Kolumban Hutter

The motion of a finite mass of granular material down a rough incline

Rock, snow and ice masses are often dislodged on steep slopes of mountainous regions. The masses, which typically are in the form of innumerable discrete blocks or granules, initially accelerate down the slope until the angle of inclination of the bed approaches the horizontal and bed friction eventually brings them to rest. The present paper describes an initial investigation which considers the idealized problem of a finite mass of material released from rest on a rough inclined plane. The granular mass is treated as a frictional Coulomb-like continuum with a Coulomb-like basal friction law. Depth-averaged equations of motion are derived; they bear a superficial resemblance to the nonlinear shallow-water wave equations. Two similarity solutions are found for the motion. They both are of surprisingly simple analytical form and show a rather unanticipated behaviour. One has the form of a pile of granular material in the shape of a parabolic cap and the other has the form of an M -wave with vertical faces at the leading and trailing edges. The linear stability of the similarity solutions is studied. A restricted stability analysis, in which the spread is left unperturbed shows them to be stable, suggesting that mathematically both are possible asymptotic wave forms. Two numerical finite-difference schemes, one of Lagrangian, the other of Eulerian type, are presented. While the Eulerian technique is able to reproduce the M -wave similarity solution, it appears to give spurious results for more general initial conditions and the Lagrangian technique is best suited for the present problem. The numerical predictions are compared with laboratory experiments of Huber (1980) involving the motion of gravel released from rest on a rough inclined plane. Although in these experiments the continuum approximation breaks down at large times when the gravel layer is only a few particle diameters thick, the general features of the development of the gravel mass are well predicted by the numerical solutions.

The dynamics of avalanches of granular materials from initiation to runout. Part I: Analysis

SummaryThis paper describes a model to predict the flow of an initially stationary mass of cohesion-less granular material down rough curved beds. This work is of interest in connection with the motion of rock and ice avalanches and dense flow snow avalanches. The constitutive behaviour of the material making up the pile is assumed to be described by a Mohr-Coulomb criterion while the bed boundary condition is treated by a similar Coulomb-type basal friction law assumption. By depth averaging the incompressible conservation of mass and linear momentum equations that are written in terms of a curvilinear coordinate system aligned with the curved bed, we obtain evolution equations for the depthh and the depth averaged velocityū. Three characteristic length scales are defined for use in the non-dimensionalization and scaling of the governing equations. These are a characteristic avalanche lengthL, a characteristic heightH, and a characteristic bed radius of curvatureR. Three independent parameters emerge in the non-dimensionalized equations of motion. One, which is the aspect ratio ε-H/L, is taken to be small. By choosing different orderings for the other two, the tangent of the bed friction angle δ and the characteristic non-dimensional curvature λ=L/R, we can obtain different sets of equations of motion which appropriately display the desired importance of bed friction and bed curvature effects. The equations, correct to order ε for moderate curvature, are discretized in the form of a Lagrangian-type finite difference representation which proved to be successful in the earlier studies of Savage and Hutter [24] for granular flow down rough plane surfaces. Laboratory experiments were performed with plastic particles flowing down a chute having a bed made up of a straight, inclined portion, a curved part and a horizontal portion. Numerical solutions are presented for conditions corresponding to the laboratory experiments. It is found that the predicted temporal-evolutions of the rear and front of the pile of granular material as well as the shape of the pile agree quite well with the laboratory experiments.

The Method of Weighted Residuals

Construction of analytical solutions to the TW-equation (2.24) subject to the no-flux boundary condition is possible only for some simple cases. Even though the equation may still be separable when written in a special coordinate system, the solution of the emerging ordinary differential equation may either only be expressible in terms of special functions which are tedious to handle, or must be obtained numerically. With realistic boundaries and non-vanishing curvature of the domain an exact solution can hardly be found. In ‘this chapter we therefore introduce a procedure, by which equation (2.24) can be solved approximately. The method consists of a reduction of the dimension of the mathematical problem by a basis (shape) function expansion and is a variant of the projection method, the spectral or modal mothod and may also be considered a generalized separation of variables procedure. Its advantage is that despite of its numerical intent, the method permits analytical techniques to be pursued farther than with classical numerical approaches.

A mathematical model of polythermal glaciers and ice sheets

Abstract A mathematical model of the flow and temperature distribution of polythermal glaciers or ice sheets is deduced. Cold ice is treated as a non-linear viscous heat conducting fluid, while temperate ice is regarded as a binary mixture of ice and water. The simplest mixture concept with two balance laws of mass but only one balance law of momentum and energy is proposed. The field equations for the ice and water content and the boundary conditions which must hold at the free surface, at the ice-water interface, at the cold-temperature transition surface and at the rock-bed are deduced. In particular it is shown that an earlier formulation of polythermal ice due to Fowler and Larson (1978) is inconsistent. No boundary value problems are solved as the emphasis is on the physical motivation and justification of the principles.

