Lowest Order Solution Research Articles

Molecular quantum dynamics in a thermal system: Fractional wave packet revivals probed by random-phase fluorescence interferometry

The method of coherence observation by interference noise (COIN) [Kinrot et al., Phys. Rev. Lett. 75, 3822 (1995)] has been shown to be a useful tool for measurements of wave packet motion at the quantum-classical border. We present the first systematic interferometric study of fractional vibrational revivals in the B state of thermal iodine (I2) vapor. Experimental COIN interferograms ranging from 200 fs to 40 ps are presented for various excitation wavelengths. The complex temporal structure of the observed fluorescence includes rapid initial damping in the short-time regime and the appearance of quarter- and half-revivals on the quantum-mechanical long-time scale. These features arise from a delicate balance between rotational and vibrational molecular coherences. The clear observation of the wave packets on the long time scale is possible due to the long-time stability of the COIN interferometer. Lowest-order perturbative solutions nicely recover the experimental results, and closed-form analytical expressions based upon the factorization approach and the Poisson summation give insights into the nature of dephasing and rephasing of vibrational wave packets subject to rotational inhomogeneous broadening.

Scalar field of nonparaxial Gaussian beams

A family of closed-form expressions for the scalar field of strongly focused Gaussian beams in oblate spheroidal coordinates is given. The solutions satisfy the wave equation and are free from singularities. The lowest-order solution in the far field closely matches the energy density produced by a sine-condition, high-aperture lens illuminated by a paraxial Gaussian beam. At the large waist limit the solution reduces to the paraxial Gaussian beam form. The solution is equivalent to the spherical wave of a combined complex point source and sink but has the advantage of being more directly interpretatable.

Atomic motion in a two-mode laser field

A modified perturbation theory is developed for the calculation of the time-dependent wave function describing the propagation of an atom in a two-mode laser field. With the mode frequencies nearly equal, a first approximation is obtained that reproduces and extends known results, for both resonant and nonresonant transitions. This lowest-order solution sums terms that describe the absorption of a photon from one mode and emission into the other, leaving the total number of photons unchanged and shifting the momentum of the atom accordingly. Corrections are generated by expansion of a resolvent from which the nearly degenerate states have been projected out. These approximate solutions are used to construct asymptotic states in a formulation of atom-atom scattering in the two-mode field. In close analogy with the soft-photon approximation, a ``soft-pair'' approximation is developed based on the dominance of those asymptotic interactions in which the transfer of energy from the field to the atom is small, allowing one to relate the scattering amplitude in the presence of the field to the field-free amplitude. When the atomic ground state and an excited state are closely coupled by the field, the asymptotic wave function can have mixed parity allowing for long-range contributions to the effective potential. In addition to an enhancement of forward-scattering probabilities, this leads to a threshold anomaly that is analyzed here, in the resonant case, with the aid of a modified effective-range theory.

Effects of the Earth’s Curvature on the Dynamics of Isolated Objects. Part II: The Uniformly Translating Vortex

Abstract Zonally propagating solutions of the primitive equations for an isolated volume of fluid are considered. In a moving stereographic projection (from the antipode of the center of mass) geometric distortion enters at O(R−2), with R the radius of the earth, whereas planet curvature effects are O(R−1). The imbalance between the centrifugal force and the poleward gravitational force, due to the drift c, is equilibrated by the average Coriolis force, proportional to β. The results are valid for both homogeneous and stratified cases and the lowest-order solution need not be an axisymmetric vortex. The classical β-plane approximation predicts correctly the leading order of c/β, but makes large errors in the O(R−1) term of the vortex structure. A method is developed to construct the correct O(R−1) term, starting from any steady solution of the f-plane equations, as the O(R0) term. The expansion is exemplified starting with a homogeneous fluid, solid body rotating at an anticyclonic rate −νf0, with 0 < ν <...

Perturbation solution for small amplitude solitary waves in two-phase fluid flow of compacting media

A perturbation solution for small-amplitude solitary waves is derived for the third-order nonlinear partial differential equation due to Scott and Stevenson which describes the one-dimensional migration of melt through the Earth's mantle. The straightforward perturbation expansion breaks down and a coordinate stretching transformation is performed to render the perturbation expansion uniformly valid. The lowest-order perturbation solution has the same form as the single-soliton solution of the Korteweg-de Vries equation. The perturbation solution is derived to second order in implicit form. It is found to be extremely accurate when compared with known exact solutions for specific values of the exponents n and m. The zero- and first-order perturbation solutions are found to be accurate when n = m. The properties of small-amplitude solitary waves are investigated using the perturbation solution.

