cellular automata model of density currents

(2007) 1–20www.elsevier.com/locate/geomorph

Geomorphology 88

Cellular automata model of density currents

T. Salles a,b,⁎, S. Lopez a, M.C. Cacas a, T. Mulder b

a Institut Français du Pétrole, Département Géologie-Géochimie-Géophysique, 4 av. de Bois Préau, 92852 Rueil Malmaison Cedex, Franceb Université Bordeaux 1, UMR CNRS 5805 EPOC, av. des Facultés, 33405 Talence Cedex, France

Received 20 June 2006; received in revised form 16 October 2006; accepted 16 October 2006Available online 14 December 2006

Abstract

Cellular automata (CA) represent an interesting approach to the modelling of dynamical systems evolving on the basis of localinteractions and internal transformations. The CA model, specially developed for simulating density currents, is described. Theobjective is to predict the formation and evolution of channels and the structure of deposits associated to the flow path. Forsimulation purposes, currents are represented as a dynamical system subdivided into elementary parts, whose state evolves as aconsequence of local interactions and internal transformations within a spatial discrete domain. The model is developed forunsteady, two-dimensional, depth-averaged, particle-laden turbulent underflows driven by gravity, acting on density gradientscreated by non-uniform and non-cohesive sediment. CA is defined as a tessellation of finite-state automata (cells). The attributes ofeach cell (substates) describe physical characteristics. The natural phenomenon is decomposed into a number of elementaryprocesses, with a particular composition that makes up the transition function of the CA. By applying this function to all the cellssimultaneously, the evolution of the phenomenon can be simulated in terms of modification of the substates. The transition functionincludes the effects of water incorporation at the suspension–ambient fluid interface, a transport equation for the particle volumeconcentration, and a toppling rule for the deposited sediments. Simple and flexible, the obtained model constitutes a first steptoward quantitative comprehension of the impact of external parameters on the turbidity current dynamics and on the organisationof the subsequent depositional sequences.© 2006 Published by Elsevier B.V.

Keywords: Cellular automata model; Density currents; Erosion/deposition processes; Deposits architecture

1. Introduction

Turbidity currents are sediment-laden flows drivenby their density difference with the ambient fluid. Theybelong to a larger class of flows known as gravity ordensity currents. Turbidity currents occur in all naturalor artificial basins (such as oceans, lakes, and reservoirs)and constitute an important process for sediment trans-

⁎ Corresponding author. Institut Français du Pétrole, DépartementGéologie-Géochimie-Géophysique, 4 av. de Bois Préau, 92852 RueilMalmaison Cedex, France.

E-mail address: [email protected] (T. Salles).

0169-555X/$ - see front matter © 2006 Published by Elsevier B.V.doi:10.1016/j.geomorph.2006.10.016

port from shallow to deep water (Parker et al., 1986).Turbidity currents in submarine canyons can reachvelocities larger than 8–14 m/s (Krause et al., 1970).They can generate substantial damage to seabed tele-graph cables and pipelines (Heezen and Ewing, 1952;Heezen et al., 1954). At the mouth of many submarinecanyons, turbidity currents formed extended sedimen-tary deposits called deep-sea turbidite systems. Turbid-ity currents are also considered to be at the origin of theformation of most deep sea canyons.

Turbidity currents are frequent, natural phenomena.For this reason, they have been studied by both hydrau-lic engineers and marine geologists for several decades.

mailto:[email protected]

http://dx.doi.org/10.1016/j.geomorph.2006.10.016

2 T. Salles et al. / Geomorphology 88 (2007) 1–20

Hydraulic engineers are particularly interested in under-sea cable or pipeline safety and reservoir sedimentation.Marine geologists are interested in turbidity currentdeposits. Some of these deposits are sandy and can beexcellent hydrocarbon reservoirs. They represent valu-able targets for oil industry exploration. Improvedexploration of these deposits requires a better under-standing of the flow and sedimentation pattern ofturbidity current, particularly the longitudinal evolutionof the deposits. Field investigations, laboratory experi-ments, and analytical and numerical simulations areparticularly valuable for this purpose. Analysis ofchannel morphology, sediment distribution, and strati-graphic architecture suggests that long duration, sub-critical turbidity currents are necessary to form andmaintain sinuous channels, particularly those extendingover hundreds of kilometers across continental marginsand abyssal plains (Babonneau et al., 2002). Initialchannel formation requires erosional turbidity currents,whereas long-term channel maintenance must beassociated with net by-pass averaged over several tur-bidity current events. Lateral channel migration seemsto occur as the result of currents that are either mainlyerosional or almost conservative. Conversely, currentsthat are mainly depositional lead to channel aggradationand infill (Pirmez, 2003). The objective of the presentpaper is to develop a numerical model of turbidity cur-rents that can predict the formation and evolution ofchannels and the structure of the deposits associated tothe flow path. The recent developments in computingscience allowed an extension of the application range ofclassical methods (commonly based on a discretisationof space and time). Moreover, innovative numericalmethods emerged from alternative computational para-digms such as cellular automata, neuronal nets, andgenetic algorithms. The models are based on an alter-native approach for modeling complex natural systemscalled cellular automata. The model is largely inspiredfrom the deterministic model SCIDDICA, speciallydeveloped for simulating debris flow by Avolio et al.(1999) and Di Gregorio et al. (1997, 1999). The model isdeveloped for unsteady, two-dimensional, depth-aver-aged, particle-laden turbid underflows driven by gravity,acting on density gradients created by non-uniform andnon-cohesive sediments. It is able to follow the evolu-tion and development of an erodible bed from sedimententrainment and deposition.

The paper is organized as follows: a brief review ofcellular automata modelling is first proposed. In Section3, the model is defined with particular attention to therules that constitute the cellular automata evolution.Then, Section 4 presents results on idealized tests.

Finally, in Section 5, we present the prospect of themodel and the future real testing case.

2. Cellular automata modelling

2.1. Introduction

A cellular automata is defined as a large tesselation ofidentical finite-state automata (cells). Each automatoncan be defined as a triplet:

b I ; S;W N ð1Þ

where I is the set of inputs, S is a set of states (both setsbeing finite), andW is the next-state function (i.e., transi-tion function) defined on input-state pairs. The set ofinputs is defined as ordered (or non-ordered) n-tuples ofthe states of a finite set of neighbouring cells. Thus, in aspatially two-dimensional cellular automata, this neigh-bourhood typically consists of four, six, or eight cells, e.g.,the adjacent cells in a square or hexagonal tessellation.

