1. Introduction
The theoretical framework of wave-generated motion in the context of water, acoustic, and plasma waves rests on the fact that under certain conditions of motion the Lagrangian particle paths do not describe closed orbits and thus, over time, the particles that make up the fluid itself will drift. Perhaps the earliest theoretical studies of this phenomenon are those of Stokes (1847) and Lord Rayleigh (1876), thus making this a research topic that is nearly 150 years old. In the 1950s Longuet-Higgins (1953) derived an asymptotic expression for the mean Lagrangian velocity in an oscillating laminar boundary layer. He also demonstrated that boundary layers, however thin, have a dramatic effect on the ensuing drift generated by waves. Further, he suggested that this type of wave-generated motion could be important in the transport of sediment and, by extension, in the transport of pollutants and nutrients. Johns (1970) and Longuet-Higgins (1958) showed that the situation is not qualitatively dissimilar in the case of a mixing-length-type boundary layer (for a review see Phillips 1978; Mei 1989, and references therein). Streaming due to the oscillation of immersed objects in a fluid or the oscillation of bounding surfaces is also well documented and of considerable interest in the engineering community; it is now a classic topic in fluid mechanics monographs and textbooks (see Telionis 1981).
On the theoretical side, a geometrical interpretation of the relation between the averages in the Lagrangian and the Eulerian frames led Andrews and McIntyre to formulate a generalized Lagrangian mean theory of wave-generated transport (Andrews and McIntyre 1978), a theory in which certain geometrical properties of the flow are preserved, such as Kelvin's circulation.
Progressive waves induce a steady transport velocity in a uniform channel (see Mei 1989). On the other hand, a standing wave field has the potential of producing a steady spatial structure in a tracer field occupying a boundary layer. This fact is exploited in models (Restrepo and Bona 1995; Restrepo 1997) for the formation and evolution of large-scale sandbar structures [see Fredsoe and Deigaard (1992) and Sleath (1984) for reviews]. Wave-generated transport has been shown to modify the dynamics of interacting flows. A consequence of momentum and continuity conservation is that short waves are dynamically modified by the mass and momentum fluxes due to finite amplitude long waves (Longuet-Higgins and Stewart 1960, 1961). The interaction of currents with the streaming induced by a wave field has been shown to generate circulation cells, which have a striking resemblance to Langmuir cells in the ocean and the atmosphere (Craik and Leibovich 1976), and has been shown to modify the Ekman boundary layer in rotating flows (Huang 1979). It has also been suggested as an important transport mechanism in oceanic flows relevant to climate dynamics (McWilliams and Restrepo 1999) and in the shallow reaches of the continental shelf where waves and currents interact in a significant way (Restrepo 2001).
In this study we consider the effect of idealized noise on the transport generated by monochromatic progressive or standing waves. Noise in this study is understood to represent underresolved processes in the external forcing. When a flow is forced by standing waves, we show that Gaussian noise, manifesting itself as statistical perturbations of the phase and amplitude of the waves, generates a nonzero total mass flux in the boundary layer. We also show that when the forcing is due to progressive waves, noise significantly diminishes the mass flux.
Elucidating ubiquitous aspects of wave-generated transport, even in an idealized setting, sheds light on the motion of tracers acted upon by waves in the physical setting. The particles affected by this transport mechanism may be passive tracers; they may be mechanically interacting particles such as sand, self-propelled biological organisms, or pollutants that interact mechanically and chemically. Often, these particles, in turn, affect the flow in the boundary layer, such as is the case in a loose sedimentary bed. Our study does not focus on the dynamics of these particle systems, although we make use of passive tracers to investigate the transport velocity itself.
Section 2 focuses on a description of the ideal wave channel and on details of the Prandtl boundary layer model and its numerical approximation. A description of how the transport velocity relates to the properties of the flow appears in section 3. In this section, we also review some salient aspects of the transport velocity in boundary layer flows forced by either progressive or standing waves in the absence of noise. The former produces a steady drift velocity with nonzero mean and no spatial structure the latter a steady drift velocity with zero mean and spatial structure. We think of the progressive wave and the standing wave cases as being two extremes in the transport they generate. The simplest external forcing that generates a nonzero mean as well as spatial structure in the steady drift is a “leaky” standing wave. In this case the external forcing can be characterized by waves with an incident and a reflecting component that has a relative amplitude 0 < R < 1. As a function of R the mass transport characteristics of this flow were explored in detail by Carter et al. (1973).
