sub-families of the geomagnetic field are independent of spherical harmonics degree n at the core–mantle boundary and exhibit flat spectra. The first is the non-zonal field, i.e., for spherical harmonics order m different from zero. The second is the quadrupole family, i.e., n+m even. The flatness of their spectra is motivated by the nearly axisymmetric time-average paleomagnetic field (for the non-zonal field) and the dominance of rotational effects in core dynamics (for the quadrupole family). We test our two expressions with two approaches using the reference case of the Earth. First we estimate at the seismic core radius the agreement between the actual spectrum and the theoretical one. Second we estimate the magnetic core radius, where the spectrum flattens. We show that both sub-families offer a better agreement with the actual spectrum compared with previously proposed analytical expressions, and predict a magnetic core radius within less than 10 km of the Earth's seismic core radius. These new expressions supersede previous ones to infer the core radius from geomagnetic field information because the low degree terms are not ignored. Our formalism is then applied to infer the radius of the dynamo regions on Jupiter, Saturn, Uranus and Neptune. The axisymmetric nature of the magnetic field of Saturn prevents the use of the non-zonal expression. For the three other planets both expressions converge and offer independent constraints on the internal structure of these planets. These non-zonal and quadrupole family expressions may be implemented to extrapolate the geomagnetic field spectrum beyond observable degrees, or to further regularize magnetic field models constructed from modern or historical observations.

Lava flow recordings of the Blake and post-Blake excursions: refining the Quaternary Geomagnetic Instability Time Scale (GITS)

Complex polarity reversals in numerical dynamos driven by thermo-chemical convection are analyzed in terms of magnetic field intensity variations, transitional field structure, and other observable characteristics. Our most Earth-like dynamos are characterized by long stable polarity chrons with dipole-dominant magnetic fields punctuated by occasional polarity reversals, and are found within a transition region of parameter space between non-reversing strongly dipolar dynamos and chaotic multi-polar-type dynamos. Dynamos in the transition region have elevated dipole terms, reduced quadrupole terms, magnetic energy that decreases slowly with spherical harmonic degree, and broadband dipole frequency spectra. Their axial dipole intensity histograms are trimodal, with large modes representing the stable polarity states and an intermediate mode representing the transitional multi-polar state. The dipole family intensity exceeds the quadrupole family intensity on the core–mantle boundary during stable polarity times, but during transitions the two families are similar. A complex dynamo model reversal is compared with paleomagnetic reconstructions around the Matuyama–Brunhes polarity transition. Both start with a gradual decrease of the dipole intensity, followed by a precursor reversal and transient polarity recovery, then a rapid dipole collapse and a final reversal that initiates with reverse flux generation in one hemisphere. Virtual geomagnetic poles (VGPs) from sites near the reverse flux trace complex paths and cross the equator several thousand years before the simpler VGP paths from more distant sites, and magnetic intensity variations during the dynamo model reversal correlate with intensity variations inferred for the Matuyama–Brunhes transition.

Novel Adjustable Permanent Magnet Quadrupoles for the CLIC Drive Beam Decelerator

The CLIC Drive Beam Decelerator (DBD) requires 41,848 quadrupoles to keep the beam focused along its length. Due to the stringent heat load requirements of the CLIC tunnel, a permanent magnet (PM) design is under investigation in a collaboration between CERN and STFC Daresbury Laboratory. The design incorporates a novel method of adjustment based on moving PMs. To cover the full length of the decelerator, two families of quadrupoles have been designed. This paper outlines the challenges of building an adjustable PM quadrupole, and explains the features of the two designs and how they will meet the demanding specifications of the CLIC DBD.

Complex polarity reversals in a geodynamo model

Complex polarity reversals in numerical dynamos driven by thermo-chemical convection are analyzed in terms of magnetic field intensity variations, transitional field structure, and other observable characteristics. Our most Earth-like dynamos are characterized by long stable polarity chrons with dipole-dominant magnetic fields punctuated by occasional polarity reversals, and are found within a transition region of parameter space between non-reversing strongly dipolar dynamos and chaotic multi-polar-type dynamos. Dynamos in the transition region have elevated dipole terms, reduced quadrupole terms, magnetic energy that decreases slowly with spherical harmonic degree, and broadband dipole frequency spectra. Their axial dipole intensity histograms are trimodal, with large modes representing the stable polarity states and an intermediate mode representing the transitional multi-polar state. The dipole family intensity exceeds the quadrupole family intensity on the core–mantle boundary during stable polarity times, but during transitions the two families are similar. A complex dynamo model reversal is compared with paleomagnetic reconstructions around the Matuyama–Brunhes polarity transition. Both start with a gradual decrease of the dipole intensity, followed by a precursor reversal and transient polarity recovery, then a rapid dipole collapse and a final reversal that initiates with reverse flux generation in one hemisphere. Virtual geomagnetic poles (VGPs) from sites near the reverse flux trace complex paths and cross the equator several thousand years before the simpler VGP paths from more distant sites, and magnetic intensity variations during the dynamo model reversal correlate with intensity variations inferred for the Matuyama–Brunhes transition.

