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2
» »
4μA 1 2π 2π x 1 1 2π b
(ρ1 ’ ρ2 )gw = ’ + ’
cos tanh
» » »
sinh 2πb cosh 2πb sinh 2πb cosh 2π b
b 2π b 2π b
» » » »
(7.10)

is evaluated at the unperturbed position of the Solving Eq. (7.10) for A 1 and substituting the re-
interface (i.e. y = 0). The value of v 1 at y = 0 is sult into Eq. (7.6) gives
obtained by differentiating Eq. (7.4) with respect
‚w (ρ1 ’ ρ2 )gb
to x and evaluating the resulting expression at =w
‚t 4μ
y = 0. The result is
2π b ’1
»2
tanh 2π b ’ sinh 2π b cosh
‚w » » »
2π A 1 2π x — 2πb
= 2πb ’1
cos (7.6) »
+ sinh 2π b cosh
‚t » » » »
2πb
(7.11)
In order to constrain the value of the coef-
¬cient A 1 in Eq. (7.6), we must incorporate the Equation (7.11) describes an exponential growth
role of buoyancy on ¬‚uid motion. As the interface process with time scale „ given by
between the two ¬‚uids is disturbed downward
2π b ’1
»
(or upward) the less (more) dense ¬‚uid will dis- + sinh 2πb
cosh
4μ » »
„= 2πb
place the other. The pressure difference between ’1
(ρ1 ’ρ2 )gb 2
»
tanh 2πb ’ sinh 2πb
cosh 2π b
» » »
2πb
the two sides of the interface resulting from this
(7.12)
buoyancy force is given by
Equation (7.12) provides the necessary relation-
2(P 1 ) y=0 = ’(ρ1 ’ ρ2 )gw (7.7)
ship between the time scale of the instability and
The pressure on y = 0 in the upper layer can the wavelength ».
be found by using a force balance equation that Figure 7.3 plots the relationship between
equates the pressure gradient in the ¬‚ow with the dimensionless growth rate of a perturba-
the viscous stresses: tion (i.e. 1/„ ) and its dimensionless wavelength
‚P ‚ 2u ‚ 2u 2πb/». The fastest growing wavelength can be
’μ +2 =0 (7.8)
determined by differentiating Eq. (7.12) as a
‚x ‚ x2 ‚y
164 INSTABILITIES


ratio of meander wavelength and channel width
100
is a constant equal to about ten for these diverse
lmax= 2.568b
phenomena (Leopold et al., 1964). This similarity
suggests a common mechanism for meandering.
10
(r1 ’ r2)gbt




In this section we present a linear stability
analysis which predicts channel migration rates
4




