ThyroidVol. 26, No. 4 Letters to the Editor“Thyroid Steal”—A Historical Approach to Thyroid Surgery for Graves' DiseasePeter E. Donnelly, Denis E. Winch, and John PollardPeter E. DonnellyDepartment of Endocrinology, Royal Prince Alfred Hospital, Newtown, Australia.Search for more papers by this author, Denis E. WinchSchool of Mathematics and Statistics, University of Sydney, Sydney, Australia.Search for more papers by this author, and John PollardAnaesthetics, North Manchester AHA, Manchester, United Kingdom.Search for more papers by this authorPublished Online:1 Apr 2016https://doi.org/10.1089/thy.2015.0196AboutSectionsView articleView Full TextPDF/EPUB Permissions & CitationsPermissionsDownload CitationsTrack CitationsAdd to favorites Back To Publication ShareShare onFacebookTwitterLinked InRedditEmail View article"“Thyroid Steal”—A Historical Approach to Thyroid Surgery for Graves' Disease." Thyroid, 26(4), p. 611FiguresReferencesRelatedDetails Volume 26Issue 4Apr 2016 InformationCopyright 2016, Mary Ann Liebert, Inc.To cite this article:Peter E. Donnelly, Denis E. Winch, and John Pollard.“Thyroid Steal”—A Historical Approach to Thyroid Surgery for Graves' Disease.Thyroid.Apr 2016.611-611.http://doi.org/10.1089/thy.2015.0196Published in Volume: 26 Issue 4: April 1, 2016Online Ahead of Print:March 16, 2016Online Ahead of Editing: February 24, 2016PDF download

We consider incompressible flows in the rapid-rotation limit of small Rossby number and vanishing Ekman number, in a bounded volume with a rigid impenetrable rotating boundary. Physically the flows are inviscid, almost rigid rotations. We interpret the Coriolis force, modified by a pressure gradient, as a linear operator acting on smooth inviscid incompressible flows in the volume. The eigenfunctions of the Coriolis operator C so defined are the inertial modes (including any Rossby modes) and geostrophic modes of the rotating volume. We show C is a bounded operator and that -iC is symmetric, so that the Coriolis modes of different frequencies are orthogonal. We prove that the space of incompressible polynomial flows of degree N or less in a sphere is invariant under C. The symmetry of -iC thus implies the Coriolis operator is non-defective on the finite-dimensional space of spherical polynomial flows. This enables us to enumerate the Coriolis modes, and to establish their completeness using the Weierstrass polynomial approximation theorem. The fundamental tool, which is required to establish invariance of spherical polynomial flows under C and completeness, is that the solution of the polynomial Poisson-Neumann problem, i.e. Poisson's equation with a Neumann boundary condition and polynomial data, in a sphere is a polynomial. We also enumerate the Coriolis modes in a sphere, with careful consideration of the geostrophic modes, directly from the known analytic solutions.

Nighttime hourly mean values of D, H and Z (or X , Y and Z in a few cases) from 113 observatories for the interval 1964.0 to 1966.0 have been analyzed to determine the semi-annual variation. Results from the 84 observatories with dip latitudes between ±60° have been subjected to spherical harmonic analysis to determine the coefficients of the internal and external parts. Only those coefficients that are found to be significantly different from zero at the 5 per cent level have been included. One of the main objectives is to obtain a reliable estimate, with confidence limits, of the internal/external ratio at a very low frequency for constraining estimates of the deep conductivity of the mantle. It is shown that a model that includes only the principal P 1 0 term can lead to a seriously misleading internal/external ratio.

The paper describes the differences in the magnetic disturbance effect on horizontal geomagnetic field (H) and declination (D) at Alibag (India), Lunping (Taiwan), Chichijima (Japan) and Kakioka (Japan). It is suggested that the trapped particles in the earth’s magnetic field lines are guided by the dipole declination and not by the ground declination and follow the direction towards the geomagnetic pole. The disturbance vector is given by sin (ψ - D) where ψ and D are dipole and ground declinations of the station respectively. The mean disturbance daily variation (SD) of H field is similar at all stations but SD of eastward field (Y) is negative at Alibag but positive at Lunping, Chichijima and Kakioka. Similarly, the daily mean values of H field systematically decrease with increasing ring current index (decreasing value of Dst index) for all stations. But declination values are positively related with Dst index at Alibag and negatively related at Lumping, Chichijima and Kakioka. The SSC amplitude in Y field shows negative values at Alibag but positive values at Kakioka. The storm-time variations Dst(H) show negative excursion during the main phase of the storm at Alibag, Lunping, Chichijima and Kakioka. On the other hand, the Dst variations in declination during the main phase show negative excursions at Alibag but positive excursions at Lumping and Chichijima. This shows a very significant longitudinal inequality in storm-time behavior of eastward geomagnetic field at any place on the earth.

