Finite amplitude bars in mixed bedrock-alluvial channels

Authors


Abstract

We present a nonlinear asymptotic theory of fully developed flow and bed topography in a wide channel of constant curvature to describe finite amplitude perturbations of bottom topography, subject to an inerodible bedrock layer. The flow field is evaluated at the leading order of approximation as a slowly varying sequence of locally uniform flows, slightly perturbed by a weak curvature-induced secondary flow. Using the constraint of constant fluid discharge and sediment flux, we calculate an analytical solution for the cross-sectional profile of flow depth and bed topography, and we determine the average slope in the bend necessary to transport the sediment supplied from a straight, alluvial, upstream reach. Both fully alluvial bends and bends with partial bedrock exposure are shown to require a larger average slope than a straight upstream reach; the relative slope increase is much larger for mixed bedrock-alluvial bends. Curvature and sediment supply are shown to have a strong effect on the characteristics of the point bars in mixed bedrock-alluvial channels. Higher curvature bends produce bars of larger amplitude and more bedrock exposure through the cross section, and increasing the sediment supply leads to taller and wider point bars. Differences in the relative roughness of sediment and bedrock have a smaller, secondary effect on point bar characteristics. Our analytical approach can potentially be extended to the case of arbitrary, yet slowly varying, curvature, and should ultimately lead to an improved understanding of the formation of meanders in bedrock channels.

1 Introduction

Bedrock rivers play an important role in the evolution of active landscapes. Bedrock channel incision communicates feedback between regional tectonics, climate, and landscape evolution [Molnar and England, 1990; Tucker and Bras, 1998; Willett, 1999; Whipple and Tucker, 2002; Whipple, 2004]. One of the primary incisional mechanisms driving bedrock morphodynamics is abrasion by saltating sediment, as suggested from field observations [e.g., Cook et al., 2012], experiments [Sklar and Dietrich, 2001; Finnegan et al., 2007; Johnson and Whipple, 2007, 2010], and numerical modeling [Chatanantavet and Parker, 2009; Nelson and Seminara, 2011]. Because sediment can act as both “tools” that erode bedrock and “cover” that shields bedrock from abrasion [e.g., Gilbert, 1877; Sklar and Dietrich, 1998, 2001, 2004; Turowski et al., 2007; Turowski and Rickenmann, 2008; Turowski, 2009], it is necessary to understand how sediment becomes spatially distributed in mixed bedrock-alluvial rivers if we want to be able to predict their morphodynamic evolution.

Unlike fully alluvial channels, mixed bedrock-alluvial rivers characteristically have a sediment supply that is less than the channel's sediment transport capacity [Howard, 1994; Turowski et al., 2008]. Experiments in straight flume channels [Davis et al., 2005; Demeter et al., 2005; Finnegan et al., 2007; Johnson and Whipple, 2007; Chatanantavet and Parker, 2008; Johnson and Whipple, 2010] have suggested that bed load sediment tends to become spatially concentrated and form preferential transport paths. This phenomenon was explored theoretically by Nelson and Seminara [2012], who performed a linear stability analysis to show that this concentration of sediment emerges spontaneously due to feedback between the roughness differences between bed load sediment and bare bedrock and local sediment transport rates.

Experimental and theoretical work aimed at understanding that the distribution of sediment in mixed bedrock-alluvial rivers has focused on straight channels. Channel curvature, however, will provide additional mechanisms (e.g., beyond differential roughness) for distributing sediment that can lead to characteristic patterns of deposition and the concomitant feedback on channel erosion and morphodynamic evolution. Curvature is known to play a major role in the morphodynamic patterns that develop in alluvial rivers. Curvature-induced secondary flows that are directed inward at the bed [e.g., Smith and McLean, 1984] lead to deposition on the inner bend and the formation of point bars, the presence of which feeds back on the flow field through topographic steeringeffects [e.g., Dietrich and Smith, 1984; Dietrich, 1987; Dietrich and Whiting, 1989; Nelson and Smith, 1989]. The effects of channel curvature on the flow field are also thought to have a sorting effect on sediment mixtures, as gravitational forces opposing the curvature-induced secondary circulations are stronger for coarse grains than for fine grains [Bridge and Jarvis, 1976, 1982; Parker and Andrews, 1985; Seminara et al., 1997; Clayton and Pitlick, 2007; Nelson et al., 2010]. Linear and weakly nonlinear theories aimed at understanding bed topography in curved alluvial channels have provided insight on the initial instabilities responsible for the development of bars and meandering [Blondeaux and Seminara, 1985; Johannesson and Parker, 1989; Seminara and Tubino, 1992], but they assume that perturbations of bottom elevation associated with curvature effects are small relative to the unperturbed flow depth. Seminara and Solari [1998], and later Bolla Pittaluga et al. [2009], overcame this assumption by developing a nonlinear asymptotic theory of flow and bed topography in curved channels wherein a slowly varying sequence of locally uniform flows is slightly perturbed by a weak curvature-induced secondary flow. This approach does not require the assumption of small perturbation in bottom topography, thereby allowing calculation of the finite amplitude bed topography.

Although considerable progress has been made in understanding flow and bed topography in curved alluvial channels, the problem of curved mixed bedrock-alluvial channels has remained largely unexplored. To that end, in this study we ask the question: How does channel curvature influence sediment deposition and bedrock exposure in mixed bedrock-alluvial rivers? Because an analytical approach provides physical insight into the basic mechanisms at work, we adapt the approach of Seminara and Solari [1998] and Bolla Pittaluga et al. [2009], which, despite providing great insight on fully alluvial channels, cannot be applied to a channel where the sediment supply is less than the transport capacity or where bedrock is exposed.

