Bjørn Gjevik

Run-up of solitary waves

A numerical model based on a Lagrangian description has been developed for studying run-up of long water waves governed by a set of Boussinesq equations. The performance of the numerical scheme has been tested by comparing with analytical solutions and experimental data. Simulations of the run-up of solitary waves on relatively steep planes (inclination angle > 20°) show surface displacements and run-up heights in good agreement with experiments. For waves with relatively large amplitude the simulations reveal the development of a breaking bore during the backwash. Results for run-up heights in converging and diverging channels are also presented.

Model simulations of the tides in the Barents Sea

A large-scale numerical model covering the Nordic Seas and the Arctic Ocean with 25–12.5-km grid resolution has been applied to study the tides in the Barents Sea. Tidal charts for elevation and phase are presented for the four tidal constituents M2 S2, N2, and K1. Computed harmonic constants for elevation and current have been compared with observations from 46 coastal and pelagic tidal stations. The agreement between model and field data is good for most stations with an overall standard deviation of ±2–5 cm for amplitude, ±18°–30° for phase, and ±1–3 cm/s for current speed. Calculated tidal residual currents show anticyclonic eddy circulation over the Svalbard Bank and around Bear Island and Hopen. Particle trajectories and streaklines due to the tidal flow over Svalbardbanken have been calculated, and the computed streaklines are found to agree well with observed rifts in the ice cover due to stranded icebergs.

A Numerical Model for Long Barotropic Waves and Storm Surges along the Western Coast of Norway

Abstract A finite-difference scheme is used in order to study the generation and propagation of long barotropic waves and storm surges along the western coast of Norway. The performance of the numerical scheme is investigated by comparing with analytical solutions for a model with a straight coastline and a continental shelf of uniform depth and width. Simulations with a model of the west coast of Norway show that the wind stress and the atmospheric pressure are of about equal importance for the largest storm surges. The maximum elevation of the sea surface occurs at the coast and the sea level decreases nearly linearly over the shelf. The surge amplitude at the coast agrees well with observations. The sea level changes outside the shelf are small and for the most part due to the pressure. Shelf waves are mainly generated by the wind stress and Kelvin waves are mainly generated by the pressure field.

Modeling tsunamis from earthquake sources near Gorringe Bank southwest of Portugal

The Azores-Gibraltar fracture zone with the huge bathymetric reliefs in the area southwest of Portugal is believed to have been the source of large historic tsunami events. This report describes simulations of tsunami generation and propagation from sources near the Gorringe Bank. The well-documented 1969 tsunami event is examined both with a ray-tracing technique and with finite difference models based on various shallow water equations. Both methods show that the most likely source location is southeast of the Gorringe Bank near the epicenter location determined from seismic data. The tsunami source is calculated by formulas given by Okada [1985] for surface deformation of an elastic half-space caused by faulting. Observed wave amplitude and travel time and values computed from an initial wave field according to Okada [1985] formulas show acceptable agreement for most stations along the coast of Portugal and Spain. However, in order to explain a large primary wave with downward displacement observed on the coast of Morocco, an alternative source model with a larger area of downward displacement has been introduced. This also leads to a better overall fit with observed travel time. Implications for disastrous events, as the one in 1755, are also discussed. Linear hydrostatic shallow water models are used for most of the simulations, but the importance of nonlinearity and dispersion is examined with the Boussinesq equations. The sensitivity of the solution to changes in the location and the strength of the source is discussed, and a series of grid refinement studies are performed in order to assess the accuracy of the simulations.

Waves and Nonlinear Processes in Hydrodynamics

Preface. Part I: Review Articles. A Brief Historical Account of Hydrodynamic Research at the University of Oslo A. Eliassen. Hydrodynamics in Industry. Some Examples with Special Emphasis on Marine Applications F.G. Nielsen. Part II: Marine Hydrodynamics. Two Fragments of a Method for Fully Nonlinear Simulations of Water Waves X. Cai, E. Mehlum. On the Motion of a Rigid Body in Non-Uniform Flows A.R. Galper, T. Miloh. Interaction between Waves and Slowly Rotating Floating Bodies J. Grue. Experiences from Comparing Theory with Environmental and Platform Data T. Marthinsen. Nonlinear Scattering of Long Waves by a Vertical Cylinder J.N. Newman. Mathematical Aspects of the Cauchy-Poisson Problem in Three Dimensions F. Ursell. Part III: Nonlinear Waves. Finite Elements for the Boussinesq Wave Equations H.P. Langtangen, G. Pedersen. Continuous Bores on a Viscous Fluid Down an Incline J.-J. Lee, C.C. Mei. Refraction of Solitons and Wave Jumps G. Pedersen. Effect of Film Elasticity on Drift Currents Induced by Growing Waves O Saetra, J.E. Weber. A Resonating Triad of Gravity-Capillary Waves on a Long Gravity Wave K. Trulsen, C.C. Mei. Surface Tension Gradient-Driven Wave Motion in Shallow Liquid Layers M.G. Velarde, et al. Part IV: Nonlinear Stability and Convective Flows. Bifurcation Sequences in Problems of Thermal Convection and of Plane Couette Flow F.H. Busse, R.M. Clever. On the Steady Flow in a Cell Created by a Double-Diffusive Convection Instability K.B. Dysthe. Double-Diffusive Experiments A. Foldvik, B. Rudels. Role of Non-Wave-Like Disturbances in Transition M.T. Landahl. Behaviour of a Nonlinear Convective Oscillator as Modified by Sub-and Super-Critically Unstable Hopf Bifurcations P.A. Lundberg. Waves and Secondary Flows in Stratified Gas/Liquid Duct Flow M. Nordsveen, A.F. Bertelsen. Transient Free Convection in a Horizontal Porous Cylinder with a Sudden Change in Wall Temperature H.O. Sundfor, P.A. Tyvand. Rayleigh-Benard Convection with Weak Shear Flow: Absolute and Convective Instabilities M. Tveitereid, H.W. Muller. Part V: Geophysical Fluid Dynamics. Experiments on Turbulence in Stratified Rotating Flows S.B. Dalziel, et al. The Flow Generated in a Stratified Fluid by the Motion of a Flat Horizontal Disk P.A. Davies, et al. Models of Drift and Dispersion in Tidal Flows B. Gjevik. Self-Organization Phenomena in 2D-Flows G.J.F. van Heijst. A New Paradigm for River Plume Mixing T.A. McClimans. Wave Attenuation due to Bottom Vegetation M. Mork. Modelling Mesoscale Features in the Ocean L.P. Roed. Alphabetical List of Contributors.

