Helfrich Karl R.

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Helfrich

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Karl R.

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Large amplitude internal waves in the coastal ocean (Preface)

(Copernicus Publications on behalf of the European Geosciences Union and the American Geophysical Union, 2011-10-07) Grimshaw, Roger H. J. ; Helfrich, Karl R. ; Scotti, Alberto

The flow in the coastal ocean, and especially on the continental shelf and slope is often characterized by the presence of very large-amplitude internal waves. These are waves which occur in the interior of the ocean, and propagate horizontally with a concentration of their energy around the oceanic pcynocline. They are usually generated by the interaction of the barotropic tide with the shelf break, topographic sill or with other prominent bottom features. This leads to the formation of an internal tide, which then deforms and evolves into a train of very large-amplitude internal waves, with associated large pycnocline displacements and strong currents. They are highly significant for sediment transport and for the biology on the continental shelf, their associated currents cause strong forces on marine platforms and submersibles, the associated strong distortion of the density field has a severe impact on acoustic signaling and their capacity to break and form microstructure has major consequences for the understanding of interior ocean mixing.
Estimating the predictability of an oceanic time series using linear and nonlinear methods

(American Geophysical Union, 2004-08-03) Yuan, Guo-Cheng ; Lozier, M. Susan ; Pratt, Lawrence J. ; Jones, C. K. R. T. ; Helfrich, Karl R.

This study establishes a series of tests to examine the relative utility of nonlinear time series analysis for oceanic data. The performance of linear autoregressive models and nonlinear delay coordinate embedding methods are compared for three numerical and two observational data sets. The two observational data sets are (1) an hourly near-bottom pressure time series from the South Atlantic Bight and (2) an hourly current-meter time series from the Middle Atlantic Bight (MAB). The nonlinear methods give significantly better predictions than the linear methods when the underlying dynamics have low dimensionality. When the dimensionality is high, the utility of nonlinear methods is limited by the length and quality of the time series. On the application side we mainly focus on the MAB data set. We find that the slope velocities are much less predictable than shelf velocities. Predictability on the slope after several hours is no better than the statistical mean. On the other hand, significant predictability of shelf velocities can be obtained for up to at least 12 hours.
Hydraulic adjustment to an obstacle in a rotating channel

(Cambridge University Press, 2000-09-08) Pratt, Lawrence J. ; Helfrich, Karl R. ; Chassignet, Eric P.

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.
Gravity currents from a dam-break in a rotating channel

(Cambridge University Press, 2005-07-26) Helfrich, Karl R. ; Mullarney, Julia C.

The generation of a gravity current by the release of a semi-infinite region of buoyant fluid of depth $H$ overlying a deeper, denser and quiescent lower layer in a rotating channel of width $w$ is considered. Previous studies have focused on the characteristics of the gravity current head region and produced relations for the gravity current speed $c_{b}$ and width $w_b$ as a functions of the local current depth along the wall $h_b$, reduced gravity $g^\prime$, and Coriolis frequency $f$. Here, the dam-break problem is solved analytically by the method of characteristics assuming reduced-gravity flow, uniform potential vorticity and a semigeostrophic balance. The solution makes use of a local gravity current speed relation $c_{b} \,{=}\, c_b(h_b,\ldots)$ and a continuity constraint at the head to close the problem. The initial value solution links the local gravity current properties to the initiating dam-break conditions. The flow downstream of the dam consists of a rarefaction joined to a uniform gravity current with width $w_b$ (${\le}\, w$) and depth on the right-hand wall of $h_b$, terminated at the head moving at speed $c_b$. The solution gives $h_b$, $c_b$, $w_b$ and the transport of the boundary current as functions of $w/L_R$, where $L_R \,{=}\, \sqrt{g^\prime H}/f$ is the deformation radius. The semigeostrophic solution compares favourably with numerical solutions of a single-layer shallow-water model that internally develops a leading bore. Existing laboratory experiments are re-analysed and some new experiments are undertaken. Comparisons are also made with a three-dimensional shallow-water model. These show that lateral boundary friction is the primary reason for differences between the experiments and the semigeostrophic theory. The wall no-slip condition is identified as the primary cause of the experimentally observed decrease in gravity current speed with time. A model for the viscous decay is developed and shown to agree with both experimental and numerical model data.
Synthetic Aperture Radar observations of resonantly generated internal solitary waves at Race Point Channel (Cape Cod)

(American Geophysical Union, 2008-11-20) da Silva, Jose C. B. ; Helfrich, Karl R.

