Azimuthal Part Research Articles

Thermal convection in a rotating porous medium layer saturated by a nanofluid under a helical magnetic field

In recent years, experiments with flows of liquid metals in a helical magnetic field have been actively carried out. The study of the processes of mixing and crystallization of a liquid metal is of practical importance for metallurgy. With the development of nanotechnology, more and more new types of nanofluids (hybrid, ternary nanofluids) are being synthesized, and the thermophysical characteristics of which can compete with liquid metals. This circumstance served as a motivation for conducting this theoretical study. In this study, the criterion for the onset of convection in a Darcy–Brinkman porous medium layer saturated by an electrically conductive nanofluid under a helical magnetic field is investigated. The Brownian motion and thermophoresis effects are combined in the model for nanofluids, whereas the Darcy–Brinkman model is used for porous media. Instead of prescribing the nanoparticle volume fraction on the borders, we adopted a boundary condition in which the nanoparticle flow is considered to be zero. In the absence of a temperature gradient, a new type of instability in a helical magnetic field in a thin layer of a nanofluid is considered. The growth rate and the region of the development of this instability are numerically obtained depending on the profile of the azimuthal magnetic field (the magnetic Rossby number Rb) and the radial wave number k. In the presence of temperature, the stationary regime of nonuniformly rotating magnetoconvection is studied. The accurate analytical equation for the critical Rayleigh–Darcy number in terms of various non-dimensional parameters is determined using the linear stability theory. The results show that rotation and the axial (vertical) part of the helical magnetic field retard the onset of convection. While the azimuthal part of the helical magnetic field has a destabilizing effect at positive Rossby numbers Rb. The conditions for stabilization and destabilization of stationary convection in a helical magnetic field are determined for metal oxide, metallic, and semiconductor nanofluids.

Analytical solutions of the Proca equation for modified Manning–Rosen potential with centrifugal term and direct coupling approaches by hypergeometric method

The Proca equation is one of the fundamental relativistic wave equations of massive vector (spin 1) boson fields. Proca equation is used to explain a relatively small particle that has a speed approaching the speed of light. The effect of photon imaginary mass to the quantum system that was coupled by noncentral potential consisting of radial, polar, and azimuthal modified Manning–Rosen potentials was investigated. The vector potential was coupled to the photon relativistic energy and the quadratic scalar potential was coupled directly to the quadratic photon mass. In the case of free space and when the vector potential was equal to scalar potential, therefore, the time-independent Proca equation reduced to a three-dimensional variable separable Schrodinger-like equation. By using variable separation method, the three-dimensional Schrodinger-like equation was resolved into three one-dimensional radial, polar, and azimuthal parts of Schrodinger-like equations. Each part of one-dimensional Schrodinger-like equation was solved using the hypergeometric differential equation through variables and wave function substitutions and resulted in the energy and wave function equations of massive photons. The relativistic energy equation of massive photons was a function of imaginary mass photon, the potential parameters, and the quantum numbers in radial, polar, and azimuthal variables. By using the Maxwell field approach, we got the correction factor for the energy function of this spin-1 particle.

Optical needles with arbitrary homogeneous three-dimensional polarization.

We propose a new method to generate optical needles by focusing vector beams comprised of radially polarized component and azimuthally polarized vortex components. The radial part can generate longitudinal polarization, while the azimuthal parts can generate left- and right-handed polarization. Hence, an arbitrary 3D polarization can be obtained. To our knowledge, it may be the first time that arbitrarily polarized optical needles whose transverse sizes are under 0.5λ have been achieved. The polarized homogeneity of the needles is beyond 0.97.

Construction of solvable non-central potential using vector superpotential: a new approach

We introduce here vector superpotential which is useful to find general potential form using supersymmetric quantum mechanics (SUSY QM) approach. Using the vector superpotential, we reconstruct different solved central potentials and non-central shape variant or shape invariant potentials. To construct and reconstruct the central potentials and non-central potentials, appropriate choice of radial part of the superpotential as well as angular and azimuthal parts of the superpotential is required. Our main aim is to construct angular and azimuthal parts of potential directly from the appropriate choice of vector superpotential. We have reconstructed few potentials which are solved by different researchers using different methods of solution and also constructed new non-central potentials.

