## Abstract

An introduction is given to the title theme, in general, and the specific topics treated in detail in the articles of this theme issue of the *Philosophical Transactions*. They fit into the following broader subjects: (i) dense, dry and wet granular flows as avalanche and debris flow events, (ii) air-borne particle-laden turbulent flows in air over a granular base as exemplified in gravity currents, aeolian transport of sand, dust and snow and (iii) transport of a granular mass on a two-dimensional surface in ripple formations of estuaries and rivers and the motion of sea ice.

## 1. Background

There are numerous occurrences of granular and particle-laden flows in nature. Examples are rock, ice and snow avalanches; water-saturated debris and mud flows; hot pyroclastic flows from volcanoes, i.e. so-called lahars; and creeping and catastrophic movements of soil in landslides. Granular and particle-laden flows also arise as sediment transport in rivers and along lake and ocean shores, both at the bed and in suspension. The aeolian transport of desert sand leads to dune formation and dune migration. Underwater turbidity currents and the turbulent air-borne flow of snow in powder snow avalanches or of volcanic ash in pyroclastic hot gravity currents are further examples with great socioeconomic and environmental impact. The motion of sea ice in the polar ocean is a two-dimensional flow of an assemblage of a large number of ice floes bouncing into each other while moving on the ocean surface. It plays an important role as a component in the prediction of the annual and inter-annual variation of the climate on Earth, and, similarly, the disk-like forms and the extents of planetary rings, such as that of Saturn, are the result of the perpetual collisions of the large number of icy particles, forming these rings subject to the gravitational field due to the central body. Although these examples might seem to be of very disparate nature, they in fact have many similarities. Consequently, it is beneficial to outline some of these similarities from a more conceptual point of view.

Granular materials are extremely complex in their physical behaviour and demand subtle theoretical descriptions and mathematical models. It is commonly observed that granular materials are capable of both fluid-like and solid-like behaviour. The situation becomes far more complicated when grains are mixed with a fluid. Dense, dry sand can support static loads, and therefore enjoy solid-like behaviour. Its load-bearing is strongly enhanced by the addition of small amounts of water. With the amount of interstitial water increasing, the sand may become fluidized. In slow creeping or moderately fast moving flows, such a fluidized mixture behaves much like a (very) viscous non-Newtonian fluid. In a debris or mud flow, the particles and fluid may interact strongly, resulting in turbulence on several different length-scales. In the rapid motion of sand down inclines and the dynamics of planetary rings, the perpetual collisions of the particles and the vigorous fluctuating motion induced by these give rise to an averaged motion that is akin to the motion of a dense gas, in which the particles play the role of the molecules. Indeed, the model equations for this gaseous limit of behaviour are derived from statistical principles that are borrowed from the kinetic theory of a dense molecular monatomic gas.

Thus, the dynamics of granular and particle-laden flow may occur in different behavioural regimes. Different flow regimes typically occur in the same geophysical event. In a powder snow avalanche, or in an underwater turbidity current, the majority of the material may be subject to a forcing above a relatively narrow region of intense shearing in which particles interact violently. In sediment transport, particles are incorporated from a dense bed, become water-borne, and interact with the bed and with each other until at distance a two-phase boundary layer is established. Much the same occurs at the surface of a desert sand dune or the snow drift on the surface of an ice sheet. In a mixed flow powder-snow avalanche this transition from the saltation layer to the turbulent suspension regime occurs at the outer edge of the bottom-most flow-avalanche layer. In all these examples, the regimes of the granular or particle-laden flows are initiated and maintained by the motion of the particles that were originally parts of the ground and broke loose from the deposits by external forcing mechanisms—winds, currents, instabilities. The growth, maintenance and decay of these flows are often caused by internal mechanisms, not further triggered by outside ‘agents’. For instance, particle-laden gravity currents, such as snow avalanches, turbidity currents and debris and mud flows often grow and eventually cease because of the nourishment and wastage, respectively, of entrained and deposited granular mass.

