Difference between revisions of "Wave set-up"

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==Notes==
 
==Notes==
 
Wave transformation in shallow water generates changes in the mean water level, called set-up for an upward change and set-down for a downward change.  
 
Wave transformation in shallow water generates changes in the mean water level, called set-up for an upward change and set-down for a downward change.  
The changes in mean sea level are related to the so-called "radiation stress". The radiation stress represents an effective momentum transfer from wave motion to steady motion that takes place when the wave amplitude changes along the direction of propagation. Wave set-down <math>\eta_d</math> occurs in the shoaling zone where the wave amplitude increases; wave wet-up <math>\eta_u</math> occurs in the surf zone where the wave amplitude decreases. Analytical expressions of the wave set-down at the breakpoint and the wave set-up at the shoreline can be derived for a monochromatic wave, see [[Shallow-water wave theory]]:
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The changes in mean sea level are related to the so-called [[Shallow-water wave theory#Radiation Stress (Momentum Flux)|radiation stress]]. The radiation stress is defined as the excess flow of momentum due to wave orbital motions. Gradients in the radiation stress induce an effective momentum transfer from wave motion to steady motion that takes place when the wave amplitude changes along the direction of propagation. Wave set-down <math>\eta_d</math> occurs in the shoaling zone where the wave amplitude increases; wave wet-up <math>\eta_u</math> occurs in the surf zone where the wave amplitude decreases. Analytical expressions of the wave set-down at the breakpoint and the wave set-up at the shoreline can be derived for a monochromatic wave, see [[Shallow-water wave theory]]:
  
 
<math>\eta_d \approx -\frac{\gamma}{16} H_b , \qquad \eta_u \approx \eta_d +\frac{3 \gamma}{8} H_b  \approx \frac{5 \gamma}{16} H_b,</math>
 
<math>\eta_d \approx -\frac{\gamma}{16} H_b , \qquad \eta_u \approx \eta_d +\frac{3 \gamma}{8} H_b  \approx \frac{5 \gamma}{16} H_b,</math>
  
where <math>\gamma = H_b/h_b</math> is the breaker index (values in the range 0.6-1.2), <math>H_b</math> is the wave height at the breakpoint and <math>h_b</math> the depth at the breakpoint.  
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where <math>\gamma = H_b/h_b</math> is the [[breaker index]] (values in the range 0.6-1.2), <math>H_b</math> is the wave height at the breakpoint and <math>h_b</math> the depth at the breakpoint.  
  
 
Assumptions are:  
 
Assumptions are:  
* a constant seabed slope <math>\beta</math>, i.e. depth <math>h(x) = \beta x</math>;
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* a constant seabed slope <math>m = \tan\beta</math>, i.e. depth <math>h(x) = m \, x</math>;
 
* shallow water, <math>2kh_b < 1</math> (<math>k</math> is the wave number);  
 
* shallow water, <math>2kh_b < 1</math> (<math>k</math> is the wave number);  
* depth-limited wave amplitude <math>H/h=\gamma</math> throughout the surf zone.   
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* depth-limited wave amplitude <math>H/h=\gamma \;</math> throughout the surf zone (saturated wave breaking).   
  
Observations at different field locations suggest that the wave set-up depends on the beach slope <math>\beta</math> following the empirical relationship<ref>Raubenheimer, B., Elgar, S. and Guza, T. 2001. Field observations of wave-driven setdown and setup. J. Geophys. Res. 106: 4629-4638</ref>
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Observations at different field locations suggest that the wave set-up depends on the beach slope <math>m</math> following the empirical relationship<ref>Raubenheimer, B., Elgar, S. and Guza, T. 2001. Field observations of wave-driven setdown and setup. J. Geophys. Res. 106: 4629-4638</ref>
  
<math>\eta_u \approx (0.02 + 0.003 / \beta) H_b .</math>
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<math>\eta_u \approx (0.02 + \large\frac{0.003}{m}\normalsize) \, H_b .</math>
  
  

Revision as of 11:11, 5 April 2021

Definition of Wave set-up:
Elevation of the mean water level at the shoreline due to wave breaking in the surf zone.
This is the common definition for Wave set-up, other definitions can be discussed in the article


Notes

Wave transformation in shallow water generates changes in the mean water level, called set-up for an upward change and set-down for a downward change. The changes in mean sea level are related to the so-called radiation stress. The radiation stress is defined as the excess flow of momentum due to wave orbital motions. Gradients in the radiation stress induce an effective momentum transfer from wave motion to steady motion that takes place when the wave amplitude changes along the direction of propagation. Wave set-down [math]\eta_d[/math] occurs in the shoaling zone where the wave amplitude increases; wave wet-up [math]\eta_u[/math] occurs in the surf zone where the wave amplitude decreases. Analytical expressions of the wave set-down at the breakpoint and the wave set-up at the shoreline can be derived for a monochromatic wave, see Shallow-water wave theory:

[math]\eta_d \approx -\frac{\gamma}{16} H_b , \qquad \eta_u \approx \eta_d +\frac{3 \gamma}{8} H_b \approx \frac{5 \gamma}{16} H_b,[/math]

where [math]\gamma = H_b/h_b[/math] is the breaker index (values in the range 0.6-1.2), [math]H_b[/math] is the wave height at the breakpoint and [math]h_b[/math] the depth at the breakpoint.

Assumptions are:

  • a constant seabed slope [math]m = \tan\beta[/math], i.e. depth [math]h(x) = m \, x[/math];
  • shallow water, [math]2kh_b \lt 1[/math] ([math]k[/math] is the wave number);
  • depth-limited wave amplitude [math]H/h=\gamma \;[/math] throughout the surf zone (saturated wave breaking).

Observations at different field locations suggest that the wave set-up depends on the beach slope [math]m[/math] following the empirical relationship[1]

[math]\eta_u \approx (0.02 + \large\frac{0.003}{m}\normalsize) \, H_b .[/math]


Related articles

Shallow-water wave theory
Wave run-up


References

  1. Raubenheimer, B., Elgar, S. and Guza, T. 2001. Field observations of wave-driven setdown and setup. J. Geophys. Res. 106: 4629-4638