Focusing of non-linear eccentric waves in astrophysical discs

ABSTRACT We develop a fully non-linear approximation to the short-wavelength limit of eccentric waves in astrophysical discs, based on the averaged Lagrangian method of Whitham. In this limit there is a separation of scales between the rapidly varying eccentric wave and the background disc. Despite having small eccentricities, such rapidly varying waves can be highly non-linear, potentially approaching orbital intersection, and this can result in strong pressure gradients in the disc. We derive conditions for the steepening of non-linearity and eccentricity as the waves propagate in a radially structured disc in this short-wavelength limit and show that the behaviour of the solution can be bounded by the behaviour of the WKB solution to the linearized equations.

Scale effects of velocity dispersion and attenuation (Q−1) in layered viscoelastic medium

We have used numerical modeling of normal incidence P-waves in periodic and nonperiodic viscoelastic layered media to help improve understanding of the scale effects of heterogeneity on velocity dispersion and attenuation. The improved understanding of these effects facilitates better interpretation and integration of data acquired at different scales, such as seismic data, well-log data, and laboratory measurements. We developed a direct method for estimating velocity and attenuation for viscoelastic media, the viscoelastic Kennett-Frazer (vKF) method, which is an invariant imbedding (reflectivity) method that uses reflection and transmission transfer functions. The vKF method is used to estimate rigorous dispersion and attenuation curves for periodic and nonperiodic cases. The results from our studies validate and quantify the intuitive qualitative understanding that dispersion and attenuation for a layered viscoelastic medium depend on the ratio of wavelength of the waves ([Formula: see text]) and the spatial period of the medium ([Formula: see text]), similar to the elastic case. We also decoupled the total effective attenuation obtained from the viscoelastic case into scattering attenuation that can be modeled from the elastic case and the intrinsic effective attenuation that is present in addition to the scattering attenuation in the viscoelastic case. The calculated intrinsic effective attenuation curves matched the theoretical values at the ray theory and effective medium theory limits. We derived analytical expressions for the long-wavelength limit of velocity and attenuation in a viscoelastic medium. Our expressions can be used directly for the upscaling of well logs to seismic scale considering viscoelastic effects. The expressions indicate the coupling between effective velocity and effective intrinsic attenuation in the long-wavelength limit. Finally, we use the derived expressions to determine the difference between elastic versus viscoelastic upscaling and to highlight the impact on traveltime and amplitude by properly considering viscoelastic information, especially for quantitative seismic interpretation workflows.