Spherical Polar Coordinate System Research Articles

Modeling mantle convection in the spherical annulus

Most methods for modeling mantle convection in a two-dimensional (2D) circular annular domain suffer from innate shortcomings in their ability to capture several characteristics of the spherical shell geometry of planetary mantles. While methods such as rescaling the inner and outer radius to reduce anomalous effects in a 2D polar cylindrical coordinate system have been introduced and widely implemented, such fixes may have other drawbacks that adversely affect the outcome of some kinds of mantle convection studies. Here we propose a new approach that we term the “spherical annulus,” which is a 2D slice that bisects the spherical shell and is quantitatively formulated at the equator of a spherical polar coordinate system after neglecting terms in the governing equations related to variations in latitude. Spherical scaling is retained in this approximation since the Jacobian function remains proportional to the square of the radius. We present example calculations to show that the behavior of convection in the spherical annulus compares favorably against calculations performed in other 2D annular domains when measured relative to those in a fully three-dimensional (3D) spherical shell.

A non-parametric vessel detection method for complex vascular structures.

Modern medical imaging techniques enable the acquisition of in vivo high resolution images of the vascular system. Most common methods for the detection of vessels in these images, such as multiscale Hessian-based operators and matched filters, rely on the assumption that at each voxel there is a single cylinder. Such an assumption is clearly violated at the multitude of branching points that are easily observed in all, but the most focused vascular image studies. In this paper, we propose a novel method for detecting vessels in medical images that relaxes this single cylinder assumption. We directly exploit local neighborhood intensities and extract characteristics of the local intensity profile (in a spherical polar coordinate system) which we term as the polar neighborhood intensity profile. We present a new method to capture the common properties shared by polar neighborhood intensity profiles for all the types of vascular points belonging to the vascular system. The new method enables us to detect vessels even near complex extreme points, including branching points. Our method demonstrates improved performance over standard methods on both 2D synthetic images and 3D animal and clinical vascular images, particularly close to vessel branching regions.

WE-D-351-03: NURBS-Based Head and Eye Dosimetry Models for Ocular Radiosurgery

Purpose: Age‐related macular degeneration (AMD) is a leading cause for vision loss for people over the age of 65 in the United States. There are a number of treatments used to help slow and in some cases stabilize the process of AMD, but these require frequent invasive injections into the eye. This paper presents a potential radiotherapytreatment used to destroy the leaky vasculature while minimizing dose to surrounding tissues, allowing for combination therapy to treat AMD. Method and Materials: An adult male head phantom was modeled with Rhinoceros 4.0 using ICRP Publication 89 reference values. The head phantom was voxelized and modeled using the MCNPX radiation transport code with a photon beam treatment operated at 100 kVp. Doses to the macula target, lens, optic nerve, brain, thyroid, and salivary glands were tabulated using MCNPX at 8 beam angles. These angles can be characterized using a spherical 3D polar coordinate system with the macula at the origin, the z‐axis as the axis of vision, a radius of 13 cm, a polar angle of 30 degrees, and azimuth angles in 8 increments of 45 degrees. Results: For each beam angle, the doses to the organs of interest were several orders of magnitude less than the 8 Gy dose to the macula target and significantly lower than the thresholds for serious implications. Conclusion: The preliminary results bode well for the proposed radiotherapytreatment.Dose Volume Histograms (DVH) will be tabulated using MCNPX so that the maximum dose to localized portions of each organ of interest can be determined. The safest beam angle for treatment will be determined from the results of this study.

Modeling fluid mixing within the crystalline lens during accommodation

Abstract Purpose: Historically, cataract and presbyopia are treated as separate entities. However, it has recently been hypothesized that as the ageing lens looses its ability to change shape, this reduces the mixing and transport of nutrients and antioxidants within the lens that normally occurs as the young, flexible lens alters its shape during accommodation. Such a loss of antioxidant mixing may render the lens more susceptible to cataract formation. Methods: We have developed a 3D cubic Hermite finite element model of the human lens to investigate the effect of increasing lens stiffness on fluid movement. Data obtained from previous studies are used to construct an anatomically accurate model. A spherical polar coordinate system is used to provide an accurate physical representation and as a framework to examine mechanics. Polynomial functions are used to define the lens surfaces. Lens capsule thickness and material properties of lens tissue are taken from the literature. Tension is applied via zonules to simulate accommodation. Results: Preliminary results verify that the change in shape of the lens surfaces (and the change in optical power) associated with displacement of the lens equator are similar to results from other published models and fall within the physiological range (i.e. using parameters for 30 y.o. lens, approx 25 dioptres per mm change in equatorial radius). We are currently investigating the fluid movements associated with these changes. Conclusions: Our lens model provides a basis for investigating fluid mixing in the young lens during simulated accommodation and also the reduction of fluid movement in stiffer, older lenses.

