Abstract
Sediment transport from mountainous rivers into a quiescent ambient with simultaneous formation of deltas is reviewed. The focus is restricted to flow in vertical cross-sections with no changes perpendicular to the plane of the flow. The bed load transport in the river is derived for quasi-steady situations using sediment mass balance and the Mohr-Terzaghi shear stress-pressure relation with the angle of internal friction, \(\varphi \), as the essential frictional parameter. The emerging model is a diffusion equation for the upper surface of the moving sediment layer and corresponding boundary conditions. Its diffusivity is expressible in terms of the hydraulic discharge, the densities of the sediment and the turbid water, the angle of internal friction and a parameter characterizing the bed-parallel sediment velocity in terms of the average velocity in the turbid layer. When this river flow enters quiescent water, two different classes of deltas can be formed. When the entering water is either neutrally buoyant or lighter than the ambient water, the sudden reduction in tractive force along the bed generates a conspicuous avalanching flow to depth. This leads to steep-sloped foreset deposits with delta fronts inclined by the angle of internal friction. Such so-called Gilbert-type deltas are governed by a jump requirement of the sediment flux across the shore line and the geometry of the receiving basin. When the inflowing discharge is denser than the receiving ambient water, it will dive down as a turbulent under-current. The basal sediment transport in this subaqueous density current is analogous to the subaerial case and again described by a diffusion equation with similarly determined diffusivity. The combined dual subaerial-subaqueous sedimenting process is mathematically very similar to a (generalized) Stefan problem, e.g. the freezing of an ice cover on a lake. We present (mostly analytical) solutions for (1) bedrock-alluvial transitions, (2) overtopping failure of a dam, (3) topset-foreset diffusion processes for hypo- and hyper-pycnal deltas. Laboratory experiments demonstrate the adequacy of the models.
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This chapter is a near-reprint of ‘A tutorial on prograding and retrograding hypo- and hyper-pycnal deltaic formations into quiescent ambients’ [25]. Hutter thanks Prof. P. Rutschmann for granting permission of reproduction.
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- 1.
- 2.
Equation (31.1) is used and \([(1-n_0)\rho _s+n_0\rho _w]\simeq 0 \) is assumed.
- 3.
- 4.
We compute this indefinite integral as follows:
$$\begin{aligned} \begin{array}{l} \displaystyle {\int _{\varXi }^{A}}\left( {\frac{1}{x^2}}\right) \exp (-x^2)\text {d}{x} = {\int _{\varXi }^{A}}{\frac{\text {d}}{\text {d}x}}\left( -{\frac{1}{x}}\right) \exp (-x^2)\text {d}x\\ = \displaystyle {\int _{\varXi }^{A}}{\frac{\text {d}}{\text {d}x}}\left[ \left( -{\frac{1}{x}} \right) \exp (-x^2)\right] \text {d}x - {\int _{\varXi }^{A}}\left( -{\frac{1}{x}} \right) (-2x)\exp (-x^2)\text {d}x\\ = -{\frac{1}{x}}\exp (-x^2){|_{\varXi }^{A}} - 2 \displaystyle {\int _{\varXi }^{A}}\exp (-x^2)\text {d}x\\ = -{\frac{1}{A}}\exp (-A^2) + {\frac{1}{\varXi }}\exp (-\varXi ^2) + 2\underbrace{{\int _{A}^{\varXi }}\exp (-x^2)\text {d}x}_{{\int _{A}^{0}}+ {\int _{0}^{\varXi }}}\\ = {\frac{\exp (-\varXi ^2}{\varXi })} + \underbrace{2{\int _{0}^{\varXi }}\exp (-x^2)\text {d}x}_{2\sqrt{\pi }\text {erf}(\varXi )} - \underbrace{{\frac{\exp (-A^2)}{A}} - 2{\int _{0}^{A}}\exp (-x^2) \text {d}x}_{C}\\ = {\frac{\exp (-\varXi ^2)}{\varXi }} + 2\sqrt{\pi } \text {erf}(\varXi ) + C , \nonumber \end{array} \end{aligned}$$which agrees with (31.25).