## 2. Modelling the physics

Conceptually, the different regimes of physical behaviour require different mathematical models. Methods of nonlinear continuum mechanics and non-equilibrium statistical mechanics are both employed. Discrete element methods that are the analogue of the molecular dynamics techniques have been used to inform modelling of the solid, fluid and gaseous states, including the transition from one to the other. Concepts from continuum mixture theory and turbulence are required to describe the interaction between the fluid and the grains in both dense granular flows with interstitial fluid, as well as in flows of dilute suspensions. Scaling arguments are often helpful in guiding theoretical formulations. For instance, scaling may give rise to the introduction of the thin film approximation as in shallow water or lubrication theory, or it may suggest asymptotic approximations in which flow regimes of different behaviour are matched or patched together. As an example, the avalanche and debris flow models treated in this volume are based on the scalings , , where *L* is a typical avalanche length or travel distance, *H* its typical depth and *V* and *T* are a typical velocity and time. Different from the classical shallow water scaling, *V* and *T* are formed with *L* (and not *H* as in the shallow water theory). The reason is that the ‘free fall velocity’ and ‘free fall time’ in a gravity field are the typical velocity and time-scales of avalanching flows. Similarly, the non-dimensionalizations in the saltation layer and the suspension regime of an aeolian transport or snow drift model require scalings and corresponding matched asymptotics that guarantee the mass flux from the granular bed into the suspension layer. The theoretical descriptions that result from such analyses are nonlinear, initial boundary-value problems, often with free boundaries. They are sometimes not well posed or of changing type. They are usually not amenable to analytic treatment, so numerical methods are required.

The basic physical principles on which the equations are based are the balance laws of mass and momenta (and energy), often complemented by model equations for the microstructural/subgrid processes such as turbulence (Reynolds stress; *k*−*Z* equations; *k*, turbulent kinetic energy; *Z*, a second scalar turbulent field, usually the turbulent dissipation rate), fluctuating motion due to collisions (granular temperature) and constitutive hypotheses for the closure variables (i.e. stresses, fluxes and dissipation rate, etc.). These equations, together with boundary conditions, are then subjected to scaling analyses and usually also to smearing operations to reduce the spatial dimension of the resulting equations. Often in such a process, additional simplifications are introduced, sometimes rationally motivated, sometimes ad hoc and a priori guessed, so that the final model equations are the result of a sequence of reductions of originally well-founded unquestioned equations. Obviously, the validity of these final equations is constrained by the assumptions they are based upon; and the equations gain their trustworthiness by comparison with corresponding experimental findings. Alternative models, based on a different chain of simplifying hypotheses, may be claimed to be valid under similar situations, but may yield conflicting results for particular aspects of the models. For example, when depth-averaging the balance laws of mass and momentum in an avalanche model, the depth dependence of the velocity field is replaced by an averaged velocity or the transport so that the information of the sliding and shearing contributions to the flow is lost. This fact is sometimes used as an argument to question the physics of depth-integrated models, especially when flow over bumpy beds is studied, because these flows may be considerably sheared. Similarly, the literature knows different models of aeolian transport for the description of the transport of granular mass from the deposit through the saltation to the suspension layer. Such models still require scrutiny of their equivalence.

## 3. Numerical techniques

The implementation of the appropriate numerical techniques is as challenging as is the formulation of the models. Often, initial boundary-value problems of nonlinear partial differential equations must be solved in domains, the boundaries of which are part of the solution. Many problems lead to hyperbolic systems of equations for which mathematical existence proofs are not yet available. Solutions for these are susceptible to shock formations, requiring shock-capturing numerical integration techniques. The governing equations for debris and particle-laden flows are generally parabolic because of the diffusive nature of the interstitial fluid pressure or the particle-phase dispersion. Strongly advective geophysical flows in complex terrain are associated with steep gradients of the field variables. Integration usually requires sufficient numerical diffusion to reduce possible unphysical instabilities, but a minimization of it when processes are smooth. This implies that shock-capturing, total variation diminishing and non-oscillatory integration techniques are indispensable for a reliable numerical prediction of granular particle-laden and catastrophic geophysical flows. Furthermore, problems with free boundaries require adaptive mesh generation in Eulerian and Lagrangian finite-difference, finite-element and finite-volume formulations, and usually involve complicated handling of marginal elements through special front-tracking procedures.

