6.7.2: Energetics approach for combination of waves and currents

Bagnold derives formulations for the immersed weight suspended load \(I_s\) and bed load transport \(I_b\) (see text below Eq. 6.4.1.2 for the definition of immersed weight transport rates) for uni-directional flow. In his derivations, it was hypothesised that the fluid acts as a machine expending energy at a prescribed efficiency rate to offset the work done in transporting sediment. His reasoning is here briefly demonstrated for bed load transport.

The immersed weight of a mass of bed load \(m_b\) is given by \(W = (\rho_s - \rho)/\rho_s gm_b\). The component hereof perpendicular to the sloping bed - with bed slope \(\tan \alpha \) - is \(W \cos \alpha\), whereas the component parallel to the bed is \(W \sin \alpha\). Hence, the frictional resistance for downslope transport is equal to \(W (\mu \cos \alpha - \sin \alpha)\). Here, the friction coefficient \(\mu = \tan \varphi_r\) with \(\varphi_r\) is the angle of repose (Sect. 6.2). ‘Work’ is defined by the exerted force times the distance travelled in the direction of the force. Therefore, the work done (per unit time) in maintaining bed load in motion is equal to the force required to overcome frictional resistance times the mean speed \(U_b\) at which the grains travel. Further, since \(U_b \cos \alpha\) is the horizontal component of the grain velocity in a vertical section, the immersed weight transport rate through a vertical section is defined as \(I_b = WU_b \cos \alpha\). We can now write:

\[\text{work done per unit time} = W(\mu \cos \alpha - \sin \alpha ) U_b = I_b (\tan \varphi_r - \tan \alpha )\]

Some fraction \(\varepsilon_b\) of the dissipated fluid power \(\omega\) is expended to offset the work done in maintaining the bed load.

Hence:

\[I_b = \dfrac{\varepsilon_b \omega}{(\tan \varphi_r - \tan \alpha)}\label{eq6.7.2.2}\]

A similar derivation for suspended load leads to:

\[I_s = \dfrac{\varepsilon_s \omega}{(w_s/U_s - \tan \alpha)}\label{eq6.7.2.3}\]

Eqs. \(\ref{eq6.7.2.2}\) and \(\ref{eq6.7.2.3}\) are for uni-directional flow along a downsloping bed. Bowen (1980) rewrites these equations for a cross-shore situation with normally incident waves.

The formulations for the instantaneous transport rates then read:

\[I_b = \dfrac{\varepsilon_b \rho c_f u^3}{\tan \varphi_r - u/|u| \tan \alpha}\label{eq6.7.2.4}\]

\[I_s = \dfrac{\varepsilon_s c_f \rho u^3 |u|}{w_s - u \tan \alpha}\label{eq6.7.2.5}\]

where:

\(\varepsilon_b, \varepsilon_s\)	efficiencies for bed and suspended load	-
\(c_f\)	friction coefficient	-
\(w_s\)	sediment fall velocity	\(m/s\)
\(\tan \varphi_r\)	tangent of angle of repose of the sediment	-
\(\tan \alpha\)	bed slope	-
\(\rho\)	water density	\(kg/m^3\)

The time-dependent velocity \(u\) is defined as positive seawards, the direction of \(x\) positive. The bed load and suspended load velocity of the grains (\(U_b\) and \(U_s\)) are represented by \(u\). Further, the dissipated fluid power \(\omega\) (work done per unit time) is computed as the energy dissipation rate, \(D_f\), due to bottom friction. \(D_f\) is given by the product of bed shear stress \(\tau\) and the near-bottom time-varying flow \(u\). The bed shear stress is assumed to be described by a quadratic friction law \(\tau_b = \rho c_f |u|u\), so that \(\omega = D_f = \rho c_f u^2|u|\). Note that in this way only the dissipation in the wave boundary layer is taken into account. For applications in the surf zone, Roelvink and Stive (1989) add to this the energy dissipation due to turbulence near the bottom induced by wave breaking. The modulus signs in Eqs. \(\ref{eq6.7.2.4}\) and \(\ref{eq6.7.2.5}\) are chosen such as to properly account for the direction of transport in terms of velocity and slope. Hence, for seaward transport (\(u\) positive) the denominators are reduced; material is therefore more easily transported downslope.