Uniaxial Diffusional Narrowing of NMR Lineshapes for Membrane Proteins Reconstituted in Magnetically Aligned Bicelles and Macrodiscs

Structure and dynamics of membrane proteins can be effectively studied by oriented-sample solid-state nuclear magnetic resonance (NMR) techniques when the lipid bilayers are macroscopically aligned with respect to the main magnetic field. Magnetic alignment of the protein-containing membrane bilayer results from the negative susceptibility anisotropy of the lipid hydrocarbon interior yielding perpendicular sample alignment. At this orientation, while the uniformity of alignment represents an essential prerequisite for obtaining high-quality NMR spectra, further line narrowing is obtained by uniaxial motional averaging of the azimuthal parts of the chemical shift anisotropies and dipolar couplings. The motional averaging is brought about by uniaxial rotational diffusion of the protein molecules about the normal to the membrane surface, which is perpendicular to the magnetic field. Uniaxial averaging is efficient when the motion about the axis of alignment becomes sufficiently fast (on the timescale of the dipolar couplings and chemical shift anisotropies). Line narrowing under uniaxial rotation can be theoretically modeled using the stochastic Liouville equation. In this mini-review, we illustrate the method of uniaxial averaging for the relatively small Pf1 coat protein which exhibits excellent resolution in magnetically aligned bicelles due to its fast uniaxial diffusion and even superior resolution in large (30 nm) nanodiscs (macrodiscs) stabilized by a belt peptide. Spectra of Pf1 coat protein in polymer-stabilized macrodiscs, an alternative and more robust alignment media, are presented. We also report on preliminary spectra of a much larger protein—uniformly 15N labeled M1-M4 domain for the human acetylcholine receptor. While some spectral resolution is apparent, significantly broader linewidths emphasize the need for creating fast rotating discoidal membrane mimetics.

Spherical integral transforms of second-order gravitational tensor components onto third-order gravitational tensor components

New spherical integral formulas between components of the second- and third-order gravitational tensors are formulated in this article. First, we review the nomenclature and basic properties of the second- and third-order gravitational tensors. Initial points of mathematical derivations, i.e., the second- and third-order differential operators defined in the spherical local North-oriented reference frame and the analytical solutions of the gradiometric boundary-value problem, are also summarized. Secondly, we apply the third-order differential operators to the analytical solutions of the gradiometric boundary-value problem which gives 30 new integral formulas transforming (1) vertical-vertical, (2) vertical-horizontal and (3) horizontal-horizontal second-order gravitational tensor components onto their third-order counterparts. Using spherical polar coordinates related sub-integral kernels can efficiently be decomposed into azimuthal and isotropic parts. Both spectral and closed forms of the isotropic kernels are provided and their limits are investigated. Thirdly, numerical experiments are performed to test the consistency of the new integral transforms and to investigate properties of the sub-integral kernels. The new mathematical apparatus is valid for any harmonic potential field and may be exploited, e.g., when gravitational/magnetic second- and third-order tensor components become available in the future. The new integral formulas also extend the well-known Meissl diagram and enrich the theoretical apparatus of geodesy.

Slow magnetic Rossby waves in the Earth's core

AbstractThe westward drift component of the secular variation is likely to be a signal of waves riding on a background mean flow. By separating the wave and mean flow contributions, we can infer the strength of the “hidden” azimuthal part of the magnetic field within the core. We explore the origin of the westward drift commonly seen in dynamo simulations and show that it propagates at the speed of the slow magnetic Rossby waves with respect to a mean zonal flow. Our results indicate that such waves could be excited in the Earth's core and that wave propagation may indeed play some role in the longitudinal drift, particularly at higher latitudes where the wave component is relatively strong, the equatorial westward drift being dominated by the mean flow. We discuss a potential inference of the RMS toroidal field strength within the Earth's core from the observed drift rate.

Walsh modes and radial quantum correlations of spatially entangled photons

Orthogonal sets of 2D transverse modes are key to controlling the spatial degree of freedom of light in a classical or quantum context. In contrast to the azimuthal part, which is easily accessible using orbital angular momentum modes, control of the radial part is more difficult. We show here that simple sets of orthogonal binary sequences, the Walsh functions, provide a workable solution for exploration of the radial space with phase-only spatial light modulation. We demonstrate this by measuring "sequency" quantum correlations between different radial Walsh modes of spatially entangled photon pairs and for numerically optimized versions thereof.

