7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011 1

Internal hydraulic jump formation in a deep water, continuously-stratified, unsteady channel flow

Matthew D. Rayson, Nicole L. Jones, Gregory N. Ivey and Oliver B. Fringer*

School of Environmental Systems Engineering and the Oceans Institute,

University of Western Australia, *Department Civil and Environmental Engineering, Stanford University

rayson@sese.uwa.edu.au

Abstract

Scott Reef is an isolated coral atoll system that lies on the edge of the Australian North-West Shelf, a region dominated by a large barotropic tide. We have used the three-dimensional, non-hydrostatic code SUNTANS to simulate the flow response in a 500 m deep channel separating North and South Scott Reef. Model results indicate that the flow accelerates through the convergent section of the channel and decelerates in the divergent section forming an internal hydraulic jump. This jump released an undular bore on the turn of the tide creating conditions favourable for mixing. Due to the physical uniqueness of the site, this is one of the first studies reporting an internal hydraulic jump generated in response to barotropic tidal flow forced through a narrow gap between two islands. This process has important implications for the vertical flux of nutrients from deep water up into the coral reef system. 1. Introduction

Abyssal channels and straits are regions where the direct transfer of energy from the mean flow to mixing may occur as the flows can be hydraulically controlled and may exhibit flow instability (e.g., Thorpe, 2010). Measurements of dissipation in deep ocean channels, such as the Lucky Strike region (St Laurent and Thurnherr, 2007), have indicated elevated turbulent dissipation rates and it was postulated that this was due to the formation of a hydraulic jump. Internal hydraulic jumps which lead to intense mixing are a common feature in unsteady shallow flows over sills such as the Knight Inlet and the Strait of Gibraltar (see examples in Farmer and Armi, 1999). Enhanced mixing due to the interaction of tidally driven flow with topography leads to enhanced vertical nutrient fluxes which can, in turn, lead to increased primary productivity (Sharples et al., 2009). Studies on flow through a horizontal contraction have mainly focused on steady exchange flows between basins with different density characteristics (see reviews by Baines, 1995 and Pratt and Whitehead, 2008). The time-varying flow response in a tidally driven stratified flow through a horizontal contraction at Scott Reef, Western Australia is the focus of this paper. The key differences between Scott Reef and other channel sites are that the reef is located in the open ocean, the water column is continuously stratified, and, as opposed to a horizontal density gradient, a large amplitude barotropic tide drives the flow through the channel. Difficulties with obtaining high-resolution measurements and the lack of any analytical solution to this highly nonlinear problem led us to pursue a numerical approach to study the flow response in these environments.

7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011 2

2. Study site and methods

2.1 Scott Reef

Scott Reef is an island atoll reef system that lies on the edge of the continental shelf and rises to the surface from 500 m deep (Figure 1). It is one of several offshore reef systems on the Australian North West Shelf – Timor Sea region which have continued to grow despite being situated on a subsiding shelf (Collins and Testa, 2010). The topography of the reef system consists of two near circular reefs: North Scott Reef and South Scott Reef, which are 50 km from North to South, 30 km from east to west, and are separated by a 2 – 6 km wide, 500 m deep channel. South Scott Reef, the larger of the two, has a 20 - 50 m deep lagoon that is open at the northern entrance to the channel (Figure 2). The channel is approximately 15 km from east to west and 2 km wide at its narrowest point. Flow through this channel is the focus of this paper. The oceanography of the region is dominated by the large semi-diurnal tide with a tidal range at Scott Reef of 4 m and the semi-diurnal barotropic tidal currents on the shelf around the reef are in the range 0.15 – 0.20 m s-1 at 500 m deep (Rayson et al., 2011). Field measurements and previous numerical modelling have revealed the presence of an energetic internal tide along the outer flanks of the reef generated by the interaction of the barotropic tide with the steep topography (Rayson et al., 2011). The reef biota is diverse with large assemblages of soft and hard corals covering over 50% of the reef area along surveyed passages (Smith et al., 2008). A strong La Nina event in 1998 advected anomalously warm waters over the reef triggering a mass bleaching event which led to an 80% decrease in relative coral cover in some regions (Smith et al., 2008). While of significant ecological importance, the physical oceanographic factors influencing the exchange of water between the lagoon and the ocean are poorly understood. 2.2 Numerical Model

