1

LECTURE 11INTRODUCTION TO TURBIDITY CURRENT

MORPHODYNAMICS

CEE 598, GEOL 593TURBIDITY CURRENTS: MORPHODYNAMICS AND DEPOSITS

0.3

0.4

0.5

0.6

after 16 runs

0.3

0.4

0.5

0.6

after 20 runs

0.3

0.4

0.5

0.6

after 25 runs

0.3

0.4

0.5

0.6

after 30 runs

1.5 2 2.5 3 3.5 40.3

0.4

0.5

0.6

after 33 runs

Top: photo showing the deposit of lightweight plastic sediment formed by the repeated passage of saline underflows (analogs of turbidity currents).

Left: time evolution of the bed.

From Spinewine et al. (submitted)

bed

saline underflowambient fresh water

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THE CASE OF RUPERT INLET

From Poling et al. (2002)

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.

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THE CASE OF RUPERT INLET contd.

Photo: http://gateway.uvic.ca/archives/featured_collections/esa/fonds_island_copper_mines/default.html

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.

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THE REAL-TIME CONSTRUCTION OF A MINI-SUBMARINE 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).

From Hay (1987a)

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THE TURBIDITY CURRENT FORMED AN EXTENDED MEANDERING CHANNEL

From Hay (1987b)meander bends

channel axis

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LONG PROFILE, RELIEF AND WIDTH OF THE CHANNEL

Relief ~ vertical distance from levee top to channel bottom ~ channel depth.

From Hay (1987b)

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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)

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THE ACOUSTIC IMAGING SHOWED MORPHODYNAMICS IN ACTION!

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.

channel bed

fish!

approximate interface of turbidity current

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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 ballistictrajectory just above bed.role of turbulence is indirect. Suspended load:feels direct dispersive effect of eddies.may be wafted high into the water column.

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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.

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x = nearly horizontal boundary-attached “streamwise” coordinate [L]

z = nearly vertical coordinate upward normal from boundary [L]

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.

xy

h

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1D EXNER EQUATION FOR THE CONSERVATION OF BED SEDIMENT: SOME PARAMETERS

Parameters:

qs = volume suspended load transport rate per unit width [L2T-1] = UCHqb = 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]

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1D EXNER EQUATION FOR THE CONSERVATION OF BED SEDIMENT; DERIVATION

s p

s b b s s sx x x

(1 ) x 1t

q q 1 x 1

h D E

Es = vsEs = volume rate per unit time per unit bed area that sediment isentrained 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].

bp s s

q(1 )

t x

h

D E-

h

qb qb

x x xx

1

Es Ds

bed sediment + pores

turbidity current

u

Time rate of change of sediment mass in control volume =deposition rate from suspension – erosion rate into suspension + net inflow rate of bedload

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REDUCTION OF THE EXNER EQUATION

Since Es = vsE and Ds =vsroC, the equation reduces to:

Compare this relation with the equation of consevation of suspended sediment:

Since qs = UCH, the Exner equation can be rewritten as:

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)

bp s o s

q(1 ) v (r C E )

t x

h

-

s s o

CH UCHv (E r C)

t x

b sp

q q CH(1 )

t x x t

h

-

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COUPLING OF THE EXNER EQUATION TO THE EQUATIONS GOVERNING THE FLOW

2 22

f

w

s s o

UH U H 1 CHRg RgCHS C U

t x 2 xH UH

e Ut xCH UCH

v (E r C)t x

Example: 3-equation model:

bp s o s

q(1 ) v (r C E )

t x

h

-

w 1 2

fs 2 2

s s

2b b f

3s

RCgHe fn ( ) ,

U

C UuE fn fn

v v

q C Ufn ( ) ,

RgD RgDRgDD

Ri Ri

where the closure relations are:

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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):

The quasi-steady assumption cannot be used for flows that develop rapidly in time, such as a surge-type turbidity current.

