Uniform Perturbation Research Articles

Parabolic equation formulation via a singular perturbation technique and its application to scattering from irregular surfaces

This paper consists of two parts. In the first part, the solution of the Helmholtz equation under forward-scattering or propagation conditions is sought as a uniform asymptotic perturbation expansion using the method of multiple scales. It is then shown that the parabolic wave equation (PWE) solution is the zeroth-order term in this expansion. In the second part, the electric-field integral equation and the magnetic-field integral equation, derived under the PWE approximation, are solved for surface currents induced on a sinusoidal surface. The scattered fields produced by these currents are then calculated using the appropriate radiation integrals. Results are compared to those obtained using the method of ordered multiple interactions developed by Kapp and Brown.

A study of general circulation model climate feedbacks determined from perturbed sea surface temperature experiments

The response of a general circulation model (GCM) to global perturbations in sea surface temperatures (SSTs) is examined. The feedback strengths in the model are diagnosed by the response of top of atmosphere (TOA) radiative fluxes determined after substitution of fields from the “perturbed” climate into the “control.” Total feedback is divided into terms due to water vapour, lapse rate, surface temperature, and clouds (in turn analysed in terms of cloud amount, height and types). The “standard experiment” prescribes a globally uniform SST perturbation with fixed soil moisture. Four additional experiments vary the number of model vertical levels, the pattern of SST changes, the convection scheme, and the soil moisture. The SST pattern change chosen follows that of an equilibrium 2×CO2 experiment, which shows polar amplification of the surface warming. Variations in the clear sky sensitivity of the model are shown to depend primarily on changes in the long wave response due to competing (positive) water vapor and (generally negative) lapse rate feedbacks. Results here indicate that these feedbacks may be very different for differing experimental boundary conditions. The long wave feedback due to cloud amount changes is negative in all experiments, due to a very consistent decrease in high and middle cloud fractions. Conversely, cloud height feedback is positive due to a general increase in the altitude of (particularly high) cloud. Cloud height feedback is very sensitive to the choice of the convection scheme and to the change in vertical resolution. Greatest changes in the strength of the short wave cloud feedback results from modifications to the soil moisture specification and the convection scheme. The results here indicate that large differences in cloud feedback may be diagnosed from a single model, even without changes being made to the cloud parametrization. The value of the sensitivity can thus be expected to be a function not only of the physical parametrizations chosen for the model (e.g. the penetrative convection scheme), but also of the details of the manner in which the experiment was performed (e.g. SST and soil moisture specifications). The TOA radiation perturbation analysis method proves to be a powerful technique for diagnosing and understanding the physical processes responsible for the range in climate sensitivity found between the experiments.

An atlas of the distribution of calcium carbonate in sediments of the deep sea

Historical observations of the concentration of calcium carbonate in global deep sea sediments are compiled and compared with a new gridded field of seawater CO3= concentration to reveal regional variations in the calcite lysocline. The most obvious mode of variability of the calcite lysocline is the thickness of the lysocline (defined here as the difference in overlying water carbonate saturation, ΔCO3=, between high and low calcite sediments) with a thicker lysocline in the Atlantic than in the Pacific. I attribute this variation to differences in the delivery rate of terriginous material. A recent model for the lower glacial atmospheric pCO2 proposed to change the relationship between the depth of the lysocline and the ΔCO3= of the water column by changing the rain rate ratio of organic carbon to calcite production (the “rain ratio model”: Archer and Maier‐Reimer, 1994). I search the data set for analogs to the proposed glacial world, by looking for a link between the regional climate at the sea surface and the depth of the lysocline below. The ΔCO3= at the carbonate compensation depth (CCD) in the tropics appears to be 10–20 μmol kg−1 ΔCO3= more undersaturated than in high latitudes, but this is smaller than the ∼40 μmol kg−1 shift required by the model. In addition, the general resemblance of the glacial lysocline to the present day requires that the proposed shift in ΔCO3= at the CCD be globally uniform rather than locally variable, as climate forcing would probably generate. I conclude that the rain ratio model would probably require some globally uniform perturbation during glacial time, such as a change in ocean Si content, if it is to explain the entire pCO2 decrease observed in the glacial atmosphere. Finally, I grid the sedimentary data to estimate that the inventory CaCO3 which is available to neutralize fossil fuel CO2 is approximately 1600 Gt carbon, a quantity which may be exceeded by fossil fuel release in the next several centuries.

