2.1. Governing equations

The nonlinear shallow water equations including bed shear stress described by a drag law (Soulsby Reference Soulsby1997) are utilised to describe the flow in the swash zone,

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle {\hat{h}}_{\,\hat{t}}+{\hat{u}}{\hat{h}}_{\,\hat{x}}+{\hat{h}}{\hat{u}}_{\,\hat{x}}=0, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle {\hat{u}}_{\,\hat{t}}+{\hat{u}}{\hat{u}}_{\,\hat{x}}+g{\hat{h}}_{\,\hat{x}}+g\hat{B}_{\,\hat{x}}=-\frac{c_{d}|{\hat{u}}|{\hat{u}}}{{\hat{h}}}, & \displaystyle\end{eqnarray}$$

where $\hat{x}$ represents cross-shore distance $(\text{m})$ , $\hat{t}$ is time (s), ${\hat{h}}$ represents the water depth ( $\text{m}$ ), ${\hat{u}}$ is a depth-averaged horizontal velocity ( $\text{m s}^{-1}$ ), $\hat{B}$ is the bed level ( $\text{m}$ ), $g$ is the acceleration due to gravity ( $\text{m s}^{-2}$ ) and $c_{d}$ is a dimensionless drag coefficient. In figure 1 we illustrate the situation that is considered.

The bed-evolution (sediment conservation) equation including both bed and suspended load is

(2.3) $$\begin{eqnarray}\hat{B}_{\,\hat{t}}+{\it\xi}\hat{q}_{b\hat{x}}={\it\xi}(\hat{D}-{\hat{E}}),\end{eqnarray}$$

where $\hat{q}_{b}$ is sediment flux due to bed load ( $\text{m}^{2}\ \text{s}^{-1}$ ), which, in general, is strongly dependent on ${\hat{u}}$ and a weak function of ${\hat{h}}$ , $\hat{D}$ is the dimensional deposition rate ( $\text{m s}^{-1}$ ) and ${\hat{E}}$ is the dimensional erosion (or entrainment) rate ( $\text{m s}^{-1}$ ). Here, ${\it\xi}=1/(1-p)$ , where $p$ is the bed porosity.

The governing equation for the transport of suspended sediment is

(2.4) $$\begin{eqnarray}{\hat{c}}_{\,\hat{t}}+{\hat{u}}\,{\hat{c}}_{\,\hat{x}}=\frac{1}{{\hat{h}}}({\hat{E}}-\hat{D}),\end{eqnarray}$$

where ${\hat{c}}$ is volumetric concentration, and ${\hat{h}}{\hat{c}}$ therefore represents the volume of sediment per unit area of seabed ( $\text{m}$ ).

The Meyer-Peter–Müller formula (see Yalin Reference Yalin1977; Soulsby Reference Soulsby1997), which is commonly used in engineering problems, is employed,

(2.5) $$\begin{eqnarray}\hat{q}_{b}={\hat{A}}\left(\frac{{\hat{u}}^{2}-{{\hat{u}}_{crb}}^{2}}{{\hat{u}}_{0}^{2}}\right)^{3/2}\frac{\mid {\hat{u}}\mid }{{\hat{u}}},\end{eqnarray}$$

where ${\hat{u}}_{0}$ is a representative velocity scale and ${\hat{A}}$ is a dimensional ( $\text{m}^{2}\ \text{s}^{-1}$ ), empirically determined, representative bed-load sediment transport rate. Here, ${\hat{u}}_{crb}$ is the threshold velocity for sediment motion as bed load ( $\text{m s}^{-1}$ ). It should be noted that ${\hat{A}}=A{\hat{u}}_{0}^{3}$ , where $A$ is the equivalent dimensional constant of KD10.

We employ the entrainment model in Pritchard & Hogg (Reference Pritchard and Hogg2003) and Pritchard & Hogg (Reference Pritchard and Hogg2005), taking the entrainment rate ${\hat{E}}=\hat{m}_{e}\left(({\hat{u}}^{2}-{\hat{u}}_{crs}^{2})/{\hat{u}}_{0}^{2}\right)$ , with $\hat{m}_{e}$ the parameter of the sediment entrainment rate ( $\text{m s}^{-1}$ ) of suspended load and ${\hat{u}}_{crs}$ the threshold velocity for sediment motion as suspended load ( $\text{m s}^{-1}$ ). It should be noted that we set ${\hat{u}}_{crb}={\hat{u}}_{crs}=0$ in all the simulations in the present work for simplicity. We anticipate that this simplification will not significantly affect beach morphodynamics except, perhaps, for shingle beaches, whereon permeability effects not considered here are also significant; see appendix A.

The deposition rate of suspended load is (also following Pritchard & Hogg Reference Pritchard and Hogg2005) $\hat{D}={\hat{w}}_{s}{\hat{c}}$ , with ${\hat{w}}_{s}$ the effective settling velocity of suspended sediment ( $\text{m s}^{-1}$ ).

Therefore, (2.3) and (2.4) become

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{B}_{\,\hat{t}}+3{\it\xi}\frac{{\hat{A}}}{{\hat{u}}_{0}^{3}}{\hat{u}}^{2}{\hat{u}}_{\,\hat{x}}={\it\xi}\left({\hat{w}}_{s}{\hat{c}}-\hat{m}_{e}\frac{{\hat{u}}^{2}}{{\hat{u}}_{0}^{2}}\right), & \displaystyle\end{eqnarray}$$

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle {\hat{c}}_{\,\hat{t}}+{\hat{u}}{\hat{c}}_{\,\hat{x}}=\frac{1}{{\hat{h}}}\left(\hat{m}_{e}\frac{{\hat{u}}^{2}}{{\hat{u}}_{0}^{2}}-{\hat{w}}_{s}{\hat{c}}\right). & \displaystyle\end{eqnarray}$$

Figure 1.

Schematic diagram for a general swash.