2 22

f

w

s s o

UH U H 1 CHRg RgCHS C U

t x 2 xH UH

e Ut xCH UCH

v (E r C)t x

bp s o s

q(1 ) v (r C E )

t x

h

-

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FLOW OF CALCULATION USING THE QUASI-STEADY APPROXIMATION

The bed profile h(x) is known at time t:

Compute S = - h/x

Compute the flow over this bed by solvingthe equations below

Once U, C and H are known, computethe new bed profile at t + t by solvingthe Exner equation:

bp s o s

q(1 ) v (r C E )

t x

h

-

2 22

f

w

s s o

U H 1 CHRg RgCHS C U

x 2 xUH

e Ux

UCHv (E r C)

x

h

bed at time tu

bed at time t + t

h

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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:

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:

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 the bed surface is Fi

x

h

z'

qbi qbi La

Esi Dsi

Fi

qsiqsi

N

T ii 1

C C

N Nsii

si i si sT si i Ti 1 i 1T sT

qCq UHC , p , q q UH C UHC

C q

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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

The volume rates per unit time per unit bed area Esi and Dsi of erosion into suspension and deposition from suspension are given as

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.

x

h

z'

qbi qbi La

Esi Dsi

Fi

qsiqsi

Nbi

bi bT bii 1bT

qp , q q

q

si si i usi si si oi iv FE , v r C E D

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1D EXNER EQUATION FOR MIXTURES

The equation takes the form

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,:

it is found that:

x

h

z'

qbi qbi La

Esi Dsi

Fi

qsiqsi

si si i usi si si oi iv FE , v r C E D

bip Ii a i a i i

q(1 ) f ( L ) FL

t t x

h D E

p Ii a i a

bisi oi i i si

(1 ) f ( L ) FLt t

qv (r C FE )

x

h

surface layer

substrate

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REDUCTION OF THE EXNER EQUATION FOR MIXTURES

By definition,

Summing

over all grain sizes yields the Exner formulation for bed evolution.

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:

N N N N

i si bi Iii 1 i 1 i 1 i 1

F p p f

bip Ii a i a si oi i si

q(1 ) f ( L ) FL v (r C E )

t t x

h

N NbT

p T T T si oi T si i usii 1 i 1i

q(1 ) , v r C , v FE

t x

h

D E D E

a bi bTip a i Ii Ii si oi i i si Ii T T

L q qF(1 ) L F f f v (r C FE ) f ( )

t t x x

D E

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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.

ai z L

Ii

i bi si

f , 0tf

F (1 )(p p ) , 0t

h

h h

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.

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THE PARAMETER Eusi

Garcia and Parker (1991) generalized their relation for entrainment in rivers to sediment mixtures. The relation for mixtures takes the form

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:

where Se is an energy slope. Both these relations apply only to non-cohesive sediment, and have not been verified for turbidity currents.

0.25i i0.6si ui s i

usi ui m pi pi5i si 50ui

7m

RgD DE AZ u DE , Z ,

AF v D1 Z0.3

1 0.298 , A 1.3x10

Re Re

0.2

0.6 0.08s iui m pi e

si 50

u DZ S

v D

Re 710x8.7A

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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:

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 (1972):

1.46

oisi

ur 1 31.5

v

ciiciibi 17q

4.0D

Dfor

DD

19log

)19log(

4.0D

Dfor

D

D843.0

sg

i

2

sg

i

sg

i

1

sg

i

scg

ci05.0scg

N

1iiissg F,2D s

2bi b

bi is i ii i

q uq ,

RgD RgDRgD D

where

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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:

In the 4-equation model, the equation for K generalizes to:

LINKAGE TO THE EQUATIONS OF MOTION

222T

T f

w

i isi i usi oi i

C HUH U H 1Rg RgC HS C U

t x 2 xH UH

e Ut xCH UCH

v (FE r C )t x

2 3w o

N

si i w si i usi oi ii 1

KH UKH 1u U U e H

t x 21 1

RgH (v C ) RgCHUe RgH [v (FE r C )]2 2

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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), 2975-2881.

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_e-book.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.

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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.