On the ignition of geostrophically rotating turbidity currents

On the ignition of geostrophically rotating turbidity currents

P. W. EMMSDepartment of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK(E-mail: [email protected])

ABSTRACT

Two models of a geostrophically rotating turbidity current are examined to

compare predictions for ignition with the catastrophic state. Both models

describe the current as a tube of sediment-laden water traversing along and

down a uniform slope. The ®rst (four-equation) model neglects the energy

required to lift the sediment from the seabed into suspension. The second (®ve-

equation) model recti®es this shortcoming by introducing a turbulent kinetic

energy equation and coupling the bottom stress to turbulence in the plume.

These models can be used to predict the ignition, path and sediment deposition

of a geostrophically rotating turbidity current. The criteria for ignition in the

four-equation model can be described by a surface in three-dimensional phase

space (for a non-entraining current). This surface lies near the geostrophic

equilibrium state. For a turbidity current occurring in the Greenland Sea,

velocities above 0á053 m s±1 or volumetric concentrations of sediment above

2á7 ´ 10±5 lead to ignition. In general, if the tube is started pointing downslope,

then ignition is more likely than if it is initially directed alongslope. However,

there exists a set of initial conditions in which the current ignites if started

along or downslope, but deposits if started at an intermediate angle. The ®ve-

equation model requires a larger initial velocity (greater than 1á6 m s±1) to

ignite than does the four-equation model. Ignition is determined qualitatively

by the geostrophic state and the initial normal Froude number. Solutions show

a tendency to travel further alongslope during ignition, re¯ecting the restriction

that the energy budget places on the sediment load. A qualitative difference to

phase space in the ®ve-equation model is the existence of a region in which the

tube has insuf®cient energy to support the sediment. Turbulence dies rapidly

in this region, and so the sediment is deposited almost immediately.

Keywords Catastrophe, deposition, extinction, geostrophy, turbiditycurrent.

INTRODUCTION

Turbidity currents sometimes occur on such alarge spatial scale that the Coriolis force maybecome important. This impacts on both the paththat the current takes down the continental slope(Nof, 1996) and the consequent distribution ofsediment. Although present process models ofturbidity currents (Pantin, 1979; Parker, 1982;Dade & Huppert, 1995) are useful for understand-ing the dynamics of sediment driven ¯ow, theyonly consider downslope currents. A more recentnumerical model (Zeng & Lowe, 1997) has soughtbetter quantitative agreement between model

predictions and observation, but this too concen-trates on non-geostrophically rotating currents.The present paper focuses on how the earth'srotation alters the progress of turbidity currentsdown a slope (the vorticity about the current'smean path is not studied).

Experiments have revealed that entrainment ofsediment into a turbidity current increases withthe mean speed of the ¯ow (Garcia & Parker,1993). Consequently, as the ¯uid travels faster, sothe density contrast increases, which can causethe current to accelerate downslope like anavalanche. Pantin (1979) and Parker (1982)described the point at which the plume starts to

Sedimentology (1999) 46, 1049±1063

Ó 1999 International Association of Sedimentologists 1049

accelerate as `ignition' and determined the crite-ria that led to ignition for non-rotating plumes.Their criteria for ignition were based upon atemporally unstable steady state to their modelcalled the ignitive state. Parker (1982) found that,after ignition, the current tends towards anotherstable steady state rather than accelerating indef-initely down the slope. This is called the cata-strophic state and represents an along-streamequilibrium between gravity and bottom friction.

It is the turbulent motion and the presence ofavailable sediment that allows a turbidity currentto form. Einstein (1968) conducted experimentswith a turbid current on a gravel bed, so that therecould be no entrainment, and found that clari®ca-tion of the turbid ¯ow always occurred, i.e. thecurrent always deposited. Parker (1982) suggestedthat turbulence counters the ¯ux of grains towardsthe bed by producing a net upward ¯ux of grainsentrained from the bed. These ideas allow a simpleparameterization of erosion to be formulated, andthis is incorporated in the present modelling.

Recently, Emms (1998) studied two rotatingturbidity current models that bring together therotational and sediment exchange processes. Bothmodels are derived using a streamtube approxi-mation, which assumes that the current can bedescribed as a steady tube of sediment-laden ¯uidwith quiescent water above. The two modelsdiffer in the form adopted for the bottom stress.The ®rst model uses a quadratic drag law,whereas the second uses a parameterization ofbottom friction based on the turbulence in theplume and an additional turbulent energy equa-tion. Emms (1998) found that, under identicalinitial conditions, the second model led to a delayin the onset of ignition. The present paperconsiders the dynamics of both models by study-ing their phase space, that is the relationshipbetween the dependent variables of the current(such as velocity or concentration of sediment).Such a diagram allows examination of a multi-tude of initial conditions and prediction of thequalitative nature of the solution without theneed for numerical integration. This work com-plements that of Parker et al. (1986) by consider-ing the dynamics of turbidity currents that arerotating geostrophically. In addition, it showshow the initial tube direction affects ignition andhow approximate model solutions are representedin phase space. Solutions are also found forwhich there is insuf®cient energy available forsuspension at all. This has a profound effect onthe predictions of the model.

The paper proceeds as follows: in the nextsection, the non-dimensional turbidity currentmodels are introduced. Both models use animproved parameterization for erosion of sedi-ment. The following section determines underwhat conditions solutions of the four-equationmodel proceed to ignition. The ignition criteriontakes the form of a surface in phase space, whichseparates those solutions that ignite and those thatdeposit, depending on initial conditions. Thissurface and the accompanying phase diagram arecalculated using parameters for a turbidity currentin the Greenland Sea. The next section describesphase space for the ®ve-equation model andcompares the ®ndings with the four-equationmodel. Again, phase space is separated dependingon the ®nal state (far downstream) of solutions.Finally, some conclusions are drawn concerningthe qualitative nature of turbidity currentsdescribed by streamtube models.