Numerical integration of the field equations of granular and particle-laden flows plays a central role because analytical solutions are hardly constructible for initial boundary-value problems that are of physical relevance. Comparison of such solutions with results from laboratory experiments and observations in the fields or forecasts of particular processes such as avalanches and debris flows, require the best possible integration techniques to ascertain that computational results indeed represent solutions of the equations. Now, depending on the intended goals, ‘compromises’ are frequently made, i.e. further approximations are introduced to facilitate the integration process. For instance, in avalanche and debris flow models, the governing equations are, more often than not, referred to a Cartesian coordinate system *Oxyz*, where *Oxy* is horizontal while *Oz* is vertical, and the shallowness scaling is introduced as referred to this system rather than with respect to a topography following metric. This would actually forbid the use of these simplified model equations for flows on steep slopes without further verification. Moreover, numerical integration techniques applied to this situation provide only limited information about the differential geometric properties of the topography over which the granular flow takes place. The fact that nothing explicitly is said about these does not make the equations automatically valid for flows over arbitrarily curved and twisted beds. On the other hand, equations for which such limitations are explicitly spelled out, are sometimes used beyond these limits. In either case, careful justification is mandatory.

## 4. Laboratory experiments and field observations

The verification of the predictions of models against in situ observations is often difficult or impossible because of aggravated accessibility and unpredictability of the time and location of catastrophic events, the dangers involved with on-site experimental campaigns or extreme meteorological–climatological conditions. Furthermore, external geophysical conditions are uncontrollable. So, measured quantities are necessarily fraught with such uncontrollable information. Consequently, data acquisition must account for such uncertainties or unpredictabilities, and is necessarily followed by analyses which eliminate these and isolate the statistical coherences between the time-series of the different measured quantities.

Because of these difficulties, laboratory tests are welcome alternatives. They can be performed under well-defined and well-controlled conditions. However, laboratory experiments are almost always conducted for processes of reduced size, both geometrically and dynamically. This raises questions regarding the significance of scale effects. It means that subprocesses comprising as a total the entire physical phenomenon, respond with different proportions at different sizes of the physical model. For instance, it is known that the run-out distances of dry rock avalanches show a clear mass dependence provided the mass of the avalanche is larger than approximately 10^{6} m^{3}. On the one hand, this reveals that the mass dependence of such run-outs probably can never be studied by laboratory tests. On the other hand, laboratory avalanche tests with dry grains are meaningful since for sufficiently small particle diameters (as compared with the avalanche depth) size effects are probably not significant. The depth integrated Savage–Hutter avalanche model and many of its extensions are described by scale invariant equations as long as its frictional behaviour in the bulk and at the base is founded on Mohr–Coulomb sliding alone. Its good performance and coincidence with laboratory experiments is at least partial corroboration that dry avalanche processes at small size are scale independent.

Water-saturated debris flows, on the other hand, are known to be scale dependent by the pore pressure of the interstitial fluid, and run-out distances indeed depend on the ratio of the amount of fluid to that of the solids that are present. Similarly, the particle-laden flow in aeolian transport of sand and snow drift are equally scale dependent, since different processes prevail in the reptation, saltation and suspension layers. In all these cases, experimental verification of the mathematical models is episodic. Verification of the model equations on prototype-size observations is vital in these cases if complete corroboration is sought.

It is evident that controlled experimental tests or field observations in natural events are difficult because of the complexity of the theoretical models, the physical phenomena, and—for field observations—the often hostile and dangerous conditions of the environment. In the laboratory, particle image velocimetry, including photographic and photogrammetric techniques and nuclear magnetic resonance are applied. Wind tunnel particle-laden flows and water flows in flumes with movable, sediment-covered beds are employed. In the field, when snow drift or aeolian transport is studied, usual meteorological stations, equipped with anemometers, temperature, humidity and radiation sensors at several levels of the atmospheric boundary layer are used. They are complemented by sensors capable of measuring the particle flux and particle diameters as functions of the distance from the ground. Pressure gauges, geophones and seismometers, as well as remote sensing such as acoustic and radar Doppler techniques, are used to obtain information about velocity and density distribution in laboratory or field avalanche events. The conduction of further measuring campaigns, the development of additional experimental expertise and an increased appreciation for experiment also by theoreticians is, however, vital.

Clearly, progress in understanding granular and particle-laden flows involves the interaction of scientists who are concerned with modelling, numerical analysis, laboratory experimentation and field observation. Each of these activities obviously benefits from a unification of approach, the development of a common language and a catalogue of concepts. This is needed to reach a more fundamental and advanced understanding of the many models derived in a case by case manner.

The articles in this special issue of the *Philosophical Transactions* describe particular aspects of the scientific concepts outlined above and provide explicit demonstration of the difficulties.

