A Mechanism for Sediment Resuspension by Internal Solitary
D. J. Bogucki and L. G. Redekopp
Department of Aerospace and Mechanical Engineering
University of Southern California
Los Angeles, CA 90089-1191
Convincing evidence is emerging that internal solitary waves and other longwave packets propagating in shallow seas can frequently stimulate remarkably elevated rates of resuspension of sedimentary material. The accumulating evidence suggests that, at least under some wave conditions, an interesting and effective coupling exists between the wave-induced velocity field and the associated boundary layer forming adjacent to the bottom surface. This note is directed toward revealing the peculiar dynamics associated with this coupling under a particular class of waves and to argue that the consequent dynamics underlies the observed bursts of resuspension under the footprint of many long wave features.
Long internal waves propagating within a waveguide in shallow seas can appear either as waves of elevation (crests containing elevated isopycnal lines) or waves of depression (crests containing depressed isopycnal lines). The particular wave polarity realized in any application is determined by the vertical structure of the density stratification and the background velocity shear of the water column. Now, the induced velocity field associated with a solitary wave or other longwave feature, and its coupling with the bottom boundary layer, will differ depending on the wave polarity. Under the crest of a wave of elevation, the induced velocity along the bottom surface (relative to a frame fixed to the wave) will experience a spatially-varying acceleration. Fluid parcels near the front of the wave will undergo a deceleration (adverse pressure gradient) followed by an acceleration downstream of the wave crest. Fluid parcels will experience a similar deceleration-acceleration sequence under troughs of a packet of waves of depression. In either case, the strength of the deceleration-acceleration will increase with wave amplitude (actually, in proportion to the 3/2 power of the wave amplitude based on KdV theory). Hence, one anticipates that boundary layer separation, the creation of a boundary-trapped vortex or recirculation region, will appear in the footprint of larger amplitude waves. Furthermore, since the wave-boundary layer interaction is dependent of the vertical proximity to the peak wave-induced velocity to the bottom surface (i.e., on the eigenfunction), the strength of the interaction is probably quite dependent on environmental conditions.
The presentation here focuses primarily on the specific wave-boundary layer interaction associated with a solitary wave of elevation propagating against a large-scale shear flow. This configuration closely mimics the observed features on the Palos Verdes Shelf described by Bogucki et al . The flow state arises as the transient longshore current experiences a resonant interaction with cross-shelf topographic features. The topographic resonance stimulates solitary wave packets which propagate counter to the sheared, longshore current. Instrumental records from the mooring reveal multiple instances of episodically-elevated resuspension rates as the solitary wave packets move slowly upstream past mooring positions.
The results presented below are based on a direct numerical simulation of the primitive equations of motion for a model flow that approximates the field experiment. The numerical model presupposes a base state consisting of a uniform shear flow (Couette flow) between parallel, horizontal boundaries defining the bottom of the wave guide and the level of the density interface in a two-layer stratification. The mean flow velocity and pressure along the upper boundary vary in the streamwise direction in accordance with an isolated KdV solitary wave. No-slip boundary conditions are applied on the bottom, impermeable surface. The wave-induced motion satisfies stress-free boundary conditions on the upper, moving surface. The speed of the mean flow along the upper surface (alternatively, the interface of a two-layer model) is prescribed according to the relation
The parameter controls the amplitude and wave number of the solitary wave (alternatively, the streamwise deceleration-acceleration of the flow and the associated pressure gradient felt by the bottom surface). The other parameter that enters the problem is the Reynolds number based on the ambient flow velocity along the upper surface and the depth of the (lower) layer. The unsteady simulations are time-accurate and permit interrogation of any ensuing spatio-temporal dynamics as the wave amplitude and Reynolds number are varied.
For purposes of brevity, and in order to expose the onset of a particular dynamics termed a global instability as the wave amplitude increases, we only present results for a fixed Reynolds number of 104. The stationary flow (steady streamline pattern) in the boundary layer under a wave with amplitude is shown in Figure 1. The flow in the boundary layer is unstable based on a local instability analysis, but the spatially-developing nature of the flow inhibits any streamwise growth of unstable disturbances. This is not to say that if some disturbance field (noise) were convected through the flow domain that some spatial amplification would not occur within the separation bubble. However, no intrinsic dynamics is observed. As shown in Figure 2, however, a completely spontaneous dynamics appears quite abruptly when the wave amplitude is increased to . Figure 2 shows the spatial structure of the streamline pattern at different phases of the synchronous motion that accompanies the onset of the global instability. That the supercritical dynamics is synchronous is clearly evident from Figure 3 where frequency spectra at several streamwise positions within the separated region are superimposed. Here red and blue colors correspond to time series of w velocity and their power spectra at two streamwise positions separated by one-half the solitary wave length. The bottommost panel shows the coherence between these two points. The oscillations are clearly coherent for the base frequency and its harmonics. The global instability appears suddenly, is entirely intrinsic to the separation bubble, is a stable spatio-temporal finite-amplitude state saturated by nonlinearity, and, based on other work, is robust in the presence of disturbances that are convected through the flow domain.
The appearance of separated flow and consequent global instability as the wave amplitude increases comprise the basic hydrodynamical features of the proposed mechanism for enhanced resuspension by internal waves in shallow seas. As is evident from Figures 2 and 3, the global instability endows the flow with a spatio-temporal coherence that extends over the entire extent of the separated flow under the solitary wave. The coherent stress fields acting on the bottom surface, together with the coherent dynamics in the field extending considerably beyond the boundary layer proper, creates the environment to both resuspend particles and to transport significant fractions to elevated levels. The latter statement is, of course, conjectural at this stage. However, we are preparing additional simulations where particles are introduced at the lowest level in order to quantify any preference for enhanced vertical particle transport by the global instability and attempt to place the conjecture on firm footing. The corresponding separated flow under the trough of a wave packet of depression is also being examined for its susceptibility to global instability and its possible contribution to resuspension by the same fundamental mechanism.
This work was supported by ONR Code 322PO.
Bogucki, D., T. D. Dickey and L. G. Redekopp. 1997. Sediment resuspension and mixing by resonantly generated internal solitary waves. J. Phys. Ocean., 27: 1181-1196.
This is work in progress; more up-to-date version can be viewed at:
Redekopp@spock.usc.edu or DBogucki@spock.usc.edu