Examination of these examples leads to an understanding of how noise affects the transport when it is present in the external forcing. The situation when noise is present is considered in section 4. Conclusions relevant to wave-generated transport derived from this work appear in section 5.
2. Description of the ideal wave channel
The horizontal coordinate in the computational domain, or “channel,” is denoted by x and the vertical coordinate by z. The z = 0 plane coincides with the bottom of the channel, which is assumed to be ideally smooth and rigid. The channel is taken to be periodic in x and three wavelengths long.
The size of kh is a determining factor in the penetration depth of these waves; in particular, if kh is small, we expect the penetration depth of the waves to be significant. This is relevant to mass transport in the boundary layer because its strength is determined, to a large extent, by the amplitude of the forcing immediately outside the layer. A small kh value implies that the waves are long compared with the depth of the water column. In the simulations to be discussed presently, kh ≈ 0.2534. The period P of the waves was 8 s, producing a wavenumber 0.2534 m–1 corresponding to waves with a wavelength of approximately λ = 24.79 m.
a. Boundary layer model and its numerical approximation
The dimensionless numbers ϵ = 1/S, where S = ωλ/Um, and Re = ωλ2/ν characterize the flow. These are S ∼ 103 and Re ∼ 107 in all the experiments reported in this study.
The flow was made periodic at the far and near ends of the channel. A no-slip boundary condition was applied at the channel wall, and the velocity at the edge of the boundary layer matched the external forcing velocity U(x, t). In the numerical experiments the flow was initialized by using the analytical solution to the linear Prandtl equations, that is, the solution to the mass and momentum conservation equations with all terms proportional to ϵ in (5) set to zero.
A full description of the discretization of the equations appears in appendix A. The appendix also details how the results derived by using the Prandtl model and its numerical approximation compare with those obtained by using a numerical solution of the Navier–Stokes equations under the same physical conditions. Since running averages will play an important role as a diagnostic tool in what follows, in appendix B we provide the particulars of our tests on the accuracy of the algorithm used. Running averages were computed after discarding the first three periods of the simulation, thereby removing the effect of the initial state of the channel flow on the resulting averages. The simulations that produced the data for the analysis were carried out for a sufficiently long time to establish an approximation of the asymptotic behavior of the flow and diagnostic quantities. In each group of simulations the actual length of the runs is explicitly given; however, figures featuring the drift or the transport in space and time will show the flow only during the last four time periods of the experiment. To assay the characteristics of the transport velocity under different wave forcing and noise conditions, we also used an algorithm to advect tracers, thus obtaining approximations of the Lagrangian paths of ideal tracers. The advection algorithm is described in detail in appendix C.
3. The transport velocity
Since we are studying aspects of the transport in a thin boundary layer whose aspect ratio is very small when compared with the horizontal spatial scales, we will speak of the horizontal component of the transport velocity as the transport velocity itself.
Let us suppose for the moment that the DC is zero and that steady running average conditions prevail. Then tracers at locations where the residual transport is zero do not move. We call these locations traps. An accumulating trap, or node, is one in which there is a zero in the residual and, in addition, the transport locally has a negative slope that persists over time. These traps attract moving particles, and accumulation ensues at these sites. If the DC component is not zero, however, the net transport is a result of the DC and the residual components. In this case, particles accumulate at the traps only if the tracers have some threshold of motion higher than the mean but lower than the maximum magnitude of the transport.
a. Transport induced by progressive and standing waves
In what follows we summarize some qualitative differences between the transport velocity induced by progressive and standing wave external forcing in the absence of noise.