## 5. Dense, dry and wet granular flows

Such flows arise in avalanches of snow, rock slides, debris flows and mud flows, as, for example in subaquatic turbidity currents, flows of hot ash down mountainsides from volcanoes, soil slope instabilities during heavy rainfall, and many other instances.

A popular depth-integrated dynamic model for the motion of a finite mass of cohesionless granular material is provided by Savage & Hutter (1989). It was derived from first principles using the three-dimensional balance laws of mass and momentum of a density preserving continuum at Coulomb plastic yield and subject to Coulomb dry basal friction. Introducing an aspect ratio parameter of typical avalanche depth to length, an asymptotic analysis to linear order in it leads to a hyperbolic set of model equations for avalanche thickness and surface-parallel velocity components. The model also involves a number of additional ad hoc assumptions and has been extended to two-dimensional curved and twisted channelized topographies (see Hutter *et al.* (2005) and references therein). The explicit identification of the simplifications and omissions that are introduced has led to criticisms of its validity with regard to its overstretched use beyond both the physical basis and geometric prerequisites. The first paper in this volume *The Savage–Hutter avalanche model: how far can it be pushed?* reiterates the underlying assumptions of the model and gives a balanced account on the delineation of its validity.

An ever reiterated topic of discussion in avalanche research is the form of the sliding law. Is it composed of Coulomb and velocity-dependent components? If so, can the two components be adequately identified? The article by Ancey (2005) is an attempt to shed light on this question by analysing 173 snow avalanche events from seven paths in the French Alps. Statistical analyses give support for the statement that, while some velocity-dependent contribution cannot be ruled out entirely, Coulomb behaviour is dominant. The back analysis suggests that the Coulomb friction coefficient is a random variable, the distribution of which can be approximated by a normal distribution with a volume-dependent mean. This result is interesting: on the one hand it provides independent support for the restriction by Savage & Hutter (1989) and many others to Coulomb type sliding; on the other hand it draws attention to a mass dependence of the drag coefficient. This sheds some light on the limitation of laboratory experiments.

The article by Pudasaini & Hutter (2005) is a contribution to the computational use and environmental implications of a topography following extended Savage and Hutter model: the equations allow estimation of the influence of the geometry of the basal topography—curvature and twist and cross-sectional profile—on the geometry of the avalanche. It is demonstrated that the geometric evolution and the forms and locations of the depositions depend as much on the geometric peculiarities of the avalanche track as on the physical parameters such as (internal and) bed friction angles.

Whereas the models discussed so far are applicable to dry and slightly wet cohesionless dense granular flows, Pitman & Le (2005) proposes a set of equations for the catastrophic flow of a saturated mixture of solid particles and a fluid. Debris flow models were already previously proposed: Pudasaini & Hutter (2005) simply used the Savage–Hutter (SH) equations and made the basal friction angle dependent on the interstitial pressure; Iverson (1997) and Iverson & Denlinger (2001) start with a two-phase formulation and give order of magnitude arguments to motivate that the fluid and solid velocities differ negligibly from one another, while the solid and fluid behave constitutively as a plastic Coulomb material subject to yield and Newtonian viscous material, respectively. The model equations are reduced to a ‘virtual’ solid material whose stress tensor is given by the sum of the solid and fluid stresses. The two-fluid model of Pitman & Le presented here does not make use of these simplifications and does not identify the solids with fluid velocities. Beyond the solids and fluid constitutive properties, the model also accounts for the fluid–solid interaction force. Much of the details of the computations that make use of the shallowness assumption referred to a horizontally–vertically oriented Cartesian coordinate system, are concerned with the depth averaging and involve a number of ad hoc assumptions. Since the fluid dissipation is taken into account by the velocity-dependent interaction force, the fluid stress may here be assumed as ideal. Beyond this difference to the Iverson–Denlinger model, the authors also present a reduced model in which the fluid inertial terms are ignored, which then exhibits complete Darcy structure.

## 6. Air-borne particle-laden flows

Particle-laden flows arise in the geophysical context as turbulent boundary layer currents in powder snow, dust and ash flows and as drift currents of sand and snow subject to strong winds. Early models were treating these as density- and gravity-driven currents; today, at a closer look, these flows are modelled as a combination of interacting reptation, saltation and suspension layers such that mass and momentum exchanges are properly accounted for. We give examples on both levels of complexity.