Synthetic Aperture Radar images revealed the two-dimensional propagation characteristics of short-period internal solitary waves in Race Point Channel in Massachusetts Bay. The images and in situ measurements of the flow in the channel are used to infer the likely generation mechanism of the waves. The solitary waves are generated during the ebb phase of the tide within the channel. On some occasions, two trains of internal waves are generated presumably at the same location but at slightly different phases of the ebb tide. The main characteristics of the (two-layer) flow are described based on the criticality of the Froude number. It is suggested that these two individual packets of waves result from flow passage through resonance (where the Froude number is one). One packet is generated as the flow passes through the transcritical regime during the acceleration phase of the (ebb) tidal current, and another packet is generated during the deceleration phase. Both packets propagate upstream when the tide slacks, but with slightly different propagation directions.
A model for large-amplitude internal solitary waves with trapped cores

(Copernicus Publications on behalf of the European Geosciences Union and the American Geophysical Union, 2010-07-15) Helfrich, Karl R. ; White, Brian L.

Large-amplitude internal solitary waves in continuously stratified systems can be found by solution of the Dubreil-Jacotin-Long (DJL) equation. For finite ambient density gradients at the surface (bottom) for waves of depression (elevation) these solutions may develop recirculating cores for wave speeds above a critical value. As typically modeled, these recirculating cores contain densities outside the ambient range, may be statically unstable, and thus are physically questionable. To address these issues the problem for trapped-core solitary waves is reformulated. A finite core of homogeneous density and velocity, but unknown shape, is assumed. The core density is arbitrary, but generally set equal to the ambient density on the streamline bounding the core. The flow outside the core satisfies the DJL equation. The flow in the core is given by a vorticity-streamfunction relation that may be arbitrarily specified. For simplicity, the simplest choice of a stagnant, zero vorticity core in the frame of the wave is assumed. A pressure matching condition is imposed along the core boundary. Simultaneous numerical solution of the DJL equation and the core condition gives the exterior flow and the core shape. Numerical solutions of time-dependent non-hydrostatic equations initiated with the new stagnant-core DJL solutions show that for the ambient stratification considered, the waves are stable up to a critical amplitude above which shear instability destroys the initial wave. Steadily propagating trapped-core waves formed by lock-release initial conditions also agree well with the theoretical wave properties despite the presence of a "leaky" core region that contains vorticity of opposite sign from the ambient flow.
The skirted island : the effect of topography on the flow around planetary scale islands

(Sears Foundation for Marine Research, 2009-09) Pedlosky, Joseph ; Iacono, Roberto ; Napolitano, Ernesto ; Helfrich, Karl R.

The flow around planetary scale islands is examined when the island possesses a topographic skirt representing a steep continental shelf. The model is barotropic and governed by the shallow water equations and the motion is driven by a wind stress with a constant curl. The presence of the strong topographic "skirt" around the island vitiates the elegant Island Rule of Godfrey and the closed potential vorticity contours around the island produced by the topography allow a geostrophic, stationary mode to resonate with an amplitude that is limited only by dissipation. In the limit of weak forcing the outline of the outermost closed potential vorticity isoline essentially replaces the island shape and determines the flow beyond that contour. Stronger nonlinearity produces substantial changes in the flow pattern as well as the transports trapped on the closed contours and the transport between the island and the basin boundary. Laboratory experiments, numerical calculations and analytical results are presented describing the structure of the flow. A western standing meander at the edge of the island's topography involves a rapid change in the direction of flow and this feature, predicted by analytical and numerical calculations is confirmed in laboratory experiments. As the measure of nonlinearity is increased beyond a threshold that depends on the ratio of the inertial boundary layer thickness to the Munk layer thickness the flow becomes time dependent and a strong eddy field emerges. The transports on the closed contours and the inter-basin exchange outside the closed potential vorticity contours show an enhancement over the linear analytical approximation as nonlinearity increases.
Hydraulic control of flow in a multi-passage system connecting two basins

(Cambridge University Press, 2022-04-05) Tan, Shuwen ; Pratt, Lawrence J. ; Voet, Gunnar ; Cusack, Jesse M. ; Helfrich, Karl R. ; Alford, Matthew H. ; Girton, James B. ; Carter, Glenn S.

