V. Evolution of Depositional Alluvial River Profiles ...

12.163/12.463 Surface Processes and Landscape Evolution

K. Whipple September, 2004

V. Evolution of Depositional Alluvial River Profiles

A. Recap: Essentials from Flow Mechanics and Sediment Transport

1. Conservation of Momentum, Steady Uniform Flow

ghSb

!" =dx

dzS !=

2uC

fb!" =

21

*

!

=f

Cu

u

!

"b

u =*

Note: Unsteady: t

u

!

!

x

uu!

!

is usually small; exceptions: dam break floods, etc.

Non-Uniform: ; x

hg!

!" -- important in flow around bends, over point bars,

across abrupt changes in channel slope or width

2. Velocity Profiles

( )o

z

z

k

uzu ln

*=

hz

o

uz

h

k

uu

37.

*1ln

==!!

"

#$$%

&'=()

3. Sediment Transport

Shield’s Stress

( )gDs

b

!!

""

#=

*

steady-uniform flow:

( )( )DhS

s!!!

"#

=*

Critical Shear Stress for Initiation of Motion

Shield’s diagram

(explicit particle Reynold’s number)

For Gravel:

( )epcr

Rf=*

!

06.003.0*

!"cr

#

1



Non-Dimensional Sediment Transport

w

Qq

s

s=

qs* =qs

!s " !( ) !( )gDD=

qs

Rep#

Bedload Transport (gravel)

( ) 23

***8

crsq !! "=

Total Load Sand Transport

qs* =0.05

Cf

!*

2.5

B. Exner Equation (Erosion Equation): Conservation of Mass (Sediment)

Derivation of conservation of mass – the erosion equation

SKETCH: Control reach, width Δy, length Δx, qs_in at x, qs_out at x + Δx

If more sediment comes in than out, bed elevation goes up – deposition

If more sediment goes out than in, bed elevation goes down – erosion

per unit time Δt, sediment volume in = qs_inΔtΔy

per unit time Δt, sediment volume out = qs_outΔtΔy

Change in sediment volume per unit time

ytqytqVoutsinss

!!"!!=!__

2



Change in bed volume per unit time

p

outsins

p

s

bed

ytqytqVzyxV

!! "

##"##=

"

#=###=#

11

__

For change in bed elevation, divide through by ΔxΔyΔt:

( ) ( )!"#

$%&''

((=!!

"

#$$%

&'(

(=

''

x

q

x

qq

t

zs

p

outsins

p)) 1

1

1

1 __

( ) x

q

t

zs

p!

!

""=

!

!

#1

1

Note from basics of sediment transport that

Accordingly, we can expand the erosion equation using the chain

rule:

( )bs

fq !=

( ) x

q

t

zb

b

s

p!

!

!

!

""=

!

! #

#$1

1

Thus we can see immediately that erosion will occur in regions of

increasing shear stress (i.e., not necessarily in regions of high

shear stress) and deposition will occur in regions of decreasing

shear stress (not necessarily low shear stress).

C. Channel Width Closure

Problem: all relations above derived in terms of flow depth, shear stress,discharge per unit width, but channel width changes downstream with: changing Qw, changing slope (S), changing D50, changing vegetation, etc.:To solve problem of alluvial river profile evolution we must specify how channelwidth evolves downstream.

1. Hydraulic Geometry (Leopold et al, 1950s)

Empirical: 5.0

Qw!

2. Equilibrium (Threshold) Straight, Gravel-bed Channels (Parker, 1978)

Concept: channel will widen until banks are just stable, just below the

threshold for motion (erosion = widening)

SKETCH

3



Critical or Equilibrium Condition: ; ( )c**

1 !"! += 4.02.0 !="

Summary Condition at Bankfull flow: Mobile bed, stable banks, generally low transport stage

Thus, for given Qw, D50, S, Cf, Width

( )c**

1 !"! +=

(w) increases downstream such thath reduces to establish

3. Sandy, Meandering Channels

Theory not well developed, but some evidence indicates near constantshield’s stress:

( )c**

1 !"! += ; 4.12.1 !="

Otherwise, generally the assumption that c**

!! >> is often reasonably

accurate.

D. Relations for Alluvial Plain Slope (Paola, 1992; with corrections)

DEFINITION SKETCH

4



1. Conservation of Mass (Water)

hwuQ =

!huV

whuq

V

Q

w

v

w

=== ;

wV

w!"

Note qv denotes water discharge per unit valley width, as opposed to q which we

have used previously for water discharge per unit channel width.

Parallel drainage: Vw = constant; no lateral inflows of water or sediment, no loss

of water to infiltration or evaporation.

0=!

!

x

Q

2. Conservation of Mass (Sediment)

wqQsS

=

!s

w

ssv

w

sq

V

wqq

V

Q===

where qsv is sediment transport per unit valley width [m2/s]

x

q

t

zsv

p!

!

""=

!

!

#1

1

3. Conservation of Momentum

5



ghSb

!" =dx

dzS !=

2uC

fb!" =

*

21

u

uC

f=

!

4. Sediment Transport (Bedload = gravel)

( )( )( ) 23

***8

cr

s

s

s

DgD

qq !!