In this paper, we develop a nonlinear asymptotic theory of flow and bed topography for mixed bedrock-alluvial channels, where we explicitly account for three novel features: (i) the existence of an interface between the bedrock and alluvial material, (ii) differential roughness effects of bedrock and sediment, and (iii) sediment supply that is less than the local sediment transport capacity. Additionally, to make the scaling of the problem more physically justifiable, the mathematical formulation of this system is scaled by average values in the channel bend (rather than upstream values), and we present an iterative procedure to arrive at a solution using this scheme.

We use this new modeling approach to investigate the mechanisms controlling the shape of the finite amplitude bed topography, the distribution of sediment and bedrock through the cross section, and the effects of roughness, curvature, and topography on the flow field through the bend. This work constitutes an initial step in developing a full understanding of the morphodynamics of meandering bedrock channels.

2 Nonlinear Theory of Fully Developed Flow and Sediment Transport in a ConstantCurvature Bedrock-Alluvial Channel Bend

In this section, we develop a mathematical model that determines the fully developed flow field and the finite amplitude bed topography in a bend of constant curvature for a channel that may have sediment supply less than the transport capacity and therefore partial bedrock exposure. The basic idea of the model is as follows: we consider a “wide” channel bend with a curvature that is “small” relative to the channel width (Figure 1). In the bend, curvature effects cause perturbations to the flow and sediment transport fields and produce a point bar at the inner bank. This bar forms atop a layer of bedrock (illustrated as a flat interface in Figure 1, but which can have an arbitrary shape), and the fluid discharge and sediment supply from upstream determine whether and how much bedrock is exposed, as well as the shape of the bar. The primary outputs of the model are the three-dimensional flow field in the bend, the shape of the channel cross section (including the point bar and possible exposed bedrock), and the slope and depth in the bend necessary to transport the sediment supply from upstream.

Figure 1.

Sketch and notation. (top right) The bend may be fully alluvial, where the maximum bedrock elevation remains below the deepest point of scour, or (bottom right) the bend may be mixed bedrock-alluvial, where some bedrock becomes exposed in the channel. The curvature parameter for this channel (ν) is approximately 0.05. Note that because we are assuming the channel is “wide” (β≫ 1), the analysis neglects the sidewalls, and the region spanning 2B does not extend fully to the banks.

In section 2.1, we formulate the morphodynamic problem by nondimensionalizing the governing equations of fluid mass and momentum and of sediment continuity, determining the boundary conditions for the flow and sediment transport fields and formulating integral constraints that enforce constant discharge and sediment flux from upstream to downstream. In section 2.2, a perturbation expansion is performed where the dimensionless variables are expanded in powers of a small parameter δ that is a function of the channel curvature, and in subsequent sections likewise powers of δ are equated to solve for the flow field and finite amplitude topography. Each order of approximation provides additional insight to the system. In section 2.2.1, the momentum equation in the downstream direction is solved at the leading order of approximation math formula to show that the downstream component of the velocity takes the form of the classical logarithmic distribution corrected by a wake function. In section 2.2.2, the cross-stream momentum and sediment continuity equations are solved at math formula to calculate the leading order curvature-induced secondary flow, the cross-stream water surface slope (i.e., the superelevation in the bend), and the leading order component of the depth (and therefore cross-sectional shape), accounting for the presence of bedrock. Here we also calculate the leading order component of the slope in the bend necessary to transport the sediment supplied from upstream. The curvature-induced secondary flow (calculated in section 2.2.2) perturbs the longitudinal flow field by transporting longitudinal momentum outward near the free surface and inward near the bed. This perturbation to the longitudinal velocity is calculated in section 2.2.3 by solving the downstream momentum equation at math formula. Finally, in section 2.2.4, the cross-stream momentum equation is solved at math formula to account for the effects of that longitudinal velocity correction on the centrifugal forces and the effects of depth and velocity perturbations on the eddy viscosity. Here we also solve the sediment continuity equation at math formula to determine the second-order correction to the depth, and we use the integral constraints to determine the first-order correction to the slope and depth in the bend necessary to transport the sediment supply.

2.1 Formulation of the Problem

Let us consider a wide channel bend of constant curvature downstream of a straight reach (Figure 1). We will refer the flow field to the intrinsic orthogonal curvilinear coordinate system (s, n, z) (we use asterisk superscripts to denote dimensional variables), where s denotes a longitudinal coordinate defined along the channel centerline, n is a transverse coordinate orthogonal to s, and z is a nearly vertical coordinate orthogonal to the (s,n) plane pointing upward.

We assume the channel centerline of the bend to be a circular helix with a constant radius of curvature math formula. The channel is assumed to have a constant width 2B through both the straight and curved reaches.

Let us then consider the steady and fully developed flow of constant discharge Q through the curved reach. Fully developed conditions are described mathematically by the condition that derivatives f/s=0, such that any function f describing a property of the flow field, sediment transport field, or bed topography does not vary in the downstream direction through the curved reach. As discussed in Seminara and Solari [1998], experimental observations [Kikkawa et al., 1976] suggest that fully developed conditions are reached in constant curvature channels about 6.5 channel widths downstream from the reach entrance.

The upstream straight reach is assumed to have a constant slope Su, and math formula, math formula, and Cfu denote the average speed, flow depth, and friction coefficient of the uniform flow field of the upstream straight reach. The centerline of the constant curvature bend is assumed to have a constant slope Sb. The flow field in the bend is characterized by uniform conditions math formula, math formula, and Cfb, which are perturbed by the effects of curvature, bed topography, and roughness as discussed below.

