(Eq. 6.168) to obtain the lower flow depth corresponding to the base level of flow
resistance.
The ratio ðh c =h
0
c Þ after some manipulation can be written as a function of the
relative roughness (D 84 /h), or the inverse of relative submergence, and grain sorting
(D 84 /D 50 )
h c
h
0
c
¼
a 0
a 1
3=2 D 84
D 50
1=4
1 þ
a 1
a 2
2 D 84
h
5=3
"
# 3=4
ð6:172Þ
This equation predicts that as relative submergence increase, ðh=h
0
Þ decline
towards asymptotic values between 1.4 and 2 depending on bed sorting, for deep
flows. From a set of values of (h c /D 84 ), the correspondent sets of values of ðh c =h
0
c Þ
and ðh c =h
0
c Þ can be obtained from Eq. (6.172) and Eq. (6.165). The corresponding
set of channel slopes can be obtained using the definition given above for the
Shields parameter (Eq. 6.152)
S ¼
ðs À 1Þh c D 50
h c
¼
ðs À 1Þ h c =h
0
c
À
Á
h
0
c
h c =D 84
ð
Þ D 84 =D 50
ð
Þ
ð6:173Þ
Equation (6.173) allows obtaining critical Shields parameter with trial values of
(h c /D 84 ).
Ferguson (2012) follows the dimensionless form of Parker et al. (2011) in
Eq. (6.155), for derivation and comparison of relationships between dimensionless
critical specific stream power x * , and e.g., channel slope or median grain size D 50 .
A dependency between x
Ã
c and (U/u * ) was proposed, e.g., by Eaton and Church
(2011) as follows:
x c
Ã
¼ h C
3=2 U
u Ã
ð6:174Þ
with ratio (U/u * ) obtained from Eq. (6.168). Equation (6.172) was also explored by
Ferguson (2012) for prediction of curves for critical shear stress variation, h c
obtained from Eq. (6.174) or (6.153), with channel slope, different grain sizes, and
with trends from mechanistic models of Lamb et al. (2008) relating critical Shield
shear stress as a function of channel flow.
Ferguson (2005) proposed a theoretical based approach for the estimation of
critical stream power, x c , as follows:
x c ¼
2:3q
j
h c ðs À 1ÞgD S
½
3=2 log
30h c
expð1Þj s
!
ð6:175Þ
with j being the von Karman constant of 0.4, s the relative sediment density of
about 2.65, d the grain size, h c the critical water depth at the onset of motion, and j s
the roughness height of a 2D s order of magnitude.
6.5 Mass Transfer
231
resistance.
The ratio ðh c =h
0
c Þ after some manipulation can be written as a function of the
relative roughness (D 84 /h), or the inverse of relative submergence, and grain sorting
(D 84 /D 50 )
h c
h
0
c
¼
a 0
a 1
3=2 D 84
D 50
1=4
1 þ
a 1
a 2
2 D 84
h
5=3
"
# 3=4
ð6:172Þ
This equation predicts that as relative submergence increase, ðh=h
0
Þ decline
towards asymptotic values between 1.4 and 2 depending on bed sorting, for deep
flows. From a set of values of (h c /D 84 ), the correspondent sets of values of ðh c =h
0
c Þ
and ðh c =h
0
c Þ can be obtained from Eq. (6.172) and Eq. (6.165). The corresponding
set of channel slopes can be obtained using the definition given above for the
Shields parameter (Eq. 6.152)
S ¼
ðs À 1Þh c D 50
h c
¼
ðs À 1Þ h c =h
0
c
À
Á
h
0
c
h c =D 84
ð
Þ D 84 =D 50
ð
Þ
ð6:173Þ
Equation (6.173) allows obtaining critical Shields parameter with trial values of
(h c /D 84 ).
Ferguson (2012) follows the dimensionless form of Parker et al. (2011) in
Eq. (6.155), for derivation and comparison of relationships between dimensionless
critical specific stream power x * , and e.g., channel slope or median grain size D 50 .
A dependency between x
Ã
c and (U/u * ) was proposed, e.g., by Eaton and Church
(2011) as follows:
x c
Ã
¼ h C
3=2 U
u Ã
ð6:174Þ
with ratio (U/u * ) obtained from Eq. (6.168). Equation (6.172) was also explored by
Ferguson (2012) for prediction of curves for critical shear stress variation, h c
obtained from Eq. (6.174) or (6.153), with channel slope, different grain sizes, and
with trends from mechanistic models of Lamb et al. (2008) relating critical Shield
shear stress as a function of channel flow.
Ferguson (2005) proposed a theoretical based approach for the estimation of
critical stream power, x c , as follows:
x c ¼
2:3q
j
h c ðs À 1ÞgD S
½
3=2 log
30h c
expð1Þj s
!
ð6:175Þ
with j being the von Karman constant of 0.4, s the relative sediment density of
about 2.65, d the grain size, h c the critical water depth at the onset of motion, and j s
the roughness height of a 2D s order of magnitude.
6.5 Mass Transfer
231