Spherical Wedge Research Articles

EFFECTS OF ENHANCED STRATIFICATION ON EQUATORWARD DYNAMO WAVE PROPAGATION

We present results from simulations of rotating magnetized turbulent convection in spherical wedge geometry representing parts of the latitudinal and longitudinal extents of a star. Here we consider a set of runs for which the density stratification is varied, keeping the Reynolds and Coriolis numbers at similar values. In the case of weak stratification, we find quasi-steady dynamo solutions for moderate rotation and oscillatory ones with poleward migration of activity belts for more rapid rotation. For stronger stratification, the growth rate tends to become smaller. Furthermore, a transition from quasi-steady to oscillatory dynamos is found as the Coriolis number is increased, but now there is an equatorward migrating branch near the equator. The breakpoint where this happens corresponds to a rotation rate that is about 3-7 times the solar value. The phase relation of the magnetic field is such that the toroidal field lags behind the radial field by about $\pi/2$, which can be explained by an oscillatory $\alpha^2$ dynamo caused by the sign change of the $\alpha$-effect about the equator. We test the domain size dependence of our results for a rapidly rotating run with equatorward migration by varying the longitudinal extent of our wedge. The energy of the axisymmetric mean magnetic field decreases as the domain size increases and we find that an $m=1$ mode is excited for a full $2\pi$ azimuthal extent, reminiscent of the field configurations deduced from observations of rapidly rotating late-type stars.

Coronal influence on dynamos

AbstractWe report on turbulent dynamo simulations in a spherical wedge with an outer coronal layer. We apply a two-layer model where the lower layer represents the convection zone and the upper layer the solar corona. This setup is used to study the coronal influence on the dynamo action beneath the surface. Increasing the radial coronal extent gradually to three times the solar radius and changing the magnetic Reynolds number, we find that dynamo action benefits from the additional coronal extent in terms of higher magnetic energy in the saturated stage. The flux of magnetic helicity can play an important role in this context.

New Balancing Principles Applied to Circumsolids of Revolution, and to n-Dimensional Spheres, Cylindroids,and Cylindrical Wedges

Archimedes’ mechanical balancing methods led him to stunning discoveries concerning the volume of a sphere, and of a cylindrical wedge. This paper introduces new balancing principles (different from those of Archimedes) including a balance-revolution principle and double equilibrium, that go much further. They yield a host of surprising relations involving both volumes and surface areas of circumsolids of revolution, as well as higher-dimensional spheres, cylindroids, spherical wedges, and cylindrical wedges. The concept of cylindroid, introduced here, is crucial for extending to higher dimensions Archimedes’ classical relations on the sphere and cylinder. We also provide remarkable new results for centroids of hemispheres in n-space. Throughout the paper, we adhere to Archimedes’ style of reducing properties of complicated objects to those of simpler objects.

Influence functions, integral formulas, and explicit solutions for thermoelastic spherical wedges

In this study, new exact Green’s functions and a new exact Green-type integral formula for a boundary value problem (BVP) in thermoelasticity for some spherical wedges with mixed homogeneous mechanical boundary conditions are derived. The thermoelastic displacements are subjected to a heat source applied in the inner points of the spherical wedges and to a mixed non-homogeneous boundary heat conditions. When the thermoelastic Green’s function is derived, the thermoelastic displacements are generated by an inner unit point heat source, described by Dirac’s δ-function. All results are obtained in elementary functions that are formulated in a special theorem. Exact solutions in elementary functions for two particular BVPs of thermoelasticity for spherical wedges also are included. In these particular BVPs, the thermoelastic displacements are subjected to a constant temperature (in the first particular BVP) or to a constant heat source (in the second particular BVP). In both BVPs, the constant temperature or the constant heat source is given on the segment of the radius of the quarter-space. On the boundary half-planes of the quarter-space zero temperature and zero heat flux are prescribed.

