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 (see Swash zone dynamics) 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.

By waves is meant: 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].

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

where [math]L = g T^2/(2 \pi)[/math] is the offshore wave length, [math]\beta[/math] is the beach slope and [math]T[/math] is the wave period. The horizontal wave incursion is approximately given by [math] R / \tan \beta[/math].

Many other 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 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} \, ) , \; \; \xi \ge 0.3 , \qquad R_2= 0.043 \; \sqrt{HL} , \; \; \xi \lt 0.3 , \qquad (3) [/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.

From an inventory of run-up formulas by Gomes da Silva et al. (2020[5]), it appears that for steep beaches ([math]\tan \beta \gt 0.1[/math]) the run-up increases with increasing beach slope (approximately linear dependance[6]), while for gently sloping dissipative beaches ([math]\tan \beta \lt 0.1[/math]) the dependence on beach slope is weak or absent[6][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[7][8]).

The general applicability of empirical formulas of run-up based on simple parametric representations of beach and shoreface is limited due to the influence of the more detailed characteristics of the local shoreface bathymetry[9]. This is similar to the limited applicability of empirical formulas for the wave set-up, which is a substantial component of the run-up. Accurate estimates of the wave run-up require in-situ observations or detailed numerical models.

Related articles

Swash zone dynamics
Wave set-up


  1. Hunt, I.A. 1959. Design of seawalls and breakwaters. J. Waterw. Harbors Division ASCE 85: 123–152
  2. Holman, R.A. and Sallenger, A.H. 1985. Setup and swash on a natural beach. J. Geophys. Res. 90: 945–953
  3. 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. 4.0 4.1 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. 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
  6. 6.0 6.1 Nielsen, P. and Hanslow, D.J. 1991. Wave runup distributions on natural beaches. J. Coast. Res. 7: 1139–1152
  7. 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
  8. 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
  9. Fiedler, J.W., Becker, J.M., Merrifield, M.A. and Guza, R.T. 2022. Estimating runup with limited bathymetry. Coastal Engineering 172, 104055