Abstracts of Invited Talks

Non-hydrostatic Effects on Rotating Gravity Currents S. Allen, U.B.C.

One of the necessary considerations of laboratory modeling is the scaling-up to the ``real'' world. Many rotating flow experiments have, in particular, large aspect ratio distortions. In many situations this is justified but for non-hydrostatic flows it is questionable. At the front of a gravity current the flow turns down in a localized, non-hydrostatic region. Here we will contrast laboratory observations of gravity currents with recently published observations of the Columbia River Plume. in the laboratory, the speed, width, depth and velocity structure of a surface gravity current in a rotating frame was measured using video techniques. In addition to the well documented dependence on the wave speed, the speed of the head of the gravity current showed a higher order correction proportional to the hydrostatic number (the ratio of the Coriolis parameter to the Brunt-Vaisala frequency). The speed decreases as the rotation rate increases. Incorporating the effect of the hydrostatic number, comparable to the aspect ratio of the system, should allow more accurate interpolation from laboratory experiments to the field.

Steady, non-dissipative theory of gravity currents in rotating channels J. N. Hacker, U. Cambridge

The propagation of high Reynolds number gravity currents in ambient fluids of similar density is essentially controlled by the requirement that the gravity current provides sufficient hydrostatic-head to accelerate the ambient fluid out its path. The nature of these dynamics, for the case of a gravity current in a nonrotating channel, were investigated by Benjamin(1), who showed that although the task of computing the complete flow is a difficult one, the upstream depth and propagation speed of the current may be calculated, without need to compute the details of the nonhydrostatic flow about the head, by using a control volume analysis based on Bernoulli's Theorem and a momentum integral.

In this paper we extend Benjamin's analysis to the case of gravity currents in rotating channels. The parameter expressing the strength of the background rotation in this situation is the ratio of the width of the channel to the Rossby radius of deformation. It is shown that for prescribed upstream vorticity conditions in the current, unique, non-dissipative solutions for the propagation speed and upstream structure of the current may be determined for all levels of background rotation. The predictions of the theory are compared with the results of laboratory experiments and are shown to be in good agreement.

1. Benjamin, T. B. (1968) J. Fluid Mech. Vol. 31, pp. 209-248

Buoyancy-driven flow in a pipe - effective pipe purging P. F. Linden, U.C.S.D.

The displacement of fluid within a horizontal pipe by the introduction of fluid with a different density is produced by a combination of the rate at which new fluid is introduced and by the buoyancy forces acting on the two fluids. At low input flowrates the buoyancy-driven flow dominates and the input fluid flows along the pipe as a gravity current. As the flowrate increases the depth of this current increases until it reaches a maximum depth (determined by the shape of the pipe). For higher flowrates the input fluid can not be carried as a gravity current and the input fluid then displaces all of the resident fluid within the pipe. Expressions are given for the movement of the input fluid over the whole range of input flowrates and momentum fluxes, and the effects of mixing produced by a high momentum flux source input are discussed. The results of small scale laboratory measurements using salt solution introduced into a circular pipe of fresh water, and large scale flows produced by the introduction of methane or propane into air-filled circular pipes, are compared with the theoretical predictions.

The non-Boussinesq lock exchange problem and related phenomena J. W. Rottman, U. Delaware

A critical review is given of two-layer hydraulic theory for the non-Boussinesq lock exchange problem and for the closely related dam break and pipe purging problems. An essential feature of this review is the hydraulic theory for two-layer bores in non-Boussinesq flows. Based on this review a hydraulic model is developed that is a generalization to arbitrary density differences of the models developed earlier by ROTTMAN & SIMPSON [J. Fluid Mech. 135, 95-110 (1983)] and extended by KLEMP, ROTUNNO & SKAMAROCK [ J. Fluid Mech. 269, 169-198 (1994)]. The results of this model are compared with the non-Boussinesq lock-exchange experiments of KELLER & CHYOU [J. Appl. Math. Phys., 42, 874-909 (1991)] and BELBAUER, FANNEL & BRITTER [J. Fluid Mech. 250, 669-687 (1993)], as well as with some more recent pipe purging data. The main result is that high density gravity currents propagating into lighter fluids behave much differently than low density gravity currents propagating into heavier fluids. This difference apparently is due to the fact that for non-Boussinesq fluids the dissipation primarily occurs in the heavier fluid.

Propagation of Internal Bores R. Rotunno, N.C.A.R.

According to classical hydraulic theory, the energy losses within an external bore must occur within the expanding layer. However, the application of this theory to describe the propagation of internal bores leads to contradiction with accepted gravity-current behavior in the limit as the depth of the expanding layer ahead of the bore becomes small. In seeking an improved expression for the propagation of internal bores, we have rederived the steady front condition for a bore in a two-layer Boussinesq fluid in a channel under the assumption that the energy loss occurs within the contracting layer. The resulting front condition is in good agreement with available laboratory data and numerical simulations, and has the appropriate behavior in both the linear long-wave and gravity-current limits. Analysis of an idealized internal bore assuming localized turbulent stresses suggests that the energy within the expanding layer should, in fact, increase. Numerical simulations with a 2-D nonhydrostatic model also reveal a slight increase of energy within the expanding layer and suggest that the structure of internal bores is fundamentally different from classical external bores, having the opposite circulation and little turbulence in the vicinity of the leading edge.

