# 8.2.1: General transport formulations

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Longshore sediment transport is the net movement of sediment particles through a fixed vertical plane perpendicular to the shoreline. The direction of this transport is parallel to the shoreline and the depth contour lines. Without further specifications it is often meant to be the total or bulk transport in the alongshore direction (e.g. the total transport along a coast in the entire active zone).

Wave-driven longshore sediment transport depends amongst other things on the hydrodynamics in the breaker zone (see Ch. 5) and on the sediment properties (see Sect. 6.2). However, regardless of the strength of the transport-generating hydraulic forces, sediment transport will only occur if moveable sediment is available in a certain area, either in the bed or in the water column through supply from an adjacent area. If the seabed is fixed (for instance by bottom protection, bio-chemical processes or vegetation or in the case of absence of sediment), erosion is prevented and the *actual transport* may be smaller than the local transport capacity based on the hydrodynamics. The sediment transport formulations presented in this section all reflect the *longshore transport capacity* assuming that the material is actually present and available for transport.

The computation of the sediment transport parallel to the coast can, in principle, be handled with any of the general sediment transport models as introduced in Ch. 6. When these formulations are applied to compute longshore sediment transport rates, we can simplify the problem quite a bit by assuming that the transport in the along-shore direction is by a time-invariant longshore current.

Let us, for instance, consider the (suspended) sediment transport description based on velocity multiplied by concentration, Eq. 6.6.1.2. In principle the current velocity in the alongshore direction \(v(z, t)\) at a certain position in the surf zone is a function of \(z\) and \(t\). It consists of a mean and oscillatory component: \(v(z,t) = V(z) + \tilde{v} (z, t)\). The short wave motion acts mainly in the cross-shore direction (\(\varphi\) is small due to refraction) and thus nearly perpendicular to the sediment transport direction. In practice it is therefore often assumed that the alongshore oscillatory velocity \(\tilde{v} (z, t)\) is small and that during a period with no significant change in the offshore wave conditions, a relatively time-invariant current velocity \(v(z, t) \approx V(z)\) is present. This assumption implies that the mean water motion is the main contributor to the longshore sediment transport. Note that the sediment concentration, in principle, fluctuates on the timescale of the waves and can – just as the velocity – be seen as the summation of a mean component and an oscillatory component, viz. \(c(z, t) = C(z) + \approx (z, t)\). The purely oscillatory component of the sediment concentration however will not give rise to a net transport since \(V(z) \cdot \tilde{c} (z, t)\) is purely oscillatory and therefore has a time-mean equal to zero. Hence, (suspended load) transport models based on solving the velocity and concentration distribution can be reduced to the current-related part only (cf. Eqs. 6.6.1.4 and 6.6.1.5):

\[\langle s_y \rangle = \int_{a}^{h} V(z) \cdot C(z) dz\]

where:

\(s_y\) | net longshore sediment transport excl. pores | \(m^3/m/s\) |

\(V(z)\) | longshore current velocity at height \(z\) above the bottom | \(m/s\) |

\(C(z)\) | time-mean sediment concentration at height \(z\) | \(m^3/m^3\) |

\(a\) | thickness of bed load layer | \(m\) |

\(h\) | local (still) water depth | \(m\) |

Note that such a computation yields the longshore (suspended) sediment transport at a certain cross-shore location. If we like to quantify the total transport in e.g. the surf zone, we must integrate the outcome of the equation over the width of the surf zone.

The longshore current velocity \(V(z)\) can have many different driving forces, but for most beaches this current is driven predominantly by breaking waves which approach the coast at an angle. The longshore current is concentrated more or less in the surf zone and occurs regardless of whether there is sediment transport or not. Methods to compute this longshore current were outlined in Sect. 5.5.5.

We have concluded that the role of the oscillatory wave motion in transporting the sediment in the alongshore direction is limited. What then is the role of waves – be- sides generating the longshore current – in the longshore sediment transport? Their role is to enhance the amount of sediment in suspension \(C(z)\). As a result of turbulence, generated in the wave boundary layer and at the surface under breaking waves, considerable amounts of sediment are brought into suspension:

- Due to the orbital motion (in mainly the cross-shore direction), the magnitude of the bed shear stress varies over the wave cycle and peaks twice every wave cycle. Especially during these peaks a lot of sediment is mobilised and entrained;
- Breaking waves strongly increase the turbulence in the water column and therefore relatively easily bring suspended sediments into the upper part of the flow.

Summarizing, since the wave motion in the breaker zone is nearly perpendicular to the resulting current, the major influence of waves is to stir more material loose from the beach and keep it in suspension, thereby increasing the sediment concentration. It is the (wave-induced) longshore current (and not the oscillatory wave motion) that is mainly responsible for the net movement of material along the coast.

We can also demonstrate these findings using a transport relation of the form Eq. 6.5.3.1 or Eq. 6.7.2.9 and 6.7.2.10. With the orbital motion approximately cross-shore and the bed shear stress according to Eq. 5.5.5.1, we have:

\[\langle s_y \rangle = m_1 \langle |\vec{\tau}_b|^{(n - 1)/2} \rangle V = m_2 \langle |\vec{u}|^{n - 1} \rangle V\label{eq8.2.1.2}\]

where:

\(s_y\) | longshore sediment transport (per unit width) | \(m^3/m/s\) |

\(V\) | longshore current velocity at some level above the bed (e.g. depth mean velocity) | \(m/s\) |

\(|\vec{\tau}_b|\) | magnitude of the bed shear stress vector due to combined wave-current motion | \(N/m^2\) |

\(|vec{u}|\) | magnitude of the combined wave-current velocity vector | \(m/s\) |

\(m_1,m_2\) | (dimensional) coefficients | - |

\(n\) | power (typically 3 to 5) | - |

In Eq. \(\ref{eq8.2.1.2}\), \(m_1 |\vec{\tau}_b|^{(n - 1)/2} = m_2 |\vec{u}|^{n - 1}\) describes the sediment load that is transported by the longshore current velocity \(V\). The sediment load is stirred up by the combination of longshore current and wave orbital motion (see Fig. 6.13 for the non-linear enhancement of the bed shear stress in the combined wave-current motion). Especially the wave orbital motion is important because of the small boundary layer thickness and hence large shear stresses (see Sect. 5.4.3). Substitution of \(|\vec{u}| = \sqrt{u^2 + V^2}\) (see Fig. 5.38), with \(u = \hat{u} \cos \omega t\) the cross-shore time-varying orbital motion, yields for \(n = 3\):

\[\langle s_y \rangle = \underbrace{m_2 \langle u^2 + V^2 \rangle}_{\begin{array} {c} {\text{sediment load stirred by}} \\ {\text{wave-current motion}} \end{array}} \underbrace{V}_{\begin{array} {c} {\text{longshore current}} \\ {\text{responsible for transport}} \end{array}} = \dfrac{1}{2} m_2 \hat{u}^2 V + m_2 V^3\]

Assuming the stirring due to the short wave motion is dominant, the longshore transport is approximately proportional to \(\langle s_y \rangle \propto \hat{u}^2 V\). This again reflects that the effect of short waves is to mobilise the material which is consequently transported by the long-shore current. In Sect. 8.2.3, specifically in Intermezzo 8.1, we will see that commonly used bulk longshore transport formulations can be interpreted in a similar way.