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1
El Sawi, M. « On the WKBJ approximation ». Journal of Mathematical Physics 28, no 3 (mars 1987) : 556–58. http://dx.doi.org/10.1063/1.527640.
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Goyal, I. C., Sukhdev Roy, A. K. Ghatak et R. L. Gallawa. « Anharmonic oscillator analysis using modified Airy functions ». Canadian Journal of Physics 70, no 12 (1 décembre 1992) : 1218–21. http://dx.doi.org/10.1139/p92-197.
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We have applied the modified Airy function (MAF) method to the analysis of an anharmonic oscillator, characterized by the potential V(x) = k x2/2 + a x4, k > 0, a > 0. The MAF method gives an accurate closed-form expression for the wave function as well as very accurate eigenvalues with much less numerical effort than the five-term (eighth-order) WKBJ approximation. The application of the first-order perturbation correction to the MAF eigenvalues makes them even more accurate than those obtained by the five-term WKBJ approximation.
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Ioannou, Petros, et Richard S. Lindzen. « WKBJ Approximation of the Stability of a Frontal Mean State ». Journal of the Atmospheric Sciences 47, no 23 (décembre 1990) : 2825–28. http://dx.doi.org/10.1175/1520-0469(1990)047<2825:waotso>2.0.co;2.
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Monkewitz, Peter A., Patrick Huerre et Jean-Marc Chomaz. « Global linear stability analysis of weakly non-parallel shear flows ». Journal of Fluid Mechanics 251 (juin 1993) : 1–20. http://dx.doi.org/10.1017/s0022112093003313.
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The global linear stability of incompressible, two-dimensional shear flows is investigated under the assumptions that far-field pressure feedback between distant points in the flow field is negligible and that the basic flow is only weakly non-parallel, i.e. that its streamwise development is slow on the scale of a typical instability wavelength. This implies the general study of the temporal evolution of global modes, which are time-harmonic solutions of the linear disturbance equations, subject to homogeneous boundary conditions in all space directions. Flow domains of both doubly infinite and semi-infinite streamwise extent are considered and complete solutions are obtained within the framework of asymptotically matched WKBJ approximations. In both cases the global eigenfrequency is given, to leading order in the WKBJ parameter, by the absolute frequency ω0(Xt) at the dominant turning pointXtof the WKBJ approximation, while its quantization is provided by the connection of solutions acrossXt. Within the context of the present analysis, global modes can therefore only become time-amplified or self-excited if the basic flow contains a region of absolute instability.
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Zhang, Li Gang, Hong Zhu, Hong Biao Xie et Jian Wang. « Love Wave in an Isotropic Half-Space with a Graded Layer ». Applied Mechanics and Materials 325-326 (juin 2013) : 252–55. http://dx.doi.org/10.4028/www.scientific.net/amm.325-326.252.
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This work addresses the dispersion of Love wave in an isotropic homogeneous elastic half-space covered with a functionally graded layer. First, the general dispersion equations are given. Then, the approximation analytical solutions of displacement, stress and the general dispersion relations of Love wave in both media are derived by the WKBJ approximation method. The solutions are checked against numerical calculations taking an example of functionally graded layer with exponentially varying shear modulus and density along the thickness direction. The dispersion curves obtained show that a cut-off frequency arises in the lowest order vibration model.
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Cao, Di, et Wafik B. Beydown. « An approximate elastic f‐k Green’s function within WKBJ (high‐frequency) approximation ». Journal of the Acoustical Society of America 87, no 4 (avril 1990) : 1397–404. http://dx.doi.org/10.1121/1.399540.
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Luo, Songting, Jianliang Qian et Hongkai Zhao. « Higher-order schemes for 3D first-arrival traveltimes and amplitudes ». GEOPHYSICS 77, no 2 (mars 2012) : T47—T56. http://dx.doi.org/10.1190/geo2010-0363.1.
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In the geometrical-optics approximation for the Helmholtz equation with a point source, traveltimes and amplitudes have upwind singularities at the point source. Hence, both first-order and higher-order finite-difference solvers exhibit formally at most first-order convergence and relatively large errors. Such singularities can be factored out by factorizing traveltimes and amplitudes, where one factor is specified to capture the corresponding source singularity and the other factor is an unknown function smooth near the source. The resulting underlying unknown functions satisfy factored eikonal and transport equations, respectively. A third-order Lax-Friedrichs scheme is designed to compute the underlying functions. Thus, highly accurate first-arrival traveltimes and reliable amplitudes can be computed. Furthermore, asymptotic wavefields are constructed using computed traveltimes and amplitudes in the WKBJ form. Two-dimensional and 3D examples demonstrate the performance of the proposed algorithms, and the constructed WKBJ Green’s functions are in good agreement with direct solutions of the Helmholtz equation before caustics occur.
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LeBras, Ronan, et Robert W. Clayton. « An iterative inversion of back‐scattered acoustic waves ». GEOPHYSICS 53, no 4 (avril 1988) : 501–8. http://dx.doi.org/10.1190/1.1442481.
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The application of the Born approximation to the scattered wave field, followed by a WKBJ and far‐field approximation on the propagation Green’s function for a slowly varying background medium, leads to a simple integral relation between the density and bulk‐modulus anomalies superimposed on the background medium and the scattered wave field. An iterative inversion scheme based on successive back‐projections of the wave field is used to reconstruct the two acoustic parameters. The scheme, when applied to data generated using the direct integral relation, shows that the variations of the parameters can be reconstructed. The procedure is readily applicable to actual data, since every iterative step is essentially a prestack Kirchhoff migration followed by the application of the direct Born approximation and far‐field operator.
