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Submesoscale coherent vortices (SCVs) are a ubiquitous feature of topographic wakes in the extratropical oceans. Recent studies demonstrate a mechanism wherein high vorticity bottom boundary layers (BBLs) on the slopes of the topography separate (forming shear layers), undergo instabilities, and subsequently merge in the horizontal and align in the vertical to form vertically coherent, columnar, SCVs (i.e. with low vertical shear). Background rotation is critical to the vertical alignment of unstable vortical filaments into coherent SCVs. In the tropics, however, the weakening of rotation prevents this alignment. Employing an idealized framework of steady barotropic flow past an isolated seamount in a background of constant stratification N and rotation rate f , we examine the wake structure for a range of f values spanning values from the poles to the tropics. We find a systematic increase in the interior vertical shear with decreasing f that manifests as a highly layered wake structure consisting of vertically thin, ‘pancake’ SCVs possessing a high vertical shear. A monotonic increase in the wake energy dissipation rate is concomitantly observed with decreasing f . By examining the evolution equations for the vertical shear and vertical enstrophy, we find that the interior shear generation is an advective process, with the location of peak shear generation approximately colocated with maximum energy dissipation. This leads to the inference that high wake dissipation in tropical tropographic wakes is caused by parameterized shear instabilities induced by interior advective generation of vertical shear in the near wake region.

Uniformly stratified flows approaching long and dynamically tall ridges develop two distinct flow components over disparate time scales. The fluid upstream and below a “blocking level” is stagnant in the limit of an infinite ridge and flows around the sides when the ridge extent is finite. The streamwise half-width of the obstacle at the blocking level arises as a natural inner length scale for the flow, while the excursion time over this half-width is an associated short time scale for the streamwise flow evolution. Over a longer time scale, low-level horizontal flow splitting leads to the establishment of an upstream layerwise potential flow beneath the blocking level. We demonstrate through numerical experiments that for sufficiently long ridges, crest control and streamwise asymmetry are seen on both the short and long time scales. On the short time scale, upstream blocking is established quickly and the flow is well described as a purely infinite-ridge overflow. Over the long time scale associated with flow splitting, low-level flow escapes around the sides, but the overflow continues to be hydraulically controlled and streamwise asymmetric in the neighborhood of the crest. We quantify this late-time overflow by estimating its volumetric transport and then briefly demonstrate how this approach can be extended to predict the overflow across nonuniform ridge shapes.

The dynamical connection between topographic control and wave excitation aloft is investigated theoretically and numerically in the idealized setting of two-dimensional stratified flow over an isolated ridge. We consider a constant far upstream inflow with uniform stratification except for a sharp density step located above the height of the ridge crest. Below this step, the stratification is sufficiently strong that the low level flow is blocked upstream and a hydraulically controlled flow spills over the crest. Above the density step, the flow supports upward radiating waves. In the inviscid limit, a bifurcating isopycnal separates the hydraulically controlled overflow from the wave field aloft. We show that, depending on the height of the density step, the sharp interface can either remain approximately flat, above the controlled downslope flow, or plunge in the lee of the obstacle as part of the controlled overflow itself. Whether the interface plunges or not is a direct consequence of hydraulic control at the crest. The flow above the crest responds to the top of the sharp density step as if it were a virtual topography. We find that a plunging interface can excite a wave field aloft that is approximately six times as energetic, with 15 % higher pressure drag, than that in a comparable flow in which the interface remains approximately flat.

We investigate the dynamic stability of stratified flow configurations characteristic of hydraulically controlled downslope flow over topography. Extraction of the correct ‘base state’ for stability analysis from spatially and temporally evolving flows that exhibit instability is not easy since the observed flow in most cases has already been modified by nonlinear interactions between the instability modes and the mean flow. Analytical studies, however, can yield steady solutions under idealized conditions which can then be analysed for stability. Following the latter approach, we study flow profiles whose essential character is determined by recently obtained solutions of Winters & Armi (J. Fluid Mech., vol. 753, 2014, pp. 80–103) for topographically controlled stratified flows. Their condition of optimal control necessitates a streamline bifurcation which then naturally produces a stagnant isolating layer overlying an accelerating stratified jet in the lee of the topography. We show that the inclusion of the isolating layer is an essential component of the stability analysis and further clarify the nature and mechanism of the instability in light of the wave-interaction theory. The spatial stability problem is also briefly examined in order to estimate the downstream location where finite-amplitude features might be manifested in streamwise slowly varying flows over topography.

Current-topography interactions in the ocean give rise to eddies spanning a wide range of spatial and temporal scales. Latest modeling efforts indicate that coastal and underwater topography are important generation sites for submesoscale coherent vortices (SCVs), characterized by horizontal scales of (0.1 – 10) km. Using idealized, submesoscale and BBL-resolving simulations and adopting an integrated vorticity balance formulation, we quantify precisely the role of bottom boundary layers (BBLs) in the vorticity generation process. In particular, we show that vorticity generation on topographic slopes is attributable primarily to the torque exerted by the vertical divergence of stress at the bottom. We refer to this as the Bottom Stress Divergence Torque (BSDT). BSDT is a fundamentally nonconservative torque that appears as a source term in the integrated vorticity budget and is to be distinguished from the more familiar Bottom Stress Curl (BSC). It is closely connected to the bottom pressure torque (BPT) via the horizontal momentum balance at the bottom and is in fact shown to be the dominant component of BPT in solutions with a well-resolved BBL. This suggests an interpretation of BPT as the sum of a viscous, vorticity generating component (BSDT) and an inviscid, ‘flow-turning ’ component. Companion simulations without bottom drag illustrate that although vorticity generation can still occur through the inviscid mechanisms of vortex stretching and tilting, the wake eddies tend to have weaker circulation, be substantially less energetic, and have smaller spatial scales.