Stable topological modes in two-dimensional Ginzburg-Landau models with trapping potentials

Complex Ginzburg-Landau (CGL) models of laser media (with cubic-quintic nonlinearity) do not contain an effective diffusion term, which makes all vortex solitons unstable in these models. Recently, it has been demonstrated that the addition of a two-dimensional periodic potential, which may be induced by a transverse grating in the laser cavity, to the CGL equation stabilizes compound (four-peak) vortices, but the most fundamental ``crater-shaped'' vortices (CSVs), alias vortex rings, which are essentially squeezed into a single cell of the potential, have not been found before in a stable form. In this work we report on families of stable compact CSVs with vorticity $S=1$ in the CGL model with the external potential of two different types: an axisymmetric parabolic trap and the periodic potential. In both cases, we identify a stability region for the CSVs and for the fundamental solitons ($S=0$). Those CSVs which are unstable in the axisymmetric potential break up into robust dipoles. All the vortices with $S=2$ are unstable, splitting into tripoles. Stability regions for the dipoles and tripoles are identified, too. The periodic potential cannot stabilize CSVs with $S\ensuremath{\geqslant}2$ either; instead, families of stable compact square-shaped quadrupoles are found.

A dynamo cascade interpretation of the geomagnetic dipole decrease

SUMMARY We propose a spectral transfer model for the secular variation of the geomagnetic core field to explain the simultaneous decrease in dipole field intensity and the increase in non-dipole field intensity from 1840 to the present in terms of a dynamo cascade process. The main assumption of this model is that magnetic energy is transferred between adjacent spherical harmonic degrees in the Mauersberger–Lowes spectrum of the geomagnetic field. The key parameters are a set of coefficients that indicate the rate and direction of magnetic energy transfer through the spectrum. Applying the spectral transfer model to the historical period, we find that the quadrupole family of the core field can be characterized by a persistent inverse magnetic energy cascade from higher towards lower spherical harmonics. In the dipole family of the core field, we find cascade behaviour generally from lower to higher spherical harmonics, consistent with axial dipole decrease, but with a high level of time variability that correlates with variations in the dipole family intensity. During time intervals when the dipole family intensity rapidly decreases, energy appears to cascade towards higher spherical harmonics, beyond the limit of the observable part of the core field spectrum. During time intervals when the dipole family intensity is nearly constant, a more limited forward cascade appears to trap energy at intermediate spherical harmonics. Similar fluctuations in the rate and direction of spectral transfer are also seen in the Mauersberger–Lowes spectrum of a numerical dynamo model during a dipole decrease event that led to a polarity excursion. We discuss the possibility of this scenario for the current geomagnetic dipole decrease.

Influence of a pipe tool on borehole modes

A sonic measurements-while-drilling tool consists of acoustic transmitters and receivers mounted on a drill collar that has the form of a thick steel pipe. This mandrel is designed to meet the strength requirements of tough logging conditions encountered in deviated and horizontal drilling. A primary goal of sonic measurement is to estimate formation compressional and shear slownesses that are not affected by the presence of a drill collar. Understanding the basic physics of monopole, flexural, and quadrupole modes in a cylindrically layered structure consisting of a concentric steel pipe in a water-filled borehole can help in estimating formation compressional and shear slownesses from at least one branch, either of the monopole, dipole, or quadrupole family of modes. Results show that coupling between the pipe and formation modes is strongly dependent on formation properties relative to the borehole-fluid compressional slowness. Thepresence of a steel pipe tool in a borehole causes a significant increase in the Stoneley dispersion at low frequencies. However, the Stoneley slownesses at high frequencies remain essentially the same as those in the open hole without a mandrel. The leaky compressional dispersion in slow formations produced by a monopole source also is perturbed at frequencies where the open-hole leaky compressional and Stoneley-wave dispersions are close to each other. Although there is a strong interference between the pipe and formation flexural modes in fast formations, such interference in slow formations occurs at very low frequencies, and the formation flexural dispersion is less affected by the presence of a drill collar at higher frequencies. Interestingly, the presence of a drill collar in a borehole causes rather small changes in the lowest-order quadrupole dispersion from the case of an open hole without a mandrel. Consequently, quadrupole logging is a preferred choice for estimating the formation shear slowness.

Centering of quadrupole family

A procedure for finding the individual centers for a family of quadrupoles fed with a single power supply is described. The method is generalized for using the correctors adjacent to the quadrupoles. Theoretical background is presented as well as experimental data for the NSLS rings. The method accuracy is also discussed.