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Zhang, Li Gang, Hong Zhu, Hong Biao Xie et Lin Yuan. « P Wave Propagation in the Functionally Graded Materials ». Advanced Materials Research 706-708 (juin 2013) : 1685–88. http://dx.doi.org/10.4028/www.scientific.net/amr.706-708.1685.
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The P wave propagation in the functionally graded materials (FGM) is studied. The differential equation with varied-coefficient of wave motion in the FGM is established. By using of the WKBJ approximation method, the differential equation with varied-coefficient is solved, and the closed-analytical solutions of displacement in the FGM are obtained. The properties of the FGM whose shear modulus and mass density are gradually varying in exponential form are calculated; the curves of P wave velocity and amplitude, and the general properties of the P wave in the FGM are analyzed.
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Fröman, Nanny, et Per Olof Fröman. « Comments on the paper ‘‘On the WKBJ approximation’’ [J. Math. Phys. 28, 556 (1987)] ». Journal of Mathematical Physics 29, no 4 (avril 1988) : 912. http://dx.doi.org/10.1063/1.527988.
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Riedinger, Xavier, et Andrew D. Gilbert. « Critical layer and radiative instabilities in shallow-water shear flows ». Journal of Fluid Mechanics 751 (23 juin 2014) : 539–69. http://dx.doi.org/10.1017/jfm.2014.303.
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AbstractIn this study a linear stability analysis of shallow-water flows is undertaken for a representative Froude number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}F=3.5$. The focus is on monotonic base flow profiles $U$ without an inflection point, in order to study critical layer instability (CLI) and its interaction with radiative instability (RI). First the dispersion relation is presented for the piecewise linear profile studied numerically by Satomura (J. Meterol. Soc. Japan, vol. 59, 1981, pp. 148–167) and using WKBJ analysis an interpretation given of mode branches, resonances and radiative instability. In particular surface gravity (SG) waves can resonate with a limit mode (LM) (or Rayleigh wave), localised near the discontinuity in shear in the flow; in this piecewise profile there is no critical layer. The piecewise linear profile is then continuously modified in a family of nonlinear profiles, to show the effect of the vorticity gradient $Q^{\prime } = - U^{\prime \prime }$ on the nature of the modes. Some modes remain as modes and others turn into quasi-modes (QM), linked to Landau damping of disturbances to the flow, depending on the sign of the vorticity gradient at the critical point. Thus an interpretation of critical layer instability for continuous profiles is given, as the remnant of the resonance with the LM. Numerical results and WKBJ analysis of critical layer instability and radiative instability for more general smooth profiles are provided. A link is made between growth rate formulae obtained by considering wave momentum and those found via the WKBJ approximation. Finally the competition between the stabilising effect of vorticity gradients in a critical layer and the destabilising effect of radiation (radiative instability) is studied.
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Moore, G. W. Kent, et W. R. Peltier. « On the Accuracy of the WKBJ Approximation to the Nonseparable Quasi-geostrophic Baroclinic Instability Problem ». Journal of the Atmospheric Sciences 47, no 23 (décembre 1990) : 2829–31. http://dx.doi.org/10.1175/1520-0469(1990)047<2829:otaotw>2.0.co;2.
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Wu, Bo, Li Zhao et Xiaofei Chen. « Uniformly asymptotic eigensolutions of the Earth's toroidal modes ». Geophysical Journal International 228, no 1 (16 août 2021) : 250–58. http://dx.doi.org/10.1093/gji/ggab329.
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SUMMARY A novel approach to computing the Earth's normal modes is presented. With the Langer approximation incorporated into the framework of the theory of the generalized reflection/transmission coefficients, our method, when applied to calculate the toroidal modes, produces results that agree extremely well with the exact results from Mineos and successfully overcomes the limitations of the WKBJ analysis. Given any 1-D earth model, regardless of the number of discontinuities, our method can perform the calculations reliably with satisfactory accuracy at high frequencies. The success achieved with the toroidal modes encourages us to tackle in a future study the asymptotic computation of the spheroidal modes, especially those high-frequency trapped modes for which the accuracy of Mineos is demonstrably inadequate.
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Feng, Bo, Ru-Shan Wu et Huazhong Wang. « A generalized Rytov approximation for accurate calculation of phase variation in strong perturbation media ». Geophysical Journal International 219, no 2 (26 juillet 2019) : 968–74. http://dx.doi.org/10.1093/gji/ggz338.
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SUMMARY In the case of long-range propagation of forward scattering, due to the accumulation of phase changes caused by the velocity perturbations, the validity of the Born approximation will be violated. In contrast, the phase-change accumulation can be handled by the Rytov approximation, which has been widely used for long-distance propagation with only forward scattering or small-angle scattering involved. However, the weak scattering assumption (i.e. small velocity perturbation) in the Rytov approximation limits its scope of application. To address this problem, we analyse the integral kernel of the Rytov transform using the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) approximation and we demonstrate that the integral kernel is a function of velocity perturbation and scattering angle. By applying a small scattering angle approximation, we show that the phase variation has a linear relationship with the slowness perturbation, no matter how strong the magnitude of perturbation is. Therefore, the new integral equation is then referred to as the generalized Rytov approximation (GRA) because it overcomes the weak scattering assumption of the Rytov approximation. To show the limitations of the Rytov approximation and the advantages of the proposed GRA method, first we design a two-layer model and we analytically calculate the errors introduced by the small scattering angle assumption using plane wave incidence. We show that the phase (traveltime) variations predicted by the GRA are always more accurate than the Rytov approximation. Particularly, the GRA produces accurate phase variations for the normal incident plane wave regardless of the magnitude of velocity perturbation. Numerical examples using Gaussian anomaly models demonstrate that the scattering angle has a crucial impact on the accuracy of the GRA. If the small scattering angle assumption holds, the GRA can produce an accurate phase approximation even if the velocity perturbation is very strong. On the contrary, both the first-order Rytov approximation and the GRA fail to get satisfying results when the scattering angle is large enough. The proposed GRA method has the potential to be used for traveltime modelling and inversion for large-scale strong perturbation media.