When a fluid stream in a conduit splits in order to pass around an obstruction, it is possible that one branch will be critically controlled while the other remains not so. This is apparently the situation in Pacific Ocean abyssal circulation, where most of the northward flow of Antarctic bottom water passes through the Samoan Passage, where it is hydraulically controlled, while the remainder is diverted around the Manihiki Plateau and is not controlled. These observations raise a number of questions concerning the dynamics necessary to support such a regime in the steady state, the nature of upstream influence and the usefulness of rotating hydraulic theory to predict the partitioning of volume transport between the two paths, which assumes the controlled branch is inviscid. Through the use of a theory for constant potential vorticity flow and accompanying numerical model, we show that a steady-state regime similar to what is observed is dynamically possible provided that sufficient bottom friction is present in the uncontrolled branch. In this case, the upstream influence that typically exists for rotating channel flow is transformed into influence into how the flow is partitioned. As a result, the partitioning of volume flux can still be reasonably well predicted with an inviscid theory that exploits the lack of upstream influence.
Testing the physical oceanographic implications of the suggested sudden Black Sea infill 8400 years ago

(American Geophysical Union, 2004-03-17) Siddall, M. ; Pratt, Lawrence J. ; Helfrich, Karl R. ; Giosan, Liviu

We apply a shock-capturing numerical model based on the single-layer shallow water equations to an idealized geometry of the Black Sea and the Sea of Marmara in order to test the implications of a suggested sudden Black Sea infill 8400 years ago. The model resolves the two-dimensional flow upstream and downstream of the hydraulic jump provoked by the cascade of water from the Sea of Marmara into the Black Sea, which would occur during a sudden Black Sea infill. The modeled flow downstream of the hydraulic jump in the Black Sea would consist of a jet that is in part constrained by bathymetric contours. Guided by the Bosporus Canyon, the modeled jet reaches depths of up to 2000 m and could explain the origin of the sediment waves observed at this depth. At a late stage of the infill the modeled jet is attached to the coast and might account for the course of a submerged channel at the mouth of the Bosporus. The preservation of continuous barrier-washover-lagoonal fill systems occurring on the Black Sea shelf is, however, not easily reconcilable with the large flows over the southwest Black Sea shelf predicted by the model. Intensified flow in the upstream basin (Sea of Marmara) is restricted to the immediate vicinity of the Bosporus, suggesting that a sudden reconnection need not have disturbed sediments in the wider Sea of Marmara.
Experimental study of the effect of rotation on nonlinear internal waves

( 2013-03-01) Grimshaw, Roger H. J. ; Helfrich, Karl R. ; Johnson, Edward R.

Large amplitude internal waves are commonly observed in the coastal ocean. In the weakly nonlinear long wave r egime, they are often modeled by the Korteweg-de Vries equation, which predicts that the long-time outcome of generic localised initial conditions is a train of internal solitary waves. However, when the e ect of background rotation is taken into account, it is known from several theoretical and numerical studies that the formation of solitary waves is inhibited, and instead nonlinear wave packets form. In this paper, we report the results from a laboratory experiment on the Coriolis platform which describes this process.
Strongly nonlinear, simple internal waves in continuously-stratified, shallow fluids

(Copernicus Publications on behalf of the European Geosciences Union and the American Geophysical Union, 2011-02-14) Ostrovsky, Lev A. ; Helfrich, Karl R.

Strongly nonlinear internal waves in a layer with arbitrary stratification are considered in the hydrostatic approximation. It is shown that "simple waves" having a variable vertical structure can emerge from a wide class of initial conditions. The equations describing such waves have been obtained using the isopycnal coordinate as a variable. Emergence of simple waves from an initial Gaussian impulse is numerically investigated for different density profiles, from two- and three-layer structure to the continuous one. Besides the first mode, examples of second- and third-mode simple waves are given.
Internal solitary wave generation by tidal flow over topography

( 2017-12-21) Grimshaw, Roger H. J. ; Helfrich, Karl R.