and the proportionality between meander wave-
10 length and channel width using a very simple
geometric approach that applies to both river
channels and lava channels. Most models of allu-
vial channel meandering stress the importance of
10
102
100 101
10 the cross-sectional circulation or secondary ¬‚ow
2pb/l
in meander initiation (Callander, 1978; Rhoads
Fig 7.3 Plot of the inverse of the dimensionless time scale „ and Welford, 1991). This model does not replace
as a function of the dimensionless wavelength for the
those more sophisticated models, but rather es-
Rayleigh“Taylor instability (Eq. (7.12)).
tablishes suf¬cient conditions for meandering to
occur in a wide variety of channel ¬‚ows. Mean-
function of „ and setting the result equal to zero. ders of the correct scaling form when the bank
shear stress (in the case of sediment transport)
Turcotte and Schubert (2002) give the result of
or temperature (in the case of thermal erosion)
that equation as
is proportional to the curvature of the bank and
»max = 2.568b (7.13)
cross-sectional variations in shear stress or tem-
perature have a linear form. For alluvial chan-
where »max is the fastest-growing wavelength. Per-
nels, this curvature dependence arises from the
turbations of this wavelength have a growth time
centripetal force of ¬‚uid rounding a bend, and
scale given by
in thermal erosion it arises through the excess
13.04μ
„= latent heat of a curved bank. A linear stability
(7.14)
(ρ1 ’ ρ2 )gb analysis predicts that channels are unstable to
growth at all wavelengths larger than a critical
Equation (7.12) predicts that the fastest growing
wavelength given by » = 2π w, where w is chan-
wavelength is equal to about two and half times
nel width, and stable below it (provided that
the layer thickness. The Rayleigh--Taylor instabil-
some channel ¬‚ows exceed the critical condition
ity provides a classic example of a linear stability
required for erosion). The wavelength of fastest
analysis. We will use the linear stability analysis
growth predicted by this analysis is equal to
procedure again in Section 7.5, to explore the in- √
2 3πw or 10.88w, which is very similar to ob-
stability mechanism that creates oscillating arid
servations from alluvial and lava channels.
alluvial channels.
In alluvial rivers, the dependence of channel
migration on the local curvature of the bed has
been stressed in measurements of channel mi-
7.3 A simple model for river
gration by Nanson and Hickin (1983) and in the-
meandering oretical studies by Begin (1981) and Howard and
Knutson (1984). Nanson and Hickin (1983) found
Alluvial rivers, supraglacial meltwater streams, that channel migration rates were a maximum
within a narrow interval of R /w near 3, where
Gulf Stream meanders, and lava channels on
Earth, the Moon, and Venus (Komatsu and Baker, R is the radius of curvature of the meander,
1994) all exhibit meandering with similar propor- and w is the channel width. We will compare
tionality between meander wavelength and chan- our theoretical results to their data. Begin (1981)
nel width (Leopold et al., 1964; Stommel, 1965; showed, by computing the centripetal force nec-
Parker, 1975; Komatsu and Bakerm, 1994). The essary to accelerate ¬‚ow around a bend, that the
7.3 A SIMPLE MODEL FOR RIVER MEANDERING 165



v
t Fig 7.4 Schematic diagram of the
w2 meander instability model
y calculation.
w
0




l

shear stress at the bed is proportional to the cur- Eqs. (7.15) and (7.17) for the cross-sectional and
vature. Howard and Knutson (1984) developed a along-channel variation in shear stress gives
simulation model of meandering where the mi- 2
„ (x, y) = „b,s (1 + w K ) 1 ’ y + „c (7.18)
gration rates were proportional to local curvature w
up to some maximum value. They obtained real-
The shape of the curved-channel bank within
istic meandering channels with their model.
this linear-stability approach is given by yb =
In Figure 7.4 we present the geometry to be
sin(2π x/»). The curvature K is given by
considered in our model. We assume symmetry
d yb /dx 2 . Substituting the sinusoidal bank shape
2
within our 2D model, i.e. that the forces and ero-
into Eq. (7.18) gives
sion on one bank are equal in magnitude and
2
opposite in direction to those on the other bank. 2 2π
„ (x, y) = „b,s 1’ y 1+w
»
First we consider the distribution of boundary w
shear stress in a straight channel of width w. The 2π x
— sin + „c (7.19)
fundamental assumption of this analysis is that »
the cross-sectional shear stress pro¬le is linear,
We further assume that spatial variations in bed
decreasing from a maximum value at the bank
shear stress produce erosion and accretion of the
(y = w/2) to a minimum value at the channel
bank proportional to the variation in shear stress.
centerline:
In other words, the bank will erode in locations
2
„ (y) = „b,s 1 ’ y + „c where the variation in shear stress downriver is
(7.15)
w positive, and a point bar will prograde in places
where „b,s is the maximum shear stress at the where variations in shear stress are negative. The
bank for a straight channel and „c is the mini- rate of migration, therefore, is proportional to
the derivative of „ (x, y) with respect to x, evalu-
mum shear stress at the centerline. In a curving
ated at y = w/2:
channel, empirical data indicate that the bank
shear stress is equal to (Richardson, 2002) ‚„ d 2π x
= sin
‚x »
dt
w 2
y=w/2
„b,c = „b,s +1 (7.16)
2R 2
w 2π 2 2π x
= „b,s ’ sin
where „b,c refers to the bank shear stress for a » »
2 w
curved channel, and R is the radius of curvature (7.20)
of the channel centerline. For incipient mean-
The normalized growth rate of perturbations of
ders, the radius of curvature of the centerline
the bed is given by
is much larger than the channel width. As such,
we can use the approximation (1 + )2 ≈ 1 + 2 2
™ 2π„b,s 2 w 2π
= ’ (7.21)
to rewrite Eq. (7.16) as » »
w 2
w
„b,c ≈ „b,s 1 + = „b,s (1 + w K ) (7.17) which can be simpli¬ed to
R
2
™ 1 2π w
where K is the planform curvature of the cen-
∝ 1’ (7.22)
» »
terline (equal to the inverse of R ). Combining
166 INSTABILITIES


is dif¬cult, however, to compare migration rates
0.8
from many different bends without ¬rst quantify-
lmax = 10.88w
ing the effects of bank-material texture. Lumping
migration rate (m/yr)



