On a Babcock-Leighton dynamo model with a deep-seated generating layer for the toroidal magnetic field, II

In a previous paper (Paper I), we studied a dynamo model of the Babcock-Leighton type (i.e., the surface eruptions of toroidal magnetic field are the source for the poloidal field) that included a thin, deep seated, generating layer (GL) for the toroidal field, B. Meridional motions (of the order of 12 m s^–1 at the surface), rising at the equator and sinking at the poles were essential for the dynamo action. The induction equation was solved by approximating the latitudinal dependence of the fields by Legendre polynomials. No solutions were found with _p = _f where _p and _f are the fluxes for the preceding and following spot, respectively. The solutions presented in Paper I, had _p = –0.5 _f , were oscillatory in time, and large radial fields, B, were present at the surface.Here, we resume the study of Paper I with a different numerical approach allowing for a much higher resolution in , the polar angle. The time dependent partial differential equations for the toroidal and poloidal field are solved with the help of a second order, time and space centered, finite difference scheme. Oscillatory solutions with _p = _f are found for various values of the meridional motions and diffusivity coefficients. The surface values of B, while considerably smaller than those of Paper I, are still unacceptably large, specially at the poles. The reason can be traced to the eruption of toroidal field at high latitudes. It appears that in order to obtain small values for the radial field in the polar regions, high latitude sources ( smaller than /4, say), must reach their maximum below the surface. Weaker meridional motions near the poles than in the equatorial region are also suggested.     

A relation in families of periodic solutions

We show the existence of a general relation between the parameters of periodic solutions in dynamical systems with ignorable coordinates. In particular, for time-independent systems with an axis of symmetry, the relation takes the form T/A=–/E, whereT is the period,A is the angular momentum, is the angle through which the system has rotated after one period, andE is the energy.     

Isotropic flat space cosmology in Jordan-Brans-Dicke theory

The process of re-escalation of the scalar field as R ³, the energy density as R ³, and the pressurep aspR ³P, lends itself to obtain a reduced equation that represents, for a wide variety of equations of state, the cosmological evolution of an homogeneous and isotropic, flat Universe. A particular solution to this equation is presented.     

A novel methodology for radiative transfer in a planetary atmosphere

A novel methodology for evaluating the field of anisotropically scattered radiation within a homogeneous slab atmosphere of arbitrary optical thickness is provided. It departs from the traditional radiative transfer approach in first considering that the atmosphere is illuminated by an isotropic light source. From the solution of this problem, it subsequently proceeds to that for the more conventional case of monodirectional illumination. The azimuthal dependence of the field is separated in the usual manner by an harmonic expansion, leaving a problem in four dimensions (=optical depth, ₀=thickness, , =directions of incidence and scattering) which, as is well known, is numerically extremely inconvenient. Two auxiliary radiative transfer formulations of increasing dimensionality are considered: (i) a transfer equation for the newly introduced functionb ^m(,,₀) with Sobolev's function ^m(,₀) playing the role of a source-function. Because the incident direction does not intervene, ^m is simply expressed as a single integral term involvingb ^m. For bottom illumination, an analogous equation holds for the other new functionh ^m(,,₀). However, simple reciprocity relations link the two functions so that it is only necessary to considerb ^m; (ii) a transfer equation for the other new functiona ^m(,,,₀) with a source-function provided by Sobolev's functionD ^m(,,₀). For bottom illumination, another functionf ^m(,,,₀) is introduced; by a similar argument using reciprocity relations,f ^m is reduced toa ^m rendering necessary only the consideration ofa ^m. However, a fundamental decomposition formula is obtained which shows thata ^m is expressible algebraically in terms of functions of a single angular variable. The functions ^m andD ^m are shown to be the values in the horizontal plane ofb ^m anda ^m, respectively. The other auxiliary functionsX ^m andY ^m are also expressed algebraically in terms ofb ^m. These results enable one to proceed to the final step of evaluating the radiation field for monodirectional illumination. The above reductions toalgebraic relations involving only the functionb ^m appear to be more advantageous than Sobolev's (1972) recent approach; they also circumvent some basic numerical difficulties in it. We believe the present approach may likewise prove to be superior to most (if not all) other methods of solution known heretofore.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory under Contract No. NAS-7-100 sponsored by the National Aeronautics and Space Administration.     