Typically, the automata are simple in the sense thatthey have very few states (Ulam and Von Neumann,1945; Von Neumann, 1966; Burks, 1970). For a smallneighbourhood, the next-state function therefore con-sists of a fairly small number of rules (Gardner, 1970).

2.2. Density flow modelling using CA

Macroscopic phenomena are usually describedwithin a continuum approach; and most past attemptsto apply CA to modelling these physical phenomenadeal with a microscopic approach, where the state vari-ables can take only a limited number of different discretevalues (Toffoli, 1984; Frisch et al., 1986; Margoluset al., 1986; Toffoli and Margolus, 1987; Fonstad,2006). In the microscopic approach, the system is con-sidered to be composed of a number of discrete particles,which move in the lattice space and interact according toappropriate laws. Then the laws that rule the system at amacroscopic level are obtained by averaging over alarge number of particles.

The approach adopted here is different. Consideringthe cells as portions of space, a complex state space isassigned to each cell. This state is the cartesian productof different subspaces. As physical quantities are usuallyexpressed with continuous variables referred to a pointin space, the cell size must be chosen small enough inorder to properly approximate the considered cell space.On the other hand, the size of the cell must be largeenough in order to allow for a macroscopic approach.The system behavior is expressed in terms of local laws.

3T. Salles et al. / Geomorphology 88 (2007) 1–20

The system complexity emerges from the interactions ofits cells by applying those simple local rules. Our modelis consistent with the attributes of the family ofSCIDDICA cellular automata models (Di Gregorioet al., 1997, 1999; Avolio et al., 2000): cellular automatais considered as a two-dimensional space (the cellularspace), divided into hexagonal cells of equal size. Eachcell is defined with an identical computational devicecalled the finite automaton. Input for each finite autom-aton is given by the states of the finite automata locatedin the neighbouring cells. The geometrical pattern of theneighborhood condition is constant both with time andover all cells.

At the initial time, the finite automata are in arbitrarystates. They describe the initial conditions of the system.The cellular automata evolves by simultaneously chang-ing all the states, at discrete times, by applying thetransition function to the cellular space (Segre andDeangeli, 1995; Di Gregorio et al., 1999).

In continuum models, a macroscopic approach is alsotaken; it is assumed that a representative volume ele-ment exists, which is small enough (with respect to thescale length of the phenomena of interest) to allow ameaningful limiting operation, ΔV→0, yet largeenough to allow the use of average quantities, thatvary smoothly in space. The PDEs are obtained byperforming mass and momentum balance on a volumeelement (ΔV ) for a small time (Δt), and then lettingboth ΔV and Δt tend to zero. Constitutive equations arealso used. If it were possible to integrate the Navier–Stokes equation, this would not have been necessary;but the disordered nature of turbidity flows and the lackof information on its internal structure makes it neces-sary to resort to empirical laws.

The major difference with respect to the discreteapproach lies on the limiting operation. This has theadvantage of eliminating any dependence on the spacescale and the shape of the cell. If analytical solutionscould be found, the advantage of the continuum ap-proach would be overwhelming. However, for systemsas complex as those considered here, this is not the case;so that in the continuum approach, a limit ΔV, Δt→0 isfollowed by a discretization with finite time and volumeincrements. These combined operations will not likelyprovide any particular advantage with respect to discretemodelling.

2.3. General frame for the model

This paragraph introduces the general frame of ourmodels. The cellular automata model we use in thispaper for density flows simulation can be represented as

a two-dimensional plane, divided into hexagonal cellsof uniform size. Each cell includes an identical finiteautomata ( fa). Each cell represents a part of spacewhose specification (state) describes the significantcharacteristics (substates) of the corresponding piece ofspace. Input for each cell is given by the states of theneighboring cells, where neighborhood conditions aredetermined by a pattern constant with time and over thecells. At the initial time, the cells are in arbitrary states andthe Cellular Automata evolves changing the state of allcells simultaneously at discrete times, according to thetransition function of the Automata (Avolio et al., 2000).

The cellular automata model can be defined asfollows:

bR;X ;Q;P; rN ð2Þ

where

(i) R={(x, y)∈Z2|0≤x≤ lx, 0≤y≤ ly}, identifies thehexagonal cellular space; Z is the set of integernumbers.

(ii) X={(0, 0), (1, 0), (0, 1), (0, −1), (−1, 0), (−1, 1),(1, −1)} is the geometrical pattern of the neigh-borhood of the cell, given by the central cell andits six adjacent cells. Indexes are attributed to theneighboring cells in order to specify the rules ofthe transition function: 0 identifies the central cell,1–6 identify the adjacent cells (Fig. 1).

(iii) Q=Qa×Qth×Qv×Qd×Qcj×Qcbj×Qo6, 1≤ j≤n is

the finite set of the states of the fa, given by thecartesian product of the sets of the consideredsubstates (Table 1). The value of the substate x inthe cell is expressed by qx∈Qx. n represents thenumber of lithologies simulated in the model.

(iv) P is the set of the global (physical and empirical)parameters (Table 2).

(v) σ : Q7→Q is the deterministic state transitionfunction for the cells in R.

Internal transformations and local interactions con-stitute the rules for the CA model and define the transi-tion function. The first ones will modify cell substatesvalues without taking into account the neighboringsubstates. The second ones simulate the impact of theneighbors on the considered cell substates. The transi-tion function is constituted by the following elementaryprocesses, listed in the order of application:

(i) water entrainment (T1), internal transformation;(ii) erosion and deposition rules (T2), internal

transformation;

Table 2Global parameters

Parameters Meaning

pc Apothem of the cell

Fig. 1. The neighborhood adopted in the models. The central cell isindividuated by the index “0”; indexes “1–6” identify the neighboringcells.


(iii) turbidity current outflows (I1), local interaction;(iv) update of current thickness and concentration (I2),

local interaction;(v) update of turbidity flow velocity (I3), local

interaction;(vi) toppling rule for the deposited sediments (I4),

local interaction.