Study and optimal correction of a systematic skew quadrupole field in the Tevatron

Increasing demands for luminosity in existing and future colliders have made lattice design and error tolerance and correction critical to achieving performance goals. The current state of the Tevatron collider is an example, with a strong skew quadrupole error present in the operational lattice. This work studies the high-order performance of the Tevatron and the strong nonlinear behavior introduced when a significant skew quadrupole error is combined with conventional sextupole correction, a behavior still clearly evident after optimal tuning of available skew quadrupole circuits. An optimization study is performed using different skew quadrupole families, and, importantly, local and global correction of the linear skew terms in maps generated by the code COSY INFINITY [M. Berz, COSY INFINITY version 8.1 user's guide and reference manual, Department of Physics and Astronomy MSUHEP-20704, Michigan State University (2002). URL http://cosy.pa.msu.edu/cosymanu/index.html ]. Two correction schemes with one family locally correcting each arc and eight independent correctors in the straight sections for global correction are proposed and shown to dramatically improve linearity and performance of the baseline Tevatron lattice.

Measurement of global betatron coupling with skew quadrupole modulation

Measurement of the residual betatron coupling with skew quadrupole modulation is a new diagnostics technique. It was developed and tested at the Relativistic Heavy Ion Collider (RHIC) as a promising method for measuring coupling on the ramp. By modulating the strengths of skew quadrupole families, the two tunes' responses are precisely measured with the phase lock loop system. The projection ratio of the residual coupling coefficient onto the coupling modulation direction can be determined. In this article, the analytical solution to the skew quadrupole modulation is given. Dedicated beam studies were carried out in RHIC Run'04 and the results are presented. The ability to measure coupling on the ramp opens the possibility of continuously correcting coupling during acceleration.

Secular variation of the Late Archean–Early Proterozoic geodynamo

We report new paleomagnetic data from ∼2.5 Ga dikes of the Superior Province and derive estimates of paleosecular variation. When combined with data from coeval dikes of the Karelia Craton (Russia) and older (∼2.7 Ga) volcanics of the Pilbara Craton, these estimates suggest a latitudinal variation of secular variation similar to that of the last 5 million years, but with potentially smaller quadrupole family contributions. The observations are consistent with some numerical models of the field that suggest a more dipolar field when the Earth had a much smaller solid inner core.

Dispersion and radial depth of investigation of borehole modes

ABSTRACTSonic techniques in geophysical prospecting involve elastic wave velocity measurements that are performed by placing acoustic transmitters and receivers in a fluid‐filled borehole. The signals recorded at the receivers are processed to obtain compressional‐ and shear‐wave velocities in the surrounding formation. These velocities are generally used in seismic surveys for the time‐to‐depth conversion and other formation parameters, such as porosity and lithology. Depending upon the type of transmitter used (e.g. monopole or dipole) and as a result of eccentering, it is possible to excite axisymmetric (n= 0), flexural (n= 1) and quadrupole (n= 2) families of modes propagating along the borehole. We present a study of various propagating and leaky modes that includes their dispersion and attenuation characteristics caused by radiation into the surrounding formation. A knowledge of propagation characteristics of borehole modes helps in a proper selection of transmitter bandwidth for suppressing unwanted modes that create problems in the inversion for the compressional‐ and shear‐wave velocities from the dispersive arrivals. It also helps in the design of a transmitter for a preferential excitation of a given mode in order to reduce interference with drill‐collar or drilling noise for sonic measurements‐while‐drilling. Computational results for the axisymmetric family of modes in a fast formation with a shear‐wave velocity of 2032 m/s show the existence of Stoneley, pseudo‐Rayleigh and anharmonic cut‐off modes. In a slow formation with a shear‐wave velocity of 508 m/s, we find the existence of the Stoneley mode and the first leaky compressional mode which cuts in at approximately the same normalized frequency ωa/VS= 2.5 (a is the borehole radius) as that of the fast formation. The corresponding modes among the flexural family include the lowest‐order flexural and anharmonic cut‐off modes. For both the fast and slow formations, the first anharmonic mode cuts in at a normalized frequency ωa/VS= 1.5 approximately. Cut‐off frequencies of anharmonic modes are inversely proportional to the borehole radius in the absence of any tool. The borehole quadrupole mode can also be used for estimating formation shear slownesses. The radial depth of investigation with a quadrupole mode is marginally less than that of a flexural mode because of its higher frequency of excitation.