TURBIDITY CURRENT MODELS

The turbidity current models are based on erosionand deposition laws coupled to a streamtubemodel of a geostrophically rotating gravity cur-rent (Smith, 1975; Emms, 1997, 1998). Referringto Fig. 1, consider a tube of ¯uid on a uniformslope at an angle S to the horizontal and with xand y axes positioned so that y points downslope.The centre-line of the tube is given by x � Xp,y � Yp, and this is related to an along-tubeco-ordinate n by

dXp

dn� cos b;

dYp

dn� sin b �1�

where b is the angle the centre-line makes withthe x-axis.

The erosional parameterization is based uponthe experimental study of Garcia & Parker (1993),which improved upon the parameterization usedpreviously by Parker et al. (1986). For a sub-merged turbidity current, Garcia & Parker (1993)found that, if vsEs denotes the sediment ¯ux intothe current (where vs is the settling velocity), then

Es � AeZ5u

�1� Ae

EsmZ5

u��2�

gave a reasonable ®t to their experimental data.Here, Ae is a constant equal to 1.3 ´ 10±7 and Esm

is the maximum erosion rate taken as 0.3. The

1050 P. W. Emms

Ó 1999 International Association of Sedimentologists, Sedimentology, 46, 1049±1063

parameter Zu is a measure of the strength of the¯ow and is de®ned by

Zu � U

vsR0:6

p �3�

where U is the shear velocity and Rp is theparticle Reynolds number. In fact, Garcia & Parker(1993) looked at a slight generalization of Eq. 3but, in the current work, the simpler expressionwill suf®ce. This is a different erosion law to thatused in Parker et al. (1986) and Emms (1998) andleads to some quantitative differences from theirresults; however, the qualitative behaviour of themodels is similar. For a depositional law, it issupposed that the sediment ¯ux onto the ocean¯oor is proportional to the sediment concentra-tion. Ambient water entrainment is assumed to beproportional to the mean speed of the ¯ow,following Ellison & Turner (1959).

Emms (1998) derived the two turbidity currentmodels in some detail, and this is not reproducedhere. The governing equations express incom-pressibility, conservation of mass, conservation ofmomentum along and across the tube and turbu-lent energy conservation. In terms of the cross-sectional area A, the along-tube velocity V, thevolumetric concentration C, the angle b and levelof the turbulence K, these relations can beexpressed as

�AV�n � ewwV �4a�

�AVC�n � vsw�Es�Zu� ÿ r0C� �4b�

�AV2�n � ÿwU2 � RgSAC sin b �4c�

V2bn � RgSC cos bÿ fV �4d�

�AVK�n �1

2ewwV3 �wU2V ÿ cwK3=2

ÿ RgvsAC ÿ 1

2RgvsA�Es�Zu� ÿ r0C�

ÿ 1

2RgewACV

�4e�

where ew is the entrainment rate of ambientwater, w is the tube width, r0 is a depositionalcoef®cient, R is the immersed speci®c gravity ofthe sediment, f is the Coriolis parameter and c is adissipative constant.

The terms on the right-hand side of Eq. 4erepresent: the transfer of kinetic energy fromthe mean ¯ow to the turbulence (®rst twoterms); viscous dissipation; the turbulent energyexpended in maintaining the sediment; the loss/gain of energy resulting from erosion/deposition;and the energy used in raising the centre ofgravity of the existing load, caused by dilation ofthe ¯ow by water entrainment. Viscous dissipa-tion is modelled using a K3/2 law (see Launder &Spalding, 1972). It is also assumed that the meanheight of the tube is proportional to the width sothat w µ A1/2. Alternatively, a spreading lawbased on the Ekman number (Price & Baringer,1994) could be chosen, but this does not alter thedynamics of the model signi®cantly.

Non-dimensionalization is carried out usingthe following scales for concentration of sedi-ment, mean velocity, tube width, cross-sectionalarea, length, shear velocity and turbulence:

C� � Esm

r0; V� � aQ��RgC�S�2

C2f

" #1=5

; w2� �

Q�aV�

;

A� � Q�V�

; L � aw�Cf

; U� � C1=2f V�; K� � V2

�d

�5�

where a is the aspect ratio of the tube (de®ned asthe mean tube height divided by the tube width),Q* is the initial volume ¯ux and Cf and d are

Fig. 1. Schematic diagram of a geostrophically rotatingturbidity current spreading down and moving along auniform slope. The aspect ratio a of the tube, which isassumed to be constant, is de®ned as the mean tubeheight divided by the tube width w.

Geostrophically rotating turbidity currents 1051


frictional coef®cients. The frictional parameter Cf

should not be confused with a sediment quantity.Length and velocity scales are set by balancinginertia, friction and gravity along the tube (inEq. 4c) ± this leads to scales that are appropriateto ¯ow down the slope. The width scale is chosenso that the initial non-dimensional volume ¯ux isQ � 1.

Two models are considered, which differ in theparameterization they adopt for the shear velocity.The ®rst model uses a quadratic drag law so that

U2 � Cf V2 �6�

and the turbulence Eq. 4e uncouples. Using thescales given in Eq. 5, the non-dimensional four-equation model is

�AV�n � EA1=2V �7a�

�AVC�n � BA1=2 Es�aV�Esm

ÿ C

� ��7b�

�AV2�n � ÿA1=2V2 �AC sin b �7c�

V2bn � C cos bÿ V=Ro (7d)

There are four parameters in Eq. 7: the entrain-ment of ambient water E, a sediment transferparameter B, the erosional capacity a and theRossby number Ro. These are given by

E � ew

Cf; B � r0vs

Cf V�; a �

R0:6p C0:5

f V�vs

; Ro � V�fL

�8�

If there is no mixing (E � 0), then the volume¯ux Q � AV � 1, and Eq. 7 reduces to threeequations for V, C and b. If B � 0, then there isno sediment exchange, and these are the classicalstreamtube equations of Smith (1975). If Ro � ¥,(corresponding to no rotation), then the three-equation model of Parker et al. (1986) is recov-ered, with a modi®ed sediment entrainment lawand hydrostatic pressure (see later).