Figure 3 shows the space–time plot of the horizontal component of the transport velocity. Figure 3a corresponds to PW and Fig. 3b to SW forcing, respectively. The velocity was measured roughly a Stokes layer depth above the wall. The figures depict the contours of the velocity amplitude, with the dark and light patches corresponding to low and high velocities, respectively. The amplitude of the velocity may be inferred from Fig. 5. At any particular position along the channel, the transport (which is defined in terms of a running average) exhibits oscillatory behavior and a slowly decaying envelope. This is illustrated in Fig. 4, for the PW case. For these monochromatic, time-harmonic waves, the mean transport is numerically equal to the value of the transport itself at times commensurate with the period of the forcing. Our calculations show that the mean transport agrees with standard theory (see Batchelor 1981, 353–364; Mei 1989, 420–434). That is, for waves progressing from left to right, the mean transport for PW forcing is steady in time and uniform in space. Its magnitude is nonzero and positive. Figures 5a and 5c portray the transport at the end of the experiment, for the PW forcing case, in the absence of noise. Figure 5c shows the superposition of the transport velocity for the last few periods. The spatially fluctuating traces in this figure depict the transport velocity at times incommensurate with the period. Traces that are constant across the entire channel correspond to the mean transport. These latter ones are colinear and depict the mean transport velocity.
For SW forcing, the transport is steady in time. It is composed of spatial cells with horizontal dimensions half the wavelength of the forcing. The spatial mean of the transport for the SW case in the absence of noise is zero. The SW transport with no noise present is shown in Fig. 3b.
With the aid of these figures we can construct a qualitative description of how ideal or passive tracers3 would respond to the transport velocity under PW forcing in the absence of noise. The space and time symmetries induced by the sinusoidal forcing imply that the mean transport must be constant. Intuitively, this reflects the fact that, as the wave train progresses down the channel, it imparts the same effect on the flow everywhere along the channel. If the transport at some given height above the wall, locally, is found to have a specific value, symmetry implies that the transport measured anywhere else along the channel must be the same, provided that the averaging process required in computing the transport is well defined. Indeed, this fact is reflected by our computation. The transport is constant and is always positive. Since the transport is positive, passive tracers would advect in the direction of wave motion, toward the end of the channel. Clearly, the tracer flux is proportional to the spatially averaged transport, which in this case is equal to the transport itself, and hence the flux is also positive and constant. Furthermore, since the mean wall shear stress is proportional to the drift velocity, the bottom of the tank would experience, on average, a positive and uniform mean shear stress. These findings agree with the standard asymptotic theory.
For the SW case, the situation can be summarized as follows. The spatial mean of the transport is found to be machine zero; therefore, the mean flux is zero as well. However, the SW flow produces a transport with steady spatial structure, proportional to sin(2kx). Thus, passive tracers will move away from locations of high transport and tend to concentrate at locations where the transport velocity amplitude is zero, forming the cells that the asymptotic theory predicts. While the total flux is zero, for the noiseless SW case, ideal particles with initial position corresponding to one of the transport cells will stay in that cell for all time.
When the flow is forced by monochromatic progressive waves, nodes are not found in the transport since the residual transport is zero. This follows from the observation that, for progressive waves in a uniform channel, the model is translationally invariant. Tracers will move in the direction of the progressive waves because the DC component is nonzero and positive. In the SW case, however, nodes do occur because the DC component is zero and the residual has time-steady traps with negative slopes.
4. The effect of noise in the forcing on the transport
Noise usually is present both in the laboratory and in the natural setting. Since wave-generated transport is typically a relatively weak phenomenon, one expects noise to have a significant effect on it, especially over the long time frames required for significant transport to occur. Our investigation focuses on how the transport and, by extension, the mass flux and the mean shear stress are affected by noise derived from unresolved physical processes.
In the context of the boundary layer problem it is reasonable to suggest that noise perturbations affect either the phase or the amplitude of the wave forcing and that the boundary layer itself would respond according to (5) and (6). At any instant in time, noise perturbations would induce an uncertainty in the measurements of the actual location of maxima and minima in the wave, as well as an uncertainty in the measurement of the wave amplitude.
Models for the PW and SW forcing that incorporate the above qualities are (3) and (4). From these expressions it is clear that a nonzero γ(t) will affect both the phase and the amplitude of the forcing. With γ(t) a Gaussian noise process, with zero mean, the resulting model is one in which the uncertainty in the phase and the amplitude of the wave grows proportional to the square root of time. The adoption of Gaussian statistics is done here in an ad hoc way. However, this assumption is often invoked (and sometimes abused) in order to characterize the statistics of a great many oceanic processes.
