A numerical investigation into the bottom boundary layer flow and vertical structure of internal waves on a continental slope

Continental Shelf Research 18 (1998) 31 — 65

A numerical investigation into the bottom boundary layer flow and vertical structure of internal waves on a continental slope Peter E. Holloway*, Belinda Barnes1 School of Geography and Oceanography, University College, The University of New South Wales, Australian Defence Force Academy, Canberra, ACT 2600, Australia Received 23 April 1996; received in revised form 6 January 1997; accepted 21 May 1997

Abstract A numerical investigation is made into the vertical structure and boundary-layer flow of semi diurnal tidal-period internal waves. The Princeton Ocean primitive equation Model, incorporating the Mellor—Yamada level 2.5 turbulence closure scheme, is used to study the flow and turbulence associated with propagating internal waves in a stratified ocean. Effects of stratification, seabed slope and a background barotropic tide are investigated. Of particular interest is the intensification of currents near the seabed that occurs for critical or near-critical slopes when the topographic slope is comparable to the slope of the internal wave characteristics. In the absence of friction, infinite intensification of currents is predicted near the seabed, but realistic bottom friction and vertical eddy viscosity reduce the bottom currents to about twice the speed of the surface currents. Over critical slopes, maximum current shear and vertical eddy viscosity are achieved. Strong asymmetry occurs between upslope and downslope flows. Downslope flows near the seabed are stronger and have thinner boundary layers than the upslope flow. Observations are presented of semi-diurnal internal tides from around the shelf break on the Australian North West Shelf. Bottom intensification of currents and asymmetry between upslope and downslope flows are observed, in qualitative agreement with the model. ( 1998 Elsevier Science Ltd. All rights reserved

1. Introduction Tidally forced internal waves are a common feature on continental slopes and shelves (Huthnance, 1989). These waves are often energetic and potentially capable of * Corresponding author. 1 Present Address: School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia 0278—4343/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved PII S 0 2 78 — 43 4 3( 9 7 ) 00 0 67 — 8

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enhanced mixing, particularly near the seabed where current shear can be large. The increased momentum mixing can be important for sediment transport (Heathershaw et al., 1987), and possibly enhanced biological productivity by mixing nutrients vertically into the photic zone (the region where light intensity is sufficient for photosynthesis). In order to understand the nature of turbulence associated with these internal waves, it is necessary to describe the structure of the flow in the bottom boundary layer and to estimate the bottom friction and turbulence induced by these waves. Considerable effort has been expended in modelling the current structure and bottom boundary layer flow associated with barotropic tides (Prandle, 1982; Soulsby, 1983; Davies and Jones, 1991; Xing and Davies, 1995, 1996), but few studies have considered internal waves, and many have neglected stratification altogether. However, laboratory experiments have been undertaken to investigate internal waves over critical bottom slopes (Ivey and Nokes, 1989; Taylor, 1993). These studies show strong intensification of near seabed currents with asymmetry in the flow between upslope and downslope directions. Downslope flows exhibit a shallower boundary layer and higher velocity than upslope flows. High levels of turbulence are generated near the boundary. This study aims to describe the vertical structure of tidal period internal waves, in particular the structure in the bottom boundary layer, and to estimate the resulting vertical eddy viscosity and potential for vertical mixing of momentum. A primitive equation model (the Princeton Ocean Model in Blumberg and Mellor (1978)) is used to investigate the vertical structure of tidal period internal waves as they propagate through the model domain. Production of turbulence and the momentum mixing coefficients are described using the Mellor Yamada level 2.5 turbulence closure scheme (Mellor and Yamada, 1982). The study investigates internal waves on both horizontal and sloping seabeds, considers the influence of vertically uniform and non-uniform stratification, and the effects of a barotropic tidal flow. Some effects of bottom friction on internal wave modes have been investigated by Brink (1988) using an analytical model for a horizontal seabed with constant buoyancy frequency and linear bottom friction. Similarly, Craig (1991) included the effects of bottom friction and constant vertical eddy viscosity in a simple model. While these models predict a horizontal decay scale for the waves and a small increase in the horizontal wavenumber as friction and mixing increase, they do not model the structure in the bottom boundary layer and they predict only minor changes to the vertical modal structure compared to inviscid solutions. Wunsch (1978) found an analytic solution for the vertical structure of internal waves in a constant buoyancy frequency ocean over a sloping seabed in the absence of friction. His solutions showed bottom intensification of the currents and production of strong shear, particularly as the seabed slope approached the slope of the internal wave characteristics (described in Section 2). In his model, bottom intensification becomes infinite as the seabed slope approaches that of the characteristics. Friction would prevent this singularity in the solution, and one aim of the present study is to examine the maximum intensification that arises when realistic friction is included. In Section 2 of this paper, the numerical model is described along with an outline of the turbulence model and method for calculating bottom friction. In Section 3, model