It is convenient to work with dimensionless quantities, so we scale the intrinsic coordinates, the local mean velocity averaged over turbulence u=(u,v,w)T, the flow depth D, the free surface elevation h, the mean pressure p, the eddy viscosity math formula, and the sediment flux per unit width math formula by the unperturbed uniform conditions within the channel bend:

display math(1)

Here ρ is the density of water, s is the relative particle density (s=ρs/ρ, where ρs is the density of the sediment), math formula is the particle diameter (assumed to be uniform), g is gravitational acceleration, βb is the aspect ratio of the channel bend, and Fb is the Froude number in the bend. We assume that small-scale bed forms are not present throughout the cohesionless portion of the bed. Hence Cfb is a friction coefficient taking the logarithmic form [Einstein, 1950]:

display math(2)

where the absolute bottom roughness height z0 is estimated as equal to math formula [Engelund and Hansen, 1967]. The dimensionless parameters βb and Fb are defined by

display math(3)

Note that the vertical component of the flow field is divided by βb to make it an order 1 quantity, a scaling suggested by flow continuity.

Using the above notation, we can write the governing equations for the conservation of fluid mass and momentum [e.g., Smith and McLean, 1984] and sediment continuity in dimensionless form:

display math(4a)
display math(4b)
display math(4c)
display math(4d)
display math(4e)

The longitudinal derivatives have all been set equal to zero in the equations of fluid continuity (4a), sediment continuity (4e), and the Reynolds equations (4b)(4d) to account for the fully developed conditions. Here ν is a curvature parameter and hs is a metric coefficient, which are defined as

display math(5)

We will assume that the channel is “wide” and “weakly curved,” that is

display math(6a)
display math(6b)

Assumption (6a) allows us to concentrate on the central region of the flow field and neglect the role of the sidewalls, while assumption (6b) allows us to treat the flow field as a slight perturbation to flow through a straight channel. This assumption does not require perturbations of bottom topography relative to a flat surface to be small, however.

Equations (4a)(4e) are subject to the following dimensionless boundary conditions:

display math(7a)
display math(7b)
display math(7c)
display math(7d)
display math(7e)

Equation (7a) imposes the no-slip condition at the bed roughness height z0, equation (7b) imposes conditions of no stress at the free surface, equation (7c) imposes the requirement that the free surface be a material surface, and equations (7d) and (7e) impose the constraint that the flux of both water and sediment vanishes at the banks.

To close the problem, we require closure relationships for the turbulent eddy viscosity νT and the sediment transport rate per unit width q. Because we have assumed that the channel is wide and the flow is fully developed, the flow field and bed topography are slowly varying in the lateral direction. We therefore may assume that the turbulent structure is in quasi-equilibrium and that the flow field is only slightly perturbed by weak curvature effects. Thus, the eddy viscosity can be represented as

display math(8)

where τ is the tangential stress vector at the bottom

display math(9)

D(n) is the local value of the dimensionless flow depth, and ζ is a normalized vertical coordinate that reads

display math(10)

With ζ0 the normalized roughness height and ζ=1 at the free surface, ζ will therefore occupy a range of values ζ0ζ≤1. The vertical distribution of eddy viscosity N(ζ) takes the classical parabolic distribution characteristic of uniform flows, corrected by Dean's [1974] wake function:

display math(11)

where k is the von Karman constant (k=0.41).

The closure for sediment transport follows the well-established semiempirical construct wherein the bed load flux rate is proportional to the excess dimensionless shear stress:

display math(12)

Here q=(qs,qn) is the dimensionless vectorial sediment transport rate, Ψ and α are empirical parameters (Ψ=8 and α=1.5 for the well-known Meyer-Peter and Müller [1948] relation), and θc is a critical value of a dimensionless boundary shear stress θ below which no significant sediment transport occurs. The dimensionless shear stress is defined as

display math(13)

Because we are considering fully developed conditions, only the cross-stream component of the sediment transport rate comes into the analysis (equations (4e) and (7e)). The direction of sediment transport deviates from the direction of the boundary shear stress because of gravitational forces acting on particles due to nonzero cross-stream bed slopes. Here we adopt the well-known structure [e.g., Parker, 1984]:

display math(14)

Here φ is defined in equation (12). The second term inside the brackets describes the gravitational effect of the cross-stream bed slope; although there is some uncertainty in the appropriate parameterization of this term [e.g., Sekine and Parker, 1992; Francalanci and Solari, 2008, Schuurman et al., 2013], here we adopt the commonly used value of 0.56 for r as suggested by the experimental study of Talmon et al. [1995].

Finally, the problem formulated above is subject to two integral constraints. The first constraint is that the fluid discharge in the straight upstream reach must equal the fluid discharge through the bend, that is

display math(15)

The second constraint imposes the condition that the total sediment flux through the bend must equal the rate of sediment supplied from upstream:

display math(16)

Here Cu is the average dimensionless areal concentration of sediment in the straight upstream reach [e.g., Nelson and Seminara, 2012]. For the fully alluvial upstream reach depicted in Figure 1, Cu=1.