2.2. Non-dimensionalisation

The non-dimensional variables are

(2.8a−f ) $$\begin{eqnarray}x=\frac{\hat{x}}{{\hat{h}}_{0}},\quad t=\frac{\hat{t}}{{\hat{h}}_{0}^{1/2}g^{-1/2}},\quad h=\frac{{\hat{h}}}{{\hat{h}}_{0}},\quad u=\frac{{\hat{u}}}{{\hat{u}}_{0}},\quad B=\frac{\hat{B}}{{\hat{h}}_{0}}\quad \text{and}\quad c=\frac{{\hat{c}}}{{\hat{c}}_{0}},\end{eqnarray}$$

where ${\hat{h}}_{0}$ is a length scale, ${\hat{c}}_{0}$ is a reference concentration and ${\hat{u}}_{0}=(g{\hat{h}}_{0})^{1/2}$ .

Substitution of (2.8a−f ) into the governing equations (2.1), (2.2), (2.6) and (2.7) gives

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle h_{t}+uh_{x}+hu_{x}=0, & \displaystyle\end{eqnarray}$$

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle u_{t}+uu_{x}+h_{x}+B_{x}=-\frac{c_{d}|u|u}{h}, & \displaystyle\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle B_{t}+\frac{3{\it\xi}{\hat{A}}}{{\hat{h}}_{0}(g{\hat{h}}_{0})^{1/2}}u^{2}u_{x}=\frac{{\it\xi}}{(g{\hat{h}}_{0})^{1/2}}({\hat{w}}_{s}{\hat{c}}_{0}c-\hat{m}_{e}u^{2}), & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle c_{t}+uc_{x}=\frac{1}{(g{\hat{h}}_{0})^{1/2}{\hat{c}}_{0}}\frac{1}{h}(\hat{m}_{e}u^{2}-{\hat{w}}_{s}{\hat{c}}_{0}c). & \displaystyle\end{eqnarray}$$

Let ${\hat{c}}_{0}=\hat{m}_{e}/{\hat{w}}_{s}$ , ${\it\sigma}={\it\xi}{\hat{A}}/({\hat{h}}_{0}(g{\hat{h}}_{0})^{1/2})$ , $M={\it\xi}(\hat{m}_{e}/(g{\hat{h}}_{0}^{1/2}))$ and ${\tilde{E}}={\hat{w}}_{s}/(g{\hat{h}}_{0}^{1/2})$ , so that (2.11) and (2.12) become

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle B_{t}+3{\it\sigma}u^{2}u_{x}=M\left(c-u^{2}\right), & \displaystyle\end{eqnarray}$$

(2.14) $$\begin{eqnarray}\displaystyle & \displaystyle c_{t}+uc_{x}=\frac{1}{h}{\tilde{E}}\left(u^{2}-c\right). & \displaystyle\end{eqnarray}$$

It should be noted that here ${\tilde{E}}=E\tan {\it\alpha}$ , where $E$ is the exchange rate parameter of Pritchard & Hogg (Reference Pritchard and Hogg2005) and $\tan {\it\alpha}$ is the beach slope. Therefore ${\tilde{E}}$ governs the response time of the suspended sediment to changes in flow conditions. Here, $M$ represents bed mobility with regard to suspended-load transport and ${\it\sigma}$ the equivalent quantity for bed-load transport.

When the sediment concentration $c$ is only determined by instantaneous $u$ , it is denoted $c_{eq}$ to represent suspended sediment concentration in an instantaneous equilibrium state (i.e. when there is neither erosion nor deposition); here, $c_{eq}=u^{2}$ . In (2.14), when ${\tilde{E}}/h\rightarrow \infty$ , the adjustment of $c$ to $c_{eq}$ becomes immediate. It should be noted that $c_{eq}=c_{eq}(x,t)$ .

The vector form of these four non-dimensional governing equations is

(2.15) $$\begin{eqnarray}\boldsymbol{U}_{t}+\boldsymbol{A}(\boldsymbol{U})\boldsymbol{U}_{x}=\boldsymbol{S},\end{eqnarray}$$

with

(2.16) $$\begin{eqnarray}\displaystyle \boldsymbol{U}=\left[\begin{array}{@{}c@{}}h\\ u\\ B\\ c\end{array}\right],\quad \boldsymbol{A}(\boldsymbol{U})=\left[\begin{array}{@{}cccc@{}}u & h & 0 & 0\\ 1 & u & 1 & 0\\ 0 & 3{\it\sigma}u^{2} & 0 & 0\\ 0 & 0 & 0 & u\end{array}\right],\quad \boldsymbol{S}=\left[\begin{array}{@{}c@{}}0\\ -\displaystyle \frac{c_{d}|u|u}{h}\\ M(c-u^{2})\\ \displaystyle \frac{1}{h}{\tilde{E}}(u^{2}-c)\\ \end{array}\right]. & & \displaystyle\end{eqnarray}$$

The eigenvalues of $\boldsymbol{A}$ are the roots of the polynomial equation

(2.17) $$\begin{eqnarray}({\it\lambda}-u)({\it\lambda}^{3}-2u{\it\lambda}^{2}+(u^{2}-3{\it\sigma}u^{2}-h){\it\lambda}+3{\it\sigma}u^{3})=0.\end{eqnarray}$$

The polynomial equation (2.17) has four roots, one of which, ${\it\lambda}_{4}\equiv u$ , corresponds to the transport of suspended load. The other three roots of (2.17) are denoted ${\it\lambda}_{1}$ , ${\it\lambda}_{2}$ and ${\it\lambda}_{3}$ , such that ${\it\lambda}_{1}\leqslant {\it\lambda}_{3}\leqslant {\it\lambda}_{2}$ . It should be noted that if ${\it\sigma}\neq 0$ and $h\neq 0$ , when $u>0$ , we have ${\it\lambda}_{1}<{\it\lambda}_{3}<{\it\lambda}_{4}<{\it\lambda}_{2}$ ; when $u<0$ , ${\it\lambda}_{1}<{\it\lambda}_{4}<{\it\lambda}_{3}<{\it\lambda}_{2}$ . Moreover, ${\it\lambda}_{1}<0$ and ${\it\lambda}_{2}>0$ as long as ${\it\sigma}\neq 0$ . For the solution of ${\it\lambda}_{1}$ , ${\it\lambda}_{2}$ and ${\it\lambda}_{3}$ we refer to Kelly & Dodd (Reference Kelly and Dodd2009, Reference Kelly and Dodd2010).