Oceanic internal solitary waves are typically generated by barotropic tidal flow over localised topography. Wave generation can be characterised by the Froude number F = U/c(0), where U is the tidal flow amplitude and c(0) is the intrinsic linear long wave phase speed, that is the speed in the absence of the tidal current. For steady tidal flow in the resonant regime, Delta(m) < F - 1 < Delta(M), a theory based on the forced Korteweg-de Vries equation shows that upstream and downstream propagating undular bores are produced. The bandwidth limits Delta(m,M) depend on the height (or depth) of the topographic forcing term, which can be either positive or negative depending on whether the topography is equivalent to a hole or a sill. Here the wave generation process is studied numerically using a forced Korteweg-de Vries equation model with time-dependent Froude number, F(t), representative of realistic tidal flow. The response depends on Delta(max) = F-max - 1, where F-max is the maximum of F(t) over half of a tidal cycle. When Delta(max) < Delta(m) the flow is always subcritical and internal solitary waves appear after release of the downstream disturbance. When Delta(m) < Delta(max) < Delta(M) the flow reaches criticality at its peak, producing upstream and downstream undular bores that are released as the tide slackens. When Delta(max) > Delta(M) the tidal flow goes through the resonant regime twice, producing undular bores with each passage. The numerical simulations are for both symmetrical topography, and for asymmetric topography representative of Stellwagen Bank and Knight Inlet.
Continuously stratified nonlinear low-mode internal tides

(Sears Foundation for Marine Research, 2008-05) Helfrich, Karl R.

A model for hydrostatic, fully nonlinear, low-mode internal tides is extended to continuously stratified conditions. Periodic inertia-gravity solutions of permanent form are shown to exist only for a limited range of amplitudes for a given stratification and frequency. As found in an earlier two-layer model, the solutions fall into two classes. In one, the waves take on a corner-shape as the limiting amplitude is approached. In the other, the waves remain continuous at the limiting amplitude, but have a lobate shape. Numerical investigation using the Euler equations shows that both classes of nonlinear solutions are robust to weak nonhydrostatic effects representative of oceanic conditions. The numerical solutions are also used to explore the evolution of an initial sinusoidal internal tide. It is demonstrated that the presence of the nonlinear solutions may limit the disintegration of the initial tide into shorter solitary-like waves. The nonlinear tide solutions and the disintegration process are briefly explored for conditions of the northeastern South China Sea where large internal tides and solitary waves are observed.
Combined effect of rotation and topography on shoaling oceanic internal solitary waves

(American Meteorological Society, 2014-04) Grimshaw, Roger H. J. ; Guo, Chuncheng ; Helfrich, Karl R. ; Vlasenko, Vasiliy

Internal solitary waves commonly observed in the coastal ocean are often modeled by a nonlinear evolution equation of the Korteweg–de Vries type. Because these waves often propagate for long distances over several inertial periods, the effect of Earth’s background rotation is potentially significant. The relevant extension of the Kortweg–de Vries is then the Ostrovsky equation, which for internal waves does not support a steady solitary wave solution. Recent studies using a combination of asymptotic theory, numerical simulations, and laboratory experiments have shown that the long time effect of rotation is the destruction of the initial internal solitary wave by the radiation of small-amplitude inertia–gravity waves, and the eventual emergence of a coherent, steadily propagating, nonlinear wave packet. However, in the ocean, internal solitary waves are often propagating over variable topography, and this alone can cause quite dramatic deformation and transformation of an internal solitary wave. Hence, the combined effects of background rotation and variable topography are examined. Then the Ostrovsky equation is replaced by a variable coefficient Ostrovsky equation whose coefficients depend explicitly on the spatial coordinate. Some numerical simulations of this equation, together with analogous simulations using the Massachusetts Institute of Technology General Circulation Model (MITgcm), for a certain cross section of the South China Sea are presented. These demonstrate that the combined effect of shoaling and rotation is to induce a secondary trailing wave packet, induced by enhanced radiation from the leading wave.
The reduced Ostrovsky equation : integrability and breaking

( 2012-05-10) Grimshaw, Roger H. J. ; Helfrich, Karl R. ; Johnson, Edward R.