Exact solutions of the Dirac equation with Pöschl—Teller double-ring-shaped Coulomb potential via the Nikiforov—Uvarov method

Exact analytical solutions of the Dirac equation are reported for the Pöschl—Teller double-ring-shaped Coulomb potential. The radial, polar, and azimuthal parts of the Dirac equation are solved using the Nikiforov—Uvarov method, and the exact bound-state energy eigenvalues and corresponding two-component spinor wavefunctions are reported.

Spherical-separability of Non-Hermitian Hamiltonians and Pseudo- $\mathcal{PT}$ -symmetry

Non-Hermitian but \(\mathcal{P}_{\varphi }\mathcal{T}_{\varphi }\) -symmetrized spherically-separable Dirac and Schrödinger Hamiltonians are considered. It is observed that the descendant Hamiltonians H r , H θ , and H φ play essential roles and offer some “user-feriendly” options as to which one (or ones) of them is (or are) non-Hermitian. Considering a \(\mathcal{P}_{\varphi }\mathcal{T}_{\varphi }\) -symmetrized H φ , we have shown that the conventional Dirac (relativistic) and Schrödinger (non-relativistic) energy eigenvalues are recoverable. We have also witnessed an unavoidable change in the azimuthal part of the general wavefunction. Moreover, setting a possible interaction V(θ)≠0 in the descendant Hamiltonian H θ would manifest a change in the angular θ-dependent part of the general solution too. Whilst some \(\mathcal{P}_{\varphi }\mathcal{T}_{\varphi }\) -symmetrized H φ Hamiltonians are considered, a recipe to keep the regular magnetic quantum number m, as defined in the regular traditional Hermitian settings, is suggested. Hamiltonians possess properties similar to the \(\mathcal{PT}\) -symmetric ones (here the non-Hermitian \(\mathcal{P}_{\varphi }\mathcal{T}_{\varphi }\) -symmetric Hamiltonians) are nicknamed as pseudo- \(\mathcal{PT}\) -symmetric.

Dynamical Friction of a Circular‐Orbit Perturber in a Gaseous Medium

We investigate the gravitational wake due to, and dynamical friction on, a perturber moving on a circular orbit in a uniform gaseous medium using a semianalytic method. This work is a straightforward extension of the work of Ostriker, who studied the case of a straight-line trajectory. The circular orbit causes the bending of the wake in the background medium along the orbit, forming a long trailing tail. The wake distribution is thus asymmetric, giving rise to the drag forces in both opposite (azimuthal) and lateral (radial) directions to the motion of the perturber, although the latter does not contribute much to orbital decay. For subsonic motion, the density wake with a weak tail is simply a curved version of that in Ostriker and does not exhibit the front-back symmetry. The resulting drag force in the opposite direction is remarkably similar to the finite-time, linear-trajectory counterpart. On the other hand, a supersonic perturber is able to overtake its own wake, possibly multiple times, and develops a very pronounced tail. The supersonic tail surrounds the perturber in a trailing spiral fashion, enhancing the perturbed density at the back, as well as the far front, of the perturber. We provide the fitting formulae for the drag forces as functions of the Mach number, whose azimuthal part is surprisingly in good agreement with Ostriker's formula, provided Vpt = 2Rp, where Vp and Rp are the velocity and orbital radius of the perturber, respectively.

On hydromagnetic instabilities and the mean electromotive force in a non-uniformly stratified Earth’s core affected by viscosity

Linear magnetoconvection in a model of a non-uniformly stratified horizontal rotating fluid layer with a toroidal magnetic field is investigated for no-slip and finitely electrically conductive boundaries and with very thin stably stratified upper sublayer. The basic parabolic temperature profile is determined by the temperature difference between the boundaries and by the homogeneous heat source distribution in the layer. This results in a density pattern, in which a stably stratified upper sublayer is present. The developed diffusive perturbations (modes) are strongly affected by the complicated coupling of viscous, thermal and magnetic diffusive processes. The calculations were performed for various values of Roberts number (q ≪ 1 and q = O(1)). The mean electromotive force produced by the developed hydromagnetic instabilities is investigated to find the modes, which can be appropriate for creating the α-effect. It was found that the azimuthal part of the EMF is dominant for westward modes when the Elsasser number Λ ≲ O(1).