Surface readjustment owing to a subcrustal phase trnsition

Abstract With time-dependent loads acting on the surface, the motion induced by a phase transition in an otherwise homogeneous mantle model is studied. An analytical solution of the initial value problem shows that when a surface load disappears, the thermal properties of the mantle material and of the phase transition process determine a relaxation time for the adjustment of the surface. Numerical results indicate that the phase transitions which are beleived to exist in the mantle, lead to a relaxation time of the order of magnitude observed in the postglacial upheaval of Fennoscandia.

A high resolution tidal model for the area around The Lofoten Islands, Northern Norway

A depth-integrated numerical model with grid resolution 500 m has been used to simulate tides around the Lofoten Islands in northern Norway. The model spans more than 31 latitude and covers a sea area of approximately 1:2 � 10 5 km 2 : The fine spatial resolution resolves the important fine scale features of the bottom topography on the shelf and the complex coastline with fjords and islands. Boundary conditions at the oceanic sides of the model domain are obtained by interpolation from a large-scale tidal model covering the Nordic Seas and implemented with the flow relaxation scheme (FRS). The semi-diurnal components M2; S2 and N2 and the diurnal component K1 are simulated. Harmonic constants for sea level are compared with observations from 21 stations. The best fit is found for the M2 component with a standard deviation between the observed and modelled amplitude and phase of 2:3 cm and 2:51; respectively. The standard deviation for the other smaller components ranges between 1.5–2:8 cm and 5.3–16:71: Current fields from the model are compared with observations in four locations: the Moskenes sound,the Gims y channel,the Tjeldsund channel and the Sortland channel. In the Sortland channel,the model predicts a dominant diurnal K1 current in agreement with observations. r 2002 Published by Elsevier Science Ltd.

On the Non-linear Stability of Plane Couette Flow

Plane Couette flow is defined as the flow which takes place between two parallel planes in the case of no pressure gradient in the flow direction. When no disturbance occurs, the velocity will be a linear function of the vertical coordinate z.

Idealized model simulations of barotropic flow on the Catalan shelf

Abstract The effect of along shelf variation in shelf width on barotropic shelf edge flows and topographic shelf waves are investigated with a numerical model. The model topography represents a gradual transition from a narrow straight to a broader straight shelf region. It is designed primarily to model the Catalan shelf on the northeastern Mediterranean coast of Spain, but the results obtained here are of general validity and will also apply to shelf flow under similar conditions elsewhere. The numerical experiments are carried out with flow imposed, both steady and oscillatory, at the upstream end of the narrow shelf region. With a prescribed shelf edge flow there is a strong tendency for topographic steering in the transition zone. The adjustment is found to take place on a short time scale of 2–5 days set by propagating wave modes rather than by advection. Bottom friction effects do not lead to significant on-shelf leakage of the flow. Flow instability and eddy formation are found to occur, even in cases where the potential vorticity for the imposed flow does not have a local maximum on the shelf slope. The instability leads to development of anticyclonic eddies on the narrow shelf which undergo strong non-linear adjustment at the transition zone from a narrow to a broader shelf. On the broad section of the shelf near the transition zone there is a tendency for formation of a large anticyclonic eddy. Bottom friction is found to influence the flow instability and the formation of eddies in cases where the flow is marginal for instability and growth of eddies. Particle tracking has been performed in order to study cross shelf mixing and transport. Finally, we have studied the current response due to a concentrated wind stress jet acting perpendicular to the coastline in the transition zone between the narrow and the broad shelf regions. A dipolar eddy structure is found to develop with the axis of the dipole tilting about 30° with the axis of the wind jet and with strong current shear in the zone near the dipole axis.

Simulations of shelf sea response due to travelling storms

Abstract A layered ocean model has been implemented for the numerical simulation of baroclinic shelf sea response, due to travelling storms. Atmospheric forcing by wind stress and surface pressure gradients is simulated by a circular cyclone model, corresponding to typical extratropical and tropical storms. Both Eulerian and Lagrangian coordinate descriptions have been used. The latter method enables also a more direct tracking of density interfaces, intersecting with the bottom topography. Three test examples have been investigated; storm response on a continental shelf and an escarpment at high latitudes; and the response of a tropical storm at a narrow shelf in low latitudes. The first two examples may simulate typical conditions over northwest European shelves: the latter may simulate hurricane response along the western coast of Mexico and the Gulf of California. The results of the simulations demonstrate typical features of storm response in these shelf regions, as narrow-banded barotropic shelf wave oscillations, near inertial period, and long periodic baroclinic wave modes. The performances of different numerical schemes have been investigated, in order to assess the accuracy of the numerical results.