Now, the governing equations for fluid and sediment (equations (4a)(4e)) may be rewritten in terms of (s,n,ζ), using the chain rules:

display math(17a)
display math(17b)

The equation for fluid continuity (4a) can then be integrated from the bottom to the free surface to obtain an expression for the vertical component of velocity as a function of depth, w(ζ):

display math(18)

When equation (18) is substituted into the continuity equation (4a) and the Reynolds equations (4b)(4c), we eliminate w from the equations and arrive at the final form of the integrodifferential equations:

s momentum

display math(19)

n momentum

display math(20)

Depth-averaged continuity

display math(21)

In equations (19)(21) the parameter δ is defined as

display math(22)

2.2 Solution for Channels of Constant Curvature

Following Seminara and Solari [1998] and Bolla Pittaluga et al. [2009], we now expand the solution in the neighborhood of the solution for flow in a straight channel with unknown bed cross-sectional shape, described by some unknown function D(n). Unlike those studies, which used characteristic upstream values (e.g., math formula) to make the governing equations dimensionless, we have scaled our variables using average values in the bend (e.g., math formula), which themselves are subject to the perturbation associated with the effects of channel curvature. We therefore perturb both the dimensional scaling parameters (math formula, and βb) and the dimensionless hydrodynamic variables (u,v,w,h, and D) around their locally uniform values:

display math(23)
display math(24)

Here δ is a small parameter defined by uniform conditions in the bend

display math(25)

Note that the parameter δ, which uses the leading order components of βb and Cfb, differs from δ (equation (22)). Because the scaling parameters are perturbed, it becomes necessary to adopt an iterative approach to solving the system of equations that emerge from the analysis, as discussed below.

The expansion in (23) and (24) is then substituted into the governing differential problems (19)(22), (4e), (7a)(7e), (8), and (12)(16). We then equate likewise powers of δ to obtain a sequence of differential problems to be solved in terms of the unknown function D.

2.2.1 Leading Order

At the leading order of approximation math formula, the s momentum equation (19) describes uniform flow in a straight channel with an unknown cross-stream distribution of flow depth D0(n):

display math(26a)
display math(26b)

We set

display math(27a)
display math(27b)

and we find

display math(28a)
display math(28b)

Equations (28a) and (28b) can be integrated immediately to obtain a solution for math formula, which describes the classical logarithmic velocity distribution corrected by a wake function

display math(29)

Here ζ0 is the normalized form of the no-slip roughness height:

display math(30a)
display math(30b)

where

display math(31)

2.2.2 First Order: n Momentum

At math formula, the n momentum equation (20) can be solved to give the leading order approximation of curvature-induced secondary flow. We find

display math(32a)
display math(32b)

We set

display math(33a)
display math(33b)

where math formula is the solution to the following problem:

display math(34a)
display math(34b)

A solution for math formula may be written in the form

display math(35)

where

display math(36)

and gi (i=0,1,2) is the solution to the following differential systems:

display math(37)

where

display math(38)

The system (37) has been solved analytically, but for the sake of brevity the solutions are not reported here.

In order to determine a1(n), we integrate the depth-averaged fluid continuity equation (21) at math formula over the cross section and apply the boundary condition w1|ζ=0=0 and find

display math(39)

Applying (39) to (35), we find

display math(40)

We now may determine the function D0(n) by solving the sediment continuity equation (4e) at leading order subject to the boundary condition (7e), which requires the cross-stream sediment flux to vanish everywhere (i.e., qn=0 for all n). We can therefore set the expression for qn (14) equal to zero to arrive at a differential equation for D0 at leading order:

display math(41)

Equation (41) will be solved subject to the math formula components of the expanded integral conditions for flow and sediment continuity (equations (15) and (16)) through the reach:

display math(42a)
display math(42b)

Here the leading order component of the dimensionless sediment transport capacity φb0 (using an exponent of n=3/2 in equation (12)) is

display math(43)

where

display math(44)

and

display math(45)

Equation (41) must be solved numerically, and this is done as follows. First, the leading order components of the mean dimensional depth math formula and the centerline slope Sb0 in the bend are estimated and used to compute the parameters Cfb0 (equation (31)), math formula (equation (45)), math formula, math formula, and ζb (equation (30b)). Then a local value of D0 at the inner bank is guessed, and a fourth-order Runge-Kutta scheme is used to calculate D0 from equation (41) by marching in n across the cross section.

During the calculation, a condition may arise in which the computed bed elevation is lower than the local bedrock elevation; this indicates a transition from alluvial bed material to exposed bedrock. Let n=ne denote the cross-stream coordinate of this transition at math formula (Figure 1), that is,

display math(46)

At this interface, the depth D can be expressed as

display math(47)

Taylor expansions can be used to determine the depth, water surface elevation, and bedrock elevation at n=ne:

display math(48a)
display math(48b)
display math(48c)

Combining (48) with (23) and (24), we arrive at the following expressions for D|ne and h|ne:

display math(49a)
display math(49b)

Equation (47) can now be written at leading order and first order:

display math(50)
display math(51)

Thus, at leading order, n0 can be determined while equation (41) is solved, and beyond this transition, the local value of D0 defaults to the math formula depth to bedrock, that is

display math(52)

Having calculated D0 for the entire cross section, the fluid integral condition (42a) is checked and used to update the initial guess for D0 at the inner bank. The process is repeated until the integral condition (42a) is met to within an acceptable tolerance.

At this point, equations (43)(45) are used to find the leading order component of the bed slope Sb0 in the bend necessary to satisfy the sediment integral constraint (42b), with the condition that locations where bedrock is exposed have no sediment flux. A new value for math formula is then calculated by solving the equation:

display math(53)

The procedure iterates by solving again for D0 and Sb0 until the changes from one iteration to the next become negligible.