Figure 2 illustrates the variation of ${\it\lambda}_{1,2,3}$ characteristics with Froude number when ${\it\sigma}=0.01$ . It should be noted that individual characteristics, ${\it\lambda}_{1,2,3}$ , can ‘behave’ as hydro- or morphodynamic characteristics, depending on the Froude number. It can be seen that there are sudden changes in these roles at critical flow conditions. From simple wave theory (Jeffrey Reference Jeffrey1976), a shock exists when the characteristics of the same family intersect, and it is a ${\it\lambda}_{i}$ shock if it is the ${\it\lambda}_{i}$ characteristics that intersect.

Figure 2.

Variation of dimensionless wave velocities with Froude number ( $F_{r}=u/\sqrt{h}$ ) for a system with bed-load transport $q=u^{3}$ ( ${\it\sigma}=0.01$ ).

2.3. Numerical method

The specified time interval method of characteristics (STI MOC) (Kelly & Dodd Reference Kelly and Dodd2009, Reference Kelly and Dodd2010), which can resolve shocks very accurately, is used to solve (2.9), (2.10), (2.13) and (2.14) simultaneously. As the ${\it\lambda}_{1,2,3}$ characteristic fields associated with (2.9), (2.10) and (2.13) are genuinely nonlinear, and the ${\it\lambda}_{4}$ associated with (2.14) is genuinely linear, (2.9), (2.10) and (2.13) are combined to obtain total derivatives (i.e. the characteristic form) of $h$ , $u$ and $B$ with respect to time, and (2.14) is used to find the total derivative of $c$ with respect to time. Thus,

(2.18) $$\begin{eqnarray}\text{Re}^{(k)}={\it\lambda}_{k}\frac{\text{d}u}{\text{d}t}+\frac{{\it\lambda}_{k}}{{\it\lambda}_{k}-u}\frac{\text{d}h}{\text{d}t}+\frac{\text{d}B}{\text{d}t}=-{\it\lambda}_{k}\frac{c_{d}|u|u}{h}+M(c-u^{2}),\quad \text{along }\frac{\text{d}x}{\text{d}t}={\it\lambda}_{k},\quad k=1,2,3,\end{eqnarray}$$

and (2.14) is written as the total derivative of $c$ with respect to time,

(2.19) $$\begin{eqnarray}\frac{\text{d}c}{\text{d}t}=\frac{1}{h}{\tilde{E}}(u^{2}-c),\quad \text{along }\frac{\text{d}x}{\text{d}t}={\it\lambda}_{4}=u.\end{eqnarray}$$

These four equations of (2.18) and (2.19) are solved numerically to obtain $h$ , $u$ , $B$ and $c$ in the combined load system.

2.4. Shock conditions

For derivations of shock conditions for mass and momentum conservation we refer to Kelly & Dodd (Reference Kelly and Dodd2010) and Zhu et al. (Reference Zhu, Dodd and Briganti2012); the shock conditions are

(2.20) $$\begin{eqnarray}\displaystyle & h_{R}u_{R}-h_{L}u_{L}-(h_{R}-h_{L})W=0, & \displaystyle\end{eqnarray}$$

(2.21) $$\begin{eqnarray}\displaystyle & \displaystyle W(h_{R}u_{R}-h_{L}u_{L})-\left(h_{R}u_{R}^{2}+\frac{h_{R}^{2}}{2}-h_{L}u_{L}^{2}-\frac{h_{L}^{2}}{2}\right)-\frac{1}{2}(B_{R}-B_{L})(h_{R}+h_{L})=0,\qquad & \displaystyle\end{eqnarray}$$

where $L$ and $R$ represent variables on the left and right sides of a shock, and $W$ is the shock velocity.

For the bed evolution, the change of bed level in the fixed domain [ $x_{1}$ , $x_{2}$ ] is balanced by the net sediment flux into (out of) that domain, and also the net sediment settlement onto (or entrainment from) it, thus

(2.22) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\int _{x_{1}}^{x_{2}}B\text{d}x+[{\it\sigma}u^{3}]_{x_{1}}^{x_{2}}=\int _{x_{1}}^{x_{2}}M(c-u^{2})\text{d}x.\end{eqnarray}$$

Here, it is also assumed that a shock located at ${\it\zeta}(t)$ lies between $x_{1}$ and $x_{2}$ , i.e. $x_{1}<{\it\zeta}<x_{2}$ . In the limit $x_{1}\rightarrow {\it\zeta}$ and $x_{2}\rightarrow {\it\zeta}$ , (2.22) becomes

(2.23) $$\begin{eqnarray}-W[B]_{x_{1}}^{x_{2}}+[{\it\sigma}u^{3}]_{x_{1}}^{x_{2}}=0.\end{eqnarray}$$

It is found that suspended load has no contribution to the shock condition for the bed evolution.