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Spigler, Renato, et Marco Vianello. « WKBJ-type approximation for finite moments perturbations of the differential equation y″ = 0 and the analogous difference equation ». Journal of Mathematical Analysis and Applications 169, no 2 (septembre 1992) : 437–52. http://dx.doi.org/10.1016/0022-247x(92)90089-v.
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Yedlin, M. J., B. R. Seymour et B. C. Zelt. « Truncated asymptotic representation of waves in a one‐dimensional elastic medium ». GEOPHYSICS 52, no 6 (juin 1987) : 755–64. http://dx.doi.org/10.1190/1.1442342.
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A new time‐domain method has been developed for solving for the stress and displacement of normally incident plane waves propagating in a smoothly varying one‐dimensional elastic medium. Both the Young’s modulus E and the density ρ are allowed to vary smoothly with depth. The restriction of geometrical optics, that the wavelength be much less than the material stratification length, is not required in this new method. We truncate the infinite geometrical‐optics asymptotic expansion after n terms (n = 2 in this paper), which imposes a condition on the acoustic impedance I for exact solutions to exist. The resultant expansion is uniform and exact for three general classes of impedance functions. Results are calculated for the case of a medium with a linear velocity gradient (for which there is an exact solution in the frequency domain); the results are compared with a two‐term WKBJ approximation and the new truncated expansion method. Since a linear velocity gradient is not one of the foregoing classes of impedance functions, a curve‐fitting approach is necessary. The results show that the new method compares favorably to both the WKBJ results and the exact solution and is accurate to within the error of the required curve fit. Two classes of synthetic seismograms are then calculated for smooth velocity and density variations. The same impedance as a function of traveltime is used for both classes. In the first class the principal variation in impedance is due to velocity, while in the second it is mainly due to density. The amplitudes in both classes of synthetic seismograms are very similar, but, as expected, the traveltime curves for each class are widely separated.
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ADAMOU, ALEXANDER T. I., R. V. CRASTER et STEFAN G. LLEWELLYN SMITH. « Trapped edge waves in stratified rotating fluids : numerical and asymptotic results ». Journal of Fluid Mechanics 592 (14 novembre 2007) : 195–220. http://dx.doi.org/10.1017/s0022112007008361.
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The existence of trapped edge waves in a rotating stratified fluid with non-constant topography is studied using asymptotic and numerical techniques. A refinement of the classical WKBJ method is employed that is uniform at both the shoreline and caustic, where the classical approximation is singular, and is also uniform over long distances from the shore. This approach requires the use of comparison equations and it is shown that the two used previously in the literature are asymptotically equivalent in terms of the wave amplitude, but have small differences in the predicted wave frequencies. These asymptotic results, and results using shallow-water theory, are then compared to results from a careful numerical study of the nonlinear differential eigenvalue problem, allowing their range of practical applicability to be assessed. This numerical approach is also used to investigate whether trapping occurs in non-trivial and realistic geometries in the internal gravity wave band, which has been an open question for some time.
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LE DIZÈS, STÉPHANE, et CHRISTOPHE MILLET. « Acoustic near field of a transonic instability wave packet ». Journal of Fluid Mechanics 577 (19 avril 2007) : 1–23. http://dx.doi.org/10.1017/s0022112006004472.
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We consider the problem of acoustic radiation generated by a spatial instability wave on a weakly developing shear flow. Assuming a local WKBJ approximation for the instability wave near its maximum, we compute the acoustic pressure field by using a Fourier transform along the streamwise direction. When the instability wave is close to transonic near its maximum amplitude, approximations for this pressure field are obtained by a steepest descent method. A branch cut and several saddle points are shown possibly to contribute to the approximation. A detailed analysis of these contributions is provided. The modifications of the acoustic field when we pass from subsonic to supersonic are examined. In particular, the superdirective character of the acoustic field of subsonic instability waves and the directivity pattern of supersonic waves are shown to be both compatible with our mathematical description and associated with a single saddle-point contribution.The acoustic near field is also shown to possess a caustic around which a specific approximation is derived. In a large region of the physical space, the near field is composed of two saddle-point contributions. Close to the shear flow, one of these contributions degenerates into a branch-point contribution which always becomes dominant over the instability wave downstream of a location that is computed. An interesting phenomenon is observed in certain regions downstream of the maximum: the transverse behaviour of the instability wave has to be exponentially growing far from the shear layer to match the acoustic field. We demonstrate that this phenomenon neither requires a branch-point contribution nor a supersonic instability wave.
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Laprise, J. P. René. « An Assessment of the WKBJ Approximation to the Vertical Structure of Linear Mountain Waves : Implications for Gravity-Wave Drag Parameterization ». Journal of the Atmospheric Sciences 50, no 11 (juin 1993) : 1469–87. http://dx.doi.org/10.1175/1520-0469(1993)050<1469:aaotwa>2.0.co;2.
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DUNSTER, T. M., MATTHEW YEDLIN et KIM LAM. « RESONANCE AND THE LATE COEFFICIENTS IN THE SCATTERED FIELD OF A DIELECTRIC CIRCULAR CYLINDER ». Analysis and Applications 04, no 04 (octobre 2006) : 311–33. http://dx.doi.org/10.1142/s0219530506000796.