The reduced Ostrovsky equation is a modi cation of the Korteweg-de Vries equation, in which the usual linear dispersive term with a third-order deriva- tive is replaced by a linear non-local integral term, which represents the e ect of background rotation. This equation is integrable provided a certain curvature constraint is satis ed. We demonstrate, through theoretical analysis and numeri- cal simulations, that when this curvature constraint is not satisfi ed at the initial time, then wave breaking inevitably occurs.
Vertical structure of barotropic‐to‐baroclinic tidal energy conversion on a continental slope

(American Geophysical Union, 2022-09-15) Liu, Z. ; Zhang, W. G. ; Helfrich, K. R.

Horizontal distribution of the vertically integrated barotropic‐to‐baroclinic energy conversion has been widely studied to examine the generation of internal tides at steep topography. The vertical structure of the energy conversion that provides insights into the associated dynamics, however, is masked by the often used depth‐integrated approach. Here, we reveal the vertical profile of barotropic‐to‐baroclinic energy conversion by employing an idealized ocean model in a slope‐shelf context forced by M2 barotropic tidal flow. The model shows two vertically separated hotspots of energy conversion, one near the sloping bottom and the other at the thermocline, resulting from the stronger vertical velocity and enhancement of the density perturbation, respectively. Isolation of the hotspots demonstrates that baroclinic energy generated in the bottom layer radiates toward onshore and offshore primarily in the form of internal wave beams, whereas that generated at the thermocline propagates away in the form of internal wave modes. Although energy converted at the thermocline contributes to only a small portion of the total energy conversion, it plays an important role in onshore baroclinic energy radiation and can be significantly affected by the internal wave activity at the bottom layer. With a fixed bottom topography, the percentage of baroclinic energy generated at the thermocline is linearly related to a body force exerted by the barotropic tidal flow over topography that can be estimated analytically. This provides a convenient way to estimate the overall barotropic‐to‐baroclinic energy conversion over a continental slope in the real ocean by measuring the energy conversion in the thermocline only.Plain Language SummaryInternal waves propagating in the interior of the stratified ocean are linked to energy redistribution and mixing. They are often generated when the surface tides flow over sloping topography. In this process, a portion of tidal energy is converted to internal waves. The horizontal distribution of this converted energy has been well‐studied, while its vertical structure has not. Here, we reveal the vertical profile of the energy conversion by employing an idealized ocean model to provide insights into the associated dynamics. The model shows two vertically separated hotspots of energy conversion, one near the seafloor and the other at the thermocline where the temperature and salinity change dramatically in the vertical direction. Although the upper hotspot contributed to only a small portion of the total energy conversion, it plays an important role for the baroclinic energy radiating onto the shallow continental shelf. By isolating the hotspots and comparing the modeled energy conversion with theoretical predictions, we find a linear relation between them. This provides a convenient way for energy conversion estimation in the real ocean that only the parameters in the thermocline need to be measured, instead of the whole water column.Key PointsVertical structure of barotropic‐to‐baroclinic energy conversion shows two hotspots, near the sloping bottom and at the thermoclineNear thermocline hotspot provides a small portion of the total energy conversion but is essential in onshore baroclinic energy radiationTotal energy conversion can be estimated from a linear relation between the thermocline energy conversion and an analytical prediction
Optimal transient growth in thin-interface internal solitary waves

( 2018-01-10) Passaggia, Pierre-Yves ; Helfrich, Karl R. ; White, Brian L.