Intuitively, it seems natural to think that phase noise acts as an effective decorrelator of running averages since the noise would act as a diffusive process. Indeed, our results bear this out. However, it is difficult to envisage in what way noise actually affects such aspects of the flow as the transport, the wall shear stress, and the mass flux. For example, does the transport slowly decay, completely disappear, destabilize, or develop a totally different structure?
Two properties characterize the noise fluctuations: the amplitude and the frequency. These become parameters in our experimental study on the effect of noise on transport. The amplitude is related to the wave uncertainty in phase. The frequency is related to how often a noise perturbation occurs. It affects the coherence of the external forcing over time; the lower the frequency, the higher the degree of spatial coherence, but the longer it takes for quantities calculated via running averages to nominally settle. There is a space–time symmetry so that noise perturbations in time, affecting the whole extent of the wave, correspond to perturbations in space, affecting the wave for a long period of time. Hence, the effect of the noise over time at a single place has a corresponding description in terms of the effect of noise over some span of the channel at some particular instant in time.
In each experiment the code was run for 150 periods. Measurements, however, were made only in the last 100 periods of the run. We summarize below the outcome of experiments in which the noise amplitude was varied in a range 0–π. Coverage of the noise frequency parameter range was less complete. We did, however, test noise frequencies that were both higher and lower than the frequency of external forcing.
For the PW case, noise caused the DC, and thereby the mass flux, to drop appreciably, compared with the noiseless case. The DC was no longer steady. The residual was also modified by the presence of noise, becoming amplified significantly. In fact, ignoring the DC component, we note an interesting qualitative change on the drift velocity due to noise, namely, the appearance of nodes in the residual. These nodes are not traps for ideal tracers, however, since the residual is everywhere positive. Hence, although the transport is structured, a nonzero flux will be present everywhere along the channel. If ideal tracers were present, they would eventually reach the end of the tank, regardless of where in the tank they were placed. On the other hand, if the tracers had some threshold of motion slightly above the DC level, it would then be possible for tracers to accumulate at the nodes.
The most salient structural changes in the drift due to noise are evidenced in Fig. 5. In Fig. 5a contours of the drift velocity, noise absent, are portrayed in space and time. For comparison, Fig. 5b shows the case when noise is present. To emphasize their symmetry, we have not shaded the contours; however, it is understood that in Fig. 5a they would be consistent with Fig. 3a and that in Fig. 5b the contours would appear as alternate shaded and unshaded regions. In fact, from Fig. 5b it is seen that, when noise is present, the residual flow has a nearly steady structure in time. Figures 5c and 5d shows a superposition of the drift for the last ten periods of the run, as a function of position. Figure 5d, which corresponds to the case in which noise is present, clearly shows that the running averages are nonuniform in space and nearly steady in time. Examination of the residual in both cases depicted in Fig. 5 reveals that the transport in the noisy case has a significantly lower DC component than does the noiseless counterpart, and a residual that is not present in the noiseless case.
Thus we find that for the PW case, the presence of noise has two significant effects on the drift. First, the magnitude of the mean drift is greatly reduced. Second, the residual flow acquires a nearly steady spatial structure. These effects are not as sensitive to the amplitude or frequency of the noise but, rather, to the existence of a noise perturbation.
For the SW case, noise has a less dramatic but nevertheless significant effect. Whether noise is present or not, the space–time picture of the transport looks very similar to that of Fig. 3. This is certainly true for small but reasonable amplitudes of the noise. The residual is affected by the appearance of fluctuations inside the trapping cells. The cases with noise and with no noise present are depicted in Figs. 6a and 6b, respectively. Of note is the absence of the periodic appearance of a reverse flow in each of the cells of the standing wave structure in the noisy case. This is evidenced by comparing it with the noiseless case in Fig. 6. Examination of the data reveals that in the absence of noise the DC is zero and thus the total flux is zero. With noise present, however, the flux is nonzero and unsteady, sometimes even becoming negative. Summarizing the findings on the SW experiments, we find that noise induces a small nonzero total flux.
Next we quantify the dependence of the transport velocity on the noise amplitude and frequency. Specifically, we calculate the Lagrangian trajectories (for details, see appendix C), using both the vertical and horizontal components of the Eulerian velocity. As Figs. 7–10 indicate, the vertical component of the velocity is very small compared with the horizontal component. Furthermore, by virtue of (6) we can infer that for the PW case the vertical velocity forms cells of upward and downward directed motion and that in these cells the vertical velocity must be monotonic. Thus we expect that particles will exhibit both a transverse and an upward or downward trend in their motion, when acted upon by the velocity field in the layer.