- 5.
Had we chosen \(C\) differently from \(-1\), then (31.36)\(_{1}\) would read \(A = S_{2} + \sqrt{\pi }(1 + C)\) and the initial value for \(s\) at \(t =0\) would no longer be zero. Requesting that \(s(0) = 0\) would in this case fix \(C\) to be again \(-1\).
- 6.
\(s^{\pm }(t) = s(t)\pm \varepsilon ,\,\varepsilon >0,\,\varepsilon \rightarrow 0\).
- 7.
An alternative derivation of formula (31.42) when \(\sigma = 0\) is given in Appendix A.
- 8.
From now on \(\phi \) is counted as positive.
- 9.
We regard the denotation ‘equilibrium’ as introduced by geologists as a misnomer, since the graded state is not a thermodynamic equilibrium.
- 10.
With \(\alpha _{1} = \alpha _2\) and \(\rho _1 = 2100\) kg m\(^{-3}\) and \(\rho _{\infty } = 1100\) kg m\(^{-3}\) one obtains \(D_{2}/D_{1} = 0.48\) (foreset conditions). Alternatively, with \(\rho _{1} = 1200\) kg m\(^{-3}\), \(\rho _{\infty } = 1100\) kg m\(^{-3}\), we get \(D_2/D_1 = 0.083\).
- 11.
Note that the primes in these functions designate differentiations with respect to different variables.
- 12.
Lai and Capart [34] choose \(d = [\![D]\!]S^\mathrm{min}\), assuming that \(S_{1}^\mathrm{min}=S_{2}^\mathrm{min}\). They say that this choice may be too restrictive, but it is clear from above that they did not restrict the flexibility of the model by this choice.
- 13.
Similar, somewhat reminiscent alterations of sediment regimes have frequently occurred in the Alps during the Middle Ages when ice avalanches, formed from hanging glaciers, dammed riverine valleys, Roethliberger (1978), [58]. As long as the ice deposit existed, an ice-dammed lake formed and changed the upslope and downslope sediment flows. More significantly, floods due to sudden dam break caused devastating debris flows. Other, related processes are artificially formed when barrages are built for valley reservoirs. They change the upstream sediment regimes and slowly fill the reservoir, thereby reducing the power-generating capacity. Through a base opening in the barrage or side channel and judicious flushing operations, in which the discharge and the lake level are monitored, the sediment deposit is partly removed, a process which affects the sediment regimes in the lake and its topset as well as the sediment flow in the river stretch below the barrage.
- 14.
Unsteady situations can also be analysed, but may need pure numerical solution techniques.
- 15.
\(h_{0}\) is the water depth on the assumption that the water flux leaving the lake is the same as the in-flux.
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Appendices
Appendix A: Derivation of the Sediment Flux Boundary Condition at the Plunge Point of a Gilbert-Type Delta
In this appendix an explicit derivation of the flux boundary condition (31.42) at the plunge point of a hypopycnal delta will be given. The derivation follows Kostic and Parker [30] but in the notation of this chapter.
Consider Fig. 31.36. With reference to this figure the front surface of the foreset delta can be described as
If this is evaluated at the toe of the alluvial deposit,
is obtained.
If conservation of mass is formulated for a sediment element as shown in the inset of Fig. 31.36, then one may deduce
or, since the solid volume fraction is assumed to be constant,
In the above, \(\bar{q}_{s}\) is the sediment flux at a certain volume fraction, whereas \(q_{s}\) is the corresponding effective flux.
