Wave-Current Interaction

Introduction

Tidal fronts (Figure 1) are a common feature of the coastal ocean. They are formed by the interaction of tidal currents with topography and are characterized by surface convergence areas with strong gradients in current speed and density (see also "flow-topography interaction"). Waves -- or even small disturbances formed at low wind speed -- that travel into these areas are slowed down by the currents and tend to steepen and break (Figure 2).

Wave energy is therefore concentrated in the frontal area leading to wave-wave interaction as well as the enhanced air-sea gas exchange in that region due to the formation of gas bubbles by breaking waves. Because these processes occur even in calm conditions and because the location of wave breaking is highly predictable, tidal fronts form an ideal natural laboratory for studying wave-current interactions

Figure 1: Aerial view of the tidal front at Battleship Island.
Figure 2: Wave breaking due to wave-current interaction in a tidal front.
Model of Wave Action Conservation

A model of wave action conservation [Bretherton and Garrett, 1969] is used to describe the behavior of a wave travelling on an opposing current.

where E is the wave energy, w the wave frequency, u the current speed, and cg the group speed. The result is plotted for d/dt=0 in Figure 3 as the gray curve: the dimensionless energy E/E0 of a wave of initial steepness k0a0 increases as the wave travels against the current. At a current speed u/c0=-1/4, the wave is stopped by the currents. Due to wave breaking, the amplitude does not go towards infinity, but follows the dashed curves. The location at which the wave breaks depends on its initial steepness.
Figure 3: Wave energy as a function of current speed. Current speed and amplitude are scaled with the values for a medium at rest (u=0). The wave energy in the absence of wave breaking (solid gray curve) is plotted together with the wave energy for a breaking wave (dashed black curves) of initial steepness k0a0 (black numbers).
Energy Dissipation by Wave Breaking

Let us now consider a wave that is at time t=0 at location x=0 (on curve G1). One period later it has moved one wave length L and the wave packet 2L to the right. If there was no wave breaking during this time, the shape of the wave packet would be given by curves G1 and U2. But because of breaking, the ``excess" energy dE=U2(L)-U1(L) is lost between x=0 and x=L. From Figure 4 it also follows that U2(L)=G2(L). Together with

G2(L) / G1(L)=G2(L/2) / G1(L/2)= const. and G2(L/2)=U1(L/2)

we yield an equation for the energy loss by wave breaking

dE=U1(L/2) G1(L) / G1(L/2) - U1(L)

Figure 4: Behavior of a quasi-monochromatic wave on an opposing current. The normalized energy E/E0 is given for the case without breaking (black curves Gi) and with breaking (gray curves Uj). dE is the energy lost by wave breaking.
Behavior of a Wave Packet on a Current

Figure 5 shows a sketch of the break behavior of a group of waves on a current. At time t=0 (a), the first wave is steeper than k a>1/2 and breaks, which also changes the form of the wave packet (b). After one period (c), the first wave has moved two wave lengths to the right (relative to the wave packet) so that its amplitude is reduced and its steepness is less than the critical value. It stops breaking. After two periods (d), the next wave has reached its critical steepness and breaks -- with an amplitude that is smaller than the first wave. Because the shape of the wave packet is altered by wave breaking, at a later stage (e), the waves are continuously pushed into the region of critical steepness and hence break continuously.

Figure 5: Sketch of the break behavior of a wave packet on a current. For explanations see text
References

B. Baschek, Air-sea gas exchange in tidal fronts. PhD thesis, University of Victoria, Canada, 2003.

B. Baschek, 2005: Wave-current interaction in tidal fronts. 14th 'Aha Huliko'a Winter Workshop 2005: Rogue Waves, ed. P. Muller and C. Garrett.

F.P. Bretherton and C.J.R. Garrett, Wavetrains in inhomogeneous moving media. Proc. Roy. Soc., 302, A, 529-554, 1969.

M. Donelan, M.S. Longuet-Higgins, and J.S. Turner, Periodicity in Whitecaps. Nature, 239, 449-451, 1972.

D.M. Farmer, R. Pawlowicz, and R. Jiang, Tilting separation flows: a mechanism for intense vertical mixing in the coastal ocean, Dyn. Atmos. Ocean, 36, 43--58, 2002.

C.J. Galvin, Breaker type classification on three laboratory beaches. J. Geophys. Res., 73, 12, 3651-3659, 1968.

M.S. Longuet-Higgins, On wave breaking and the equilibrium spectrum of wind-generated waves. Proc. Roy. Soc., 310, A, 151-159, 1969.

O.M. Phillips, The dynamics of the upper ocean. 2nd ed., Cambridge, Cambridge University Press, 1977.