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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
dz
dx
" b = !ghS
S =!
" b = !C f u 2
u
!1 2
= Cf
u*
u* =
"b
!
!u
is usually small; exceptions: dam break floods, etc.
!t
!u
!h
Non-Uniform: u
; "g
-- important in flow around bends, over point bars,
!x
!x
Note: Unsteady:
across abrupt changes in channel slope or width
2. Velocity Profiles
u (z ) =
u*
z
ln
k zo
)u ( =
#
u* & h
$$ ln ' 1!! = u
k % zo "
"* =
"b
(! s # ! )gD
z =.37 h
3. Sediment Transport
Shield’s Stress
steady-uniform flow:
"* =
hS
((! s # ! ) ! )D
Critical Shear Stress for Initiation of Motion
Shield’s diagram ! *cr = f (Rep )
(explicit particle Reynold’s number)
For Gravel: # *cr " 0.03 ! 0.06
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September, 2004
Non-Dimensional Sediment Transport
qs =
qs* =
Qs
w
(( !
qs
s
" ! ) ! )gD D
=
qs
Rep#
Bedload Transport (gravel)
q s* = 8(! * " ! cr* )
32
Total Load Sand Transport
qs* =
0.05 2. 5
!
Cf *
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
!Vs = qs _ in !t!y " qs _ out !t!y
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Change in bed volume per unit time
#Vbed = #x#y#z =
q #t#y " qs _ out #t#y
#Vs
= s _ in
1 " !p
1 " !p
For change in bed elevation, divide through by ΔxΔyΔt:
'z
1 & qs _ in ( qs _ out
$
=
't (1 ( ) p )$%
'x
#
1 & 'qs #
!! = (
(1 ( ) p )$% 'x !"
"
!z
1 !qs
="
(1 " # p ) !x
!t
Note from basics of sediment transport that qs = f (! b )
Accordingly, we can expand the erosion equation using the chain
rule:
!z
1 !qs !# b
="
(1 " $ p )!# b !x
!t
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 channel
width evolves downstream.
1. Hydraulic Geometry (Leopold et al, 1950s)
Empirical: w ! Q 0.5
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
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September, 2004
Critical or Equilibrium Condition: ! * = (1 + " )! *c ; " = 0.2 ! 0.4
Summary Condition at Bankfull flow: Mobile bed, stable banks, generally low transport stage
Thus, for given Qw, D50, S, Cf, Width (w) increases downstream such that
h reduces to establish ! * = (1 + " )! *c
3. Sandy, Meandering Channels
Theory not well developed, but some evidence indicates near constant
shield’s stress:
! * = (1 + " )! *c ; " = 1.2 ! 1.4
Otherwise, generally the assumption that ! * >> ! *c is often reasonably
accurate.
D. Relations for Alluvial Plain Slope (Paola, 1992; with corrections)
DEFINITION SKETCH
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September, 2004
1. Conservation of Mass (Water)
Q = u hw
Q
w
= qv = u h
= u h!
Vw
Vw
; "!
w
Vw
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.
!Q
=0
!x
2. Conservation of Mass (Sediment)
QS = q s w
Qs
w
= q sv = q s
= qs !
Vw
Vw
where qsv is sediment transport per unit valley width [m2/s]
!z
1 !q sv
="
!t
1 " # p !x
3. Conservation of Momentum
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" b = !ghS
S =!
" b = !C f u 2
Cf
!1 2
September, 2004
dz
dx
u
u*
=
4. Sediment Transport (Bedload = gravel)
qs
q s* =
((" s # " ) " )gD D
= 8(! * # ! cr* )
32
Dimensional sediment flux per unit channel width:
qs = 8
qs =
"
&
) ' ) cr #
!!
% (( s ' ( )gD "
((( s ' ( ) ( )gD D$$
32
32
8
(! # ! cr )3 2
((" s # " ) " )g
5. Channel width closure:
Braided, gravel-bedded channel --
! = (1 + " )! cr
& ( #
) ' ) cr = ) $
!
%1+ ( "
Meandering, sand-bedded channel –
! " ! cr = !
6. Relation for sediment flux using channel closure rule(s)
qs =
"
32
8c w
!32
((" s # " ) " )g
3
& ' #2
cw = $
!
%1 + ' "
Braided, gravel-bedded channels
cw = 1
Meandering, sand-bedded channels
Write this relation in terms of slope and water discharge per unit valley width, qv
! 3 2 = !! 1 2
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Using two relations for conservation of momentum:
" 3 2 = $ !gh
#z
(!C f u 2 )1 2
#x
Using conservation of mass for water:
% 3 2 = " $ 3 2 g C f hu
q !z
!z
= "$ 3 2 g C f v
!x
# !x
Substitute into relation for sediment flux (per unit channel width):
qs = "
!z
$ ((# s " # ) # ) !x
8c w q v C f
qs = " K f
!z
!x
; Kf =
8c w q v C f
" ((! s # ! ) ! )
(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)
!z
1 !q sv
="
!t
1 " # p !x
recall qsv is sediment transport per unit valley width
q sv = #q s = " #K f
!z
!x
'z
1 ' &
'z #
=
$ )K f
!