Stimulated Stokes and anti-Stokes Raman scattering in liquid acetone with a Bessel beam

The first to third Stokes, and first and second anti-Stokes components of stimulated Raman scattering in acetone generated with a conical J 0 Bessel–Gauss beam were investigated experimentally and theoretically as a function of pump cone angle ϑ P. From the observed first Stokes spot and cone the latter is caused by gain guiding. All other Stokes and anti-Stokes cones are generated in cascaded four-wave mixing processes. Cone angles are discussed in terms of longitudinal and transverse wave vector matching. Experiments where an azimuthal part of the cone of the pump beam was blocked demonstrate exact non-planar phase matching over a wide range of pump cone angles for the first and second anti-Stokes cones.

Quantum flux-based resonances in solitonic vortices

Working with the Nielsen–Olesen Lagrangian in static cylindric coordinates, we derive the system of coupled field equations and perform a first-order perturbative approach, pointing out an interesting contribution connected to the London–Heitler current. For an r, θ-depending scalar boson, evolving in a constant or zero magnetic field we get, besides the flux quantization and the Landau energy levels, a less expected structure of the scalar modes whose radial and azimuthal parts are decoupled by the presence of the quantized magnetic flux.

Nonplanar phase-matching of stimulated anti-Stokes Raman scattering pumped by a Bessel beam

We have studied the behaviour of anti-Stokes emission of stimulated Raman scattering (SRS) in hydrogen gas pumped with a Bessel beam of variable cone angle. We demonstrate that for the generation of anti-Stokes emission via a four-photon process including pump and Stokes photons two types of phase-matching exist: the usual planar phase-matching involving two collinear pump photons and nonplanar phase-matching involving two noncollinear pump photons with wave vectors on the surface of the cone of the Bessel beam. Both types of phase-matching are distinguished experimentally by their different behaviour when a small azimuthal part of the pump wave vectors is blocked.

Volume and surface rf power absorption in a helicon plasma source

We consider the excitation and absorption of waves in a plasma cavity with parameters typical for helicon sources. rf power is shown to be transferred into the plasma via two channels. The first is realized by weakly damping helicon waves which are excited directly by the azimuthal parts of the antenna and penetrate into the bulk plasma. Electrostatic waves arising at a plasma edge owing to the linear mode conversion form the second channel. Strongly damped electrostatic waves can reach a plasma core at low magnetic fields only, while at high fields they deposit energy at the periphery of the plasma column. A principal fraction of the rf power is transferred into the plasma via the electrostatic channel, and so the power input turns out to be volume at low magnetic fields and surface at high fields. An enhanced volume input is possible at high fields in special anti-resonance regimes when the excitation of the electrostatic wave is suppressed. The effective collision frequency is introduced to describe effective damping of helicon waves arising due to their conversion into electrostatic waves. The results obtained permit explanation of both measured field profiles and the high absorption efficiency in helicon sources.

Electromagnetic energy density around a superconducting cosmic string

The electromagnetic energy density w(r) is calculated outside an infinitely long and straight superconducting cosmic string of finite radius a. The feature distinguishing this situation from the analogous situation in Minkowski space is the presence of the factor in the azimuthal part of the line element, G being the gravitational constant and the string mass per unit length. We regularize w(r) by subtracting (i) one volume term arising from the (boundary independent) volume terms in the two scalar Green's functions, and (ii) one term giving the energy density for the same geometrical configuration but with the gravitational field switched off. The remaining regularized energy density is calculated and illustrated graphically. The expression is finite everywhere for r > a, but diverges near the surface as . This is the typical kind of divergence encountered for the electromagnetic field near a curved boundary even in Minkowski space. The physical reason for this divergence is the assumed idealized boundary conditions at the surface.

Path integral quantization of certain noncentral systems with dynamical symmetries

Path integral quantization is done for the five classes of potentials appearing in the systematic search for nonrelativistic systems with dynamical symmetries done by Makarov, Smorodinsky, Valiev, and Winternitz [Nuovo Cimento A 52, 1061 (1967)]. By an iterated application of Bateman’s series formula to the polar coordinate path integral, an expansion is obtained of the Feynman kernel or the Green’s function, whichever is possible, in terms of hypergeometric functions of the polar and azimuthal parts and a radial path integral is obtained whose evaluation yields the energy eigenvalues and the normalized wave functions. Special cases include the Hartmann potential and the ring-shaped oscillator.