The result of calculating the Lagrangian trajectories, over nine periods of forcing, appears in Fig. 7. The particles were placed initially at equally spaced horizontal intervals, roughly one Stokes layer above the wall. The path traced by a particle in the noiseless PW case appears in Fig. 7a, its noisy counterpart in Fig. 7b. The noise amplitude and frequency were 0.5π and 20/P, respectively. The dissipation inherent in the advection scheme is apparent in the figures by a trend in the orbits to get smaller in radii. This causes the particle to get deflected slightly in the vertical direction. The effect is fairly small, however, when compared with the particles' reaction to the total velocity field. Overall, the effect of dissipation is slight: note the aspect ratio of the vertical and horizontal coordinates on the plots.
When the particle paths are sampled at times commensurate with the period and their tracks connected by lines, we obtain Fig. 8a and Fig. 8c for the noiseless and noisy case, respectively. Figures. 8b and 8d show the starting and final position particle position along the channel for these two cases. From this figure it is clear that the particles are not traveling as far along the channel in the presence of noise and that the horizontal component of the velocity is greatly affected, thus making the relative effects of the vertical component of the velocity on the particle motion more prominent.
How this manifests itself in the flux can be assessed by the direct calculation of the flux at some given location in the channel. By counting particles traveling from left to right across x = 1.2λ, say, as positive and those traveling from right to left across x = 1.2λ as negative, we can arrive at a net flux value. The choice of location was arbitrary in this calculation. Figures 9a and 9b show the flux of tracers for the PW noiseless and noisy case, respectively. Clearly, the noise reduces the net flux significantly.
As we said previously, the most salient effect of noise on the SW case is the creation of a small net flux. This is borne out in the Lagrangian trajectories shown in Figs. 10a and 10b, which correspond to the noiseless and noisy case, respectively. As is evident from the noisy case, tracers leak from cell to cell.
Next we show how the flux is affected by the noise amplitude and the frequency. The quantitative dependence of the flux on the presence of noise is examined by the following proxy calculation. We calculate the statistical mean displacement of a particle over a fixed time. We then use this data to calculate a transport velocity and thus obtain the flux. Ensemble averages of the Lagrangian trajectory are computed. Each member of the ensemble corresponds to the trajectory of 1 of 33 different release times, over the period of the forcing wave. Hence, each particle is released at a different phase value of the wave. Furthermore, for any given Lagrangian path identified by a given initial phase, we calculate a path under three different noise time series. Among these three realizations the parameters of noise amplitude and frequency are the same. When no noise is present, the three trajectories are identical. With noise, the three trajectories starting with an initial phase have no commonality in their time series. In fact, removal of the deterministic component of all the trajectories shows that the 99 Lagrangian paths are unique realizations that share only statistical properties. As before, the total time of each run is nine periods, and settings for the experiments are the same as in Fig. 8, save for variations in the noise frequency and amplitude.
The results of this experiment are tabulated in Table 1. In the PW case the table shows that noise reduces the flux by significantly diminishing, on average, particle advection. A fit of the data shows that the distance drops nearly linearly with noise amplitude. The drop in the displacement is strikingly similar to the distance, indicating that the particle, on average, gets further away the longer the forcing is active, regardless of the amount of noise, and that on average the maximum displacement is well captured by the distance itself. In the SW case, the data indicates that the presence of noise increases the average distance and displacement of ideal tracers as the noise amplitude increases, provided that the noise amplitude is small. When the amplitude of the noise is large, however, the distance and the displacement drop. Since the noise has zero mean and is Gaussian, the results indicate that for large noise levels the noise itself largely determines the flux in the SW case. Moreover, the discrepancy between the displacement and the distance indicates that, on average, particles do not attain maximum displacement at the final time. Transport in the large noise amplitude regime is then diffusion dominated.