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results are presented for a horizontal seabed and, in Section 4, results are presented for a variety of sloping seabeds. In Section 5, observations of internal waves, that show detailed bottom boundary layer flow, are presented from the vicinity of the shelf break on the Australian North West Shelf (NWS). Finally, a discussion of the work is presented in Section 6.

2. The numerical model The bottom boundary layer and vertical structure of propagating internal waves is investigated using the Princeton Ocean Model, a primitive equation, stratified and non-linear numerical model. Features of the model include a sigma-coordinate transformation for the vertical grid; a free surface, allowing for the specification of tides; and a level 2.5 Mellor Yamada turbulence closure scheme (here after MY) used to model the turbulence and to specify the coefficients of vertical mixing of momentum, the vertical eddy viscosity (K ), and eddy diffusivity of heat and salt (K ). The model . ) uses the hydrostatic assumption which is valid for internal wave frequencies low compared to the buoyancy frequency. The model has been used by Holloway (1996) to model generation, propagation and dissipation of internal tides over continental shelf/slope topography. Although the model is fully three dimensional, in the present study there is no variability in the along-shelf direction so that the model is effectively two dimensional. The MY turbulence model is imbedded in the dynamic model and solves two coupled equations for q2 and q2l where q2/2 is the turbulent kinetic energy and l is a mixing length of the turbulence. There is no specification for the mixing length as in lower-order schemes. The mixing model is described by Mellor and Yamada (1982) and is summarised in Appendix A. The model uses a number of empirical constants derived from experiments with density homogeneous flows and this is a potential weakness for applications to stratified flows, as discussed by Kantha and Clayson (1994). However, Simpson et al. (1996) have compared direct measurements of energy dissipation in stratified tidal flows and several of the Mellor Yamada turbulence closure models. The models, using a specified mixing length, performed reasonably in predicting turbulence levels but had some problems diffusing turbulence upwards from the bottom boundary layer. Simpson et al. did not use the level 2.5 scheme. The calculation of bottom stress and specification of the velocity nearest the seabed (u , v ) is an important aspect of the modelling. The two components of the bottom " " stress are defined as Lu Lv q "o K , q "o K "x 0 . Lz "y 0 . Lz

(1)

where the stress can also be calculated from a quadratic friction law q "o C (u2#v2 )1@2 u , q "o C (u2#v2)1@2 v . " " " " "y 0 $ " "x 0 $ "

(2)

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the drag coefficient is the maximum of 0.0025 and

G

H

i 2 , C" $ ln (h/z ) 0

(3)

z is the roughness height, and h is the distance between the seabed and the grid point 0 where (u , v ) are calculated. A simple empirical relationship (z (m)"0.01#1/H) is " " 0 used to estimate z that takes account of increased roughness for shallow water where 0 wave action is increased. For example, when H"200 m, z "1.5 cm and when 0 H"100 m, z "2 cm. Equating Eqs. (1) and (2) gives 0 Lv Lu K "C (u2#v2)1@2 u , K "C (u2#v2)1@2 v " " . Lz " " . Lz $ " $ "