At the beginning of each simulation, the states of allcells in R must be specified by defining the initialcellular automata configuration. Subscripts used foreach substate are referred in Table 1. Initial values aregiven to the considered substates as follows:

(i) qa is equal to the considered bathymetry;(ii) qth is zero everywhere — except for the source

area where the current thickness is specified;(iii) qv is zero everywhere— except for the source area

where the turbidity current thickness is specified;(iv) qcj is zero everywhere — except for the source

area where the volume concentration of eachsediment present in the current is specified;

(v) qcbj is the volume fraction of each sedimentpresent in the bed;

Table 1Substates

Substates Meaning

Qa Cell altitudeQth Thickness of the turbidity currentQv Velocity of the turbidity currentQcj jth current sediment volume concentrationQcbj jth bed sediment volume fractionQd Thickness of the soft sediment coverQo Density current outflow

(vi) qd is the thickness of the soft sediment cover,which can be eroded by the flow along the path;and

(vii) qo is zero everywhere.

These initial values are the boundary conditions ofthe model. The transition function σ is then applied, stepby step, to all the cells in R; and the cellular automataconfiguration changes: in this way, the evolution of thesimulation is obtained. The geometrical regularity of thecellular space allows for some computational simplifi-cations: i.e. the thickness can be used to describe thevolume of turbidity current in a given cell. Accordingly,the elements of Qa are expressed as length; as is thesame for the elements of Qth and Qd. The elements ofQv are expressed in velocity dimensions. The elementsof Qcj and Qcbj are, respectively, the volume concentra-tion and volume fraction of the jth sediment. Qo isexpressed in terms of length for reasons of computa-tional homogeneity.

The following conventions are adopted in this text:(i) indexes, specifying the neighboring cells, added be-tween brackets to qx∈Qx, when the substates of all theneighborhood are considered: i.e. the value of substate xof the cell with index a is given by qx(a); (ii) the sixvalues qo of the substate outflows need further specifica-tions: qo(a, b) is the value of the outflow from cell atoward cell b of the neighborhood (e.g. the value of theinflow into cell b from cell a).

3. Mathematical model

This model is developed for two-dimensional, depth-averaged, particle-laden turbid underflows driven bygravity, acting on density gradients created by non-

pt Time correspondence of a CA steppadh Unmovable amount of density currentpf Height threshold (related to friction angle)ptoppling Height threshold (related to toppling)f Darcy–Weisbach friction coefficienta Empirical coefficientDsj jth sediment diameterυsj jth sediment fall velocityρj jth sediment densityg Gravitational accelerationγ PorositycD Bed drag coefficientν Water kinematic viscosity

Fig. 2. Example of potential energy in the CA context: a currentcolumn (base A, height h, run-up r and mass m) is shown on the planez=0 (from D'Ambrosio et al., 2003).


uniform and non-cohesive sediments. Some rules in-spired from classical models allow us to track the evo-lution and development of an erodible bed fromsediment entrainment and deposition. The followingparagraphs introduce the mathematical assumptions anddescribe the transition function (σ).

3.1. The problem of modelling turbidity flows

Submarine fan and channel formation from densitycurrent activity are a real three-dimensional process.Hence, analytical solutions to the differential equations(i.e., the Navier–Stokes equations) governing densityflows are a major challenge, except for a few simple,unrealistic cases. Particular difficulties in simulationsarise from the complexity of the topography and fromthe difficulties in defining closure equations. Inaddition, the computational time required to numericallysolve the hydrodynamic equations is not justified as theavailability and accuracy of field data is usually toosparse to run too many accurate models.

Approximated numerical methods for the solution ofdifferential equations, accurate resolution of boundarylayers, and turbulence scales in the solution of governingequations are problematic because of the large compu-tational resources necessary to obtain well-approximatedsolutions (even though recent developments in computerscience extended their applicability by raising the com-puting power).

3.2. General considerations

When the topography where the sediment gravityflow moves is not completely flat, it might have a majorinfluence on deposition (Pickering et al., 1989; Knellerand McCaffrey, 1999), either by controlling the non-uniformity of the flow or by confining it, either partiallyor completely. The behavior of a turbidity flow aroundan obstacle varies according to the forward velocity ofthe current, the obstacle height, and the densitystratification within the current (Muck and Underwood,1990; Lane-Serff et al., 1995). This topographic impacthas major implications for the spatial distribution ofsediment in the deep sea, for the interactions ofunconfined currents with the intrabasinal highs andbasin margins, and for the interactions of channelledcurrents with channel flanks and sedimentary levees.

3.3. Run-up heights

The run-up height is the maximum height that can bereached by a flow for a given velocity (Kneller and

Buckee, 2000). Using the run-up height instead of thecurrent thickness in the distribution computation allowsto simulate the hydrodynamic pressure linked to thekinetic energy of the turbidity flow.

For obstacles that are much larger than the current,oceanographic data suggest that run-up distances innature may be several hundred meters long (Muck andUnderwood, 1990). According to Lane-Serff et al.(1995), a finite volume of fluid related to the head mayflow over the obstacle if H⁎ (the ratio of the obstacleheight to the current body thickness) is b4 or 5 or if theobstacle height is b1.5 times the height of the head ofthe current, according to (Muck and Underwood, 1990).Both studies reported the results of experiments withsaline currents. However, it seems likely that inrelatively poorly stratified currents, the maximum run-up height is probably dependent on the bulk Froudenumber of the current (Rottman et al., 1985):

hk ¼ 12U2

g V ð3Þ

where U is depth-averaged downstream velocity and g′is the reduced gravity.

In the cellular automata context, these points can helpto calculate the velocity. Accordingly, the run-up (r) canbe defined as the height reached by the flow: r=h+hk,where h is the thickness of the flow.

In the following description, energetic considerationsare referred to potential energy and its variations in thelocal context of the cellular automata (Di Gregorio et al.,1999; D'Ambrosio et al., 2003). For this purpose, let'sconsider a column of base A, mass m, and height h

Fig. 3. Example of transition: on the left, the situation at step t; on the right, at step t+1. A case of turbidity current outflow from one cell towardanother one (this latter characterised by a greater height) is shown (from D'Ambrosio et al., 2003).


on the plane z=0 (Fig. 2). Its potential energy is givenby

Up ¼Z h

0qcg

VAzdz ð4Þ

where(i) ρc is the density of the current, which is defined in

the case of n sediments as follows:

qc ¼ qa 1−Xnj¼1

qcj

!þXnj¼1

qjqcj ð5Þ

(ii) g′ is the reduced gravity with ρj jth sedimentdensity and ρa ambient fluiddensity:

g V¼ gXnj¼1

qcjqj−qaqa

ð6Þ

and

Up ¼ qcgVA

2h2 ð7Þ

Up ¼ gA qa 1−Xnj¼1

qcj

!þXnj¼1

qjqcj

!

