scholarly journals Determination of the instantaneous geostrophic flow within the three-dimensional magnetostrophic regime

2018 ◽  
Vol 474 (2218) ◽  
pp. 20180412 ◽  
Author(s):  
Colin M. Hardy ◽  
Philip W. Livermore ◽  
Jitse Niesen ◽  
Jiawen Luo ◽  
Kuan Li
Keyword(s):  
Rotation Axis ◽  
Logarithmic Singularity ◽  
Three Dimensional ◽  
Outer Core ◽  
Necessary Condition ◽  
Small Subset ◽  
Dynamo Action ◽  
Geostrophic Flow ◽  
Geostrophic Flows

In his seminal work, Taylor (1963 Proc. R. Soc. Lond. A 274 , 274–283. ( doi:10.1098/rspa.1963.0130 ).) argued that the geophysically relevant limit for dynamo action within the outer core is one of negligibly small inertia and viscosity in the magnetohydrodynamic equations. Within this limit, he showed the existence of a necessary condition, now well known as Taylor's constraint, which requires that the cylindrically averaged Lorentz torque must everywhere vanish; magnetic fields that satisfy this condition are termed Taylor states. Taylor further showed that the requirement of this constraint being continuously satisfied through time prescribes the evolution of the geostrophic flow, the cylindrically averaged azimuthal flow. We show that Taylor's original prescription for the geostrophic flow, as satisfying a given second-order ordinary differential equation, is only valid for a small subset of Taylor states. An incomplete treatment of the boundary conditions renders his equation generally incorrect. Here, by taking proper account of the boundaries, we describe a generalization of Taylor's method that enables correct evaluation of the instantaneous geostrophic flow for any three-dimensional Taylor state. We present the first full-sphere examples of geostrophic flows driven by non-axisymmetric Taylor states. Although in axisymmetry the geostrophic flow admits a mild logarithmic singularity on the rotation axis, in the fully three-dimensional case we show that this is absent and indeed the geostrophic flow appears to be everywhere regular.

The structure of Taylor's constraint in three dimensions

2008 ◽  
Vol 464 (2100) ◽  
pp. 3149-3174 ◽  
Author(s):  
Philip W Livermore ◽  
Glenn Ierley ◽  
Andrew Jackson
Keyword(s):  
Lorentz Force ◽  
Inner Core ◽  
Rotation Axis ◽  
Analytic Form ◽  
Spectral Expansion ◽  
Three Dimensions ◽  
Necessary Condition ◽  
Dynamo Action ◽  
Azimuthal Component ◽  
Electrically Conducting

In a 1963 edition of Proc. R. Soc. A , J. B. Taylor (Taylor 1963 Proc. R. Soc. A 9 , 274–283) proved a necessary condition for dynamo action in a rapidly rotating electrically conducting fluid in which viscosity and inertia are negligible. He demonstrated that the azimuthal component of the Lorentz force must have zero average over any geostrophic contour (i.e. a fluid cylinder coaxial with the rotation axis). The resulting dynamical balance, termed a Taylor state, is believed to hold in the Earth's core, hence placing constraints on the class of permissible fields in the geodynamo. Such states have proven difficult to realize, apart from highly restricted examples. In particular, it has not yet been shown how to enforce the Taylor condition exactly in a general way, seeming to require an infinite number of constraints. In this work, we derive the analytic form for the averaged azimuthal component of the Lorentz force in three dimensions after expanding the magnetic field in a truncated spherical harmonic basis chosen to be regular at the origin. As the result is proportional to a polynomial of modest degree (simply related to the order of the spectral expansion), it can be made to vanish identically on every geostrophic contour by simply equating each of its coefficients to zero. We extend the discussion to allow for the presence of an inner core, which partitions the geostrophic contours into three distinct regions.

scholarly journals Vector Products As A Tool For The Investigation Of The Isometric Transformation Of Objects

2015 ◽  
Vol 98 (1) ◽  
pp. 60-71
Author(s):  
Ryszard Józef Grabowski
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Necessary Condition ◽  
Vector Product ◽  
Isometric Transformation ◽  
Check Points ◽  
Normal Vectors ◽  
Definition Of ◽  
Three Dimensional Space