Ejections of Magnetic Structures Above a Spherical Wedge Driven by a Convective Dynamo with Differential Rotation

We combine a convectively driven dynamo in a spherical shell with a nearly isothermal density-stratified cooling layer that mimics some aspects of a stellar corona to study the emergence and ejections of magnetic field structures. This approach is an extension of earlier models, where forced turbulence simulations were employed to generate magnetic fields. A spherical wedge is used which consists of a convection zone and an extended coronal region to $\approx1.5$ times the radius of the sphere. The wedge contains a quarter of the azimuthal extent of the sphere and $150\degr$ in latitude. The magnetic field is self-consistently generated by the turbulent motions due to convection beneath the surface. Magnetic fields are found to emerge at the surface and are ejected to the coronal part of the domain. These ejections occur at irregular intervals and are weaker than in earlier work. We tentatively associate these events with coronal mass ejections on the Sun, even though our model of the solar atmosphere is rather simplistic.

Solar-like differential rotation and equatorward migration in a convective dynamo with a coronal envelope

AbstractWe present results of convective turbulent dynamo simulations including a coronal layer in a spherical wedge. We find an equatorward migration of the radial and azimuthal fields similar to the behavior of sunspots during the solar cycle. The migration of the field coexist with a spoke-like differential rotation and anti-solar (clockwise) meridional circulation. Even though the migration extends over the whole convection zone, the mechanism causing this is not yet fully understood.

CYCLIC MAGNETIC ACTIVITY DUE TO TURBULENT CONVECTION IN SPHERICAL WEDGE GEOMETRY

We report on simulations of turbulent, rotating, stratified, magnetohydrodynamic convection in spherical wedge geometry. An initially small-scale, random, weak-amplitude magnetic field is amplified by several orders of magnitude in the course of the simulation to form oscillatory large-scale fields in the saturated state of the dynamo. The differential rotation is solar-like (fast equator), but neither coherent meridional poleward circulation nor near-surface shear layer develop in these runs. In addition to a poleward branch of magnetic activity beyond 50 degrees latitude, we find for the first time a pronounced equatorward branch at around 20 degrees latitude, reminiscent of the solar cycle.

A NEW CODE FOR NONLINEAR FORCE-FREE FIELD EXTRAPOLATION OF THE GLOBAL CORONA

Reliable measurements of the solar magnetic field are still restricted to the photosphere, and our present knowledge of the three-dimensional coronal magnetic field is largely based on extrapolation from photospheric magnetogram using physical models, e.g., the nonlinear force-free field (NLFFF) model as usually adopted. Most of the currently available NLFFF codes have been developed with computational volume like Cartesian box or spherical wedge while a global full-sphere extrapolation is still under developing. A high-performance global extrapolation code is in particular urgently needed considering that Solar Dynamics Observatory (SDO) can provide full-disk magnetogram with resolution up to $4096\times 4096$. In this work, we present a new parallelized code for global NLFFF extrapolation with the photosphere magnetogram as input. The method is based on magnetohydrodynamics relaxation approach, the CESE-MHD numerical scheme and a Yin-Yang spherical grid that is used to overcome the polar problems of the standard spherical grid. The code is validated by two full-sphere force-free solutions from Low & Lou's semi-analytic force-free field model. The code shows high accuracy and fast convergence, and can be ready for future practical application if combined with an adaptive mesh refinement technique.

Effects of stratification in spherical shell convection

AbstractWe report on simulations of mildly turbulent convection in spherical wedge geometry with varying density stratification. We vary the density contrast within the convection zone by a factor of 20 and study the influence of rotation on the solutions. We demonstrate that the size of convective cells decreases and the anisotropy of turbulence increases as the stratification is increased. Differential rotation is found to change from anti‐solar (slow equator) to solar‐like (fast equator) at roughly the same Coriolis number for all stratifications. The largest stratification runs, however, are sensitive to changes of the Reynolds number. Evidence for a near‐surface shear layer is found in runs with strong stratification and large Reynolds numbers (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Reynolds stress and heat flux in spherical shell convection