�Xnj¼1

qcjqj−qaqa

� �h2

2

ð8Þ

The run-up effect can be inserted by “virtually”incrementing the height of the column from h to r. Asthe mass must be conserved, a new density ρc′ is derived:

qc′ ¼hrqcbqc ð9Þ

and the following “potential energy”-like formulaexpresses the energy increase:

U Vp ¼

Z r

0qc′ g ′Azdz ð10Þ

U Vp ¼ gA qa 1−

Xnj¼1

qcj

!þXnj¼1

qjqcj

!

�Xnj¼1

qcjqj−qaqa

� �hr2

ð11Þ

Up′ represents the energy of the flow and h representsqth. In Fig. 3, an example of outflow toward a cellcharacterised by greater height (given by altitude plusturbidity current thickness) is shown. The describeddistribution of current is allowed by the assumed ener-getic context.

3.4. Numerical stability

The implementation of the cellular automata withcontinuous state variables makes the explicit time ad-vancement scheme only conditionally stable. We ob-served, for example, the appearance of an oscillatoryinstability pattern. This can occur at any given time step.When occurring, an exceeding amount of sediment istransferred from a cell to its neighbor(s). If the materialis then not drained away rapidly enough, then at thefollowing time step the slope between these cells maygive a reverse result, so that the first outflow returns intothe original cell. If the mechanism is allowed to repeatitself, the back and forth transfer of sediment gives rise


to persistent oscillations, in which odd and even cells arealternately depleted and filled. The best solution to thisproblem is a refinement of the time step that scales downthe elementary outflows.

The fact that we do not include an energy dissipationconstraint explain this instability. This energy dissipa-tion is linked to the transfer of a portion of the flowtoward the adjacent cells in one hand and to the tur-bulent support of the particles in suspension in the otherhand. As a parameter characterising such dissipation isdifficult to define the actual criterion for stability isformed using the following description: consider all thecells ( j) which contain at time t a flow thickness (i.e.,qthN0). We determine the run-up with the previousequations for each cell (r( j)). Then, a maximumrelaxation time is defined with the following equation:

Prmaxð jÞ ¼ pcffiffiffiffiffiffiffiffiffiffiffiffiffiffi2rð jÞg V

p ð12Þ

The value for the time step of the considered iterationpt is equal to the minimum value of all the cells ( j):

pt ¼ Dt ¼ minjðPrmaxðjÞÞ ð13Þ

3.5. The transition function

3.5.1. I1: turbidity current outflowsThe local interaction

I1 : Q7a � Q7

th � Qv � QcjYQ6o ð14Þ

determines the outflows from the central cell toward itsadjacent cells.

It is based on an opportune minimisation algorithmderived from the “minimisation of the differences” pro-posed by Di Gregorio et al. (1999). In order to accountfor the run-up effects, the height of the column in thecentral cell is “virtually” incremented from h=qth(0) tor=qth(0)+hk(0). Obtained outflows have to be normal-ised by a factor �nf ¼ h

r.In the context of this minimisation algorithm,

qð0Þ ¼ qað0Þ þ padh; p ¼ r−padh

qðiÞ ¼ qaðiÞ þ qthðiÞ; 1ViV6; ð15Þ

f ðiÞ; 1ViV6

are the not-normalised outflows. Set A comprises thecells that may receive flows; at the beginning of thealgorithm, A includes all the neighbouring cells.

According to D'Ambrosio et al. (2003), the mini-misation algorithm is composed of the following steps:

(i) angles βi, specified by the differences in heightbetween the central cell (q(0)+p=qa(0)+ r) and the ad-jacent cell i (q(i)=qa(i)+qth (i), 1≤ i≤6) are computed;the cell i with βibpf is eliminated from A.

(ii) the following average is computed, consideringthe set A of the not-eliminated cells (where Card(A) isthe cardinality of the set A):

Average ¼ pþXiaA

qðiÞ !

=CardðAÞ ð16Þ

(iii) cell i with q(i)≥Average is eliminated from A; ifany cell is eliminated, we go back to step 2.

(iv) the not-normalised outflows f(i), 0≤ i≤6 towardthe adjacent cells are computed as follows:

f ðiÞ ¼ Average−qðiÞ; ðiaAÞ

f ðiÞ ¼ 0; ðigAÞ ð17Þ

(v) the six values qo(0, i), 1≤ i≤6 of the substate“outflows” from the central cells are obtained, consid-ering the normalisation factor υnf and the relaxation ratepr:

qoð0; iÞ ¼ υnfprf ðiÞ; ð1ViV6Þ ð18ÞHere, the value of the relaxation rate is not constant

for all iterations and over all cells but depends on thevalue of pt:

pr ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2rð0Þg V

pc

spt ð19Þ

3.5.2. I2: update of current thickness and concentrationThe local interaction

I2 : ðQth � Qcj � Q6oÞ7YQth � Qcj ð20Þ

updates the values qth and qcj of the substate turbiditycurrent thickness and concentration.

The new value of turbidity flow thickness (nqth ) isobtained by considering flow thickness variations becauseof outflows and inflows from/into the central cell:

nqth ¼ qthð0Þ þX6i¼1

ðqoði; 0Þ−qoð0; iÞÞ ð21Þ

The new value of the volume concentration of thejth sediment (nqcj) is obtained by considering volume


concentration variations from outflows and inflowsfrom/into the central cell:

nqcj ¼qthð0Þ−

P6i¼1

qoð0; iÞ� �

qcjð0Þ þP6i¼1

ðqoði; 0ÞqcjðiÞÞnqth

ð22Þ

3.5.3. I3: update of turbidity flow velocityThe local interaction

I3 : Q7a � Q7

th � Q7o � QciYQv ð23Þ

updates the values qv of the substate turbidity currentvelocity.