(4)

which are solved implicitly, using Eq. (3), to find (u ,v ). These bottom velocities " " are calculated at the second to bottom grid point which is half a grid cell above the seabed. Flow at the bottom grid point, half a grid cell below the seabed, is set to zero. The numerical model is run with zero gradients in the y-direction (alongshore component) and thus results are only functions of x, z and t. The eastern boundary is a coast, with zero flow across the boundary. The northern and southern boundaries are open with zero gradient conditions on all variables. In most model runs, the run is terminated prior to the internal waves propagating to the eastern boundary so that wave reflection is not a concern. Forcing is applied at the open western boundary and is a combination of oscillating surface elevation and/or an internal wave modal structure for the onshore (x-component) velocity. In the case of barotropic forcing, the elevation at the boundary is specified as f" f sin(ut), where f is the amplitude and u the frequency, representing a sinusoidal 0 0 semi-diurnal tide. Also at the boundary, the total velocity is set equal to the depth averaged flow, i.e. there is zero baroclinic flow, and temperatures are held constant at their initial values. Further, a sponge is applied to the baroclinic velocity, temperature and salinity so that the solution is relaxed to boundary values and any internal waves are damped out as they propagate across the western boundary. The sponge layer is 5 grid points wide with a stretched grid from 3.5 km at the boundary reducing to 3.0, 2.5, 2.0, 1.5 and to 1 km out of the sponge layer. The sponge is not applied to the depth averaged velocity or elevation. In some runs, an internal wave is specified at the boundary and allowed to propagate into the model domain. In this case, the onshore velocity is given by the vertical modal function for the internal wave in an inviscid ocean of constant depth and the flow is sinusoidal in time. No sponge layers are imposed as these would damp the internal waves that are being propagated into the domain. The modal function is obtained from the solution of the linear internal wave equation,

G

H

d2f N2(z)!u2 #k2 f"0 dz2 u2!f2

(5)

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with f"0 at z"0 and at z"!H and where u df u" , k dz

(6)

u is the wave frequency and f the Coriolis parameter. In some runs, the propagation of an internal wave through the boundary is combined with surface elevation forcing. In all runs, a radiation condition is applied to the depth averaged onshore velocity (u6 ), at the western boundary, of the form Lu6 Lu6 !c "0 Lt Lx

(7)

where c"JgH is the propagation speed of long gravity waves, g gravitational acceleration and H the water depth on the boundary, t is time and x the onshore spatial coordinate. An upstream advection scheme is used to set the values of the alongshore velocity component, as well as the depth averaged alongshore velocity component, on the western boundary. The grid spacing for the x-coordinate has a resolution of 1 km in the region of interest away from boundaries, i.e. on the slope in the cases of sloping bathymetry, and a stretched grid of up to 5 km on the coastal side of the domain. Internal waves propagate eastwards through the central part of the model domain from which model output is analysed. A non-uniform vertical grid with 61 levels is used giving fine resolution near the seabed. Layer thicknesses are, for 100 m water depth, 0.2 m in the lowest two layers increasing to 0.4, 0.9 and 1.8 m thereafter.

3. Internal waves over a uniform shelf 3.1. Internal wave forcing The first model runs consider an ocean of constant depth (H"100 m) with a linear temperature gradient from 30°C at the surface to 20°C at 100 m giving rise to an approximately constant buoyancy frequency of 0.016 s~1, representative of summer stratification on the NWS. The model is forced by specifying a first mode internal wave at the western boundary and allowing the wave to propagate into the model domain. For a constant buoyancy frequency, Eqs. (5) and (6) give the horizontal velocity as u (z, t)"u cos (nz/H) sin (ut) (8) 0 where the amplitude is given a representative value of u "0.3 m s~1 and the M tidal 0 2 period (u"1.41]10~4 s~1) is used. In all runs, the Coriolis parameter is set to f"!5.0]10~5 s~1, corresponding to a latitude for the NWS of 20°S. The model is run for 3 d, with forcing spun-up from zero over the first 24 h. Results are considered from day 3. The waves are damped as they propagate through the domain, due to

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bottom friction and vertical mixing of momentum. Results are considered at a mid-model grid point when the boundary layer has fully developed but the wave is not too weak. The input modal function, which has a maximum in velocity at the seabed, is modified as the wave propagates through the model domain to produce a velocity profile with a region of strong shear near the seabed, as given by Eq. (4). The nature of this boundary layer flow depends on the specification of the vertical eddy viscosity. A time sequence of velocity profiles over a wave period from the model using the standard MY scheme are plotted in Fig. 1 along with profiles of the coefficient of vertical mixing of momentum (K ). An approximate cosine distribution of velocity . with depth with a zero crossing and change in phase at mid-depth is seen, although the depth of zero velocity changes with the phase of the wave due to its finite amplitude.