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The classical modal expansion for the scattered field of a plane wave from a circular dielectric cylinder is studied. A new uniform asymptotic approximation is presented for the late coefficients in this expansion, in the case of a fixed relative dielectric constant εr, both real and complex. These new approximations for the mode values are not based on the scattering matrix but rather the classical WKBJ approximations for the Bessel functions, and are valid for the entire region exterior to the cylinder, including the transition region. Furthermore, a precise asymptotic form for the location of a certain critical Regge pole is obtained. It is shown that this pole can lead to at least one dramatic resonant modal term at certain critical values, and the exponential nature of the mode in question is determined explicitly. This is followed by an extension to complex values of εr with new uniform asymptotic approximations for the modes also being obtained, and these in turn demonstrate a heavy damping of the resonant mode.
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Dunster, T. M. « Liouville-Green expansions of exponential form, with an application to modified Bessel functions ». Proceedings of the Royal Society of Edinburgh : Section A Mathematics 150, no 3 (29 janvier 2019) : 1289–311. http://dx.doi.org/10.1017/prm.2018.117.
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AbstractLinear second order differential equations of the form d2w/dz2 − {u2f(u, z) + g(z)}w = 0 are studied, where |u| → ∞ and z lies in a complex bounded or unbounded domain D. If f(u, z) and g(z) are meromorphic in D, and f(u, z) has no zeros, the classical Liouville-Green/WKBJ approximation provides asymptotic expansions involving the exponential function. The coefficients in these expansions either multiply the exponential or in an alternative form appear in the exponent. The latter case has applications to the simplification of turning point expansions as well as certain quantum mechanics problems, and new computable error bounds are derived. It is shown how these bounds can be sharpened to provide realistic error estimates, and this is illustrated by an application to modified Bessel functions of complex argument and large positive order. Explicit computable error bounds are also derived for asymptotic expansions for particular solutions of the nonhomogeneous equations of the form d2w/dz2 − {u2f(z) + g(z)}w = p(z).
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Langston, Charles A., et Chang-Eob Baag. « The validity of ray theory approximations for the computation of teleseismic SV waves ». Bulletin of the Seismological Society of America 75, no 6 (1 décembre 1985) : 1719–27. http://dx.doi.org/10.1785/bssa0750061719.
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Abstract Teleseismic SV waves have been generally ignored in wave propagation and source studies because of known complications in wave propagation for structure near the source and near the receiver. The validity of common optic ray and WKBJ seismogram methods for computing SV synthetic seismograms is examined by computing synthetic seismograms using these techniques and comparing them to SV synthetics produced from a wavenumber integration technique. Both ray methods give a poor approximation to the wave propagation for distances less than 60°. Diffracted Sp and the SPL wave interfere with near-source phases, such as S, pS, and sS for a shallow seismic source, producing anomalously high amplitudes and complex waveforms in agreement with observational experience. Because of the Moho Sp and diffracted Sp phases, the vertical component of motion shows greater distortion, relative to the ray theory result, than does the radial component of motion. Ray theory appears to be appropriate for the initial 20 sec of the SV wave train from a shallow source for ranges greater than 60°. SV waves from deep sources are less affected by diffracted Sp and SPL than SV from shallow sources.
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Park, J., V. Prat et S. Mathis. « Horizontal shear instabilities in rotating stellar radiation zones ». Astronomy & ; Astrophysics 635 (mars 2020) : A133. http://dx.doi.org/10.1051/0004-6361/201936863.
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Context. Rotational mixing transports angular momentum and chemical elements in stellar radiative zones. It is one of the key processes for modern stellar evolution. In the past two decades, an emphasis has been placed on the turbulent transport induced by the vertical shear instability. However, instabilities arising from horizontal shear and the strength of the anisotropic turbulent transport that they may trigger remain relatively unexplored. The weakest point of this hydrodynamical theory of rotational mixing is the assumption that anisotropic turbulent transport is stronger in horizontal directions than in the vertical one. Aims. This paper investigates the combined effects of stable stratification, rotation, and thermal diffusion on the horizontal shear instabilities that are obtained and discussed in the context of stellar radiative zones. Methods. The eigenvalue problem describing linear instabilities of a flow with a hyperbolic-tangent horizontal shear profile was solved numerically for a wide range of parameters. When possible, the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) approximation was applied to provide analytical asymptotic dispersion relations in both the nondiffusive and highly diffusive limits. As a first step, we consider a polar f-plane where the gravity and rotation vector are aligned. Results. Two types of instabilities are identified: the inflectional and inertial instabilities. The inflectional instability that arises from the inflection point (i.e., the zero second derivative of the shear flow) is the most unstable when at a zero vertical wavenumber and a finite wavenumber in the streamwise direction along the imposed-flow direction. While the maximum two-dimensional growth rate is independent of the stratification, rotation rate, and thermal diffusivity, the three-dimensional inflectional instability is destabilized by stable stratification, while it is stabilized by thermal diffusion. The inertial instability is rotationally driven, and a WKBJ analysis reveals that its growth rate reaches the maximum value of √f(1 − f) in the inviscid limit as the vertical wavenumber goes to infinity, where f is the dimensionless Coriolis parameter. The inertial instability for a finite vertical wavenumber is stabilized as the stratification increases, whereas it is destabilized by the thermal diffusion. Furthermore, we found a selfsimilarity in both the inflectional and inertial instabilities based on the rescaled parameter PeN2 with the Péclet number Pe and the Brunt–Väisälä frequency N.
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Gholami, V., H. Hamzehloo, C. La Mura, M. R. Ghayamghamian et G. F. Panza. « Simulation of selected strong motion records of the 2003 MW = 6.6 Bam earthquake (SE Iran), the modal summation-ray tracing methods in the WKBJ approximation ». Geophysical Journal International 196, no 2 (7 novembre 2013) : 924–38. http://dx.doi.org/10.1093/gji/ggt405.