The noise frequency dependence of the displacement and the distance is straightforward, given a fixed noise amplitude. It speaks more about the measurement process itself than about the physics. Our comments are limited to the effect of the noise frequency on the outcome, for frequencies significantly higher than the forcing frequency. Our findings indicate that the transport magnitude and shape are not influenced greatly by the change in noise frequency, regardless of whether the forcing is PW or SW. The slow convergence of the running averages, however, means that, as the frequency of the noise decreases, the running averages of the monitored quantities take longer to converge to within some small tolerance. When the duration of the experiments is increased in indirect proportion to the noise frequency, the results become statistically similar, particularly in the PW case. Significant changes occur because of the presence or absence of noise, rather than the noise frequency value itself.
5. Conclusions
The transport velocity and the mass flux are associated with the time-averaged Lagrangian velocity and thus are important in the dynamics of tracers in a fluid flow. For waves of finite amplitude, the transport velocity may be approximated by the superposition of the mean Eulerian velocity and the Stokes drift. In this study we explored how the transport velocity in a laminar boundary layer, bounded by an ideal straight wall, under the action of sinusoidal progressive or standing waves, is affected by noise. The highly idealized setting of our experimental configuration enabled us to compare the effect of the noiseless situation—which is well known—with that of the noisy situation.
We specifically addressed how noise from underresolved dynamics affects the transport, the mean shear stress, and the fluxes in a laminar boundary layer. For our numerical simulations we purposely used measuring techniques that are commonly used in a laboratory or field setting. The boundary layer asymptotic theory, though of limited validity, is entirely appropriate for the perfectly smooth channel that we used in our experiments. Nevertheless, as a further check, we benchmarked our results qualitatively to a direct numerical simulation of the Navier–Stokes equations in the channel and found very good agreement. In summary, our results may show (acceptable) quantitative differences with what we think of a generally agreed upon model for fluids, and may show only mild qualitative differences.
The noise perturbation applied in this problem was Gaussian with zero mean. The noise perturbation produces monotonic slow growth in the uncertainties in the wave amplitude and phase that accumulate over time.
We found that noise has a dramatic and somewhat unexpected effect on the transport velocity in the boundary layer and, by extension, on the mean wall shear stress and the mass flux. Our general organizing principle has been to examine how noise affects two naturally occurring situations: when the forcing external to the layer is a progressive wave (PW) and when the forcing is a standing wave (SW). The asymptotic theory for the deterministic problem shows that these two forcing cases may be viewed as two extremes of the outcome on the transport in the natural setting. In fact, with these extremes it is easy to envision, at least qualitatively, how the situation for the leaky wave will play out [in this endeavor we are helped by the calculations of the deterministic case by Carter et al. (1973)].
The findings in this study suggest that the transport resulting from PW and SW external forcing subject to noise due to underresolved processes may be characterized by two components: a deterministic one and a diffusive one. Their relative effect on the transport depends on whether the external flow is PW or SW.
In the PW case, noise in the forcing decreases the mean and the total mass flux. The spatial mean wall shear and the transport velocity also decrease. At the same time, there is an enhancement of the spatially nonhomogeneous portion, or residual, of the transport velocity and thus of the wall shear stress. The mean flux drops almost linearly with the noise amplitude. Nevertheless, the mean flux is still positive, and thus ideal tracers present in the flow still will move in the direction of the waves, but at a reduced rate compared with the noiseless case. In the event that these tracers have a threshold of motion greater than the spatial mean of the transport but smaller than the maximum transport, the tracers may produce a nonuniform spatial distribution along the channel. With respect to the frequency of perturbations, the results are more telling about the measurement process itself. For noise frequencies that are higher than the forcing frequency, the running averages of quantities such as drift velocity and transport tend to require more time in reaching an adequate level of convergence. The reason is that these running averages converge at a rate inversely proportional to time, weighted by the period of the forcing. If the running averages are taken over commensurably longer time periods, the lower the noise frequency is made, the closer they reach statistical equivalence, all other things remaining the same. This assumes, of course, that the time steps in the numerical calculation are sufficiently small that the noise signal is adequately resolved.