The local complex of high-velocity clouds (geometry and kinematics)

Many HVC - high velocityHi clouds - are added to the known combination of diffuse soft X-rays and radioloops. Among these HVC there are previously examined complexes A, C, Magellanic Stream, etc. Their common composition has a clear geometry and kinematics.Two powerful long HVC filaments form the bridges between Loop I and double Loop II + III. The Bridges join the polarX _max regions in C band. The bridges ends at Loop I side are bounded by the systems of shorter filaments withX _max spots in M₁ and larger energy bands. There are no HVC in the spots themselves. The opposite ends of bridges reach Loops II and III centres. The two structures have the features of rotational symmetry.The isolines X-ray count rates pass along HVC chains with the same _r -velocities. The frequent alternation of different _r on the same filament is better explained by the turbulent gas motion than by the differences of HVC distance.All of the above-mentioned, as well as the _r signs of dependence on the galactical longitude, are the result of HVC interaction with X-rays. The whole composition from HVC-loops-X-rays is a component of the local system (Gould Belt).     

Linear current instability in loop prominences

By use of the dispersion equation given by Song, Wu, and Dryer (1987) for a cylinder plasma with mass motion and gravity included, we investigate the linear current instabilities developed in loop prominences. The results indicate that the mode of linear instability depends mainly on whetherv _s ² > or not, wherev _s is the sonic velocity at heightz, =GM/(R +z) is the gravity potential,G the gravitational constant,M andR the mass and the radius of the Sun respectively. Ifv _s ² > , then the sausage instability will be dominant. Otherwise, the kink instability will be more important. A possible explanation of knot structure, which appears sometimes in solar loop prominences has been given.     

On the A-transform of the exponential integrals

In this paper, a technique of recursive analysis is developed for the integral transform A of the exponential integral functionsE _n which is denoted as _n (). The main result of this analysis enables us to establish a two-term recurrence formula for _n (0) and a three-term recurrence formula for _n (); 0. A computational algorithm based on these formulae is also constructed and its numerical results forn=2(1)25 are presented to 15-digit accuracy.     