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Amundsen, Lasse. « The propagator matrix related to the Kirchhoff‐Helmholtz integral in inverse wavefield extrapolation ». GEOPHYSICS 59, no 12 (décembre 1994) : 1902–10. http://dx.doi.org/10.1190/1.1443577.
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The Kirchhoff‐Helmholtz formula for the wavefield inside a closed surface surrounding a volume is most commonly given as a surface integral over the field and its normal derivative, given the Green’s function of the problem. In this case the source point of the Green’s function, or the observation point, is located inside the volume enclosed by the surface. However, when locating the observation point outside the closed surface, the Kirchhoff‐Helmholtz formula constitutes a functional relationship between the field and its normal derivative on the surface, and thereby defines an integral equation for the fields. By dividing the closed surface into two parts, one being identical to the (infinite) data measurement surface and the other identical to the (infinite) surface onto which we want to extrapolate the data, the solution of the Kirchhoff‐Helmholtz integral equation mathematically gives exact inverse extrapolation of the field when constructing a Green’s function that generates either a null pressure field or a null normal gradient of the pressure field on the latter surface. In the case when the surfaces are plane and horizontal and the medium parameters are constant between the surfaces, analysis in the wavenumber domain shows that the Kirchhoff‐Helmholtz integral equation is equivalent to the Thomson‐Haskell acoustic propagator matrix method. When the medium parameters have smooth vertical gradients, the Kirchhoff‐Helmholtz integral equation in the high‐frequency approximation is equivalent to the WKBJ propagator matrix method, which also is equivalent to the extrapolation method denoted by extrapolation by analytic continuation.
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Amundsen, Lasse. « A theoretical contribution to the 1D inverse problem of reflection seismograms ». GEOPHYSICS 86, no 4 (1 juin 2021) : R351—R368. http://dx.doi.org/10.1190/geo2020-0257.1.
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Least-squares full-waveform inversion (FWI) is considered in the frequency domain for a set of noise-free observations of time length [Formula: see text] at the surface obeying the 1D wave equation, with a known source. The initial model is of constant velocity. The first iteration, which equals the constant-velocity migration inversion (CVMI), is thoroughly analyzed. In CVMI, for the unit source power spectrum, it is within reach to analytically derive and interpret the mathematical formulas of the first-order partial derivatives of the modeled observations (Jacobian), and the gradient and Gauss-Newton Hessian of the objective function, and learn what information the calculation requires to obtain a successful physical result (i.e., velocity update). We recognize the gradient elements, except the last one, to be sums of reflection-amplitude weighted band-limited sign functions and the Hessian elements, except along the last column and row, to be band-limited, diagonal-centered triangle functions, which for infinite bandwidth reduces to the Kronecker delta function. When the fundamental frequency [Formula: see text] is lacking in the observations, the gradient loses information of the low-wavenumber trend of the velocity update. The Hessian becomes close to singular, and any stabilized inverse has no chance to repair the deficiencies of the gradient caused by any missing low frequency in the observations. FWI is started by applying CVMI. First, Jacobians are modeled by classic reflectivity modeling. Second, the diagonal Hessians can be used for estimating discrete velocity updates. Third, the Jacobian can be modeled in the first-order Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) approximation and by neglecting transmission effects. Finally, single-frequency and low-frequency seismograms can be inverted by using broadband Hessians. The main mathematical findings are developed by simple numerical models and data.
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Mestel, L., et K. Subramanian. « Galactic Dynamos and Density Wave Theory ». Symposium - International Astronomical Union 140 (1990) : 135. http://dx.doi.org/10.1017/s0074180900189764.
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A steady density wave in the stellar background of a disk–like galaxy is supposed to force a spiral shock wave in the interstellar gas. The jump in vorticity across the shock leads to a locally enhanced helicity, and so to an α–effect which is steady but azimuth–dependent in the frame rotating with the angular velocity ω of the density wave. This is simulated by the adoption of the form for the local dynamo growth rate arising when the standard kinematic dynamo equation is treated by the thin–disk approximation (Ruzmaikin et al 1988). The global magnetic field is proportional to the function Q satisfying where η is the turbulent resistivity (for simplicity assumed uniform) and is the laminar angular velocity of the gas in the inertial frame. We look for solutions of the form where is a global eigen-value, and the non-vanishing of couples all odd or all the even m-values. Anticipating that the strong differential rotation will ensure that in the modes with the largest growth-rate the higher-m parts are weak, the equations are truncated, leaving just a pair in q1, q-1, to describe a basically bisymmetric (m = 1) mode. Approximate treatment by the WKBJ technique suggests that a corotating growing mode (with Γ real and positive) will differ significantly from zero over the range between the points where Numerical solutions have been found for a set of illustrative parameters with corotation occurring at 6.67 kpc, and the turbulence parameters close to those in the M51 mode studied by Ruzmaikin et al which extends over = 1 kpc. Three growing corotating modes were found, the fastest extending for ~ 3 kpc, the other two for over 4 kpc. The first two grow 2-3 times faster, the third somewhat slower, than the M51 mode.
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Margrave, Gary F. « Direct Fourier migration for vertical velocity variations ». GEOPHYSICS 66, no 5 (septembre 2001) : 1504–14. http://dx.doi.org/10.1190/1.1487096.
