CEE 598, GEOL 593
TURBIDITY CURRENTS: MORPHODYNAMICS AND DEPOSITS
LECTURE 11
INTRODUCTION TO TURBIDITY CURRENT
MORPHODYNAMICS
after 16 runs
0.6
0.5
0.4
ambient fresh water
0.3
saline underflow
after 20 runs
0.6
0.5
bed
0.4
0.3
after 25 runs
0.6
0.5
after 30 runs
Top: photo showing the deposit of
lightweight plastic sediment formed by the
repeated passage of saline underflows
(analogs of turbidity currents).
after 33 runs
Left: time evolution of the bed.
0.4
0.3
0.6
0.5
0.4
0.3
0.6
0.5
From Spinewine et al. (submitted)
0.4
0.3
1.5
2
2.5
3
3.5
4
1
THE CASE OF RUPERT INLET
The Island Copper Mine, Vancouver Island, British Columbia, was in
operation from 1970 to 1995. To deal with the massive amounts of mine
tailings (= waste crushed rock) produced, Island Copper Mine discharged
around 400 million tons of tailings through an outfall at 50 m depth into the
adjacent Rupert Inlet.
2
From Poling et al. (2002)
THE CASE OF RUPERT INLET contd.
The tailings (ground up rock), were ~ 40% fine to very fine sand, and ~
60% silt, with a median size ~ 30 m. They were disposed continuously to
form a turbidity current that was sustained for decades.
3
Photo: http://gateway.uvic.ca/archives/featured_collections/esa/fonds_island_copper_mines/default.html
THE REAL-TIME CONSTRUCTION OF A MINISUBMARINE FAN
Monitoring of the tailings disposal allowed for one of the first cases where
the evolution of morphology due to turbidity currents was monitored in real
time (Hay, 1987a,b).
4
From Hay (1987a)
THE TURBIDITY CURRENT FORMED AN EXTENDED
MEANDERING CHANNEL
channel
axis
From Hay (1987b)
meander bends
5
LONG PROFILE,
RELIEF AND WIDTH
OF THE CHANNEL
Relief ~ vertical distance from
levee top to channel bottom ~
channel depth.
From Hay (1987b)
6
THE CHANNEL SHOWED WELL-DEVELOPED
CONSTRUCTIONAL LEVEES
The acoustic image shows
the channel cross-section at
site 67, located below. Flow
direction is out of the page.
From Hay (1987b)
7
THE ACOUSTIC IMAGING SHOWED
MORPHODYNAMICS IN ACTION!
fish!
approximate interface
of turbidity current
channel bed
From Hay (1987b)
The turbidity current is overbanking
due to superelevation at the outside of
a bend. This overbanking has caused
the outer bank to become higher than
the inner bank. Flow direction is out
of the page.
8
BEDLOAD AND SUSPENDED LOAD
Bed material load is that part of the sediment load that exchanges with
the bed (and thus contributes to morphodynamics).
Wash load is transported through without exchange with the bed.
In rivers, material finer than 0.0625 mm (silt and clay) is often
approximated as wash load.
Bed material load is further subdivided into bedload and suspended
load.
Bedload:
sliding, rolling or saltating in ballistic
trajectory just above bed.
role of turbulence is indirect.
Suspended load:
feels direct dispersive effect of eddies.
may be wafted high into the water column.
9
TURBIDITY CURRENTS MAY CARRY BEDLOAD, BUT
THEY MUST BE DOMINATED BY SUSPENDED LOAD
Rivers are driven by the downstream pull of gravity on the water. The
water then pulls the sediment with it. The sediment can move
predominantly as bedload, predominantly as suspended load or some
combination thereof.
Turbidity currents are driven by the downstream pull of gravity on the
suspended sediment. The suspended sediment then pulls the water
with it. The resulting flow can then move bedload as well.
A turbidity current cannot be driven by bedload alone, because the
bedload is a) supported essentially by collisions with the bed, not
turbulence and b) moves in a very thin layer very close to the bed.
10
BOUNDARY-ATTACHED COORDINATE SYSTEM
We assume a bed that is sloping only modestly in the streamwise
direction. The parameter x is parallel to the bed and the parameter z is
upward normal to the bed.
y
x
h
x = nearly horizontal boundary-attached “streamwise” coordinate [L]
z = nearly vertical coordinate upward normal from boundary [L]
11
1D EXNER EQUATION FOR THE CONSERVATION OF
BED SEDIMENT: SOME PARAMETERS
Parameters:
qs = volume suspended load transport rate per unit width [L2T-1] = UCH
qb = volume bedload transport rate per unit width [L2T-1]
s = sediment density [ML-3]
vs = sediment fall velocity
h = bed elevation [L]
p = porosity of sediment in bed deposit [1]
(volume fraction of bed sample that is holes rather than sediment:
0.25 ~ 0.55 for noncohesive material, larger for cohesive material)
g = acceleration of gravity [L/T2]
t = time [T]
12
1D EXNER EQUATION FOR THE CONSERVATION OF
BED SEDIMENT; DERIVATION
Es = vsEs = volume rate per unit time per unit bed area that sediment is
entrained from the bed into suspension [LT-1].
Ds = vsroC = volume rate per unit time per unit bed area that sediment is
deposited from the water column onto the bed [LT-1].
Time rate of change of sediment mass
in control volume =
deposition rate from suspension –
erosion rate into suspension + net
inflow rate of bedload