Fig. 1. Vertical profiles of velocity and of vertical eddy viscosity over a wave period for a constant depth of H"100 m, internal wave forcing and using the MY turbulence closure scheme.

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The velocity reaches a maximum at about 10 m above the seabed, then decreases to zero through the boundary layer. The transition at the top of the boundary layer is discontinuous rather than smooth, particularly at times of maximum flow. Vertical mixing of momentum is restricted to the bottom 5—12 m, depending on the phase of the flow, and is shut off at the top of the boundary layer where buoyancy forces are too large for shear to continue vertical mixing. The discontinuous nature of the velocity profile is a result of the shut-off of K . . The sensitivity of the velocity profiles to the specification of K is investigated by . making model runs with K and K set to zero and then to constant values of . ) K "2]10~3 and K "3]10~3 m2 s~1 (Fig. 2), values similar to those found in the . ) above model runs. These runs provide two extreme cases for specification of vertical mixing since there are many alternative parametrisations for K that could be used .

Fig. 2. Vertical profiles of velocity over a wave period for a constant depth of H"100 m, internal wave forcing and using K "0"K and using constant values of K "2]10~3 and K "3]10~3 m2 s~1. . ) . )

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(Nunes Vaz and Simpson, 1994). With zero mixing, there is only a very thin boundary layer (1—2 m) with extreme shear at the seabed. In contrast, the finite K and . K produce a smooth rounded velocity profile with a deep boundary layer, 15—20 m ) thick. There is also much less dissipation of the signal with K and K equal to zero. . ) The constant K removes the discontinuity in the velocity. . The final specification of vertical mixing is based on the MY scheme but with a weak (compared to the peak value of approximately 3]10~3 m2 s~1) background value of K "0.1]10~3 m2 s~1 added throughout the water depth and . for all time. The velocity profiles (Fig. 3) are only slightly modified from the MY scheme (Fig. 1) but the transition at the top of the boundary layer is smoother. The extent of this smoothing depends on the magnitude of the background value. In all subsequent model runs, this background value is added to the MY computed K . . Evident in all of the results is asymmetry in the velocity profiles with time where the negative flow (opposing the direction of wave propagation) is stronger at the seabed than the positive flow (in the direction of wave propagation). This is due to weak non-linearity in the flow, arising from the wave’s finite amplitude. The waveform is distorted to have long flat crests and short deeper troughs. 3.2. Barotropic tidal forcing Internal tides on continental shelves and slopes interact with the barotropic (surface) tide. One consequence of this is increased turbulence beyond that generated by the internal tide alone. Before looking at the combined effects of barotropic and internal tides, the structure of the barotropic tide in the presence of stratification is investigated.

Fig. 3. Vertical profiles of velocity over a wave period for a constant depth of H"100 m, internal wave forcing and using the MY turbulence closure scheme plus a background value of 0.1]10~3 m2 s~1.

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The model is forced with a 1.0 m amplitude M period surface elevation at the 2 western end. This produces barotropic currents of approximately 0.2 m s~1 in the middle of the model domain. Fig. 4 shows modelled profiles of velocity and eddy viscosity (K ). A 10—15 m thick boundary layer with some overshoot of the velocity at . the top of the boundary layer occurs. During weak flow (slack water), flow within the boundary layer is stronger than, and in the opposite direction to, flow above the boundary layer. Maximum velocity occurs at about 4 m height. At all times, flow above the boundary layer exhibits little or no shear. The maximum in K . is around 7]10~3 m2 s~1, twice the values found for internal waves. The velocity profile through the boundary layer is not as smooth and continuous as shown by models of the tidal boundary layer in unstratified water (Davies and Jones,

Fig. 4. Vertical profiles of velocity and of vertical eddy viscosity over a wave period for a constant depth of H"100 m, barotropic tidal forcing and using the MY turbulence closure scheme plus a background value of 0.1]10~3 m2 s~1.