Texte intégralRésumé :
The Stolt f‐x migration algorithm is a direct (i.e. nonrecursive) Fourier‐domain technique based on a change of variables, or equivalently a mapping, that converts the unmigrated spectrum to the migrated spectrum. The algorithm is simple and efficient but limited to constant velocity. A v(z) f‐k migration method, capable of very high accuracy for vertical velocity variations, can be formulated as a nonstationary filter that avoids the change of variables. The result is a direct Fourier‐domain process that, for each wavenumber, applies a nonstationary migration filter to a vector of input frequency samples to create a vector of output frequency samples. The filter matrix is analytically specified in the mixed domain of input frequency and migrated time. It can be moved to the full‐Fourier domain of input frequency and output frequency by a fast Fourier transform. When applied for constant velocity, the v(z) f‐k algorithm is slower than the Stolt method but without the usual artifacts related to complex‐valued frequency‐domain interpolation. Vertical velocity variations, through an rms‐velocity (straight‐ray) assumption, are handled by the v(z) f‐k method with no additional cost. Greater accuracy at slight additional expense is obtained by extending the method to a WKBJ phase‐shift integral. This has the same accuracy as recursive phase shift and is similar in cost. For constant velocity, the full‐Fourier domain migration filter is a discrete approximation to a Dirac delta function whose singularity tracks along a hyperbola determined by the migration velocity. For variable velocity, the migration filter has significant energy between hyperbolic trajectories determined by the minimum and maximum instantaneous velocities. The full‐Fourier domain offers interesting conceptual parallels to Stolt’s algorithm. However, unless a more efficient method of calculating the Fourier filter matrix can be found, the mixed‐domain method will be faster. The mixed‐domain nonstationary filter moves the input data from the Fourier domain to the migrated time domain as it migrates. It is faster because the migration filter is known analytically in the mixed domain.
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VanDecar, John C., et Roel Snieder. « Obtaining smooth solutions to large, linear, inverse problems ». GEOPHYSICS 59, no 5 (mai 1994) : 818–29. http://dx.doi.org/10.1190/1.1443640.
Texte intégralRésumé :
It is not uncommon now for geophysical inverse problems to be parameterized by [Formula: see text] to [Formula: see text] unknowns associated with upwards of [Formula: see text] to [Formula: see text] data constraints. The matrix problem defining the linearization of such a system (e.g., [Formula: see text]m = b) is usually solved with a least‐squares criterion [Formula: see text]. The size of the matrix, however, discourages the direct solution of the system and researchers often turn to iterative techniques such as the method of conjugate gradients to obtain an estimate of the least‐squares solution. These iterative methods take advantage of the sparseness of [Formula: see text], which often has as few as 2–3 percent of its elements nonzero, and do not require the calculation (or storage) of the matrix [Formula: see text]. Although there are usually many more data constraints than unknowns, these problems are, in general, underdetermined and therefore require some sort of regularization to obtain a solution. When the regularization is simple damping, the conjugate gradients method tends to converge in relatively few iterations. However, when derivative‐type regularization is applied (first derivative constraints to obtain the flattest model that fits the data; second derivative to obtain the smoothest), the convergence of parts of the solution may be drastically inhibited. In a series of 1-D examples and a synthetic 2-D crosshole tomography example, we demonstrate this problem and also suggest a method of accelerating the convergence through the preconditioning of the conjugate gradient search directions. We derive a 1-D preconditioning operator for the case of first derivative regularization using a WKBJ approximation. We have found that preconditioning can reduce the number of iterations necessary to obtain satisfactory convergence by up to an order of magnitude. The conclusions we present are also relevant to Bayesian inversion, where a smoothness constraint is imposed through an a priori covariance of the model.
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Bleistein, Norman, Jack K. Cohen et Frank G. Hagin. « Two and one‐half dimensional Born inversion with an arbitrary reference ». GEOPHYSICS 52, no 1 (janvier 1987) : 26–36. http://dx.doi.org/10.1190/1.1442238.
Texte intégralRésumé :
Multidimensional inversion algorithms are presented for both prestack and poststack data gathered on a single line. These algorithms both image the subsurface (i.e., give a migrated section) and, given relative true amplitude data, estimate reflection strength or impedance on each reflector. The algorithms are “two and one‐half dimensional” (2.5-D) in that they incorporate three‐dimensional (3-D) wave propagation in a medium which varies in only two dimensions. The use of 3-D sources does not entail any computational penalty, and it avoids the serious degradation of amplitude incurred by using the 2-D wave equation. Our methods are based on the linearized inversion theory associated with the “Born inversion.” Thus, we assume that the sound speed profile is well approximated by a given background velocity, plus a perturbation. It is this perturbation that we seek to reconstruct. We are able to treat the case of an arbitrary continuous background profile. However, the cost of implementation increases as one seeks to honor, successively, constant background, depth‐only dependent background, and, ultimately, fully lateral and depth‐dependent background. For depth‐only dependent background, the increase in CPU time is quite modest when compared to the constant‐background case. We exploit the high‐frequency character of seismic data ab initio. Therefore, we use ray theory and WKBJ Green’s functions in deriving our inversion representations. Furthermore, our algorithms reduce to finding quantities by ray tracing with respect to a background medium. In the constant‐background case, the ray tracing can be eliminated and an explicit algorithm obtained. In the case of a depth‐only dependent background, the ray tracing can be done quite efficiently. Finally, in the general 2.5-D case, the ray‐tracing procedure becomes the principal issue. However, the robustness of the inversion allows for a sparse computation of rays and interpolation for intermediary values. The inversion techniques presented here cover the cases of common‐source gather, common‐receiver gather, and common‐offset gather. Zero offset is a special case of the last of these. For offset data, the reflection coefficient is angle‐dependent, so parameter extraction is more difficult than in the zero‐offset case. Nonetheless, we are able to determine the unknown angle pointwise and derive parameter estimates at the same time as we produce the image. For each reflector, this estimate of the output is based on the Kirchhoff approximation of the upward‐scattered data. Thus, it is constrained to neither small discontinuities in sound speed at the reflector nor to small offset angle as would be the case for a strict “Born approximation” of the reflection process. The prestack algorithms presented here are inversions of single gathers. The question of how best to composite or “stack” these inversions is analogous to the question for any migration scheme and is not treated here.