An earlier comprehensive analysis of geomagnetic tides by Winch is revisited to display changes in the current systems with season and longitude. The data used come from the quiet Sun years 1964–1965. We choose to present some total equivalent current systems, being the sum of the external and internal parts derived in the spherical harmonic analysis. The latitudes and local times of the current system foci are generally in agreement with other workers. The amplitudes of the current system vortices follow an annual variation with maximum in summer. The longitude variations of the vortex amplitudes vary as the inverse of the magnetic field strength at both equinoxes but have different variations at the solstices. Other longitude maxima which have been reported in the equatorial electrojet intensity were not found. We examine in detail the “invasions” of the summer current systems across the equator, identifying these as signatures of field‐aligned currents (FACs). The tilting of current contours with respect to the equator is interpreted as being due to midday FACs. As others have found, the identification of afternoon FACs is more difficult. The seasonal swap‐over of FAC directions occurs not in September but in October–November.

We investigate the orthogonality of the potential distributions that are the basis solutions of Laplace's equation appropriate to 3-D ellipsoidal (including spheroidal) coordinate systems, and also the orthogonality of the corresponding vector gradient fields, both over the surface of the ellipsoid, and for integration over the volume of the annular shell between two confocal ellipsoids. The only situation for which there is orthogonality is for the vector gradients when integrated over the annular shell. In the other three cases (potential over surface or annulus, and field over surface) orthogonality can be restored by using an appropriate geometrical weighting factor applied to the integrand; it is therefore still possible to perform the equivalent of a classical spherical harmonic analysis. In the special case of the sphere, there is real orthogonality in all four cases; in effect the weighting factors are all unity. In geodesy, spheroidal harmonic analysis is done using a method that relies on a particular result valid only for potential; it cannot be extended to the corresponding vector field, or to ellipsoidal geometry. The lack of orthogonality over the surface means that care must be taken when interpreting conventional geomagnetic power spectra, and geodetic degree variance, as these no longer correspond exactly to the mean-square values over the actual ellipsoidal surface. We illustrate some of the problems by comparing different versions of the power spectrum for a spheroidal analysis of the global lithospheric magnetic field. We use only simple vector algebra, and do not need to know the details of the actual basis solutions, only that they are the product of three functions, one for each coordinate and involving only that coordinate, and that they satisfy Laplace's equation. Similarly, our results do not depend on the normalization used in the basis functions.

The lithospheric contribution to the geomagnetic field arises from magnetized rocks in a thin shell at the Earth's surface. The lithospheric field can be calculated as an integral of the distribution of magnetization using standard results from potential theory. Inversion of the magnetic field for the magnetization suffers from a fundamental non-uniqueness: many important distributions of magnetization yield no potential magnetic field outside the shell. We represent the vertically integrated magnetization (VIM) in terms of vector spherical harmonics that are new to geomagnetism. These vector functions are orthogonal and complete over the sphere: one subset (I) represents the part of the magnetization that produces a potential field outside the shell, the observed field; another subset (epsilon) produces a potential field exclusively inside the shell; and a third, toroidal, subset (T) produces no potential field at all. epsilon and T together span the null space of the inverse problem for magnetization with perfect, complete data. We apply the theory to a recent global model of VIM, give an efficient algorithm for finding the lithospheric field, and show that our model of magnetization is dominated by epsilon, the part producing a potential field inside the shell. This is largely because, to a first approximation, the model was formed by magnetizing a shell with a substantial uniform component by an potential field originating inside the shell. The null space for inversion of lithospheric magnetic anomaly data for VIM is therefore huge. It can be reduced if the magnetization is assumed to be induced by a known inducing field, but the null space for susceptibility is not so easily recovered.