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1991). Without stratification, K is large and non-zero throughout the water column . giving rise to a thick boundary layer, extending tens of meters, and a smooth velocity profile. 3.3. Combined barotropic tide and internal wave forcing The barotropic tide provides an oscillatory flow that is very close to constant in phase across the model domain. In contrast, the internal wave has several wavelengths across the domain so results from different locations exhibit different phase relations between the barotropic and baroclinic currents. Fig. 5 shows vertical profiles of velocity (component flowing across the model domain), K and temperature at . a location where the barotropic currents are in phase with the near-surface internal wave currents and Fig. 6 shows profiles of variables 12 km away where the barotropic velocity is in phase with the near-bottom internal wave currents. When the currents are out of phase at the seabed, weak total bottom only currents and a small K (1]10~3 m2 s~1) result. When currents are in phase, strong seabed currents and . larger K (7]10~3 m2 s~1) result. Temperature profiles (Figs. 5 and 6) show vertical . movement of the stratification in the middle of the water column but, in the bottom boundary layer, stratification is invariant so that variability in K is determined by . variability in the velocity shear alone. The thickness of the boundary layer varies according to the level of turbulence, increasing for an increase in near bottom velocities. 3.4. Internal wave forcing and non-uniform stratification The situations considered above, with constant buoyancy frequency, produce a symmetrical velocity distribution with depth in an inviscid fluid. Non-uniform stratification produces vertical asymmetry in the velocity profiles. Background shear flow, not considered in this study, also has this effect. If the maximum buoyancy frequency is in the lower half of the water column then, for an inviscid ocean, maximum currents will occur at the seabed. In order to investigate the interaction of this asymmetric modal structure with the bottom boundary layer, the model is run with internal wave forcing for an ocean of constant depth and a temperature profile that has a weak gradient through the upper 50 m, becoming stronger towards the seabed (Fig. 7). Fig. 8 shows the time sequence of profiles of modelled velocity, K and temperature. The zero crossing in the velocity is at 60 m with the . maximum in the currents about 5 m above the seabed double that of the surface currents. 3.5. Modal functions In addition to time sequences of velocity profiles, amplitude and phase functions are calculated in order to compare model results with inviscid solutions. Model profile time series of horizontal and vertical velocities from the last 24 h of the model run are

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Fig. 5. Vertical profiles of velocity (º), vertical eddy viscosity (K ) and temperature (¹ ) at grid point . I"40 over a wave period for a constant depth of H"100 m, with combined barotropic tide and internal wave forcing and using the MY turbulence closure scheme plus a background value of 0.1]10~3 m2 s~1. Barotropic tidal currents are in phase with the internal wave currents at the surface.

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Fig. 6. Vertical profiles of velocity (º), vertical eddy viscosity (K ) and temperature (¹ ) at grid point . I"52 over a wave period for a constant depth of H"100 m, with combined barotropic tide and internal wave forcing and using the MY turbulence closure scheme plus a background value of 0.1]10~3 m2 s~1. Barotropic tidal currents are in phase with the internal wave currents near the seabed.

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Fig. 7. Vertical profiles of input density and corresponding buoyancy frequency for model run with internal wave forcing and non-uniform stratification.

subject to tidal harmonic analysis (Foreman, 1978). Vertical displacement amplitudes are then computed from vertical velocity (w) as DfD"DwD/u, phase is 90° plus the phase of w, and amplitudes are normalised using the surface value of horizontal velocity and the maximum value of vertical displacement. Fig. 9a shows resulting vertical profiles for the model with uniform stratification as well as inviscid solutions, and Fig. 9b shows the corresponding results for non-uniform stratification. For uniform stratification, there is little difference in the elevation distributions for the inviscid and model solutions, but the phase decreases by about 20° in the bottom boundary layer where it is uniform for the inviscid solution. Velocity profiles are similar except near the boundary layer where there are significant differences in amplitude and flow in the boundary layer leads that immediately above the boundary layer. With the nonuniform stratification, the differences are more pronounced. The point of maximum elevation, and minimum horizontal current, is raised from 35 m above the seabed for the inviscid solution to 40 m for the model solution. The velocity profile is significantly altered throughout the lower half of the water column by viscosity and bottom friction. In both stratification cases, the numerical model shows a gradual change in phase around the depth of zero velocity whereas the inviscid model predicts a discontinuity in phase of 180°.

