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.
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 . 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 , 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 , , 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 , , 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 , and the sediment flux per unit width by the unperturbed uniform conditions within the channel bend:
Here ρ is the density of water, s is the relative particle density (s=ρs/ρ, where ρs is the density of the sediment), 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]:
where the absolute bottom roughness height z0 is estimated as equal to [Engelund and Hansen, 1967]. The dimensionless parameters βb and Fb are defined by
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:
(4a) (4b) (4c) (4d) (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
We will assume that the channel is “wide” and “weakly curved,” that is
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:
(7a) (7b) (7c) (7d) (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
where τ∗ is the tangential stress vector at the bottom
D(n) is the local value of the dimensionless flow depth, and ζ is a normalized vertical coordinate that reads
With ζ0 the normalized roughness height and ζ=1 at the free surface, ζ will therefore occupy a range of values ζ0≤ζ≤1. The vertical distribution of eddy viscosity N(ζ) takes the classical parabolic distribution characteristic of uniform flows, corrected by Dean's  wake function:
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:
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  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
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]:
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. .
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
The second constraint imposes the condition that the total sediment flux through the bend must equal the rate of sediment supplied from upstream:
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:
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):
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:
In equations (19)–(21) the parameter δ′ is defined as
2.2 Solution for Channels of Constant Curvature
Following Seminara and Solari  and Bolla Pittaluga et al. , 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., ) to make the governing equations dimensionless, we have scaled our variables using average values in the bend (e.g., ), which themselves are subject to the perturbation associated with the effects of channel curvature. We therefore perturb both the dimensional scaling parameters (, and βb) and the dimensionless hydrodynamic variables (u,v,w,h, and D) around their locally uniform values:
Here δ is a small parameter defined by uniform conditions in the bend
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.
2.2.2 First Order: n Momentum
At , the n momentum equation (20) can be solved to give the leading order approximation of curvature-induced secondary flow. We find
where is the solution to the following problem:
A solution for may be written in the form
and gi (i=0,1,2) is the solution to the following differential systems:
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 over the cross section and apply the boundary condition w1|ζ=0=0 and find
Applying (39) to (35), we find
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:
Equation (41) will be solved subject to the components of the expanded integral conditions for flow and sediment continuity (equations (15) and (16)) through the reach:
Here the leading order component of the dimensionless sediment transport capacity φb0 (using an exponent of n=3/2 in equation (12)) is
Equation (41) must be solved numerically, and this is done as follows. First, the leading order components of the mean dimensional depth and the centerline slope Sb0 in the bend are estimated and used to compute the parameters Cfb0 (equation (31)), (equation (45)), , , 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 (Figure 1), that is,
At this interface, the depth D can be expressed as
Taylor expansions can be used to determine the depth, water surface elevation, and bedrock elevation at n=ne:
(48a) (48b) (48c)
Combining (48) with (23) and (24), we arrive at the following expressions for D|ne and h|ne:
Equation (47) can now be written at leading order and first order:
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 depth to bedrock, that is
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 is then calculated by solving the equation:
The procedure iterates by solving again for D0 and Sb0 until the changes from one iteration to the next become negligible.
2.2.4 Second Order
The n momentum equation (20) at O(δ2) includes many terms that correct the secondary flow. If we set
we can derive the following problem for which is the solution:
where R1 and R2 are defined in (56a) and (56b), and
The solution for takes the form
and g2j(j=0,5) are the solutions to the following differential systems:
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 , one finds that
Combining this relation with (65), one finds
The sediment continuity equation requires the component of qn (equation (14)) to be zero. Therefore, the expression for D1 becomes
Equation (72) is solved subject to the components of the integral conditions (15) and (16):
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 , and this value is used to compute the parameters
(76a) (76b) (76c) (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 , 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
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 , the parameters (76a)–(76d) are recalculated, and the procedure is repeated until changes are negligible.