Nonequilibrium distribution function formalism for diffusion-influenced bimolecular reactions: Beyond the superposition approximation

We introduce an improved and systematic procedure for solving the diffusion-influenced reaction kinetics problem based on a hierarchical set of kinetic equations for many-particle distribution functions. As a representative example, we apply the formalism to the reversible association kinetics problem. The new method gives an approximate analytic solution which is in almost perfect agreement with the Brownian dynamics simulation results for the whole time range. It is shown that results of previous nonequilibrium distribution function theories are reproduced as lower order solutions in the present formalism.

Asymptotic theory of laminated circular conical shells

An asymptotic theory is presented for the analysis of laminated circular conical shells. The formulation begins with the basic equations of three-dimensional elasticity. By means of proper nondimensionalization and asymptotic expansion, the equations of three-dimensional elasticity can be decomposed into recursive sets of differential equations at various levels. After integrating these equations through the thickness direction in succession, we obtain the recursive sets of governing equations for the bending of a laminated circular conical shell. Note that the stiffness coefficients in the formulation are functions of the longitudinal coordinate. This involves the mathematical complexities in the formulation and the use of the existing analytical approach is restricted. The method of differential quadrature (DQ) is adopted for solving the problems of various orders. The formulation reveals that the differential operators corresponding to the governing equations of various orders remain the same. The nonhomogeneous terms of the higher-order problems are related to the lower-order solutions. Solution procedure of the DQ method for the leading order can be repeatedly applied for the solution to the higher-order level. In view of the efficiency and accuracy of the DQ method, the asymptotic solution of the present study is obtained readily and asymptotically approaches the three-dimensional solution. The illustrative examples are given to demonstrate the performance of the theory.

Complete binary collision approximation for the gas transport coefficients via the time correlation formulation

The connection between the low density limit of the time correlation equations for the transport coefficients and the solution of the Boltzmann equation to lowest order approximation appear to have been made in essentially two different ways. Either the time correlation function is evaluated by using the time dependent (linearized) Boltzmann equation, or by utilizing a resummation of an expansion in the reciprocal of a convergence parameter. As well, the connection is often made to only the lowest order solution of the Boltzmann equation, ignoring the possible importance of higher order moments (Sonine polynomials) in the solution of the Boltzmann equation. The present work uses a projection operator method and a subsequent binary collision expansion of the time correlation function to retain all contributions to the transport coefficient from binary collisions. This explicitly avoids an expansion in a divergent parameter and reproduces all Sonine polynomial contributions to the transport coefficient. Gas transport coefficients for a binary mixture are obtained in a similar manner.

Self-consistent models of incompressible stationary reconnection by weakly two dimensional extension of magnetic annihilation

Weakly two-dimensional self-consistent solutions of the stationary resistive magnetohydrodynamic equations are presented. The solutions are found by applying an asymptotic expansion procedure to the stationary resistive MHD equations with the annihilation solution as lowest order solution. Two different approximation schemes are investigated. Whereas the first scheme allows the introduction of a normal magnetic field component in the lowest order, the second scheme also introduces a weak variation of the current density along the current sheet in the lowest order. This is closely related to the fact that in the first case the inflow velocity does not vary along the current sheet, whereas in the second case the inflow velocity may vary weakly along the current sheet.

Accurate 3D viscous incompressible flow calculations with the FEM

A second-order-accurate (in both time and space) formulation is developed and implemented for solution of the three-dimensional incompressible Navier–Stokes equations involving high-Reynolds-number flows past complex configurations. For stabilization, only a fourth-order-accurate artificial dissipation term in the momentum equations is used. The finite element method (FEM) with an explicit time-marching scheme based on two-fractional-step integration is used for solution of the momentum equations. The element-by-element (EBE) technique is employed for solution of the auxiliary potential function equation in order to ease the memory requirements for matrix. The cubic cavity problem, the laminar flow past a sphere at various Reynolds numbers and the flow around the fuselage of a helicopter are successfully solved. Comparison of the results with the low-order solutions indicates that the flow details are depicted clearly even with coarse grids. © 1997 John Wiley & Sons, Ltd.