We used a three-dimensional non-hydrostatic numerical circulation model, SUNTANS (Fringer et al., 2006) to simulate the stratified tidally-driven dynamics at Scott Reef. SUNTANS solves the Reynolds-averaged, non-hydrostatic Navier-Stokes equations using the Boussinesq approximation. The equations are solved using the finite volume formulation on an unstructured horizontal grid and fixed vertical z-coordinates. We used a horizontal unstructured grid, which spanned the entire northern section of the North-West Shelf from Rowley Shoals in the south, to Ashmore Reef in the north (Figure 1 inset b). The distance between cell centres, dx, near the outer boundaries was 10 km and the resolution was gradually decreased down to 75 m at Scott Reef itself. The final grid was composed of 244,619 horizontal grid cells. The vertical coordinate was discretized with 150 fixed layers; the vertical spacing in the upper 50 m was 2-5 m, 5-15 m between 50 and 500 m, and 15-90 m between 500 and 3000 m. We initialised the model with a horizontally homogenous density profile (Figure 2 b) typical of summer stratification conditions (Rayson et al., 2011) and forced the open boundaries with barotropic tidal velocities from the OSU TPXOv7.2 global tide solution (Egbert and Erofeeva, 2002). We used a constant vertical eddy viscosity (νv = 0.001 m2 s-1) and the vertical tracer diffusivity was set to zero. To parameterise bottom drag we used a quadratic friction term with a drag coefficient Cd = 0.0025. We ran the model during a spring-tide period for three M2 tidal cycles with the non-hydrostatic solver turned on.

7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011 3

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Figure 1: Contour map of Scott Reef with the channel location marked by the red box. Inset a)

Location of the Browse Basin in Australia. Inset b) Map showing the SUNTANS grid extents. The black dots represent the triangular mesh nodes.

3. Along-channel flow response

3.1 Geometric and dynamic properties

The important geometric parameters of the channel are the depth H, length scale L, and the width at both the surface b0(x) and through the water column, b(x, z). Representative values for H and L are 450 m and 15 km, respectively. The surface width varies between 2000 and 6000 m with the narrowest section between easting 379000 and 382000 m (Figure 2 c). The channel width varies with depth and can best be approximated by a parabolic profile such that 0.5

0 ( )( )( ) /, x zb Hx z b . The width of the channel is asymmetric and b0(x) diverges more

rapidly at the eastern exit than the western exit (∂b0/∂x ≈ 1.0 and 0.4, respectively). The channel also bends at the point of maximum contraction with a radius of curvature R ≈ 5000 m. The dynamic parameters influencing the flow response in the channel were derived from the tidal flow rate q0, tidal period T and the stratification N(z). The model derived peak flow rate through the channel was between 5(3.2 3.5) 10 m3 s-1. The peak depth-averaged velocity of U0 = 0.8 m s-1 occurred at the narrowest point along the channel. The dominant velocity profile in the narrowest section, as inferred from the first vertical empirical orthogonal function (EOF) scaled by its maximum amplitude, was an asymmetric sine profile with a peak value of 1.0 m s-1 between 300 and 400 m (Figure 2 b) where 96% of the variance was

7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011 4

accounted for by the first EOF. The shape of the velocity profile is a response to the parabolic shape of the channel due to the change of width with depth. Following Helfrich, (1995) the two important dynamic parameters are: the internal Froude number, Fr = U/cn, where U is the tidal (barotropic) flow speed and cn is the group velocity of the nth baroclinic mode; and the time-dependence parameter γ=c1T/L, where T is the tidal period (12.42 h) and L is the length scale of the channel (15 km). Taking /nc NH n and U as the maximum barotropic velocity

along the thalweg of the channel, the maximum Fr for mode-one was 0.5 at the narrowest part of the channel (Figure 2 d). Baroclinic mode two and greater modes were supercritical along the narrowest section of the channel and transitioned to subcritical at easting 378000 and 384000, at both the west and east entrance of the channel, respectively. Open-channel hydraulic theory predicts that an internal hydraulic jump is likely to form in these transition regions (Baines, 1995). The estimated value of γ was 5, indicating that the distance a mode-one baroclinic wave will propagate over a tidal period is greater than the length scale of the channel and therefore the effects of time-dependency on the flow are very important (Helfrich, 1995).