Abstract The identification of isometric displacements of studied objects with utilization of the vector product is the aim of the analysis conducted in this paper. Isometric transformations involve translation and rotation. The behaviour of distances between check points on the object in the first and second measurements is a necessary condition for the determination of such displacements. For every three check points about the measured coordinate, one can determine the vector orthogonal to the two neighbouring sides of the triangle that are treated as vectors, using the definition of the vector product in three-dimensional space. If vectors for these points in the first and second measurements are parallel to the studied object has not changed its position or experienced translation. If the termini of vectors formed from vector products treated as the vectors are orthogonal to certain axis, then the object has experienced rotation. The determination of planes symmetric to these vectors allows the axis of rotation of the object and the angle of rotation to be found. The changes of the value of the angle between the normal vectors obtained from the first and second measurements, by exclusion of the isometric transformation, are connected to the size of the changes of the coordinates of check points, that is, deformation of the object. This paper focuses mainly on the description of the procedure for determining the translation and rotation. The main attention was paid to the rotation, due to the new and unusual way in which it is determined. Mean errors of the determined parameters are often treated briefly, and this subject requires separate consideration.

A new Legendre-type polynomial and its application to geostrophic flow in rotating fluid spheres

2010 ◽  
Vol 466 (2120) ◽  
pp. 2203-2217 ◽  
Author(s):  
Xinhao Liao ◽  
Keke Zhang
Keyword(s):  
Asymptotic Analysis ◽  
Rotation Axis ◽  
Spherical Geometry ◽  
Polar Coordinates ◽  
Rotating Fluid ◽  
Geostrophic Flow ◽  
Spherical Boundary ◽  
Essential Properties ◽  
Geostrophic Flows ◽  
Fluid Spheres

In rapidly rotating spheres, the whole fluid column, extending from the southern to northern spherical boundary along the rotation axis, moves like a single fluid element, which is usually referred to as geostrophic flow. A new Legendre-type polynomial is discovered in undertaking the asymptotic analysis of geostrophic flow in spherical geometry. Three essential properties characterize the new polynomial: (i) it is a function of r and θ but takes a single argument , which is restricted by 0≤ r ≤1 and 0≤ θ ≤ π , where ( r , θ , ϕ ) denote spherical polar coordinates with θ =0 at the rotation axis; (ii) it is odd and vanishes at the axis of rotation θ =0, and (iii) it is defined within—and orthogonal over—the full sphere. As an example of its application, we employ the new polynomial in the asymptotic analysis of forced geostrophic flows in rotating fluid spheres for small Ekman and Rossby numbers. Fully numerical analysis of the same problem is also carried out, showing satisfactory agreement between the asymptotic solution and the numerical solution.

scholarly journals Synthesis, crystal structure determination of a novel phosphate Ag1.64Zn1.64Fe1.36(PO4)3 with an alluaudite-like structure

2020 ◽  
Vol 76 (9) ◽  
pp. 1491-1495
Author(s):  
Jamal Khmiyas ◽  
Abderrazzak Assani ◽  
Mohamed Saadi ◽  
Lahcen El Ammari
Keyword(s):  
Structural Model ◽  
Rotation Axis ◽  
Three Dimensional ◽  
Iron Phosphate ◽  
X Ray Diffraction ◽  
Conventional Solid State Reaction ◽  
Electrical Neutrality ◽  
Edge Sharing Octahedra ◽  
Bond Valence Sum

Single crystals of Ag1.64Zn1.64Fe1.36(PO4)3 [silver zinc iron phosphate (1.64/1.64/1.36/3)] have been synthesized by a conventional solid-state reaction and structurally characterized by single-crystal X-ray diffraction. The title compound crystallizes with an alluaudite-like structure. All atoms of the structure are in general positions except for four, which reside on special positions of the space group, C2/c. The Ag+ cations reside at full occupancy on inversion centre sites and at partial occupancy (64%) on a twofold rotation axis. In this structure, the unique Fe3+ ion with one of the two Zn2+ cations are substitutionally disordered on the same general position (Wyckoff site 8f), with a respective ratio of 0.68/0.32 (occupancies were fixed so as to ensure electrical neutrality for the whole structure). The remaining O and P atoms are located in general positions. The three-dimensional framework of this structure consists of kinked chains of edge-sharing octahedra stacked parallel to [10\overline{1}]. These chains are built up by a succession of [MO6] (M = Zn/Fe or Zn) units. Adjacent chains are connected by the PO4 anions, forming sheets oriented perpendicular to [010]. These interconnected sheets generate two types of channels parallel to the c axis, in which the Ag+ cations are located. The validity and adequacy of the proposed structural model of Ag1.64Zn1.64Fe1.36(PO4)3 was established by means of bond-valence-sum (BVS) and charge-distribution (CHARDI) analysis tools.