The suspended sediment in the water column in the domain [ $x_{1}$ , $x_{2}$ ] is balanced by the net suspended sediment flux inflow across the $x_{1}$ and $x_{2}$ sections and the entrainment from (settlement onto) the bed, thus

(2.24) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\int _{x_{1}}^{x_{2}}(hc)\text{d}x+[huc]_{x_{1}}^{x_{2}}=\int _{x_{1}}^{x_{2}}M(u^{2}-c)\text{d}x & \displaystyle \nonumber\\ \displaystyle & \Rightarrow \ [-hcW+hcu]_{x_{1}}^{x_{2}}=0. & \displaystyle\end{eqnarray}$$

(2.25) $$\begin{eqnarray}\displaystyle & (B_{R}-B_{L})W-{\it\sigma}(u_{R}^{3}-u_{L}^{3})=0, & \displaystyle\end{eqnarray}$$

(2.26) $$\begin{eqnarray}\displaystyle & h_{R}c_{R}(u_{R}-W)-h_{L}c_{L}(u_{L}-W)=0. & \displaystyle\end{eqnarray}$$

It should be noted that the shock condition of the bed evolution (2.25) in the combined load system is identical to that without suspended load in Kelly & Dodd (Reference Kelly and Dodd2010).

For a ${\it\lambda}_{1,2,3}$ shock, if $h_{L}\neq 0$ , $h_{R}\neq 0$ , $u_{L}\neq W$ and $u_{R}\neq W$ , from (2.20), the shock condition for the transport of suspended load (2.26) is simplified as

(2.27) $$\begin{eqnarray}c_{R}-c_{L}=0.\end{eqnarray}$$

This implies that the sediment concentration across a ${\it\lambda}_{1,2,3}$ shock is continuous, and it is determined by the genuinely linear characteristic field associated with the transport of suspended sediment.

For a ${\it\lambda}_{4}$ shock, which is a contact wave, we have $h_{L}=h_{R}$ , $u_{L}=u_{R}$ , $B_{L}=B_{R}$ and $c_{L}\neq c_{R}$ .

For the shock fitting method, we refer the reader to Kelly & Dodd (Reference Kelly and Dodd2010), who have described the technique in great detail.

2.5. Model testing

The model is verified against both suspended- and bed-load-only models. In appendix B, we compare with the PH05 results, which are for suspended load, on a nearly fixed bed. Testing of the bed-load-only model is presented in Zhu et al. (Reference Zhu, Dodd and Briganti2012).

3.1. Suspended load

A suspended-load-only simulation is achieved by setting the bed mobility parameter for bed load ${\it\sigma}=0$ in the combined load model.

The final bed changes after one swash event for various $M$ values ( ${\it\sigma}=0$ ) with and without pre-suspended sediment are shown in figure 4(a,b). The effect of varying $M$ can be seen. For ${\it\sigma}=0$ , (2.13) shows the linear relationship between bed change and suspended-load mobility $M$ : as long as the flow remains largely unaffected by the bed change ( ${\tilde{E}}$ is constant), $c$ will remain similarly unaffected by varying $M$ , and a change in $M$ results in a similar but amplified pattern of erosion and deposition, i.e. ${\rm\Delta}B\propto M$ (where ${\rm\Delta}B$ is the change in the bed level). The swash flow is largely unaffected by suspended load, see figure 22 in appendix D. This point is illustrated in figure 4(c,d), which shows the normalised beach changes against $M$ , both with and without pre-suspended sediment. Only for $M=0.005$ do we see a noticeable change in the resulting beach change profile (see figure 4 c,d).

Figure 4.

(a,b) Bed change after one PW01 swash event for various $M$ values ( ${\tilde{E}}=0.001$ ), for (a) only locally entrained sediment and (b) both locally and pre-suspended sediment ( $c_{0}=1$ ). (c,d) Bed changes for the same simulations shown, respectively, in (a) and (b), normalised by $M$ . (e,f) Bed changes after one PW01 swash event for various ${\tilde{E}}$ values ( $M=1\times 10^{-4}$ ), for (e) only locally entrained sediment and ( f) both locally and pre-suspended sediment ( $c_{0}=1$ ) .

The effect of ${\tilde{E}}$ on the net sediment flux was discussed in depth by Pritchard & Hogg (Reference Pritchard and Hogg2005). In figure 4(e,f), we show a variety of resulting profiles for different values of ${\tilde{E}}$ , now with a fully mobile bed. As ${\tilde{E}}$ increases there is less net movement overall. This is due to the erosion/deposition term in (2.13) ( $c-u^{2}$ ) being near zero for larger ${\tilde{E}}$ (consistent with PH05) for most of the swash event, because $c\approx c_{eq}$ (because of the more rapid adjustment as ${\tilde{E}}$ increases). Therefore, overall beach change is reduced (for fixed $M$ ) (see also figure 21 for verification against PH05 in terms of net fluxes). This is illustrated in figure 5.

Figure 5.

Contour plots for $c-u^{2}$ for the simulations in figure 4: (a) ${\tilde{E}}=0.001$ ; (b)  ${\tilde{E}}=0.03$ .

Additionally, when pre-suspended sediment is present (figure 4 f) the net change increasingly favours deposition at the base of the swash. This is again consistent with PH05, and stems from the fast response time for larger ${\tilde{E}}$ causing initially entrained sediment to be immediately deposited (in the lower swash).

When no pre-suspended sediment is present (figure 4 e) the proportion of deposited to eroded sediment volume (deposited volume/eroded volume) for $x>0$ varies from 0.06 ( ${\tilde{E}}=0.03$ ) to 0.05 ( ${\tilde{E}}=0.001$ ), peaking at approximately 0.07 ( ${\tilde{E}}\approx 0.01$ ) in between. It should be noted that Pritchard & Hogg (Reference Pritchard and Hogg2005) also observed a peak in the maximum positive net flux across the swash zone caused by locally entrained sediment, which essentially is a measure of the total amount of deposition in the swash zone. Moreover, the work of Pritchard & Hogg (Reference Pritchard and Hogg2005) (see figure 11a,c of PH05) has further indicated a peak value in the proportion of deposited to eroded sediment flux/volume. For ${\tilde{E}}\rightarrow \infty$ , this ratio tends to $0$ , as expected.