3.2 Flow field over a tidal cycle

A sequence of vertical slices of the model velocity and isopycnals reveals the flow response in the channel over a complete tidal cycle (Figure 3). This sequence was taken from the last tidal cycle in the non-hydrostatic model simulation. We begin our description of the flow response in the channel during maximum flood tide when the acceleration term was minimal. The key response to the channel constriction during the flood phase was an acceleration and deceleration of the horizontal velocity in the convergent and divergent sections of the channel, respectively (Figure 3 a). The accelerating flow also affected the vertical displacement of isopycnals along the channel. Along the upstream (convergent) end of the channel, the isopycnals were lifted about their mean (i.e., tidally averaged) position in response to the flow; effectively generating an upstream region of high pressure and hence a positive baroclinic pressure gradient in the direction of the flow. In the divergent section of the channel, the flow decelerated and the isopycnals sharply returned to their mean position in the form of an internal hydraulic jump. This steep jump-like feature developed between easting 380 000 and 385 000 m and a depth of 150-300 m (Figure 3 a). The local vertical displacement of the isopycnals was approximately 30 m and the horizontal length scale was approximately 2000 m. As the tide slackened (t=3T/8), the jump steepened and there was some local overturning around the same location. The tide then slackened (t=T/2), and the jump released an internal undular bore that subsequently propagated (or was advected) west back through the channel (Figure 3 b). As the ebb tide then intensified (t= 3T/4) another jump formed but now at the western end of the channel at easting 375000 m (Figure 3 c). Our previous Froude number analysis along the channel predicted that the flow would undergo a hydraulic transition in this region, supporting the interpretation of this feature as a hydraulic jump (Figure 2 d). On the ebb phase of the tide, a similar response to flood tide occurred at the opposite end of the channel although at slightly different depths (Figure 3 c). We attribute this variation in response, to the asymmetry in the variation of width and depth along either side of the main contraction. Many numerical and observational studies have reported internal bore formation when a flow contracts vertically due to a sill (see Farmer and Armi, 1999), and in our case this happens due to a horizontal contraction. The key differences here are the unsteadiness due to the tidal forcing and the lack of any external horizontal density gradient driving the flow.

7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011 5

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the bathymetry along the thalweg. The vertical density profile and the mode-one velocity EOF are indicated. c) Cross-sectional area (blue) and channel width at the surface (red) as a function of

distance. d) Mode-one (black), mode-two (red) and mode-three (blue) Froude number as a function of distance.

We have presented results along the streamwise direction of the flow; there was however, considerable cross-channel variability in the currents and density perturbations associated with the curvature of the channel. Model results indicated that re-circulating eddies and secondary flows formed in the cross-channel direction, particularly during slack tide periods (not shown). The presence of these secondary flow structures may influence mixing (Seim and Gregg, 1997) and total volume fluxes through the channel (Klymak and Gregg, 2001).

7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011 6

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(t=3T/4) phase of the tide.

3.3 Dissipation and mixing

One important motivation for studying the flow in the channel separating North and South Scott Reef was to predict when and where elevated mixing rates might occur, as this will have significant implications for the vertical flux of nutrients and other tracers of importance to the local ecology. We used the model vertical shear, S2= (∂u/∂z+∂v/∂z)2, and buoyancy frequency, N, to estimate the local Richardson number, Ri =N2/S2, and thereby identified likely regions for increased mixing due to shear instabilities (Figure 4). During maximum flood tide, a region of low Ri (<0.25) developed throughout much of the water column in the divergent section of the channel (Figure 4). When the tide slackened and the jump was released, the region of low Ri also formed upstream in the narrowest part of the contraction (not shown). During maximum ebb tide, a region of Ri < 0.25 formed throughout the water column in the

7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011 7

divergent section of the channel (easting 375 000 m) and also near the steeply sloping seabed (not shown). This analysis indicates that the internal hydraulic jump and subsequent internal bore are likely to trigger elevated mixing rates in the channel. To determine whether the magnitude of dissipation and mixing in the water column was significant, we compared model estimates of dissipation, 2( / + / )v u z v z , integrated through the volume of the channel, with

estimates of energy loss due to bottom drag ( 3

beddC U ). The results revealed that the tidally

averaged energy loss through the water column was equal in magnitude to the direct loss through bottom drag (≈ 2–3 MW); emphasizing the importance of the internal shear flow to the system. Any mixing estimates from our three-dimensional, non-hydrostatic solution are limited by our choice of grid resolution (dx ~ (10) m, dz ~ (1) m) and zeroth-order

turbulence closure (constant eddy viscosity); but in the absence of turbulence measurements within the channel for comparison, we were unable to conduct model sensitivity studies using either higher-order closure schemes or finer grid resolution.

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corresponding period to Figure 3 a. Values have been multiplied by four and are on a log10 scale; purple regions indicate likely regions of instability.

4. Conclusions

Whilst the location, geometry and tidal dynamics of Scott Reef make it a particularly unique site, the dynamic parameters Fr and γ make the interpretation of the dynamics comparable to other stratified straits and channels driven by oscillatory, barotropic flows. To our knowledge, this is one of first the studies showing evidence of the formation of an internal undular bore near an offshore island atoll. Quantifying the mixing rate associated with the passage of these bores is important for determining the vertical flux of nutrients and other biological tracers into the reef ecosystem and should form the basis for future studies.

7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011 8

Acknowledgements

The Western Australian Marine Science Institute (WAMSI) and Woodside Energy Ltd jointly funded this study. Computing was performed on the Western Australian Interactive Virtual Environments Centre’s WASP node (iVEC).

References

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