"""#=

#=

Dimensional sediment flux per unit channel width:

( )( )( )

23

8 !!"

#$$%

&''

'=gD

DgDq

s

cr

ss (())

(((

( )( )( ) 23

23

8

cr

s

s

gq !!

""""#

#=

5. Channel width closure:

Braided, gravel-bedded channel --

( )cr

!"! += 1

!"#

$%&

+='

((

)))1

cr

Meandering, sand-bedded channel –

!!! ="cr

6. Relation for sediment flux using channel closure rule(s)

( )( )23

23

8!

"""" g

cq

s

w

s

#=

2

3

1!"#

$%&

+=

''

wc

1=w

c

Braided, gravel-bedded channels

Meandering, sand-bedded channels

Write this relation in terms of slope and water discharge per unit valley width, qv

2123!!! =

6



Using two relations for conservation of momentum:

( ) 21223uC

x

zgh

f!!"

#

#$=

Using conservation of mass for water:

x

zqCg

x

zuhCg

v

ff

!

!"=

!

!"=

#$$% 232323

Substitute into relation for sediment flux (per unit channel width):

( )( ) x

zCqcq

s

fvw

s

!

!

""=

###$

8

x

zKq

fs

!

!"=

( )( )!!!" #=

s

fvw

f

CqcK

8

; (fluvial transport coefficient)

No dependence on grainsize? Why? -- Channel width closure

accounts for grainsize, ie. channel width adjusts according to

grainsize.

Note qs is sediment transport per unit channel width.

7. Conservation of Mass (sediment) (from above)

x

q

t

zsv

p!

!

""=

!

!

#1

1

recall qsv is sediment transport per unit valley width

x

zKqq

fssv

!

!"== ##

!"#

$%&

''

''

(=

''

x

zK

xt

zf

p

)*1

1

Assume: constant Vw, Cf, q (and note β is not constant, but cancels out)

(diffusion equation)

Effective Diffusivity:

2

2

1 x

zK

t

z

p

f

!

!

"=

!

!

#

$

7



( )( )ps

fvw

p

f

f

CqcKD

!""

"

!

#

$$=

$=

1

8

1

2

2

x

zD

t

zf

!

!=

!

!

Sediment inflow (delivered from upstream erosional source area) is Qso.

Steady-state condition:

x

q

t

zsv

!

!==

!

!0

Thus sediment flux per unit valley width qsv is constant at steady state:

( )w

so

xsvsv

V

Qqxq ==

=0

Whether steady-state or not, slope at inflow must be sufficient to carry all

sediment:

00 ==

!

!"==

xf

w

so

xsv

x

zK

V

Qq #

Thus inlet slope must be (under all conditions):

( )( ) ( )

Q

Q

CcCqcw

Q

wK

Q

KV

Q

x

zso

fw

s

fvw

sso

f

so

fw

so

x

!!

"

#

$$

%

& ''=

''='='=

((

=

)

)))))** 88

0

where the term in brackets collects physical constants. Inlet slope is linearly

dependent on the ratio Qso/Q. Note that valley width and sediment porosity do not

influence the inlet slope.

Steady state condition:

2

2

10

x

zK

t

z

p

f

!

!

"==

!

!

#

$

f

so

x

wK

Q

x

zconst

x

z!=

"

"==

"

"=0

(linear profile, slope = inlet slope)

SKETCHES

8



8. Effect of subsidence (σ) on steady-state profile

2

2

1 x

zK

t

z

p

f

!

!

"+"=

!

!

#

$%

At steady state, the rate of aggradation must balance subsidence, here uniform in

space.

Boundary conditions

; soxs

QQ ==0 seLxs

QQ ==

Gives

; linear decline from Qso to Qse ( )!!"

#

$$%

&''(

)**+

,--=

L

x

Q

QQxQ

so

se

sos11

And since at steady state, deposition must balance subsidence, the total volume

rate of deposition ((Qso – Qse)/(1-λp)) divided by alluvial plain area (LVw) equals

the subsidence rate:

( ) LV

QQ

wp

seso

!"

#

#=

1

9



Another way to think about this is to ask, what sets the length of the alluvial plain

(L)? The answer is obtained by solving the above relation for L. Basically the

total volume rate of deposition, valley width, and subsidence rate set the length of

the alluvial plain, or the distance of gravel progradation into a depositional basin:

( ) !"wp

seso

V

QQL

#

#=

1; for the case of gravel progradation, Qse = 0 would be the

condition of interest – all gravel is deposited in the gravel wedge.

So we have resolved the controls on the size of the system, and the pattern of

Qs(x) down the system. Now we can ask, ‘what sets the system slope and

longitudinal profile?’ In the case of zero aggradation (same as Qse = Qso), the

alluvial plain had a constant slope, linear profile. Intuitively, in the case of

subsidence balanced by aggradation, will the profile be concave-up or convex-up?

Recall that sediment flux declines linearly down the system in this case. For the

case of a gravel plain, Qse = 0 we have:

( ) !"