s (1  p )h x  1 
t
s qb x  qb x x   1  s  Ds  Es  x  1
turbidity current
Ds
Es
u
qb
qb
h
bed sediment + pores
1
qb
h
(1  p )
 Ds  Es
t
x
x
x
x x
13
REDUCTION OF THE EXNER EQUATION
Since Es = vsE and Ds =vsroC, the equation reduces to:
qb
h
(1  p )
 v s (roC  Es )
t
x
Compare this relation with the equation of consevation of suspended
sediment:
CH UCH

 v s (Es  roC)
t
x
Since qs = UCH, the Exner equation can be rewritten as:
qb qs CH
h
(1  p )


t
x
x
t
The last term can be usually neglected because the mass stored as
suspended sediment per unit volume is negligible compared to the
mass of sediment stored per unit volume in the bed (C << 1)
14
COUPLING OF THE EXNER EQUATION TO THE
EQUATIONS GOVERNING THE FLOW
Example: 3-equation model:
UH U2H
1
CH2

  Rg
 RgCHS  Cf U2
t
x
2
x
H UH

 e wU
t
x
CH UCH

 v s (Es  roC)
t
x
qb
h
(1  p )
 v s (roC  Es )
t
x
where the closure relations are:
e w  fn1(Ri) , Ri 

 u 
Es  fn2    fn2 

 vs 

qb
RgD D

 fn3 (  ) ,
RCgH
U2
Cf U 

v s 
b
Cf U2
 

sRgD RgD

15
THE QUASI-STEADY ASSUMPTION
Turbidity currents are dilute suspensions of sediment. As a result, the
volume suspended sediment discharge per unit width qs = UHC is much
smaller than the water discharge per unit width qw = UH (since C << 1).
Under these conditions, the morphodynamics of sustained turbidity
currents can often be simplified using the quasi-steady approximation
(de Vries, 1965):
UH U2H
1
CH2

  Rg
 RgCHS  Cf U2
t
x
2
x
H UH

 e wU
t
x
CH UCH

 v s (Es  roC)
t
x
qb
h
(1  p )
 v s (roC  Es )
t
x
The quasi-steady assumption cannot be used for flows that develop
rapidly in time, such as a surge-type turbidity current.
16
FLOW OF CALCULATION USING THE QUASI-STEADY
APPROXIMATION
The bed profile h(x) is known at time t:
u
Compute S = - h/x
Compute the flow over this bed by solving
the equations below
bed at time t
h
U2H
1
CH2
  Rg
 RgCHS  Cf U2
x
2
x
UH
 e wU
x
UCH
 v s (Es  roC)
x
Once U, C and H are known, compute
the new bed profile at t + t by solving
the Exner equation:
qb
h
(1  p )
 v s (roC  Es )
t
x
bed at time t + t
h
17
GENERALIZATION OF THE FORMULATION FOR
SEDIMENT SIZE MIXTURES
We divide the range of grain sizes into N bins I = 1 to N. The volume
concentration of suspended sediment in each bin is Ci, so that the total
concentration CT is given as:
N
C T   Ci
i1
The volume suspended load transport rate per unit width qsi and the fraction
of sediment in the suspended load in the ith grain size range psi are:
qsi  UHCi
qsi
Ci
, psi 