A numerical study of laminar free convection heat transfer between inner sphere and outer vertical cylinder

The effects of height and radius ratio with a Newtonian fluid have been investigated numerically to determine heat transfer by natural convection between the sphere and vertical cylinder with isothermal boundary conditions. The inner sphere and outer vertical cylinder were heated and cooled in a steady change of temperature. Calculations were carried out systematically for a range of the Rayleigh numbers to determine the average Nusslet numbers which are affected by the geometric ratio parameters ( HR and RR) on the flow and temperature fields. The governing equations, in terms of vorticity, stream function and temperature are expressed in a spherical polar coordinate system. Results of the parametric study conducted further reveal that the heat and flow fields are primarily dependent on the Rayleigh number and height and radius ratio, for a Prandtl number of 0.7, with the Rayleigh number ranging from 10 3 to 10 6, and the height and radius ratio varying from 1.2 to 5.0. Above all, the specification of different convective configurations has a significant effect on the average heat transfer rate across the composite annulus gap.

Thermal effects of the micropolar fluid parameters on the laminar free convection in spherical annuli with mixed boundary conditions

The effects of micro-rotation and vortex viscosity in micropolar fluids have been investigated numerically to determine heat transfer by natural convection between concentric and vertically eccentric spheres with specified mixed boundary conditions. Calculations were carried out systematically for several different eccentricities and a range of modified Rayleigh numbers to determine the average Nusslet numbers which are affected by the micropolar parameters (F) of the flow and temperature fields. The skin friction stress on the walls has also been studied and discussed. The governing equations, in terms of vorticity, stream function, temperature and angular momentum are expressed in a spherical polar coordinate system. Results were obtained for steady heat-transfer in spherical annuli at a Prandtl number of 0.7, with the modified Rayleigh number ranging from 103 to 5 × 105, for a radius ratio of 2.0 and eccentricities varying from −0.625 to +0.625. Comparisons are attempted between the Newtonian fluid and micropolar fluid.

Transient natural convection of micropolar fluids between concentric and vertically eccentric spheres

Micropolar fluids in transient analysis have been investigated numerically to determine heat transfer by natural convection between concentric and vertically eccentric spheres with specified isothermal boundary conditions. Calculations were carried out systematically for several different eccentricities and a range of Rayleigh numbers to determine the average Nusselt numbers which are affected by the micropolar parameters ( F) on the flow and temperature fields. The skin friction stress on the walls has also been studied and discussed. The governing equations, in terms of vorticity, stream function, temperature and angular momentum are expressed in a spherical polar coordinate system. Results were obtained for steady and transient heat-transfer in vertically eccentric spheres at a Prandtl number of 0.7, with the Rayleigh number ranging from 10 3 to 5 × 10 5, for a radius ratio of 2.0 and eccentricities varying from −0.625 to +0.625, and for the value of micropolar parameters are 0, 1 and 5 respectively. Comparisons are attempted between the Newtonian fluid and micropolar fluid.

NUMERICAL STUDY OF THERMAL WALL BOUNDARY EFFECTS FOR TRANSIENT NATURAL CONVECTION BETWEEN CONCENTRIC AND VERTICALLY ECCENTRIC SPHERES

Transient analysis has been investigated numerically to determine heat transfer by natural convection between concentric and vertically eccentric spheres with specified constant heat flux wall boundary conditions. The governing equations, in terms of vorticity, stream function and temperature are expressed in a spherical polar coordinate system. The alternating direction implicit method and the successive over-relaxation techniques are applied to solve the finite difference form of governing equations. Results were obtained for steady and transient heat-transfer in vertically eccentric spheres at a Prandtl number of 0.7, with the modified Rayleigh number ranging from 10 3 to 5 2 10 5 , for a radius ratio of 2.0 and eccentricities varying from m 0.625 to +0.625. Comparisons are attempted between the isothermal boundary condition and the present computations for constant heat flux boundary condition.

