Numerical treatment of the unsteady hydromagnetic thermal boundary layer problem

This paper presents a suitable numerical method for the treatment of the unsteady hydromagnetic thermal boundary layer problem for flows past an infinite porous flat plate, the motion of which is governed by a general time-dependent law, under the influence of a transverse externally set magnetic field. The normal velocity of suction/injection at the plate is also assumed to be time-dependent. The results obtained on the basis of numerical approximations seem to compare favourably with earlier results (Pandeet al., 1976; Tokis, 1978). Analytical approximations are given for the cases of a plate (i) generally accelerated and (ii) harmonically oscillating. The direct numerical treatment is obviously advantageous since it allows, handling of cases where the known methods for analytical approximations are not applicable. This problem is closely related to the motions and heat transfer occurring locally on the surfaces of stars.     

Constraints and numerical integration

In an autonomous Hamiltonian system, one constraint always exists, namely the energy integral or a constant magnitude of the 4-velocity in relativistic dynamics. The constraint should bring better numerical stability if it can be kept at every step of the numerical integration. In Newtonian mechanics, the order of the equations of motion can not be reduced by use of the constraint in most cases, because its kinetic energy is usually a quadratic form of the elliptic type and one would meet difficulty when trying the order reducing. However, the metric in general relativity is hyperbolic. In particular, when the spacetime bears some symmetries there exists a global transformation so that at least one element in its main diagonal vanishes. As a result, the constraint can be solved for a certain velocity or a momentum without any difficulty, and so reducing the order. Similarly, this technique can also be applied to the evolution of the Mixmaster universe. It is shown that this technique can raise the precision and improve the numerical stability dramatically even when a classical integrator is used, although it might not keep the symplectic structure of the system.     

约束条件和数值积分

自治的哈密顿系统存在约束条件，例如能量积分或广义相对论中的4速度大小为常数，它能否在数值积分过程中始终满足将直接影响数值稳定性．在牛顿力学中哈密顿系统的动能一般为椭圆型，直接运用约束条件对方程进行降阶存在开平方判断正负号的困难，导致应用高精度的经典数值积分器时能量存在耗散．然而相对论力学的度规为双曲型，利用约束条件有可能实行方程降阶．在时空具有一定对称性的情况下，能够找到整个时空的一个全局变换使变换后的度规的主对角线某一元素为零，于是从约束方程中不需开平方能够解出某一动量，顺利实现运动方程的降阶．相对论力学中另一个可以降阶的模型是Mixmaster宇宙模型．数值实验表明将经典算法用于降阶后的运动方程能够严格地满足约束，但不一定能保持辛结构。     

Cases of a star's planar motion integrated in the gravitational field of a rotating system

The paper deals with some special cases of the existence of a local integral of motion in a two-dimensional potential field rotating with constant angular velocity. In this case the trajectories may be completely determined, which is not always possible in other cases with a local integral, in contrast to the cases with a true integral. Some cases where the trajectories can be determined analytically are trivial but there are also some new nontrivial cases.     

Chaos in N-body systems

Here is a selection of applications of what is now called theory of dynamical systems in galactic dynamics and N-body systems. The study of chaotic motions in potentials used as a model for elliptical galaxies is a first example of these applications. The interest in this problem stems from the fact that there are now many theoretical and observational evidences that the overall potentials of galaxies are indeed non-integrable. There are classes of objects, for example small and intermediate luminosity elliptical galaxies, for which the presence of the famous third integral is not necessary or others in which we observe peculiarities in their photometry or kinematics. We address here some of these issues and their implications in modifying our current understanding of the structure and evolution of galaxies.More in general, there is the natural question of how the systems we see have settled to their present status and what would happen if some external cause perturbs it. This issue is related to the question of the stochasticity involved in the general N-body dynamics, especially when N is very large. An N-body dynamical system is definitely chaotic, as shown by several numerical investigations, at least for N not very large. However, this statement must be reconciled with the picture of non-collisional equilibrium of big systems. The second part of this review presents a survey of numerical experiments and an interpretation of the results obtained using standard chaoticity indicators.     

Analytic results for screened non-resonant nuclear reaction rates

This paper deals with the closed-form representation of the non-resonant nuclear reaction rate taking into account electron screening effects for the reacting particles. The basic physical principles concerning nuclear reactions in dense astrophysical plasmas are applied to derive the representation of the screened nuclear reaction rate integral. Taking advantage of the theory of Meijer'sG-function it is shown that both the reaction rate integrals with and without screening corrections can be represented in closed form. Mathematical approximations for the reaction rate integral, till now considered as inevitable in the literature, are avoided and exact computable representations given.     

