Wave run-up

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Definition of Wave run-up:
Wave run-up is the maximum onshore elevation reached by waves, relative to the shoreline position in the absence of waves.
This is the common definition for Wave run-up, other definitions can be discussed in the article


Fig. 1. Definition sketch wave set-down, wave set-up and wave run-up.

Wave run-up is the sum of wave set-up and swash uprush and must be added to the water level reached as a result of tides and wind set-up (Fig. 1). Wave run-up on a beach is generally due to so-called swash bores: the uprush of waves after final collapse on the beach. Wave run-up is an important parameter for assessing the safety of sea dikes or coastal settlements.

Waves refer to waves generated by wind (locally or on the ocean) or waves generated by incidental disturbances of the sea surface such as tsunamis, seiches or ship waves. Wave run-up is often indicated with the symbol [math] R [/math].

Empirical formulas

For waves collapsing on the beach, a first order-of-magnitude estimate is given by the empirical formula of Hunt (1959) [1][2][3],

[math]R \sim \eta_u + H \xi , \qquad (1)[/math]

where [math]\eta_u \sim 0.2 H[/math] is the wave set-up, [math]H[/math] is the offshore significant wave height and [math]\xi[/math] is the surf similarity parameter,

[math]\xi = \Large\frac{\beta}{\sqrt{H/L}}\normalsize = T \, \beta \, \Large\sqrt{\frac{g}{2\pi H}}\normalsize , \qquad (2)[/math]

where [math]\tan \beta \approx \beta[/math] is the beach slope and [math]T[/math] is the peak wave period. The offshore wavelength [math]L[/math] is given by [math]L = g T^2/(2 \pi)[/math], assuming that [math]L / (2 \pi) [/math] is smaller than the local depth [math]h[/math]. The horizontal wave incursion is approximately given by [math] R / \beta[/math].

The maximum run-up depends on the mean water level at the shoreline. This water level is increased by wave-induced setup [math]\eta_0[/math], see Wave set-up. The following empirical relationship has been derived for the sum of maximum wave setup and swash runup[4]:

[math] R \approx 0.73 \; \beta \; \sqrt{HL} . \qquad (3)[/math]

Many empirical formulas have been proposed for the run-up. A popular formula for the run-up [math] R_2[/math] exceeded by only 2 % of the waves for given [math]H, L[/math] has been developed by Stockdon et al. (2006[4]), based on a large dataset:

[math] R_2 = 1.1 \; (\eta_u + 0.5 \sqrt{S_w^2 + S_{ig}^2} \, ) , \qquad \xi \ge 0.3 , \qquad R_2= 0.043 \; \sqrt{HL} , \qquad \xi \lt 0.3 , \qquad (4) [/math]

where [math]\eta_u = 0.35 H \xi[/math] is the wave set-up, [math]S_w=0.75 H \xi[/math] is the swash uprush related to incident waves and [math]S_{ig}=0.06 \sqrt{HL}[/math] is the additional uprush related to infragravity waves. The factor 1.1 takes into account the non-Gaussian distribution of run-up events.

An empirical expression taking the medium sediment grain size [math]d_{50}[/math] [m] into account is[1][5],

[math]R = 0.4 \beta \,T \, \sqrt{gH} \, \exp(-10 \, d_{50}^{0.55}) \, . \qquad (5) [/math]

This formula is less suitable for dissipative beaches, where swash is dominated by infragravity waves.

From an inventory of run-up formulas by Gomes da Silva et al. (2020[6]), it appears that for steep beaches ([math] \beta \gt 0.1[/math]) the run-up increases with increasing beach slope (approximately linear dependance[7]), while for gently sloping dissipative beaches ([math] \beta \lt 0.1[/math]) the dependence on beach slope is weak or absent[7][4]. In these latter cases, run-up is dominated by infragravity waves, that yield a small run-up that increases with increasing wave height (approximately linear dependence[8][9]). Field observations[10] and numerical models[11] point to a dependence of infragravity swash on the frequency spread and the directional spread of incident waves. The largest infragravity swash has been observed for incident waves with a small directional spread and a large frequency spread.

Run-up on cobble revetments

Blenkinsopp et al. (2022[12]) observed that the significant swash height increases substantially as the swash zone moves from the lower dissipative sand beach to a higher reflective gravel berm during rising tide. In cases where the backshore berm or foredune of such composite beaches is (artificially) protected with gravel or cobbles, the run-up [math] R_2[/math] exceeded by only 2 % of the waves when the water level during storms reaches higher than the toe of the cobble berm, can be estimated from the approximate empirical formula

[math] R_2 \approx 0.26 + 0.19 H + 3.1 H_{toe} \, \beta_{berm} \, . \qquad (6) [/math]

The first two terms represent the wave setup; [math]\beta_{berm}[/math] is the berm slope and [math]H_{toe}[/math] is the significant wave height at the toe of the berm. For saturated (depth-limited) surf zones, the significant wave height at the toe of the berm can be estimated from the breaker index [math]\gamma_b[/math] and the water depth [math]h_{toe}[/math] at the toe, [math]H_{toe} = \gamma_b h_{toe}[/math]. See also the article Wave overtopping.