Parker et al. (1986) found that models of theform of Eq. 7 could raise more sediment than wasenergetically possible and remedied this by con-sidering an additional turbulent kinetic energyequation for the plume (Eq. 4e). More sophisti-cated turbulence models have also been studied

by Eidsvik & Brors (1989), who added a model ofturbulence to a non-rotating turbidity currentmodel. However, this introduces additional tur-bulent parameters, which are dif®cult to quantify.The present study follows Parker et al. (1986) andconsiders only a simple model of turbulence.

The second model supposes that the shearvelocity is

U2 � dCf K �9�

The constants in Eq. 9 ensure that both modelshave the same velocity scale. To reduce thenumber of free parameters further, the approachof Parker et al. (1986) is adopted, which setsc � Cfd

3/2. This forces both models to yieldsimilar uniform solutions (Emms, 1998) and aidscomparison. Consequently, the non-dimensional®ve-equation model is

�AV�n � EA1=2V �10a�

�AVC�n � BA1=2 Es�aK1=2�Esm

ÿ C

� ��10b�

�AV2�n � ÿA1=2K �AC sin b �10c�

V2bn � C cos bÿ V=Ro �10d�

�AVK�n �d

1

2EA1=2V3 �A1=2KV

ÿA1=2K3=2 ÿ GAC ÿ 1

2BkA

Es

Esmÿ C

� �ÿ 1

2EkACV

!�10e�

The additional parameters required here are theBagnold number G and the frictional parameter kgiven by

G � vs

SV�; k � Cf

S�11�

The details of the parameterizations andapproximations that are part of both these modelscan be found in Emms (1998). Typical values forall the parameters used in the models are shownin Table 1 for a medium-grained silt (particle

1052 P. W. Emms


diameter � 20 lm) and a coarse-grained silt (par-ticle diameter � 63 lm). These parameter valuesare representative of the Kveitehola out¯ow,which occurred in the Greenland Sea, and aretaken from Fohrmann et al. (1998), who haverecently described a numerical simulation of thisturbidity current.

Both models (Eqs 7 and 10) are completed byspecifying initial conditions at n � 0, where theinitial disturbance occurs:

A � As; V � Vs; C � Cs; b � bs; K � V2s

�12�The last condition is chosen so that, in both

models, the tube experiences the same initialfrictional drag, which is a useful condition inorder to compare the two models.

By specifying initial conditions, no informationfrom downstream of the source can in¯uence themodel solutions. In the streamtube equations, aterm of the form 1

2(CA3/2)n is neglected, whichrepresents the change in hydrostatic pressurecaused by changes in tube height and sedimentconcentration (Emms, 1997). It is precisely thisterm that, in the full time-dependent equations,models waves travelling on the tube interface andtransmitting information both upstream anddownstream. Thus, the turbidity current modelsneglect information propagating backwards alongthe tube, which is reasonable if the ¯ow issupercritical (i.e. the ¯ow velocity is greater thanthe wave speed). However, this is not so reason-able if the ¯ow is subcritical, because down-stream topography will then in¯uence the steadystate, if it exists.

To study subcritical ¯ow rigorously, the fulltime-dependent equations should be integrated.However, this is a formidable problem to solveeven numerically (Zeng & Lowe, 1997). Thus, inorder to make further progress, a compromisemust be made by neglecting this interfacial wavemotion ± the energetics of subcritical tubes canthen be examined, and this yields some novelresults. Also, streamtube models have been suc-cessfully compared against real oceanic out¯owswhen the ¯ow is subcritical (Smith, 1975; Kill-worth, 1977; Price & Baringer, 1994), so there issome justi®cation for this assumption.

IGNITION OF THE FOUR-EQUATIONMODEL

Uniform (no n variation) solutions of Eq. 7 for noentrainment satisfy

V50 �

V20

Ro2ÿ C2

0 � 0; C0 � Es�aV0�Esm

;

cosb0 �V0

RoC0

�13�

Emms (1998) found that there are two solutionsto Eq. 13. One is the geostrophic alongslope stategiven approximately by

V0 � RoC0; C0 � Es�aV0�Esm

; b0 � 0 �14�

while the other is the catastrophic downslopestate given approximately by

Table 1. The non-dimensional parameters for typical medium and coarse silts taken from Emms (1998).

Entrainment E 2.7Rossby number Ro 33.5Relationship between turbulent and mean velocity scales d 100.0Friction k 5.88 ´ 10)2

Medium CoarseParticle diameter Ds (lm) 20 63Particle Reynolds number Rp 0.20 1.12Deposition rate B 2.91 ´ 10)2 0.29Erosional capacity a 6.62 ´ 102 1.88 ´ 102

Bagnold number G 1.07 ´ 10)3 1.06 ´ 10)2

Downslope velocity scale V* (m s)1) 11.0Cross-sectional area scale A* (m2) 1.37 ´ 103

Length scale L (km) 2.34Volumetric concentration scale C* 0.188

The ®gures for the erosional capacity a differ slightly from Emms (1998) because of the change in erosional para-meterization. To aid re-dimensionalization of the model, typical scales are also shown.



