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Conditions for the ignition of catastrophically erosive turbidity currents
Marine Geology, 46 (1982) 307--327
307
Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
CONDITIONS FOR THE IGNITION OF CATASTROPHICALLY EROSIVE TURBIDITY CURRENTS
GARY P A R K E R
St. Anthony Falls Hydraulic Laboratory, Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55414 (U.S.A.) (Received January 22, 1981; revised and accepted July 22, 1981)
ABSTRACT Parker, G., 1982. Conditions for the ignition of catastrophically erosive turbidity currents. Mar. Geol., 46: 307--327. The basic impediment to a clear understanding of eroding and depositing turbidity currents has been the lack of a proper formulation of bed sediment entrainment. This problem is addressed herein: a detailed analysis indicates that the Bagnold criterion is necessary but insufficient for self-sustaining turbidity currents. The analysis reveals two possible equilibrium states, a relatively low-velocity ignitive state, and a high-velocity, dense catastrophic state. A stability analysis indicates that the ignitive state is unstable; flows below it invariably die out, and flows above it "ignite", i.e. accelerate and entrain sediment to the catastrophic state, which is stable. The ignitive state thus defines the criterion for a self-sustaining turbidity current. Estimates of the catastrophic state suggest that it is highly erosive and competent to scour out submarine canyons. INTRODUCTION
One of the greatest mysteries concerning turbidity currents is their apparent ability to erode through shelf sediments and carve out magnificent submarine canyons. Even in the essentially depositional environment of the fan below the canyon, enough erosive power often remains to maintain a well-defined channel often bounded by natural levees. In comparison to subaerial rivers, the down-slope impelling force of gravity acts on the solid phase only and is vastly reduced in the subaqueous environment. In addition, a true turbidity current operates on a constraint more severe than its simpler cousin, the density current; it must generate enough turbulence to hold its sediment in suspension. If it cannot, the sediment will deposit and the current, having lost its raison d'etre, must vanish. It is easy to imagine self-sustaining turbidity currents, the solid phase of which is mostly cohesive clay; at sufficiently low concentrations the fall velocity of such material is so low that it is not readily deposited. Such currents have been observed in reservoirs and have been reproduced experimentally (Stefan, 1973). They probably also occur in the ocean (Stow and Bowen, 1980). 0025--3227/82/0000--0000/$02.75 © 1982 Elsevier Scientific Publishing Company
308
However, abundant evidence on submarine canyons and fans suggests that highly erosive turbidity currents occur even in environments where the only available material is medium to coarse silts and sands, or coarser (Shepard and Dill, 1966). Such currents have even been created artificially by man in Lake Superior (Normark and Dickson, 1976). Many experiments have demonstrated that turbulence alone is n o t sufficient to keep these non-cohesive materials in suspension. For example, Einstein (1968) observed the clarification of water laden with silt (grain size Ds -~ 0.01 mm) flowing over a gravel bed. Material readily fell into the gravel matrix and could n o t be re-entrained subsequently. A non-cohesive turbidity current must have the ability to reentrain sediment from the vicinity of the bed. Such a current is essentially depositive if it deposits more than it entrains; in the opposite case it is essentially erosive. The state of the literature on the mechanics of turbidity currents must be viewed in this light. The analysis in most studies is hardly removed from that appropriate for density currents, i.e. little more than lip service is paid to the mechanics of the sediment phase. Bagnold (1962) did consider the solid phase and was able to obtain a simple approximate energy constraint on turbidity currents. Where U is mean down-channel flow velocity, S is b o t t o m slope (or more properly energy slope), and vs is sediment fall velocity, a current is supplied energy at a rate larger than that consumed by maintaining the sediment in suspension only if: US -->1 us
(1)
Bagnold concluded that if eq.1 is satisfied, a self-sustaining "auto-suspension" can occur.
Bagnold's work is of considerable significance, b u t while eq.1 may be a necessary constraint, it is by no means sufficient, for there is no mechanistic guarantee that the energy available to suspend the sediment can in fact be contributed toward this end. For example, the analogy to eq.1 for sedimentladen river flow is: US - -
R C vs
>1
(2)
where C is mean volumetric sediment concentration and R is sedimentsubmerged specific gravity (usually near 1.65). In Einstein's experiments on clarification, the number on the left-hand side of eq.2 becomes very large at low concentrations, y e t eventually the water clarifies completely. The reason for this is that t h e energy actually expended to suspend grains is turbulent energy, and turbulence does not suspend specific grains. It counters the flux of grains to the bed due to their fall velocity by producing a net upward wafting of grains entrained from the bed. If each grain, upon settling, cannot be re-entrained at the bed, the stock of grains available for suspension by turbulence gradually diminishes and the water clarifies. In simplified terms,
309
energy available to suspend is contributed to this end only to the extent that grains are available to be suspended. Clearly the missing feature of turbidity currents is a description of entrainment and deposition at the bed. Clearly the morphology of submarine canyons and fans is crucially d e p e n d e n t on this feature. Nevertheless it is precisely this feature that has been left o u t of nearly every treatment of the fluid mechanics of the subject. The comprehensive experimental and analytical study of Ashida and Egashira {1975) pays considerably more attention to the stratifying role of sediment than can be done in the c o n t e x t of the layer-integrated treatment used herein, but is still limited to the case for which neither net deposition nor entrainment of sediment occurs. Recently Chu et al. (1979 and 1980) have taken the important step of recognizing the problem, b u t their attempts to resolve it have been either qualitative (1979) or cumbersome and involving erroneous assumptions such as "Once the sediment particle is in suspension, it will always remain in suspension" (1980}. The present analysis seeks a simple b u t mechanistically well-founded treatment of erosion and deposition. In particular a critical "ignition" condition for the onset of a fast, highly erosive state is identified. This state is conjectured to be responsible for submarine canyons. A HEURISTIC MODEL FOR "IGNITION"
Consider an idealized continuous turbidity current o f infinite longitudinal extent, flowing down a slope with angle 0, that is uniform in the lateral and downstream directions b u t is free to vary in time (Fig.l). At time t = 0, the current is assumed to have initial layer-averaged velocity and volumetric concentration Uo and Co and is flowing over a bed of loose non-cohesive material with grain size Ds similar to that in the turbidity current. What will be the future of this current? Suspended sediment is constantly falling o u t of the current at the rate (flux} vs cb, where Cb is a mean volumetric concentration of suspended sedim e n t near the bed. Suppose the turbidity current is so slow that the stress it exerts on the bed is less than that required to entrain sediment into suspension, or at least so that it produces a volumetric rate of sediment entrainment per
x
h
ERODIBLE BED
Fig.1. Definition diagram.
310
unit area from the bed Xs which is less than the settling rate vs c b. In time, then, the turbidity current experiences a net loss of sediment, and layeraveraged concentration C decreases. Consequently the impelling force per unit fluid mass due to the downstream c o m p o n e n t of gravity acting on the current g R C sin 0 decreases, and the current decelerates. Bed stress is lowered further, and xs droPs further and more of the remaining sediment settles out, reinforcing the deceleration and causing the current to eventually vanish. On the other hand, suppose that initially Xs > v~ co. The current entrains more sediment, C and the impelling force g R C sin 0 increase, the current accelerates, bed stress increases and more sediment is entrained in a selfreinforcing cycle that allows for the development of a high-speed, highly erosive turbidity current. This "catastrophic" current would eventually be limited by other factors such as any limit on the entrainment rate, which w o u l d l i m i t the growth of the impelling force, or perhaps by damping of the turbulence at high concentrations. The implication is the existence of a set of critical values for velocity and concentration below which the turbidity current dies and above which it grows. The terminology "ignition c o n d i t i o n " is introduced herein in analogy to the critical values of oxygen and fuel concentration and temperature necessary for the onset of combustion. DYNAMIC EQUATIONS
The present analysis is intended to be extremely simplified in its formulation so as not to obscure the fundamental role of erosion and deposition in the process of ignition. It is assumed that the mean concentration of sediment c may be treated as a parameter sufficiently smaller than unity to allow the use of the Boussinesq approximation. Layer-averaged mean velocity U and volumetric concentration C can be defined in terms of local mean values u and c as follows: h
h
u=-ff o
o
where y denotes the coordinate normal to the bed and h is defined such that u ( h ) / U and c ( h ) / C are appropriately small parameters, e.g., the largest of the
two equal to 0.01 (Fig.l). Forms for the layer-integrated equations of energy and fluid mass balance reduce with the aid of the boundary layer approximations to: -- ~1 - - U 2 h + - ~ 2
0t
2
0x
U
U2
=---oL3 UgRcosO--
2
+ o~4 g R C h U sin 0 -- (u2, + u2,i) U -- R C g vsh cos 0
Oh --
3t
O +
--
3x
Uh
=
e
U
0x
Ch 2
(3)
(4)
311 for flow of constant width down a straight channel of constant slope. In the above, x denotes the downstream coordinate, 0 denotes bed angle, u. and u.i denote bed and interfacial friction velocities, respectively, and e is a coefficient of fluid (rather than sediment) entrainment. The parameters a l to a4 are order-one shape factors; e.g.:
-h Ol 1
u2dy
--
U2
so t h a t all shape factors are equal to u n i t y for the case where u is uniform in the vertical. The equation of sediment mass balance takes the layer-integrated form: OC --
0t
+
~
ax
as
UC
= F b
(5)
where as is another shape factor. In the above, F b = F ( b ) where: F ( y ) = c' v' - - v s c
(6)
denotes the mean vertical flux of suspended sediment and y = b is an elevation conveniently close to the bed. In eq.6, c' and v' denote the instantaneous turbulent fluctuations of concentration and vertical velocity about their mean values at level y; the overbar denotes averaging. F ( y ) is seen to be the net result of two components due to turbulent mixing (upward in this case) and downward flux due to the fall velocity. Where it is necessary to attach numerical values to the shape factors, all are set equal to u n i t y as a convenient expression of ignorance. A formulation of the term Fb for eroding or depositing flows was developed by Parker (1978); t h a t formulation provided the original idea for this paper. Therein it is assumed t h a t the term c' v' evaluated near the bed, which specifically describes the rate of sediment entrainment, can be represented as a function ×s of bed shear velocity u. and other parameters: c' v'l b --- Xs
(7)
An evaluation of ×s can be obtained as follows from data on steady uniform suspensions. For each flows, F = 0 everywhere and it follows that: ×~ = v~ cb
(8)
Thus defining a dimensionless sediment entrainment function E~ such t h a t vs E~ = ×~ it follows that: Es = Cb implying t h a t entrainment just equals deposition.
