Modified Stokes drift due to resonant interactions between surface waves and corrugated sea floor with and without a mean current

In this paper, we show that Stokes drift may be significantly affected when an incident intermediate or shallow water surface wave travels over a corrugated sea floor. The underlying mechanism is Bragg resonance: reflected waves generated via nonlinear resonant interactions between an incident wave and a rippled bottom. We theoretically explain the fundamental effect of two counterpropagating Stokes waves on Stokes drift and then perform numerical simulations of Bragg resonance using the high-order spectral method. A monochromatic incident wave on interaction with a patch of bottom ripple yields a complex interference between the incident and reflected waves. When the velocity induced by the reflected waves exceeds that of the incident, particle trajectories reverse, leading to a backward drift. Lagrangian and Lagrangian-mean trajectories reveal that surface particles near the up-wave side of the patch are either trapped or reflected, implying that the rippled patch acts as a non-surface-invasive particle trap or reflector. On increasing the length and amplitude of the rippled patch, reflection, and thus the effectiveness of the patch, increases. The inclusion of realistic constant current shows noticeable differences between Lagrangian-mean trajectories with and without the rippled patch. Theoretical analysis reveals additional terms in the Stokes drift arising from the particular solution due to mean-current–bottom-ripple interactions, irrespective of whether Bragg resonance condition is met. Our analyses may be useful for designing artificial, corrugated sea-floor patches for mitigating microplastics and other forms of ocean pollution. We also expect that sea-floor corrugations, especially in the near-shore region, may significantly affect oceanic tracer transport.

Figure 1

(a) Schematic of surface wave and bottom ripple with particles at different positions [labeled $\overline{)\mathrm{a}}-\overline{)\mathrm{h}}$] at $t=0$. Interaction of the rightward propagating incident wave (deep blue curve centered at $z=0$) with the bottom ripple (undulations centered at $z=-H$) leads to the generation of a leftward propagating reflected wave (red dashed curve centered at $z=0$). In reality, many reflected waves are generated from a finite patch, but here just one is shown for simplicity. (b) Trajectory of a particle in the absence of bottom ripple, and (c) trajectory of particle labeled $\overline{)\mathrm{d}}$ in the presence of bottom ripple. In (b) and (c), black and red filled circles, respectively, denote initial and final location of the particle. Blue curves represent clockwise (forward) trajectories while red curves represent anticlockwise (backward) trajectories. The axes in (b) and (c) denote nondimensional displacements from the initial position ($k_i$ is the wave number of the incident wave).

Figure 2

Particle trajectories resulting from two counterpropagating Stokes waves for $\theta =0$ and different values of $R$: (a) $R<1$, and (b) $R\ge 1$. Black filled circle denotes initial position, while colored filled circles (with a color corresponding to that of the respective trajectory) denote position after five time periods. The quantity $s=0.005$ for the parameters chosen.

Figure 3

Trajectories of tracer particles at different initial locations, as shown in Fig. 1. Blue colored trajectories show forward (clockwise) drift, red trajectories show backward (anticlockwise) drift, while green trajectories show the next round of forward (clockwise) drift. Black and cyan filled circles, respectively, denote initial and final particle positions. Gray trajectories denote Stokes drift for a flat-bottom topography. In this case, the final position of the particle is represented by a magenta filled circle. Note the initial position (denoted by black filled circle) for the flat topography case is the same as the one with rippled-bottom topography.

Figure 4

Horizontal particle trajectories versus time in the absence (small black dotted curves) and presence (red, blue, and green curves with color scheme same as Fig. 3) of the rippled-bottom topography. Trajectories corresponding to the particles $\overline{)\mathrm{b}}–\overline{)\mathrm{e}}$ are, respectively, shown in (a)–(d). Horizontal component of Lagrangian-mean trajectories without and with the bottom ripple are, respectively, represented by the large black and magenta dotted lines and curves; each dot denotes particle position after one time period ($T$).

Figure 5

Horizontal component of the Lagrangian-mean trajectory of particle $\overline{)\mathrm{b}}$ (initial abscissa: $x_0=5\pi /8$) versus time for (i) flat bottom (black), (ii) $a_b/H=0.1$ (red), (iii) $a_b/H=0.3$ (green), (iv) $a_b/H=0.5$ (blue), and (v) $a_b/H=0.7$ (magenta). Simulations have been performed for a continuous source of incident wave packets and patch length is held fixed at $2l=\pi /2$. Small dots represent position after one time period, while large filled circles denote initial and final positions.

Figure 6

Horizontal component of the Lagrangian-mean trajectory of particle with initial position fixed at (a) $\overline{)\mathrm{b}}$ (initial abscissa: $x_0=5\pi /8$), and (b) $\overline{)\mathrm{c}}$ (initial abscissa: $x_0=3\pi /4$) for different patch sizes ($2l$): (i) flat bottom (black), (ii) $\pi /8$ (red), (iii) $\pi /4$ (green), (iv) $\pi /2$ (blue), and (v) $\pi$ (magenta). Simulations have been performed for a wave train and bottom ripple's amplitude is held fixed at $a_b/H=0.1$. Small dots represent position after one time period, while large filled circles denote initial and final positions.

Figure 7

Comparison of trajectories of a parcel initially located at $\overline{)\mathrm{b}}$ in the presence of a constant background current $\text{Fr}=0.003$ with and without bottom ripples. (a) Lagrangian trajectory without (gray) and with (blue and red) bottom ripples. Blue trajectories denote forward (clockwise) drift, while red trajectories show backward (anticlockwise) drift. Green circle denotes the initial position of the parcel, while black and magenta circles, respectively, denote final positions without and with bottom ripples. (b) Horizontal component of Lagrangian-mean trajectories without (black dotted) and with (magenta dotted) bottom ripples. Each dot represents position after one time period.