2.2.3 First Order: s Momentum

The curvature-induced secondary flow computed in the previous section perturbs the downstream velocity by transporting longitudinal momentum outward near the free surface and inward near the bed. To calculate this effect, we consider the s momentum equation (19) at math formula. If we set

display math(54)

we can derive the following problem for which math formula is the solution:

display math(55a)
display math(55b)

where

display math(56a)
display math(56b)
display math(56c)

The solution for math formula takes the form

display math(57)

where

display math(58)

and f1j (j=0,1,2,3) are solutions to the following differential systems:

display math(59)

where

display math(60)

In the above,

display math(61)

The system (59) has been solved analytically, but again the results are not reported here for the sake of brevity.

2.2.4 Second Order

The n momentum equation (20) at O(δ2) includes many terms that correct the secondary flow. If we set

display math(62)

we can derive the following problem for which math formula is the solution:

display math(63a)
display math(63b)

where R1 and R2 are defined in (56a) and (56b), and

display math(64a)
display math(64b)

The solution for math formula takes the form

display math(65)

where

display math(66)

and g2j(j=0,5) are the solutions to the following differential systems:

display math(67)

where

display math(68)

The system (67) has been solved analytically but is not reported here for the sake of brevity.

The parameter a2 is found using a method similar to that used to determine a1. Integrating the depth-averaged fluid continuity equation (21) at math formula, one finds that

display math(69)

Combining this relation with (65), one finds

display math(70)

where

display math(71)

The sediment continuity equation requires the math formula component of qn (equation (14)) to be zero. Therefore, the expression for D1 becomes

display math(72)

Equation (72) is solved subject to the math formula components of the integral conditions (15) and (16):

display math(73a)
display math(73b)

where

display math(74)

and

display math(75)

Equation (72) is solved numerically in a manner similar to that of equation (41). First an initial guess is assigned to the first-order component of the mean depth in the bend math formula, and this value is used to compute the parameters

display math(76a)
display math(76b)
display math(76c)
display math(76d)

An initial guess is given to the value of D1 at the inner bank, and a fourth-order Runge-Kutta scheme is used to solve equation (72) by marching in n across the cross section. The cross-stream coordinate of the transition to exposed bedrock at math formula, ne=n0+δn1 can be determined by solving (51) for n1. Beyond this coordinate, D1 is set so that the local depth becomes the depth to bedrock, that is

display math(77)

When D1 has been calculated for the entire cross section, the fluid continuity integral condition (73a) is checked and used to update the initial guess for D1 at the inner bank. This is repeated until (73a) is met within the tolerance. The sediment continuity integral condition (73b) is then used to calculate a new estimate of math formula, the parameters (76a)(76d) are recalculated, and the procedure is repeated until changes are negligible.

3 Results

In the following sections, we explore the behavior of the model through example calculations of the flow field and finite amplitude bed topography. In these calculations, we have provided the model with parameters characteristic of the Bagmati River in central Nepal [Lavé and Avouac, 2001]: Su=0.028, math formula, 2B=200 m, math formula, and θu=0.121. To facilitate interpretation of the results, these figures present velocities and length scales nondimensionalized by the a priori known upstream values math formula and math formula.

3.1 Flow Field

3.1.1 Alluvial Bend

Figure 2 shows the computed flow field at leading, first, and second order for the case of an alluvial bend where the depth to bedrock exceeds the maximum scour depth. Figure 2 (top left) illustrates the leading order downstream velocity, which as described in equation (29) exhibits the classical logarithmic vertical profile. The first-order correction to the downstream flow field (Figure 2, top right) shows deceleration of the outer flow and acceleration of the inner flow. These effects are driven by the transverse component of longitudinal momentum, the metric decrease (increase) of channel slope toward the outer (inner) bank, and the topographic feedback of the finite amplitude bed deformation on the flow field.

Figure 2.

Components of the flow field computed for an alluvial bend (i.e., the bedrock elevation is lower than the maximum depth of scour) for ν=0.04, Su=0.0028, θu= 0.12, math formula, and z0r/z0s= 1. The plotted values are the dimensional components of the flow field scaled by the upstream velocity math formula. Black lines are contours where the component of velocity is zero.

The plots for the cross-stream and vertical components of the velocity in Figure 2 show the classical helical secondary flow, where the lateral flow near the bed is directed toward the inner bank and the lateral flow near the surface is directed toward the outer bank. This is further illustrated in Figure 3, which shows the downstream flow field at first order and vectors showing the secondary motion computed at second order. The high-velocity core is located toward the outer portion of the bend due to the math formulacorrection for u (Figure 2, top right).

Figure 3.

Computed downstream flow field math formula and cross-stream flow vectors for the alluvial channel bend in Figure 2, where ν=0.04, Su= 0.0028, math formula, and θu= 0.12.

3.1.2 Mixed Bedrock-Alluvial Bend

Figure 4 shows the flow field for a mixed bedrock-alluvial bend. The conditions used to generate Figure 4 are identical to those that were used to generate Figure 2, with the exception that the bedrock elevation is high enough to cause bedrock exposure on the outer portion of the bend, so comparison of the two figures illustrates the effects of bedrock exposure on the flow field through the bend.

Figure 4.

Components of the flow field computed for a mixed bedrock-alluvial bend for ν=0.04,Su= 0.0028,θu=0.12, math formula, and z0r/z0s= 1. The plotted values are the dimensional components of the flow field scaled by the upstream velocity math formula. Black lines are contours where the component of velocity is zero.