An explicit expression for \(q_{s}\) is obtained, if Eq. (31.169) is integrated from \(X = s^{-}\) to \(X = u(t)\). This integration is composed of an ‘integration’ from \(X = s^{-}(t)\) to \(X = s^{+}(t)\) plus the integration from \(X = s^{+}(t)\) to \(X = u(t)\). Thus,
Here, the integral on the far right vanishes because of continuity requirements for \(\hat{\zeta }(\cdot )\). It follows that the flux \(q_{s}\) is continuous across the plunge point. Therefore, we may write
in which integration can now be restricted to \(X > s(t)\). Moreover, it was assumed that the sediment flux at the toe of the frontal surface of the delta vanishes, which is realistic. With \(\hat{\zeta }\) as given in (31.167) one may write
in which
Here, \(\tan \alpha _{1}\) is the slope of the sediment bed in the topset of the plunge point. Substituting (31.172) into (31.171) and the resulting expression for \(\partial \,\hat{\zeta }/\partial \,t\) into (31.170) yields
The surface point \(\hat{\zeta }_{\mid s(t)}\) is given by the level of the lake surface and the water depth above the sediment as follows: \(\hat{\zeta }_{\mid s(t)} = Z_{\ell }(t) - h_{\mid s(t)}\). Consequently,
Substituting this into (31.173) yields a first variant of the final formulae for \(q_{s}\):
Sometimes it is more convenient to additionally use the trigonometric relation
We then obtain
Even though \(\tan \alpha _1 <(\ll ) \tan \phi \), it is not justified in general, to ignore the expression \(\tan \alpha _{1}/\tan \phi \) in the above formulae. However, it is justified to ignore the term associated with \((\tan \phi )^{\cdot }\). Ignoring also \((\partial \,h/\partial \,t)_{\mid s(t)}\) and \(\dot{Z}_{\ell }(t)\) yields
Formula (31.42) with \(\sigma = 0\) is obtained from (31.175), if the river water depth is ignored, \((\partial \,h/\partial \,t)_{\mid s(t)}\simeq 0\) and \(\tan \alpha _{1}\) is ignored in comparison to \(\tan \phi \).
Appendix B: Characteristics of Error Functions
In this Appendix we collect a number of properties of mathematical expressions which are connected to the error function. These have been collected and/or derived in Carslaw and Jaeger (1959) [8]
-
Definition of the error function and complementary error function
$$\begin{aligned} \mathrm{{erf}}(x)&= {\frac{2}{\sqrt{\pi }}}\int _{0}^{x} \exp (-\xi ^2)\mathrm{{d}}\xi , \end{aligned}$$(31.179)$$\begin{aligned} \mathrm{{erfc}}(x)&= 1 - \mathrm{{erf}}(x) = {\frac{2}{\sqrt{\pi }}}\int _{x}^{\infty } \exp (-\xi ^2)\mathrm{{d}}\xi , \end{aligned}$$(31.180) -
The above definitions imply
$$\begin{aligned} \begin{array}{lll} \mathrm{{erf}}(0) = 0, \quad &{} \mathrm{{erf}}(\infty ) = 1, \quad &{} \mathrm{{erf}}(-x) = -\mathrm{{erf}}(x),\\ \mathrm{{erfc}}(0) = 1, \quad &{} \mathrm{{erfc}}(\infty ) = 0, \quad &{} \mathrm{{erfc}}(-x) = 2 - \mathrm{{erfc}}(x). \end{array} \end{aligned}$$(31.181) -
Both,