(9)
312 Itakura and Kishi (1980) have developed a relation for c b uniform suspensions in open channels; it takes the form: Es = k,
k2 R~ #2
(=Es)for steady, (10)
where R~ #2
1
k3 exp ( - - 1 a 2) A(a) =
exp ( - - l x 2)
dxJ
o
and v~ Rf
-
V/('-~g
Ds
(11)
U~ # = -vs
(12)
The parameters k l, k2, and k3 are constants taking the values 0.008, 0.14, and 0.143, respectively. In general, then, Es is a function of # and Rf. Implicit in eq.10 is a threshold dimensionless shear velocity #t~ below which entrainment vanishes. In Fig.2 Pth is plotted versus Rf for 0.1 < Rf < 0.5 (0.056 mm < Ds < 0.35 mm at 20°C). The values are in general agreement with the criterion #th -~ 1 suggested by Engelund (1965) and Bagnold (1966). 5
,
,
,
,
I
I
I
I
P-th21 0 0. t
I 017
Rf Fig.2. Threshold dimensionless shear velocity Pth as a function of Rf.
313
Itakura and Kishi verified eq.10 with the use of measurements of Cb, with b arbitrarily set equal to 0.05 h, where h is the depth of flow. The measurements cover the range in grain size Ds from 0.08 mm to 0.35 mm, or using the plot of R~ versus Rp = R ~ s ( D s / v ) in Fig.3, 0.16 < R~ < 0.5 at 20°C (v denotes the kinematic viscosity of water). This range is appropriate for the future consideration of oceanic turbidity currents in non-cohesive material. The range of u is from near unity to a b o u t ten. Equation 10 is obviously only valid if the bed is completely covered with suspendible material. If the stock of such bed material is depleted to the point of partial exposure of non-suspendible material such as bedrock, the entrainment rate is reduced below that predicted by eq.10. Many researchers have approached the problem of non-equilibrium suspensions by using as a b o t t o m condition the assumption that Cb itself is the same function of flow conditions as found for equilibrium conditions. Parker (1978) illustrated that this is erroneous and showed that only the entrainment rate Es can be taken to be a specified function of local flow conditions for non-equilibrium suspensions, e.g., eq.10. Thus eq.5 reduces with the aid of eqs.6 and 7 to: aC --
at
a
UC =v~ [ E s - - C b ]
+ - - a s
ax
(13)
where herein Es is assumed to be given b y eq.10. Equation 13 provides a specific evaluation of the relative contribution of erosion and deposition. Certain auxiliary relations are necessary for a layer-integrated treatment. In particular, parameters such as u., u.i e, and Cb must be related to layeraveraged parameters U and C; to wit :
10 ] I I! I===== i
= ==:~=I=I
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JIIJl JII]l
I l[Illll
Illl]
...............
I
I IIIIIL I I][IIII
i I
Rf
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I IIIllll
[lll[tll
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I
[lllll
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I II
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i
lllN!
13~rl
I II111[I
I I I1[I
lJ[llil
l lltllll I III]II]ll
I I I ] ~ ~"
J
19¢111]1 J IIIIIIII
IIIlll
I IIIII IIII11
I
I
l
[1[111
111111
I I1111[ II1111IIIIII f i iiiN I~llll
I 10-]
nit
I I
I 10°
Jiiii iI i[iiiii ~ Jllll I IIII]1 Ill 101
IIIII 10 2
RF, Fig.3. Rf as a function
of Rp.
; Ill
II[IFI I1[111
10-~
i0-2
~I~III I I Illl l l J I IJll IIII
I ]
]III] I00 ,
I I
~iii;
iI iiii I
IIII
; i iiFii]i I
I IIIIIII
I IIIIII]1
I111IIII nroI1~o, lllll ....
I0 s
10 4
I II
I II
! !!!! ..... I III
1/11 10 5
314
u, 2 =lf
U2;
u 2,i =
1 ~fi U 2
(14)
(15)
Cb = r C
where f and fi are bed and interfacial friction coefficients and r relates bed concentration to layer-averaged concentrations. The parameter e is a function of Richardson n u m b e r R i = R C g h / U 2 (see e.g., Turner, 1973). The parameter f can be estimated from a knowledge of the bed resistance of alluvial rivers (Vanoni, 1975). Egashira and Ashida (1979) have recently developed comprehensive relations for estimating fi. An ad-hoc m e t h o d for estimating r for the present analysis can be developed using the well known Rousean distribution for open channel suspensions: c
where 77 = y / h , % = b / h , a n d Z = 2 . 5 / p is the Rouse number. Setting 'Tb = 0.05, then: __ 7?/ l __ Vb1 2.5/~ d~) ro:l/( i [ l--;---C-. 0.05
where the subscript denotes the fact that the relation applies to open channels. The above equation can be accurately approximated by: r0 = 1 + 31.5g -L46
(16)
f o r ~ > 0.5. Herein it is assumed that for turbidity currents: (17)
r = o~ 6 r o
where a6 is another order-one shape constant. Setting all shape factors equal to unity in a first analysis, the complete set of equations reduces to: -- - U2h + -- U ~2 t 2 3x
2
=--2
U g R c o s O - - C h 2 + R g C h U sin O 3x
1 ---2 f(1 + 7)U 3 - - R g C h v s cos 0 Z
Oh --
~t
OUh +
-
Ot
- eU
~x
OCh -
(18a)
3UCh +
Ox
where 7 = f i / f .
- v~[Es - - r o C ]
(18b)
(18c)
315 EQUILIBRIUM TURBIDITY CURRENTS When water entrainment e is negligible, eqs.18a, b and c admit solutions for steady, uniform equilibrium turbidity currents with constant thickness h. This is strictly valid only for Ri greater than about unity. Herein e is always set equal to zero for simplicity; also 7 is neglected as well, implying t h a t bed stress by far dominates interfacial stress. These simple assumptions can be replaced in future more detailed analyses. Equations 18a--18c reduce to: vs Rg Ch sin 0 = u .2 + R g C h c o s O -~-
(19)
Es = roC
(20)
Eliminating C from these two equations, the constraint for equilibrium is seen to reduce to the form: F ( p ) = FI(p ) -- FR(P)
R~ ro
--
where
~E~
FI(p) = ~ R~ ro Fa(p) = p2 +
(22a)
Es PB
(22b)
R~ro In the above: h sin 0 =~
Ds
(23)
and the Bagnold limit PB is given by:
PB = - S
(24)
where S = tan 0 is the bed slope. The term FI(p) quantifies the impelling force acting on the current; it is either positive when p is greater than Pm or zero otherwise. The term FR(p) quantifies the resistive force, with a frictional contribution p2, and a contribution equal to a fraction PB/P of the impelling force which represents that needed to keep the sediment in suspension. FR(P) is always positive. An examination of eq.21 reveals that no positive solution for p can exist when p is less than the Bagnold limit/~B given by eq.24. Not surprisingly, this relation reduces with the aid of eq.14 to the Bagnold "autosuspension" criterion, eq.1. However, this criterion is not sufficient for an equilibrium turbidity current; the extra constraint is clearly t h a t F1 = FR.
316
To describe this condition, it is of use to consider first the case where /~B = 0. F R (#) is then parabolic in p and plots as a straight line on log-log paper. On the other hand Es increases sharply with values of p only slightly greater than #th ; for higher values the increase is less steep. Thus on log-log paper FI curves downward. The implication, shown on Figs.4a and b, is that if there is any non-zero equilibrium solution for/~, there are two. In Figs.4a and 4b, Fi(p) and FR(#) are plotted as functions of g for the special case Rf = 0.5 (Ds = 0.24 m m at 20°C) at two values of @. For the lower value ~b = 250, the curves never meet and no equilibrium solution exists. For the higher value @ = 352, two equilibrium solutions to eq.21 exist; these may be labelled p, and p: where p, < P2. In this case p, = 5.5 and P2 = 10.0. In so far as U = x / ~ v~ p, C = E Jr o and bulk current density Pc equals p(1 + RC) where p is the density of water, it can easily be verified that U=, C2, and pc2 are each invariably greater than U,, C~, and Pc ,, respectively (approximating f as a constant). For this reason the upper equilibrium is termed the "catastrophic equilibrium." For reasons to be described later, the lower equilibrium is termed the "ignition equilibrium." Crudely estimating f = 0.004 (as is done t h r o u g h o u t this paper; Komar, 1977), it can be f o u n d that for the above example U~ = 3.8 m/s, C~ = 0.0215 and pc~ = 1.035 g/cm3; likewise U2 = 7.0 m/s, C: = 0.0710 and pc2 = 1.117 g/cm 3, When no solution exists, it is because for that value of ~ no rate of entrainm e n t consistent with eq.10 can supply enough sediment to maintain an impelling force large enough to overcome frictional resistance. However, it is apparent from eq.21 that for prescribed p and Rf, @ can always be increased 4XlO 2
iO3
--
/
102
102
F I ,F R
IOt
F t . FR
I
/~
Fig.4a. FR(U ) and
....
i01
qom 3XlO-J
F~ 1/~)
Io ~
FI(u ) f o r
7XlC~
I 2
R f = 0 . 5 , ~ = 2 5 0 a n d UB = 0.
F i g . 4 b . F R ( P ) a n d F I ( u ) f o r R f = 0 . 5 , ~ = 3 5 2 a n d uB = 0.
,a
I0
2XlO
317 ,,,,.