't 1 ( * p 'x %
'x "
Assume: constant Vw, Cf, q (and note β is not constant, but cancels out)
!z $K f ! 2 z
=
!t 1 " # p !x 2
(diffusion equation)
Effective Diffusivity:
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Df =
#K f
1$ !p
=
September, 2004
8c w "q v C f
(" s
$ " )(1 $ ! p )
!z
!2z
= Df 2
!t
!x
Sediment inflow (delivered from upstream erosional source area) is Qso.
Steady-state condition:
!q
!z
= 0 = sv
!t
!x
Thus sediment flux per unit valley width qsv is constant at steady state:
qsv (x )= qsv
x =0
=
Qso
Vw
Whether steady-state or not, slope at inflow must be sufficient to carry all
sediment:
qsv
x =0
=
Qso
!z
= " #K f
Vw
!x
x =0
Thus inlet slope must be (under all conditions):
(z
(x
x =0
='
& () ' ) )
Q so
Q
Q * (() s ' ) ) ) )
= ' so = ' so
= '$ s
V w *K f
wK f
$ 8c w ) C f
w8c w q v C f
%
#Q
! so
! Q
"
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:
$K f ! 2 z
!z
=0=
!t
1 " # p !x 2
"z
"z
= const =
"x
"x
x =0
=!
Qso
wK f
(linear profile, slope = inlet slope)
SKETCHES
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September, 2004
8. Effect of subsidence (σ) on steady-state profile
$K f ! 2 z
!z
= "% +
!t
1 " # p !x 2
At steady state, the rate of aggradation must balance subsidence, here uniform in
space.
Boundary conditions
Qs
x =0
= Qso
;
Qs
x=L
= Qse
Gives
& , Q
Qs (x ) = Qso $1 - **1 - se
%$ + Qso
) x#
'' !
( L "!
;
linear decline from Qso to Qse
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:
"=
Qso # Qse
(1 # ! p )Vw L
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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:
L=
Qso # Qse
(1 # " p )Vw!
; 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:
& x#
Qs (x ) = Qso $1 ' ! ;
% L"
with L =
Qso
(1 # " p )Vw!
Qs (x ) = Qso # x(1 # " p )Vw! ; which gives Qs = 0 at x = L, as required.
From above, we know that locally the slope is linearly dependent on sediment
flux:
& () ' ) )
Q
(z
= ' s = '$ s
(x
wK f
$ 8c w ) C f
%
#Q
! so
! Q
"
; thus substituting the relation for Qs(x):
Qso $ x(1 $ # p )Vw"
x(1 $ # p )"
Q
Q
%z
=$ s =$
=$ s +
%x
wK f
wK f
wK f
!K f
$
(1 $ # p )" x
%z
Q
= S = so $
%x
wK f
!K f
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Naturally, we can find the same result by direct integration of the conservation of
mass (or erosion-transport) equation:
Steady state condition
$K f ! 2 z
!z
= 0 = "% +
!t
1 " # p !x 2
;
%=
$K f ! 2 z
1 " # p !x 2
Integrate once:
"z (1 ! % p )$
=
x + const
"x
#K f
const =
"z
"x
x =0
=!
Qso
wK f
Q
"z (1 ! % p )$
=
x ! so
"x
#K f
wK f
S =$
(1 $ # p )" x
Q
%z
= so $
%x wK f
!K f
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.
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9. Effect of Uplift on Steady-State Profile
$K f ! 2 z
!z
=U +
!t
1 " # p !x 2
Directly analogous, opposite sign on source term, so jump to solution:
S =#
(1 # " p )U x
Q
$z
= so +
$x wK f
!K f
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 pieshape with apex angle θf.
SKETCH
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September, 2004
Boundary conditions are:
Qs
r =0
= Qso
;
Qs
r =L
= Qse
Gives
& , Q
Qs (r ) = Qso $1 - **1 - se
$% + Qso
), r )
''* '
(+ L (
2
#
!
!"
;
non-linear decline from Qso to Qse because fan
Area grows with square distance downfan, r2
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:
#=
Qso $ Qse
(1 $ " p )! f (L2 2)
Solving for steady-state fan size:
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K. Whipple
L=
Qso $ Qse
(1 $ # p )" f (! 2)
September, 2004
.
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:
& , r )2 #
Qs (r ) = Qso $1 - * ' ! ;
$% + L ( !"
Qs (r ) = Qso $
(1 $ # )" ! r
p
2
f
with L =
Qso
(1 $ # )" (! 2)
p
2
f
; which gives Qs = 0 at r = L, as required.
From above, we know that locally the slope is linearly dependent on sediment
flux:
"z
Q
=! s
"r
wK f
; (no difference from alluvial plain case) thus substituting the
relation for Qs(r):
(1 $ # p )" f ! r 2
Q $ (1 $ # p )" f ! 2 2
%z
Q (r )
Q
= $ s = $ so
r =$ s +
%r
wK f
wK f
wK f
2 wK f
$
(1 $ # p )" f ! r 2
%z
Q
=S= s $
%r
wK f
2 wK f
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.
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11. System Response Time to Perturbation
Response time is well known for Diffusion equation (see Paola et al. 1992)
Df =
Teq =
"K f
1# !p
L2
Df
(Effective Diffusivity)
; 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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