RESONANCE OF LONGSHORE CURRENTS UNDER TOPOGRAPHIC FORCING

The steady perturbation caused in a longshore flow by a bottom undulat ion is considered. The bedforms are assumed to be alongshore periodic, with crests in the cross-shore direction and with a small amplitude i n order for linear theory to be applicable. The inviscid shallow-water equations are considered in order to investigate topographic resonanc e, that is, the condition under which the perturbation in the flow rea ches a maximum. Since upstream edge waves held stationary by the mean flow are solutions to the homogeneous resonance equations, the existen ce of such flows gives rise to the existence of resonances of infinite amplitude (linear, inviscid theory). For a maximum local Froude numbe r of the basic flow F of less than 1, the flow is found to behave subc ritically according to classic channel flow theory. In addition, neith er steady edge waves nor infinite amplitude resonances exist in this c ase. However, by numerical simulation, a finite maximum in the flow pe rturbation as a function of bedform wavelength is found. This topograp hic resonance is rather weak and wide banded. For a bedform height of 1% the local water depth, the perturbation on the flow;nay typically b e 4% of the mean current. The resonant wavelength is between two and t hree times the distance of the peak longshore current to the shoreline , l(V), when the current profile has a maximum at some distance offsho re, or nearly four times the cross-shore length scale of the sandbars, l, for a flow profile monotonically increasing to a constant current far offshore. For F > 1 resonances of infinite amplitude are found. Fo r every F, l(V), and l, there is an infinite set of resonant modes wit h an increasing cross-shore complexity when the mode number increases, similarly to edge waves. The resonant wavelength increases with F and with l(V). Some implications on the growth of transverse sandbar fami lies and cuspidal coast are discussed.