Wave-current interaction in morphological change
on a barred beach
Dai MIYAZAKI 1, Yusuke UCHIYAMA 1 and Hideki KAIDA 11
Department of Civil Engineering, Kobe University, Japan

Abstract
Current effect on waves (CEW) is examined for formation of rip channel systems with a barotropic modeling framework.
The model consists of an Eulerian phase-averaged shallow water equation based on a vortex-force formalism, WKB rayequations, and a bed evolution equation with the Soulsby-Van Rijn’s total sediment flux formula. CEW acts on reducing
the offshore extent of seaward rip currents through wave refraction on the currents, leading to modifying the budget ofsediment flux and associated surf-zone topography. Inclusion of CEW results in shoaling rip channels, deepening offshore
mounds, and elongating alongshore spacing of rip channels with normal incidence of offshore waves.
Keywords - wave-current interaction , rip channel, sediment transport
1

Introduction

Recently, Garnier et al. (2008) showed a self-organization of
rip channel systems arising from the deformation of a longshore
bar by using a barotropic wave-current interaction model withsediment transport. However, they did not properly consider theeffects of CEW. Kaida and Uchiyama (2012) found that CEW
is responsible for reducing the offshore extent of rip currents
through the wave refraction by currents that changes momentum
balance substantially. Therefore, we anticipate that themorphological evolution of rip channel may change with orwithout CEW. In the present study, we demonstrate a selforganization of rip channel systems arising from thedeformation of a longshore bar by a phase-averaged barotropic
numerical model with sediment transport with an attention paid
to the influence of CEW. We then quantitatively examine themodulation associated with CEW on morphological evolution ofrip channel influenced by wave, current and sediment flux.
2

Numerical ModelIn the present study, the model relies mainly on a novel
mathematical framework by Uchiyama et al. (2009) based on
the multi-scale asymptotic theory of an Euler wave-averaged
vortex force formalism (McWilliams et al., 2004), accounting
for the two-way mutual interaction between waves and currents
around the surf zone. The hydrodynamic model consists of theshallow-water equation for slowly-evolving Eulerian phaseaveraged mass and momentum conservation. The wave modelis based on WKB ray equations for spectrum-peak waves,consisting of a wave action density conservation equation and awavenumber conservation equation along with the linear
dispersion relation. The sediment transport model is composedof the depth-integrated horizontal sediment flux q based on thetotal (i.e., suspended and bed) load model (Soulsby and Van
Rijn, described in Soulsby, 1997) and the divergence of thehorizontal sediment transport.

{q = α ( u + u st ) − γ u b ∇zbβ∂zb+ ∇q = 0∂t}

3Result3.1

Mobile Bed Experiment

The first result is from the mobile bed cases where the initial
topography (Fig. 1) is set to be alongshore uniform. The size ofthe domain is chosen to be 384 m (x)  2048 m (y) with ananisotropic rectangular grid spacing of dx = 4 m and dy = 8 m.
The imposed offshore wave condition is normal incidence withHrms = 1.0 m (Hrms: the RMS wave height) and the peak wave
period of 10 s. A grid-size (4  8 m) bottom perturbation zb(zb: upward bed level perturbation measured from the initialresting depth h) is placed at (x, y) = (48, 1024) m with a heightof 10 cm as a “seed” of instability on the topographic model day
5 (the hydrodynamic model hour 72) when alongshore-uniform
waves and littoral currents are fully developed. The present
model successfully demonstrates that an alongshore-uniformbarred topography evolves into a rhythmic rip channel systemthrough intrinsic instability triggered by the small topographic
disturbance. Normal incidence of waves result in (nearly) steadyrip channels, whereas alongshore topography migration occurs
with retaining the rip channels under obliquely incident waves(not shown). CEW is found to be responsible for widening therip channel spacing and reducing sediment load in the offshoreof the rip channels, attributed to a noticeably different ripchannel system from that without CEW (Fig. 2).
The

(1)
(2)

where β is an acceleration factor (= 100) to enhance the muchslower temporal evolution of the morphological process thanthat of the current and wave fields.

Fig. 1: Initial topography of an alongshore-uniform single
barred beach mimicking a sandy beach in Duck, NC.
The bar crest is located at x = 80 m.

Fig. 2: Modeled topography on the topographic model day
100 reaching a quasi-steady state. Left: without
CEW, right: with CEW. Color: perturbation
topography zb where positive and negative values
correspond to accretion and erosion, contours: bottom
topography h* = h – zb, and vectors: barotropic
Eulerian velocity.

Fig. 3: Sediment flux divergence on the steady-state ripchannel topography without CEW (same as Fig.2left). Left: without CEW, right: with CEW. Color:
sediment flux divergence (red indicates erosion whileblue is accretion), contours: bottom topography h*.

alongshore wavelength of the rip channel spacing is estimated tobe 400 m without CEW whereas 500 m with CEW.
3.2

Immobile Bed Experiment

We then make use of the steady-state rip channel topography(Fig.2 left) frozen (immobile) for a comparative twin
experiment with and without CEW, in order to examine effectsof CEW on the rip channel formation on the identical
topography. Since waves and currents are modulated by CEW,
a resultant sediment flux imbalance occurs for the case withCEW. The sediment flux divergence shows that the steady-state
rip channel topography without CEW to change asymptoticallytowards that with CEW (Fig.3). Since the alongshore sediment
flux at x = 120 (m) is found to be largely dominated by thetransport term (Fig.4) and u >> ust (not shown), the associatedmorphology is considerably controlled by the wave-driven
Eulerian velocity u, that is modulated by CEW. In order todetermine causes of the change in u, we diagnose a momentum
budget in a simplified manner. Amplified wave height near thebar crest in the rip channels and wave ray bending (refraction)by currents through CEW lead to an increase of breaker wave
dissipation and a change in pressure gradient force throughmodified wave set-up/down distribution. All these modificationby CEW act to attenuate the offshore extent of the rip currents,
resulting in reduction of the offshore sediment transport andwidening the rip channel spacing. This result is consistent withthe findings in Kaida and Uchiyama (2012).
4Conclusion

Inclusion of CEW results in shoaling rip channels, deepening
offshore mounds, and elongating alongshore spacing of ripchannels with normal incidence of offshore waves. There is astrong correlation between the sediment flux and the nearshore
currents affected by CEW. The topographic change estimatedfrom the sediment flux divergence show that the steady-state ripchannel topography without CEW tends to change towards thatwith CEW.

Fig. 4: Terms in the alongshore sediment flux (1) at x = 120
m. Red line: alongshore total sediment flux, blue line:transport term (the first LHS term), green line:
downslope term (the second LHS term).
Acknowledgements
This work is financially supported by the Grant-in-Aid for
Scientific Research (#24560622).
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