0.6
data from meander bends with different erodi-
bilities would increase the scatter compared to
the Nanson and Hickin data, even if individual
0.4
bends closely follow a universal growth curve. Mi-
gration rate data from within individual bends
would help to resolve this question.
0.2
The analysis that led to Eq. (7.22) also ap-
plies to channels carved by thermal erosion.
0.0 Supraglacial meltwater streams (Parker, 1975)
40
30
10 20
0
l/w and lava channels on the Earth, the Moon, and
Venus (Hulme, 1973) form by thermal erosion. In
Fig 7.5 Plot of the normalized growth rate of perturbations the case of lava channels, melting of the channel
as a function of wavelength » (Eq. (7.22)). Also shown (dots) bed by hot lava (and solidi¬cation on the opposite
are the bank-migration data of Nanson and Hickin (1983)
bank) causes channel migration to occur. Curved
along the Beatton River for comparison.
banks have a melting point that depends on the
curvature of the bank. The melting temperature
of the bank is given by
Equation (7.22) is plotted in Figure 7.5 as the solid
curve. The maximum growth rate occurs where
T = Tm + c K (7.23)
the ¬rst derivative of ™/√ with respect to » is
zero. This occurs at » = 2 3πw = 10.88w, which
where T m is the melting point for a straight chan-
is close to the observed scaling relation (Leopold
nel, K is the planform curvature of the bank, and
et al., 1964).
c is a constant that depends on the latent heat of
For comparison, Figure 7.5 shows observed mi-
the bank material. The temperature required to
gration rate data for the Beatton River from Nan-
melt a curved bank is elevated because eroding
son and Hickin (1983). These authors employed
the bank by a given amount requires melting a
dendrochronological techniques to estimate the
larger volume of material per unit surface area
migration rates for sixteen bends of that river.
than for a straight channel. As such, the tempera-
They expressed their data as a function of the
ture boundary condition in lava channels has the
average radius of curvature of the bend. To com-
same curvature dependence as the shear stress
pare their results to our theoretical growth curve,
boundary condition in alluvial channels. In ad-
we used the relation » = π R which is applica-
dition, turbulent eddies will transport heat away
ble for an ideal shape of fully developed me-
from the hot center of the ¬‚ow towards the cool
anders: the sine-generated curve (Langbein and
sides of the ¬‚ow in the same way that turbu-
Leopold, 1966). Their results show the same de-
lent eddies transport shear stress through an al-
pendence with meander wavelength predicted
luvial channel, resulting in an approximately lin-
by our linear stability analysis. The migration
ear temperature pro¬le. These underlying simi-
rate rises rapidly for small »/w, reaches a maxi-
larities between alluvial and lava channels sug-
mum value, and then decreases. Migration rates
gests that the same geometrical instability mech-
decrease rapidly for tightly curved bends with
anism is at work in both cases.
»/w < 2π , consistent with our model™s prediction
that channel bends become stable below that
wavelength. It should be noted that some stud- 7.4 Werner™s model for eolian
ies following Nanson and Hickin (1983) showed
dunes
that migration rates are far less systematic that
the results obtained for the Beatton River. Most
of those studies have concluded that the Nanson Eolian ripples, dunes, and megadunes obey a
and Hickin results are not generally applicable. It striking periodicity. Many attempts have been
7.4 WERNER™S MODEL FOR EOLIAN DUNES 167


