1、Coastal Hydrodynamics,Chapter 4 NEARSHORE CURRENTS,Stating types of currents,Stating concept of radiation stress,Stating phenomena of wave set-up and wave set-down,Stating characteristics of longshore currents,1/33,Definition,Chapter 4,4.2 Radiation Stress,2. Expressions,3. Applications,2/33,Chapter

2、 4,There is a stress field in water waves, which is similar to the Reynolds stress in turbulence. This force is usually called momentum flux, since it is resulted from the exchange of momentum among the water particles.,1. Definition,3/33,Chapter 4,Radiation stress is defined as the excess momentum

3、flux induced by the existence of wave motion. It equals the difference between the total momentum fluxes and the hydrostatic pressure in the absence of waves.,4/33,Chapter 4,The radiation stress across the plane x=constant in the direction of wave advance is,The radiation stress of y-momentum across

4、 the plane y=constant,The transverse radiation stress in the direction of waves is,5/33,Chapter 4,If the x-axis is placed in the direction of wave advance and the y-axis parallel to the wave crests, then there are two nonzero components to the radiation stress: the x- and y- fluxes of x-momentum and

5、 y-momentum.,2. Expressions,6/33,Chapter 4,In the more general case, when a linear wave is propagating at some angle to the x-axis, the radiation stress can be expressed as,7/33,Chapter 4,Radiation stress has been proved to be a very powerful tool in the study of a variety of oceanographic phenomena

6、. In the context of littoral processes, it has been used to predict changes in the mean water level (set-up and set-down) in the nearshore region and to analyze the generation of longshore currents. Other applications have been to the generation of surf beat, the interaction of waves with steady cur

7、rents.,3. Applications,8/33,Wave set-down,Chapter 4,4.3 Wave Set-down and Wave Set-up,2. Wave set-up,9/33,Chapter 4,If one considers waves arriving at the beach with their crests parallel to the shoreline, there will be a shoreward flux of momentum. This momentum flux entering the nearshore zone wit

8、h the incoming waves cannot simply disappear but rather must be balanced by opposing forces which will dissipate the momentum.,10/33,Chapter 4,Consider a train of waves encountering the coast with normal incidence. For a short distance dx, a force balance can be developed.,The pressure gradient of t

9、he sloping water surface balances the change of the incoming momentum. There is therefore a change in mean water surface slope whenever there is a change in the radiation stress.,11/33,Chapter 4,Wave set-down is a phenomenon that the mean sea level falls below the still water level from offshore to

10、the breaking point. Based on the radiation stress concept, the amount of wave set-down is found to be,1. Wave set-down,12/33,Chapter 4,This relationship describes a gradual reduction of the mean water level as the shoreline is approached. The mean displacement in water level from the still water lev

11、el is always negative, becoming more so as the wave enters shallow water until breaking commences. That is to say, the mean sea level falls below the still water level.,13/33,Chapter 4,At the breaking point, the maximum amount of wave set-down is,The set-down is less than 5% of the breaking depth be

12、cause the breaking criterion equals 0.78.,14/33,Chapter 4,Wave set-up is a phenomenon that the mean sea level rises above the still water level inside the surf zone. Inside the surf zone, based on a spilling breaker model, the amount of wave set-up can be expressed as,2. Wave set-up,15/33,Chapter 4,

13、The above expression indicates that the mean water surface displacement increases linearly with depths as the shore is approached. In the surf zone, the mean sea level slope is constant and proportional to the beach slope. The water surface slope provides a hydrostatic pressure gradient directed off

14、shore to counter the change of wave momentum by breaking across the shoreline.,16/33,Chapter 4,The maximum value of wave set-up at the shoreline is,For=0.8, the maximum amount of set-up is about 15% of the breaker depth or about 25% of the breaker height.,17/33,Chapter 4,Wave set-down and set-up,18/

15、33,Mean velocity,Chapter 4,4.4 Longshore Currents,2. Horizontal velocity distribution,19/33,3. Numerical modeling,Chapter 4,19/33,Longshore current & longshore drift,Chapter 4,Bowen (1969) has suggested that the longshore current due to an oblique wave approach may be generated by the longshore comp

16、onent of the radiation stress.,1. Mean velocity,20/33,Chapter 4,Following Bowens suggestion and development, Longuet-Higgins (1970) derived the following relationship,This derivation is based on the assumption that Sxy is dissipated in the nearshore zone by the production of the longshore current an

17、d its associated bottom drag.,21/33,Chapter 4,The sand transport studies of Komar and Inman (1970) had earlier suggested that,This was prompted by the agreement between two seemingly independent estimates of the littoral sand transport, the agreement only being possible if the longshore current due

18、to an oblique wave approach is given by the relationship of the above equation.,22/33,Chapter 4,It was found that the remaining laboratory measurements as well as all the field data are in agreement with the above equation. Since the agreement is much better than with the other existing theories, an

19、d since it has the best theoretical basis, being derivable from radiation stress concepts, this equation offers the best available prediction of the longshore current velocity at mid-surf generated by waves breaking at an angle to the shoreline.,23/33,Chapter 4,In order to establish a relationship b

20、etween the longshore current and coastal sediment phenomena, it is necessary to know the horizontal and vertical longshore current velocity distributions in the nearshore area. Extensive analyses have been carried out on the horizontal distribution. However, the vertical distribution is still a prob

21、lem to be solved in the future.,2. Velocity distribution,24/33,Chapter 4,Consider the local spatial change of Sxy as the driving thrust, and include a horizontal eddy viscosity, a horizontal mixing effect which produces an onshore-offshore transfer of the momentum. Again using the radiation stress a

22、pproach leads to the solution,25/33,Chapter 4,where X and V are non-dimensional variables, representing the relative distance and relative velocity,26/33,Chapter 4,A family of longshore current profiles across the surf zone,27/33,Chapter 4,The dimensionless parameter P reflects the significance of the horizontal eddy transfer: the larger the value of P the more important this effect.,is a non-dimensional parameter representing the relative importance of the horizontal mixing and bottom friction.,28/33,Chapter 4,Compar