$$\begin{aligned} \mathrm{{erf}}\left( \frac{x}{2\sqrt{Dt}}\right) \quad \mathrm{{and}}\quad \mathrm{{erfc}}\left( \frac{x}{2\sqrt{Dt}}\right) \end{aligned}$$satisfy the diffusion equation
$$\begin{aligned} \frac{\partial \,f}{\partial \,t} - D \frac{\partial ^{2}f}{\partial \,x^{2}} = 0. \end{aligned}$$ -
The \(n\)th integral complementary error functions are defined as
$$\begin{aligned} \mathrm{i}^{n}\mathrm{{erfc}}(x)&\qquad = \qquad \int _{x}^{\infty }\mathrm{i}^{n-1}\mathrm{{erfc}}\,\xi \mathrm{{d}}\xi , \quad n = 2, 3, 4, \dots \nonumber \\ \mathrm{i}^{0}\mathrm{{erfc}}(x)&\qquad = \qquad \mathrm{{erfc}}(x), \end{aligned}$$(31.182)$$\begin{aligned} \mathrm{i}\mathrm{{erfc}}(x)&\qquad = \qquad \int _{x}^{\infty }\mathrm{{erf}}\,\xi \,\mathrm{{d}}\xi \nonumber \\&\mathop {=}\limits ^{\text {integr. by parts}} \xi \mathrm{{erfc}}(\xi )|_{x}^{\infty } - \frac{1}{\sqrt{\pi }}\int _{x}^{\infty }\underbrace{(-2\xi )\exp (-\xi ^2)}_{\frac{\mathrm{{d}}}{\mathrm{{d}}\xi } \left( \exp (-\xi ^2)\right) }\mathrm{{d}}\xi \end{aligned}$$(31.183)$$\begin{aligned}&\qquad = \qquad - x\,\mathrm{{erfc}}(x) + \frac{\exp \,(-x^2)}{\sqrt{\pi }} . \end{aligned}$$(31.184) -
The function \(\mathrm{{erfc}}(x)\) exhibits the following properties:
-
(1)
$$\begin{aligned}&\hat{\zeta }_{s}^{(2)}(x, t) = \frac{A}{D}\sqrt{Dt}\,\mathrm{{ierfc}}\left( \frac{x}{2\sqrt{Dt}}\right) , \quad x \geqslant 0,\nonumber \\ - \qquad&\lim _{x \rightarrow \infty }\hat{\zeta }_{s}^{(2)}(x, t) \rightarrow 0, \nonumber \\ - \qquad&\frac{\partial \,\hat{\zeta }_{s}^{(2)}}{\partial \,x} = \frac{A}{2D}\left( \mathrm{{erf}}\left( \frac{x}{2\sqrt{Dt}}\right) - 1\right) \\ - \qquad&\zeta _{s}^{2}(x, t) \quad {\text {satisfies the diffusion equation}} \nonumber \\&\frac{\partial \,\hat{\zeta }_{s}^{(2)}(x, t)}{\partial \,t} - D \frac{\partial ^{2}\,\hat{\zeta }_{s}^{(2)}(x, t)}{\partial \,x^{2}} = 0,\nonumber \end{aligned}$$(31.185)
-
(2)
$$\begin{aligned}&\hat{\zeta }_{s}^{(3)}(x, t) = \frac{A}{D}\sqrt{Dt}\,\mathrm{{ierfc}}\left( \frac{|x|}{2\sqrt{Dt}}\right) , \quad x \leqslant 0,\nonumber \\ - \qquad&\lim _{x \rightarrow -\infty } \hat{\zeta }_{s}^{(3)}(x, t) \rightarrow 0 ,\nonumber \\ - \qquad&\frac{\partial \,\hat{\zeta }_{s}^{(3)}(x, t)}{\partial \,x} = \frac{A}{2D}\left( -\mathrm{{erf}}\left( \frac{|x|}{2\sqrt{Dt}}\right) + 1\right) \\ - \qquad&\hat{\zeta }_{s}^{(3)}(x, t) \quad {\text {satisfies the diffusion equation}} \nonumber \\&\frac{\partial \,\hat{\zeta }_{s}^{(3)}(x, t)}{\partial \,t} - D \frac{\partial ^{2}\,\hat{\zeta }_{s}^{(3)}(x, t)}{\partial \,x^{2}} = 0. \nonumber \end{aligned}$$(31.186)
-
(1)
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Hutter, K., Wang, Y., Chubarenko, I.P. (2014). Prograding and Retrograding Hypo- and Hyper-Pycnal Deltaic Formations into Quiescent Ambients. In: Physics of Lakes. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-00473-0_31
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