ALBERT

Description: Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution April 1982

Description: The dynamics of steady and unsteady channel flow over large obstacles is studied analytically and numerically in an attempt to determine the applicability of classical hydraulic concepts to such flows. The study is motivated by a need to understand the influence of deep ocean straits and sills on the abyssal circulation. Three types of channel flow are considered: nonrotating one dimensional (Chapter 2); semigeostrophic, constant potential vorticity (Chapter 3); and dispersive, zero potential vorticity (Chapter 4). In each case the discussion centers around the time-dependent adjustment that occurs as a result of sudden obtrusion of an obstacle into a uniform initial flow or the oscillatory upstream forcing of a steady flow over topography. For nondispersive (nonrotating or semigeostrophic) flow, nonlinear adjustment to obstacle obtrusion is examined using a characteristic formulation and numerical results obtained from a Lax-Wendroff scheme. The adjustment process and asymptotic state are found to depend upon the height of the obstacle bO in relation to a critical height bc and a blocking height bb. For bO 〈 bc 〈 bb, isolated packets of nondispersive (long gravity or Kelvin) waves are generated which propagate away from the obstacle, leaving the far field unaffected. For bc 〈 bO 〈 bb, a bore is generated which moves upstream and partially blocks the flow. In the semigeostrophic case, the potential vorticity of the flow is changed by the bore at a rate proportional to the differential rate of energy dissipation along the line of breakage. For bb 〈 bO the flow is completely blocked. Dispersive results in the parameter range bO 〈 bc are obtained from a linear model of the adjustment that results from obstacle obtrusion into a uniform, rotating-channel flow. The results depend on the initial Froude number Fd (based on the Kelvin wave speed). The dispersive modes set up a decaying response about the obstacle if Fd 〈 1 and (possibly resonant) lee waves if Fd 〉 1. However, the far-field upstream response is found to depend on the behavior of the nondispersive Kelvin modes and is therefore nil. Nonlinear steady solutions to nondispersive flow are obtained through direct integration of the equations of motion. The characteristic formulation is used to evaluate the stability of various steady solutions with respect to small disturbances. Of the four types of steady solution, the one in which hydraulic control occurs is found to be the most stable. This is verified by numerical experiments in which the steady, controlled flow is perturbed by disturbances generated upstream. If the topography is complicated (contains more than sill, say), then controlled flows may become destabilized and oscillations may be excited near the topography. The transmission across the obstacle of energy associated with upstream-forced oscillations is studied using a reflection theory for small amplitude waves. The theory assumes quasi-steady flow over the obstacle and is accurate for waves long compared to the obstacle. For nonrotating flow, the reflection coefficients are bounded below by a value of 1/3. For semigeostrophic flow, however, the reflection coefficient can be arbitrarily small for large values of potential vorticity. This is explained as a result of the boundary-layer character of the semigeostrophic flow.

Description: Author Posting. © Cambridge University Press, 2008. This article is posted here by permission of Cambridge University Press for personal use, not for redistribution. The definitive version was published in Journal of Fluid Mechanics 602 (2008): 241-266, doi:10.1017/S0022112008000827.

Description: The stability of a hydraulically driven sill flow in a rotating channel with smoothly varying cross-section is considered. The smooth topography forces the thickness of the moving layer to vanish at its two edges. The basic flow is assumed to have zero potential vorticity, as is the case in elementary models of the hydraulic behaviour of deep ocean straits. Such flows are found to always satisfy Ripa's necessary condition for instability. Direct calculation of the linear growth rates and numerical simulation of finite-amplitude behaviour suggests that the flows are, in fact, always unstable. The growth rates and nonlinear evolution depend largely on the dimensionless channel curvature κ=2αg′/f2, where 2α is the dimensional curvature, g′ is the reduced gravity, and f is the Coriolis parameter. Very small positive (or negative) values of κ correspond to dynamically wide channels and are associated with strong instability and the breakup of the basic flow into a train of eddies. For moderate or large values of κ, the instability widens the flow and increases its potential vorticity but does not destroy its character as a coherent stream. Ripa's condition for stability suggests a theory for the final width and potential vorticity that works moderately well. The observed and predicted growth in these quantities are minimal for κ≥1, suggesting that the zero-potential-vorticity approximation holds when the channel is narrower than a Rossby radius based on the initial maximum depth. The instability results from a resonant interaction between two waves trapped on opposite edges of the stream. Interactions can occur between two Kelvin-like frontal waves, between two inertia–gravity waves, or between one wave of each type. The growing disturbance has zero energy and extracts zero energy from the mean. At the same time, there is an overall conversion of kinetic energy to potential energy for κ〉0, with the reverse occurring for κ〈0. When it acts on a hydraulically controlled basic state, the instability tends to eliminate the band of counterflow that is predicted by hydraulic theory and that confounds hydraulic-based estimates of volume fluxes in the field. Eddy generation downstream of the controlling sill occurs if the downstream value of κ is sufficiently small.

Description: This work was supported by the National Science Foundation (Grant OCE- 0525729).

Description: Author Posting. © Cambridge University Press, 2000. This article is posted here by permission of Cambridge University Press for personal use, not for redistribution. The definitive version was published in Journal of Fluid Mechanics 404 (2000):117-149, doi:10.1017/S0022112099007065.