V0 � 1; C0 � 1; b0 � p=2 �15�

The non-dimensionalization has been chosen tomatch to the catastrophic uniform state. Emms(1998) found the geostrophic state to be spatiallyunstable, so that model solutions tended to eitherthe catastrophic uniform or the non-uniformdepositional state, depending on the initial condi-tions (see Fig. 2). It is important to realize thedistinction between temporal and spatial stability.Temporal stability refers to whether a steady statewill remain steady. Spatial stability of a stateindicates the behaviour of steady solutions asn ® ¥ for initial conditions `near' that state.

Temporal stability is not considered here.Rather, this paper describes the dynamics ofsteady ¯ow in the belief that the full time-dependent equations will reach a quasi-steadyequilibrium. Speci®cally, the analysis determineswhich initial conditions lead to steady cata-strophic ¯ow.

For a non-rotating tube, phase space is two-dimensional (ignoring entrainment), and thedependent variables are the mean velocity Vand sediment concentration C. By consideringthe domain of attraction of the catastrophicequilibrium point, a line can be found thatseparates those initial conditions that lead todeposition (the subsiding ®eld) and those thatlead to catastrophe (the igniting ®eld). Parkeret al. (1986) called this line the autosuspensiongeneration line (AGL).

For a rotating turbidity current modelled by thefour-equation model, a surface separates the two®elds (for no entrainment) ± the autosuspensiongeneration surface (AGS). The dependent vari-ables are V, C and the tube angle b. To calculatethis surface, those initial conditions that just leadto ignition must be found. The numerical proce-dure is to ®x the initial tube concentration Cs andtube angle bs and then use a bisection method todetermine which Vs just leads to ignition. Alter-natively, Vs and bs can be ®xed and Cs varied, andthis might be more appropriate depending on theorientation of the surface. The initial tube angle bs

varies from 0 (along the slope) to p/2 (down theslope): values outside this range seem implausi-ble.

Fig. 2. Qualitative description of the possible solutionsof the turbidity current models.

Fig. 3. The autogeneration surface(AGS) for the four-equation modelcontaining medium-grained silt.Six equally spaced contours of bags

are shown. Above this value of b, atube always ignites. The b-axis isat right angles to the V and C axesand points out of the page.

1054 P. W. Emms


The AGS for the four-equation model is illus-trated in Fig. 3 with parameters chosen for amedium-grained silt (Table 1). Six equally spacedcontours of bags are shown. For ®xed Vs and Cs,initial values of bs > bags tend to the catastrophicstate. For large V and C, the contours arereasonably spread out on the log-log plot, indi-cating that the initial tube angle has some impacton whether ignition takes place. Tubes starteddownslope are more likely to ignite than thosetravelling alongslope, in the sense that a smallervelocity or concentration is required for ignition.For small Vs, the contours converge, and theinitial tube angle is unimportant. For small Cs, thecontours cross, which means the surface foldsover. The behaviour of the solutions in this case isillustrated by three solutions started atVs � 6 ´ 10±3, Cs � 10±6 (shown as a solid circlein Fig. 3) and bs � 0, p/4, p/2.

The tube travelling initially alongslope ignites(Fig. 4a) as V, C ® 1 as n ® ¥. The tube started atb � p/4 deposits (Fig. 4b) as V, C ® 0 as n ® ¥,and that travelling initially downslope ignites(Fig. 4c) again. Thus, there is an intermediaterange of b in which the tube will deposit. Thisrange, however, occurs over a small interval ofvelocities and may be dif®cult to see in practice.The AGS is next superimposed onto the phasediagram for a medium-grained silt, and Fig. 5shows the projection of phase curves onto theV±C plane. With the chosen data, the range ofvelocities considered is about 10±3 m s±1 to10 m s±1, while the range of concentrations startsat very low-density turbidity plumes (C ~ 10±6)and ends at mud (C ~ 1). The phase curves areshown as thin solid lines: catastrophic solutionsoriginate on the co-ordinate axes. For V > 1 orC > 1, all the phase curves ignite. Two represen-tative depositional solutions are also shown, andthe periphery of the AGS is shown as two thickcontours. The geostrophic uniform solution pro-vides a good indicator as to the magnitude of Vs

and Cs required for ignition. Those solutions thatdo ignite tend to a straight line on the plot asn ® ¥, which is indicative that the along-tubebalance is between gravity and friction. FromEqs 7a and 7c, V ~ C2/5, and so the line hasgradient 2/5 in a log-log plot. This is shown as adotted line in Fig. 5. Depositional solutions circleon the slope as a result of the Coriolis force (Fig. 2and see Emms, 1998), and so the projected phasecurves oscillate.

To illustrate how the parameters affect themodel, the phase plane for a coarse silt is shownin Fig. 6. There is no folding of the AGS for this

grain size, at least in the portion of the planeconsidered, but the phase plane is qualitativelysimilar. In comparison with the medium-grainedsilt, a larger initial concentration and velocity isneeded for ignition: a coarse silt has a smallererosional capacity so that a larger velocity isneeded for the same erosion of sediment. Also,the circling of the tube on the slope is not sopronounced for depositional ¯ow. This indicatesthat coarse silt settles more rapidly as the Rossbynumber Ro, which determines the period ofoscillation, is the same for both plots.