3.2. Suspended and bed load

If we include both suspended and bed load we can now examine their effects on the beach during this event. The final bed changes in three combined load simulations are shown in figure 6, and correspond to beaches in which either bed or suspended load is dominant, or they are approximately equal. Also shown are the bed changes caused by the equivalent bed-load-only and suspended-load-only simulations, and that due only to those same components of the combined load simulation. It should be noted that, for the parameters chosen here, each mode of sediment transport has little effect on the other, and each has a distinct morphodynamical signature, and these are consistent with the results of Pritchard & Hogg (Reference Pritchard and Hogg2005) and Kelly & Dodd (Reference Kelly and Dodd2010).

Figure 6.

Bed changes in the combined load simulations and comparisons with bed-load-only and suspended-load-only simulations after one single PW01 swash: (a) bed-load dominance; (b) bed- and suspended-load approximate parity; (c) suspended-load dominance.

4.1. Solitary wave event

A solitary wave of height $H_{w}=0.6$ on a still water depth of $h_{st}=1$ , with its crest located at $x=-22$ when $t=0$ , is considered, see figure 3. In the region $x\leqslant -10$ the bed is flat ( $\tan {\it\alpha}_{off}=0$ ), while for $x\geqslant -10$ the beach is of a uniform slope $\tan {\it\alpha}=0.0667$ . For $x\geqslant -10$ , $h(x,t=0)=1-\tan {\it\alpha}(x+10)$ , $u(x,t=0)=0$ and $B(x,t=0)=\tan {\it\alpha}(x+10)$ . In the region $x<-10$ , the water depth is determined according to the surface profile for a solitary wave given by Mei (Reference Mei1990), with $h(x<-10,t=0)=1+H_{w}\ \text{sech}^{2}(0.3\left((3H_{w})/(4h_{st}^{3})\right)^{1/2}(x+22))$ . The water velocity is then determined by the hydrodynamic Riemann invariant along the backward characteristic: $u(x<-10,t=0)=2(\sqrt{h(x,t=0)}-1)$ . The bed level is adjusted to the water flow with $B(x<-10,t=0)={\it\sigma}((u(x,t)^{3})/\sqrt{h(x,t)})$ . However, the water flow is assumed not to be affected by the bed change at the initial time. Across the domain, $c(x,t=0)=c_{eq}(x,t=0)=u^{2}(x,t=0)$ is assumed.

4.2. Simulation without bed shear stress

The flow structure as a solitary wave travels shorewards over an erodible beach ( ${\it\sigma}=0.01$ , $M=0.001$ and ${\tilde{E}}=0.01$ ) simulated by the combined load model is shown in figure 7. When the solitary wave travels shorewards, it breaks and forms a shock (bore) travelling to the shore. The shock then collapses at the initial shoreline position ( $x=5$ ), and then the water climbs up the dry beach. The water velocity reaches its maximum when the shock collapses, and then decreases when the flow climbs up the dry beach. The flow (in the region $x>5$ ) is similar to that in the PW01 swash (cf. figures 7 a and 22). This is because the wave breaks at the base of the swash with little water momentum behind it, similarly to the PW01 event (see Guard & Baldock Reference Guard and Baldock2007).

Figure 7.

Contour plots for the combined load solitary wave swash simulation over an erodible beach ( ${\it\sigma}=0.01$ , $M=0.001$ and ${\tilde{E}}=0.01$ ): (a) $h$ ; (b) $u$ ; (c) ${\rm\Delta}B$ ; (d) $c$ ; (e) instantaneous deposition/erosion distribution due to suspended load. The thick black dashed line in (e) represents the backwash bore path.

The bore collapse at $x=5$ causes considerable entrainment into the water column, which is then moved onshore. Erosion prevails at the base throughout the computation time; see figure 7(c). In the early uprush, $c<c_{eq}$ in the lower swash, and so sediment is transported to the upper swash (figure 7 d,e). Conversely, in the later stages of the uprush, $c>c_{eq}$ , and suspended sediment begins to settle out and $c$ decreases. The overall picture is similar to that depicted for moderate/small ${\tilde{E}}$ values by Pritchard & Hogg (Reference Pritchard and Hogg2005).

It should be noted that at the shoreline $c=c_{eq}=u_{s}^{2}$ and at the maximum inundation it is 0. This is because as the tip is approached ${\tilde{E}}/h\rightarrow \infty$ , so that there is an instantaneous adjustment of $c$ to $c_{eq}$ . For the whole backwash (for $x>5$ ), $c<c_{eq}$ , and sediment is entrained into the water column and moved seawards, and the beach is eroded; see figure 7(d,e).

Bed-load transport results in erosion everywhere (not shown) (because ${q_{b}}_{x}>0$ everywhere), except at the shoreline, where there is instantaneous increase (decrease) in $B$ due to the sediment bore advancing (receding) (see Kelly & Dodd Reference Kelly and Dodd2010; Zhu et al. Reference Zhu, Dodd and Briganti2012).

The shoreline reaches the maximum inundation $x\approx 42.6$ at $t\approx 50.3$ . The accelerating offshore flow rapidly becomes supercritical, and, flowing into slowly retreating deeper water, results in a backwash bore (Hibberd & Peregrine Reference Hibberd and Peregrine1979; Guard & Baldock Reference Guard and Baldock2007; Antuono & Hogg Reference Antuono and Hogg2009; a hydraulic jump), which is also found in the morphodynamic bed-load-only simulations for the HP79 swash event (Zhu et al. Reference Zhu, Dodd and Briganti2012). The development of a backwash bore can be seen in figure 7(a,b). The strength gradually increases, with increasing differences in $h$ , $u$ and $B$ on the two sides of the shock, and therefore leads to the development of a bed step coincident with the bore; this shock moves offshore and gradually slows. The shoreline eventually catches up with this shock, leading to a fully developed bed step at the base of a dry beach; see § 4.3.

Across the shock $u$ is discontinuous, and is $c$ continuous, (2.20), (2.21) and (2.27), which results in entrainment and erosion (deposition and accretion) on the shoreward (seaward) side of the bed step as (from) suspended load (see (2.13)). This continues until the sediment concentration $c$ on the shoreward side exceeds $c_{eq}$ , at which point the boundary between suspended-load erosion and accretion departs from the shock path (figure 7 e).