In the SW case it is found that noise induces a very small mass flux. This mass flux can be in either direction along the horizontal coordinate of the channel. Hence, the spatial mean of the transport tends to increase slightly at the expense of the residual. Overall, however, the SW case shows that the structure and the magnitude of the transport changes when noise is present, but is relatively insensitive to increases in the noise amplitude or noise frequency. Compared to the PW case, the Lagrangian path of the SW case displays more statistical scatter. That is to say, in the PW case it is found, on average, the mean orbit displacement tends to be closely related to the beginning and average ending particle position, over a specified time span. In contrast, the maximum displacement for the SW case does not usually agree with the average ending particle position.
Acknowledgments
This work was supported in part by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38. This work was performed while JMR was a summer Faculty Research Program visitor of the Mathematics and Computer Science Division at Argonne National Laboratory and a Computational and Applied Mathematics/Program in Computing visiting faculty member at UCLA. Funding of the Beowulf-class cluster on which the calculations were performed was made possible by a grant from the University of Arizona.
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APPENDIX A
Numerical Approximation of the Wave Channel
Figure A1 shows cross sections of the computed horizontal velocity u as a function of z at a fixed location xj. The units of the z axis in the figure are given in terms of boundary layer thicknesses δ. The external velocity was a progressive wave of dimensionless amplitude 0.006. Figure A1a shows the velocity, as approximated by using a uniform grid (p = 1 following (7). The K = 20 case is depicted by crosses, the K = 40 case by squares, and the K = 80 case by dots in the figure. All cases were computed with M = 90 and Δt = 2π/800. The graph shows the computed solutions converging in accordance with our expectations of what a typical layer profile should look like. Figure A1b shows the computed velocity profiles at the same position in xj and same time as in Fig. A1a, but these solutions were computed on a variable mesh [again following (A1)] with p = 1.75. It is clear from these figures that the nonuniform mesh adequately captures the solution, using fewer discretization points than would otherwise be required with a uniform grid. Furthermore, Δx = λ/66, and the time step was set to τ/400, where τ is the period of the forcing.
APPENDIX B
Testing the Running-Average Algorithm
Running averages were used extensively as a diagnostic tool in this study. In particular, running averages were used to calculate the mean Eulerian velocity. The running averages were tested as follows. A sinusoidal progressive wave cos(kx − ωt) replaced the velocity field within the Prandtl code. All other aspects of the code remained the same, including the running average routine itself. The running average output of the code was then examined at times commensurate with the period, over many tens of thousands of time steps. At these times, the outcome for the running average of this sinusoidal signal was machine zero throughout the channel length, as expected.
APPENDIX C
The Advection Scheme
In addition to characterizing the flow under different forcing and noise conditions, we were also interested in assessing how the flow itself affects the distribution of passive tracers present in the boundary layer. To do this, we used a tracer tracking algorithm. The algorithm yields the eventual distribution of a collection of ideal tracers that are initially placed uniformly along the length of the channel and then subjected to advection by the transport velocity.
Specifically, the computed Eulerian velocity u(xj, zi, tn), n = 0, 1, · · · , and tn = nΔt, was used to compute UT(x, z, tn), a first-order splined semidiscrete interpolation of the advection velocity. This advection velocity was assumed fixed for a time interval TΔt, where T ≥ 1. Np particles were distributed uniformly along the channel, at locations Xk(t0), where k = 1, 2, · · · , Np, at some height zi, at the initial time t0. We then allowed the particles to advect according the the local velocity UT. Hence, particles moved to a new position X(tn + TΔt) = TΔtU(X
Since the magnitude of the velocities was small, freezing both the particle velocity and position for T steps yielded a saving in computational expense. The value of T in the experiments reported here was 5. This value increases the dissipation inherent in the advection algorithm. Hence, particle orbits, even for a perfectly linear acoustic field, would exhibit drift and thus not close on themselves. Drift, in turn, leads to overestimates in the particle transport. Hence, particle advection experiments must be short in duration and particle flux comparisons made in relative, rather than absolute, terms.