(abridged) Context. Turbulent fluxes of angular momentum and heat due to rotationally affected convection play a key role in determining differential rotation of stars. Here we perform a systematic comparison between Cartesian and spherical geometries as a function of the rotation rate. Aims. We extend the earlier studies by using spherical wedges to obtain turbulent angular momentum and heat transport as functions of the rotation rate from stratified convection. We compare results from spherical and Cartesian models in the same parameter regime. In particular, we want to clarify whether the sharp equatorial profile of the horizontal Reynolds stress found in earlier Cartesian models is reproduced in spherical models. Methods. We employ direct numerical simulations of turbulent convection. In order to reach as high spatial resolution as possible in the spherical runs, we model only parts of the latitude and longitude. The rotational influence, measured by the Coriolis number, is varied from zero to roughly seven, which is the regime that is likely to be realised in the solar convection zone. Cartesian simulations are performed in overlapping parameter regimes. Results. For slow rotation we find that the radial and latitudinal turbulent angular momentum fluxes are directed inward and equatorward, respectively. In the rapid rotation regime the radial flux changes sign in contradiction with theory. The latitudinal flux remains mostly equatorward and develops a maximum close to the equator. In Cartesian simulations this peak can be explained by the strong `banana cells'. The latitudinal heat flux is mostly equatorward for slow rotation but changes sign for rapid rotation. The rotation profiles vary from anti-solar (slow equator) for slow and intermediate rotation to solar-like (fast equator) for rapid rotation. The solar-like profiles are dominated by the Taylor--Proudman balance.

Alleviatingαquenching by solar wind and meridional flows

We study the ability of magnetic helicity expulsion to alleviate catastrophic $\alpha$-quenching in mean field dynamos in two--dimensional spherical wedge domains. Motivated by the physical state of the outer regions of the Sun, we consider $\alpha^2\Omega$ mean field models with a dynamical $\alpha$ quenching. We include two mechanisms which have the potential to facilitate helicity expulsion, namely advection by a mean flow ("solar wind") and meridional circulation. We find that a wind alone can prevent catastrophic quenching, with the field saturating at finite amplitude. In certain parameter ranges, the presence of a large-scale meridional circulation can reinforce this alleviation. However, the saturated field strengths are typically below the equipartition field strength. We discuss possible mechanisms that might increase the saturated field.

Convective dynamos in spherical wedge geometry

AbstractSelf‐consistent convective dynamo simulations in wedge‐shaped spherical shells are presented. Differential rotation is generated by the interaction of convection with rotation. Equatorward acceleration and dynamo action are obtained only for sufficiently rapid rotation. The angular velocity tends to be constant along cylinders. Oscillatory large‐scale fields are found to migrate in the poleward direction. Comparison with earlier simulations in full spherical shells and Cartesian domains is made (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Errata for “Flow-Boundary Effects on Critical Submergence of Intake Pipe” by Nevzat Yıldırım, Fikret Kocabaş, and Salih Cem Gülcan

The authors have unrealistically approximated that all common corner blockages of flow boundaries on the critical spherical sink surface CSSS are in the forms of complete spherical sectorsurfaces, which are spherical wedge surfaces in reality. Therefore, the geometrical terms representing, but not equal to, the common corner blockages of the flow boundaries in a rectangular canal with or without a dead-end wall , JH, K1Z1, K2Z2, K3Z3, K4Z4, and their equalities given in the paper must be corrected as explained in the following.

Dynamic behavior of oscillatory plastic flow in a smectic liquid crystal

Dynamic surface force measurements are used to study the response of a smectic-$A$ liquid crystal under layer-normal stress. The smectic $A$ is confined in a spherical wedge between crossed cylindrical surfaces having a minimum gap spacing of 0.5--4 \ensuremath{\mu}m. The force transmitted between the surfaces by the liquid crystal is measured vs surface spacing using a capacitance micrometer-based surface force apparatus. Above a threshold stress plastic flow results, consisting of individual layers being excluded or included. Each layer flow event has an intriguing dynamical structure, beginning with an enhanced drift rate, which can last for many minutes, accelerating to a rapid separation change of \ensuremath{\sim}1 or 2 s duration wherein the bulk of the relaxation occurs, and tapering off to a background drift rate over a period of a 100 s or more. The single-layer nature of the observed jumps in liquid crystal thickness indicates that they are topological in origin, i.e., slippage events in the phase of the smectic-$A$ order parameter that must necessarily involve edge or screw dislocations. A model based on the Glaberson-Clem-Oswald-Kl\'eman helical instability in screw dislocations is the only one found to explain the data, the layering events arising from a cascade of these helical instabilities sweeping radially outward through the smectic-$A$ sample. The slow precursor acceleration is due to the nucleation of a few helices in the thin central portion of the sample. As time goes on, the force relieved is transferred to the rest of the sample, pushing larger and larger amounts of the area into the unstable regime, and a type of chain reaction occurs whereby the bulk of a layer is removed. In the end only the material at the edge of the droplet, where the thickness is largest, is left to slowly continue to nucleate, producing a long-term tail.

