ALBERT

Description: This paper is an extension of the single-phase cohesionless dry granular avalanche model over curved and twisted channels proposed by Pudasaini and Hutter (2003). It is a generalisation of the Savage and Hutter (1989, 1991) equations based on simple channel topography to a two-phase fluid-solid mixture of debris material. Important terms emerging from the correct treatment of the kinematic and dynamic boundary condition, and the variable basal topography are systematically taken into account. For vanishing fluid contribution and torsion-free channel topography our new model equations exactly degenerate to the previous Savage-Hutter model equations while such a degeneration was not possible by the Iverson and Denlinger (2001) model, which, in fact, also aimed to extend the Savage and Hutter model. The model equations of this paper have been rigorously derived; they include the effects of the curvature and torsion of the topography, generally for arbitrarily curved and twisted channels of variable channel width. The equations are put into a standard conservative form of partial differential equations. From these one can easily infer the importance and influence of the pore-fluid-pressure distribution in debris flow dynamics. The solid-phase is modelled by applying a Coulomb dry friction law whereas the fluid phase is assumed to be an incompressible Newtonian fluid. Input parameters of the equations are the internal and bed friction angles of the solid particles, the viscosity and volume fraction of the fluid, the total mixture density and the pore pressure distribution of the fluid at the bed. Given the bed topography and initial geometry and the initial velocity profile of the debris mixture, the model equations are able to describe the dynamics of the depth profile and bed parallel depth-averaged velocity distribution from the initial position to the final deposit. A shock capturing, total variation diminishing numerical scheme is implemented to solve the highly non-linear equations. Simulation results present the combined effects of curvature, torsion and pore pressure on the dynamics of the flow over a general basal topography. These simulation results reveal new physical insight of debris flows over such non-trivial topography. Model equations are applied to laboratory avalanche and debris-flow-flume tests. Very good agreement between the theory and experiments is established.

Description: We present the results of numerical experiments performed with the use of a fully non-linear non-hydrostatic numerical model to study the baroclinic response of a long narrow tank filled with stratified water to an initially tilted interface. Upon release, the system starts to oscillate with an eigen frequency corresponding to basin-scale baroclinic gravitational seiches. Field observations suggest that the disintegration of basin-scale internal waves into packets of solitary waves, shear instabilities, billows and spots of mixed water are important mechanisms for the transfer of energy within stratified lakes. Laboratory experiments performed by D. A. Horn, J. Imberger and G. N. Ivey (JFM, 2001) reproduced several regimes, which include damped linear waves and solitary waves. The generation of billows and shear instabilities induced by the basin-scale wave was, however, not sufficiently studied. The developed numerical model computes a variety of flows, which were not observed with the experimental set-up. In particular, the model results showed that under conditions of low dissipation, the regimes of billows and supercritical flows may transform into a solitary wave regime. The obtained results can help in the interpretation of numerous observations of mixing processes in real lakes.

Description: We present a viscoelastic constitutive relation which describes transient creep of a modified second grade fluid enhanced with elastic properties of a solid. The material law describes a Rivlin-Ericksen material and is a generalization of existing material laws applied to study the viscoelastic properties of ice. The intention is to provide a formulation tailored to reproduce the viscoelastic behaviour of ice ranging from the instantaneous elastic response, to recoverable deformation, to viscous, stationary flow at the characteristic minimum creep rate associated with the deformation of polycrystalline ice. We numerically solve the problem of a slab of material shearing down a uniformly inclined plate. The equations are made dimensionless in a form in which elastic effects and/or the influence of higher order terms (i.e., strain accelerations) can be compared with viscous creep at the minimum creep rate by means of two dimensionless parameters. We discuss the resulting material behaviour and the features exhibited at different parameter combinations. Also, a viable range of the non-dimensional parameters is estimated in the scale analysis.

Description: Results of an experimental and theoretical study of the interaction of a first mode internal solitary wave with a localised bottom topography (sill) are presented. Laboratory experiments have been performed in a 10m long and 0.33m wide channel filled with a stratified fluid. The interface between the two layers (fresh and salt water) is diffuse and has a finite thickness. Soliton-type disturbances of the interface having characteristics of the first baroclinic mode are generated at one channel end. They move along the channel and encounter an underwater obstacle (sill) in the middle of the channel, where they break into reflected and transmitted waves. Two types of internal waves are produced by the interaction: a fast first mode internal soliton and a slower (by a factor of approximately 3) second mode soliton-like wave. A numerical model, based on the two-dimensional Navier-Stokes equations in the Boussinesq approximation, is used tore produce the laboratory experiment. The detailed analysis of the horizontal and vertical structures of transmitted and reflected waves showed that the fast reflected and transmitted waves observed in the experiment can be interpreted as a first mode internal solitary wave whose characteristics are very close to those of the K-dV solitons. It is also demonstrated that the slow speed waves, generated during the interaction behind the first fast wave have vertical and horizontal structures very close to the second mode internal K-dV solitons.