The new value of the turbidity current velocity (nqv)is obtained by considering the outflows from the centralcell. Moreover, the velocity determination is inspiredfrom Middleton's formula (1966). The velocity of theturbidity current is given by a Chezy-type equation:

U ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8g V/

f ð1þ aÞ hss

ð24Þ

where U represents the mean flow velocity, ϕ sedimentvolume concentration, s bottom slope, f the Darcy–Weisbach friction coefficient (∼0.04), and a an empiricalcoefficient (∼0.43 as Ri ∼1). This formula, analogous tothat found in Daly (1936) for river flows, is obtainedsimply bywriting the balance between the apparentweight,g′ϕhs, and the steady friction resistance f

8 ð1þ aÞU 2.In order to account for the outflows, we determine the

velocity Uk, 1≤k≤6 as follows:

Uk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8g VPni¼1

qcið0Þf ð1þ aÞ qoð0; kÞsð0; kÞ

vuuut ð25Þ

where the slope s(0, k) is specified by the difference inheight between the central cell (qa(0)+qth (0)) and theadjacent cell (qa(k)+qth(k)).

The mean velocity U (equal to the substate qv ) of thecurrent is then computed thanks to the obtained Uk.

3.5.4. I4: toppling rule for the deposited sedimentsThe local interaction

I4 : Q7a � Q2

cbj � Q2dYQ2

a � Qcbj � Q2d ð26Þ

determines the new values nqa, nqcj, and nqd of thealtitude, the volume fraction of each sediment present inthe bed, and the thickness of soft sediment cover, whichcan be eroded by the flow.

As sediment particles do not have an infinite valuefor cohesion, we can realistically consider the followingtoppling rule: when a cell altitude is higher than atoppling threshold (ptoppling) with respect to its adjacentcells and if the depth of soft, erodible sediment above ispositive, toppling occurs. The toppling consists inmoving by half the quantity of the deposited particles(Chopard et al., 2000).

The toppling rule is composed of the following steps:(i) If the value of the thickness of the soft sediment

cover is positive, toppling can occur. In this case,computing goes directly to step 2.

(ii) The differences in height δi between the centralcell (qa(0), 1≤ i≤6) are computed; the cell itpg withδtpg=Maxi=1

6 δi is chosen and the toppling occurs ifδtpgNptoppling;

(iii) The new values nqa, nqd and nqcbj for the centralcell (0) and the toppling cell (itpg) are obtained byconsidering the value of qd(0):

nqcbjð0Þ ¼ qcbjð0Þ; nqað0Þ ¼ qað0Þ− qdð0Þ2

; nqdð0Þ

¼ qdð0Þ− qdð0Þ2

nqcbjðitpgÞ ¼ qdðitpgÞqcbjðitpgÞ þ qcbjð0Þ qdð0Þ2

� �=nqdðitpgÞ;

nqaðitpgÞ ¼ qaðitpgÞ þ qdð0Þ2

; nqdðitpgÞ

¼ qdðitpgÞ þ qdð0Þ2

ð27Þ

3.5.5. T1: water entrainmentThe internal transformation:

T1 : Qa � Qth � Qcj � QvYQcj � Qth ð28Þdetermines the new values nqcj and nqth of the volumeconcentration of sediment and turbidity current thick-ness from ambient fluid incorporation.

Considering the mean velocity of the flow U as acharacteristic of the phenomenon, the rate of seawaterincorporation, Ew, can be written as

Ew ¼ UE⁎w ð29Þ

where Ew⁎ is a dimensionless incorporation rate, a

function of the Richardson number Ri. This dimension-less number is defined by

Ri ¼ g VhU 2

ð30Þ

Fig. 4. Tailing deposit bathymetry in November 1976 during the meandering channel phase. Contours in meters. The dashed lines are acousticsounding lines (from Hay, 1987a,b).


It is the ratio of gravity to inertia. It characterizes thestability of the flow interface.

The law determining the dimensionless incorporationrate of ambient fluid by the flow Ew

⁎ is an importantproperty. It distinguishes gravity flows from density andriver flows. The water incorporation increases thevolume of the flow and reduces the concentration insediment particles in the flow. The following expression(Parker et al., 1987) is used in the present model:

E⁎w ¼ 0:075ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 718Ri2:4

p ð31Þ

The new value of turbidity flow thickness (nqth) isobtained by simply adding the determined rate of sea-water incorporation to the considered cell Ew(0):

nqth ¼ qthð0Þ þ Ewð0Þpt ð32Þ

The new values of sediment volume concentration(nqcj) are obtained by considering the value of nqth:

nqcj ¼ qcjð0Þqthð0Þnqth

ð33Þ

3.5.6. T2: erosion and deposition rulesThe internal transformation

T2 : Qa � Qth � Qcj � Qcbj � QvY

Qa � Qd � Qcj � Qcbj

ð34Þ

determines the erosion of the soft sediment cover andparticles deposition and its effects.

A flow is able to erode, along its lower surface,sediments in its path. Moreover, sediments in suspen-sion fall out of the flow by gravitational effect within theviscous sublayer.

The deposition rate Dj of the jth sediment is specifiedin terms of the settling velocity υsj and the near-bedconcentration of suspended sediment cnbj. Neitherhindered settling nor the formation of flocs is consideredin the model. The near-bed concentration cnbj is relatedto the layer-averaged concentration value cj using theexpression developed by Garcia and Parker (1993) forpoorly sorted sediments:

cnbj ¼ cj 0:40Dsj

Dsg

� �1:64

þ1:64

!ð35Þ

where Dsg is the geometric mean size of the suspendedsediment mixture, and Dsj is the diameter of the con-sidered sediment. As a consequence, the deposition ratecan be written as

Dj ¼ υsjcnbj ð36Þ

where the fall velocity υsj is calculated by using therelation of Dietrich (1982).