PyP216. Pyroelectric response of piezoelectrics

Abstract It has been originally shown that uniform elastic or thermal perturbation may induce volume piezoeffect or pyroelectricity correspondingly in all 20 piezoelectric classes of crystals if they are partially clamped. The last of these effects may be named “thermopiezoelectric” response - TPER. It has the same value of magnitude as the usual pyroelectric response but. TPER can be essentially amplified in the vicinities of SHF-electromagnetic or (and) electromechanical resonances. Piezoelectric crystals significantly extend the choice of pyroelectric materials and the possibilities of its applications. For instance, the semiconductors of GaAs type gives rise to a new generation of pyroelectric sensors being microelectronics processors in which thermal transducers, amplifiers and commutation system could be integrated in the space of the same crystal.

Intercomparison and interpretation of surface energy fluxes in atmospheric general circulation models

We have analyzed responses of the surface energy budgets and hydrologic cycles of 19 atmospheric general circulation models to an imposed, globally uniform sea surface temperature perturbation of 4 K. The responses of the simulated surface energy budgets are extremely diverse and are closely linked to the responses of the simulated hydrologic cycles. The response of the net surface energy flux is not controlled by cloud effects; instead, it is determined primarily by the response of the latent heat flux. The prescribed warming of the oceans leads to major increases in the atmospheric water vapor content and the rates of evaporation and precipitation. The increased water vapor amount drastically increases the downwelling infrared radiation at the Earth's surface, but the amount of the change varies dramatically from one model to another.

An improved perturbation technique for eigenvalues of continuous systems

AbstractA technique that improves the accuracy of the eigenvalue predictions as computed by a first‐order uniform expansion perturbation scheme for a class of eigenvalue problems associated with linear ordinary and partial differential equations is introduced. As suggested by a comparison with the classical Ritz method, the computational accuracy of a usual perturbation technique such as strained parameters can be effectively increased by simply replacing the usual M‐norms appearing in the denominator of the expressions for the first‐order changes in the eigenvalues by the corresponding MP‐norms, where MP is the perturbed M operator. With increasing sizes of the perturbations of the operators L and M of the class of boundary‐value problems considered, this improved perturbation technique produces results of substantially increased accuracy relative to the usual first‐order uniform expansion perturbation techniques. Numerical examples are presented.

Phase space and path integral methods in computational acoustics

The development and application of microscopic phase space methods and global path integral constructions to both direct and inverse problems in acoustics are reviewed. The Helmholtz equation is considered for (1) one‐way direct wave propagation, (2) two‐way direct wave propagation, and (3) inverse formulations. The one‐way theory concentrates on the construction of exact, numerical, and uniform perturbation solutions to the Weyl composition equation and their application to the further development and testing of the phase space marching algorithm. Specific comparisons will be made with the IFD codes for one‐way wave equations derived from formal operator rational approximations. The two‐way theory considers the evaluation of the Feynman/Garrod approximate path integral, the construction of an explicit path integral representation for the homogeneous Helmholtz equation, and the combination of transmutation and quantum field theoretic variable‐dimensionality ideas to incorporate backscatter. The inverse analysis seeks to exploit the Fourier integral operator structure of both the finite and infinitesimal propagators and the semigroup property of the one‐way Helmholtz equation to exactly solve the refractive index profile reconstruction problem for transversely inhomogeneous environments. [Work supported by NSF, AFOSR, ONR.]

Transport in the Frenkel-Kontorova model. III. Thermal conductivity

For pt.II see ibid., vol.18, p.4903 (1985). The authors use molecular dynamics simulation to study the thermal conductivity lambda of the undamped Frenkel-Kontorova model. The aims of the work are to provide evidence for the existence of lambda in this one-dimensional system, to investigate a variety of simulation techniques for calculating it, and to obtain numerical results for lambda as a function of temperature and winding number. They have examined three techniques: (i) direct determination of the energy flux caused by a temperature gradient; (ii) use of the Kubo formula to obtain lambda from the autocorrelation function of the energy flux; (iii) the calculation of lambda from the response of the energy flux to a suitable chosen uniform external perturbation. They show that the results given by the three approaches support the existence of lambda and are in satisfactory agreement with each other. The dependence of lambda on temperature and winding number can be explained by simple physical arguments.

