Ecological Archives E087-067-A1

Per R. Jonsson, Lena Granhag, Paula S. Moschella, Per Åberg, Stephen J. Hawkins, and Richard C. Thompson. Year. Interactions between wave action and grazing control the distribution of intertidal macroalgae. Ecology VOL: pp-pp.

Appendix A. Model of wave breaking and maximum flow speed.

Shoaling of off-shore waves (from wave buoy data) was simulated to the site of the breakwaters using linear wave theory (Denny 1988) and allowing for the local tide:

Equation 1
(A.1)

    
where H0 and Ht are the off- and inshore wave heights and k0 and kt are the off- and inshore wave numbers 2π/L where L is the wave length:

Equation 2
(A.2)


d0 and dt are the off- and inshore water depths, and g is the acceleration due to gravity. The height of shoaling waves was then compared to the expected height for breaking waves (Hb) as waves approached the breakwaters calculated following Goda (1985) as:

Equation 3
(A.3)

where β is the beach slope. The beach slope was set to 0.02 ending in the step-like breakwater of height 5 m. If Ht > Hb waves will break before reaching the breakwaters. This sets the upper height limit of deep water waves that will break on the breakwaters. The maximum flow velocity (umax) associated with fully breaking waves of height Hb is then calculated from Gaylord (1999) as:

Equation 4
(A.4)

Figure A1 (below) shows the modelled maximum velocities in breaking waves from wave-buoy data using Eqs. A.1–A.4.

Figure 1
 
   FIG. A1. Maximum flow speed in waves breaking on the Elmer breakwaters modelled from wave-buoy data (BODC Owers; 50° 37’N, 0° 40.8’W).

 

Predicted maximum flow speeds from Eqs. 1–4 were validated against the empirically measured flow speeds using spring-loaded wave gauges. Here deep-water wave data (significant wave height and period) covering the same time periods as the wave-gauge measurements were collected from buoy BODC 62305 (50° 24’ N,  0° 0’ E) which is ca 60 km south of the Elmer breakwaters (Fig. A2, below).

 

Figure 2
 
  FIG. A2. Relation between modelled maximum flow speed of breaking waves and off-shore wave height. The open squares indicate flow speeds estimated with drag force gauges in the field.



LITERATURE CITED

Denny, M. W. 1988. Biology and the mechanics of the wave-swept environment. Princeton University Press, Princeton, New Jersey, USA.

Gaylord, B. 1999. Detailing agents of physical disturbance: wave-induced velocities and accelerations on a rocky shore. Journal of Experimental Marine Biology and Ecology 239:85–124.

Goda, Y. 1985. Random sea and design of maritime structures. University of Tokyo Press, Tokyo, Japan.

[Back to E087-067]