Solution of a Riemann–Hilbert problem for a new expression of Chandrasekhar’s H-function in radiative transfer

The linear singular integral equation derived from the nonlinear integral equation of Chandrasekhar’s H-function in radiative transfer is considered here to develop a new form of H-function as a solution of a Riemann–Hilbert problem using Plemelj and Cauchy integral formulae for complex domain. This new form of H-function is a simple integral of known functions. Forms of H-function both for conservative and nonconservative cases are obtained. Their numerical evaluations are made by Simpson’s one-third rule to arrive at an accuracy to ninth places of decimals.     

The general analytical expression for computation of generalized relativistic Fermi-Dirac functions

In this paper, general sufficiently analytical formulae are developed for the arbitrary order generalized relativistic Fermi-Dirac (FD) functions. Analytical assessment of relativistic FD function is very important for various fields of physics especially in the theory of relativistic nondegenerate and degenerate electron gas systems. One of the more appropriate and correct approximations is based on a binomial expansion method and incomplete Gamma functions that have been used in the calculations of the generalized relativistic FD functions. Note that, the established expression in special cases of specific values of parameters becomes the evaluation formulae of other type FD functions. Calculation results of the generalized relativistic FD functions are compared with the other approximations methods and available numerical approaches and demonstrated satisfactory agreement.     

About the third integral of charged particle motion in strongly curved magnetic fields

Charged particle motion is studied in magnetic fields with an inversion of field direction and a strong curvature in the reversal region, that means field structures which are expected to play a crucial role in energetics and dynamics of space plasma. Investigations are performed in a typical field model. Due to its symmetry and stationarity two of the integrals of motion of the resulting Hamiltonian immediately arise. The question is analyzed whether the system is integrable (this means regular solutions exist) and of what kind the third integral is. Two different third integrals are found, which are valid in different parts of the phase space. Their validity is estimated analytically for particles, crossing the neutral plane (z = 0) -- in addition the action integral I_z previously unusual in plasma applications is verified numerically. Further numerical research establishes that there are no more third integrals -- all remaining parts of the phase space are filled with chaotic solutions. Because in strongly curved field reversals the conservation of the magnetic moment as an integral of motion is shown to be restricted to very small energies or very large pitch angles, the unusual I_z-integral will be an important tool for solution of plasma problems of cosmical current sheets and plasma boundaries.     

基于Broyden方法的不对称与对称周期解延拓算法与应用

考虑周期解的数值延拓问题并提出基于Broyden拟牛顿法来延拓周期解的一种有效算法,先后以布鲁塞尔振子、平面圆型限制性三体问题(Planar Circular Restricted Three-Body Problem, PCRTBP)的周期解为例进行了验证.这里的Broyden方法包含线性搜索、正交三角分解求线性方程组的步骤.对一般的周期解,周期性条件方程组中含有周期作为待延拓参数,可用周期来决定积分时长,将解代入周期性条件得到积分型的非线性方程组,利用Broyden方法迭代延拓直至初值收敛.根据两次垂直通过一个超平面的轨道是对称周期轨道的性质,可采用插值的方法求得再次抵达超平面的解分量,得到周期性条件方程组,再用Broyden方法求解.结合哈密顿系统的对称性和PCRTBP周期轨道的一些分类,对2/1、3/1的内共振周期解族进行了数值研究.最后,对算法和计算结果做了总结和讨论.     

Is the third integral a function of the Hamiltonian?

Numerical evidence is presented which indicates that, although the third integral is tangent to the Hamiltonian (energy integral) along some periodic orbits (as has been shown by Goudas), it is not tangent to it along non-periodic orbits; therefore it is not a function of the Hamiltonian. The set of periodic orbits is probably dense in general, but a given form of the third integral is valid in the neighbourhood of a limited number of them; no form of the third integral is valid for all periodic orbits, except in integrable cases.     

Symmetric theory of probe-plasma interactions

The theory of interactions between a probe and the surrounding plasma at rest is developed in a spherically and in a cylindrically symmetric model (probe theory). The theory is based on the Vlasov-Poisson system; a general numerical program was developed to solve this system by means of an iterative procedure. Various ambient plasma and charged particle emission properties are described by the complete set of boundary conditions for the distribution functions in the phase space. By use of this numerical method, potential and space charge density in the whole surroundings of the probe as well as the current densities of all plasma constituents are calculated self-consistently.Furthermore, the regions of the phase space with particle trajectories of the same kind can be approximated depending on the plasma properties. Then, the current densities can be estimated analytically. This approach to the problem yields self-consistent approximations and is the only stringent derivation of the thick sheath and of the thin sheath approximation of the classical Langmuir theory. These approximations are generalized with respect to the charged particle emission from the surface.The symmetric probe theory is applied to the following problems of spacecraft environment and spacecraft charging: (i) a spacecraft in the ionosphere with very negative surface potential, (ii) a spacecraft in the solar wind with strong photoelectron emission, and (iii) a spacecraft in the transition region of comet Halley with very strong secondary plasma emission.     