Indeed, the common fall velocity is rearranged to geta dimensionless sphere settling velocity

υsj ¼ υsj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðqgðqj−qÞ=m2Þ3

qð37Þ

where υsj is the sphere settling velocity of the jthsediment. The relation for the erosion rate of the bedsediment Ej is taken to be that of Garcia and Parker(1991) but modified for fine or light sediments by usingthe results of Garcia and Parker (1993). The modifiedrelation becomes

Ej ¼ Esjυsj ¼1:3x10−7Z5

mj

1þ 4:3x10−7Z5mj

ð38Þ

Table 3First conceptual case: channel, sediment, bed, and currentcharacteristics

Characteristics Values

Studied area 60×160 cellsApothem pc 1 mChannel depth 3.5 mChannel width 6 mChannel slope 5°Sinuous channel amplitude 20 mSinuous channel wave length 100 mVery fine sediments Dvfs=85 μm;

ρvfs=2450 kg/m3

Fine sediments Dcs=740 μm;ρcs=2400 kg/m3

Coarse sediments Dcs=1.1 mm;ρcs=2650 kg/m3

Very fine sediments bed volume concentration 0.4Fine sediments bed volume concentration 0.3Coarse sediments bed volume concentration 0.3


where Zmj ¼ jffiffiffiffiffiffiffiU 2⁎

qυsjf ðRpjÞ and κ is a straining param-

eter defined as

j ¼ 1−0:288r/ ð39Þ

where σϕ is the standard deviation of the grain sizedistribution based on the ϕ scale, ϕ= log2Ds. Thefunction f is dependent on the particle Reynolds

Rpj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigðqj−qÞDsj

q

qDsj

m , i.e.,

f ðRpjÞ ¼R0:6pj ; Rpjz3:5

0:586R1:23p j ; 1bRp jb3:5

8<: ð40Þ

Another important consideration is that turbiditycurrents are driven by gravity acting on density

Thickness of erodible soft sediment 1.5 mptoppling 30°Bed drag coefficient 0.003Porosity 44 0.3Flow thickness 1.5 mFlow velocity 0.2 m/s

gradients created by the suspended sediments, whichcan be gained through entrainment and lost throughdeposition. Therefore, a turbidity current has theability to modify the bed over which it flows, whichmay substantially affect its hydrodynamics (Garciaand Parker, 1993). In order to simulate this interac-tion, the equations governing the hydrodynamics havebeen coupled to the evolution of the bed through abed-sediment conservation equation (Bradford andKatopodes, 1999). The bed continuity equation isneeded to keep track of the amount of loose sedimenton the bed.

The bed-sediment conservation equation has thefollowing form:

ð1−gÞAzAt

¼Xnj¼1

ðDj−pjEjÞ ð41Þ

where z is the bed elevation, γ is the bed porosity(assumed constant in the model), and pj is thevolume fraction of sediment j present in the bed; pj

Fig. 5. The 200-KHz acoustic image of the channelized dischargeplume along line 78. See previous figure for line location (from Hay,1987a,b).

may also change with time. In order to compute itsvariation, each grain size may be considered indi-vidually, i.e.,

ð1−gÞ pjAzAt

þ zApjAt

� �¼ Dj−pjEj ð42Þ

Finally, we obtain

ð1−gÞzApjAt

¼ fsj−pj fs ð43Þ

where

fs ¼Xnj¼1

fsj ¼Xnj¼1

Dj−pjEj ð44Þ

In the model, the previous equations help to deducethe new values of altitude, erosion of the soft sedimentcover and variation in particle concentrations in the bedand in the flow with the following considerations:

cnbj ¼ 0:4Dsj

Dsg

� �1:64

þ1:64

!qcjð0Þ ð45Þ

U⁎ ¼ cDqυð0Þ ð46Þwhere cD is the bed drag coefficient.

Fig. 7. Color scale of the sections at different time steps for the firstconceptual case.


The bed-sediment conservation equation gives:

nqa ¼ qað0Þ þ pt

Xnj¼1

ðDj−qcbjEjÞ

ð1−gÞ ð47Þ

nqd ¼ qdð0Þ þ pt

Xnj¼1

ðDj−qcbjEjÞ

ð1−gÞ ð48Þ

nqcj ¼ qcjð0Þ þ ptðDj−qcbjEjÞð1−gÞqthð0Þ ð49Þ

nqcbj ¼ qcbjð0Þ

þ pt

Dj−qcbjEj−qcbjXnj¼1

ðDj−qcbjEjÞ

ð1−gÞqdð0Þ ð50Þ

4. Numerical results

4.1. Turbidity currents and submarine channel forma-tion in Rupert Inlet

Hay published two articles (1987a, 1987b) aboutsubmarine channel formation in Rupert Inlet. The chan-

Fig. 6. Erosion deposition maps for different time steps (2000, 3000, 500respectively (color scale : −0.5 m→0.5 m).

nel formation resulted from mine tailing discharge intoRupert Inlet. The current flows partially within a sub-marine channel, which hooks to the left and meanders.The Rupert Inlet study provides a case history of sub-marine channel development, including observationssurge-type and continuous flow and identification ofturbidites in the deposits (Hay, 1987a,b).

4.1.1. Channel characteristicsThe bathymetry of the mine tailing deposit in Rupert

Inlet in November 1976 is shown in Fig. 4. The sub-marine channel in this figure is in its so-called mean-dering channel phase. From the start of the channel nearthe outfall, the upper reach extends across the inlet,hooking slightly to the left, and then enters the meanderreach, which is located in the central trough of the inlet.

0, 9000). Red and blue zones are the deposited and eroded zones,

Fig. 8. Cross sections at different time steps for the straight part of the channel.


The straight lower reach is below the meander reach(Hay, 1987b).

The axial slope of the upper reach is between 9.5 and12° at the top, decreasing to 1.9° at the bottom; while theslopes of the meander and lower reaches are 0.91° and0.47°, respectively.

4.1.2. Channel pattern persistenceDetailed surveys of the meandering channel were

conducted between 1976 and 1977. Essentially the samechannel pattern as that in Fig. 4 persisted throughout thisinterval. Additional surveys with less accurate naviga-tion show that the channel was present for a period ofabout 3 yr. Whether the meanders persisted that long isnot known. However, we do know from turbiditiesfound in cores that surge-type turbidity currentsoccurred at 1- to 5-d intervals. The submarine channel

Fig. 9. Cross sections at different tim

in Fig. 4 is therefore taken to be in a slowly changing, orquasi-steady, state of equilibrium with the continuousflow and surge-type turbidity currents responsible for itsformation (Hay, 1987b).