CT qsT
N
N
i1
i1
, qsT   qsi  UH Ci  UHCT
Esi
Using the active layer concept
introduced in Chapter 4 of Parker (2004;
e-book), the bed is divided into a surface
active layer of thickness Ls and a
substrate below. The surface has no
vertical structure: the fraction of
sediment in the ith grain size range in
z'
the bed surface is Fi
Dsi
qsi
qsi
qbi
qbi
h
x
La
Fi
18
GENERALIZATION OF THE FORMULATION FOR
SEDIMENT SIZE MIXTURES contd.
We further define the volume bedload transport per unit and the fraction
bedload in the ith grain size as qbi and pbi, where
q
pbi  bi
qbT
N
, qbT   qbi
i1
The volume rates per unit time per unit bed area Esi and Dsi of erosion into
suspension and deposition from suspension are given as
Esi  v siFE
i usi
,
Dsi  v siroiCi
Esi
qsi
qsi
qbi
where vsi is the fall velocity for the ith
grain size range, Eusi is a unit
entrainment rate for the ith grain size
range, and roi = cbi/Ci, where cbi is the
near-bed concentration in the ith grain
size range.
qbi
h
x
z'
Dsi
La
Fi
19
1D EXNER EQUATION FOR MIXTURES
The equation takes the form
qbi

 

(1  p )  fIi (h  La )  FL
 Di  Ei
i a   
t
x
 t

where fIi denotes the fraction in the ith range of the sediment that
interchanges between the surface layer and the substrate below as the bed
aggrades or degrades. Reducing with the forms below,:
Esi  v siFE
i usi
,
Dsi  v siroiCi
Dsi
qsi
qsi
qbi
it is found that:

 

(1  p )  fIi (h  La )  FL

i a  
t
 t

q
 bi  v si (roiCi  FE
i si )
x
Esi
qbi
La
surface layer
h
Fi
x
z'
substrate
20
REDUCTION OF THE EXNER EQUATION FOR MIXTURES
By definition,
N
N
F  p
i1
i
i1
si
N
N
i 1
i 1
  pbi   fIi
Summing
qbi

 

(1  p )  fIi (h  La )  FL


 v si (roiCi  Esi )
i a 
t
x
 t

over all grain sizes yields the Exner formulation for bed evolution.
q
h
(1  p )
  bT  DT  ET
t
x
N
,
DT   v siroiC
i1
i
N
,
ET   v siFE
i usi
i1
Between the second and third equations above the following equation can
be derived for the time evolution of the grain size distribution of the surface
layer:
L 
q
q
 F
21
(1  p ) La i  Fi  fIi  a    bi  fIi bT  v si (roiCi  FE
i si )  fIi ( DT  ET )
t 
x
x
 t
INTERFACIAL EXCHANGE FRACTIONS fIi
Closure relations for fIi, roi Esi and qbi need to be specified in order to
implement the formulation for mixtures. The substrate fractions below the
surface layer are denoted as fi. Note that fi can vary as a function of
elevation within the substrate z, so reflecting the stratigraphic architecture
of the deposit.
h

f
,
0
i z hL

a
t
fIi  
F  (1   )(p  p ) , h  0
bi
si
 i
t
where 0    1 (Hoey and Ferguson, 1994; Toro-Escobar et al., 1996).
That is:
The substrate is mined as the bed degrades.
A mixture of surface and bedload material is transferred to the substrate as
the bed aggrades, making stratigraphy.
Stratigraphy (vertical variation of the grain size distribution of the substrate)
needs to be stored in memory as bed aggrades in order to compute
subsequent degradation.
22
THE PARAMETER Eusi
Garcia and Parker (1991) generalized their relation for entrainment in
rivers to sediment mixtures. The relation for mixtures takes the form
5
ui
us
0.6  Di 
Zui  m
Repi 

v si
 D50 
E
AZ
Eusi  si 
,
Fi 1  A Z5
0.3 ui
m  1  0.298 , A  1.3x10 7
0.2
, Repi 
RgDi Di

where Di denotes the characteristic grain size of the ith range and D50 is a
median size of the sediment in the active layer.
Wright and Parker (2004) amended the above relation so as to apply to
larger scale. as well as the types previously considered by Garcia and
Parker (1991). The relation is the same as that of Garcia and Parker
(1991) except for the following amendments:
 us
0.6  0.08  Di 
Zui  m 
Repi  Se 

v
D
 si

 50 
0.2
A  7.8x107
where Se is an energy slope. Both these relations apply only to noncohesive sediment, and have not been verified for turbidity currents.
23
THE PARAMETERS roi AND qbi
The parameter roi is not very well constrained for turbidity currents. In the
lack of an alternative, the relation given in Lecture 8 can be generalized
to mixtures as:
1.46
 u 
roi  1  31.5  
 v si 
This relation was introduced by Parker (1982) based on the vertical
distribution of suspended sediment in a river proposed by Rouse (1939).
A review of bedload transport relations for sediment mixtures is given in
Parker (2004, e-book). A sample relation is that of Ashida and Michiue
1
(1972):