Numerical stabilization of all laws of conservation in the many body problem

When a system of differential equations admits a first integral (e.q. the law of energy), the value of that integral may be used as a check during the numerical integration. Often this check is satisfied with poor accuracy since the existence of the first integral is unknown to the computer. The aim of the paper is to show how such a first integral can be satisfied with better accuracy and in a stabilized manner by adding an appropriate control term to the differential system. The accuracy of the numerical integration is thereby improved. The basic idea is applied to the problem ofN bodies.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

Transient Flow of a Magnetised Plasma due to Thermal perturbation

At time t<0, a steady stationary condition of a fully ionised gas exists in space, such that the velocity components, the induced magnetic field and a uniform temperature are given by (0,0,0),(0, 0,H_o) and Θ, respectively, in the rectangular coordinate (x,y,z). At t =0, a sudden increase of temperature is applied at z = 0. If the Eckert terms are neglected, an integral transform technique is employed to solve for the z–axis related transient flow (U,V,0),(H_o), and temperature Θ (z,t). For large applied magnetic field, H_o the flow is observed to exhibit disturbance modes some of which are oscilatory, and some streaming modes are seen accompanied by the expected decays. For a thermally perturbed plasma the flow is seen to be largely governed by conduction parameters if viscous terms are not neglected under the MHD approximations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Bidirectional reflectance spectroscopy: 3. Correction for macroscopic roughness

A mathematically rigorous formalism is derived by which an arbitrary photometric function for the bidirectional reflectance of a smooth surface may be corrected to include effects of general macroscopic roughness. The correction involves only one arbitrary parameter, the mean slope angle $\text{θ}$, and is applicable to surfaces of any albedo. Using physically reasonable assumptions and mathematical approximations the correction expressions are evaluated analytically to second order in $\text{θ}$. The correction is applied to the bidirectional reflectance function of B. Hapke (1981, J. Geophys. Res.86, 3039–3054). Expressions for both the differential and integral brightnesses are obtained. Photometric profiles on hypothetical smooth and rough planets of low and high albedo are shown to illustrate the effects of macroscopic roughness. The theory is applied to observations of Mercury and predicts the integral phase function, the apparent polar darkening, and the lack of limb brightness surge on the planet. The roughness-corrected bidirectional reflectance function is sufficiently simple that it can be conveniently evaluated on a programmable hand-held calculator.     

Further aspects of weak shock theory applied to the solar chromosphere

The low chromosphere now seems definitely to require mechanical heating, and dissipation of initially acoustic waves by shocking is one of the most promising possibilities. Results of recent calculations indicate that the weak shock theory may be applicable here, but discrepancies exist among various applications of this theory, and the explanations offered to date are not completely satisfactory. It is shown here that the different approximations used by different authors to evaluate the mechanical flux integral play an important role in producing these discrepancies, in addition to the already well known effects of the density scale heights and the wave periods. Arguments are presented favoring Ulmschneider's method for evaluation of this flux integral.     

Analytical and numerical analysis of the generalized Shkarofsky function

The main analytical properties of the generalized Shkarofsky function and a numerical code for its computation are discussed. The results of a numerical analysis are compared with the results of an asymptotic analysis for parameter values relevant to the problem of whistler-mode propagation in the Earth's magnetosphere. This comparison allows us to specify the range of applicability of different approximations to the generalized Shkarofsky function, which have been used for the analysis of relativistic effects on whistler-mode propagation and instability.     

A numerical investigation of the one-dimensional Newtonian three-body problem

The one-dimensional Newtonian three-body problem is known to have stable (quasi-)periodic orbits when the masses are equal. The existence and size of the stable region is discussed here in the case where the three masses are arbitrary. We consider only the stability of the periodic (generalized) Schubart's (1956) orbit. If this orbit is linearly stable it is almost always surrounded by a region of stable quasi-periodic orbits and the size and shape of this stable region depends on the masses. The three-dimensional linear stability of the periodic orbits is also determined. Final results show that the region of stability has a complicated shape and some of the stable regions in the mass-plane are quite narrow. The non-linear three-dimensional stability is studied independently by extensive numerical integrations and the results are found to be in agreement with the linear stability analysis. The boundaries of stable region in the mass-plane are given in terms of polynomial approximations. The results are compared with a similar work by Héenon (1977).We thank the referee for pointing out this reference to us.     

Toy Stars in two dimensions

Toy Stars are gas masses where the compressibility is treated without approximations but gravity is replaced by a force which, for any pair of masses, is along their line of centres and proportional to their separation. They provide an invaluable resource for testing the suitability of numerical codes for astrophysical gas dynamics. In this paper, we derive the equations for both small-amplitude oscillations and non-linear solutions for rotating and pulsating Toy Stars in two dimensions, and show that the solutions can be reduced to a small number of ordinary differential equations. We compare the accurate solutions of these equations with Smoothed Particle Hydrodynamics (SPH) simulations. The two-dimensional Toy Star solutions are found to provide an excellent benchmark for SPH algorithms, highlighting many of the strengths and also some weaknesses of the method.