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Liberato, M. L. R., J. M. Castanheira, L. de la Torre, C. C. DaCamara et L. Gimeno. « Wave Energy Associated with the Variability of the Stratospheric Polar Vortex ». Journal of the Atmospheric Sciences 64, no 7 (1 juillet 2007) : 2683–94. http://dx.doi.org/10.1175/jas3978.1.
Texte intégralRésumé :
Abstract A study is performed on the energetics of planetary wave forcing associated with the variability of the northern winter polar vortex. The analysis relies on a three-dimensional normal mode expansion of the atmospheric general circulation that allows partitioning the total (i.e., kinetic + available potential) atmospheric energy into the energy associated with Rossby and inertio-gravity modes with barotropic and baroclinic vertical structures. The analysis mainly departs from traditional ones in respect to the wave forcing, which is here assessed in terms of total energy amounts associated with the waves instead of heat and momentum fluxes. Such an approach provides a sounder framework than traditional ones based on Eliassen–Palm (EP) flux diagnostics of wave propagation and related concepts of refractive indices and critical lines, which are strictly valid only in the cases of small-amplitude waves and in the context of the Wentzel–Kramers–Brillouin–Jeffries (WKBJ) approximation. Positive (negative) anomalies of the energy associated with the first two baroclinic modes of the planetary Rossby wave with zonal wavenumber 1 are followed by a downward progression of negative (positive) anomalies of the vortex strength. A signature of the vortex vacillation is also well apparent in the lagged correlation curves between the wave energy and the vortex strength. The analysis of the correlations between individual Rossby modes and the vortex strength further confirmed the result from linear theory that the waves that force the vortex are those associated with the largest zonal and meridional scales. The two composite analyses of displacement- and split-type stratospheric sudden warming (SSW) events have revealed different dynamics. Displacement-type SSWs are forced by positive anomalies of the energy associated with the first two baroclinic modes of planetary Rossby waves with zonal wavenumber 1; split-type SSWs are in turn forced by positive anomalies of the energy associated with the planetary Rossby wave with zonal wavenumber 2, and the barotropic mode appears as the most important component. In respect to stratospheric final warming (SFW) events, obtained results suggest that the wave dynamics is similar to the one in displacement-type SSW events.
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Huang, Guangtan, Xiaohong Chen, Jingye Li, Cong Luo, Hang Wang et Yangkang Chen. « Prestack seismic inversion using a Rytov-WKBJ approximation ». Geophysical Journal International, 20 juillet 2021. http://dx.doi.org/10.1093/gji/ggab281.
Texte intégralRésumé :
Summary Approaches to seismic modeling using integral methods, notably Born-WKBJ (Wentzel-Kramers-Brillouin-Jeffreys), have seen an increase in applications in geophysical prospecting and near-surface exploration. Moreover, due to its linearity characteristic, which can fast and efficiently simulate full-waveform information, the Born-WKBJ based method performs well in seismic inversion. However, the Born approximation is the linearized wavefield simulation by Taylor expansion of the wavefield and omitting the high-order term, which cannot simulate the wavefiled accurately, including the amplitude, phase, and waveform information. For the seismic inversion, especially the amplitude variation with angle/offset (AVA/AVO) inversion, the amplitude is the most important element for estimating the parameters. In this paper, a Rytov-WKBJ approximation-based method, which can simulate the seismic amplitude information more accurately, is introduced to prestack seismic inversion. Besides, in order to improve the resolution of the inversion results, the ℓ1 − 2-norm regularized basis pursuit inversion (BPI) is introduced to the inversion algorithm. Then, we demonstrate the superiority of the proposed method with the zero-offset and angle-dependent seismic data simulation. The model tests show that the proposed method performs better than the conventional method significantly both on amplitude and phase of the seismic data. Finally, the inversion tests of synthetic and field seismic data indicate that the proposed method can obtain more accurate results with a higher resolution.
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Kováč, Juraj, et Václav Klika. « Liouville-Green approximation for linearly coupled systems : Asymptotic analysis with applications to reaction-diffusion systems ». Discrete and Continuous Dynamical Systems - S, 2022, 0. http://dx.doi.org/10.3934/dcdss.2022133.
Texte intégralRésumé :
<p style='text-indent:20px;'>Asymptotic analysis has become a common approach in investigations of reaction-diffusion equations and pattern formation, especially when considering generalizations of the original model, such as spatial heterogeneity, where finding an analytic solution even to the linearized equations is generally not possible. The Liouville-Green approximation (also known as WKBJ method), one of the more robust asymptotic approaches for investigating dissipative phenomena captured by linear equations, has recently been applied to the Turing model in a heterogeneous environment. It demonstrated the anticipated modifications to the results obtained in a homogeneous setting, such as localized patterns and local Turing conditions. In this context, we attempt a generalization of the scalar Liouville-Green approximation to multicomponent systems. Our broader mathematical approach results in general approximation theorems for systems of ODEs without turning points. We discuss the cases of exponential and oscillatory behaviour first before treating the general case. Subsequently, we demonstrate the spectral properties utilized in the approximation theorems for a typical Turing system, hence showing that Liouville-Green approximation is plausible for an arbitrary number of coupled species outside of turning points and generally valid for fast growing modes as long as the diffusivities are distinct. Note that our line of approach is via showing that the solution is close (using suitable weight functions for measuring the error) to a linear combination of Airy-like functions.</p>
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Katti, Aavishkar. « Bright optical spatial solitons in a photovoltaic photorefractive waveguide exhibiting the two photon photorefractive effect ». Revista Mexicana de Física 69, no 2 Mar-Apr (1 mars 2023). http://dx.doi.org/10.31349/revmexfis.69.021301.