Description: We study how a coastal obstruction (peninsula or coastal island) affects the three-dimensional barotropic currents in an oblong rectangular basin with variable bathymetry across the basin width. The transverse depth profile is asymmetric and the peninsula or island lies in the middle of the long side of the rectangle. A semi-spectral model for the Boussinesq-approximated shallow water equations, developed in Haidvogel et al. and altered for semi-implicit numerical integration in time in Wang and Hutter, is used to find the steady barotropic state circulation pattern to external winds. The structural (qualitative) rearrangements and quanti2tative features of the current pattern are studied under four principal wind directions and different lengths of the peninsula and its inclination relative to the shore. The essentially non-linear relationships of the water flux between the two sub-basins (formed by the obstructing peninsula) and the corresponding cross-sectional area left open are found and analysed. It is further analysed whether the depth-integrated model, usually adopted by others, is meaningful when applied to the water exchange problems. The flow through the channel narrowing is quantitatively estimated and compared with the three-dimensional results. The dynamics of the vortex structure and the identification of the up-welling/down-welling zones around the obstruction are discussed in detail. The influence of the transformation of the peninsula into a coastal island on the global basin circulation is considered as are the currents in the channel. The geometric and physical reasons for the anisotropy of the current structure which prevail through all obtained solutions are also discussed.Key words: Oceanography: general (limnology; numerical modeling) - Oceanography: physical (currents)

Description: For many lakes the nonlinear transfer of energy from basin-scale internal waves to short-period motions, such as solitary internal waves (SIW) and wave trains, their successive interaction with lake boundaries, as well as over-turning and breaking are important mechanisms for an enhanced mixing of the turbulent benthic boundary layer. In the present paper, the evolution of plane SIWs in a variable depth channel, typical of a lake of variable depth, is considered, with the basis being the Reynolds equations. The vertical fluid stratification, wave amplitudes and bottom parameters are taken close to those observed in Lake Constance, a typical mountain lake. The problem is solved numerically. Three different scenarios of a wave evolution over variable bottom topography are examined. It is found that the basic parameter controlling the mechanism of wave evolution is the ratio of the wave amplitude to the distance from the metalimnion to the bottom d. At sites with a gentle sloping bottom, where d is small, propagating (weak or strong) internal waves adjust to the local ambient conditions and preserve their form. No secondary waves or wave trains arise during wave propagation from the deep part to the shallow water. If the amplitude of the propagating waves is comparable with the distance between the metalimnion and the top of the underwater obstacle ( d ~ 1), nonlinear dispersion plays a key role. A wave approaching the bottom feature splits into a sequence of secondary waves (solitary internal waves and an attached oscillating wave tail). The energy of the SIWs above the underwater obstacle is transmitted in parts from the first baroclinic mode, to the higher modes. Most crucially, when the internal wave propagates from the deep part of a basin to the shallow boundary, a breaking event can arise. The cumulative effects of the nonlinearity lead to a steepening and overturning of the rear wave face over the inclined bottom and to the formation of a turbulent jet propagating upslope. Some time later, after the breaking event, a new stable stratification is formed at the site of wave destruction. The breaking criterion of ISWs is discussed.Key words. Oceanography: general (limnology; numerical modeling) – Oceanography: physical (internal and inertial waves)

Description: We report results gained with a three-dimensional, semi-implicit, semi-spectral model of the shallow water equations on the rotating Earth that allowed one to compute the wind-induced motion in lakes. The barotropic response to unidirectional, uniform winds, Heaviside in time, is determined in a rectangular basin with constant depth, and in Lake Constance, for different values and vertical distributions of the vertical eddy viscosities. It is computationally demonstrated that both the transitory oscillating, as well as the steady state current distribution, depends strongly upon the absolute value and vertical shape of the vertical eddy viscosity. In particular, the excitation and attenuation in time of the inertial waves, the structure of the Ekman spiral, the thickness of the Ekman layer, and the exact distribution and magnitude of the upwelling and downwelling zones are all significantly affected by the eddy viscosities. Observations indicate that the eddy viscosities must be sufficiently small so that the oscillatory behaviour can be adequately modelled. Comparison of the measured current-time series at depth in one position of Lake Constance with those computed on the basis of the measured wind demonstrates fair agreement, including the rotation-induced inertial oscillation.Key words. Oceanography: general (limnology) – Oceanography: physical (Coriolis effects; general circulation)

Description: Lake Onega is located in the southern part of Karelia, in the north-west of Russia. We report data taken by various limnigraphs and current meters and isolate, by spectral analysis, the most conspicuous barotropic periods of these signals. The analysis that follows is based on the linearized shallow-water equations, that are solved for the free oscillations and thus identify the eigenperiods and corresponding mode structures for this lake. Computational results are presented for a finite difference representation for these surface-seiche equations applied to the entire Lake Onega including its bays. The grid consists of quadratic elements of 1000 m side length corresponding to 9344 active cells. The emerging matrix eigenvalue problem exceeds the storage and compuational capacity of standard PCs or workstations and thus requires use of the approximate Lanczos procedure to isolate the first ten eigenperiods and corresponding mode structures of the barotropic seiches, lying between 12.1 and 2.2 h. Comparison of computational results and inferences from the water level and current-meter records, disclose satisfactory agreement between theory and observation.

Description: We present results of various circulation scenarios for the wind-induced three-dimensional currents in Lake Constance, obtained with the aid of a semi-spectral semi-implicit finite difference code developed in Haidvogel et al. and Wang and Hutter. Internal Kelvin and Poincaré-type oscillations are demonstrated in the numerical results, whose periods depend upon the stratification and the geometry of the basin and agree well with measured data. By solving the eigenvalue problem of the linearized shallow water equations in the two-layered stratified Lake Constance, the interpretation of the oscillations as Kelvin and Poincaré-type waves is corroborated.Key words: Oceanography: general (limnology; numerical modeling) – Oceanography: physical (internal and inertial waves)