In the lunar tide-producing potential, the largest terms are the main lunar semi-diurnal constituent M2, the main lunar diurnal constituents OI and K1 and the lunar elliptic semidiurnal constituent N2. Unfortunately, the geomagnetic daily variations associated with K1 are completely obscured by the seasonal modulation of So, and hence K1 is not included in the following discussion. Geomagnetic daily variations associated with N2 will be the subject of another paper. According to Doodson(1921), the explicit forms for the terms M2 and O1 are:

We present a new method in which the magnetic field at satellite altitude is found by solving an inverse problem using our magnetisation estimates as data. This avoids the need for magnetisation estimates on a uniform grid and allows proper estimation of error propagation. A vector spherical harmonic formulation allows proper estimation of the annihilators, those parts of the magnetisation that produce internal and non-potential fields. These yield zero potential field at satellite altitude for perfect data (i.e. perfect and complete magnetisation estimates) but will contaminate the satellite field when the magnetisation estimates are inaccurate and incomplete.

Spherical cap harmonic analysis (SCHA) has been applied to geomagnetic data from an array of magnetometers deployed across the Australian mainland during 1989–90 in order to examine features of the ionospheric S q current system. The external contours, corresponding to ionospheric currents, are generally in good agreement with the data in that the direction of current flow is perpendicular to the measured horizontal magnetic field. The derived internal (induced) current systems are less reliable but exhibit fairly consistent behaviour on some occasions. As the S q current system whorl passes over Australia, the reversed internal current whorl lags by a bit less than an hour. When the external system moves off the continent into the Indian Ocean, the derived internal system sometimes appears to remain over the continent for about two hours but this is found to be caused by the large coastal effect in the west. The variations obtained from the analysis at particular sites are compared with the original data and give generally good agreement. Internal and external variations at several sites are derived and behave as expected to some extent. In particular we find that the internal component of the vertical element adds to the external component on the west coast but subtracts from it on the east coast. Further analyses are performed using data from the CM4 model. The SCHA gives a separation of internal and external components in good agreement with the original for the horizontal CM4 magnetic fields while the agreement is not so good for the vertical component.

Regular magnetic daily quiet time (Sq) variations in total intensity of about 30 nT amplitude are determined in Universal Time (UT) from satellite magnetic field measurements. The CHAMP satellite traverses all hours of local time in 132 days and the Sq variations in total intensity are therefore calculated as an average over the 132 days for each hour of UT. Results are compared with the Sq daily variations in total intensity for the region above the ionosphere calculated from Malin's (1973) spherical harmonic analysis of the Sq Fourier coefficients for hourly mean value magnetic data from a global distribution of ground-based magnetic observatories. From the reasonable agreement between the two calculations, we conclude that low-Earth orbit satellites that traverse all hours of local time can determine Sq variations in total intensity above the ionosphere.

Spherical harmonic analysis of the main magnetic field of the Earth and its daily variations is the numerical determination of coefficients of solid spherical harmonics in the mathematical expressions used for the magnetic scalar potential of fields of internal and external origin. The coefficients are determined from vector components of the field and their purpose is to represent the vector field, not to reconstruct the magnetic scalar potential. An alternative interpretation of the spherical harmonic analysis is presented: namely the determination of the coefficients of a series representation of the magnetic vector field on a spherical surface in orthonormal real vector spherical harmonics, which correspond to the internal and external fields, and an additional non-potential toroidal field. The numerical values of the coefficients of an orthonormal vector spherical harmonic series have a direct physical significance, which is not obscured by some arbitrary normalization of the vector spherical harmonics. Therefore, we propose a Schmidt vector normalization to be used in conjunction with the Schmidt quasi-normalization of associated Legendre functions. A property of orthonormalized functions is that the standard deviations of the coefficients determined by the method of least squares from ideal data, which are uniformly accurate and uniformly globally distributed, are constant for all coefficients. The real vector spherical harmonic analysis of the geomagnetic field is extended to a spherical shell and conditions that restrict the radial dependence of the vector spherical harmonic coefficients are examined. In particular, two hypotheses for the current systems deriving from the non-potential toroidal component of the magnetic field over the surface of a sphere are presented, namely, Earth-air currents and field-aligned currents.

The records of an array of magnetometers set up across the Australian mainland are examined. In addition to a well-defined current whorl corresponding to the ionospheric Sq current system, another system of eastward flowing currents is often found in the early morning. The system is most easily identified at observatories poleward of the focus of the Sq system, where a morning reversal from eastward to westward currents can be seen. The time of the reversal is usually later, sometimes up to 12h local noon, in June (Southern Winter) than in other seasons. There is some evidence of a similar current system at other longitudes and in the Northern Hemisphere. An important outcome of the study is that it enables identification of which features of a daily variation of the northward magnetic field ΔX relate to an Sq current whorl and which must be attributed to some other current system.