Texte intégralRésumé :
We investigate for the first time, photorefractive solitons in a two photon photorefractive waveguide which also exhibits the bulk photovoltaic effect. The dynamical evolution equation of such solitons has been obtained under the paraxial ray approximation along with the Wentzel-Kramers-Brilluoin Jefferys (WKBJ) approximation. The existence curve for the solitons is derived and four distinct regions of power have been identified in the absence of waveguiding depending upon the threshold power for self trapping. Bistable states have been observed to be present. We have studied the effect of the planar waveguide and found that it enhances the self trapping nonlinearity and hence results in a reduction of the threshold power required for formation of the soliton. The propagation of the light beam is studied for various different strengths of the waveguide. A beam which would not have normally been self trapped can now become a soliton by virtue of the planar waveguide structure. Finally, we investigate the linear stability of these solitons by both, the Lyapunov method and numerical simulations.
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Bhat, Pallavi, et Kandaswamy Subramanian. « Fluctuation dynamos at finite correlation times using renewing flows ». Journal of Plasma Physics 81, no 5 (13 juillet 2015). http://dx.doi.org/10.1017/s0022377815000616.
Texte intégralRésumé :
Fluctuation dynamos are generic to turbulent astrophysical systems. The only analytical model of the fluctuation dynamo, due to Kazantsev, assumes the velocity to be delta-correlated in time. This assumption breaks down for any realistic turbulent flow. We generalize the analytic model of fluctuation dynamos to include the effects of a finite correlation time, ${\it\tau}$, using renewing flows. The generalized evolution equation for the longitudinal correlation function $M_{L}$ leads to the standard Kazantsev equation in the ${\it\tau}\rightarrow 0$ limit, and extends it to the next order in ${\it\tau}$. We find that this evolution equation also involves third and fourth spatial derivatives of $M_{L}$, indicating that the evolution for finite-${\it\tau}$ will be non-local in general. In the perturbative case of small-${\it\tau}$ (or small Strouhal number), it can be recast using the Landau–Lifschitz approach, to one with at most second derivatives of $M_{L}$. Using both a scaling solution and the WKBJ approximation, we show that the dynamo growth rate is reduced when the correlation time is finite. Interestingly, to leading order in ${\it\tau}$, we show that the magnetic power spectrum preserves the Kazantsev form, $M(k)\propto k^{3/2}$, in the large-$k$ limit, independent of ${\it\tau}$.
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« On the instability of the flow in a squeeze lubrication film ». Proceedings of the Royal Society of London. Series A : Mathematical and Physical Sciences 430, no 1879 (8 août 1990) : 347–75. http://dx.doi.org/10.1098/rspa.1990.0094.
Texte intégralRésumé :
When two parallel plates move normal to each other with a slow time-dependent speed, the velocity field developed in the intervening film of fluid is approximately that of plane Poiseuille flow, except that the magnitude of the velocity is dependent on time and on the coordinate parallel to the planes. This fact is intrinsic to Reynolds’ lubrication theory, and can be shown to follow from the Navier-Stokes equations when both the modified Reynolds number ( Re M ) and an aspect ratio ( δ ) are small. The modified Reynolds number is the product of δ and an actual Reynolds number ( Re ), which is based on the gap between the planes and on a characteristic velocity. The occurrence of flow instability and of turbulence in the film depend on Re . Typical values of Re , which are known to be required for the linear instability of plane Poiseuille flow, are of order 6000. This condition can be achieved, even if Re M is of order 1, provided that δ is of order 10 -4 . Such parameter values are typical of lubrication problems. The Orr-Sommerfeld equation governing flow instability is derived in this paper by use of the WKBJ technique, δ being the approximate small parameter to represent the small length-scale of the disturbance oscillations compared with the larger scale of the basic laminar flow. However, the coefficients in the Orr-Sommerfeld equation depend on slow space and time variables. Consequently the eigenrelation, derivable from the Orr-Sommerfeld equation and the associated boundary conditions, constitutes a nonlinear first-order partial differential equation for a phase function. This equation is solved by use of Charpit’s method for certain special forms of the time-dependent gap between the planes, followed by detailed numerical calculations. The relation between time-dependence and flow instability is delineated by the calculated results. In detail the nature of the instability can be described as follows. We consider a disturbance wave at or near a particular station, the initial distribution of amplitude being gaussian in the slow coordinate parallel to the planes. In the context of the Orr-Sommerfeld equation and its eigenrelation, the particular station implies an equivalent Reynolds number, while the initial distribution of the disturbance wave implies an equivalent wavenumber. As time increases, the disturbance wave can be considered to move in the instability diagram of equivalent wavenumber against Reynolds number, in the sense that these parameters are time- and space-dependent for the evolution of the disturbance-wave system. For our detailed calculations we use a quadratic approximation to the eigenrelation, an approximation which is quite accurate. If the initial distribution implies a point within the neutral curve, when the plates are squeezed together the equivalent wavenumber falls while the equivalent Reynolds number rises, and amplification takes place until the lower branch of the neutral curve is nearly crossed. If the plates are pulled apart (dilatation) the equivalent wavenumber rises, while the Reynolds number drops, and amplification takes place until the upper branch of the neutral curve has been just crossed. In the case of dilatation the transition from amplification to damping takes place more quickly than for the case of squeezing, in part due to the geometry of the neutral curve.