Coordinate systems for the radiative transfer equation in curvilinear media

Spatial/directional coordinate systems are presented on which general directional boundary conditions may be satisfied for the solution of the radiative transfer equation in curvilinear, participating media. The general directional boundary conditions described herein may be applied to curvilinear media without regard to spatial symmetry, although their application depends upon use of a local coordinate system, defining the intensity direction, which has a polar axis directed normal to any nearby surfaces. In the most commonly used spatial/directional coordinate systems for cylindrical and spherical media, the orientation of the directional polar axis indicates that the non-symmetric boundary conditions may be applied only to, respectively, parallel disk and spherical annulus boundary geometries. Application of the general directional boundary conditions to non-symmetric curvilinear media of these geometries, referred to only partially or obliquely in previous works, is demonstrated completely and consisely in the paper. Also, new spatial/directional coordinate systems are derived, based on reorientation of the local directional coordinate system in spatially cylindrical and spherical geometries. These new coordinate systems allow application of the general directional boundary conditions to other curvilinear geometries, such as a cylindrical annulus, cylindrical wedge, conical annulus, and spherical wedge. For each coordinate system, expressions for the pathlength derivative of radiation intensity are given in simple, conservative, and discrete ordinates form. The techniques illustrated in the derivation of these spatial/directional coordinate systems for non-symmetric curvilinear media may be applied to other, more complicated boundary geometries as well.

THE PLASMA ANVIL IN INERTIAL CONFINEMENT FUSION

We examine theoretically an inertial confinement fusion (ICF) target consisting of a spherical wedge embedded in a relatively nondeformable "anvil". Questions such as heat loss to the anvil, optimum wedge angle, liner and anvil materials, anvil deformations and deleterious 2-D shock effects on D-T burn and compression symmetry are discussed.

Fault detection device

A U-shaped tuning fork is fixed to a supporting block connected to a body in which a fault is to be detected. A spring loaded rod is movably supported at one end on the supporting block and extends through a spherical wedge sandwiched between two legs of the tuning fork. The tuning fork resonates at a frequency at which the detected body vibrates when a fault occurs in it. This permits the wedge and therefore the rod to move indicating the occurrence of the fault. Simultaneously a microswitch is operated to indicate electrically the existence of the fault.

Large elastic deformation of a spherical wedge

A large deformation analysis is presented for the distortion of an elastic spherical wedge. Assuming isotropy, homogeneity, and incompressibility, the formulation is reduced to that of four non-linear ordinary differential equations in four unknowns. Two special cases are considered in which specific analytic solutions are obtained, and the possible use of one of these for determining mechanical properties is suggested.

Deformation of an Elastic Spherical Wedge

Analytical solutions are presented for two separate modes of deformation of an infinite elastic spherical wedge. In part, these solutions take the form of the stresses on the boundaries of the wedge caused by prescribed surface displacements. Full discussion is given to the means of utilizing these two solutions, along with corresponding tests, to determine the complete mechanical-properties characterization of a linear isotropic viscoelastic solid.

Virtual source luneberg lenses

The portion of a spherical Luneberg lens contained between two plane reflectors has been investigated as a lens of reduced size and weight. If the reflectors pass through the center of the sphere, the resulting system produces several perfectly focused radiation beams each appearing to originate from a virtual source on the surface of the full sphere. The virtual source positions and the position, beamwidth, and gain of the beams are accurately predicted from the spherical wedge angle and the source position. When the wedge angle is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\pi/p</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> is an integer, rays with p reflections form the beam having the greatest gain at a displacement from the wedge bisector equal to the source displacement. For applications in which only this principal beam is desired, the gain of the unwanted beams can be reduced by absorption, reflection, or illumination taper. Scanning is achieved by moving the feed along the surface of the spherical wedge; if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> is an odd integer, scanning can be obtained by moving the wedge past a fixed feed. Experimental data were taken on a two-dimensional <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> -band model having a value of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p=1</tex> . Good agreement was found with the predicted performance regarding beam position, beamwidth, and gain. The single, undesired beam was minimized by the use of absorbing material.