As in the alluvial case, the downstream component of the flow field at leading order shows the classical logarithmic vertical distribution. The first-order correction to the downstream velocity causes deceleration in the outer bend and acceleration in the inner bend, but because the finite amplitude bed deformation is smaller than in the fully alluvial case (that is, because the effects of topographic steering are reduced), the magnitude of the deceleration in u1 is lower than for the alluvial case and the region of deceleration in the outer bend extends further into the cross section. This leads to an inward shift of the high-velocity core relative to the fully alluvial case, which is illustrated in Figure 5b. As in the alluvial case, secondary flows in the mixed bedrock-alluvial bend exhibit a helical structure with inward flow near the bed and outward flow near the bank (Figures 4 and 5b); however, the magnitude of the second-order correction to these secondary flows is lower than for the alluvial case because of the weakened topographic feedback on the flow.

Figure 5.

Computed downstream flow field math formula and cross-stream flow vectors for the mixed bedrock-alluvial bend in Figure 4 with different ratios of bedrock roughness to sediment roughness z0r/z0s. All three figures are for conditions where ν = 0.04, Su = 0.0028, math formula, and θu = 0.12.

While Figure 4 illustrates the flow field for conditions where the bedrock roughness is equal to the sediment roughness (i.e., z0r/z0s=1), Figure 5 provides example results showing the effect of different bedrock and sediment roughnesses. Because differences in bedrock and sediment roughnesses have a minor effect on the finite amplitude bed topography in the bend (e.g., Figure 9a), topographic feedback on the flow fields plotted in Figure 5 are minimal. Figure 5 shows that the main effect of different bedrock and sediment roughnesses is on the location of the high-velocity core; when the rock is smoother than the sediment (Figure 5a), the high-velocity core shifts outward, and when the rock is rougher than the sediment (Figure 5c), the high-velocity core shifts inward. The overall pattern of secondary circulation is negligibly affected by the roughness difference.

3.2 Finite Amplitude Bed Topography

Figures 6-9 explore the effects of curvature (ν), upstream shear stress (θu), bedrock elevation (math formula), and relative bedrock roughness (z0r/z0s) on the finite amplitude topography in the bend.

Figure 6.

Finite amplitude topography calculated to math formula for an alluvial bend (i.e., the bedrock elevation is lower than the maximum depth of scour) for varying upstream dimensionless shear stress θu and constant curvature parameter ν. (a) The predicted cross-sectional bed profile in the bend (solid lines) and predicted free surface (dashed lines). (b) The leading order math formula (Sb0 and math formula) and the sum of the leading order and first-order math formula (Sb and math formula) contributions to the mean slope and depth in the bend, scaled by their upstream counterparts.

Figure 7.

Finite amplitude topography calculated to math formula for an alluvial bend (i.e., the bedrock elevation is lower than the maximum depth of scour) for constant upstream dimensionless shear stress θu and different values of the curvature parameter ν. (a) The predicted cross-sectional bed profile in the bend (solid lines) and predicted free surface (dashed lines). (b) The leading order math formula (Sb0 and math formula) and the sum of the leading order and first-order math formula (Sb and math formula) contributions to the mean slope and depth in the bend, scaled by their upstream counterparts.

Figure 8.

Finite amplitude topography calculated to math formula for different bedrock elevations with constant values of the curvature parameter ν and upstream dimensionless shear stress θu. (a) The predicted cross-sectional bed profile in the bend (solid lines) and predicted free surface (dashed lines). (b) The leading order math formula (Sb0 and math formula) and the sum of the leading order and first-order math formula (Sb and math formula) contributions to the mean slope and depth in the bend, scaled by their upstream counterparts.

Figure 9.

Finite amplitude topography calculated to math formula for different ratios of bedrock roughness to sediment roughness (z0r/z0s) for constant dimensionless bedrock elevation zbr, curvature parameter ν, and upstream dimensionless shear stress θu. (a) The predicted cross-sectional bed profile in the bend (solid lines) and predicted free surface (dashed lines). (b) The leading order math formula (Sb0 and math formula) and the sum of the leading order and first-order math formula (Sb and math formula) contributions to the mean slope and depth in the bend, scaled by their upstream counterparts. (c) The dimensionless unit sediment transport rate φb across the channel for two calculations, z0r/z0s = 0.1 and z0r/z0s = 1.0.

Figures 6a and 7a show the predicted finite amplitude cross-stream bed and water surface profiles in the downstream bend, for conditions where the bedrock elevation math formula is lower than the depth of maximum scour, so that the bend is fully alluvial. Increases in the upstream shear stress and the bend curvature are both shown to increase the amplitude of the point bar. Figure 6a shows that for the same bend curvature (i.e., constant ν), when the upstream dimensionless shear stress (θu) becomes larger, the magnitude of both deposition and scour in the bend increases, that is, the point bar becomes larger and the pool on the outer bend becomes deeper. Figure 7a shows the predicted finite amplitude cross-stream bed and water surface profiles in the bend for different bend curvatures (i.e., varying ν) and constant upstream dimensionless boundary shear stress θu. As the curvature increases (i.e., as the radius of curvature r0 becomes smaller and ν increases) the magnitude of scour and deposition increases. That is, tighter bends are predicted to have deeper pools and larger point bars.

When the bedrock elevation math formula is high enough that it exceeds the scour depth of an otherwise alluvial bend, the scour depth is limited to that elevation and bedrock becomes exposed on the outside portion of the bend. Figure 8a shows the predicted cross-sectional bed and water surface profiles in the bend for different bedrock elevations, where the curvature and upstream bed shear stress parameters ν and θu are held constant and the roughness of the bedrock and the sediment is the same (z0r/z0s=1). Figure 8a shows that as the bedrock interface elevation becomes higher, the dimensionless depth to the top of the bar on the inner bend and the cross-stream profile of the bar remain essentially the same as the fully alluvial case. The presence of bedrock forces the bar to occupy a smaller proportion of the bend cross-sectional area, however. Figure 9 illustrates the effect of bedrock roughness height z0r that differs from the roughness height of the sediment z0s. In Figure 9a, several cross-sectional bed and water surface profiles for bend have been calculated for different z0r/z0s ratios spanning an order of magnitude, while ν, θu, and zbr have been held constant. Although the differences between very smooth and fairly rough bedrock are subtle, in general, as the bedrock becomes more smooth relative to the sediment, the point bar becomes taller and wider, occupying more of the cross section of the bend.