D 
D

bi


i

ci
q  17   


i

ci
  

ci
scg
scg  0.05
Dsg  2
s
N
,
 s    iFi
i1
where
qbi 
qbi
RgDi Di
,
 0.843 i  for i  0.4
D 
Dsg

 sg 

2


 

 log(19) 
Di
for
 0 .4
 

D

sg
 log19 Di  
  Dsg  

 
b
u2
 

sRgDi RgDi

i
24
LINKAGE TO THE EQUATIONS OF MOTION
In order to link to the Exner formulation for sediment mixtures, the
equations of motion need to be modified in a straightforward way. In the
case of the 3-equation model, the equations become:
CTH2
UH U2H
1

  Rg
 RgCTHS  Cf U2
t
x
2
x
H UH

 e wU
t
x
CH
UCH
i
i

 v si (FE
i usi  roiCi )
t
x
In the 4-equation model, the equation for K generalizes to:
KH UKH
1

 u2U  U3 e w  oH
t
x
2
N
1
1
RgH (v siCi )  RgCHUe w  RgH [v si (FE
i usi  roiCi )]
2
2
i1
25
REFERENCES
Ashida, K. and M. Michiue, 1972, Study on hydraulic resistance and bedload transport rate in
alluvial streams, Transactions, Japan Society of Civil Engineering, 206: 59-69 (in Japanese).
García, M., and G. Parker, 1991, Entrainment of bed sediment into suspension, Journal of
Hydraulic Engineering, 117(4): 414-435.
Hay, A. E., 1987, Turbidity currents and submarine channel formation in Rupert Inlet, British
Columbia, Canada 1. Surge observations. Journal of Geophysical Research, 92(C3), 29752881.
Hay, A. E., 1987, Turbidity currents and submarine channel formation in Rupert Inlet, British
Columbia, Canada 1. The roles of continuous and surge-type flow. Journal of Geophysical
Research, 92(C3), 2883-2900.
Hoey, T. B., and R. I. Ferguson, 1994, Numerical simulation of downstream fining by selective
transport in gravel bed rivers: Model development and illustration, Water Resources
Research, 30, 2251-2260.
Parker, G., 1982, Conditions for the ignition of catastrophically erosive turbidity currents. Marine
Geology, 46, pp. 307-327, 1982.
Parker, G., 2004, ID Sediment Transport Morphodynamics, with applications to Fluvial and
Subaqueous Fans and Fan-Deltas, http://cee.uiuc.edu/people/parkerg/morphodynamics_ebook.htm .
Poling, G. W., Ellis, D. V., Murray, J. W., Parsons, T. R. and Pelletier, C. A., 2002, Underwater
tailing placement at Island Copper Mine: A Success Story. SME, 216 p.
Rouse, H., 1939, Experiments on the mechanics of sediment suspension, Proceedings 5th
International Congress on Applied Mechanics, Cambridge, Mass,, 550-554.
26
REFERENCES contd.
Spinewine, B., Sequeiros, O. E., Garcia, M. H., Beaubouef, R. T., Sun, T., Savoye, B. and Parker,
G., Experiments on internal deltas created by density currents in submarine minibasins. Part
II: Morphodynamic evolution of the delta and associated bedforms. submitted 2008,
Sedimentology.
Toro-Escobar, C. M., C. Paola, G. Parker, P. R. Wilcock, and J. B. Southard, 2000, Experiments
on downstream fining of gravel. II: Wide and sandy runs, Journal of Hydraulic Engineering,
126(3): 198-208.
de Vries, M. 1965, Considerations about non-steady bed-load transport in open channels.
Proceedings, 11th Congress, International Association for Hydraulic Research, Leningrad:
381-388.
Wright, S. and G. Parker, 2004, Flow resistance and suspended load in sand-bed
rivers: simplified stratification model, Journal of Hydraulic Engineering, 130(8), 796-805.
27