Based on: ``On the propagation of internal bores'' by Klemp, Rotunno and Skamarock, J. Fluid Mech. (1997), vol. 331, pp. 81-106.

Gravity Currents Everywhere J. E.Simpson, U. Cambridge

Gravity currents are produced by forces between two fluid of different density. These fluids may be gases or liquids of fluidised particulates, and the currents may be along horizontal surfaces or down slopes. The density differences may be due to changes in composition, temperature or concentration of solute or suspended materials. The distance of travel may be small, or it may be large enough for the current to be affected by the earth's rotation In this paper, examples of gravity currents in various situations in the environment are discussed. In the atmosphere, examples being thunderstorm outflows, sea-breeze fronts and escapes of dense gases In the ocean, gravity currents occur, and have been studied at the saline interfaces in estuaries. In the earth sciences, avalanches of snow or other material are important examples. Volcanoes produce viscous gravity currents of basaltic lava and avalanches of pyroclastic flows. Some recent problems in the understanding of gravity currents are discussed and the importance of laboratory experimental work is stressed.

Excitation of trapped internal waves by gravity currents B. R. Sutherland, U. Alberta

Interfacial gravity currents, or intrusions, may occur naturally along an inversion in the atmosphere as a consequence, for example, of cold thunderstorm outflows or due to rapid mixing by the collision of two fronts. Depending on the relative density of the upper and lower layers and the density of the intrusion, solitary waves or bores may be generated ahead of the current, as demonstrated in laboratory experiments (e.g. Rottman and Simpson (1989)). Less well studied is the dynamics of trapped internal waves in the wake of the gravity current. In laboratory experiments we show that, if the interfacial thickness is small compared with the width of the gravity current, the excitation of trapped internal waves is weak. However, if the interfacial thickness is comparable to, but smaller than the width of the gravity current, large amplitude trapped internal waves are excited. In most experiments these waves are sinuous and propagate with horizontal phase speeds much slower than that of the intrusion. These large amplitude waves extract significant momentum from the current, and in some circumstances, the current itself is observed to stop propagating and a solitary wave is generated. Theory of a three-layer fluid is presented to explain this transitional behaviour in terms of a resonant coupling between the gravity current and trapped internal waves.

Dynamics of radiating cold domes on a sloping bottom G. E. Swaters, U. Alberta

Numerical simulations of benthic gravity-driven currents along continental shelves suggest they exhibit considerable time and spatial variability and tend to organize themselves into large scale bottom-intensified cold domes or eddies. Attempts to derive simple relations governing the evolution of the spatial moments of the mass equation for baroclinic eddies have failed because it is not clear how to express the form or wave drag stresses associated with the excited (topographic) Rossby wave field in the surrounding fluid in terms of the eddy moments. We develop a simple model for the leading order time evolution of a cold dome configuration which initially nearly satisfies the Mory-Stern isolation constraint. As the topographic Rossby wave field in the surrounding fluid interacts with the cold dome, higher azimuthal modes are excited within the cold dome which develop into spiral-like filamentary structures on the eddy boundary. The trajectory followed by the position of the maximum height of the cold dome corresponds to sub-inertial along and cross slope oscillations superimposed on a mean along slope drift (well described by the Nof velocity). Nevertheless, the theory suggests that there are no oscillations (at least to second order) in the horizontal spatial moments of the eddy height, that is, the centre of mass of the eddy moves steadily in the along and down slope direction (i.e., ``southwestward'' relative to the topographic beta-plane). The theoretical analysis is in good agreement with a nonlinear numerical simulation which we present.

Gravity Currents in Rotating Fluids J. A. Whitehead, W.H.O.I.

Laboratory experiments and theory have revealed many different types of gravity currents in rotating fluids. Formulas for width, volume flux, and velocity scales of the currents have been known since the 1970's. Some issues concerning currents upstream of a control section are clarified. Laboratory experiments reveal a current preferably on the left hand side (looking upstream, for counterclockwise fluid rotation). Effects of drag are evident in the laboratory results which make detailed agreement with inviscid theory impossible. Transient experiments reveal that a Kelvin wave initiates a starting current on the right hand wall. Ekman suction removes fluid from this current and builds up the interior fluid depth, making the current on the left hand wall increase until only it remains. A new theory is presented which allows calculation of constant potential vorticity flow through a passage of arbitrary shape. Critically controlled flows are missing from present numerical models, and the need for their inclusion is revealed by a survey of nine important ocean sill flows. Independent calculations by colleagues may show how to include features of real gravity currents in numerical models of the ocean.

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