9
FIG. 5. (Color online) Velocity probes for the initial time interval 0 < t < 20 t0. (a-c) probes for the first- and second-order
velocity (a) vy1(0, 0), (b) vy2(w/4, 0), and (c) vz2(0, h/4). (d-f) Running time-average Eq. (30) on an interval one oscillation
period wide of the velocity probes in (a-c). The thick lines show the oscillating velocity probes while the thin lines emphasize
the envelopes of the oscillations.
FIG. 6. (Color online) The velocity probes from Fig. 5, but now extended to the long time interval 0 < t < 1500 t0, showing
the convergence towards a periodic state. The dashed lines in (e) and (f) indicate the magnitude of the steady time-averaged
second-order velocity from the frequency-domain simulation Eqs. (11) and (12).