The geostrophic state is a good indicator of thevalues of V and C required for ignition. However,the parameterization of erosion signi®cantlyalters this predictor in the case that there is norotation. In Fig. 7, V5 + V2/Ro2 and V5 are plottedagainst V. In addition, the concentration isplotted against C0

2 � Es2/Esm

2 for two erosionalparameterizations: that given by Eq. 2 and thatadopted by Parker et al. (1986) and Emms (1998).The intersection of these curves gives the uniformstates. With the parameterization of erosion usedby Parker et al. (1986), the ignitive velocity andconcentration do not change signi®cantly if rota-tion is neglected (Ro � ¥). However, with theparameterization given in Eq. 2, the uniformignitive velocity changes by an order of magni-tude, and the ignitive composition changes byfour orders of magnitude when looking at therotating and non-rotating cases. The problem isthat the parameterization (Eq. 2) does not forceEs � 0 below some critical value of the ¯owstrength Zu. It is the determination of this criticalvalue that sets the ignitive state and, consequent-ly, when ignition will occur. For a rotatingturbidity current, there is not a signi®cant quan-titative problem because rotation changes thebalances in Eq. 13. Figure 7 also shows that theignitive velocity is larger in a rotating turbiditycurrent, thus implying that more energy isrequired for ignition.

The effect of rotation on ignition can be seen bylooking at the approximate expression of thegeostrophic state V0. Eliminating C0 from Eq. 14gives

V0

R0� Es�aV0�

Esm�16�

Consequently, the geostrophic state is givenwhere the straight line, with gradient Ro±1, crossesthe normalized erosional parameterization. As themagnitude of the Coriolis parameter, f, increaseswith latitude, the Rossby number decreases, and



Fig. 4. Solutions in phase spacestarting at Vs � 6 ´ 10±3, Cs � 10±6,and (a) bs � 0, (b) bs � p/4, (c)bs � p/2. In (a) and (c), the tubeignites, whereas in (b), there isdeposition. The parameters arechosen to correspond to a medium-grained silt.

1056 P. W. Emms


so the gradient of the line on the left-hand side ofEq. 16 increases. It therefore follows that theignition velocity increases with latitude and thatthe way it increases depends sensitively on theerosional parameterization.

A quantitative prediction for the ignition of theKveitehola turbidity current (Fohrmann et al.,1998) is given by numerical values for the ignitive

state: V0 � 4.82 ´ 10±3, C0 � 1.44 ´ 10±4. Multi-plying by the scales in Table 1, the ignitivevelocity and concentration are

Vig � 0:0053 m sÿ1; Cig � 2:7� 10ÿ5 �17�

Without rotation, the erosional parameterization(Eq. 2) leads to a small ignitive velocity andconcentration (see Fig. 7): Vig � 0.0088 m s±1,Cig � 3.4 ´ 10±9.

Fig. 5. Phase diagram for the four-equation modelcontaining medium-grained silt. The thin solid lines areprojections of the three-dimensional phase trajectoriesonto the V±C plane (the b-axis comes out of the page).Arrows denote the direction of increasing n. The geo-strophic solution is denoted by s and the catastrophicsolution by h. The thick solid curves are the contoursbags � 0 and bags � p/2.

Fig. 6. Phase diagram for the four-equation modelcontaining coarse-grained silt. Thin solid lines arephase curves, while the thick lines denote the edges ofthe AGS.

Fig. 7. Uniform states for two dif-ferent parameterizations of erosionEs. Equilibrium states (geostrophy,catastrophe) in this paper are givenwhere V5 + V2/Ro crosses C2 (new).Equilibrium states using the Parkeret al. (1986) parameterization forerosion are given where V5 + V2/Rocrosses C2 (old). Equilibrium statesfor a non-rotating tube are givenwhere V5 crosses C2. The experi-mental range indicates the values atwhich the new parameterizationof sediment erosion agrees withexperiment.



THE FIVE-EQUATION MODEL

For the ®ve-equation model, the tube can alsodeposit or ignite. However, there is anotherpossibility: the tube may have insuf®cient energyto support a given concentration of sediment. Inthis case, turbulence decreases rapidly in theplume, and so sediment is deposited almostimmediately (see Fig. 2). Extinction differs fromdeposition in that K � 0 at some ®nite n � nK

downstream, whereas the mean velocity V andsediment concentration C both remain non-zero.In the laboratory, this behaviour corresponds tocurrents with a ®nite run-out (Bonnecaze et al.,1993; Dade & Huppert, 1995). Emms (1998) onlyconsidered one example solution of an extinguish-ing tube: the full signi®cance of this behaviour canonly be seen by examining the phase diagram.

There are now four dependent variables, ifentrainment is ignored: V, C, b and K, and soprojections of phase curves onto the V±C planeare more complicated. For simplicity, supposethe initial tube angle is bs � p/2; it is expectedthat variations of this parameter will not affect theresults substantially.

To calculate the qualitative behaviour of solu-tions in phase space, the V±C plane is griddedand the equations integrated from each grid-pointto determine the ®nal model state. Contours arethen determined that separate phase space intoregions in which the tube deposits, ignites orextinguishes. For a medium-grained silt, thisdiagram is plotted in Fig. 8. The boundary ofthe extinction region gives the initial energy

required (in other words the value of Vs) tomaintain sediment of concentration Cs in suspen-sion. If there is suf®cient energy, then the tubeeither deposits or ignites. The initial velocity andsediment concentration must be higher than inthe four-equation model for the tube to ignite,because more energy is required to maintainsuspension. The diagram is also divided intosupercritical and subcritical ¯ows by the lineFr � 1, where the Froude number of the tube isFr � V/(RgCh)1/2 (Parker et al., 1986). Here,h � aw is the tube height. In terms of the presentnon-dimensional variables

Fr � V5=4��kCp �18�

where self-similarity is assumed (so w � A1/2),and ambient water entrainment is negligible (soA � 1/V). The dynamics of the model do notchange through the critical Froude numberbecause of the supposition that information canonly propagate from the source (see the discus-sion after Eq. 12).