When dry snow on a mountainside breaks loose, the moving snow mass sometimes becomes an air-borne turbulent gravity current with a somewhat larger head and particle concentrations of the order of 10^{−3}–10^{−2}, depths of more than 100 m and velocities as large as 100–150 m s^{−1}. Early models of such particle-laden flows make use of density current concepts in which the interior pressure is supposed to be hydrostatic. This is oversimplified and does not allow determination of the velocity field in the front and within the head. Neither can the shape of the front approximately be computed. McElwaine's (2005) article is devoted to the determination of the angle between the inclination of the front and the slope, and the fluid velocities in the vicinities ahead and behind it. The calculations are extensions of those by von Kármán (1940); they are done here analytically for an arbitrary inclination angle of the slope.

The article by Pasini & Jenkins (2005) concentrates on the flow within the saltation layer of aeolian transport with wind so strong, that, apart from saltation, collisions between particles are inevitable. ‘The particles are suspended in the gas against gravity by a gradient in the particle pressure associated with the collisional momentum transfer and, through viscous drag, by the turbulent velocity fluctuations of the gas. The average drag of the particles upon the gas influences the local shear stress in the gas, while the total shear stress in the mixture is independent of position…’. The authors phrase a two-point boundary-value problem for the variation with distance above the bed of the average particle concentration, average horizontal particle and fluid velocities, and the average measures of the strength of the particle and fluid velocity fluctuations. Boundary conditions at the bed are uncritical, but those at the top of the collisional–saltation flow are tricky and require transition statements to the suspension layer. However, comparison with observations suggests amendments.

A similar problem, but now emphasizing experiments, is treated by Nishimura & Nemoto (2005). On a 30 m high mast, a blowing snow observation system including a snow particle counter, which can also sense the particle diameter, was installed; continuous recording was maintained for approximately a 2 month period. Through this atmospheric boundary layer, particle size distributions showed a particle size segregation with increasing distance from the saltation to the suspension regime. The differential equation for the horizontal wind velocity profile is affected by the presence of the particles and vice versa. Simulated and measured vertical profiles of the horizontal mass flux show close agreement and demonstrate the validity of the model. The aims of this article are very similar to those of the previous one, but methods of approach are surprisingly disjoint.

## 7. Granular transport in a surface

Ripples and sea ice are taken as examples describing surface transport of grains in a particle-laden and a dense form, respectively. Riverine and estuarine beds, and snow and sand formations subject to currents and wind from above are prone to forming ripples. Obviously, their formation necessitates that the granular material is set in motion by a flow above it. Thus, the ripple-formation process is closely associated with soil erosion or sediment transport by streams of air or water. Their theoretical description must, consequently, involve the reptation and saltations layers, subject to the shear velocity from above. It is seen that the description of sand ripples lies at the heart of the description of particle-laden transport that is generated by particle flow from the granular deposit. The article by Thomas & Zoueshtiagh (2005) is a contribution to the experimental study of ripples in the laboratory. The thesis is that the scaling law expressing the dependence of the ripple-pattern wavelength on the governing experimental parameters does not depend on the rotation of the system. This may facilitate experimental investigations—perhaps beyond the immediate goal of ripple formation.

Sea ice dynamics can be described by the flow equations of a two-dimensional granular material, the ‘grains’ being the ice floes, i.e. floating ice plates of approximately 0.5–4 m thickness and horizontal extent of tens to hundreds of meters, and the interstices, called leads, being the water masses that can exchange mass with the ocean. At high concentrations, the floes are in rubbing contact, and the corresponding continuous constitutive behaviour is visco-plastic; at low concentration the floes bounce into each other, and the corresponding continuum description is that of a two-dimensional dense granular gas. Close to the free ice edge, this second behaviour prevails, in confined regions, the visco-plastic behaviour is more adequate. The marginal ice zone (MIZ) is the transition region from the free ice edge to the region of compacted ice. Here, the constitutive behaviour is a combination of the two behaviours. Obviously, boundary layer structure is suggested. In the article by Feltham (2005) a steady solution of the MIZ is constructed that leads to the edge-parallel boundary-layer flow, which is also observed, and results as a natural consequence of the granular nature of sea ice.

## Acknowledgments

The first four sections of this manuscript are based on an unpublished text by the author, written in 2002 that was criticized by J. T. Jenkins. The author equally acknowledges reading and critical remarks by I. Chubarenko, S. P. Pudasaini and Y. Wang.

## Footnotes

One contribution of 11 to a Theme ‘Geophysical granular and particle-laden flows’.

- © 2005 The Royal Society