Reanalysis of Saturn's magnetospheric field data view of spin‐periodic perturbations

Periodic perturbations with their period close to that of planetary rotation are observed in most of the magnetic field data from Saturn's magnetosphere. These data arise from the three spacecraft encounters with Saturn (Pioneer 11, Voyager 1 and 2). The long‐held view that no planetary spin‐periodic modulation was present in the magnetic field observations is thus not true. Here we present several new pieces of information obtained from a careful analysis of the magnetic field data in view of this peculiar periodic feature. First, by simple considerations of the magnetic field morphology, we argue that these perturbations cannot be directly due to the planetary intrinsic field. Also, we analyze by means of two‐dimensional (2‐D) hodographs the rotation of the magnetic field vector, expressed in an inertial planetocentric spherical polar coordinate system, and obtain a definitive argument against the possibility of a dipole tilt signature. In addition, we find that the inbound and outbound Pioneer 11 observations of the perturbations are nearly in phase (once the spatial distance is accounted for), indicating that this periodic feature is of a global nature. Finally, we discuss the fact that in the same data set, the magnetopause position in the dawn sector seems to be modulated in phase with the radial component of the perturbation magnetic field.

Using satellite technology (global positioning system) to teach the spherical polarcoordinate system

Students discover the equivalence between plausible geographic variables(co-latitude and longitude) of the Earth and less-familiar geometric variables(polar and azimuthal angles) of spherical polar coordinates by collecting andanalysing positional data recorded during field exercises with a handheld globalpositioning system (GPS) receiver. This pedagogical approach to teachingspherical geometry, based on field experience rather than classroom theory, usesGPS technology to capture the curiosity of contemporary students, technical orotherwise, who might not normally find the topic of the spherical polar coordinatesystem comprehensible, relevant or interesting.

Unexpected periodic perturbations in Saturn's magnetic field data from Pioneer 11 and Voyager 2

A direct comparison between the three magnetic field data sets from Saturn's magnetosphere is carried out. These data arise from the three spacecraft encounters with Saturn (Pioneer 11, Voyager 1 and 2). A standard planetocentric spherical polar coordinate system is used for the analysis. The inbound observations of the magnetic field from Pioneer 11 and Voyager 2 show a periodic perturbation of a few nanoTeslas (with a period close to the planetary rotation period) in the azimuthal component of the field, from the inbound magnetopause crossing to closest approach (∼ 17 hours). The outbound observations present an equivalent feature only in the case of Pioneer 11, where the radial and azimuthal components of the field are perturbed throughout the whole outbound pass (∼ 40 hours). A dipole tilt and/or current sheet crossings are unlikely causes for such periodic perturbations. A large scale ballooning of the magnetic field and plasma is a possible generating mechanism, and is used for the time-dependent scenario proposed here to explain the observed feature. The Pioneer 11 outbound magnetopause crossings may provide some evidence for such ballooning.

Periodic perturbations in Saturn's magnetic field

Analysis of all the existing magnetic field data from Saturn's magnetosphere is carried out. These data arise from the three spacecraft encounters with Saturn (Pioneer 11, Voyager 1 and 2). In order to carry out a direct comparison between the three magnetic field data sets, they are all expressed in the same planetocentric spherical polar coordinate system. In the cases of Pioneer 11 (whole encounter) and Voyager 2 (inbound) a surprising perturbation of a few nanoteslas is observed (with a period close to Saturn's rotation period) in the radial and/or azimuthal components of the magnetic field. It is shown that a dipole tilt and/or current sheet crossings are unlikely causes for such periodic perturbations. This is in agreement with the negligible magnetic dipole tilt resulting from the existing models of Saturn's magnetic field.

Reflection of equatorial Kelvin waves at eastern ocean boundaries Part I: hypothetical boundaries

Abstract. A baroclinic shallow-water model is developed to investigate the effect of the orientation of the eastern ocean boundary on the behavior of equatorial Kelvin waves. The model is formulated in a spherical polar coordinate system and includes dissipation and non-linear terms, effects which have not been previously included in analytical approaches to the problem. Both equatorial and middle latitude response are considered given the large latitudinal extent used in the model. Baroclinic equatorial Kelvin waves of intraseasonal, seasonal and annual periods are introduced into the domain as pulses of finite width. Their subsequent reflection, transmission and dissipation are investigated. It is found that dissipation is very important for the transmission of wave energy along the boundary and for reflections from the boundary. The dissipation was found to be dependent not only on the presence of the coastal Kelvin waves in the domain, but also on the period of these coastal waves. In particular the dissipation increases with wave period. It is also shown that the equatorial β-plane approximation can allow an anomalous generation of Rossby waves at higher latitudes. Nonlinearities generally have a small effect on the solutions, within the confines of this model.Key words. Oceanography: general (equatorial oceanography; numerical modeling) · Oceanography: physical (eastern boundary currents)