Comparison of uniform perturbation and numerical solutions for some potential flows past slender bodies

Approximate solutions for potential flow past an axisymmetric slender body and past a thin airfoil are calculated using a uniform perturbation method and then compared with either the exact analytical solution or the solution obtained using a purely numerical method. The perturbation method is based upon a representation of the disturbance flow as the superposition of singularities distributed entirely within the body, while the numerical (panel) method is based upon a distribution of singularities on the surface of the body. It is found that the perturbation method provides very good results for small values of the slenderness ratio and for small angles of attack. Moreover, for comparable accuracy, the perturbation method is simpler to impelement, requires less computer memory, and generally uses less computation time than the panel method. In particular, the uniform perturbation method yields good resolution near the regions of the leading and trailing edges where other methods fail or require special attention.

Uniform expansions for nearly locked coupled Josephson weak links

A uniform perturbation calculation is carried out for two interacting Josephson weak links in the resistively shunted junction model, particular attention being paid to the region of near voltage locking. Good agreement is achieved with numerical simulation. Phenomena observed experimentally are also predicted. A comparison with other theoretical calculations is made.

Contradirectional frequency-selective couplers for guided-wave optics

Frequency-selective optical couplers in which a periodic perturbation in refractive index induces contradirectional power transfer in parallel single-mode waveguides are analyzed. Expressions for the spectral response are obtained by solving the coupled-mode equations. It is shown that by varying the amplitude of the periodic perturbation along he direction of propagation, the sidelobes present for the case of a uniform perturbation can be substantially reduced. Calculations are made of the mode propagation constants and coupling parameters for two rectangular-core waveguides with a periodic surface corrugation in the region between them. By using these results, design parameters and theoretical performance characteristics for a coupler with high transfer efficiency, low cross talk, and low reflection efficiency in the primary waveguide are determined.

Dynamic behaviour of a phase boundary under non-uniform surface loads

Summary. A method is outlined to determine the dynamic behaviour of a phase boundary in the Earth when non-uniform time-varying pressure and temperature conditions are assumed at the Earth's surface. An integral equation describing the phase boundary motion is derived and it is solved under a linearizing assumption. The solution is obtained in the form of a double integral transform. Short and long timeexpansions of the solution can be obtained from series expansion and integration of the Laplace trans- form along a branch cut. The method is illustrated by considering a stepwise change in surface pressure conditions. For short times, the solution exhibits the same type of time dependence (i.e. the first-order term is in t') as the one obtained in the one-dimensional case (i.e. uniform pressure perturbation at the Earth's surface). For long times, it is shown that the time dependence of the phase boundary motion is almost identical to the one derived for the one- dimensional case if the wavenumber kL of the surface excitation is such that K kt7 g 1 (where T is the relaxation time associated with the one-dimensional phase boundary motion and K is the thermal diffusivity). If K kt~ > 1, then the relaxation time for the phase boundary motion in two dimensions is of the order of K -' ki2. When considering parameters that would be appropriate for a basalt to eclogite phase transition at Moho depth, the latter situation is met only when the load wavelength is smaller than 3 5 km.

Three dimensional model of seismic velocity variation in the Earth's mantle

Three dimensional velocity models for the earth's mantle were obtained satisfying P, S and some PcP and ScS travel time anomalies from deep focus earthquakes. A method of successive approximation was used to compute the uniform relative velocity perturbation over three dimensional blocks of size 10°×10° in longitude and latitude, and 500 km in thickness. Features of these velocity perturbations indicated that lateral heterogeneity is the most pronounced in the upper mantle and near the core‐mantle boundary. The upper mantle anomalies are correlated with surface tectonic features, but these are not correlated with deep mantle anomalies.

Nonlinear transverse vibrations of beams with properties that vary along the length

The method of multiple scales is used to analyze the nonlinear transverse vibrations of inhomogeneous beams including the effects of transverse shear and rotary inertia. First-order uniform perturbation expansions are derived for the displacements, shear force, and frequency in terms of the ratios of maximum transverse displacement to cross-sectional radius of gyration and cross-sectional radius of gyration to wave length.