The bed slope is limited by the following two conditions:

\[\tan \alpha \to \tan \varphi_r \ \ \ \text{ giving avalanching or slumping}\label{eq6.7.2.6}\]

\[\tan \alpha \to \tan w_s/u \ \ \ \ \text{ giving autosuspension}\label{eq6.7.2.7}\]

Autosuspension refers to the fact that for a certain bed slope (and hence a certain amount of gravitational energy) the sediment ‘suspends itself’. Later it has been argued (Bailard, 1981) that the condition in Eq. \(\ref{eq6.7.2.7}\) needs to be supplied with an efficiency factor as well (such that \(\varepsilon_s u\) instead of \(u\) should be used).

Note that the model lacks an initiation of transport condition like the critical Shields parameter and that both the bed load and the suspended sediment transport respond instantaneously to the flow field (as opposed to the diffusion approach of Sect. 6.6).

In a parallel development, Bailard (Bailard, 1981, 1982; Bailard & Inman, 1981) has generalised the Bagnold model to a total load model of time-varying sediment transport over a plane sloping bed. In the Bailard formulation both the bed load and suspended load transport rate vectors are composed of a velocity-induced component directed parallel to the instantaneous velocity vector and a gravity-induced component directed downslope. Now alongshore and onshore-offshore currents as well as oscillatory wave-induced orbital motion with a local angle of incidence are considered.

Here we do not go into the details of the derivation and the final expressions, but follow Roelvink and Stive (1989) who write the Bailard formulation in general terms (for the simplified case of the velocity aligned with the local bed slope):

\[S(t) = \underbrace{C_1 u(t) |u(t)|^{n - 1}}_{\text{quasi-steady response to time-varying flow}} + \underbrace{C_2 |u(t)|^m \tan \alpha}_{\text{response to downslope gravity force}}\]

Here \(\tan \alpha\) is the local bed slope and \(u(t)\) is the near-bottom time-varying cross-shore flow. For bed load, the powers are \(n = m = 3\), and for suspended load the powers are \(n = 4\) and \(m = 5\). After time-averaging (indicated by brackets) we find that:

• the bed load transport \(\langle S_b \rangle\) is proportional to the odd moment \(\langle u|u|^2 \rangle\) and the even moment \(\langle |u|^3 \rangle\); and
• the suspended load transport \(\langle S_s \rangle\) is proportional to the odd moment \(\langle u|u|^3 \rangle\) and the even moment \(\langle |u|^5 \rangle\)

Note that for the even moments, the terms within brackets (i.e. before time-averaging) are positive for every moment in time, whereas for the odd moments these terms have the sign of the instantaneous velocity. The transport terms containing the odd moments \(\langle u|u|^2 \rangle\) and \(\langle u|u|^3 \rangle\) reflect the quasi-steady bed load and suspended sediment load transport respectively due to the time-varying flow. The terms containing the even moments \(\langle |u|^3 \rangle\) and \(\langle |u|^5 \rangle\) reflect the downslope directed gravity driven transport and are usually an order of magnitude smaller than the terms containing the odd velocity moments.

We could therefore make the following approximate statement:

\[\langle S_b \rangle \propto \langle u|u|^2 \rangle\]

\[\langle S_s \rangle \propto \langle u|u|^3 \rangle\]

Note that in Sect. 6.5.5, we found similar dependencies for bed load transport, viz. with \(n = 3\) to 4. There we also found that net sediment transport is either due to net currents or due to skewed oscillatory velocity signals. This distinction is comparable to the distinction between current-related and wave-related suspended sediment transport as discussed in Sect. 6.6.1. We will further explore these different transport contributions in Sect. 7.5 (for cross-shore sediment transport) and in Sect. 9.7.2 (for tide-induced sediment transport).

As said before, a criterion for initiation of motion was not taken into account in the above energetics approach. This can be expected to increase the asymmetry in sediment stirring. Another simplification is the assumed quasi-steady approach for suspended load transport. Coarse sediment can be thought to respond instantaneously to the flow velocity, but for finer material this may not be the case. Fine material in this respect is defined as having a diameter such that \(u_*/w_s > 1\), where \(u_*\) is the shear velocity and \(w_s\) is the fall velocity (see also Sect. 6.6.3). The larger this parameter the more time the suspended particles need to settle. As a result of these time-lags in the vertical sediment distribution, the transport rates may be reduced (since the maximum concentrations do not coincide with the maximum velocities any more at every height above the bed). An example of a sediment transport formulation in which such time-lag effects are taken into account is the transport formulation of Dibajnia and Watanabe (1993).