4.1.3. Continuous flow observationsAn acoustic image of the discharge plume in the

upper reach is shown in Fig. 5. The channel appears as adownstream view that is on the right with the higherwest levee. The backscattered signal from the dischargeplume is concentrated on the right-hand side of thechannel and can be seen spilling out of the channelbeyond the crest of the higher west levee. This profile istypical of those acquired in this portion of the upperreach (Hay, 1987b). All exhibit the same tendency onthe part of the plume to hug the right bank and spill overthe right levee, independent of the phase of the tide.

e steps for the transition zone.

Table 4Second conceptual case: channel, sediment, bed, and currentcharacteristics

Fig. 10. Cross sections at different time steps for the sinuous part of the channel.


4.2. First conceptual case: straight to sinuous channeltransition

4.2.1. CharacteristicsThe considered bathymetry, in this case, is quite

similar to the geometry of the upper reach in Hay's(1987b) description of the Rupert Inlet channelformation. The geometry is an 80 m straight channelthat evolves in a sinuous channel downstream. Thecharacteristics of the channel are given in Table 3.

The model allows us to take into account a largenumber of different grain size classes. We have chosenthree classes which are defined in Table 3. The volumefraction of each grain size present in the bed at the initialstep is also defined in the Table 3.

The ambient fluid is supposed to have a density equalto 1000 kg/m3. The source area is composed of two

Fig. 11. Initial bathymetry of the second conceptual case.

cells, and the fluid injection is located in the middle ofthe channel. The injection is made at each step duringthe first 10000 iterations. The characteristics of theinjected flow are given in Table 3.

4.2.2. Bed evolutionThe bed evolution is a function of the internal trans-

formation of the flow T2 and the local interaction of theflow I4. This evolution is computed using the erosionand deposition equations and the toppling of the depos-ited sediments, respectively.

Fig. 6 shows the bed evolution. The first phase ischaracterised by a strong sediment entrainment from the

Characteristics Values

Studied area 60×160 cellsApothem pc 1 mChannel depth 2.75 mChannel width 6 mChannel slope 0.7°Sinuous channel amplitude 3 mSinuous channel wave length 80 mClay Dclay=5 μm; ρclay=2600 kg/m3

Silt Dsilt=60 μm; ρsilt =2600 kg/m3

Fine sand Dsand=135 μm; ρsand=2600 kg/m3

Clay bed volume concentration 0.8Silt bed volume concentration 0.15Sand bed volume concentration 0.05Thickness of erodible soft

sediment0.5 m

ptoppling 30°Bed drag coefficient 0.003Porosity 45 0.3Flow thickness 2.5 mFlow velocity 0.2 m/s


channel bed. This entrainment generates self-accelerat-ing turbidity current. Each sediment class has its ownmeans of transport.

The coarse particles move thanks to a traction orsaltation transport. These particles are deposited rapidly inthe surroundings of the channel axis. In Fig. 6, we can seein the first iteration steps a red zone inside the channel.This deposited zone corresponds to the sedimentation ofthe coarsest sediments just eroded upstream.

The finest particles are transported as a suspendedload and remain in the body of the current over a longdistance before settling. They are deposited by over-spilling outside the channel. During the first steps, theseparticles formed the levees on the sides of the channel.In the straight part of the channel, the levees aresymmetrical. Because of entrainment of surroundingfluid, the current thickness increases downstream.

The transition between the straight and sinuous part isa critical zone, which allows us to study the flowbehavior. In this zone, the flow is able to spill out because

Fig. 12. Erosion deposition maps for different time steps (5000, 10000, 150respectively (color scale: −0.3 m→0.3 m).

of the small difference between current and ambient fluiddensity (Fig. 6). Dissymmetrical levees are formed witha higher and wider levee on the external bank in thistransition zone (cf. bottom pictures in Fig. 6).

In the sinuous bottom part, the same observations canbe seen. The flow tends to spill over the external leveesof the channel, and the deposits are thicker on theselevees.

4.2.3. Cross section studyIn order to show the channel evolution under the

action of the turbidity current, three cross sections weredrawn. The color scale of the cross sections for thecomputed iterations is given in Fig. 7.

Straight part cross sections: Fig. 8 shows that theflow starts to erode the channel bed right from the start ofthe injection. The incision is continuous and the thalwegis hemmed in more and more by the levees. When theflow intensity decreases, the erosion is localised in thedeepest part of the channel. For iterations 9000 and

00, 20000). Red and blue zones are the deposited and eroded zones,

Fig. 13. Color scale of the sections at different time steps for the secondconceptual case.


10,000, we can see a thin deposit inside the channel fromthe setting of very fine sediments.

The overspill of the flow creates symmetrical levees.Both thickness and width of these levees increase duringthe simulation. Moreover, these levees have a tendencyto migrate away from the channel axis. In fact, the flowgenerates instabilities on channel banks. These instabil-ities cause the toppling of the steepest slope bankinducing the levees' migration.

Transition zone cross sections: Fig. 9 shows theresults for the cross sections in the transition zonebetween the straight and sinuous channel.

The flow preferentially erodes the internal flank ofthe channel. In fact, the flow tends to smooth thelongitudinal profile in order to reach an equilibriumstate. The talweg migrates laterally on the internalbank.

The external levee begins to grow as soon as the flowstarts to overspill. This external levee is higher andwider than the internal levee and grows continuouslyduring the simulation.

Sinuous zone cross sections: In the sinuous part ofthe channel, the descriptions are identical as in thetransition zone: Fig. 10 shows an erosion of the internalbank of the channel. This erosion is lower than in thetransition zone but remains uniform. The flow still tendsto reach an equilibrium state.

Fig. 14. Cross sections at different ti

The external levee is the first to appear. Similar to thetransition zone, this levee is higher and wider than theinternal one and migrates during the simulation.

The model simulates correctly the natural deposits,asymmetry resulting from the passage of a turbiditycurrent in a sinuous channel.

4.3. Second conceptual case: sinuous channel to non-channelled transition

4.3.1. CharacteristicsIn this case, the considered bathymetry (Fig. 11) is

quite similar to the geometry of the lower reach in Hay's(1987b) description of the Rupert Inlet channel formation.

The geometry is a 130 m sinuous channel thatevolves into a non-channelled zone downstream. Thecharacteristics of the channel are given in Table 4. Wehave chosen three classes which are defined in Table 4.The volume fraction of each grain size present in the bedand in the current are also defined in this table.