It is the bed load that is most closely linked with the bed-step development, however, as this responds instantly to all flow changes in the vicinity of the shock. Across the shock the massively different bed-load transport rates ( $q_{b}$ ) result in accretion immediately seaward of the shock, and development and offshore propagation of the bed step. It should be noted, however, that shoreward of the shock there is also local accretion due to convergence of bed-load transport. This is caused by corresponding initially small gradients in $u$ (figure 7 b) shoreward of the shock, which instantly affect $q_{b}$ (unlike $q_{s}$ ), leading to local deposition. Thus, the bed step both advances seawards and grows in height.

It should be noted that although suspended load can affect a moving bed discontinuity through changed erosion/accretion rates across the shock, and by modifying the beach profile generally, the bed-step height and velocity are directly linked to bed-load transport (as long as $W\neq 0$ ; see (2.25)).

4.3. Analysis of when the shoreline catches up with the backwash bore

In the simulation without bed shear stress, the shoreline eventually catches up with the backwash bore. As there is a sediment bore at the shoreline, this is a shock–shock collision. In this situation there are three regions, with the rightmost being dry, and in which the right shock (the shoreline) is wet–dry and the left (backwash bore/bed step) is a wet–wet shock; see figure 8.

Figure 8.

Schematic diagram for two-shock collision.

As the two shocks come closer, the middle region gradually vanishes, and in the limiting case this region converges to one point, such that the values of dependent variables on the left and right extremities of this region become equal. Therefore, at the moment of the two shocks colliding, the middle region is assumed to disappear and the flow has only one newly formed discontinuity; however, the shock conditions are then usually not satisfied at this new discontinuity, which is therefore not stable and collapses, with the resulting states found by solution of a local Riemann problem. This is not pursued here.

As the middle region width tends to $0$ , $h_{R}\rightarrow 0$ and $u_{R}\rightarrow \text{d}x_{s}/\text{d}t=u_{s}$ . As the numerical code cannot proceed when the two shocks are very close, the numerical solution is only an approximation for the case of zero middle region width. The analytical solution for the backwash bore (appendix E), of which one side is a nearly dry bed but with water of finite velocity, can be utilised to obtain the limit flow structure right before the shock collision.

From appendix E, when the shoreline approaches the backwash bore

(4.1) $$\begin{eqnarray}B_{R}-B_{L}\rightarrow h_{L}-h_{R}\rightarrow h_{L}.\end{eqnarray}$$

Across the sediment bore at the shoreline we have $B_{D}=B_{R}-{\it\sigma}u_{s}^{2}$ . Thus, as the shoreline approaches the bed step

(4.2) $$\begin{eqnarray}B_{D}-B_{L}=B_{D}-B_{R}+B_{R}-B_{L}=h_{L}-{\it\sigma}u_{s}^{2}.\end{eqnarray}$$

Therefore, the resulting (post-collision, unstable) bed-step height will be slightly lower than the water depth on the seaward side $h_{L}$ . The water on the seaward side may overtop the bed step, depending on the velocity and also bed mobility.

4.4. The inclusion of bed shear stress

The contour plots for the solitary wave simulation with bed shear stress ( ${\it\sigma}=0.01$ , $M=0.001$ , ${\tilde{E}}=0.01$ and $c_{d}=0.01$ ) are shown in figure 9. It should be noted that the shoreline does not recede in the backwash (Antuono, Soldini & Brocchini Reference Antuono, Soldini and Brocchini2012; Zhu & Dodd Reference Zhu and Dodd2013), because of bed friction, and the swash zone is always wet once wetted. It should be noted that the corresponding swash period is much longer than for $c_{d}=0$ . The flow structure in the uprush is similar to that in the simulation without bed shear stress, with an overall deeper flow and smaller velocity. However, bed shear stress greatly reduces the seaward velocity of the backwash flow, and the backwash duration exceeds that of the uprush. The bed shear stress also modifies the flow structure in the backwash, and more shocks are formed in the backwash. The shock paths (including inception and termination points) are illustrated in figure 9(a–c). It should be noted that the first shock to form (a ${\it\lambda}_{1}$ shock) does so at about $t=23.5$ . It quickly propagates offshore, and has no influence on the subsequent bed-step development, so we ignore it hereafter. It should be noted that the values of $c$ in figure 9(d) are considerably reduced compared with those in figure 7(d), especially during the backwash, consistent with reduced backwash velocities.

Figure 9.

Contour plots for solitary wave simulations with bed shear stress over an erodible beach ( ${\it\sigma}=0.01$ , $M=0.001$ , ${\tilde{E}}=0.01$ and $c_{d}=0.01$ ): (a) $h$ ; (b) $u$ ; (c) ${\rm\Delta}B$ ; (d) $c$ . In (a–c) we show the shock paths of the ${\it\lambda}_{3}$ and ${\it\lambda}_{2}$ shocks, with their points of inception and termination. Dotted line: ${\it\lambda}_{1}$ shock; dot-dashed line: first ${\it\lambda}_{3}$ (weak) shock; dashed line: second ${\it\lambda}_{3}$ shock (backwash bore); solid line: ${\it\lambda}_{2}$ shock.

In the backwash, a bore ( ${\it\lambda}_{3}$ shock) develops (the backwash bore), which initially travels seawards (see figure 9). This is a robust feature of the backwash (see also figure 7 a). At $t\approx 61.4$ , another weaker ${\it\lambda}_{3}$ shock collides with the backwash bore. Here, we treat the shock collision by neglecting the weaker shock when it is in close proximity to the backwash bore. This approach shows good agreement with the idealised Riemann solution. The collision increases the backwash bore ( ${\it\lambda}_{3}$ ) shock strength slightly; see figure 10. Thereafter, the backwash bore slows, and decreases in strength (see figure 10).

Figure 10.