Development and Production of Open-Type Quadrupole Magnets With Advanced Techniques for a 2.5 GeV SRS at CAT

The quadrupole magnets for 2.5 GeV SRS, Indus-2, are characterized by high gradient and stringent magnetic field quality. This requires precise control on magnet geometry during its manufacturing. One type of quadrupole family needs to be an open type to take out SR beam lines. This feature introduces further complexity in the engineering design of the magnets. The magnet cores are made from 1.5 mm thick decarburized steel sheets and their excitation coils are made from hollow OF copper conductors. The core is an assembly of four pole pieces, which are made by laser-cut laminations and consolidated by welding. The pole tip profile was finish machined by wire-EDM. The excitation coils are made using a semi-automatic coil-winding machine. The wound coils are consolidated with vacuum pressure impregnation (VPI) followed by epoxy encapsulation for ground insulation. A number of open type quadrupole magnets have been made successfully and the magnetic characterization has been done. The measurement results meet the stringent beam dynamics specifications. The advanced manufacturing techniques followed for these magnets, their geometrical accuracies and test results are discussed in this paper.

Experimental determination of storage ring optics using orbit response measurements

The measured response matrix giving the change in orbit at beam position monitors (BPMs) with changes in steering magnet excitation can be used to accurately calibrate the linear optics in an electron storage ring [1–8]. A computer code called LOCO (Linear Optics from Closed Orbits) was developed to analyze the NSLS X-Ray Ring measured response matrix to determine: the gradients in all 56 quadrupole magnets; the calibration of the steering magnets and BPMs; the roll of the quadrupoles, steering magnets, and BPMs about the electron beam direction; the longitudinal magnetic centers of the orbit steering magnets; the horizontal dispersion at the orbit steering magnets; and the transverse mis-alignment of the electron orbit in each of the sextupoles. Random orbit measurement error from the BPMs propagated to give only 0.04% rms error in the determination of individual quadrupole gradients and 0.4 mrad rms error in the determination of individual quadrupole rolls. Small variations of a few parts in a thousand in the quadrupole gradients within an individual family were resolved. The optics derived by LOCO gave accurate predictions of the horizontal dispersion, the beta functions, and the horizontal and vertical emittances, and it gave good qualitative agreement with the measured vertical dispersion. The improved understanding of the X-Ray Ring has enabled us to increase the synchrotron radiation brightness. The LOCO code can also be used to find the quadrupole family gradients that best correct for gradient errors in quadrupoles, in sextupoles, and from synchrotron radiation insertion devices. In this way the design periodicity of a storage ring's optics can be restored. An example of periodicity restoration will be presented for the NSLS VUV Ring. LOCO has also produced useful results when applied to the ALS storage ring [8].

Quasi-isochronous storage ring for enhanced FEL performance

Abstract A compact storage ring is designed to be used as driver for a free electron laser (FEL). This ring can be operated very close to zero momentum compaction factor (α) to increase the electron density and thus the gain of the FEL. In order to control α with zero dispersion in the straight sections we use an inverted dipole located between the bending magnets and 4 families of quadrupoles. By using 3 families of sextupoles we can control the 2 transverse chromaticities and 2nd order momentum compaction. We find that the ring has sufficient dynamic aperture for good performance.

On the interpretation of virtual geomagnetic pole (VGP) scatter curves

The present paper aims at assessing the information contained in the so-called virtual geomagnetic pole (VGP) scatter curves which are commonly used in palaeosecular variation studies. As the corresponding curve derived for the last 5 Myr has been compared to a very similar “equivalent” VGP scatter curve derived from the present geomagnetic field (McFadden et al., 1988), we also address the meaning of these “equivalent” curves. We especially investigate the way the two types of curve are related to each other within the statistical formalism first introduced by Constable and Parker (1988). A number of simple and useful (but approximate) properties are derived for that purpose. In particular, a close formal link is established between the two quantities. This allows us to first study the behaviour of the equivalent VGP scatter curve (this curve is shown to be highly time dependent and mainly controlled by the degree and order (1, 1), (2, 1), (3, 1) and (4, 1) terms) and then proceed with some suggestions regarding the interpretation of palaeomagnetic scatter curves. These curves are shown to be (albeit approximately) the quadratic sum of a mean field contribution and of a variance and cross-correlation contribution, which cannot be distinguished from one another unless additional information is being used. In the case of the VGP scatter of the last 5 Myr, available mean field models show that the mean field contribution is very weak. But assuming an isotropic-with-no-correlations statistical model of the field cannot account for the corresponding VGP scatter curve, whatever the spectrum of the non-dipole field. Anisotropy and/or cross-correlations are thus required. We show examples of statistical models incorporating such ingredients. It appears that the separation into dipole and quadrupole (antisymmetric and symmetric) families along the line suggested by McFadden et al. (1988) is not the most appropriate. Rather, it is likely that the pre-eminence of the order 1 terms suggested by the observation of the historical field is responsible for the VGP scatter curve of the past 5 Myr. This study somewhat follows and completes the very recent work of Kono and Tanaka (1995).