Figures 6b, 7b, 8b, and 9b show how the slope of the channel centerline in the bend Sb and the mean flow depth in the bend math formula compare with their upstream counterparts. In these figures, we present both the leading order math formula (Sb0 and math formula) contributions and the sum of the leading order and first-order math formula (Sb and math formula).

At leading order, alluvial bends (Figures 6b and 7b) require a higher average depth and a lower slope than upstream in order to satisfy both integral constraints of constant fluid and sediment discharges (equations (42a) and (42b)). On the contrary, in mixed bedrock-alluvial bends (Figures 8b and 9b), the opposite behavior is found: a lower average depth and higher slope are required to satisfy both integral constraints. The interpretation of this result is readily obtained by noting that the fluid discharge integral constraint (42a) depends crucially on the lateral distribution of the flow depth at leading order D0(n): depending on the shape of this distribution (which is markedly different in the two cases), the average flow depth math formula may be larger or smaller than math formula. The ratio (Sb0/Su), according to equation (53), is proportional to a negative power of math formula; hence, values of (Sb0/Su) larger than one are associated with values of math formula smaller than one and vice versa.

At first order, the corrections math formula and (Sb1/Su) are, respectively, negative and positive for both alluvial and mixed bedrock-alluvial bends. The physical explanation for this trend becomes evident upon examination of the form of the perturbation of the Shields stress θ1 (equation (75)). Three contributions appear in the right-hand side of (75): the first is associated with momentum redistribution, the second with the correction of the average flow depth, and the third with the correction of the friction coefficient. Using (76c), the third contribution is readily seen to behave like the second, being proportional to math formula. The integral constraint (73b), with (74), then implies that the sign of math formula arises from the integrated effect of the redistribution term. Figures 2 and 4 suggest that this term (i.e., u1/ζ at ζ=ζ0) is negative in the outer region and positive in the inner region, but its intensity is larger in the outer region. This explains why math formula is invariably negative.

The magnitudes of these adjustments of the mean depth and slope in the bend increase with increasing shear stress (Figure 6b) and curvature (Figure 7b), and, for mixed bedrock-alluvial bends, increasing the elevation of the bedrock layer (Figure 8b). These adjustments are not really affected, however, by the roughness ratio z0r/z0s (Figure 9b). Figure 9c shows the distribution of dimensionless sediment transport rate φb across the bend for the case of equal bedrock and sediment roughness (z0r/z0s=1) and for the case where the bedrock is much smoother than the sediment (z0r/z0s=0.1). Relative to the equal roughness case, the reduced friction on the bedrock for the z0r/z0s=0.1 conditions reduces the shear stress on the point bar and consequently the unit sediment flux rates over the bar are lower. This is partially compensated by the larger dimensions of the point bar at steady state, so the net effect on the mean depth and slope in the bend is negligible.

In summary, in the mixed bedrock-alluvial case, the math formula correction simply enhances the trends in depth and slope obtained at leading order; in the alluvial case, the math formula correction is able to reverse the trend obtained at leading order. In other words, in both cases, the cumulative effect of the curvature-induced perturbations to the flow and sediment transport fields is such that bends exhibit lower average depths and higher slopes than the upstream reach.

4 Discussion

A comparison of results from the present model with those from the earlier work of Nelson and Seminara [2012] can provide insight on the relative effects of differential bedrock and sediment roughness and curvature on mixed bedrock-alluvial morphodynamics. Nelson and Seminara [2012] performed a linear stability analysis for a straight mixed bedrock-alluvial channel in which the only feedback between the sediment transport and the flow field was through differences in roughness of bedrock and sediment, and they showed that this roughness difference was responsible for sediment to begin to become locally concentrated on the channel bed. In addition to this differential roughness, the present model incorporates the effects of curvature and topographic steering on the flow, and the model results of Figures 6 and 9 show that these mechanisms have much greater influence on the equilibrium morphology of the bars than the difference in roughness between bedrock and sediment.

Our model results suggest that in mixed bedrock-alluvial bends, there is a characteristic relationship between the degree of curvature, the amount of sediment supplied from upstream, the morphology of the point bar, and the amount of bedrock exposure.

The points in Figure 10a show model calculations of the fraction of bedrock exposed in the bend as a function of the ratio of the sediment supply Qs to the total sediment transport capacity in the bend Qc. Curvature values ranging from 0.02≤ν≤0.08 were used in the computation and account for the limited scatter present in the results. The calculations show a clear, nearly linear, relationship between Qs/Qc and the fraction of bedrock exposed. This is in close agreement with the exposure term proposed by Sklar and Dietrich [2004], Fe=1−Qs/Qt, which is shown in Figure 10a as a dashed line. Note that the cover term in Sklar and Dietrich [2004] is an integrated effect that effectively makes their model zero-dimensional; here (and in the previous work of Nelson and Seminara [2011]) we have a spatial component such that locally the bed is either covered with sediment (exposure = 0) or bedrock is exposed (exposure = 1). By integrating the bedrock exposure over the cross section, we arrive at a value comparable with their definition of Fe, which we plot in Figure 10a.

Figure 10.

