Short Story Collection Vol. 041 by Various: Recent Episodes
LibriVox
LibriVox’s Short Story Collection 041: a collection of 20 short works of fiction in the public domain read by a group of LibriVox members.
In some tsunami waves travelling over the ocean, such as the one approaching the eastern coast of Japan in 2011, the sea surface of the ocean is depressed by a small meter-scale displacement over a multi-kilometer horizontal length scale, lying in front of a positive elevation of comparable magnitude and length, which together constitute a "down-up" or breather'' wave. Shallow water theory shows that the latter travels faster than the former and, according to the extended Korteweg-de Vries model presented here, the waves undergo a transition. Firstly, the two parts coincide at a given position and time producing a maximum elevation, whose amplitude depends on the shape of the approaching wave. Typically this amplitude is larger than the initial displacement magnitude by a factor which can be as large as two, which may explain anomalous elevations of tsunamis at particular positions along their trajectories. It is physically significant that for these small amplitude waves, no wave breaking occurs and there is no excess dissipation. Secondly, following the transition, the elevation wave moves ahead of the depression wave and the distance between them increases either linearly or logarithmically with time.The implications for how thesedown-up'' tsunami waves reach beaches are considered. This is joint work with Chow & Hunt.
Some numerical methods for water waves, such as the Craig-Sulem method, involve expanding terms in the water wave evolution equations as series, truncating those series, and then simulating the resulting equations. For one such scheme, we present analytical evidence that the truncated system is in fact ill-posed; this involves further reducing the evolution equations to a model for which we can prove ill-posedness. We then present numerical evidence that the full truncated system is ill-posed, showing that arbitrarily small data can lead to arbitrarily fast growth. We present this numerical evidence for multiple levels of truncation. We are able to prove that by adding a viscosity to the system, we instead arrive at a well-posed initial value problem. This is joint work with Jerry Bona and David Nicholls.
In fluid dynamics, a "splash" singularity occurs when a locally smooth interface self-intersects in finite-time. It is now well-known that solutions to the water waves equations (and a host of other one-phase fluid interface models) has a finite-time splash singularity. By means of elementary arguments, we prove that such a singularity cannot occur in finite-time for vortex sheet evolution (or two-fluid interfaces). This means that the evolving interface must lose regularity prior to self-intersection. We give a proof by contradiction: we assume that such a singularity does indeed occur in finite-time. Based on this assumption, we find precise blow-up rates for the components of the velocity gradient which, in turn, allows us to characterize the geometry of the evolving interface just prior to self-intersection. The constraints on the geometry then lead to an impossible outcome, giving the contradiction. This is joint work D. Coutand.