The source area is composed of two cells and thefluid injection is located in the center of the channel. Theinjection is made at each step during the first 10 000iterations. The characteristics of the injected flow aregiven in Table 4.

4.3.2. Bed evolutionThe bed evolution is a function of the internal

transformations of the flow T2 and the local interactionsof the flow I4. This evolution is computed using theerosion and deposition equations and the toppling of thedeposited sediments, respectively.

Fig. 12 shows the bed evolution. We can distinguishtwo stages for the inside channel erosion deposition

me steps for the sinuous zone.

Fig. 15. Long sections at different time steps for the non-channelized zone.


variations. The first stage is defined by a progressiveerosion of the bed until the switching off of the flowsource (step 10,000). The second stage is a depositionstage inside the channel as soon as the flow stops.

Inside the channel, flow and bed particles aresuccessively deposited and eroded. The deposits inthis area are constituted by the coarser sediments.

When the current is no longer channelled, it spreadsrapidly and particles start to settle. Deposit geometry inthis zone is lobate as it is for distal lobes located in deep-sea environments.

4.3.3. Cross section studyIn order to show the channel evolution under the

action of the turbidity current, three cross sections aredrawn. The color scale of the cross sections for differentiterations is given in Fig. 13.

Fig. 16. Cross sections at different time s

Sinuous zone cross sections: As for the firstconceptual case, the current inside the sinuous zone isable to spill over. The geometry of the deposits (Fig. 14)is similar to that of the previous case. The flowpreferentially spills over the external bank and erodesthe internal one. However, the dissymmetry between thelevees is not as clear as in the first simulation because ofthe initial topography (slope, channel wave length, andamplitude) and flow parameters (concentrations, thick-ness). The comparison between the two simulationsshows the importance of topography, bed composition,and flow characteristics on the obtained sedimentarygeometries.

Non-channelled zone sections: In the distal part, theflow is not channelled and a lobate geometry forms.

Down-channel sections (Fig. 15) show a rapiddecrease in lobe thickness. The first stage (erosion)

teps for the non-channelized zone.


can be related to the maximum growth of this distal lobe.In the second stage, lobe thickness still increases but thedeposits are thinner and are constituted mainly of clays.

Fig. 16 shows the results for the cross sections in thenon-channelled area. In this area, the flow is not able toerode the bed. In addition, the current is spreadinglaterally very rapidly and deposits the largest particlesnear the channel mouth. The lobe geometry, obtained inthis second simulation, is similar to natural case studieswhere channel-levee systems open onto an abyssalplain.

Deposited sediment distribution: Fig. 17 shows thedistribution of the three different grain size classes.

In the sinuous area, only a small quantity of particlessettles inside the channel. On the levees, the figureshows a lateral segregation of the deposited sediments.The levee deposits close to the channel axis are made upof the three types of particles. When the flow movesaway from the channel axis, sand settles first, silt settlessecond, and finally, the clay settles.

For the non-channelled area, the same lateral particlesorting appears. It is associated with a longitudinalsorting. In fact, the sand starts to settle near the channelopening. Its proportion decreases rapidly downstream.

Fig. 17. Deposited sediment distribution for the iteration step 10 000. From lefproportions.

Then, the silt fraction becomes predominant. In the mostdistal part, the clay is the only grain size present and ableto settle in the flow.

5. Conclusions and perspectives

The purpose of this work was to develop an innovativenumerical approach for density flow motion, erosion,transport, and deposition. In the past, most attempts atmodeling such flows in subaqueous environments withmore classical approaches were one-dimensional models.In addition, sedimentationwas often neglected, thus severe-ly limiting the application of suchmodels to natural studies.

In order to predict the formation and evolution ofchannels and the nature and geometry of the depositsassociated to the flow path downslope, we developed amodel to simulate two-dimensional, vertically integrat-ed turbidity current. In addition, the model is dynam-ically coupled to the environment through erosion anddeposition equations.

In a cellular automata model, conservation law isrewritten in a very different context of space-time dis-cretisation. Values of model parameters cannot alwaysbe determined directly, e.g., by physical measures. They

t to right, pictures show (respectively) clay, silt, and fine sand deposited


are commonly selected iteratively by comparing theresults of simulations with the global behavior of thereal phenomena. These values are then considered onlyas the optimal combination of such parameters.

The model simplifies natural flows and, therefore,computational time is saved.

A preliminary validation, based on conceptual cases,showed that the models are able to simulate most of thefeatures of the natural processes. From these preliminaryresults, improving the models and calibrating them withfield and experimental data would be interesting. Acomplete calibration should include case studies of awell-known turbidity current event, in which the grainsize distribution at various locations and the extent ofthe current are known.

Fig. 18. Capbreton Canyon erosion deposition m

Primary simulations have been made in order toreproduce the turbulent surge that occurred during theDecember 1999 storm event (called Martin event) inCapbreton Canyon (Bay of Biscay, North AtlanticOcean) (Mulder et al., 2004). Fig. 18 shows the ero-sion deposition maps obtained with the model for twotime steps. The deposit composition still has to becorrelated to the sediment present in cores in order tospecify the parameter values. Like classical numericalmodels, the critical point is the initial flow compo-sition and duration. Hence, many simulations have tobe done in order to fit as well as possible to the fielddata.

At the same time, we are trying to simulate a series ofsmall-scale laboratory experiments on the formation of

aps (scale −1 m→1 m) at 3 h and 11 h.


subaqueous channels and lobes (experiments realized byMétivier et al. (2005). These experiments show thatsteady flow of a dense current on a bed of light particlescan induce both the spontaneous formation of channelsand spontaneous meandering (Métivier et al., 2005).

The model presented in this paper is a simplifiedrepresentation of flow and sedimentary processes builtfrom a original cellular automata approach. In most ofthe CA models, the evolution is based on a more phe-nomenological nature; indeed their major value istypically expressed in terms of how the local interac-tions through relatively simple rules may produce(sometimes complex) patterns. As the local interactionsare expressed in a more physical meaning this model isdifferent from most CA approaches in the geosciences.

It is a new tool for modelling deep-sea densitycurrents that helps to overcome some of the short-comings of the classical models and to fill gaps in ourknowledge of turbidity current behavior.

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