Accuracy of run-up estimates

The accuracy (root-mean-square deviation) of run-up predictions using empirical formulas such as (1), (3) or (4) is usually not better than [math]0.25 \, R[/math].[3] The general applicability of empirical formulas for run-up prediction based on simple parametric representations of beach and shoreface is limited due to the influence of the more detailed characteristics of the local beach and shoreface bathymetry (e.g., beach permeability and groundwater level, curvature of the beach profile, crescentic bars, or swash acceleration along the horn of a beach cusp embayment[13]). The tide level may influence the wave run-up due to wave breaking on nearshore sandbars.[14] The applicability of empirical formulas for the wave set-up, which is a substantial component of the run-up, is limited for similar reasons. Accurate estimates of the wave run-up require in-situ observations or detailed numerical models.

A more detailed discussion of wave run-up and backwash is given in the articles Swash and Swash zone dynamics.

An analytical model estimate of the runup of monochromatic non-broken waves on a uniform sloping beach is discussed in the article Waves on a sloping bed.

Run-up saturation

The empirical formulas suggest that the run-up is an ever increasing function of the incident wave height. Field and laboratory observations show that this is not the case[15][16]. Several processes limit the uprush on the beach when incident waves are very high: breaking of short waves in the surf zone (leading to surf zone saturation), breaking of infragravity waves on the beach face and collision of the uprushing swash bore with the backwash of the preceding swash bore. More details can be found in the articles Swash, Swash zone dynamics and Breaker index. The greatest wave run-up occurs for the highest waves that do not break, which are generally waves in the lower part of the wave frequency spectrum (long-period waves). Theoretical formulas for run-up saturation (for monochromatic frictionless waves, see the articles Waves on a sloping bed, Swash) have the form

[math]R \lt R_s = A \, g \, \beta^2 \, T^2 \, . \qquad (7)[/math]

Using for the coefficient [math]A[/math] the value [math]A \approx 0.14[/math], reasonable agreement is found with field and laboratory observations. [16]


Related articles

Swash zone dynamics
Wave set-up
Swash
Waves on a sloping bed
Tsunami
Breaker index


References

  1. Jump up to: 1.0 1.1 Hunt, I.A. 1959. Design of seawalls and breakwaters. J. Waterw. Harbors Division ASCE 85: 123–152
  2. Jump up Holman, R.A. and Sallenger, A.H. 1985. Setup and swash on a natural beach. J. Geophys. Res. 90: 945–953
  3. Jump up to: 3.0 3.1 Atkinson, A.L., Power, H.E., Moura, T., Hammond, T., Callaghan, D.P. and Baldock, T.E. 2017. Assessment of runup predictions by empirical models on non-truncated beaches on the south-east Australian coast. Coast. Eng. 119: 15–31
  4. Jump up to: 4.0 4.1 4.2 Stockdon, H.F., Holman, R.A., Howd, P.A. and Sallenger, A.H. 2006. Empirical parameterization of setup, swash, and runup. Coast. Eng. 53: 573–588
  5. Jump up Sunamura,T. 2004. A predictive relationship for the spacing of beach cusps in nature. Coastal Engineering 51: 697-711
  6. Jump up Gomes da Silva, P., Coco, G., Garnier, R. and Klein, A.H.F. 2020. On the prediction of runup, setup and swash on beaches. Earth-Science Reviews 204, 103148
  7. Jump up to: 7.0 7.1 Nielsen, P. and Hanslow, D.J. 1991. Wave runup distributions on natural beaches. J. Coast. Res. 7: 1139–1152
  8. Jump up Ruessink, B.G., Kleinhans, M.G. and Van Den Beukel, P.G.L. 1998. Observations of swash under highly dissipative conditions. J. Geophys. Res. 103: 3111–3118
  9. Jump up Ruggiero, P., Holman, R. A. and Beach, R. A. 2004. Wave run-up on a high-energy dissipative beach. J. Geophys. Res. 109, C06025, doi:10.1029/2003JC002160
  10. Jump up Matsuba, Y. and Shimozono, T. 2021. Analysis of the contributing factors to infragravity swash based on long-term observations. Coastal Engineering 169, 103957
  11. Jump up Guza, R.T. and Feddersen, F. 2012. Effect of wave frequency and directional spread on shoreline runup. Geophys. Res. Lett. 39: 1–5
  12. Jump up Blenkinsopp, C.E., Bayle, P.M., Martins, K., Foss, O.W., Almeida, L.-P., Kaminsky, G.M., Schimmels, S. and Matsumoto, H. 2022. Wave runup on composite beaches and dynamic cobble berm revetments. Coastal Engineering 176, 104148
  13. Jump up Kim, L.N., Brodie, K.L., Cohn, N.T., Giddings, S.N. and Merrifield, M. 2023. Observations of beach change and runup, and the performance of empirical runup parameterizations during large storm events. Coastal Engineering 184, 104357
  14. Jump up Guedes, R.M.C., Bryan, K.R., Coco, G. and Holman, R.A. 2011. The effects of tides on swash statistics on an intermediate beach. J. Geophys. Res. 116, C04008
  15. Jump up Guza, R.T., Thornton, E.B. and Holman, R.A. 1984. Swash on steep and shallow beaches. Proceedings of the 19th International Conference Coastal Engineering, ASCE, 1, pp. 708 – 723
  16. Jump up to: 16.0 16.1 Hughes, M.G., Baldock, T.E. and Aagaard, T. 2017. Swash saturation: is it universal and do we have an appropriate model? Coastal Dynamics Conf. 2017, Paper No.108, pp. 192-203


The main author of this article is Job Dronkers
Please note that others may also have edited the contents of this article.

Citation: Job Dronkers (2025): Wave run-up. Available from http://www.coastalwiki.org/wiki/Wave_run-up [accessed on 20-02-2025]