External forcing velocity U(x, t): (a) progressive wave case, (b) standing wave case. See text for a description of the forcing velocity amplitude, wavelength, and frequency
Citation: Journal of Physical Oceanography 32, 8; 10.1175/1520-0485(2002)032<2334:NEOWGT>2.0.CO;2
External forcing velocity U(x, t) with added noise, for the progressive wave case: (a) phase noise perturbations at times shorter than the period (Tn = P/20), (b) noise perturbations at times equal or longer than the period (Tn = P). The parameters are the same as in Fig. 1. Noise amplitude is 0.1π
Citation: Journal of Physical Oceanography 32, 8; 10.1175/1520-0485(2002)032<2334:NEOWGT>2.0.CO;2
Contours of the transport velocity (horizontal component) due to (a) progressive waves (PW), (b) standing waves (SW). The vertical scale is comparable to that of Fig. 4
Citation: Journal of Physical Oceanography 32, 8; 10.1175/1520-0485(2002)032<2334:NEOWGT>2.0.CO;2
Time history of the drift velocity at some particular position in the channel for the PW case with no added noise: (a) detail during the first few periods of the simulation, (b) full history. See Fig. 5 for a vertical scale for the amplitude of the transport velocity at the end of the experiment.
Citation: Journal of Physical Oceanography 32, 8; 10.1175/1520-0485(2002)032<2334:NEOWGT>2.0.CO;2
Effect of noise on the drift velocity, PW case. The space–time plot of the drift velocity: (a) no noise, (b) with noise. The superposition of the drift velocity over the last ten periods of the simulation, as a function of position along the channel: (c) no noise, (d) with noise. Noise frequency: 1/Tn = 20/P; noise amplitude: 0.5π
Citation: Journal of Physical Oceanography 32, 8; 10.1175/1520-0485(2002)032<2334:NEOWGT>2.0.CO;2
SW case. Superposition of the drift velocities, at many times corresponding to the last periods of the simulation, as a function of position along the channel: (a) no noise, (b) with noise. Noise frequency: 1/Tn = 20/P; noise amplitude: 0.5π
Citation: Journal of Physical Oceanography 32, 8; 10.1175/1520-0485(2002)032<2334:NEOWGT>2.0.CO;2
Lagrangian particle trajectories: (a) no noise, (b) noise, with frequency 20/P and amplitude 0.5π. PW case
Citation: Journal of Physical Oceanography 32, 8; 10.1175/1520-0485(2002)032<2334:NEOWGT>2.0.CO;2
Effect of noise on the Lagrangian trajectories, PW case: (a) and (b) no noise, (c) and (d) with noise. Trajectories were sampled at times commensurate with the period. Noise frequency: 20/P; noise amplitude: 0.5π. The initial and final particle positions are shown in (b) and (d)
Citation: Journal of Physical Oceanography 32, 8; 10.1175/1520-0485(2002)032<2334:NEOWGT>2.0.CO;2
Lagrangian particle fluxes at x = 1.2λ, PW case: (a) no noise, (b) noise. Note the different vertical scales. Noise frequency: 20/P; amplitude: 0.5π. Experimental conditions are the same as in Fig. 8
Citation: Journal of Physical Oceanography 32, 8; 10.1175/1520-0485(2002)032<2334:NEOWGT>2.0.CO;2
Particle trajectories sampled at times commensurate with the period, SW case: (a) no noise, (b) noise. Noise frequency: 20/P; noise amplitude: 0.5π
Citation: Journal of Physical Oceanography 32, 8; 10.1175/1520-0485(2002)032<2334:NEOWGT>2.0.CO;2
Fig. A1. The computed horizontal velocity u at a fixed location xj, as a function of depth, for different values of K; at some location x, after five periods of oscillation: (a) uniform mesh, with p = 1; (b) nonuniform mesh, with p = 1.75. Notation is crosses: K = 20; squares: K = 40; and dots: K = 80
Citation: Journal of Physical Oceanography 32, 8; 10.1175/1520-0485(2002)032<2334:NEOWGT>2.0.CO;2
Fig. A2. Wall shear stress: (a) Navier–Stokes, (b) Prandtl
Citation: Journal of Physical Oceanography 32, 8; 10.1175/1520-0485(2002)032<2334:NEOWGT>2.0.CO;2
Relative displacement (D) and distance (d) of ensemble trajectory as a function of the noise amplitude (N). Noise frequency is 1/20P, total times nine periods
These equations are more commonly known as “Stokes's second problem.” Since Stokes's name is invoked in several places and in different contexts, however, we will call these equations the linear Prandtl equations to avoid possible confusion.
We emphasize that the terms DC and residual apply only to spatial considerations, not time dependencies.
As the name implies, ideal tracers trace the Lagrangian path in a flow.