4. Internal waves over a sloping seabed In this section, the effects of a sloping seabed on the boundary layer flow and vertical structure of internal waves are investigated. The model domain consists of a 200 m deep ocean where the internal wave is propagated through the domain, followed by a linear slope to 40 m depth and a constant depth shelf. The slope is varied

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Fig. 8. Time sequence of vertical profiles of velocity (º), vertical eddy viscosity (K ) and temperature (¹ ) . for model with internal wave forcing, constant depth of H"100 m and non-uniform stratification.

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Fig. 9. Normalised amplitude and phase of horizontal currents (DºD, º ) and vertical displacements (DfD,f ) ' ' for model with internal wave forcing, constant depth of H"100 m and (a) uniform and (b) non-uniform stratification. Also shown (by dashed lines) are the modal solutions for amplitude and phase neglecting bottom friction and eddy viscosity.

for different runs. A horizontal eddy viscosity is added to provide some weak smoothing. The magnitude of the sloping seabed is measured in terms of the ratio of the seabed slope (a"dh/dx) to the slope of the internal wave characteristics, given as

G

s"$

H

u2!f 2 1@2 . N2!u2

(9)

Subcritical (a/s(1), critical (a/s"1) and supercritical (a/s'1) seabed slopes are each considered. The model is run for uniform stratification giving an approximately constant buoyancy frequency of 0.0016 s~1, although this value is modified to some extent by mixing during the model runs. With semi-diurnal frequency u"1.41]10~4 s~1 and f"!5]10~5 s~1, s"$8.2]10~3. 4.1. Subcritical seabed slope Water depth changes by 160 m over 20 km giving a"4.6]10~3 and hence a/s"0.6. Fig. 10 shows a section over the slope of vertical displacement amplitude

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Fig. 10. Section of vertical displacement amplitude and phase and onshore current amplitude and phase for internal wave forcing and a subcritical seabed slope, a/s"0.6.

and phase and onshore current amplitude and phase, calculated from tidal analysis of the model output time series. The structure is similar to that for a horizontal seabed, except ‘‘squashed’’ to accommodate shallow water as the wave propagates up the slope. Phase plots show wave propagation towards shallower water with some downward migration of phase over the slope region. The small step like features in phase are a result of the horizontal resolution. The point of maximum vertical displacement and the zero in horizontal velocity remain approximately at mid-depth. However, vertical profiles of amplitude functions, plotted in Fig. 11 halfway along the slope (water depth of 120 m), show some distortion from the horizontal seabed modal solution. There is a gradual increase in phase lag in the vertical displacements from the surface toward the seabed and the velocity shows weak bottom intensification. The approximately 180° phase change with depth is a gradual change rather than instantaneous. There is no zero in the amplitude of the horizontal velocity. A time sequence of velocity profiles are plotted in Fig. 12 at 120 m water depth from over a tidal cycle along with the profiles of K and temperature. There is some . asymmetry in the profiles with downslope velocity stronger but thinner than upslope. There is also stronger shear associated with the downslope velocity than upslope.

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Fig. 11. Vertical profiles of vertical displacement and onshore current amplitude and phase at 120 m depth for internal wave forcing and a subcritical seabed slope, a/s"0.6.

However, the vertical eddy viscosity is stronger during the upslope flow with a maximum of 25]10~3 m2 s~1 about 5 m above the seabed. During upslope flow, stratification is weaker near the seabed, from advection of cold water up the slope. Hence shear produces a larger K . The features are also seen in time series, over . 2 tidal cycles, of temperature, velocity and K at 1.5, 5.4 and 9.7 m above the seabed . (Fig. 13). There is weak asymmetry in the velocity time series with downslope flow slightly stronger than upslope. At times of upslope flow, the water is homogeneous in the lower 10 m while it is stratified at other times. Strong vertical mixing (large K ) is seen at 5.4 m above the seabed only at times of weaker upslope flow. At . other phases of the wave period, mixing at this depth shuts off. Close to the seabed (1.5 m) shear is strong and mixing occurs at times of maximum up- and downslope flows. 4.2. Critical seabed slope Depth changes by 160 m over 35 km giving a"8.0]10~3 and a/s"1.0. Fig. 14 shows a section of vertical displacement amplitude and phase and onshore current