,o4
~2~726
103 F I ,F R
-
~
120 i01 I
I0
I
I0
2
Fig.4c. FR(~u) and FI(~ ) for Rf = 0.5, ~ = 1 2 5 0 and gB-" 0.894. The dashed line denotes a more realistic qualitative shape for Fi(g) at high concentrations. I
,0 ~
v
,
,
,
i
_
o0
10 2
i0 t
m OI
i
I
Rf
i
J
0.7 L
Fig.5. ~ , as a function o f Rf and ~B.
until that equation is satisfied. Thus a lower bound 4, exists below which equilibrium turbidity currents cannot exist at any p. In Fig.5, 4, is given as a function of Rf for 0.1 < Rf < 0.7 (0.056 m m < Ds < 0.35 m m at 20°(]). In practical terms, this means that for a given grain size Ds and downstream channel inclination sin 0, current thickness h must be greater than the value h,, where from eq.23: h,-
D~ 4, sin 0
(25)
318
For example, on a slope S = 0.05 (0 = 2.86 °) for 0.18 m m (Rf = 0.4) material, h must exceed 0.79 m. The case of greater interest is the one for which two solutions exist. For any value of R~, there exists a value p, such that Pl < P. < P2 for all values of $. Indeed, it can be verified from eq.21 that/~, is the value of/~ at which = $ ,. The parameter p. thus divides the p-Rf plane into a region where only ignitive equilibrium is possible, and one where only catastrophic equilibrium is possible; this is shown in Fig.6. It can be seen from Fig.6 that for the given range of Rf, P2 can never be less than 7.4, and C2 can never be less than 0.040 in the range of catastrophic equilibrium. Indeed, for Rf = 0.7 (D s = 0.35 mm), using the previously quoted estimated for f, the lower bounds on velocity U2 and current density Pc2 are 8.7 m/s and 1.07 g/cm 3, respectively; for R~ = 0.1 (Ds = 0.056 mm) these lower bounds are 0.66 m/s and 1.10 g/cm 3. The densities in particular are rather high; at concentrations as high as this the structure of the flow may be so modified as to invalidate the use of the equations presented herein. Thus, the present t h e o r y may n o t describe the catastrophic region too accurately. Under conditions of severe stratification, for example, damping of the turbulence may cause FI(p) to flatten out rather more rapidly in p than the functional form of eq.22a; this would lead to a somewhat lower value of p~. Another p h e n o m e n o n which might also act to lower/~2 is depletion of the i
i
i
v
CATASTROPHIC
~B=5
IO #-B=I dividing line for /a.B=O IGNITIVE I
I
I
0.7
03 Rf Fig.6. Plot o f / z , as a f u n c t i o n o f R f a n d PB.
319
supply of available suspendible bed sediment by bedrock exposure, for example; this reduces Es as discussed previously. A suggestion of these effects is indicated by the dashed line in Fig.4c. (The figure is explained below). None of these modifying factors could be expected to modify the essential result of the existence of the catastrophic state. In addition, through most of the range of occurrence of the ignitive equilibrium concentrations are low enough to suggest the validity of the present treatment, and some detail in calculation is warranted. In Fig.7a the solution p~ t o eq.21 is given as a function of @ with Rf as a third parameter; from this diagram and previously presented relations auxiliary parameters such as CI, UI, and Pc1 can be computed, as was done for the example of Fig.4b. The above analysis only applies for the limiting case ~B = 0. However, all of the concepts generalize quite readily. For example, consider a current 40
I
I
I
/x 2
lI
I
Il
il
I
I
i
I
l
I
I
I I Ill
I
I
I
I
I
I III
I
I
i i i i1
0.4 0.50.6 0.7
0.3
0.2
I !--_
10
I
I I lllJ
' Rf=O.I
i
I
II
II
t
I
lI
CATASTROPHIC
/ I
!
I
I
Rf =0.1 FB=I I
,
0.2 i
,
,,,,,
IOI
,d
iO2
IO4
i0 5
o/ Fig.7a. Ignitive s o l u t i o n . 1 and c a t a s t r o p h i c s o l u t i o n u2 as f u n c t i o n s o f $ a n d R i f o r #B = 0 . 30 .Rf:OI /-L2
I
l/
I
'''[
Q3
0.2 /
/ i
/ I
IO
'1
Q4 Q50.6Q7
!/
1/
t I
1/
//
I / /
CATASTROPHIC
/
IGNITIVE
:
Rf=O.I
X
0.2
07
p-B=O i 0.4
'
I0
'
'
' ' ' '
I0
2
I ,
, , l .
,
3 IO
qs
,
,
, , J ,
4 I0
5 I0
Fig.7b. Ignitive s o l u t i o n ~, and c a t a s t r o p h i c s o l u t i o n u2 as f u n c t i o n s o f $ and R f f o r .B --1.
320 I00
I
I
I
I
I Ill
I
Rf= 0.1
I
0.2
I
I
0.3
/ /
I
/
/x 2
I
I
I
/
I
II
i
I
I
/
/
I
I
I
I I I
/I
!
//
/
I
I
I
I 1l
I
/
I
#
I
0.4 0.5 0.6 0.7 I ," / II
tI
CATASTROPHIC
IGNITIVE I0 I-Xl
/.~B= 5 I
2
I
I
I
III
[
102
I0
I
qJ
I
i
I I II
103
4
I0
Fig.7c. Ignitive solution u~ and catastrophic solution P2 as functions of ~ and Rf for PB=5.
with a thickness h = 6 m flowing down a c a n y o n slope with S = 0.05, carrying 0.24 m m sand (R~ ~ 0.5). Using f = 0.004, PB is f o u n d to be 0.894 (/,lth is 0.83); ¢ = 1250. In Fig.4c, F , ( # ) and F R (#) are pl ot t ed; again eq.21 has both ignitive and catastrophic solutions. For the ignitive solution, Pl = 1.83, a value of dimensionless bed shear velocity t hat is considerably in excess of the Bagnold limit. Values of C, Pc, u. and U for this case are 0.00131, 1.00216 g/cm 3 , 5.71 cm/s and 1.28 m/s, respectively. For the catastrophic case P2 = 72.6, a value t hat is far outside the range of validity of eq.10 and p r o b ab ly unrealistically high (it yields a flow velocity U of 50.6 m/s). The lower b o u n d ~ . for the existence of any equilibrium turbidity current is shown on Fig.5 for the cases PB = 1 and PB = 5 in addition to PB = 0; h. can again be calculated from eq.25. On Fig.6 the value of p. dividing the ignitive and catastrophic range is p l o t t e d versus Rf for PB = 0, 1, and 5. It is seen th at ~. and p, increase with #B, and thus in general increase with decreasing slope, as indicated by eq.24. The solution p, for the ignitive state is given as a funct i on of $ and Rf for PB = 1 in Fig.7b and PB = 5 in Fig.7c. A p o in t o f some later importance concerns the local exponents, or slopes on log-log paper, NI and NR of the functions F I ( p ) a n d F R (P), respectively at the ignitive and catastrophic state. These e x p o n e n t s are given by the expressions: ~/ dFa
P dFI N~ = - - - ; F x dp
NR
-fir
dp
where p = p, or P2. After some manipulation using eqs.22a and 22b, it is f o u n d th at these e x p o n e n t s take the form: N~ = M
(26a)
321
NR=2
1--
+Mf
(26b)
where
M-
P dEs
p dro.
E s dp
Ps = -P
ro d p '
(26c)
and p again equals Pl or P2. It can be shown from eqs.10 and 16 that M > 0; the Bagnold criterion requires that ~ < 1. It can be seen from Fig.4c that on log-log paper F1 is steeper than FR at the ignitive root, implying that N I -- NR > 0 or: M(1--~-)--2
--
>0
(27a)
On the other hand, at the catastrophic r o o t FR is steeper than FI, in which ease N < NR and: M(1--f)--2
--
<0
(27b)
The nature of the functional forms in eqs.22a and 22b insures that this result generalizes, so that for all values o f R f , Ps, and ~ , if both the ignitive and catastrophic states exist then the inequality (27a) holds for the former state, and (27b) holds for the latter state. These two inequalities play an important role in determining the stability of the ignitive and catastrophic equilibrium. TEMPORAL
INSTABILITY O F T H E IGNITIVE S T A T E
The significance of the ignitive state is e m b e d d e d in its name. It is of use to reconsider the idealized continuous current of the heuristic model; it is longitudinally uniform b u t free to vary in time. Suppose that at t = 0, U < U1 and C < C~ ; under these circumstances the turbidity current must gradually drop its sediment and die in time. On the other hand, if initially U > U1 and C > C1 the current will both accelerate and become denser until the catastrophic state is reached. Thus the ignitive state delineates the conditions necessary to ignite a self-sustaining turbidity current. In the language of stability, the ignitive equilibrium is unstable, and bifurcates to either the null or the catastrophic state, both of which are stable to infinitesimal disturbances. A graphical illustration of this that can be immediately appreciated is given in Fig.8. If/~ is slightly below p, both Fi(p) (impelling force) and FR(U ) are lower than their (equal) values at Pl ; however, since the FI(p) curve is steeper, i.e. Nx > NR, the impelling force FI(p) drops below the resistive force FR(P). Resistance dominates; the current decelerates, drops sediment, loses more impelling force, etc., until the null state is reached. By the same token, when N~ > NR, FI(P) > FR(P) for p > p~, a n d a cycle of acceleration and entrainm e n t occurs until the catastrophic state is reached.
322
F,,~
FR(/Z2*A/z ) F I (#2+Z~#) F I (/J,2-AU.) FR(/Z2-&/z) FI (]~1 ,5~.) FR(/J- I +A/z)
FR(/Z ~-Z~/~) F I (/J i -lk/z }
F2
/
Fig.8. Graphical indication of instability of ignitive root and stability of catastrophic root.
