The dynamics of unsteady strait and sill flow

Abstract

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.