The lower boundary of the depositional regionand part of the lower boundary of the ignitionregion follow the line Fr � k±1/2 � 4.12, which isthe initial normal Froude number (Chow, 1959).Rearranging Eq. 10c gives (at n � 0)

VVn � �kÿ1 ÿ Fr2� �19�

so that the initial change in velocity determinesthe ultimate fate of the tube. Speci®cally, below

Fig. 8. Regions of phase space forthe ®ve-equation model of a tubecontaining medium-grained silt.

1058 P. W. Emms


the line Fr � k±1/2, the velocity increases initiallyand so does the turbulence by the transfer ofkinetic energy from the mean ¯ow. This increasesthe sediment concentration in the plume througherosion and, gradually, the turbulence in the tubediminishes as more energy is required to supportthe sediment. Eventually, the turbulence reacheszero, sediment can no longer be supported andthe tube extinguishes.

The abrupt change in the boundary of theignition region at C ~ 10±3 is caused by the changein dynamic behaviour on crossing Fr � k±1/2. Inthe four-equation model, there was no possibilityof extinction, so that the boundary continueddown until it met V � 0. Consequently, there isno minimum concentration of sediment abovewhich ignition is always possible in the ®ve-equation model. As the initial velocity isincreased further towards the catastrophic point,the ignition velocity deviates away from thenormal Froude number contour (C ~ 2 ´ 10±2).In the region above the line, solutions tendrapidly to the catastrophic point: there is noreason why the domain of attraction of thecatastrophic state should be smooth. As C in-creases further, the ignition velocity reaches aminimum and then increases with C to the rightof the catastrophic point. This is not shown inFig. 8 because such concentrations are unrealis-tically high for turbidity currents.

Greater insight into the behaviour of the modelsolutions can be obtained by looking at projec-tions of phase curves onto the V±C plane, the V±Kplane and the V±b plane. Again, the V±C plane isgridded to give initial conditions, although at alower spacing, so that the phase curves can befollowed. These projections are shown for amedium-grained silt in Fig. 9a±c with bs � p/2.First, from Fig. 9a, solutions started in theextinction region die rapidly (i.e. their phasecurves ®nish abruptly) ± the velocity increasesdownstream initially, while the concentration isapproximately constant. The ignition point is areasonable predictor of ignition, as in the four-equation model, giving the minimum value of Vs

that is required. The curves in Fig. 9b all originateon K � V2, which is a straight line on the log-logplot, those solutions that extinguish moving tothe left until they reach the axis. Those solutionsthat deposit head towards the point K � V � 0.Most of the trajectories in the lower portion ofFig. 9c extinguish rapidly, and b does not changeappreciably. For larger Vs, the curves cross b � 0and begin to circle on the slope: these are thedepositional curves. For still larger Vs, the tube

Fig. 9. The phase diagram for the ®ve-equation modelcontaining medium-grained silt. Three projections areshown: (a) onto the V±C plane; (b) onto the V±K plane;and (c) onto the V±b plane.



ignites and heads towards the catastrophic point.It is evident that, for some initial velocities neargeostrophic, the tube travels alongslope for a largedistance before igniting properly, that is where Vincreases monotonically.

A quantitative prediction of the ignition veloc-ity can again be found by multiplying the geo-strophic state (V0 � 0á143) by the velocity scale inTable 1 to obtain

Vig � 1�6 m sÿ1 �20�

Approximate solutions can be found for thedifferent qualitative forms of solution in the ®ve-equation model by determining the dominantbalances in Eq. 10. For simplicity, suppose thatthe entrainment of water is negligible in all thesesolutions so that the volume ¯ux Q � AV � 1. If aparameterization of water entrainment isincluded, then the volume ¯ux increases down-stream, and the model only has quasi-equilibriumsolutions. However, as in density-driven stream-ube models (Emms, 1997), these quasi-equilibriumstates determine the dynamics of the tube.

The dominant balances for the depositionalstate are between inertia and friction in thealongstream momentum equation (Eq. 10c),between inertia and the Coriolis force in theacross-stream momentum equation (Eq. 10d) andbetween sediment change and deposition inEq. 10b. Consequently,

Vn � ÿ K

V1=2; Vbn � ÿ

1

Ro; Cn � ÿ BC

V1=2�21�

The turbulence equation (Eq. 10e) rapidly equilib-rates on a scale n ~ d±1, as d >> 1. In addition,Bk « « 1, so the dominant equilibrium balance is

KV3=2 � K3=2V1=2 � GC �22�

To make progress, a further balance is soughtbetween the generation of turbulence, KV3/2, andthe viscous dissipation, K3/2V1/2. There are nowenough equations to integrate Eq. 21:

V � K1=2; C � Csexp�ÿB=V�; V � 4=n2 �23�

using the fact that V ® 0 as n ® ¥. Consequently,in the log-log plot of the phase diagrams, deposi-tional curves tend to straight lines of gradient 1/2in Fig. 9b. The phase curves in Fig. 9a do not tendto a straight line for large n, although this is notapparent on the plot, because the relationshipbetween V and C is not a power law.

An approximate erosional solution can befound by adopting the following balances:friction with gravity in the alongstream momen-tum equation (Eq. 10c); gravity with the Coriolisforce in the across-stream momentum equation(Eq. 10d); and the change in composition witherosion. Finally, in the turbulence equation, thegeneration of turbulence is balanced with theviscous dissipation to obtain

V � K1=2; K � C

V1=2; Cn � B

V1=2; V � RoCb�

�24�

where b* � p/2±b. On integration, and applyingthe initial conditions, it can be shown that

V � C2=5; b� � C2=5=Ro; C � 6B

5n� C5=6

s

� �5=6

�25�

Thus, in the phase plots (Fig. 9), catastrophiccurves tend to a straight line of gradient 2/5 in theV±C plane and a straight line of gradient 1/2 inthe V±K plane.