On the other hand, at the catastrophic state NR > NI, i.e., F R is steeper than FI. If p is slightly less than P2, FI(p) > FR(p), and the current accelerates back to catastrophic equilibrium. A similar result holds for p slightly greater than P2. Thus this state is stable. A more formal proof of this can be obtained from eqs.18a through 18c, again assuming that e = 7 = 0; for the postulated spatially uniform but temporally varying flow of the heuristic model, t h e y reduce to: a U = g R Ch sin 0 h a--t
1 U2 vs 2f - - g R Ch cos 0 - U
aC h - - =vs [Es--ro C]
at
(28a)
(28b)
and h =h o
where h o is a constant. Herein f is also assumed to be constant. These equations are now perturbed about equilibrium; that is: U=U,~ + u ' C = C ~ + c'
where a = 1 denotes the ignitive equilibrium and a = 2 denotes the catastrophic equilibrium. Assuming the primed perturbations to be small, eqs.28a and 28b linearize to:
au' c' + f ~u' ] --f~u' ba~ ho-~=gC, ho I(I--~')~-
(29a)
323
=
h°~t
Us
(295)
Ca
where {- = PB/g~ and M is given by eq.26c evaluated at # = p~. Defining and utilizing the equilibrium eqs.19 and 20 at p = p~, dimensionless versions of eqs.29a and 29b can be obtained: c = c'/C~, u = u'/U~, T = (UJho)t
~u_f*
~T ~c
~T
2
Iv(1 -- {-) -- 2 (1 -- ~{-) u]
(30a) (305)
- (3[Mu - - c]
where: f * = f / ( 1 -- {-}, and
[3-
Vsro
us
provides a measure of the "ease of deposition" of the suspended material. For the limit of extremely fine sediment, which deposits only very slowly, approaches zero and the turbidity current becomes a simple density current. The stability of eqs.30a and 30b can be tested by assuming perturbations of the form: u = Uo e x p ( ~ r )
and
c = c0exp(~T)
(31)
where Uo and Co are constants denoting initial perturbation amplitude. Clearly if ~, < 0, u and c decrease to zero in time, and the equilibrium is stable; if ~ > 0 the perturbations grow and instability results. Inserting eqs.31 into eqs.30a and 30b, it is found that: ~1 f* ( 1 - - {")Co -- X +
--2
f*
Uo = 0 (32)
(~ + [3)Co - - [ 3 M u o = 0
The above equations can be satisfied in general only if Co = Uo = 0, yielding a trivial solution, or: 1
3
=0 X+ ~
--M~
yielding the dispersion relation: ~2 + B ~ . _ D = O
where
(33a)
324
B=[3+ D=-~f3f
1--
f)
-
=~f*
-
(NI -- NR )
(33b)
Note that eqs.26a and 26b have been used to obtain eq.33b in terms of NI and NR. The roots of eq.33a are thus: 1
+
2~f*(NI--NR) ~)
(34)
1 Two cases are apparent; that for which ~ = PB/P, < 2/3 and that for which 2/3 ~< ~ < 1. In the former case (1 -- 3/2 ~) is positive. When N1 > N R , one r o o t X is positive and instability results; when NI < NR, both roots are negative and equilibrium is stable. Recalling eqs.27a and 27b, this indicates that the ignitive state is unstable, whereas the catastrophic state is stable. The remaining case applies only to the ignitive state because as can be readily deduced from eqs.26a and 26b, NR can never exceed NI for 2/3 ~< < 1. It is seen that for NI > NR one r o o t X is positive regardless of the sign of [~ + (1 -- 3/2 ~)f*], thus again implying that the ignitive equilibrium is unstable. This completes the stability analysis with the establishment of the desired result. It may be noted that for the unstable ignitive case (X > 0), Co and Uo must have the same sign according to eq.32. Thus if Uo < 0 then Co < 0 and the ignitive instability veers toward the null state; if Uo > 0 then Co > 0 and the stable catastrophic state is approached. IMPLICATIONS FOR SUBMARINE
CANYONS
The concept of ignition can be tentatively applied to the origin of submarine canyons. Silt and sand presumably collect at the head of submarine canyons due to direct supply from rivers or littoral drift. Occasional smallscale slumping may produce discontinuous slug-like turbidity currents of the type studied in the laboratory by Middleton (1966); they do not ignite, and thus they deposit sediment on the bed of the canyon as they progress downward and decay. Such turbidity currents would thus lay down a layer of "fuel" on the canyon bed to be utilized by a catastrophic current, perhaps triggered by a massive slump or large amounts of continuous sediment infeed from an episodic river flood. In order for such a current to ignite there must be, at least initially, an extra supply of sediment on the bed to be entrained and utilized by the current to accelerate. If the bed is still essentially covered with suspendible sediment when the equilibrium catastrophic state is reached, no further entrainment will occur and the canyon bedrock is n o t eroded. The more interesting case, however, is
325
where at equilibrium most of the supply of suspendible bed sediment is entrained, leaving bedrock overlain by thin gravel deposits too coarse to suspend. The catastrophic current would likely be capable of moving the coarser material which would act as tools to corrade the bedrock. The bedrock would also be subjected to intense sandblasting. Some evidence of this erosive capability can be obtained as follows. In a previous example a current with thickness h = 6 m flowing down a slope S = 0.05, carrying 0.24 m m sand (Rf ~ 0.5) over a similar bed was considered. For this example ~ = 1250 and PB = 0.894; at ignition p, = 1.83. The values of u, and C at ignition are 5.71 cm/s and 0.00131; if all the sediment in the current were suddenly precipitated to form a loose layer on the bed with a porosity of 0.35, the deposit would have a thickness of 1.2 cm. Maximum grain size D ~ t h a t could be moved by the ignitive current can be estimated from the Shields' relation; U 2, -
-
Rgnmax
- 0.06
where the value of 0.06 is conservatively high. It is f o u n d that: Dmax = 3.4 m m As previously noted, the value of p: c o m p u t e d at the catastrophic equilibrium is excessively high and probably not realistic. A conservative estimate of the catastrophic state can be obtained by using the lowest possible value of p in the catastrophic range for Rf = 0.5. This value is p = 8.9, yielding values of C, Pc, u., U and D ~ x of, respectively, 0.0572, 1.094 g/cm 3, 27.8 cm/s, 6.21 m/s, and 79.4 mm. Sudden precipitation of the suspended sedim e n t would yield a layer 52.8 cm thick. Subtraction of this thickness from the thickness of the precipitated layer of the ignitive state yields a lower bound on the potential depth of scour in going from the ignitive to the catastrophic state (realized only if the fine material is available to scour) of 51.6 cm. A current with mean velocity of 6 m/s, a scour potential of about 50 cm, and the ability to move up to 80 m m cobbles is likely c o m p e t e n t to erode canyons. The concept of ignition would seem to have some relevance to a reported a t t e m p t by divers to " k i c k " a turbidity current into existence, which ended in a failure that may have had a salutary effect on their health (Dill, 1964). DISCUSSION
In its present form the analysis is rather crude. Only continuous currents have been considered, and a long list of simplifications such as the neglect of water entrainment and interfacial stress has been introduced. Thus, the analysis provides in quantitative terms only an estimate of the condition for ignition, and provides only qualitative results and rough quantitative lower
326
bounds for the catastrophic state. Any application should be undertaken only with a clear awareness of these limitations. For example, currents predicted to be "ignitive" by the present analysis may often be supercritical, in which case water is entrained from above. The consequent dilution would tend to decelerate the current. The condition for true ignition must be somewhat more stringent for such currents; the accelerative effect due to sediment entrainment must be stronger than the decelerative effect due to water entrainment. On the other hand, the modifications necessary to obtain a quantitative theory applicable to even the supercritical and catastrophic ranges would not seem overly difficult in light of the present knowledge of sediment transport mechanics and stratified flow. Such a theory, when cast in the form of eqs.18a to 18c, would have the capability of predicting depositing or eroding backwater (or frontwater) curves from the canyon head to the extremities of the fan. The results of such a theory would be of value to sedimentologists. ACKNOWLEDGEMENTS
Long discussions with J. Shaw and S. Egashira during the several years of contemplation before the above picture crystallized are gratefully acknowledged J. Akiyama provided a useful comment. This research was n o t funded by any agency, and thus has a benefit-cost ratio of infinity. St. A n t h o n y Falls Hydraulic Laboratory graciously denoted the support services necessary to produce this paper. REFERENCES Ashida, K. and Egashira, S., 1975. Basic study on turbidity currents. Proc. Jpn. Soc. Civ. Eng., 237: 37--50. Bagnold, R.A., 1962. Auto-suspension of transported sediment; turbidity currents. Proc. R. Soc. London, Ser. A, 205: 315--319. Bagnold, R.A., 1966. An approach to the sediment transport problem from general physics. U.S. Geol. Surv., Prof. Pap., 422-I:I1--I37. Chu, F.H., Pilkey, W.D. and Pilkey, O.H., 1979. An analytical study of turbidity current steady flow. Mar. Geol., 33: 205--220. Chu, F.H., Pilkey, O.H. and Pilkey, W.D., 1980. A turbidity current model. In: Civil Engineering in the Oceans, IV. Am. Soc. Civ. Eng., pp. 416--432. Dill, R.F., 1964. Sedimentation and erosion in Scripps Submarine Canyon head. In: R.L. Miller (Editor), Papers in Marine Geology, Shepard Commemorative Volume. MacMillan, New York, N.Y., pp.23--41. Einstein, H.A., 1968. Deposition of suspended particles in a gravel bed. Proc. Am. Soc. Civ. Eng., 94(HY5): 1197--1205. Engelund, F., 1965. Turbulent energy and suspended load. Basic Res. Progress Rep. 10, Hydraulic Lab., Tech. Univ. Denmark, pp. 1--9. Itakura, T. and Kishi, T., 1980. Open channel flow with suspended sediments. Proc. Am. Soc. Cir. Eng., 106(HY8): 1345--1352. Komar, P.D., 1977. Computer simulation of turbidity current flow and the study of deepsea channels and fan sedimentation. In: E.D. Goldberg (Editor), The Sea: Ideas and Observations on Progress in the Study of the Sea. Wiley, New York, N.Y., pp. 603--621.
327 Middleton, G.V., 1966. Experiments on density and turbidity current, I. Motion of the head. Can. J. Earth Sci., 3: 523--546. Normark, W.R. and Dickson, F.H., 1976. Man-made turbidity currents in Lake Superior. Sedimentology, 23: 815--831. Parker, G., 1978. Self-formed straight rivers with equilibrium banks and mobile bed. Part 1. The sand-silt river. J. Fluid Mech., 89: 109--125. Shepard, F.P. and Dill, R.F., 1966. Submarine Canyons and Other Sea Valleys. Rand McNally, Chicago, 460 pp. Stefan, H., 1973. High concentration turbidity currents in reservoirs. Proc. 15th Conf. Int. Assoc. Hydraul. Res., 1: 341--352. Stow, D.A.V. and Bowen, A.J., 1980. A physical model for the transport and sorting of fine-grained sediment by turbidity currents. Sedimentology, 27: 31--46. Turner, J.S., 1973. Buoyancy Effects in Fluids. Cambridge University Press, Cambridge, 367 pp. Vanoni, V., 1975. Sedimentation Engineering. Am. Soc. Cir. Eng. Man. 54, New York, N.Y.