On the periodic solutions of a particular Lame equation

We consider the Hill's equation: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% WGKbWaaWbaaSqabeaacaaIYaaaaOGaeqOVdGhabaGaamizaiaadsha% daahaaWcbeqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaWGTbGaai% ikaiaad2gacqGHRaWkcaaIXaGaaiykaaqaaiaaikdaaaGaam4qamaa% CaaaleqabaGaaGOmaaaakiaacIcacaWG0bGaaiykaiabe67a4jabg2% da9iaaicdaaaa!4973!\[\frac{{d^2 \xi }}{{dt^2 }} + \frac{{m(m + 1)}}{2}C^2 (t)\xi = 0\]Where C(t) = Cn (t, {frbuilt|1/2}) is the elliptic function of Jacobi and m a given real number. It is a particular case of theame equation. By the change of variable from t to defined by: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcaawaaOWaaiqaaq% aabeqaamaalaaajaaybaGaamizaGGaaiab-z6agbqaaiaadsgacaWG% 0baaaiabg2da9OWaaOaaaKaaGfaacaGGOaqcKbaG-laaigdajaaycq% GHsislkmaaleaajeaybaGaaGymaaqaaiaaikdaaaqcaaMaaeiiaiaa% bohacaqGPbGaaeOBaOWaaWbaaKqaGfqabaGaaeOmaaaajaaycqWFMo% GrcqWFPaqkaKqaGfqaaaqcaawaaiab-z6agjab-HcaOiab-bdaWiab% -LcaPiab-1da9iab-bdaWaaakiaawUhaaaaa!51F5!\[\left\{ \begin{array}{l}\frac{{d\Phi }}{{dt}} = \sqrt {(1 - {\textstyle{1 \over 2}}{\rm{ sin}}^{\rm{2}} \Phi )} \\\Phi (0) = 0 \\\end{array} \right.\]it is transformed to the Ince equation: (1 + · cos(2)) y + b · sin(2) · y + (c + d · cos(2)) y = 0 where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcaawaaiaadggacq% GH9aqpcqGHsislcaWGIbGaeyypa0JcdaWcgaqaaiaaigdaaeaacaaI% ZaGaaiilaiaabccacaWGJbGaeyypa0Jaamizaiabg2da9aaacaqGGa% WaaSaaaKaaGfaacaWGTbGaaiikaiaad2gacqGHRaWkcaaIXaGaaiyk% aaqaaiaaiodaaaaaaa!4777!\[a = - b = {1 \mathord{\left/{\vphantom {1 {3,{\rm{ }}c = d = }}} \right.\kern-\nulldelimiterspace} {3,{\rm{ }}c = d = }}{\rm{ }}\frac{{m(m + 1)}}{3}\]In the neighbourhood of the poles, we give the expression of the solutions.The periodic solutions of the Equation (1) correspond to the periodic solutions of the Equation (3). Magnus and Winkler give us a theory of their existence. By comparing these results to those of our study in the case of the Hill's equation, we can find the development in Fourier series of periodic solutions in function of the variable and deduce the development of solutions of (1) in function of C(t).     

Effects of Hall current on hydromagnetic free-convective flow through a porous medium

An analysis of the effects of Hall current on hydromagnetic free-convective flow through a porous medium bounded by a vertical plate is theoretically investigated when a strong magnetic field is imposed in a direction which is perpendicular to the free stream and makes an angle to the vertical direction. The influence of Hall currents on the flow is studied for various values of .Nomenclature c _p specific heat at constant pressure - e electrical charge - E Eckert number - E electrical field intensity - g acceleration due to gravity - G Grashof number - H ₀ applied magnetic field - H magnetic field intensity - (j _x , j _y , j _z ) components of current densityJ - J current density - K permeability of porous medium - M magnetic parameter - m Hall parameter - n _e electron number density - P Prandtl number - q velocity vector - (T, T _w , T ) temperature - t time - (u, v, w) components of the velocity vectorq - U ₀ uniform velocity - v ₀ suction velocity - (x, y, z) Cartesian coordinates Greek Symbols angle - coefficient of volume expansion - _e cyclotron frequency - frequency - dimensionless temperature - thermal conductivity - coefficient of viscosity - magnetic permeability - kinematic viscosity - mass density of fluid - _e charge density - electrical conductivity - _e electron collision time     

YO molecule in the sun

Vibrational transition probabilities, namely Franck—Condon factors and -centroids have been evaluated by an approximate analytical method for the (A–X), (A–X), and (A–X) system of YO molecule. Morse potential energy curves forX ²⁺,A ²₂,A²₂, andA²₂, states of YO have been constructed using the latest spectroscopic data. The value of -centroids for the band have been found to decrease linearly with the corresponding wavelengths. We show results for two new transitions of (A–X) and (A–X) and five new bands of (A–X) of YO in the umbral spectrum of the Sun.     

Horizontal-branch star models

Evolutionary tracks of 0.9M , 0.7M and 0.6M models for Population II stars have been computed in the post-red giant phase. The stars are initially composed of a helium core containing a mass fraction equal to 0.9 and of a hydrogen-rich envelope. They represent hypothetical remnants of stars after substantial mass loss in previous evolutionary phases or/and at the helium flash.u 0,9M , 0,7M 0,6M . , 90% , . , .Riassunto È stata calcolata per modelli di stelle di 0,9M , 0,7M e 0,6M di Popolazione II l'evoluzione successiva allo stadio di gigante rossa. Inizialmente i modelli constano di un nucleo di elio contenente il 90% della massa totale e di un inviluppo idrogenico. Essi possono interpretarsi come i resti di stelle che abbiano perso una frazione di massa considerevole in fasi evolutive precedenti, o all'innesco violento dell'elio al centro.     