Nonlinear Transverse Vibrations of Inhomogeneous Beams

The method of multiple scales is used to analyze the nonlinear transverse vibrations of inhomogeneous beams including the effects of transverse shear and rotary inertia. First-order uniform perturbation expansions are derived for the displacements, shear force, and frequency in terms of the ratios of maximum transverse displacement to cross-sectional radius of gyration and cross-sectional radius of gyration to wavelength.

More uniform perturbation theory of the Vlasov equation

The Vlasov—Poisson equations are reformulated by applying an arbitrary transformation to the velocity variable in such a way that perturbation theory of the transformed equations does not exhibit the customary secularity in time (or space) in second or higher order. The lowest order approximation of the new formulation is discussed and compared with conventional results. The source of the non-uniformity appears to be the divergence of Particle trajectories as calculated by perturbation methods, from the exact ones after long times. The transformation which allows one to follow the particle trajectories is a transformation to a frame of reference moving with a plasma test particle in the self-consistent field.

Evolution of the trans-Alfvénic normal shock in a gas of finite electrical conductivity

The stability of a steady, plane, one-dimensional trans-Alfvénic shock to small transverse disturbances in the velocity and magnetic field is discussed. In the undisturbed flow the magnetic field and velocity are normal to the plane of the shock which is treated as a discontinuity in an inviscid gas of zero thermal conductivity. However, the electrical conductivity of the ambient gas is taken to be finite, i.e. the magnetic diffusivity is very much larger than both the viscous diffusivity and the thermal diffusivity.A small uniform transverse perturbation of the magnetic field is imposed at zero time. A specific super-Alfvénic normal shock is chosen. Computed values of the magnetic field, in and near the shock, are given for various later times. Two diffusing Alfvén waves of the amplitudes predicted by infinite conductivity theory are shown to propagate away from the shock region, leaving behind the expected steady-state shock profile.Similar computations are carried out for a specific trans-Alfvénic normal shock. An analytic asymptotic solution, valid for large times, is also obtained. This result agrees with the computations carried out. The shock profile of the transverse quantities is one which grows linearly with time. Outside this magneto-hydrodynamic shock region which surrounds the trans-Alfvénic hydrodynamic shock (discontinuity), steady states are reached in each of which the transverse velocity and transverse magnetic field are uniform. An incident Alfvén wave, consisting of a weak diffusing current sheet, produces these same effects. The resolution of an arbitrary transverse fluctuation, in which both the magnetic field and the velocity have limiting values at large distances from the shock, is discussed. The solution for large times is found.It is shown that the trans-Alfvénic normal shock and its two limiting cases, the null switch-on and null switch-off shocks, are unstable to general transverse disturbances although there exist particular disturbances of this kind which will not destroy them. An integral condition is obtained which, together with the relevant boundary conditions, determines the profiles of the transverse quantities in the trans-Alfvénic normal shock whenever a steady state is reached. This removes the puzzling arbitrariness of these profiles.

Interaction of the Solar Wind with the Geomagnetic Field

The dynamical properties of the solar wind blowing past the geomagnetic field are investigated by considering the effective viscosity and the resulting transition layer thickness. The collision of ions in the solar wind produces a negligible viscosity in the flow past the geomagnetic field, but such an inviscid flow is shown to be unstable. The resulting disordered interface between the field and the wind yields Fermi acceleration of ions and consequently a not insignificant effective viscosity. The Fermi acceleration results in suprathermal ions which may have an energy spectrum like that observed for primary auroral protons. The auroral zones and the agitated nature of the polar geomagnetic field are shown to follow from the depth of penetration of the solar wind into the geomagnetic field. The injection of gas into the geomagnetic field is studied. The effect at Earth of the distortion of the outer boundary of the geomagnetic field is computed; no matter how unevenly and anisotropically the outer field is distorted, the effect at Earth is a nearly uniform perturbation field which is closely parallel to the geomagnetic axis. Pushing in on the outer field increases the horizontal component at Earth, and pulling out decreases it; the total increase of the horizontal component is the algebraic sum of all the pushing and pulling. The simultaneous world-wide onset and the main phase of a geomagnetic storm follow. The common tendency of large and/or violent bodies of plasma to produce suprathermal particles is noted and suggested to be a general dynamical property.