307
Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
CONDITIONS FOR THE IGNITION OF CATASTROPHICALLY EROSIVE TURBIDITY CURRENTS
GARY P A R K E R
St. Anthony Falls Hydraulic Laboratory, Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55414 (U.S.A.) (Received January 22, 1981; revised and accepted July 22, 1981)
ABSTRACT Parker, G., 1982. Conditions for the ignition of catastrophically erosive turbidity currents. Mar. Geol., 46: 307--327. The basic impediment to a clear understanding of eroding and depositing turbidity currents has been the lack of a proper formulation of bed sediment entrainment. This problem is addressed herein: a detailed analysis indicates that the Bagnold criterion is necessary but insufficient for self-sustaining turbidity currents. The analysis reveals two possible equilibrium states, a relatively low-velocity ignitive state, and a high-velocity, dense catastrophic state. A stability analysis indicates that the ignitive state is unstable; flows below it invariably die out, and flows above it "ignite", i.e. accelerate and entrain sediment to the catastrophic state, which is stable. The ignitive state thus defines the criterion for a self-sustaining turbidity current. Estimates of the catastrophic state suggest that it is highly erosive and competent to scour out submarine canyons. INTRODUCTION
One of the greatest mysteries concerning turbidity currents is their apparent ability to erode through shelf sediments and carve out magnificent submarine canyons. Even in the essentially depositional environment of the fan below the canyon, enough erosive power often remains to maintain a well-defined channel often bounded by natural levees. In comparison to subaerial rivers, the down-slope impelling force of gravity acts on the solid phase only and is vastly reduced in the subaqueous environment. In addition, a true turbidity current operates on a constraint more severe than its simpler cousin, the density current; it must generate enough turbulence to hold its sediment in suspension. If it cannot, the sediment will deposit and the current, having lost its raison d'etre, must vanish. It is easy to imagine self-sustaining turbidity currents, the solid phase of which is mostly cohesive clay; at sufficiently low concentrations the fall velocity of such material is so low that it is not readily deposited. Such currents have been observed in reservoirs and have been reproduced experimentally (Stefan, 1973). They probably also occur in the ocean (Stow and Bowen, 1980). 0025--3227/82/0000--0000/$02.75 © 1982 Elsevier Scientific Publishing Company
308
However, abundant evidence on submarine canyons and fans suggests that highly erosive turbidity currents occur even in environments where the only available material is medium to coarse silts and sands, or coarser (Shepard and Dill, 1966). Such currents have even been created artificially by man in Lake Superior (Normark and Dickson, 1976). Many experiments have demonstrated that turbulence alone is n o t sufficient to keep these non-cohesive materials in suspension. For example, Einstein (1968) observed the clarification of water laden with silt (grain size Ds -~ 0.01 mm) flowing over a gravel bed. Material readily fell into the gravel matrix and could n o t be re-entrained subsequently. A non-cohesive turbidity current must have the ability to reentrain sediment from the vicinity of the bed. Such a current is essentially depositive if it deposits more than it entrains; in the opposite case it is essentially erosive. The state of the literature on the mechanics of turbidity currents must be viewed in this light. The analysis in most studies is hardly removed from that appropriate for density currents, i.e. little more than lip service is paid to the mechanics of the sediment phase. Bagnold (1962) did consider the solid phase and was able to obtain a simple approximate energy constraint on turbidity currents. Where U is mean down-channel flow velocity, S is b o t t o m slope (or more properly energy slope), and vs is sediment fall velocity, a current is supplied energy at a rate larger than that consumed by maintaining the sediment in suspension only if: US -->1 us
(1)
Bagnold concluded that if eq.1 is satisfied, a self-sustaining "auto-suspension" can occur.
Bagnold's work is of considerable significance, b u t while eq.1 may be a necessary constraint, it is by no means sufficient, for there is no mechanistic guarantee that the energy available to suspend the sediment can in fact be contributed toward this end. For example, the analogy to eq.1 for sedimentladen river flow is: US - -
R C vs
>1
(2)
where C is mean volumetric sediment concentration and R is sedimentsubmerged specific gravity (usually near 1.65). In Einstein's experiments on clarification, the number on the left-hand side of eq.2 becomes very large at low concentrations, y e t eventually the water clarifies completely. The reason for this is that t h e energy actually expended to suspend grains is turbulent energy, and turbulence does not suspend specific grains. It counters the flux of grains to the bed due to their fall velocity by producing a net upward wafting of grains entrained from the bed. If each grain, upon settling, cannot be re-entrained at the bed, the stock of grains available for suspension by turbulence gradually diminishes and the water clarifies. In simplified terms,
309
energy available to suspend is contributed to this end only to the extent that grains are available to be suspended. Clearly the missing feature of turbidity currents is a description of entrainment and deposition at the bed. Clearly the morphology of submarine canyons and fans is crucially d e p e n d e n t on this feature. Nevertheless it is precisely this feature that has been left o u t of nearly every treatment of the fluid mechanics of the subject. The comprehensive experimental and analytical study of Ashida and Egashira {1975) pays considerably more attention to the stratifying role of sediment than can be done in the c o n t e x t of the layer-integrated treatment used herein, but is still limited to the case for which neither net deposition nor entrainment of sediment occurs. Recently Chu et al. (1979 and 1980) have taken the important step of recognizing the problem, b u t their attempts to resolve it have been either qualitative (1979) or cumbersome and involving erroneous assumptions such as "Once the sediment particle is in suspension, it will always remain in suspension" (1980}. The present analysis seeks a simple b u t mechanistically well-founded treatment of erosion and deposition. In particular a critical "ignition" condition for the onset of a fast, highly erosive state is identified. This state is conjectured to be responsible for submarine canyons. A HEURISTIC MODEL FOR "IGNITION"
Consider an idealized continuous turbidity current o f infinite longitudinal extent, flowing down a slope with angle 0, that is uniform in the lateral and downstream directions b u t is free to vary in time (Fig.l). At time t = 0, the current is assumed to have initial layer-averaged velocity and volumetric concentration Uo and Co and is flowing over a bed of loose non-cohesive material with grain size Ds similar to that in the turbidity current. What will be the future of this current? Suspended sediment is constantly falling o u t of the current at the rate (flux} vs cb, where Cb is a mean volumetric concentration of suspended sedim e n t near the bed. Suppose the turbidity current is so slow that the stress it exerts on the bed is less than that required to entrain sediment into suspension, or at least so that it produces a volumetric rate of sediment entrainment per
x
h
ERODIBLE BED
Fig.1. Definition diagram.
310
unit area from the bed Xs which is less than the settling rate vs c b. In time, then, the turbidity current experiences a net loss of sediment, and layeraveraged concentration C decreases. Consequently the impelling force per unit fluid mass due to the downstream c o m p o n e n t of gravity acting on the current g R C sin 0 decreases, and the current decelerates. Bed stress is lowered further, and xs droPs further and more of the remaining sediment settles out, reinforcing the deceleration and causing the current to eventually vanish. On the other hand, suppose that initially Xs > v~ co. The current entrains more sediment, C and the impelling force g R C sin 0 increase, the current accelerates, bed stress increases and more sediment is entrained in a selfreinforcing cycle that allows for the development of a high-speed, highly erosive turbidity current. This "catastrophic" current would eventually be limited by other factors such as any limit on the entrainment rate, which w o u l d l i m i t the growth of the impelling force, or perhaps by damping of the turbulence at high concentrations. The implication is the existence of a set of critical values for velocity and concentration below which the turbidity current dies and above which it grows. The terminology "ignition c o n d i t i o n " is introduced herein in analogy to the critical values of oxygen and fuel concentration and temperature necessary for the onset of combustion. DYNAMIC EQUATIONS
The present analysis is intended to be extremely simplified in its formulation so as not to obscure the fundamental role of erosion and deposition in the process of ignition. It is assumed that the mean concentration of sediment c may be treated as a parameter sufficiently smaller than unity to allow the use of the Boussinesq approximation. Layer-averaged mean velocity U and volumetric concentration C can be defined in terms of local mean values u and c as follows: h
h
u=-ff o
o
where y denotes the coordinate normal to the bed and h is defined such that u ( h ) / U and c ( h ) / C are appropriately small parameters, e.g., the largest of the
two equal to 0.01 (Fig.l). Forms for the layer-integrated equations of energy and fluid mass balance reduce with the aid of the boundary layer approximations to: -- ~1 - - U 2 h + - ~ 2
0t
2
0x
U
U2
=---oL3 UgRcosO--
2
+ o~4 g R C h U sin 0 -- (u2, + u2,i) U -- R C g vsh cos 0
Oh --
3t
O +
--
3x
Uh
=
e
U
0x
Ch 2
(3)
(4)
311 for flow of constant width down a straight channel of constant slope. In the above, x denotes the downstream coordinate, 0 denotes bed angle, u. and u.i denote bed and interfacial friction velocities, respectively, and e is a coefficient of fluid (rather than sediment) entrainment. The parameters a l to a4 are order-one shape factors; e.g.:
-h Ol 1
u2dy
--
U2
so t h a t all shape factors are equal to u n i t y for the case where u is uniform in the vertical. The equation of sediment mass balance takes the layer-integrated form: OC --
0t
+
~
ax
as
UC
= F b
(5)
where as is another shape factor. In the above, F b = F ( b ) where: F ( y ) = c' v' - - v s c
(6)
denotes the mean vertical flux of suspended sediment and y = b is an elevation conveniently close to the bed. In eq.6, c' and v' denote the instantaneous turbulent fluctuations of concentration and vertical velocity about their mean values at level y; the overbar denotes averaging. F ( y ) is seen to be the net result of two components due to turbulent mixing (upward in this case) and downward flux due to the fall velocity. Where it is necessary to attach numerical values to the shape factors, all are set equal to u n i t y as a convenient expression of ignorance. A formulation of the term Fb for eroding or depositing flows was developed by Parker (1978); t h a t formulation provided the original idea for this paper. Therein it is assumed t h a t the term c' v' evaluated near the bed, which specifically describes the rate of sediment entrainment, can be represented as a function ×s of bed shear velocity u. and other parameters: c' v'l b --- Xs
(7)
An evaluation of ×s can be obtained as follows from data on steady uniform suspensions. For each flows, F = 0 everywhere and it follows that: ×~ = v~ cb
(8)
Thus defining a dimensionless sediment entrainment function E~ such t h a t vs E~ = ×~ it follows that: Es = Cb implying t h a t entrainment just equals deposition.