Accurate 3D viscous incompressible flow calculations with the FEM

A second-order-accurate (in both time and space) formulation is developed and implemented for solution of the three-dimensional incompressible Navier–Stokes equations involving high-Reynolds-number flows past complex configurations. For stabilization, only a fourth-order-accurate artificial dissipation term in the momentum equations is used. The finite element method (FEM) with an explicit time-marching scheme based on two-fractional-step integration is used for solution of the momentum equations. The element-by-element (EBE) technique is employed for solution of the auxiliary potential function equation in order to ease the memory requirements for matrix. The cubic cavity problem, the laminar flow past a sphere at various Reynolds numbers and the flow around the fuselage of a helicopter are successfully solved. Comparison of the results with the low-order solutions indicates that the flow details are depicted clearly even with coarse grids. © 1997 John Wiley & Sons, Ltd.

Efficient and accurate method for radiation transfer problems

An efficient method of analysis, which utilizes trial functions based on Case's eigenvalues, is developed for solving radiation transfer in an absorbing and scattering homogeneous semi-infinite plane-parallel medium subjected to externally incident radiation. Expressions for the forward and backward intensities, reflectivity and total radiation intensity are included. Numerical results are given and compared involving different forms of the externally incident radiation on the boundary surface. It is shown that the solution converges rapidly to the exact results and that lower-order solutions predict values of the physical parameters that are accurate to five figures in all values of the single-scattering albedos in the range 0.1 ≤ ω ≤ 1. The method has been also used to get approximate formulae for calculating Chandrasekhar's characteristic H-functions and their moments.

A consistency condition for the double series approximation method

The double series approximation method of Bonnor is a means for examining the gravitational radiation from an axisymmetric isolated source that undergoes a finite period of oscillation. It involves an expansion of the metric as a double Taylor series. Here we examine the integration procedure that is used to form an algorithmic solution to the field equations and point out the possibility of the expansion method breaking down and predicting a singularity along the axis of symmetry. We derive a condition on the solutions obtained by the double series method that must be satisfied to avoid this singularity. We then consider a source with only a quadrupole moment and verify that to fourth order in each of the expansion parameters, this condition is satisfied. This is a reassuring test of the consistency of the expansion procedure. We do, however, find that the imposition of this condition makes a physical interpretation of any but the lowest-order solutions very difficult. The most obvious decomposition of the solution into a series of independent physical effects is shown not to be valid.

Approximate Solution of the Enskog Equation Far from Equilibrium

A moment method is used to solve the Enskog equation for the steady-state distribution under conditions of uniform shear flow. Comparison to nonequilibrium molecular dynamics demonstrates that the lowest order solution gives a good quantitative description of nonlinear effects such as shear thinning and normal stresses in a moderately dense fluid. The results are used as a basis in the formulation of a simple and quantitatively accurate kinetic model of the Enskog equation.

Multiphoton processes without the dipole approximation

We employ the lowest-order analytic solution of the Schrödinger equation, obtained via an asymptotic expansion of the wavefunction, to calculate the transition matrix elements of some typical multiphoton processes without making the dipole approximation.

Accuracy of Lagrangian approximations in voids

We study the behaviour of spherical Voids in Lagrangian perturbation theories L(n), of which the Zel'dovich approximation is the lowest order solution L(1). We find that at early times higher order L(n) give an increasingly accurate picture of Void expansion. However at late times particle trajectories in L(2) begin to turnaround and converge leading to the {\em contraction} of a Void, a sign of pathological behaviour. By contrast particle trajectories in L(3) are well behaved and this approximation gives results in excellent agreement with the exact top-hat solution as long as the Void is not too underdense. For very underdense Voids, L(3) evacuates the Void much too rapidly leading us to conclude that the Zel'dovich approximation L(1), remains the best approximation to apply to the late time study of Voids. The behavior of high order approximations in spherical voids is typical for asymptotic series and may be generic for Lagrangian perturbation theory.