Sobolev's Φ-function of radiative transfer in planar and spherical scattering media

In recent years Sobolev's -function of radiative transfer has been discussed in connection with the resolvent of Milne's integral equation such that it plays an important role in the determination of the radiation field in semi-infinite (or finite) atmospheres with internal sources (cf. Sobolev, 1963). In the present paper, the part of Sobolev's -function in plane-parallel and spherical, isotropically scattering, atmospheres with internal source distribution is investigated from analytical and numerical aspects. With the aid of invariant imbedding (cf. Bellmanet al., 1968), we computed Sobolev's -function of Milne's integral equation for the planar case by solving the Cauchy system for the auxiliary function and Chandrasekhar'sX- andY-functions. The corresponding -function for the spherical case is readily obtained from the for the planar case.Investigation supported by the National Science Foundation under Grant No. GP 29049, the Atomic Energy Commission, Division of Research under Contract No. AT(04-3)-113, Project 19, and the National Institutes of Health under Grant No. 16197-05.     

Spectral analysis of cyclotron radiation in anisotropic plasmas

Following up our previous analysis of cyclotron radiation in anisotropic plasmas, we derived expression for the power received at a far field point per unit frequency range along the group velocity direction dP(, )/d. We then carry out a series of numerical analysis presenting the spectral features rather than directional features of cyclotron radiation. In particular, we analyse the power received per unit solid angle per unit frequency range d² P(, )/(d d). It is expected the analysis result presented here can be compared directly with observation for parameters pertaining to astrophysical plasmas in stellar and terrestrial atmospheres.     

Isotropization in Brans-Dicke-Bianchi type-V

In this paper we find that from the exact solution for Bianchi type-V in the Brans-Dicke theory with =0 the Hubble parameters are the same for , so that the Universe will be isotropized.     

The ‘critical inclination’: Another look

The occurrence and uniqueness of the critical inclination in satellite theory is discussed.An infinite set of canonical transformations in Hill variables are shown to exist whereby the first order secular part of the disturbing function can be changed into an alternative form. As a result of such a transformation the critical inclination can become (a)any other real or complex inclination or a function independent of the satellite's orbital inclination and (b) a function dependent on the semi-major axis, eccentricity and inclination of the satellite's orbit. It is also shown that all transformations of types (a) and (b) are only valid for short intervals of time of the order of a few satellite revolutions. Furthermore if such transformations are modified so that they become valid for greater intervals of time, then the resulting solutions in all cases containno singular divisor other than the critical inclination.It is concluded that the singularity at the critical inclination is unique and that it represents an actual physical resonance rather than something resulting from the method of solution or the type of variable used in the analysis. This conclusion is supported by numerical evidence which shows that a satellite's perigee height does suffer resonant changes when the orbital inclination is equal to the critical inclination of 63.°4.     

Free convection and mass transfer effects on the hydro-magnetic oscillatory flow past an infinite vertical porous plate