(9)
312 Itakura and Kishi (1980) have developed a relation for c b uniform suspensions in open channels; it takes the form: Es = k,
k2 R~ #2
(=Es)for steady, (10)
where R~ #2
1
k3 exp ( - - 1 a 2) A(a) =
exp ( - - l x 2)
dxJ
o
and v~ Rf
-
V/('-~g
Ds
(11)
U~ # = -vs
(12)
The parameters k l, k2, and k3 are constants taking the values 0.008, 0.14, and 0.143, respectively. In general, then, Es is a function of # and Rf. Implicit in eq.10 is a threshold dimensionless shear velocity #t~ below which entrainment vanishes. In Fig.2 Pth is plotted versus Rf for 0.1 < Rf < 0.5 (0.056 mm < Ds < 0.35 mm at 20°C). The values are in general agreement with the criterion #th -~ 1 suggested by Engelund (1965) and Bagnold (1966). 5
,
,
,
,
I
I
I
I
P-th21 0 0. t
I 017
Rf Fig.2. Threshold dimensionless shear velocity Pth as a function of Rf.
313
Itakura and Kishi verified eq.10 with the use of measurements of Cb, with b arbitrarily set equal to 0.05 h, where h is the depth of flow. The measurements cover the range in grain size Ds from 0.08 mm to 0.35 mm, or using the plot of R~ versus Rp = R ~ s ( D s / v ) in Fig.3, 0.16 < R~ < 0.5 at 20°C (v denotes the kinematic viscosity of water). This range is appropriate for the future consideration of oceanic turbidity currents in non-cohesive material. The range of u is from near unity to a b o u t ten. Equation 10 is obviously only valid if the bed is completely covered with suspendible material. If the stock of such bed material is depleted to the point of partial exposure of non-suspendible material such as bedrock, the entrainment rate is reduced below that predicted by eq.10. Many researchers have approached the problem of non-equilibrium suspensions by using as a b o t t o m condition the assumption that Cb itself is the same function of flow conditions as found for equilibrium conditions. Parker (1978) illustrated that this is erroneous and showed that only the entrainment rate Es can be taken to be a specified function of local flow conditions for non-equilibrium suspensions, e.g., eq.10. Thus eq.5 reduces with the aid of eqs.6 and 7 to: aC --
at
a
UC =v~ [ E s - - C b ]
+ - - a s
ax
(13)
where herein Es is assumed to be given b y eq.10. Equation 13 provides a specific evaluation of the relative contribution of erosion and deposition. Certain auxiliary relations are necessary for a layer-integrated treatment. In particular, parameters such as u., u.i e, and Cb must be related to layeraveraged parameters U and C; to wit :
10 ] I I! I===== i
= ==:~=I=I
I I EIIIII I IIIIIII
JIIJl JII]l
I l[Illll
Illl]
...............
I
I IIIIIL I I][IIII
i I
Rf
, , i iiiiill I Illllll
I IIIllll
[lll[tll
I
I I ;;Jill
I
I I llIlIII]
I
[lllll
I I I1
Jllll I iiii~-T
I II
i i iiiiiiiiii
i
lllN!
13~rl
I II111[I
I I I1[I
lJ[llil
l lltllll I III]II]ll
I I I ] ~ ~"
J
19¢111]1 J IIIIIIII
IIIlll
I IIIII IIII11
I
I
l
[1[111
111111
I I1111[ II1111IIIIII f i iiiN I~llll
I 10-]
nit
I I
I 10°
Jiiii iI i[iiiii ~ Jllll I IIII]1 Ill 101
IIIII 10 2
RF, Fig.3. Rf as a function
of Rp.
; Ill
II[IFI I1[111
10-~
i0-2
~I~III I I Illl l l J I IJll IIII
I ]
]III] I00 ,
I I
~iii;
iI iiii I
IIII
; i iiFii]i I
I IIIIIII
I IIIIII]1
I111IIII nroI1~o, lllll ....
I0 s
10 4
I II
I II
! !!!! ..... I III
1/11 10 5
314
u, 2 =lf
U2;
u 2,i =
1 ~fi U 2
(14)
(15)
Cb = r C
where f and fi are bed and interfacial friction coefficients and r relates bed concentration to layer-averaged concentrations. The parameter e is a function of Richardson n u m b e r R i = R C g h / U 2 (see e.g., Turner, 1973). The parameter f can be estimated from a knowledge of the bed resistance of alluvial rivers (Vanoni, 1975). Egashira and Ashida (1979) have recently developed comprehensive relations for estimating fi. An ad-hoc m e t h o d for estimating r for the present analysis can be developed using the well known Rousean distribution for open channel suspensions: c
where 77 = y / h , % = b / h , a n d Z = 2 . 5 / p is the Rouse number. Setting 'Tb = 0.05, then: __ 7?/ l __ Vb1 2.5/~ d~) ro:l/( i [ l--;---C-. 0.05
where the subscript denotes the fact that the relation applies to open channels. The above equation can be accurately approximated by: r0 = 1 + 31.5g -L46
(16)
f o r ~ > 0.5. Herein it is assumed that for turbidity currents: (17)
r = o~ 6 r o
where a6 is another order-one shape constant. Setting all shape factors equal to unity in a first analysis, the complete set of equations reduces to: -- - U2h + -- U ~2 t 2 3x
2
=--2
U g R c o s O - - C h 2 + R g C h U sin O 3x
1 ---2 f(1 + 7)U 3 - - R g C h v s cos 0 Z
Oh --
~t
OUh +
-
Ot
- eU
~x
OCh -
(18a)
3UCh +
Ox
where 7 = f i / f .
- v~[Es - - r o C ]
(18b)
(18c)
315 EQUILIBRIUM TURBIDITY CURRENTS When water entrainment e is negligible, eqs.18a, b and c admit solutions for steady, uniform equilibrium turbidity currents with constant thickness h. This is strictly valid only for Ri greater than about unity. Herein e is always set equal to zero for simplicity; also 7 is neglected as well, implying t h a t bed stress by far dominates interfacial stress. These simple assumptions can be replaced in future more detailed analyses. Equations 18a--18c reduce to: vs Rg Ch sin 0 = u .2 + R g C h c o s O -~-
(19)
Es = roC
(20)
Eliminating C from these two equations, the constraint for equilibrium is seen to reduce to the form: F ( p ) = FI(p ) -- FR(P)
R~ ro
--
where
~E~
FI(p) = ~ R~ ro Fa(p) = p2 +
(22a)
Es PB
(22b)
R~ro In the above: h sin 0 =~
Ds
(23)
and the Bagnold limit PB is given by:
PB = - S
(24)
where S = tan 0 is the bed slope. The term FI(p) quantifies the impelling force acting on the current; it is either positive when p is greater than Pm or zero otherwise. The term FR(p) quantifies the resistive force, with a frictional contribution p2, and a contribution equal to a fraction PB/P of the impelling force which represents that needed to keep the sediment in suspension. FR(P) is always positive. An examination of eq.21 reveals that no positive solution for p can exist when p is less than the Bagnold limit/~B given by eq.24. Not surprisingly, this relation reduces with the aid of eq.14 to the Bagnold "autosuspension" criterion, eq.1. However, this criterion is not sufficient for an equilibrium turbidity current; the extra constraint is clearly t h a t F1 = FR.
316
To describe this condition, it is of use to consider first the case where /~B = 0. F R (#) is then parabolic in p and plots as a straight line on log-log paper. On the other hand Es increases sharply with values of p only slightly greater than #th ; for higher values the increase is less steep. Thus on log-log paper FI curves downward. The implication, shown on Figs.4a and b, is that if there is any non-zero equilibrium solution for/~, there are two. In Figs.4a and 4b, Fi(p) and FR(#) are plotted as functions of g for the special case Rf = 0.5 (Ds = 0.24 m m at 20°C) at two values of @. For the lower value ~b = 250, the curves never meet and no equilibrium solution exists. For the higher value @ = 352, two equilibrium solutions to eq.21 exist; these may be labelled p, and p: where p, < P2. In this case p, = 5.5 and P2 = 10.0. In so far as U = x / ~ v~ p, C = E Jr o and bulk current density Pc equals p(1 + RC) where p is the density of water, it can easily be verified that U=, C2, and pc2 are each invariably greater than U,, C~, and Pc ,, respectively (approximating f as a constant). For this reason the upper equilibrium is termed the "catastrophic equilibrium." For reasons to be described later, the lower equilibrium is termed the "ignition equilibrium." Crudely estimating f = 0.004 (as is done t h r o u g h o u t this paper; Komar, 1977), it can be f o u n d that for the above example U~ = 3.8 m/s, C~ = 0.0215 and pc~ = 1.035 g/cm3; likewise U2 = 7.0 m/s, C: = 0.0710 and pc2 = 1.117 g/cm 3, When no solution exists, it is because for that value of ~ no rate of entrainm e n t consistent with eq.10 can supply enough sediment to maintain an impelling force large enough to overcome frictional resistance. However, it is apparent from eq.21 that for prescribed p and Rf, @ can always be increased 4XlO 2
iO3
--
/
102
102
F I ,F R
IOt
F t . FR
I
/~
Fig.4a. FR(U ) and
....
i01
qom 3XlO-J
F~ 1/~)
Io ~
FI(u ) f o r
7XlC~
I 2
R f = 0 . 5 , ~ = 2 5 0 a n d UB = 0.
