Numerical Simulations of a Dispersive Model Approximating Free-Surface Euler Equations

In some configurations, dispersion effects must be taken into account to improve the simulation of complex fluid flows. A family of free-surface dispersive models has been derived in Fernández-Nieto et al. (Commun Math Sci 16(05):1169–1202, 2018). The hierarchy of models is based on a Galerkin approach and parameterised by the number of discrete layers along the vertical axis. In this paper we propose some numerical schemes designed for these models in a 1D open channel. The cornerstone of this family of models is the Serre – Green-Naghdi model which has been extensively studied in the literature from both theoretical and numerical points of view. More precisely, the goal is to propose a numerical method for the $$LDNH_2$$ L D N H 2 model that is based on a projection method extended from the one-layer case to any number of layers. To do so, the one-layer case is addressed by means of a projection-correction method applied to a non-standard differential operator. A special attention is paid to boundary conditions. This case is extended to several layers thanks to an original relabelling of the unknowns. In the numerical tests we show the convergence of the method and its accuracy compared to the $$LDNH_0$$ L D N H 0 model.

Estimation and geologic interpretation of regolith chargeability and superparamagnetic susceptibility in airborne electromagnetic data

All available inversion software for airborne electromagnetic (AEM) data can at a minimum fit a nondispersive conductivity model to the observed inductive secondary field responses, whether operating in the time or frequency domain. Quasistatic inductive responses are essentially controlled by the induction number, the product of frequency with conductivity and magnetic permeability. Recent research has permitted the conductivity model to be dispersive, commonly using a single Cole-Cole parameterization of the induced polarization (IP) effect; but this parameterization slows down and destabilizes any inversion, and it does not account for the need for dual or multiple Cole-Cole responses. Little has been published on inverting AEM data affected by frequency-dependent magnetic permeability, or superparamagnetism (SPM), usually characterized by a Chikazumi model. Because the IP and SPM effects are small and are usually only obvious at late delay times, the aim of our research is to determine if these IP and SPM effects can be fitted and stripped from the AEM data after being approximated with simple dispersive models. We are able to successfully automate a thin-sheet model to do this stripping. Stripped data then can be inverted using a nondispersive conductivity model. The IP and SPM parameters fitted independently to each independent measured decay to provide stripping are proven to be spatially coherent, and they are geologically sensible. The results are found to enhance interpretation of the regolith geology, particularly the nature and distribution of transported materials that are not afforded by mapping conductivity/conductance alone.

Synchronization of traveling waves in coupled dispersive systems

<p>Сoupled KdV-type equations arise in multimodal dispersive models such as the Gear – Grimshaw system which describes weakly nonlinear internal waves in neighboring pycnoclines. Coupling occurs when two or more phase speeds of different modes are close together.  This phenomenon of kissing modes is known as the Eckart resonance providing energy transfer between pycnoclines in stratified fluid. Decoupled basic equations generate separated modes of traveling waves with different phase shifts. In this context, synchronization means the existence of coupled phase-shifted solutions which can be constructed from decoupled modes by appropriate perturbation procedure.  In the present paper, we consider analytic conditions which provide the existence of periodic solutions describing synchronized cnoidal-type wave trains. Application of the Lyapunov – Schmidt method reduces this problem to the nonlinear system of implicit bifurcation equations for unknown phase shift and wave amplitude. Asymptotic analysis of these equations results sufficient condition of synchronization, which involves the Poincare – Pontryagin function depending on coupling nonlinear terms. In addition, we illustrate two different limit cases which lead to the same existence condition.  First of them corresponds to a solitary-wave limit for cnoidal waves (i.e. a nonlinear long-wave limit), and the second one is adapted to a small-amplitude limit of coupled harmonic wave packets.</p><p>This paper was supported by RFBR (grant No 18-01-00648).</p>

Fully dispersive models for moving loads on ice sheets

The response of a floating elastic plate to the motion of a moving load is studied using a fully dispersive weakly nonlinear system of equations. The system allows for an accurate description of waves across the whole spectrum of wavelengths and also incorporates nonlinearity, forcing and damping. The flexural–gravity waves described by the system are time-dependent responses to a forcing with a described weight distribution, moving at a time-dependent velocity. The model is versatile enough to allow the study of a wide range of situations including the motion of a combination of point loads and loads of arbitrary shape. Numerical solutions of the system are compared to data from a number of field campaigns on ice-covered lakes, and good agreement between the deflectometer records and the numerical simulations is observed in most cases. Consideration is also given to waves generated by a decelerating load, and it is shown that a decelerating load may trigger a wave response with a far greater amplitude than a load moving at constant celerity.