For extinction, inertia is balanced with gravityin Eq. 10c to obtain

VVn � Cssin bs �26�

where it is supposed that C and b are approxi-mately constant over the integration range, as iscon®rmed by the numerical solutions. IntegratingEq. 26 then yields

V � �2Cssin bsn� V2s �1=2 �27�

using the initial conditions at n � 0. In theturbulent energy equation (Eq. 10e), if the varia-tion in K is balanced with the energy required tokeep the sediment in suspension, then

Kn � ÿ dGCs

V�28�

which yields on integrating

K � dG

sinbs

Vs ÿ �2Csnsin bs � V2s �1=2

h i� Ks �29�

Thus, the turbulence in the tube dies out at

nK �V3

s

2CsdG

Vssinbs

dG� 2

� ��30�

1060 P. W. Emms


which is valid providing Cs >> Vs7/2/G (i.e. for

large initial concentrations). If Cs is small, thenthe generation of turbulence, dKV1/2, cannot beneglected in Eq. 28, and the equation cannot beintegrated analytically.

Phase space for the ®ve-equation model of ageostrophically rotating turbidity current is there-fore signi®cantly different compared with the four-equation model. It is then useful to consider howthe Parker et al. (1986) turbulence model of a non-rotating turbidity current changes if the term12(CA3/2)n is neglected from the governing equa-tions. Figure 10 shows this phase diagram for anon-rotating turbidity current with a medium-grained silt. Four features are noteworthy. First,the ignitive state occurs at a very small concentra-tion, which is a consequence of the erosionalparameterization (see Fig. 7). Secondly, the mini-mum ignitive velocity is two orders of magnitudesmaller than for the rotating case (V � 4 ´ 10±3),re¯ecting the balances in the uniform ignitive state.Thirdly, the extinction region no longer separatesthe ignitive and subsiding ®elds. In the rotatingcase, solutions in this region turn under theCoriolis force and travel up the slope, therebylosing kinetic energy. This loss of energy preventsthe suspension of sediment, and the turbiditycurrent therefore dies. Clearly, this mechanismcannot occur in the non-rotating case, and so thereis no separation of ®elds. Lastly, the lower boun-dary of the depositional region follows a normalFroude number contour, which indicates that it isthe initial increase or decrease in V that determinesthe ®nal state of the model. The uniform ignitive

point does not pass through this line because thenormal Froude number Frn � k±1/2 only at n � 0.

CONCLUSIONS

The qualitative behaviour of two models describ-ing geostrophically rotating turbidity currents hasbeen found using phase plane analysis. For thefour-equation model, the phase space is separatedby the AGS into two regions: for a high initialvelocity and sediment concentration, the tubeignites and proceeds to the catastrophic uniformstate, whereas for a lower initial velocity andconcentration, the tube deposits and circles onthe slope inde®nitely. The initial angle at whichthe tube starts on the slope has a relatively smallimpact on ignition over the large range of veloc-ities and concentrations considered here. Tubesstarting downslope are more likely to ignite thanthose travelling alongslope, in the sense that thereare more initial conditions that lead to catastro-phe. The geostrophic equilibrium state is a robustpredictor for the ignition of a rotating turbiditycurrent.

For the ®ve-equation model, there is an addi-tional ®nal state to the model: the turbulence candie out, and so the sediment drops almostimmediately. This suggests that turbidity currentsdo not exist for these initial conditions and,consequently, this represents a minimum energyrequirement for turbid ¯ow. The surfaces thatseparate the initial conditions that lead to the®nal state in the ®ve-equation model are more

Fig. 10. Phase space for a non-rotating tube (Ro � ¥) containingmedium-grained silt using the ®ve-equation model.



complicated than in the four-equation model,with ignition less likely. Moreover, ignitive solu-tions show a tendency to travel further alongslopein a quasi-geostrophic balance. Thus, alongslope¯ow is possible even if a turbidity current iseroding, especially if the initial conditions arenear-geostrophic.

From these simple model solutions, a morerelevant description of the path a turbiditycurrent might take down the continental slopecan be proposed. First, a minimum disturbance isrequired for the turbidity current to form. Thecurrent then ignites and proceeds down thecontinental slope in approximate catastrophicbalance. Gradually, the slope diminishes andtherefore so will the mean velocity. Eventually,the depositional balance will take effect, and thecurrent will move to the right (in the northernhemisphere) looking down¯ow.

Both models describe poorly the mechanismsthat determine the initiation of a turbidity cur-rent: an initial concentration and velocity arespeci®ed for the ¯ow, which will be dif®cult todetermine in practice. Pantin (1986) has consid-ered small periodic perturbations applied to anon-rotating current, and such modelling couldalso be incorporated here. Pantin (1990) has alsoconsidered interstitial ¯uid of a greater densitythan the ambient ¯uid, so that convection ispossible if the turbidity current deposits most ofits sediment. It is simple to modify the presentmodels to encompass these features.

To describe the distribution of sediment in acatastrophic or depositional current, a moresophisticated model is required. Recently, Sha-piro & Hill (1997) have studied a time-dependenttwo-dimensional tube model, which ultimatelyreduces to an advection±diffusion equation forthe tube height. The key simpli®cation thatallowed them to do this was the adoption of asteady linear momentum equation coupled to atime-dependent mass equation. Shapiro & Hill(1997) found some simple analytical solutions inspecial cases and also solved the full equationnumerically. It should be possible to add erosio-nal and depositional laws to this model and evenconsider more than one grain size. Alternatively,a sediment equation of the form of Eq. 7b could beadded to a channel model of a rotating ¯uid toinvestigate how eddies affect the distribution ofsediment (cf. Zeng & Lowe, 1997). Moody et al.(1998) have also recently suggested that a two-layer model should more properly be used todescribe a turbidity current. These ideas are areasof research currently being pursued.

