On the destabilization of a longshore current on a plane beach: Bottom shear stress, critical conditions, and onset of instability

Abstract

The effect on the stability of a wave‐driven longshore current of varying the bottom shear stress is investigated. The shear stress is assumed to be quadratic and parameterized (as usual) through a bottom friction coefficient (cd). The bottom shear stress affects both the mean longshore current (which it opposes) and the onset (or otherwise) of a shear instability in the longshore current through its aforementioned effect on the mean current (and therefore the current shear) and by damping instabilities directly. Reducing cd can destabilize the longshore current by increasing its strength, thereby increasing the current shear so as to overcome the damping of bottom friction and thus leading to the development of instabilities. At the same time, this decrease in cd directly reduces the damping experienced by the instabilities that develop. It is also shown that while the primary effect of bottom friction is to damp instabilities, there are other effects, notably linked to the flow curvature, which act to destabilize the flow. Both the weak‐ and the strong‐longshore‐current approximations are examined. For the weak‐current approximation it is shown that curvature effects are particularly important in providing a destabilizing mechanism. It is also shown that for both approximations a “global” cutoff frequency is likely to exist, below which no linear instability will develop. Critical conditions are identified for each approximation (and therefore for two different longshore current profiles), beyond which instabilities will start to develop. Results suggest that a critical shear will exist for any plane beach and that for Leadbetter Beach, Santa Barbara (examined in the 1980 Nearshore Sediment Transport Study (NSTS)), the current will not become unstable until an offshore shear of about 0.03 s−1 is reached. This was apparently not achieved during NSTS.