Structure Determination of Tilt Boundaries in Gold by High Resolution Electron Microscopy

1986 ◽  
Vol 82 ◽  
Author(s):  
William Krakow ◽  
David A. Smith
Keyword(s):  
High Resolution ◽  
Structure Determination ◽  
Rotation Axis ◽  
Three Dimensional ◽  
High Resolution Electron Microscopy ◽  
Transmission Microscopy ◽  
Resolution Electron ◽  
Tilt Boundaries ◽  
Growth Twin

ABSTRACTA number of tilt grain boundaries prepared from evaporated gold thin films have been investigated by high resolution transmission microscopy. When the grain boundary is parallel to the electron beam and the beam is parallel to a low index rotation axis such as [110] or [001]it is possible to identify atomic positions at the cores of these boundaries as demonstrated here by a Σ = 3 70.5°/[110], (112) growth twin. In many cases it is not possible to make a full atomistic structure determination because the specimen does not have translational periodicity in the beam direction. This may be because the boundary plane is not parallel to the beam or the specimen contains dislocations which have a component of Burgers vector parallel to the beam. Examples are given of various low angle boundary structures in goldwhere there are complications because of the three dimensional nature of their structure.

Conformation of Ribosomes from Escherichia Coli

1974 ◽  
Vol 32 ◽  
pp. 210-211
Author(s):  
M. Boublik ◽  
W. Hellmann ◽  
F. Jenkins
Keyword(s):  
Binding Sites ◽  
Ribosomal Proteins ◽  
Fluorescent Dyes ◽  
Protein Biosynthesis ◽  
Three Dimensional ◽  
Spatial Arrangement ◽  
Dimensional Structure ◽  
Complete Understanding ◽  
And Function

The present knowledge of the three-dimensional structure of ribosomes is far too limited to enable a complete understanding of the various roles which ribosomes play in protein biosynthesis. The spatial arrangement of proteins and ribonuclec acids in ribosomes can be analysed in many ways. Determination of binding sites for individual proteins on ribonuclec acid and locations of the mutual positions of proteins on the ribosome using labeling with fluorescent dyes, cross-linking reagents, neutron-diffraction or antibodies against ribosomal proteins seem to be most successful approaches. Structure and function of ribosomes can be correlated be depleting the complete ribosomes of some proteins to the functionally inactive core and by subsequent partial reconstitution in order to regain active ribosomal particles.

scholarly journals 2-Amino-5-chloropyridinium cis-diaquadioxalatochromate(III) sesquihydrate

2012 ◽  
Vol 68 (6) ◽  
pp. m824-m825 ◽  
Author(s):  
Ichraf Chérif ◽  
Jawher Abdelhak ◽  
Mohamed Faouzi Zid ◽  
Ahmed Driss
Keyword(s):  
Rotation Axis ◽  
Three Dimensional ◽  
Octahedral Geometry ◽  
Distorted Octahedral Geometry ◽  
Water Molecules ◽  
Lattice Water ◽  
Compound C ◽  
Hydrogen Bonding Interactions ◽  
Distorted Octahedral ◽  
Independent Lattice

In the crystal structure of the title compound, (C5H6ClN2)[Cr(C2O4)2(H2O)2]·1.5H2O, the CrIII atom adopts a distorted octahedral geometry being coordinated by two O atoms of two cis water molecules and four O atoms from two chelating oxalate dianions. The cis-diaquadioxalatochromate(III) anions, 2-amino-5-chloropyridinium cations and uncoordinated water molecules are linked into a three-dimensional supramolecular array by O—H...O and N—H...O hydrogen-bonding interactions. One of the two independent lattice water molecules is situated on a twofold rotation axis.

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