On internal wave dynamics in the northern basin of the lake of Lugano

Abstract Data collected from the Lake of Lugano during 3 July to 17 August 1979 are analysed for internal gravity wave motions. We demonstrate that a two-layer linear model is capable of explaining the internal wave response. A numerical finite difference procedure is used to determine the seiche periods and eigenfunctions of this model. The computed results (periods and phase differences between station pair interfacial displacements are then compared with the measured data. This comparison demonstrates that four conspicuous internal mode periods can be identified with fair to excellent statistical coherence between data-set pairs and that even higher order modes can be detected but with less statistical confidence. This identification proves that for the Lake of Lugano, no recourse has to be made to multi-layer models that would account for higher order baroclinicity.

Topographic waves in channels and lakes on the f-plane

1. Introduction.- 1.1 Preamble.- 1.2 Waves in waters.- 1.3 Observations - Their interpretations.- a) Lake Michigan.- b) Lake of Lugano (North basin).- c) Lake of Zurich.- d) Lake Ontario.- e) Other iakes and ocean basins.- 1.4 Aim of this work.- 2. Governing equations.- 2.1 Equations of adiabatic fluid flow.- 2.2 Vorticity, potential vorticity, topographic Rossby waves.- 2.3 Baroclinic coupling - the two-layer model.- a) Prerequisites.- b) Two-layer equations.- c) Approximations.- d) Scaie anaiysis.- e) Boundary conditions.- 2.4 Continuous stratification.- a) Modal equations.- b) Scaie anaiysis.- 2.5 TW-equation in orthogonal coordinate systems.- a) Preparation.- b) Cyiindricai coordinates.- c) Eiiipticai coordinates.- d) Natural coordinates.- e) Cartesian-coordinate correspondence principle.- 3. Some known solutions of the TW-equation in various domains.- 3.1 Circular basin with parabolic bottom.- 3.2 Circular basin with a power-law bottom profile.- 3.3 Elliptic basin with parabolic bottom.- 3.4 Elliptic basin with exponential bottom.- a) Basin with central island.- b) Basin without island.- 3.5 Topographic waves in infinite domains.- a) Straight channel.- b) Channel with one-sided topography.- c) Shelf.- d) Trench.- e) Single-step shelf.- f) Elliptic island.- 4. The Method of Weighted Residuals.- 4.1 Application to the TW-equation.- 4.2 Symmetrization.- 5. Topographic waves in infinite channels.- 5.1 Basic concept.- 5.2 Dispersion relation.- 5.3 Channel solutions.- 5.4 Velocity profiles.- 5.5 Alternative solution procedures.- 5.6 Hyperbolically curved channels.- 6. Topographic waves in rectangular basins.- 6.1 Crude lake model.- 6.2 Lake model with non-constant thalweg.- a) Numerical method.- b) New types of topographic waves.- c) Convergence and parameter dependence.- d) The bay-type.- 7. Reflection of topographic waves.- 7.1 Reflection at a vertical wall.- 7.2 Reflection at an exponential shore.- 7.3 Reflection at a sin2-shore.- a) Numerical method.- b) Reflection patterns.- 8. Review and outlook: Review and outlook: Restrictions of this study - A list of unsolved problems.- 8.1 A brief summary.- 8.2 Validity of TW-equations.- 8.3 Single or multivalued dispersion relation.- 8.4 Bay-type modes.- 8.5 A list of unsolved problems.- a) On the computational side.- b) On the physical side.- 8.6 Measurements, observations.- Appendix A.- Appendix B.- Appendix C.- References.- Author Index.

An extended channel model for the prediction of motion in elongated homogeneous lakes. Part 1. Theoretical introduction

Taking account of slenderness of many lakes, a hydrodynamic model is developed. The three-dimensional differential equations are formulated in a curvilinear Co-ordinate system along the ‘long’ axis of the lake. Applying the method of weighted residuals and expanding the field variables with shape functions over the cross-sections, approximate equations for the fluid motion are derived. The emerging equations form a cross-sectionally discretized set of spatially one-dimensional partial differential equations in the longitudinal lake direction. At first, these channel equations are presented for unspecified fluid properties and arbitrary shape functions, leaving applications possible for inviscid or viscous fluids with arbitrary closure conditions. The channel equations are subsequently specialized for Cauchy series as shape functions. For the free oscillation the simplest channel model is shown to reduce to the classical Chrystal equation. A first-order linear channel model is deduced. It exhibits the essential features of gravitational oscillations in rotating basins, in that it provides wave-type solutions with the characteristics of Kelvin and Poincare waves. This paper presents the derivation of the equations. Their application to ideal and real basins is deferred to several further papers.