Reflection of equatorial Kelvin waves at eastern ocean boundaries Part I: hypothetical boundaries

A baroclinic shallow-water model is developed to investigate the effect of the orientation of the eastern ocean boundary on the behavior of equatorial Kelvin waves. The model is formulated in a spherical polar coordinate system and includes dissipation and non-linear terms, effects which have not been previously included in analytical approaches to the problem. Both equatorial and middle latitude response are considered given the large latitudinal extent used in the model. Baroclinic equatorial Kelvin waves of intraseasonal, seasonal and annual periods are introduced into the domain as pulses of finite width. Their subsequent reflection, transmission and dissipation are investigated. It is found that dissipation is very important for the transmission of wave energy along the boundary and for reflections from the boundary. The dissipation was found to be dependent not only on the presence of the coastal Kelvin waves in the domain, but also on the period of these coastal waves. In particular the dissipation increases with wave period. It is also shown that the equatorial β-plane approximation can allow an anomalous generation of Rossby waves at higher latitudes. Nonlinearities generally have a small effect on the solutions, within the confines of this model.Key words. Oceanography: general (equatorial oceanography; numerical modeling) · Oceanography: physical (eastern boundary currents)

Transient natural convection between concentric and vertically eccentric spheres with mixed boundary conditions

Transient analysis has been investigated numerically to determine heat transfer by natural convection between concentric and vertically eccentric spheres with constant heat flux on the inner wall and a specified isothermal temperature on the outer wall. The governing equations, in terms of vorticity, stream function and temperature are expressed in a spherical polar coordinate system. The alternating direction implicit method and the successive over-relaxation techniques are applied to solve the finite difference form of governing equations. A physical model is introduced which accounts for the effects of fluid buoyancy as well as eccentricity of the outer sphere. Transient solutions of the entire flow field are obtained for a range of modified Rayleigh number (103<Ra⋆<5×105), for a Prandtl number of 0.7 and a radius ratio of 2.0, with the outer sphere near the top and bottom of the inner sphere (e=±0.625). Results of the parametric study conducted further reveal that the heat and flow fields are primarily dependent on the modified Rayleigh number and the eccentricity of the spherical annulus. The results of average Nusselt numbers are also compared with the results obtained for flow between two isothermal spheres.

On conical swirling flows in an infinite fluid

The steady axisymmetric flow generated in an unbounded incompressible viscous fluid, of density ρ and kinematic viscosity v, by torque-producing singularities with constant line density c along the semi-infinite line θ = 0 of a spherical polar coordinate system (r, θ, ϕ) that was investigated by Paull &amp; Pillow (1985b), is reconsidered. The numerical solution constructed revealed the following features. (i) For values of c up to about 46.9 there is only one solution where the axial component of the meridional flow is directed from θ = 0 to θ = π. This solution can be continued to all values of c. (ii) For c &gt; 46.9 the system of equations allows bifurcation and two more solutions with a single separatrix are possible. (iii) For c = ∞ one of the two branches of the separatrix asymptotes to θ = ½π and the other to θ = π. The asymptotic solution for large c constructed by Paull &amp; Pillow (1985b), where the meridional flow consists of two colliding flows, relates to the bifurcation solution where the separatrix asymptotes to ½π as c → ∞.

Steady MHD flows with an ignorable co-ordinate and the potential transonic flow equation