Shock strength and Froude number for ${\it\lambda}_{2}$ and ${\it\lambda}_{3}$ shocks in the backwash. A ○ represents the inception of a shock.

At $t\approx 67.7$ , a ${\it\lambda}_{2}$ shock develops on the shoreward side of the backwash bore/bed step, the strength of which rapidly increases (see figure 10), and which propagates shorewards, as the flow in the later backwash diminishes. Thereafter it diminishes and slows, in effect re-establishing a ‘shoreline’ boundary between the sea and water still draining back. It should be noted that, as with all morphodynamical shocks, this ${\it\lambda}_{2}$ shock is accompanied by a bed discontinuity, which travels with the shock, but which is much smaller than the bed step associated with the original ${\it\lambda}_{3}$ backwash bore/bed step; see figure 10.

At $t\approx 73.9$ , the backwash bore/bed step ( ${\it\lambda}_{3}$ shock) is brought to rest: $W=0$ . This is the point of reversal. From the shock conditions, a bed discontinuity with $W=0$ exists only when $u_{L}=u_{R}=0$ . Furthermore, $h_{L}+B_{L}=h_{R}+B_{R}$ when $W=0$ , indicating the continuous surface level across the bed step ( ${\it\lambda}_{3}$ shock) at the point of stationary state. Thus, there is at this instant only the bed step, and an associated change in depth, with other quantities being continuous. Thereafter $u_{L}$ and $u_{R}$ increase, but the backwash bore ( ${\it\lambda}_{3}$ shock) vanishes, because the characteristics on either side of the bed step diverge, thus allowing the shock conditions only to be satisfied by a non-physical shock (i.e., one with diverging characteristics). Mathematically, the structure at the former bed-step position after reversal could be derived from an idealised dam-break problem, and (seaward to shoreward) is a ${\it\lambda}_{1}$ shock, a ${\it\lambda}_{3}$ rarefaction fan and a ${\it\lambda}_{2}$ shock. The rarefaction fan is almost stationary, and the change in variables across it is much larger than that across the two (weak) shocks (which are not shown here). Thus, the previous bed discontinuity acquires a continuous structure; however, the observed bed-step profile is little changed.

This process of the original seaward moving ${\it\lambda}_{3}$ shock (backwash bore/bed step) slowing, and the creation of a fast-moving mainly hydrodynamical ${\it\lambda}_{2}$ shock, is equivalent to the reversal of a hydraulic jump on an immobile bed but here results in a bed step (Larson & Sunamura Reference Larson and Sunamura1993) feature being left, almost inert, on the lower beachface as the hydraulic jump reverses. Finally, it should be noted that all three shocks are formed at approximately critical conditions ( $F_{r}\approx -1$ ) (figure 10 d). It is for these conditions that the dispersion curves for the ${\it\lambda}_{2}$ and ${\it\lambda}_{3}$ waves (figure 2) change slope rapidly, implying that relatively small changes in flow may result in convergence of characteristics. Further, it should be noted that as an initially small shock passes through critical conditions it is likely to grow, for the same reason (it should be noted that the definition of shock strength here is the jump in characteristic slope across the shock); see also figure 10(a,d). It should be noted that the weak ${\it\lambda}_{3}$ shock does not pass through critical conditions.

Finally, in figure 11 we show the characteristic fields associated with the shock dynamics described above. The characteristic convergence and shock formation can clearly be seen.

Figure 11.

Characteristics diagram for the solitary wave simulation with bed shear stress: (a) for the whole domain; (b) for the vicinity of the reversing shock. Grey solid lines: ${\it\lambda}_{1}$ characteristics; black dashed line: ${\it\lambda}_{2}$ characteristics; black solid lines: ${\it\lambda}_{3}$ characteristics. The thick black solid, dashed, dash-dotted and dotted lines represent the paths of the backwash bore, ${\it\lambda}_{2}$ shock, ${\it\lambda}_{3}$ shock and ${\it\lambda}_{1}$ shock, respectively.

Figure 12.

Final beach changes for solitary wave simulations without bed shear stress for two different values of ${\it\sigma}$ and $M$ .

4.5. Final beach change and bed-step development

The beach changes due to different ${\it\sigma}$ , and $M$ when the shoreline catches up with the backwash bore in the simulations without bed shear stress are shown in figure 12. For all the cases examined, a bed step forms, the height of which decreases as ${\it\sigma}$ decreases. The bed-step height decreases slightly as $M$ increases, due almost entirely to the increased bed level on the seaward side of the bed step, which itself is due to greater sediment deposition for increased $M$ . Thus, bed load ( ${\it\sigma}$ ) dictates crest level (figure 12), but suspended load ( $M$ ) can affect height. This is also shown by figure 13. The effect of $M$ on the step height is not large compared with that of ${\it\sigma}$ , but becomes more pronounced for small ${\it\sigma}$ . It should be noted, however, that these heights refer to the size of the jump in $B$ ( $B_{R}-B_{L}$ ); the overall prominence of the step is dictated by ${\it\sigma}$ (figure 13). One should note also the increased erosion shoreward of the step for larger $M$ (figure 12).

Figure 13.

Bed-step heights and final beach changes for solitary wave simulations without bed shear stress for different ${\it\sigma}$ and $M$ values ( ${\tilde{E}}=0.01$ );  ${\it\sigma}=0.01$ in (b).

To examine the importance of the swash event to the bed-step generation, the final water surface profiles and bed changes for both the present solitary wave simulation and the HP79 swash (see Zhu et al. Reference Zhu, Dodd and Briganti2012) (both for $c_{d}=0$ ) are illustrated in figure 14. It should be noted that both events have the same wave height, but that the HP79 event is essentially an adjustment in water level, so that the backwash (and therefore backwash velocities) is (are) much reduced compared with the uprush. It should be noted also that the bed steps for the solitary wave case including friction (figure 15), in which backwash velocities are significantly reduced and no longer dependent solely on maximum run-up, are consistent with those for the frictionless HP79 case. The bed-step heights in both swashes are close to but slightly smaller than the water depths due to the sediment bore at the tip, consistent with the analysis in § 4.3.