#$%

&'=

L

xQxQ

sos1 ;

( ) !"wp

so

V

QL

#=

1with

( ) ( ) !"wpsos

VxQxQ ##= 1 ; which gives Qs = 0 at x = L, as required.

From above, we know that locally the slope is linearly dependent on sediment

flux:

( )

Q

Q

CcwK

Q

x

zso

fw

s

f

s

!!

"

#

$$

%

& ''='=

((

)

))

8

; thus substituting the relation for Qs(x):

( ) ( )

f

p

f

s

f

wpso

f

s

K

x

wK

Q

wK

VxQ

wK

Q

x

z

!

"#"# $+$=

$$$=$=

%

% 11

( )x

KwK

QS

x

z

f

p

f

so

!

"#$$==

%

%$

1

10



Naturally, we can find the same result by direct integration of the conservation of

mass (or erosion-transport) equation:

Steady state condition

2

2

10

x

zK

t

z

p

f

!

!

"+"==

!

!

#

$% ;

2

2

1 x

zK

p

f

!

!

"=

#

$%

Integrate once:

( )constx

Kx

z

f

p

+!

="

"

#

$%1

f

so

x

wK

Q

x

zconst !=

"

"=

=0

( )

f

so

f

p

wK

Qx

Kx

z!

!=

"

"

#

$%1

( )x

KwK

Q

x

zS

f

p

f

so

!

"#$$=

%

%$=

1

Concave-up profile with slope decreasing linearly from the inlet (i.e., a

parabola) at a rate determined by the fluvial diffusivity, subsidence rate,

sediment porosity, and the channel width to valley width ratio. Note that

the limiting case of 100% porosity converges on the zero deposition

(subsidence) case, as it must.

11



9. Effect of Uplift on Steady-State Profile

2

2

1 x

zKU

t

z

p

f

!

!

"+=

!

!

#

$

Directly analogous, opposite sign on source term, so jump to solution:

( )x

K

U

wK

Q

x

zS

f

p

f

so

!

"#+=

$

$#=

1

Convex-up profile with slope increasing linearly from the inlet at a rate

determined by the fluvial diffusivity, uplift rate, sediment porosity, and the

channel width to valley width ratio.

Note this result is unrealistic in that an incising system will dissect the alluvial

plain into a dendritic drainage pattern – the result here assumes the channel

sweeps back and forth to maintain a one-dimensional form – a tilting plain – as is

the case for an aggradational system. We will discuss incising systems later.

10. Adaptation for Alluvial Fans at Steady-state (aggradation = uniform subsidence)

Assume radially symmetrical fan, r is radial position (from 0 to L), fan is pie-

shape with apex angle θf.

SKETCH

12



Boundary conditions are:

;

Gives

sorsQQ =

=0 seLrsQQ =

=

; non-linear decline from Qso to Qse because fan

Area grows with square distance downfan, r2

( )!!"

#

$$%

&'()

*+,''(

)**+

,--=

2

11L

r

Q

QQrQ

so

se

sos

Again, at steady state deposition must balance subsidence, the total volume rate of

deposition ((Qso – Qse)/(1-λp)) divided by alluvial fan area (θf L2/2) equals the

subsidence rate:

( ) ( )212

L

QQ

fp

seso

!"#

$

$=

Solving for steady-state fan size:

13



( ) ( )21 !"#fp

sesoQQ

L$

$= .

So we have resolved the controls on the size of the system, and the pattern of

Qs(r) down the system, and how these differ due to the expanding geometry of the

alluvial fan. Now we can ask, ‘what difference does the fan geometry make to the

longitudinal profile?’ Intuitively, will the profile be more or less concave than the

alluvial plain? Recall that sediment flux declines with the square of distance

down the fan in this case. For the case of a gravel fan merging with a playa for,

for instance, Qse = 0 we have:

( )!!"

#

$$%

&'()

*+,-=

2

1L

rQrQ

sos;

( ) ( )21 !"#fp

soQ

L$

=with

( )( )

2

2

1rQrQ

fp

sos

!"#$$= ; which gives Qs = 0 at r = L, as required.

From above, we know that locally the slope is linearly dependent on sediment

flux:

; (no difference from alluvial plain case) thus substituting the

relation for Qs(r):

f

s

wK

Q

r

z!=

"

"

( ) ( ) ( )22

2

121r

wKwK

Qr

wK

Q

wK

rQ

r

z

f

fp

f

s

f

fpso

f

s!"#!"# $

+$=$$

$=$=%

%

( )2

2

1r

wKwK

QS

r

z

f

fp

f

s!"#$

$==%

%$

Thus, the concavity of alluvial fans is quite distinct from that of alluvial plains:

near the fan-head the slope remains close to constant (linear profile) as Qs(r)

decreases slowly at first, then on the lower fan slope decreases rapidly to zero.

14



11. System Response Time to Perturbation

Response time is well known for Diffusion equation (see Paola et al. 1992)

(Effective Diffusivity)

p

f

f

KD

!

"

#=

1

f

eq

D

LT

2

= ; where L is system (alluvial plain) length.

View graphs: System response to slow (T > Teq) and fast (T < Teq) perturbations

(Figures from Paola et al., 1992).

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