ACKNOWLEDGEMENTS

Part of this work was carried out at the South-ampton Oceanography Centre with the support ofthe UK Natural Environment Research Councilgrant GR3/8578. The manuscript was consider-ably improved by the review of Henry Pantin andthe editorial comments of Jim Best.

Notation

a aspect ratio of the tubeA cross-sectional area of tubeAe constant in erosional parameterizationC volumetric concentration of sedimentCf frictional coef®cientDs particle diameterew entrainment rate of ambient waterEs erosional parameterizationEsm maximum erosion ratef Coriolis parameterg acceleration due to gravityg¢ reduced gravityh mean tube heightK level of turbulenceL length scaleQ volume ¯ux through tuber0 depositional coef®cientR immersed speci®c gravity of sedimentS slope of inclined planeU shear velocityV along-tube ¯uid velocityms settling velocity of particlesw tube widthx, y distance along and downslopeXp, Yp position of the centre-lineZu ¯ow-strength

Greek notation

b angle centre-line makes with the x-axisb* angle centre-line makes with the y-axisbags the Autosuspension Generation Surfaced frictional coef®cientc viscuous dissipative constantn distance along centre-linenK extinction length

Non-dimensional parameters

a the erosional capacityB sediment transfer parameterE entrainment of ambient waterFr Froude number

1062 P. W. Emms


Frn normal Froude numberG Bagnold numberk frictional parameter in the 5-equation

modelRo Rossby numberRp particle Reynolds number

Subscripts

* dimensional scale0 uniform state (a solution independent

of n)ig dimensional values of ignitive states initial nondimensional value at n � 0

REFERENCES

Bonnecaze, R.T., Huppert, H.E. and Lister, J.R. (1993)Particle-driven gravity currents. J. Fluid Mech., 250,339±369.

Chow, V.T. (1959) Open-Channel Hydraulics. McGraw-Hill, New York.

Dade, W.B. and Huppert, H.E. (1995) A box model fornon-entraining, suspension-driven gravity currents.Sedimentology, 42, 453±471.

Eidsvik, K.J. and Brors, B. (1989) Self-accelerated tur-bidity current prediction based upon turbulence.Cont. Shelf Res., 9, 617±622.

Einstein, H.A. (1968) Deposition of suspended particlesin a gravel bed. Proc. Am. Soc. Civ. Eng., 94 (HY5),1197±1205.

Ellison, T.H. and Turner, J.S. (1959) Turbulententrainment in strati®ed ¯ows. J. Fluid Mech., 6,423±448.

Emms, P.W. (1997) Streamtube models of gravity cur-rents in the ocean. Deep-Sea Res., 44, 1575±1610.

Emms, P.W. (1998) A streamtube model of rotatingturbidity currents. J. Mar. Res., 56, 41±74.

Fohrmann, H., Backhaus, J.O., Blaume, F. and Rumohr, J.(1998) Sediments in bottom arrested gravity plumes.Numerical case studies. J. Phys. Oceanogr., 28,2250±2274.

Garcia, M. and Parker, G. (1993) Experiments on theentrainment of sediment into suspension by a

dense bottom current. J. Geophys. Res., 98,4793±4807.

Killworth, P.D. (1977) Mixing on the Weddell Seacontinental slope. Deep-Sea Res., 24, 427±448.

Launder, B.E. and Spalding, D.B. (1972) MathematicalModels of Turbulence. Academic Press, London.

Moodie, T.B., Pascal, J.P. and Swaters, G.E. (1998)Sediment transport and deposition from a two-layer¯uid model of gravity currents on sloping bottoms.Stud. Appl. Math., 100, 215±244.

Nof, D. (1996) Rotational turbidity ¯ows and the 1929Grand-Banks earthquake. Deep-Sea Res., 43,1143±1163.

Pantin, H.M. (1979) Interaction between velocity andeffective density in turbidity ¯ow: phase-planeanalysis, with criteria for autosuspension. Mar. Geol.,31, 59±99.

Pantin, H.M. (1986) Triggering of autosuspension by aperiodic forcing function. In: Proceedings of theThird International Symposium on River Sedimen-tation (Eds S.Y. Yang, H.W. Shen and L.Z. Ding),pp. 1765±1770. The University of Mississippi, Oxford.

Pantin, H.M. (1990) A model for ignitive autosuspen-sion in brackish under¯ows. In: Proceedings of theEuromech 262 Colloquium on Sand Transport inRivers, Estuaries and the Sea (Eds R. Soulsby andR. Bettess), pp. 283±290. A.A. Balkema, Rotterdam.

Parker, G. (1982) Conditions for the ignition of cata-strophically erosive turbidity currents. Mar. Geol.,46, 307±327.

Parker, G., Fukushima, Y. and Pantin, H.M. (1986) Self-accelerating turbidity currents. J. Fluid Mech., 171,145±181.

Price, J.F. and Baringer, M. (1994) Out¯ows and deepwater production by marginal seas. Prog. Oceanogr.,33, 161±200.

Shapiro, G.I. and Hill, A.E. (1997) Dynamics of densewater cascades at the shelf edge. J. Phys. Oceanogr.,27, 2381±2394.

Smith, P.C. (1975) A streamtube model for the bottomboundary currents in the ocean. Deep-Sea Res., 22,853±874.

Zeng, J. and Lowe, D.R. (1997) Numerical simulation ofturbidity current ¯ow and sedimentation: I. Theory.Sedimentology, 44, 67±84.

Manuscript received 3 June 1998;revision accepted 24 February 1999



Documents

On the ignition of geostrophically rotating turbidity currents