Description: In order to gain insight into the hydraulics of rotating-channel flow, a set of initial-value problems analogous to Long's towing experiments is considered. Specifically, we calculate the adjustment caused by the introduction of a stationary obstacle into a steady, single-layer flow in a rotating channel of infinite length. Using the semigeostrophic approximation and the assumption of uniform potential vorticity, we predict the critical obstacle height above which upstream influence occurs. This height is a function of the initial Froude number, the ratio of the channel width to an appropriately defined Rossby radius of deformation, and a third parameter governing how the initial volume flux in sidewall boundary layers is partitioned. (In all cases, the latter is held to a fixed value specifying zero flow in the right-hand (facing downstream) boundary layer.) The temporal development of the flow according to the full, two-dimensional shallow water equations is calculated numerically, revealing numerous interesting features such as upstream-propagating shocks and separated rarefying intrusions, downstream hydraulic jumps in both depth and stream width, flow separation, and two types of recirculations. The semigeostrophic prediction of the critical obstacle height proves accurate for relatively narrow channels and moderately accurate for wide channels. Significantly, we find that contact with the left-hand wall (facing downstream) is crucial to most of the interesting and important features. For example, no instances are found of hydraulic control of flow that is separated from the left-hand wall at the sill, despite the fact that such states have been predicted by previous semigeostrophic theories. The calculations result in a series of regime diagrams that should be very helpful for investigators who wish to gain insight into rotating, hydraulically driven flow.

Description: The authors have been supported by the National Science Foundation through Grants (OCE-9810599 for L.J.P. and K.R.H. and OCE-9711186 for EPC). L.J.P. also received support from the Office of Naval Research under Grant N00014-95-1-0456 and K.R.H. under grant N00014-93-1-0263.

Description: Author Posting. © Cambridge University Press, 2008. This article is posted here by permission of Cambridge University Press for personal use, not for redistribution. The definitive version was published in Journal of Fluid Mechanics 605 (2008): 281-291, doi:10.1017/S002211200800150X.

Description: A condition is derived for the hydraulic criticality of a 2-layer flow with transverse variations in both layer velocities and thicknesses. The condition can be expressed in terms of a generalized composite Froude number. The derivation can be extended in order to obtain a critical condition for an N-layer system. The results apply to inviscid flows subject to the usual hydraulic approximation of gradual variations along the channel and is restricted to flows in which the velocity remains single-signed within any given layer. For an intermediate layer with a partial segment of sluggish flow, the long-wave dynamics of the overlying and underlying layers become decoupled.

Description: The work described herein was supported by the Office of Naval Research (N00014- 07-1-0590) and the National Science Foundation (OCE-0525729).

Description: Author Posting. © Cambridge University Press, 2004. This article is posted here by permission of Cambridge University Press for personal use, not for redistribution. The definitive version was published in Journal of Fluid Mechanics 515 (2004): 415-443, doi:10.1017/S0022112004000576.

Description: The investigation involves the hydraulic behaviour of a dense layer of fluid flowing over an obstacle and subject to entrainment of mass and momentum from a dynamically inactive (but possibly moving) overlying fluid. An approach based on the use of reduced gravity, shallow-water theory with a cross-interface entrainment velocity is compared with numerical simulations based on a model with continuously varying stratification and velocity. The locations of critical flow (hydraulic control) in the continuous model are estimated by observing the direction of propagation of small-amplitude long-wave disturbances introduced into the flow field. Although some of the trends predicted by the shallow-water model are observed in the continuous model, the agreement between the interface profiles and the position of critical flow is quantitatively poor. A reformulation of the equations governing the continuous flow suggests that the reduced gravity model systematically underestimates inertia and overestimates buoyancy. These differences are quantified by shape coefficients that measure the vertical non-uniformities of the density and horizontal velocity that arise, in part, by incomplete mixing of entrained mass and momentum over the lower-layer depth. Under conditions of self-similarity (as in Wood's similarity solution) the shape coefficients are constant and the formulation determines a new criterion for and location of critical flow. This location generally lies upstream of the critical section predicted by the reduced-gravity model. Self-similarity is not observed in the numerically generated flow, but the observed critical section continues to lie upstream of the location predicted by the reduced gravity model. The factors influencing this result are explored.

Description: M. H. N. would like to thank the Danish Natural Science Research Council for financial support. L. P. and K. H. were supported by the Office of Naval Research under grant N00014-1-01-0167 and by the National Science Foundation under grant OCE-0132903.

Description: The overflow of dense water from the Nordic Seas through the Faroe Bank Channel (FBC) has attributes suggesting hydraulic control—primarily an asymmetry across the sill reminiscent of flow over a dam. However, this aspect has never been confirmed by any quantitative measure, nor is the position of the control section known. This paper presents a comparison of several different techniques for assessing the hydraulic criticality of oceanic overflows applied to data from a set of velocity and hydrographic sections across the FBC. These include 1) the cross-stream variation in the local Froude number, including a modified form that accounts for stratification and vertical shear, 2) rotating hydraulic solutions using a constant potential vorticity layer in a channel of parabolic cross section, and 3) direct computation of shallow water wave speeds from the observed overflow structure. Though differences exist, the three methods give similar answers, suggesting that the FBC is indeed controlled, with a critical section located 20–90 km downstream of the sill crest. Evidence of an upstream control with respect to a potential vorticity wave is also presented. The implications of these results for hydraulic predictions of overflow transport and variability are discussed.