An extended channel model for the prediction of motion in elongated homogeneous lakes. Part 3. Free oscillations in natural basins

The extended channel model derived and analysed in two previous articles is further developed by investigating free oscillations and forced motion in natural enclosed basins. Firstly, a zeroth-order model is analysed. In this model, field variables are expressed as a product of a single known cross-sectional shape function and an unknown function of time and the co-ordinate along the lake axis. Conditions are discussed under which this zeroth-order model is meaningful, and it is shown that under normal circumstances Coriolis effects must be ignored. Subsequently the general Nth-order channel model is applied to the Lake of Lugano. It is shown that eigedrequencies and amphidromic systems are well predicted in such channel-like lakes. The paper ends with a discussion on the selection of shape functions and with further applications and limitations of the channel model.

Strömungsdynamische Untersuchungen im Zürich- und im Luganersee Ein Vergleich von Feldmessungen mit Resultaten theoretischer Modelle

Density distribution and temperature and current data which were measured in Lake of Zürich and Lake of Lugano in two summer field programs are scrutinized and interpreted by means of hydrodynamic models. Barotropic surface seiches and baroclinic internal seiches are studied using a one-and two-layer model and measured data permit identification of the respective resonant periods. Wind induced barotropic and baroclinic circulation dynamics is also studied by means of three-dimensional finite difference models of the hydrodynamic viscous equations on the rotating earth. In general the predictions of these models agree favorably with observed data.

The surface seiches of Lake of Lugano

A computational analysis of the periods and structure of surface seiches of the southern basin of Lake of Lugano and its experimental verification from three simultaneous water gauge recordings, mounted along the shores in Agno, Morcote and Riva S. Vitale, is given. The first five theoretical modes are calculated with a finite element code of the tidal equations; it yields the eigenperiods and co-range and co-tidal lines, which are graphically displayed and discussed in detail.Experimental verification is from recordings taken during February/March 1982. Inspection by eye allows identification of the five lowest order modes, partly including interstation phase shift. Power spectral analysis of three-time series and interstation phase difference and coherence spectra allow identification of higher order modes, probably up to order 13. Agreement between the theoretically predicted and the experimentally determined periods is excellent for all calculated modes.Computational results for the four lowest modes and their structure of the northern basin of Lake of Lugano are also presented and experimentally verified with records taken from an event in August 1979. Agreement is again excellent.ZusammenfassungDie Perioden und die Struktur der Oberflächenseiches des Südbeckens des Luganersees werden rechnerisch ermittelt und mittels dreier simultan arbeitender Pegel, die dem Ufer entlang bei Agno, Morcote und Riva S. Vitale aufgebaut waren, bestätigt und erweitert. Die ersten fünf theoretischen Eigenschwingungen werden mit einem Finiten-Element-Code für die Tidegleichungen berechnet; man erhält die Seicheperioden und die Linien konstanten Hubes und konstanter Phase, die graphisch dargestellt und detailliert diskutiert werden.Die experimentelle Überprüfung erfolgt mit Zeitreihen von Oberflächenschwankungen, welche im Februar/März 1982 registriert worden sind. Die Sichtung dieser Zeitreihen gestattet die Identifikation der fünf ersten Seiches, teilweise einschliesslich der Phasendifferenz Zwischen Zeitreihen von Stationspaaren. Spektralanalyse der Energieverteilung von drei Zeitreihen sowie Spektren der Phasendifferenz und Kohärenz gestatten auch höhere Eigenschwingungen des Sees zu identifizieren, wahrscheinlich bis zur Ordnung 13. Die Übereinstimmung zwischen den theoretisch vorausgesagten und den rechnerisch bestimmten Perioden ist für alle berechneten Seiches ausgezeichnet.Rechnerische Resultate für die vier niedrigsten Seicheperioden und deren Struktur werden auch für das Nordbecken des Luganersees gegeben und experimentell mit Daten einer Episode vom August 1979 bestätigt. Die Übereinstimmung ist wiederum sehr gut.RésuméLes périodes et la structure des seiches de surface du bassin sud du lac de Lugano sont calculés et vérifiés simultanément au moyen de trois jauges installées à Agno, Morcote et Riva S. Vitale. Les cinq premières oscillations théoriques sont calculées au moyen dun code à éléments finis pour les équations de niveau; on obtient les périodes de seiches ainsi que les lignes délévation constante et de phase constante, dont la représentation graphique est discutée en détail.La vérification expérimentale est effectuée au moyen denregistrements de seiches pendant les mois de février et mars 1982. Lanalyse visuelle permet didentifier les cinq premières seiches, en partie même la différence de phase entre les enregistrements de stations opposées. Lanalyse spectrale de la distribution dénergie de trois enregistrements ainsi que les spectres de la différence de phase et de la cohérence permettent lidentification doscillation dordre supérieur, probablement jusquà 13. Laccord entre les périodes déterminées théoriquement et par mesure est excellent pour toutest les seiches considérées.Des résultats numériques concernant les quatre premiers ordres de seiches du bassin nord du lac de Lugano sont également présentés et confirmés par des données de mesures dun épisode du mois daoût 1979. Laccord est une fois de plus très bon.