Figure 14.

Beach profiles (a) and changes (b) under a solitary wave (black) and the HP79 swash (grey) with ${\it\sigma}=0.01$ , $M=0.001$ and ${\tilde{E}}=0.01$ ( $c_{d}=0$ ).

In figure 15, we show final bed changes for various $c_{d}$ values. (Profiles are deemed ‘final’ when $h_{R}\leqslant 0.006$ , where here $h_{R}$ is the water depth on the shoreward side of the shock nearest to the shoreline.) Increasing bed friction strongly influences the bed-step development, both the step size and its location, by reducing backwash velocities, and thereby the ${\it\lambda}_{3}$ shock strength and its accompanying bed-step size. Finally, it should be noted that the bed step shown in figure 15 (except that for $c_{d}=0$ ) is no longer a discontinuity, but in fact a rapid variation in $B$ .

Figure 15.

Final beach changes for solitary wave simulations with bed shear stress for various $c_{d}$ values ( ${\it\sigma}=0.01$ , $M=0.001$ and ${\tilde{E}}=0.01$ ).

5.1. Proportions of bed and suspended load

We use figures 16 and 17 to allocate a proportion of the nominal $30\ \text{kg m}^{-1}$ to each mode. These allocations are summarised in table 1, and correspond roughly to a scenario in which bed and suspended load are both significant (test 1), which therefore comprises our best estimate of reality, and two further ones in which bed (test 2) or suspended (test 3) load dominates. Finally, in test 4, we consider a case in which pre-suspended load dominates. Therefore, for test 4, we reduce the local bed- and suspended-load $\hat{Q}$ values (and therefore $M$ and ${\it\sigma}$ ) and then impose $c$ at the time of bore collapse at the shore: $t=t_{c}$ . This is to reproduce the large local concentrations that might be expected due to bore turbulence, which is not present in the mathematical model. The approach here is to impose $c(x,t=0)=c_{eq}(x,0)$ and $c(x,t_{c})=nc_{eq}(x,t_{c})$ , and to determine $n$ such that the net total mid-swash onshore flux is $30\ \text{kg m}^{-1}$ . This results in $n=1.8$ . It should be noted that pre-suspended load exists in tests 1–3 too, but only at equilibrium values (at $t=0$ ). Therefore, much of the sediment initially present falls out of suspension prior to bore collapse; see § 4.1. The volume of sand in each bed step ( $\hat{V}$ ) is also presented in table 1, and is calculated from the base on the left side ${\rm\Delta}B={\rm\Delta}B_{L}$ to the position on the right side such that ${\rm\Delta}B_{R}={\rm\Delta}B_{L}$ , see figure 19(b).

Figure 18.

Contour plots of non-scaled concentration ${\hat{c}}$ for the combined load solitary wave swash simulation for test 1 (a), test 2 (b), test 3 (c) and test 4 (d).

Figure 19.

Final beach profiles for solitary wave simulations with bed shear stress for four test cases. (a) Beach profile $B$ ; (b) schematic diagram for calculation of the sediment volume in the bed step. The grey lines in (a) represent beach profiles at ${\it\lambda}_{3}$ shock reversal ( $W=0$ ).

5.2. Concentrations and final bed profiles

The contour plots of ${\hat{c}}$ for four test cases are shown in figure 18. The suspended-load dominant case test 3 has the largest concentration with a maximum of ${\sim}0.01$ . In the uprush the effect of the pre-suspended sediment can still be seen, but not in the backwash (figure 18 d).

The corresponding final beach profiles for these four tests are shown in figure 19. In figure 19(a) we show bed changes at the point of ${\it\lambda}_{3}$ shock reversal ( $W=0$ ), at which point the bed step achieves peak magnitude. At this instant the bed-step height is directly proportional to ${\it\sigma}$ , as noted earlier, as is the volume (see table 1). Thereafter, the degree to which the bed step is modified depends on $M$ primarily, with the largest reworkings in test 3 ( $M=0.0009$ ) and test 1 ( $M=0.00066$ ); see figure 19(a).

If ${\hat{h}}_{0}=1\ \text{m}$ , we obtain average bed change magnitudes in the region shoreward of the initial shoreline position of about 2 mm, with moderately larger values including the bed step. Masselink & Hughes (Reference Masselink and Hughes1998) and Hughes et al. (Reference Hughes, Masselink and Brander1997) do not record the bed change due to individual events, but Blenkinsopp et al. (Reference Blenkinsopp, Turner, Masselink and Russell2011), on a beach with a grain diameter of $0.4\ \text{mm}$ , but of slope 0.067, record most ( ${\sim}60$ , 75, 95 % moving from seaward to shoreward in the swash) individual swash events with a zero (or at least non-measurable) net beach level change at three cross-shore locations across the swash zone. These negligible changes would appear to include beach changes similar to the average magnitudes calculated here, from the initial shoreline position (figure 7 of Blenkinsopp et al. (Reference Blenkinsopp, Turner, Masselink and Russell2011)). Nonetheless, there is a significant proportion that has non-zero net change (both positive and negative) up to an occasional extreme of ${\sim}4\ \text{cm}$ . Based on this comparison the changes recorded here are a little larger than those observed, particularly at the beach step, but only moderately so. It is not clear where the measurements of bed-level change were made relative to where a bed step might form, but it seems likely that some of them might have been made at such a location, because the measurements of Blenkinsopp et al. (Reference Blenkinsopp, Turner, Masselink and Russell2011) were made over tidal cycles.

Finally, it should be noted that the values of ${\hat{c}}$ in figure 18 are consistent with values measured in some field experiments (Masselink et al. Reference Masselink, Evans, Hughes and Russell2005; Butt et al. Reference Butt, Russell, Puleo and Masselink2005).