F i g . 4 b . F R ( P ) a n d F I ( u ) f o r R f = 0 . 5 , ~ = 3 5 2 a n d uB = 0.
,a
I0
2XlO
317 ,,,,.
,o4
~2~726
103 F I ,F R
-
~
120 i01 I
I0
I
I0
2
Fig.4c. FR(~u) and FI(~ ) for Rf = 0.5, ~ = 1 2 5 0 and gB-" 0.894. The dashed line denotes a more realistic qualitative shape for Fi(g) at high concentrations. I
,0 ~
v
,
,
,
i
_
o0
10 2
i0 t
m OI
i
I
Rf
i
J
0.7 L
Fig.5. ~ , as a function o f Rf and ~B.
until that equation is satisfied. Thus a lower bound 4, exists below which equilibrium turbidity currents cannot exist at any p. In Fig.5, 4, is given as a function of Rf for 0.1 < Rf < 0.7 (0.056 m m < Ds < 0.35 m m at 20°(]). In practical terms, this means that for a given grain size Ds and downstream channel inclination sin 0, current thickness h must be greater than the value h,, where from eq.23: h,-
D~ 4, sin 0
(25)
318
For example, on a slope S = 0.05 (0 = 2.86 °) for 0.18 m m (Rf = 0.4) material, h must exceed 0.79 m. The case of greater interest is the one for which two solutions exist. For any value of R~, there exists a value p, such that Pl < P. < P2 for all values of $. Indeed, it can be verified from eq.21 that/~, is the value of/~ at which = $ ,. The parameter p. thus divides the p-Rf plane into a region where only ignitive equilibrium is possible, and one where only catastrophic equilibrium is possible; this is shown in Fig.6. It can be seen from Fig.6 that for the given range of Rf, P2 can never be less than 7.4, and C2 can never be less than 0.040 in the range of catastrophic equilibrium. Indeed, for Rf = 0.7 (D s = 0.35 mm), using the previously quoted estimated for f, the lower bounds on velocity U2 and current density Pc2 are 8.7 m/s and 1.07 g/cm 3, respectively; for R~ = 0.1 (Ds = 0.056 mm) these lower bounds are 0.66 m/s and 1.10 g/cm 3. The densities in particular are rather high; at concentrations as high as this the structure of the flow may be so modified as to invalidate the use of the equations presented herein. Thus, the present t h e o r y may n o t describe the catastrophic region too accurately. Under conditions of severe stratification, for example, damping of the turbulence may cause FI(p) to flatten out rather more rapidly in p than the functional form of eq.22a; this would lead to a somewhat lower value of p~. Another p h e n o m e n o n which might also act to lower/~2 is depletion of the i
i
i
v
CATASTROPHIC
~B=5
IO #-B=I dividing line for /a.B=O IGNITIVE I
I
I
0.7
03 Rf Fig.6. Plot o f / z , as a f u n c t i o n o f R f a n d PB.
319
supply of available suspendible bed sediment by bedrock exposure, for example; this reduces Es as discussed previously. A suggestion of these effects is indicated by the dashed line in Fig.4c. (The figure is explained below). None of these modifying factors could be expected to modify the essential result of the existence of the catastrophic state. In addition, through most of the range of occurrence of the ignitive equilibrium concentrations are low enough to suggest the validity of the present treatment, and some detail in calculation is warranted. In Fig.7a the solution p~ t o eq.21 is given as a function of @ with Rf as a third parameter; from this diagram and previously presented relations auxiliary parameters such as CI, UI, and Pc1 can be computed, as was done for the example of Fig.4b. The above analysis only applies for the limiting case ~B = 0. However, all of the concepts generalize quite readily. For example, consider a current 40
I
I
I
/x 2
lI
I
Il
il
I
I
i
I
l
I
I
I I Ill
I
I
I
I
I
I III
I
I
i i i i1
0.4 0.50.6 0.7
0.3
0.2
I !--_
10
I
I I lllJ
' Rf=O.I
i
I
II
II
t
I
lI
CATASTROPHIC
/ I
!
I
I
Rf =0.1 FB=I I
,
0.2 i
,
,,,,,
IOI
,d
iO2
IO4
i0 5
o/ Fig.7a. Ignitive s o l u t i o n . 1 and c a t a s t r o p h i c s o l u t i o n u2 as f u n c t i o n s o f $ a n d R i f o r #B = 0 . 30 .Rf:OI /-L2
I
l/
I
'''[
Q3
0.2 /
/ i
/ I
IO
'1
Q4 Q50.6Q7
!/
1/
t I
1/
//
I / /
CATASTROPHIC
/
IGNITIVE
:
Rf=O.I
X
0.2
07
p-B=O i 0.4
'
I0
'
'
' ' ' '
I0
2
I ,
, , l .
,
3 IO
qs
,
,
, , J ,
4 I0
5 I0
Fig.7b. Ignitive s o l u t i o n ~, and c a t a s t r o p h i c s o l u t i o n u2 as f u n c t i o n s o f $ and R f f o r .B --1.
320 I00
I
I
I
I
I Ill
I
Rf= 0.1
I
0.2
I
I
0.3
/ /
I
/
/x 2
I
I
I
/
I
II
i
I
I
/
/
I
I
I
I I I
/I
!
//
/
I
I
I
I 1l
I
/
I
#
I
0.4 0.5 0.6 0.7 I ," / II
tI
CATASTROPHIC
IGNITIVE I0 I-Xl
/.~B= 5 I
2
I
I
I
III
[
102
I0
I
qJ
I
i
I I II
103
4
I0
Fig.7c. Ignitive solution u~ and catastrophic solution P2 as functions of ~ and Rf for PB=5.
with a thickness h = 6 m flowing down a c a n y o n slope with S = 0.05, carrying 0.24 m m sand (R~ ~ 0.5). Using f = 0.004, PB is f o u n d to be 0.894 (/,lth is 0.83); ¢ = 1250. In Fig.4c, F , ( # ) and F R (#) are pl ot t ed; again eq.21 has both ignitive and catastrophic solutions. For the ignitive solution, Pl = 1.83, a value of dimensionless bed shear velocity t hat is considerably in excess of the Bagnold limit. Values of C, Pc, u. and U for this case are 0.00131, 1.00216 g/cm 3 , 5.71 cm/s and 1.28 m/s, respectively. For the catastrophic case P2 = 72.6, a value t hat is far outside the range of validity of eq.10 and p r o b ab ly unrealistically high (it yields a flow velocity U of 50.6 m/s). The lower b o u n d ~ . for the existence of any equilibrium turbidity current is shown on Fig.5 for the cases PB = 1 and PB = 5 in addition to PB = 0; h. can again be calculated from eq.25. On Fig.6 the value of p. dividing the ignitive and catastrophic range is p l o t t e d versus Rf for PB = 0, 1, and 5. It is seen th at ~. and p, increase with #B, and thus in general increase with decreasing slope, as indicated by eq.24. The solution p, for the ignitive state is given as a funct i on of $ and Rf for PB = 1 in Fig.7b and PB = 5 in Fig.7c. A p o in t o f some later importance concerns the local exponents, or slopes on log-log paper, NI and NR of the functions F I ( p ) a n d F R (P), respectively at the ignitive and catastrophic state. These e x p o n e n t s are given by the expressions: ~/ dFa
P dFI N~ = - - - ; F x dp
NR
-fir
dp
where p = p, or P2. After some manipulation using eqs.22a and 22b, it is f o u n d th at these e x p o n e n t s take the form: N~ = M
(26a)
321
NR=2
1--
+Mf
(26b)
where
M-
P dEs
p dro.
E s dp
Ps = -P
ro d p '
(26c)
and p again equals Pl or P2. It can be shown from eqs.10 and 16 that M > 0; the Bagnold criterion requires that ~ < 1. It can be seen from Fig.4c that on log-log paper F1 is steeper than FR at the ignitive root, implying that N I -- NR > 0 or: M(1--~-)--2
--
>0
(27a)
On the other hand, at the catastrophic r o o t FR is steeper than FI, in which ease N < NR and: M(1--f)--2
--
<0
(27b)
The nature of the functional forms in eqs.22a and 22b insures that this result generalizes, so that for all values o f R f , Ps, and ~ , if both the ignitive and catastrophic states exist then the inequality (27a) holds for the former state, and (27b) holds for the latter state. These two inequalities play an important role in determining the stability of the ignitive and catastrophic equilibrium. TEMPORAL
INSTABILITY O F T H E IGNITIVE S T A T E
The significance of the ignitive state is e m b e d d e d in its name. It is of use to reconsider the idealized continuous current of the heuristic model; it is longitudinally uniform b u t free to vary in time. Suppose that at t = 0, U < U1 and C < C~ ; under these circumstances the turbidity current must gradually drop its sediment and die in time. On the other hand, if initially U > U1 and C > C1 the current will both accelerate and become denser until the catastrophic state is reached. Thus the ignitive state delineates the conditions necessary to ignite a self-sustaining turbidity current. In the language of stability, the ignitive equilibrium is unstable, and bifurcates to either the null or the catastrophic state, both of which are stable to infinitesimal disturbances. A graphical illustration of this that can be immediately appreciated is given in Fig.8. If/~ is slightly below p, both Fi(p) (impelling force) and FR(U ) are lower than their (equal) values at Pl ; however, since the FI(p) curve is steeper, i.e. Nx > NR, the impelling force FI(p) drops below the resistive force FR(P). Resistance dominates; the current decelerates, drops sediment, loses more impelling force, etc., until the null state is reached. By the same token, when N~ > NR, FI(P) > FR(P) for p > p~, a n d a cycle of acceleration and entrainm e n t occurs until the catastrophic state is reached.
322
F,,~
FR(/Z2*A/z ) F I (#2+Z~#) F I (/J,2-AU.) FR(/Z2-&/z) FI (]~1 ,5~.) FR(/J- I +A/z)
FR(/Z ~-Z~/~) F I (/J i -lk/z }
F2
/
Fig.8. Graphical indication of instability of ignitive root and stability of catastrophic root.