(a) Points show the fraction of bedrock exposed on the bed as a function of the ratio of the upstream sediment supply (Qs) to the total sediment transport capacity in the bend (Qc), computed for math formula and z0r/z0s = 1. The dashed line shows the linear cover relationship proposed by Sklar and Dietrich [2004]. (b) Dimensionless bar height as a function of curvature (ν) and dimensionless sediment supply (φu), computed for math formula, and z0r/z0s = 1.

Figure 10b shows, for the same set of calculations, the dimensionless bar amplitude (the difference in elevation between the bar top and the pool or bedrock surface, scaled by math formula) as a function of Qs/Qc and ν. Increasing bend curvature and increasing sediment supply both tend to cause the amplitude of point bars to increase. The characteristic relationship between sediment supply and bar morphology that emerges from Figure 10 points to a potentially useful model application in which local sediment supply in mixed bedrock-alluvial bends might be calculated simply from local measurements of bar morphology and channel curvature.

Ideally, one could compare the predictions of our model with detailed measurements of the three-dimensional flow field, water surface elevation, and bed topography collected from mixed bedrock-alluvial river bends covering a range of curvature, slope, and grain size. Unfortunately, field data of the detail necessary to fully validate the model are not available. Presently, the best available set of field observations that might be used to assess the model's performance comes from Nittrouer et al.'s [2011] study of the lower Mississippi River. Although not a classical bedrock river cutting through steep mountainous terrain, the lower Mississippi has been shown to have characteristics of a mixed bedrock-alluvial river, that is, sand is transported over a consolidated bedrock-like substratum, which becomes exposed at some locations along the river.

Nittrouer et al. [2011] do not present detailed velocity data, but they do present data for the lower 165 km of the Mississippi showing that sharp bends with low radii of curvature (what they call “tight-bend segments” where math formula, i.e., ν≥0.19) tended to have substratum (i.e., bedrock) exposed from bank to bank, bends of intermediate curvature (what they call “subtle-bend segments” where math formula, i.e., 0.029≤ν<0.19) exhibited a mixture of substratum and alluvial sand, and straight reaches (math formula, i.e., ν<0.029) were covered bank to bank with alluvial sand. We have provided the model with values characteristic of the lower Mississippi [Nittrouer et al., 2011] and varied the curvature parameter ν from 0 to 0.08; in Figure 11 we plot the fraction of the bedrock exposed as a function of curvature. The model predicts a threshold curvature value of ν=0.029 below which the bed is fully alluvial (i.e., no bedrock exposure) and above which the bedrock is partially exposed. This curvature threshold for the transition from alluvial to mixed bedrock-alluvial conditions nicely agrees with Nittrouer et al.'s [2011] observations. Furthermore, given parameters characteristic of bends in the lower Mississippi (math formula, Su=1×10−5, math formula, B=325 m, and Dbr=0.5, Cu=1), the model calculates a point bar with a finite amplitude height of about 25 m and sediment coverage spanning about 60% of the channel, which fall within the range of observations reported by Nittrouer et al. [2011]. The agreement of Figure 11 with Nittrouer et al.'s [2011] observations, along with the reasonable characteristic predicted morphology for lower Mississippi parameters, provides confidence that the model is capturing the essential dynamics of mixed bedrock-alluvial bends. Additional field and experimental studies where detailed three-dimensional flow fields, topographic data, grain sizes, and sediment transport rates for mixed bedrock-alluvial bends are measured are greatly needed to further improve our understanding of bedrock river morphodynamics.

Figure 11.

Fraction of bedrock exposed as a function of curvature for parameters typical of the lower Mississippi [Nittrouer et al., 2011]: Su=1×10−5, math formula, B = 430 m, and math formula. Bends with ν≤ 0.029 are predicted to have fully alluvial coverage, while bends with ν> 0.029 are predicted to have partial bedrock exposure.

5 Conclusions

We have developed an analytical, nonlinear model that describes the fully developed flow field and finite amplitude bed deformations for alluvial or mixed bedrock-alluvial bends of constant curvature. This model builds upon previous efforts in that (a) it allows the dimensional scaling parameters math formula, math formula, Cfb, and Sb to be themselves perturbed and (b) it allows for a finite depth of scour leading to partial bedrock exposure in the bend, which is characteristic of mixed bedrock-alluvial rivers.

Our calculations suggest that bedrock exposure in mixed bedrock-alluvial bends influences the size of point bars and the overall structure of the flow field through the bend. The point bars that develop in mixed bedrock-alluvial bends are shorter and narrower than those that form in fully alluvial bends, so the effects of topographic steering are reduced in mixed bedrock-alluvial bends and the high-velocity core shifts inward. For a given depth to bedrock, channel curvature and upstream sediment supply are the primary controls on the shape of the bar, and in principle the model may be used to infer sediment supply rates from topographic measurements in the channel bend.

This model represents a first step in a more complete and general understanding of meandering bedrock rivers. Further insight can be gained by relaxing some of the simplifying assumptions we have made in developing the model. In particular, the model could be extended to the case of arbitrary, but slowly varying, curvature [e.g., Bolla Pittaluga et al., 2009], and the assumption of uniform sediment could be relaxed to explore the effects of sorting on finite amplitude morphology [e.g., Seminara et al., 1997]. Finally, incorporation of an erosion submodel [e.g., Nelson and Seminara, 2011] would allow the prediction of bed and planform evolution.

Acknowledgments

Funding for this work was provided by a National Science Foundation International Research Fellowship to P.A.N. (grant 0965064). Constructive reviews from two anonymous reviewers, the Associate Editor (Dimitri Lague), and the Editor helped to improve the manuscript.

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