The dynamics of perturbations to large-amplitude Internal Solitary Waves (ISW) in two-layered ows with thin interfaces is analyzed by means of linear optimal transient growth methods. Optimal perturbations are computed through direct-adjoint iterations of the Navier-Stokes equations linearized around inviscid, steady ISWs obtained from the Dubreil-Jacotin-Long (DJL) equation. Optimal perturbations are found as a function of the ISW phase velocity c (alternatively amplitude) for one representative stratification. These disturbances are found to be localized wave-like packets that originate just upstream of the ISW self-induced zone (for large enough c) of potentially unstable Richardson number, Ri < 0:25. They propagate through the base wave as coherent packets whose total energy gain increases rapidly with c. The optimal disturbances are also shown to be relevant to DJL solitary waves that have been modi ed by viscosity representative of laboratory experiments. The optimal disturbances are compared to the local WKB approximation for spatially growing Kelvin-Helmholtz (K-H) waves through the Ri < 0:25 zone. The WKB approach is able to capture properties (e.g., carrier frequency, wavenumber and energy gain) of the optimal disturbances except for an initial phase of non-normal growth due to the Orr mechanism. The non-normal growth can be a substantial portion of the total gain, especially for ISWs that are weakly unstable to K-H waves. The linear evolution of Gaussian packets of linear free waves with the same carrier frequency as the optimal disturbances is shown to result in less energy gain than found for either the optimal perturbations or the WKB approximation due to nonnormal effects that cause absorption of disturbance energy into the leading face of the wave. Two-dimensional numerical calculations of the nonlinear evolution of optimal disturbance packets leads to the generation of large-amplitude K-H billows that can emerge on the leading face of the wave and that break down into turbulence in the lee of the wave. The nonlinear calculations are used to derive a slowly varying model of ISW decay due to repeated encounters with optimal or free wave packets. Field observations of unstable ISW by Moum et al. (2003) are consistent with excitation by optimal disturbances.
Generalized conditions for hydraulic criticality of oceanic overflows

(American Meteorological Society, 2005-10) Pratt, Lawrence J. ; Helfrich, Karl R.

Two methods for assessing the hydraulic criticality of an observed or modeled overflow are discussed. The methods are valid for single-layer deep flows with arbitrary potential vorticity and cross section. The first method is based on a purely steady view in which the flow at a given section is divided up into a group of “streamtubes.” A hydraulic analysis requires an extension of Gill’s functional formulation to systems with many degrees of freedom. The general form of the critical condition and associated compatibility condition for such a system are derived and applied to the streamtube model. As an aside, it is shown by example that Gill’s original critical condition can fail to capture all possible critical states, but that this problem is fixed when the multivariable approach is used. It is also shown how Gill’s method can be applied to certain dispersive or dissipative systems. The second method of assessing criticality involves direct calculation of linear, long-wave speeds using a time-dependent version of the streamtube model. This approach turns out to be better suited to the analysis of geophysical datasets. The significance of the local Froude number F is discussed. It is argued that F must take on the value unity at some point across a critical section.
Buoyant gravity currents along a sloping bottom in a rotating fluid

(Cambridge University Press, 2002-08-22) Lentz, Steven J. ; Helfrich, Karl R.

The dynamics of buoyant gravity currents in a rotating reference frame is a classical problem relevant to geophysical applications such as river water entering the ocean. However, existing scaling theories are limited to currents propagating along a vertical wall, a situation almost never realized in the ocean. A scaling theory is proposed for the structure (width and depth), nose speed and flow field characteristics of buoyant gravity currents over a sloping bottom as functions of the gravity current transport Q, density anomaly g[prime prime or minute], Coriolis frequency f, and bottom slope [alpha]. The nose propagation speed is cp [similar] cw/ (1 + cw/c[alpha]) and the width of the buoyant gravity current is Wp [similar] cw/ f(1 + cw/c[alpha]), where cw = (2Qg[prime prime or minute] f)1/4 is the nose propagation speed in the vertical wall limit (steep bottom slope) and c[alpha] = [alpha]g/f is the nose propagation speed in the slope-controlled limit (small bottom slope). The key non-dimensional parameter is cw/c[alpha], which indicates whether the bottom slope is steep enough to be considered a vertical wall (cw/c[alpha] [rightward arrow] 0) or approaches the slope-controlled limit (cw/c[alpha] [rightward arrow] [infty infinity]). The scaling theory compares well against a new set of laboratory experiments which span steep to gentle bottom slopes (cw/c[alpha] = 0.11–13.1). Additionally, previous laboratory and numerical model results are reanalysed and shown to support the proposed scaling theory.
Mixing and entrainment in hydraulically driven stratified sill flows

(Cambridge University Press, 2004-09-09) Nielsen, Morten Holtegaard ; Pratt, Lawrence J. ; Helfrich, Karl R.

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.

Helfrich Karl R.

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