Stability of spatially developing boundary layers in pressure gradients

A new formulation of the stability of boundary-layer flows in pressure gradients is presented, taking into account the spatial development of the flow and utilizing a special coordinate transformation. The formulation assumes that disturbance wavelength and eigenfunction vary downstream no more rapidly than the boundary-layer thickness, and includes all terms nominally of orderR−1in the boundary-layer Reynolds numberR. In Blasius flow, the present approach is consistent with that of Bertolottiet al.(1992) toO(R−1) but simpler (i.e. has fewer terms), and may best be seen as providing a parametric differential equation which can be solved without having to march in space. The computed neutral boundaries depend strongly on distance from the surface, but the one corresponding to the inner maximum of the streamwise velocity perturbation happens to be close to the parallel flow (Orr-Sommerfeld) boundary. For this quantity, solutions for the Falkner-Skan flows show the effects of spatial growth to be striking only in the presence of strong adverse pressure gradients. As a rational analysis toO(R−1) demands inclusion of higher-order corrections on the mean flow, an illustrative calculation of one such correction, due to the displacement effect of the boundary layer, is made, and shown to have a significant destabilizing influence on the stability boundary in strong adverse pressure gradients. The effect of non-parallelism on the growth of relatively high frequencies can be significant at low Reynolds numbers, but is marginal in other cases. As an extension of the present approach, a method of dealing with non-similar flows is also presented and illustrated.However, inherent in the transformation underlying the present approach is a lower-order non-parallel theory, which is obtained by dropping all terms of nominal orderR−1except those required for obtaining the lowest-order solution in the critical and wall layers. It is shown that a reduced Orr-Sommerfeld equation (in transformed coordinates) already contains the major effects of non-parallelism.

A multiple scales approach to sound generation by vibrating bodies

The problem of determining the acoustic field in an inviscid, isentropic fluid generated by a solid body the surface of which executes prescribed vibrations is formulated and solved as a multiple scales perturbation problem, using the Mach number, M, based on the maximum surface velocity as the perturbation parameter. Following the idea of multiple scales, new “slow” spacial scales are introduced, which are defined as the usual physical spatial scales multiplied by powers of M. The governing non-linear differential equations lead to a sequence of linear problems for the perturbation coefficient functions. However, it is shown that the higher order perturbation functions obtained in this manner will dominate the lower order solutions unless their dependence on the slow spatial scales is chosen in a certain manner. In particular, it is shown that the perturbation functions must satisfy an equation similar to Burgers’ equation, with a slow spatial scale playing the role of the time-like variable. The method is illustrated by a simple one-dimensional example, as well as by three different cases of a vibrating sphere. The results are compared with solutions obtained by purely numerical methods and some insights provided by the perturbation approach are discussed.

ASYMPTOTIC THERMOELASTIC ANALYSIS OF ANISOTROPIC INHOMOGENEOUS AND LAMINATED PLATES

On the basis of three-dimensional elasticity without a priori assumptions, we develop an asymptotic theory for the thermoelastic analysis of anisotropic inhomogeneous plates subject to general temperature variations and under the action of lateral loads. The inhomogeneities considered are in the thickness direction, and the laminated plate represents an important special case. Through reformulation of the basic equations and nondimensionalization of the field variables, we find that the method of asymptotic expansions is well suited for the problem. Upon using the asymptotic expansion, we obtain sets of recurrence equations that can be integrated successively to determine the solution for a problem. We show that the classical laminated plate theory (CLT) is merely the leading-order approximation in the asymptotic theory. Furthermore, the higher-order equations are essentially the same as the CLT equations, only with nonhomogeneous terms that are completely determined from the lower-order solutions. As a r...

Parabolic approximations for water waves in conformal coordinate systems

A general formulation of small and large-angle parabolic approximations in conformally mapped coordinate systems is introduced. The technique is applied to the study of two particular cases involving a polar coordinate system. Comparisons to data and full solutions of the governing Helmholtz equation are given. For the case of waves between diverging breakwaters, we find that distinct differences exist between the lowest order parabolic approximation and an analytic solution in the Kirchoff approximation in polar coordinates. The errors are only partially alleviated in the next higher order approximation. For the case of waves in a circular channel bend, we find a similar level of disagreement between lowest-order parabolic approximations and full solutions. The higher order approximation produces results which are reasonably accurate in this case. In both cases, we also investigate the effects of wave nonlinearity, and investigate the growth of Mach stems at the outer wall of circular channel bends.