We present in situ and remote observations of a Mississippi plume front in the Louisiana Bight. The plume propagated freely across the bight, rather than as a coastal current. The observed cross-front circulation pattern is typical of density currents, as are the small width (≈100 m) of the plume front and the presence of surface frontal convergence. A comparison of observations with stratified density current theory is conducted. Additionally, subcritical to supercritical transitions of frontal propagation speed relative to internal gravity wave (IGW) speed are demonstrated to occur. That is in part due to IGW speed reduction with decrease in seabed depth during the frontal propagation toward the shore. Theoretical steady-state density current propagation speed is in good agreement with the observations in the critical and supercritical regimes but not in the inherently unsteady subcritical regime. The latter may be due to interaction of IGW with the front, an effect previously demonstrated only in laboratory and numerical experiments. In the critical regime, finite-amplitude IGWs form and remain locked to the front. A critical to supercritical transition eventually occurs as the ambient conditions change during frontal propagation, after which IGWs are not supported at the front. The subcritical (critical) to critical (supercritical) transition is related to Froude number ahead (under) the front, consistently with theory. Finally, we find that the front-locked IGW (critical) regime is itself dependent on significant nonlinear speed enhancement of the IGW by their growth to finite amplitude at the front.

Upstream flow blocking is a distinguishing feature of stratified flows incident on dynamically tall mountain ridges. Blocking occurs as a consequence of the upstream propagation of long internal wave modes that are excited at the obstacle and which permanently modify the oncoming flow. When the ridge is infinite, the fluid upstream and below a `blocking level' is brought to stagnation. The resulting across-crest asymmetry combined with volume transport constraints causes the overflowing layer to accelerate and develop into a hydraulically controlled flow. The processes leading to the establishment of upstream blocking and hydraulic control occur on a characteristic short time scale. In the interior of a long, but finite ridge, a hydraulically controlled overflow similarly develops on a short time scale, while over a longer time scale, low-level horizontal flow splitting leads to the establishment of an upstream layer-wise potential flow beneath the blocking level. We demonstrate through numerical experiments that for sufficiently long ridges, crest control and streamwise asymmetry are seen on both the short and long time scales. We then proceed to quantify the overflow using the framework of stratified hydraulics. In a separate study, we investigate the dynamic stability of stratified flow configurations characteristic of blocked, topographically controlled downslope flows. The essential character of the base flow profiles considered is determined by the analytical solutions of Winters and Armi (2014). Their condition of optimal control necessitates a streamline bifurcation above the blocking location, which then naturally produces a stagnant isolating layer overlying an accelerating downslope flow. We show that the inclusion of the isolating layer is an essential component of the stability analysis. The spatial stability problem is also examined in order to estimate the downstream location where finite amplitude features might manifest in streamwise slowly-varying flows over topography. Finally, to expose the dynamical connection between topographic control and wave excitation aloft, we consider flow over dynamically tall ridges under stratification conditions that feature a strong density jump above crest level. We show that the height of the bifurcating streamline depends sensitively on the location of the step. Further, the question of whether or not the density interface remains flat or plunges across the crest as part of the hydraulically controlled flow is found to be directly related to the requirement of maintaining a subcritical overflow upstream. We also demonstrate that the top of the density interface acts as a `virtual topography' for the flow aloft and fundamentally controls the amplitude of the wave response there.

In this thesis, a 2D Chebyshev spectral domain decomposition method is developed for simulating the generation and propagation of internal waves over a topography. While the problem of stratified flow over topography is by no means a new one, many aspects of internal wave generation and breaking are still poorly understood. This thesis aims to reproduce certain observed features of internal waves by using a Chebyshev collocation method in both spatial directions. The numerical model solves the inviscid, incompressible, fully non-linear, non-hydrostatic Boussinesq equations in the vorticity-streamfunction formulation. A number of important features of internal waves over topography are captured with the present model, including the onset of wave-breaking at sub-critical Froude numbers, up to the point of overturning of the pycnoclines. Density contours and wave spectra are presented for different combinations of Froude numbers, stratifications and topographic slope.

The comparison theorem for a smooth projective variety \\(X\\) over \\(C\\) tells us that the Betti numbers are independent of \\(l\\). We aim to understand the \\(l\\) independence of Betti numbers for smooth projective varieties \\(X\\) over \\(k\\), where \\(k\\) is an algebraic extension of \\(F_p\\).

Let \\(X\\) be a smooth projective variety over a field of characteristic \\(0\\) and \\(L\\) an ample divisor. In this paper we study the Weak Lefschetz conjecture for Chow groups using the technique employed by Grothendieck in his study of the problem for Picard groups, and using Bloch's formula to interpret Chow groups in terms of Milnor \\(K\\)- theory we prove the infinitesimal part of the conjecture.