made to understanding the controls on eolian Folk, 1976). Wilson (1972) has proposed the same
bedform spacing in terms of both microscale pro- mechanism for ripples and megadunes with the
cesses (e.g. the trajectories of individual grains bedforms on these scales created by different
in saltation and reptation) and macroscale pro- scales of atmospheric circulation. However, care-
cesses (e.g. the interaction between a growing ful ¬eld studies have not found evidence for the
bedform, the wind ¬‚ow above it, and the result- existence of these persistent atmospheric circula-
ing pattern of erosion and deposition). Despite tions (Livingstone, 1986).
the efforts of many scientists, the mechanisms In 1995, Werner made a signi¬cant advance in
responsible for the formation of eolian ripples, eolian geomorphology in a paper that described
dunes, and megadunes and the factors responsi- a very simple but powerful model of eolian dune
ble for their size and spacing are not fully un- formation (Werner, 1995). Werner™s model begins
derstood. Perhaps the most well studied type of with a set of discrete sand slabs on a rectangular
bedform is the eolian ripple. Bagnold (1941) per- grid. Slabs may represent individual sand grains
formed the pioneering work on the formation of or a collection of grains; the number of slabs on
eolian ripples. He proposed that ripples form by the grid is limited for computational reasons, but
a geometrical instability in which a perturbation the model is not sensitive to the size of each slab.
exposes the windward side of the perturbation During each time step of the model, a sand slab is
to more impacts and faster surface creep than picked up from a randomly chosen pixel. If there
the leeward side, resulting in more grains being is no sand at that pixel, nothing happens. If a
ejected into saltation on the windward slope. In slab is present, it is picked up and moved down-
his theory, the spacing of ripples is related to a wind a distance of l grid point and deposited
characteristic saltation distance that is constant back on the surface with a probability that de-
in time. Sharp (1963), however, observed that in- pends on the presence or absence of sand at the
cipient ripples increase in spacing over time until new location. If sand is present at the new loca-
a steady-state spacing is achieved. In addition, ex- tion, the slab is deposited with probability ps . If
perimental studies of grain impacts suggest that no sand is present, it is deposited with a lower
a wide distribution of energies are imparted to probability pns . In nature, sand is more likely to
grains on the bed during each impact, resulting be deposited on a patch of sand than on a bare
in a wide distribution of saltation path lengths desert surface because the boundary layer above
(Mitha et al., 1986). As such, Bagnold™s hypothesis a sandy surface is usually rougher than above a
is not well supported by available data, and the bare, smooth surface, and because energy is ab-
controls on ripple spacing are still not well un- sorbed from the saltating grain by the granular
derstood. Anderson and Bunas (1992) developed bed. If the slab is not deposited, it is transported
a model that produced realistic ripples based on another distance l and again deposited with a
probability given by ps or pns . One additional rule
the microscale processes of grain movement and
controls the probability of deposition: if a sand
bed impact, but the spacing of ripples in their
slab lands in a ˜˜shadow zone,™™ it is deposited with
model was controlled by an ad hoc ˜˜ceiling™™ in
100% probability. A shadow zone is de¬ned as the
the model domain that does not occur in nature.
domain downwind of any topographic high that
The mechanisms controlling the formation
lies below a plane with an angle of 15—¦ to the
and spacing of eolian dunes are also incom-
horizontal. To map the shadow zone, a path is
pletely understood. Eolian dunes represent a dis-
traced from every local ridge down a 15—¦ slope
tinct bedform type from eolian ripples since
in the downwind direction until the plane in-
they occur at a larger scale with no transitional
tersects the surface. The 15—¦ angle of the shadow
bedforms. It has been proposed that dunes re-
zone can be varied, but its value should be signif-
sult from the presence of pre-existing wave-like
icantly less than the angle of repose. Physically,
motions or secondary circulations in the atmo-
the shadow zone represents the zone of recircu-
spheric boundary layer. Periodic motions intrin-
lation on the lee side of a growing dune. In this
sic to the ¬‚ow or resulting from the response
zone, air¬‚ow velocities are greatly reduced and
of the ¬‚ow to a topographic disturbance upwind
any sand in saltation will likely be deposited.
may produce periodic bedforms (Wilson, 1972;
168 INSTABILITIES



wind Fig 7.6 A schematic diagram of
l Werner™s model for eolian dune
shadow
formation and evolution. After
zone
Werner (1995).

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