Cary D. Troy

The instability and breaking of long internal waves

Laboratory experiments are carried out to determine the nature of internal wave breaking and the limiting wave steepness for progressive, periodic, lowest-mode internal waves in a two-layer, miscible density stratification. Shoaling effects are not considered. The waves investigated here are long relative to the thickness of the density interface separating the two fluid layers

Cross-shelf thermal variability in southern Lake Michigan during the stratified periods

[1] Results from a field experiment in southern Lake Michigan are used to quantify the cross-shelf nearshore variability in Great Lakes temperatures during the stratified season. The experiment was conducted along the Indiana coast of southern Lake Michigan, with temperature and velocity moorings arranged in a cross-shelf transect that extended to approximately 20 km from shore (40 m depth). The field site is noteworthy because of its location at the end of a major axis of an elliptical Great Lake, the relatively mild bathymetric slope, and local shoreline orientation that is perpendicular relative to the dominant summer winds. Measurements demonstrate that the location of the thermocline-bottom intersection is highly variable, causing a wide zone of extreme thermal variability in the nearshore region with time scales of variability ranging from hours to months. Near-inertial internal Poincare waves are shown to cause large thermocline excursions but primarily only during periods of elevated activity. Several full upwelling events were observed, but in general, they were brief, lasting only 1–2 days, and had very limited spatial extent (2.5 km or less). Nonetheless, the offshore extent of the upwelling front was shown to be reasonably estimated with a simple estimate of the cross-shelf transport caused by alongshore wind events. A persistent feature that determined the zone of elevated thermal variability (the thermocline-shelf intersection point) was the strongly tilted thermocline, which resulted in the thermocline being located very close to shore. No evidence was found to support the hypothesis that internal Kelvin waves affect thermal variability at the study location.

The breaking of interfacial waves at a submerged bathymetric ridge

The breaking of periodic progressive two-layer interfacial waves at a Gaussian ridge is investigated through laboratory experiments. Length scales of the incident wave and topography are used to parameterize when and how breaking occurs. Qualitative observations suggest both shear and convection play a role in the instability of waves breaking at the ridge. Simultaneous particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) measurements are used to calculate high resolution, two-dimensional velocity and density fields from which the local gradient Richardson number Ri g is calculated. The transition to breaking occurred when 0.2 ≤ Ri g ≤ 0.4. In these wave-ridge breaking events, the destabilizing effects of waves steepening in shallow layers may be responsible for breaking at higher Ri g than for similar waves breaking through shear instability in deep water (Troy & Koseff, J. Fluid Mech. , vol. 543, 2005 b , p. 107). Due to the effects of unsteadiness, nonlinear shoaling and flow separation, the canonical Ri g > 0.25 is not sufficient to predict the stability of interfacial waves. A simple model is developed to estimate Ri g in waves between finite depth layers using scales of the incident wave scale and topography. The observed breaking transition corresponds with a constant estimated value of Ri g from the model, suggesting that interfacial shear plays an important role in initial wave instability. For wave amplitudes above the initial breaking transition, convective breaking events are also observed.

The viscous decay of progressive interfacial waves

The viscous damping of progressive, two-layer interfacial waves is examined theoretically and experimentally. Traditional water wave theory is modified to derive the damping rates associated with interfacial wave propagation in a rectangular channel. The individual wave damping contributions are considered from the bottom, side, and interfacial boundary layers, as well as the damping associated with the wave-induced velocities within the homogenous fluid layers. These results show that for most laboratory-scale experiments, sidewall friction plays the dominant role in wave damping. Laboratory experiments are conducted to verify the damping rates for progressive two-layer internal waves in a rectangular channel. Experiments are conducted on both monochromatic and polychromatic wave trains. The results of these experiments are in good agreement with the derived damping rates, but show poorer agreement for large-amplitude waves when the sidewall boundary layers become turbulent. More work is necessary to qua...

Spatial structure of internal Poincaré waves in Lake Michigan

In this paper we examine the characteristics of near-inertial internal Poincaré waves in Lake Michigan (USA) as discerned from field experiments and hydrodynamic simulations. The focus is on the determination of the lateral and vertical structure of the waves. Observations of near-inertial internal wave properties are presented from two field experiments in southern Lake Michigan conducted during the years 2009 and 2010 at Michigan City (IN, USA) and Muskegon (MI, USA), respectively. Spectra of thermocline displacements and baroclinic velocities show that kinetic and potential baroclinic energy is dominated by near-inertial internal Poincaré waves. Vertical structure discerned from empirical orthogonal function analysis shows that this energy is predominantly vertical mode 1. Idealized hydrodynamic simulations using stratifications from early summer (June), mid-summer (July) and fall (September) identify the basin-scale internal Poincaré wave structure as a combination of single- and two-basin cells, similar to those identified in Lake Erie by Schwab, with near-surface velocities largest in the center of the northern and southern basins. Near-inertial bottom kinetic energy is seen to have roughly constant magnitude over large swathes across the basin, with higher magnitude in the shallower areas like the Mid-lake Plateau, as compared with the deep northern and southern basins. The near-bottom near-inertial kinetic energy when mapped appears similar to the bottom topography map. The wave-induced vertical shear across thermocline is concentrated along the longitudinal axis of the lake basin, and both near-bottom velocities and thermocline shear are reasonably explained by a simple conceptual model of the expected transverse variability.

Logarithmic velocity structure in the deep hypolimnetic waters of Lake Michigan

The characteristics of the bottom boundary layer are reported from a Lake Michigan field study carried out in deep hypolimnetic waters (55 m depth) during the stratified period (June–September 2012). The sandy substrate at the measurement site was densely covered with invasive quagga mussels (mean size: 1.6 cm; mean density: 10,000 mussels m). The measurements reveal a sluggish, compact bottom boundary layer, with flow speeds at 1 mab less than 5 cm s for most of the period, and a dominance of subinertial energy. A downwelling event caused the largest currents observed during the deployment (10 cm s at 1 mab) and a logarithmic layer thickness of 15 m. In spite of the weak flow, logarithmic profile fitting carried out on high-resolution, near-bed velocity profiles show consistent logarithmic structure (90% of profiles). Flow was dominated by subinertial energy but strong modified by near-inertial waves. Fitted drag coefficients and roughness values are Cd1m 5 0.004 and z0 5 0.12 cm, respectively. These values increase with decreasing flow speed, but approach canonical values for 1 mab flow speeds exceeding 4 cm s. The estimated vertical extent of the logarithmic region was compact, with a mean value of 1.2 m and temporal variation that is reasonably described by Ekman scaling, 0.07 u /f , and the estimated overall Ekman layer thickness was generally less than 10 m. Near-bed dissipation rates inferred from the law of the wall were 10210 W kg and turbulent diffusivities were 10210 ms.