Gregory A. Lawrence

Measurement of stimulated Hawking emission in an analogue system

Hawking argued that black holes emit thermal radiation via a quantum spontaneous emission. To address this issue experimentally, we utilize the analogy between the propagation of fields around black holes and surface waves on moving water. By placing a streamlined obstacle into an open channel flow we create a region of high velocity over the obstacle that can include surface wave horizons. Long waves propagating upstream towards this region are blocked and converted into short (deep-water) waves. This is the analogue of the stimulated emission by a white hole (the time inverse of a black hole), and our measurements of the amplitudes of the converted waves demonstrate the thermal nature of the conversion process for this system. Given the close relationship between stimulated and spontaneous emission, our findings attest to the generality of the Hawking process.

The stability of a sheared density interface

This study investigates the stability of stratified shear flows when the density interface is much thinner than, and displaced with respect to, the velocity interface. Theoretical results obtained from the Taylor–Goldstein equation are compared with experiments performed in mixing layer channels. In these experiments a row of spanwise vortex tubes forms at the level of maximum velocity gradient which, because of the profile asymmetry, is displaced from the mean interface level. As the bulk Richardson number is lowered from a high positive value the effects of these vortex tubes become more pronounced. Initially the interface cusps under their influence, then thin wisps of fluid are drawn from the cusps into asymmetric Kelvin–Helmholtz billows. At lower Richardson numbers increasingly more fluid is drawn into these billows. The inherent asymmetry of flows generated in mixing layer channels is shown to preclude an effective study of the Holmboe instability. Statically unstable flows (negative Richardson num...

On the hydraulics of Boussinesq and non-Boussinesq two-layer flows

Exact expressions for the internal and external Froude numbers for two-layer flows are derived from the celerities of infinitesimal long internal and external waves, without recourse to the Boussinesq approximation. These expressions are functions of the relative density difference between the layers; the relative thickness of the layers; and the stability Froude number, which can be regarded as an inverse bulk Richardson number. A fourth Froude number, the composite Froude number, has been most often used in previous studies. However, the usefulness of the composite Froude number is shown to diminish as the stability Froude number increases. The potential confusion associated with having four Froude numbers of importance has been alleviated by deriving an equation interrelating them. This equation facilitates a comprehensive understanding of the hydraulics of two-layer flows. It is demonstrated that in substantial portions of some flows (both Boussinesq and non-Boussinesq exchange flow through a contraction are presented as examples), the stability Froude number exceeds a critical value. In this case hydraulic analysis yields imaginary phase speeds corresponding to the instability of long internal waves. Various implications of this result are discussed.

The hydraulics of steady two-layer flow over a fixed obstacle

This paper reports the results of a theoretical and experimental study of steady two-layer flow over a fixed two-dimensional obstacle. A classification scheme to predict the regime of flow given the maximum height of the obstacle, the total depth of flow, and the density and flow rate of each layer, is presented with experimental confirmation. There are differences between this classification scheme and that derived for flow over a towed obstacle by Baines (1984, 1987). These differences are due to the motion of upstream disturbances in towed obstacle flows. Approach-controlled flows, i.e. flows with an internal hydraulic control in the flow just upstream of the obstacle are studied in detail for the first time. This study reveals that non-hydrostatic forces, rather than a shock solution (called an internal hydraulic drop by previous investigators), need to be considered to explain the behaviour of Approach-controlled flows.

Holmboe's instability in exchange flows

A laboratory study of the exchange of two fluids of different density through a constant-width channel with an underwater sill has enabled us to study Holmboes instability in greater detail than has been possible in mixing-layer experiments. The internal hydraulics of the exchange flow are such that we have been able to observe the initiation of instability, the development and behaviour of both symmetric and asymmetric Holmboe instabilities, and the suppression of the instability at bulk Richard-son numbers above about 0.7. A number of stability criteria resulting from previous numerical investigations have been verified experimentally. Our laboratory measurements are consistent with theoretical predictions of wave speed and wavenumber.

Estimation of wind-forced internal seiche amplitudes in lakes and reservoirs, with data from British Columbia, Canada

Analyses of observations from four lakes in British Columbia, Canada, compare estimates of the amplitude of thermocline deflections to predictions of wind-driven internal seiche amplitudes made using the Wedderburn number,W. The study sites range from the 750 m diameter Brenda Mines pit-lake to the 107 km long Kootenay Lake. Causal filtering of the wind data with a frequency cut-off based on the fundamental baroclinic time-scale is critical for correct calculation ofW. With the filtering incorporated, good comparison betweenW, its integral equivalent the Lake numberLN and the observations can be made. In all but the mine pit-lake, upwelling or near-upwelling conditions (W≈1) were encountered.

Mixing in Symmetric Holmboe Waves

Abstract Direct simulations are used to study turbulence and mixing in Holmboe waves. Previous results showing that mixing in Holmboe waves is comparable to that found in the better-known Kelvin–Helmholtz (KH) billows are extended to cover a range of stratification levels. Mixing efficiency is discussed in detail, as are effective diffusivities of buoyancy and momentum. Entrainment rates are compared with results from laboratory experiments. The results suggest that the ratio of the thicknesses of the shear layer and the stratified layer is a key parameter controlling mixing. With that ratio held constant, KH billows mix more rapidly than do Holmboe waves. Among Holmboe waves, mixing increases with increasing density difference, despite the fact that the transition to turbulence is delayed or prevented entirely by the stratification. Results are summarized in parameterizations of the effective viscosity and diffusivity of Holmboe waves.

Evolution and mixing of asymmetric Holmboe instabilities

Natural Sciences and Engineering Research Council of Canada. Canada Research Chairs Program. National Science Foundation (USA, OCE 0221057).

Non-hydrostatic effects in layered shallow water flows

This paper develops a one-dimensional extension to classical layered hydraulics that incorporates non-hydrostatic effects. General results for a homogeneous layer in a multi-layer steady flow are applied to single- and two-layer flow over a two-dimensional sill. The equation obtained for single-layer flows is the same as that obtained by Naghdi & Vongsarnpigoon (1986) using the direct theory of constrained fluid sheets, and compares very well with the laboratory measurements of Sivakumaran et al . (1983). The new equation derived for two-layer flows provides excellent agreement with the laboratory measurements of Lawrence (1993). Accurate solutions are obtained for a regime of two-layer flow whose behaviour cannot be explained, even qualitatively, using classical hydraulic theory.

Instability in Stratified Shear Flow: Review of a Physical Interpretation Based on Interacting Waves

Instability in homogeneous and density stratified shear flows may be interpreted in terms of the interaction of two (or more) otherwise free waves in the velocity and density profiles. These waves exist on gradients of vorticity and density, and instability results when two fundamental conditions are satisfied: (I) the phase speeds of the waves are stationary with respect to each other (“phase-locking“), and (II) the relative phase of the waves is such that a mutual growth occurs. The advantage of the wave interaction approach is that it provides a physical interpretation to shear flow instability. This paper is largely intended to purvey the basics of this physical interpretation to the reader, while both reviewing and consolidating previous work on the topic. The interpretation is shown to provide a framework for understanding many classical and nonintuitive results from the stability of stratified shear flows, such as the Rayleigh and Fjortoft theorems, and the destabilizing effect of an otherwise stable density stratification. Finally, we describe an application of the theory to a geophysical-scale flow in the Fraser River estuary. [DOI: 10.1115/1.4007909]