Two-dimensional unsteady free convection and mass transfer, flow of an incompressible viscous dissipative and electrically conducting fluid, past an infinite, vertical porous plate, is considered, when the flow, is subjected in the action of uniform transverse magnetic field. The magnetic Reynolds number is taken to be small enough so that the induced magnetic field is negligible. The solution of the problem is obtained in the form of power series of Eckert numberE, which is very small for incompressible fluids. Analytical expressions for the velocity field and temperature field are given, as well as for the skin friction and the rate of heat transfer for the case of the mean steady flow and for the unsteady one. The influence of the magnetic parameter,M, modified Grashof numberG _c , Schmidt numberS _c and frequency , on the flow field, is discussed with the help of graphs, when the plate is being cooled, by the free convection currents (G _r ,E>0), or heated (G _r ,E<0). A comparative study with hydrodynamic case (M=0) and the hydromagnetic one (M0) is also made whenever necessary.List of symbols B₀ applied magnetic field - |B| amplitude of the skin friction - C concentration inside the boundary layer - C concentration in the free stream - C _w concentration at the porous plate - C _p specific heat at constant pressure - D diffusion coefficient - E Eckert number - g _x acceleration due to gravity - G _c modified Grashof number - G _r Grashof number - M magnetic parameter - N _u Nusselt number - P Prandtl number - |Q| amplitude of the rate of heat transfer - S _c Schmidt number - T temperature of the fluid - T _w temperature of the plate - T temperature of the fluid in the free stream - T _r ,T _i fluctuating parts of the temperature profile - u, v velocity components in thex, y directions - u dimensionless velocity in thex direction - u ₀ mean steady velocity - u ₁ unsteady part of the velocity - u _r ,u _i fluctuating parts of the velocity profile - U dimensionless free stream volocity - U ₀ mean free stream velocity - v ₀ suction velocity - x, y co-rodinate system Greek Symbols phase angle of the skin-friction - coefficient of volume expansion - _* coefficient of expansion with concentration - phase angle of the rate of heat transfer - dimensionless co-ordinate normal to the plate - dimensionless temperature - ₀ mean steady temperature - ₁ unsteady part of temperature - k thermal conductivity - v kinematic viscocity - density of fluid in the boundary layer - density of fluid in the free stream - electrical conductivity of the fluid - skin friction - ₀ mean skin friction - frequency - dimensionless frequency     

Eruption of a helically twisted prominence

In this paper we analyze the eruption of a prominence, characterized by a helical-like structure and by a non-linear rising motion. We approximated the prominence as a cylindrical curved flux tube and estimated the behaviour of several geometrical parameters during the activation and the eruption phases. We determined that, at the onset of the activation, the number N of turns of a magnetic field line over the whole length of the prominence was 5.0, while the value of the ratio P/r ₀ between the pitch of the magnetic field lines and the prominence width was 0.45. These values are in good agreement with those predicted by the kink-mode instability. Moreover, we found a decrease of the total twist of one helical thread from 10 to 2 during the prominence eruption, indicating a relaxation of the magnetic field towards a less twisted configuration. We conclude that the prominence was initially destabilized by the kink-mode instability and, not succeeding in finding a new equilibrium configuration, it erupted.     

Approximate expressions for Gyrosynchrotron radiation in Transverse Propagation

Two sets of accurate approximate expressions for the gyrosynchrotron radiation in the transverse propagation case are presented for the first time. They contain emissivity _–/BNand absorptivity _– B/Nfor e-mode, effective temperature T _effand frequency of peak brightness _p. The expressions are designed for the range 2 to 7 of electron energy spectral index and for the ranges from 2 to 10 and 10 to 100 of harmonic numbers s(=/_B). Their statistical error is, respectively, ±18% and ±29% for _–/BNand _– B/Nfor 10/_B100, ±128% and and ±170% for 2/_B10.     

Классикаци дбиижэхий в обобщэхной плоцкой задачэ трех неподБижнΨη цэнтпов

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On the classification of motion in the generalized two-dimensional problem of three fixed centres

A qualitative analysis and classification of forms of motion in the problem under consideration have been carried out using a method (applicable to any case of integrability) due to Liouville. All the forms of the two-dimensional motions for any masses (negative and complex as well) at fixed centres corresponding to the real potential have been considered.

     

The numerical evaluation of theH-functions of radiative transfer in active amplifying media

The present paper contains extensive tables of the values of theH-functionH(z,<0) and of the moments ofQ(x) (in terms of which the moments ofH(z, ) can be determined) appropriate for transfer of radiation in active amplifying media in which<0. These values have been computed correct to the 7th decimal place for values of in the range (–10^–12)-(–10³⁰) and for values of z[0,1] with the aid of a 48-point gaussian quadrature formula.