Modeling Shallow Water Flows on General Terrains

A formulation of the shallow water equations adapted to general complex terrains is proposed. Its derivation starts from the observation that the typical approach of depth integrating the Navier-Stokes equations along the direction of gravity forces is not exact in the general case of a tilted curved bottom. We claim that an integration path that better adapts to the shallow water hypotheses follows the "cross-flow" surface, i.e., a surface that is normal to the velocity field at any point of the domain. Because of the implicitness of this definition, we approximate this "cross-flow" path by performing depth integration along a local direction normal to the bottom surface, and propose a rigorous derivation of this approximation and its numerical solution as an essential step for the future development of the full "cross-flow" integration procedure. We start by defining a local coordinate system, anchored on the bottom surface to derive a covariant form of the Navier-Stokes equations. Depth integration along the local normals yields a covariant version of the shallow water equations, which is characterized by flux functions and source terms that vary in space because of the surface metric coefficients and related derivatives. The proposed model is discretized with a first order FORCE-type Godunov Finite Volume scheme that allows implementation of spatially variable fluxes. We investigate the validity of our SW model and the effects of the bottom geometry by means of three synthetic test cases that exhibit non negligible slopes and surface curvatures. The results show the importance of taking into consideration bottom geometry even for relatively mild and slowly varying curvatures

Physical modeling of tsunamis generated by three-dimensional deformable granular landslides

Tsunamis are gravity water waves that are generated by impulsive disturbances such as submarine earthquakes, landslides, volcanic eruptions, underwater explosions or asteroid impacts. Submarine earthquakes are the primary tsunami source, but landslides may generate tsunamis exceeding tectonic tsunamis locally, in both wave and runup heights. The field data on landslide tsunami events are limited, in particular regarding submarine landslide dynamics and wave generation. Tsunamis generated by three-dimensional deformable granular landslides are physically modeled in the NEES (Network of Earthquake Engineering Simulation) 3D tsunami wave basin (TWB) at Oregon State University in Corvallis, Oregon. A novel pneumatic landslide tsunami generator is deployed to simulate natural landslide motion on a hill slope. The instrumentation consists of various underwater, above water and particle image velocimetry (PIV) cameras, numerous wave and runup gauges and a multi-transducer acoustic array (MTA). The subaerial landslide shape and kinematics on the hill slope and the surface elevation of the offshore propagating tsunami wave and runup on the hill slope are measured. The evolution of the landslide front velocity, maximum landslide thickness and width are obtained along the hill slope. The landslide surface velocity distribution is obtained from the PIV analysis of the subaerial landslide motion. The shape and the size of the submarine landslide deposit are measured with the MTA. Predictive equations are obtained for the tsunami wave amplitude, wave period and wavelength in terms of the non-dimensional landslide parameters. The generated 3D tsunami waves propagate away from the landslide source as radial wave fronts. The amplitudes of the leading tsunami waves decay away from the landslide source in radial and angular direction. The wave celerity of the leading tsunami wave may be approximated by the solitary wave speed while the trailing waves are slower due to the dispersion effects. The energy conversion rate between the landslide and the generated wave is estimated. The observed waves are weakly non-linear in nature and span from shallow water to deep water depth regime. The unique experimental data serves the validation and advancement of numerical models of tsunamis generated by landslides. The obtained predictive equations facilitate initial rapid tsunami hazard assessment and mitigation.Ph.D.Fritz, Herman

THE SEDIMENTARY EVOLUTION OF THE 'EXMOOR BASIN' DURING THE LATE EMSIAN - EARLY EIFELIAN: THE LYNTON FORMATION

An integrated investigation of the sedimentological, ichnological and syn-sedimentary tectonic aspect of the late Emsian to early Eifelian Lynton Formation (Lynton Beds of previous studies) has revealed a varied mudstone-dominated shallow marine succession which accumulated in a rapidly subsiding rifted basin. The 'Exmoor Basin' developed in response to a Devonian phase of transtension associated with dextral shear along a fundamental east-west lineament to the north (the Bristol Channel Fault Zone). The Lynmouth - East Lyn Fault (a splay off the BCFZ?) was active throughout the deposition of the Lynton Formation and strongly influenced the depositional styles developed along its length. A re-evaluation of the p/subsidence curve for the 'Exmoor Basin' using the latest biostratigraphic and lithological thickness data indicates a pattern consistent with a strike-slip basin; Carboniferous thermal phase p/subsidence values suggest only 10% crustal thinning compared to values c. 50 % claimed be previous authors (Dewey 1982, Sanderson 1984). The base of the exposed Lynton Formation is characterised by extensive intraformational slide deposits and the presence of phosphatic material which represents a highstand deposit that correlates with the eustatic transgressive T-R event Ic of Johnson et al. (1985). Following a period of gradual reduction in accomodation space the sequence was punctuated by a massive influx of sand and granule grade material deposited at the base of the Lynmouth - East Lyn Fault scarp. This material was swept together into a series of offshore sand ridges and a shoreface deposit adjacent to the fault scarp. A new process-response model has been developed to describe the offshore sand ridges that were moulded by a combination of semi-permanent trade wind induced geostrophic flow, oscillatory currents and (possibly) weak tides. The central part of the Lynton Formation records a gradual upwards increase in relative accomodation space and decrease in the influence of semi-permanent currents; dysaerobic substrates became widespread and a localised anoxic mud developed offshore. The transition into the overlying Hangman Sandstone Group was marked by the southward progradation of a sandy shoreline in the face of a period of world-wide eustatic sea-level rise. The older, more northerly shoreline was dominated by longshore currents whilst the younger shoreline preserved a mixed (lower energy) storm- and wave-dominated sequence. The rate of shoreline progradation was relatively slow and the Lynton Formation - Hangman Sandstone Group boundary is markedly diachronous; the thickness of the exposed Lynton Formation varies from 200m adjacent to the Lynmouth - East Lyn Fault, where previously unrecognised outliers of the Hangman Sandstone Group occur, to 250m some 5km down-palaeoslope. Although the ichnofauna was locally diverse, with 27 distinct ichnotaxa recognised within the Lynton Formation, the succession was dominated by a gradation between straight Palaeophycus tubulraris burrows and branching Chondrites systems reflecting the response of an organism tolerant to dysaerobic conditions. The study demonstrates the value of integrating sedimentological, ichnological and structural techniques when studying Devonian marine shelf successions which accumulated m a tectonically unstable setting