On the other hand, at the catastrophic state NR > NI, i.e., F R is steeper than FI. If p is slightly less than P2, FI(p) > FR(p), and the current accelerates back to catastrophic equilibrium. A similar result holds for p slightly greater than P2. Thus this state is stable. A more formal proof of this can be obtained from eqs.18a through 18c, again assuming that e = 7 = 0; for the postulated spatially uniform but temporally varying flow of the heuristic model, t h e y reduce to: a U = g R Ch sin 0 h a--t
1 U2 vs 2f - - g R Ch cos 0 - U
aC h - - =vs [Es--ro C]
at
(28a)
(28b)
and h =h o
where h o is a constant. Herein f is also assumed to be constant. These equations are now perturbed about equilibrium; that is: U=U,~ + u ' C = C ~ + c'
where a = 1 denotes the ignitive equilibrium and a = 2 denotes the catastrophic equilibrium. Assuming the primed perturbations to be small, eqs.28a and 28b linearize to:
au' c' + f ~u' ] --f~u' ba~ ho-~=gC, ho I(I--~')~-
(29a)
323
=
h°~t
Us
(295)
Ca
where {- = PB/g~ and M is given by eq.26c evaluated at # = p~. Defining and utilizing the equilibrium eqs.19 and 20 at p = p~, dimensionless versions of eqs.29a and 29b can be obtained: c = c'/C~, u = u'/U~, T = (UJho)t
~u_f*
~T ~c
~T
2
Iv(1 -- {-) -- 2 (1 -- ~{-) u]
(30a) (305)
- (3[Mu - - c]
where: f * = f / ( 1 -- {-}, and
[3-
Vsro
us
provides a measure of the "ease of deposition" of the suspended material. For the limit of extremely fine sediment, which deposits only very slowly, approaches zero and the turbidity current becomes a simple density current. The stability of eqs.30a and 30b can be tested by assuming perturbations of the form: u = Uo e x p ( ~ r )
and
c = c0exp(~T)
(31)
where Uo and Co are constants denoting initial perturbation amplitude. Clearly if ~, < 0, u and c decrease to zero in time, and the equilibrium is stable; if ~ > 0 the perturbations grow and instability results. Inserting eqs.31 into eqs.30a and 30b, it is found that: ~1 f* ( 1 - - {")Co -- X +
--2
f*
Uo = 0 (32)
(~ + [3)Co - - [ 3 M u o = 0
The above equations can be satisfied in general only if Co = Uo = 0, yielding a trivial solution, or: 1
3
=0 X+ ~
--M~
yielding the dispersion relation: ~2 + B ~ . _ D = O
where
(33a)
324
B=[3+ D=-~f3f
1--
f)
-
=~f*
-
(NI -- NR )
(33b)
Note that eqs.26a and 26b have been used to obtain eq.33b in terms of NI and NR. The roots of eq.33a are thus: 1
+
2~f*(NI--NR) ~)
(34)
1 Two cases are apparent; that for which ~ = PB/P, < 2/3 and that for which 2/3 ~< ~ < 1. In the former case (1 -- 3/2 ~) is positive. When N1 > N R , one r o o t X is positive and instability results; when NI < NR, both roots are negative and equilibrium is stable. Recalling eqs.27a and 27b, this indicates that the ignitive state is unstable, whereas the catastrophic state is stable. The remaining case applies only to the ignitive state because as can be readily deduced from eqs.26a and 26b, NR can never exceed NI for 2/3 ~< < 1. It is seen that for NI > NR one r o o t X is positive regardless of the sign of [~ + (1 -- 3/2 ~)f*], thus again implying that the ignitive equilibrium is unstable. This completes the stability analysis with the establishment of the desired result. It may be noted that for the unstable ignitive case (X > 0), Co and Uo must have the same sign according to eq.32. Thus if Uo < 0 then Co < 0 and the ignitive instability veers toward the null state; if Uo > 0 then Co > 0 and the stable catastrophic state is approached. IMPLICATIONS FOR SUBMARINE
CANYONS
The concept of ignition can be tentatively applied to the origin of submarine canyons. Silt and sand presumably collect at the head of submarine canyons due to direct supply from rivers or littoral drift. Occasional smallscale slumping may produce discontinuous slug-like turbidity currents of the type studied in the laboratory by Middleton (1966); they do not ignite, and thus they deposit sediment on the bed of the canyon as they progress downward and decay. Such turbidity currents would thus lay down a layer of "fuel" on the canyon bed to be utilized by a catastrophic current, perhaps triggered by a massive slump or large amounts of continuous sediment infeed from an episodic river flood. In order for such a current to ignite there must be, at least initially, an extra supply of sediment on the bed to be entrained and utilized by the current to accelerate. If the bed is still essentially covered with suspendible sediment when the equilibrium catastrophic state is reached, no further entrainment will occur and the canyon bedrock is n o t eroded. The more interesting case, however, is
325
where at equilibrium most of the supply of suspendible bed sediment is entrained, leaving bedrock overlain by thin gravel deposits too coarse to suspend. The catastrophic current would likely be capable of moving the coarser material which would act as tools to corrade the bedrock. The bedrock would also be subjected to intense sandblasting. Some evidence of this erosive capability can be obtained as follows. In a previous example a current with thickness h = 6 m flowing down a slope S = 0.05, carrying 0.24 m m sand (Rf ~ 0.5) over a similar bed was considered. For this example ~ = 1250 and PB = 0.894; at ignition p, = 1.83. The values of u, and C at ignition are 5.71 cm/s and 0.00131; if all the sediment in the current were suddenly precipitated to form a loose layer on the bed with a porosity of 0.35, the deposit would have a thickness of 1.2 cm. Maximum grain size D ~ t h a t could be moved by the ignitive current can be estimated from the Shields' relation; U 2, -
-
Rgnmax
- 0.06
where the value of 0.06 is conservatively high. It is f o u n d that: Dmax = 3.4 m m As previously noted, the value of p: c o m p u t e d at the catastrophic equilibrium is excessively high and probably not realistic. A conservative estimate of the catastrophic state can be obtained by using the lowest possible value of p in the catastrophic range for Rf = 0.5. This value is p = 8.9, yielding values of C, Pc, u., U and D ~ x of, respectively, 0.0572, 1.094 g/cm 3, 27.8 cm/s, 6.21 m/s, and 79.4 mm. Sudden precipitation of the suspended sedim e n t would yield a layer 52.8 cm thick. Subtraction of this thickness from the thickness of the precipitated layer of the ignitive state yields a lower bound on the potential depth of scour in going from the ignitive to the catastrophic state (realized only if the fine material is available to scour) of 51.6 cm. A current with mean velocity of 6 m/s, a scour potential of about 50 cm, and the ability to move up to 80 m m cobbles is likely c o m p e t e n t to erode canyons. The concept of ignition would seem to have some relevance to a reported a t t e m p t by divers to " k i c k " a turbidity current into existence, which ended in a failure that may have had a salutary effect on their health (Dill, 1964). DISCUSSION
In its present form the analysis is rather crude. Only continuous currents have been considered, and a long list of simplifications such as the neglect of water entrainment and interfacial stress has been introduced. Thus, the analysis provides in quantitative terms only an estimate of the condition for ignition, and provides only qualitative results and rough quantitative lower
326
bounds for the catastrophic state. Any application should be undertaken only with a clear awareness of these limitations. For example, currents predicted to be "ignitive" by the present analysis may often be supercritical, in which case water is entrained from above. The consequent dilution would tend to decelerate the current. The condition for true ignition must be somewhat more stringent for such currents; the accelerative effect due to sediment entrainment must be stronger than the decelerative effect due to water entrainment. On the other hand, the modifications necessary to obtain a quantitative theory applicable to even the supercritical and catastrophic ranges would not seem overly difficult in light of the present knowledge of sediment transport mechanics and stratified flow. Such a theory, when cast in the form of eqs.18a to 18c, would have the capability of predicting depositing or eroding backwater (or frontwater) curves from the canyon head to the extremities of the fan. The results of such a theory would be of value to sedimentologists. ACKNOWLEDGEMENTS
Long discussions with J. Shaw and S. Egashira during the several years of contemplation before the above picture crystallized are gratefully acknowledged J. Akiyama provided a useful comment. This research was n o t funded by any agency, and thus has a benefit-cost ratio of infinity. St. A n t h o n y Falls Hydraulic Laboratory graciously denoted the support services necessary to produce this paper. REFERENCES Ashida, K. and Egashira, S., 1975. Basic study on turbidity currents. Proc. Jpn. Soc. Civ. Eng., 237: 37--50. Bagnold, R.A., 1962. Auto-suspension of transported sediment; turbidity currents. Proc. R. Soc. London, Ser. A, 205: 315--319. Bagnold, R.A., 1966. An approach to the sediment transport problem from general physics. U.S. Geol. Surv., Prof. Pap., 422-I:I1--I37. Chu, F.H., Pilkey, W.D. and Pilkey, O.H., 1979. An analytical study of turbidity current steady flow. Mar. Geol., 33: 205--220. Chu, F.H., Pilkey, O.H. and Pilkey, W.D., 1980. A turbidity current model. In: Civil Engineering in the Oceans, IV. Am. Soc. Civ. Eng., pp. 416--432. Dill, R.F., 1964. Sedimentation and erosion in Scripps Submarine Canyon head. In: R.L. Miller (Editor), Papers in Marine Geology, Shepard Commemorative Volume. MacMillan, New York, N.Y., pp.23--41. Einstein, H.A., 1968. Deposition of suspended particles in a gravel bed. Proc. Am. Soc. Civ. Eng., 94(HY5): 1197--1205. Engelund, F., 1965. Turbulent energy and suspended load. Basic Res. Progress Rep. 10, Hydraulic Lab., Tech. Univ. Denmark, pp. 1--9. Itakura, T. and Kishi, T., 1980. Open channel flow with suspended sediments. Proc. Am. Soc. Cir. Eng., 106(HY8): 1345--1352. Komar, P.D., 1977. Computer simulation of turbidity current flow and the study of deepsea channels and fan sedimentation. In: E.D. Goldberg (Editor), The Sea: Ideas and Observations on Progress in the Study of the Sea. Wiley, New York, N.Y., pp. 603--621.
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