The paper explores the interrelationship between the generalized Grad-Shafranov equation, or trans-field force balance equation, for steady MHD flows with an ingnorable co-ordinate, and work by Imai on field-aligned MHD flows. The development of Imai, assumes at the outset that the fluid velocity V is parallel to the magnetic field B, and exploits an analogy with steady compressible irrotational flow in ordinary fluid dynamics. In Imai's analysis the magnetic induction B is written in the form B = σb, where , and MA is the appropriate Alfvén Mach number. Gauss' law Δ. B = Δ. (σb) = 0 then plays a role analogous to the mass continuity equation in ordinary fluid dynamics, where σ corresponds to the density of the pseudo-fluid. Imai's analysis leads to a transonic equation for the field potential φ defined by b = Δφ. For a restricted class of flows the trans-field force balance equation formulation also leads to the transonic potential flow equation, but the assumption of an ignorable co-ordinate allows for the possibility of non-field-aligned flows with non-zero electric field potential ΦE The characteristics of the generalized Grad—Shafranov equation are related to the Mach cone and the group velocity surface for linear magnetosonic waves. The corresponding forms of the characteristics for the potential transonic flow equation in the (x, y) plane and in the (bx, by) hodograph plane are discussed. Sample solutions of the potential transonic flow equation for radial, helical and spiral flows are obtained by means of the hodograph transformation, and are used to illustrate the differences between hyperbolic and elliptic flows. The potential transonic flow equation is obtained for the case of an ignorable co-ordinate z of a rectangular Cartesian co-ordinate system (x, y, z), and also for the case of flows with an ignorable co-ordinate of a spherical polar co-ordinate system (r, θ, ω). Astrophysical applications are briefly discussed.

Distribution of Solvent Molecules Around Apolar Side-chains in Protein Crystals

We have analysed the distribution of solvent sites within 5·0 Å of the apolar side-chains alanine, valine, leucine, isoleucine and phenylalanine based on experimental data from 24 high-resolution protein structures. Clustering of solvent molecules into specific regions can be seen superimposed on a broad background of sites. The non-random nature of these distributions is confirmed by quantitative analysis of the solvent sites according to a spherical polar (r, θ and φ) co-ordinate system with the apolar atom of interest at the centre. One of the general features of these solvent sites is that they peak at around 4·0 Å from an apolar protein carbon atom. Preferences in orientation (θ and φ) are also seen in the solvent distributions especially around the alanine CB atom and the phenylalanine ring. Most (around 75%) of the solvent sites around apolar groups are also within hydrogen bonding distance of protein (main chain) polar groups which leads to a distribution dependent on the local secondary structure. The remaining 25% of solvent sites are referred to as "non-polar" water molecules and their crystallographic temperature factors are higher than average by between 15% to 28%. For alanine and phenylalanine there are enough data to show that water molecules not within hydrogen bonding distance of protein polar atoms also cluster into specific regions. However, the main conclusion appears to be that the hydrophobic hydration in protein crystals is correlated with hydration of polar groups and thus depends on the local environment as well as on the stereochemistry of the apolar atoms.

Development of the Stokes flow field associated with a line vortex perpendicular to a plane wall

A viscous incompressible fluid occupying the space θ≤π/2 and bounded by the wall θ=π/2 of a spherical polar coordinate system (r,θ,φ), is stirred by a line vortex along the line θ=0 which is switched on at time t=0. The line vortex is perpendicular to the wall. The development of the flow configuration is considered for the case where the poloidal flow is weak and does not affect the structure of the inducing azimuthal flow. The problem is formulated in terms of the similarity variable r/2(νt)1/2 and the polar angle θ, where ν is the kinematic viscosity of the fluid. An analytical solution is constructed for the azimuthal flow. At any given station r the steady azimuthal velocity field is, practically, reached within time r2/ν. The equations governing the poloidal flow are coupled partial differential equations of mixed elliptic–parabolic type which are transformed to equations that are elliptic throughout the solution domain. These equations are solved numerically using the methods of successive overrelaxation and fast Fourier transform. The results show that the poloidal flow in a meridional plane at time t forms closed loops about the point r≊1.58(νt)1/2, θ=π/4, where the velocity has only an azimuthal component. The case of a diffusing configuration from the steady state, due to switching off at t=0 of the agent generating the flow, is also considered. For this case the poloidal field consists of open streamlines and at t=2r2/ν its intensity is a very small fraction of that associated with the steady state.

A differential game in three dimensions: The aerial dogfight scenario

In this article, an aerial dogfight in three dimensions between two aircraft was modeled, with each aircraft having combined qualitative objectives of capture with avoidance. A spherical polar-coordinate system was devised to describe the system. From a standard point-mass model of aircraft dynamics for three-dimensional flight, a kinematic model was derived, with its corresponding controls for each aircraft. This kinematic model was then used, assuming bang-bang control functions for each aircraft, in order to determine a map of the game. A technique of trajectory dissection was introduced, whereby the airplane loci were decomposed in terms of regions of zero-level and saturation-level control values, and controllable and winning regions were determined for each aircraft using a Liapunov-function approach, the winning regions being calculated using the Getz-Leitmann theorem. A map of the game was constructed, and the barrier was found to be nonvoid. The concept of the posthumous mutual-kill strategy was introduced.