Night-time measurements of the Earth’s undisturbed, quiet magnetic field at satellite altitudes provide models of the main field and its secular variation, and of the crustal magnetic anomaly field. Fields from the regular daily variations of solar and lunar origin are most pronounced in daytime data and can affect aeromagnetic and ground magnetic surveys. Satellite magnetic data in total intensity are now available across all local times, and for the first time it is possible to determine the regular daily variations in the ionosphere-magnetosphere region. We give the daily variation in total intensity at satellite altitudes for six equally spaced Universal Time epochs each covering daytime and night-time, showing clearly the equatorial electrojet and Sq current system above the ionosphere during daytime hours. Also given are crustal magnetic anomaly maps, based on high-degree residuals in main field data, for the world and the Australian region at satellite altitudes. The daily variations and crustal magnetic anomalies at satellite levels are both of the order of 20 nT.

We examine three methods of determining the latitude of the focus of the Sq current system using data from an array of more than 50 magnetometers operating on the Australian mainland from November 1989 to July 1990. The magnetometer array enables the location of the Sq focus with more certainty than usual. Chains of stations within the array are then used to test the various methods of determining the focus latitude. The most accurate method was that which first determined the time when the magnetic eastward component ΔY passes through zero and then used northward magnetic components ΔX at that time, in a least squares linear fit, to find that latitude at which ΔX went to zero. This was found preferable to an alternative method which uses maximum and minimum values of ΔX. The reliability of these methods was examined on days with some magnetic disturbance present and on days when the focus latitude lay outside the chain of stations being used. The third method uses a principal component analysis to determine eigenvector elements for the daily variations of each station in the chain and fits the associated coefficients of the eigenvector elements to a straight line. This method had several shortcomings, especially if the procedure for determining the focus position were to be automated. For all methods the value of the correlation coefficient associated with the linear fit gave a good indication of the reliability of the estimation.

Satellite magnetic data include the slowly varying main field, the static lithospheric or crustal anomaly field, and fields associated with the regular daily variations of solar and lunar origin as well as magnetic disturbances. Satellite magnetic data are now available across a range of local times, and research is now being directed to studies of the daily variations from an ‘above the ionosphere’ point of view. The equatorial electrojet is easily seen in satellite orbits crossing the magnetic (dip) equator at local noon, and results are presented here for the satellite view of daily varying fields of ionospheric and magneto spheric origin.

[1] Previous studies of the longitudinal variation of the local noon electrojet have yielded doubtful results either because of the poor data quality or because the local times of equatorial crossings occurred in the early morning or late afternoon. The recent launch of the Orsted satellite in a near-circular orbit with slow drift in local time of equatorial crossing has provided the opportunity for researchers to study the electrojet more accurately. Most studies remove the main field using a spherical harmonic model and then search the daytime equatorial passes for the distinctive electrojet trough in total intensity. The present study examines the electrojet for two consecutive 6-month periods and consequently two local time ranges. Pure signal processing is used to remove the main field directly. The residuals are binned separately for night and day passes on a 1degrees by 1degrees grid to enhance the signal to noise ratio and are bin centered by a least squares fitted linear model to compensate for the variations in satellite altitude. Thereafter, for each period the compensated night and day binned values are subtracted from each other to produce a difference set. Global plots of the subsequently spatially filtered difference sets reveal an almost constant electrojet 1/e half width of 3, as seen at satellite altitude apart from a region in the western Pacific. There are four maxima in the electrojet amplitude at 0degrees-30degreesE, 90degrees-120degreesE, 180degrees-220degreesE, and 260degrees-290degreesE in each local time range.

The scalar anomaly field determined from available Ørsted data is compared with the upward continued scalar anomaly field derived from Magsat data. Two techniques were used to remove the core field from the Ørsted satellite data. In the first method, monthly spherical harmonic core field models of degree and order 13 derived from scalar and vector data were subtracted, and in the second method, along-track high-pass filtering of scalar data only was used. In both methods, the binned residuals were interpolated to a sphere, and subsequently filtered. Monthly degree and order 13 spherical harmonic core field models were removed from Magsat vector data. The binned Magsat vector residuals were interpolated to a sphere, filtered, and upward continued by high degree spherical harmonic analysis. The corresponding Magsat scalar anomaly field at Ørsted altitude was then determined. For latitudes below 50 degrees, removal of the core field by signal processing techniques from presently available Ørsted data led to a scalar anomaly field in better agreement with that determined from Magsat data, than removal by spherical harmonic analysis.

Differentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to 'spin-weighted' associated Legendre functions, as used in studies of distributions of rotations.