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Fig. 12. Time sequence of vertical profiles of onshore currents, K and temperature at 120 m water depth . for a subcritical seabed slope, a/s"0.6.

amplitude and phase, over the slope. The signal is strongly distorted over the slope with strong intensification of both vertical displacements and cross-slope currents near the seabed. The intensification is consistent with the idea of critical slopes and a resonance of the signal along the seabed. Analytic solutions for an inviscid fluid with internal waves over a slope (Wunsch, 1978) give a singularity in the velocity profile for

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Fig. 13. Time series, over 2 wave periods, of vertical eddy viscosity (K ), velocity (º) and temperature (¹ ) . computed at 1.5, 5.4 and 9.7 m above the seabed. Results are for the sub-critical slope, a/s "0.6.

a critical slope, with infinite intensification of bottom currents. However, friction and vertical mixing keep the intensification finite. Vertical profiles of the amplitude functions, at 120 m water depth (Fig. 15), show maximum flow about 15 m above the seabed approximately double the surface current. This ratio of maximum surface to bottom currents is approximately constant along the slope, suggesting this as a maximum bottom intensification, for this stratification, the degree limited by bottom friction and vertical mixing of momentum induced by the strong shear in both the bottom boundary layer and the fluid above the boundary layer. Instantaneous velocity profiles are plotted in Fig. 16 at 120 m water depth from over a tidal cycle. Strong shear and bottom intensification of the currents are evident, with a strong asymmetry between upslope and downslope flow near the seabed. The

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Fig. 14. Section of vertical displacement amplitude and phase and onshore current amplitude and phase for internal wave forcing and a critical seabed slope, a/s"1.0.

downslope flow reaches about 0.55 m s~1 approximately 10 m above the seabed and shows a concave profile with very strong shear above and below the depth of maximum flow. The upslope flow near the seabed is weaker, reaching a maximum of 0.35 m s~1 with a thicker and less elongated velocity profile than the downslope flow. Intermediate profiles show multiple flow reversals and strong shear regions in the lower part of the water column. Profiles of model eddy viscosity (K ) and temperature . are also plotted in Fig. 16. Non-zero K are confined to the lower 10—20 m with . maxima in (27—35)]10~3 m2 s~1 occurring at times of maximum upslope and downslope flow when shear is strongest. The corresponding temperature profiles show large variations over a tidal cycle. The up-slope flow advects cold deep water up the slope creating a 15 m thick well-mixed bottom layer beneath a strong linearly stratified temperature profile. The down-slope flow has a thinner (2 m) bottom mixed-layer beneath a strongly stratified layer. Time series of temperature, velocity and K over 2 tidal cycles at 1.4, 5.1 and 11 m . above the seabed are plotted in Fig. 17. Strong asymmetry is seen in the velocities with downslope flow approximately twice the strength of the upslope flow. The phase lag between velocities is also evident with the flow closest to the seabed leading (this is consistent with models of tidal flow, e.g. Soulsby (1983). Mixing is stronger than in the

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Fig. 15. Vertical profiles of vertical displacement and onshore current amplitude and phase at 120 m depth for internal wave forcing and a critical seabed slope, a/s"1.0.

subcritical case. The water is homogeneous in the lower 5 m most of the time (note a slight difference between temperature and hence K from the first and second wave . periods). Again, strong vertical mixing occurs during upslope flow at 5.1 m above the seabed and during downslope flow, at least during the first tidal cycle. At 11 m, shear is too weak to produce significant K . . 4.3. Supercritical seabed slope Water depth changes by 160 m over 10 km giving a"16.0]10~3 and a/s"2.0. Fig. 18 shows a section over the slope of vertical displacement amplitude and phase and onshore current amplitude and phase. Compared to the critical case, vertical displacement amplitudes are larger and a reflected signal off the slope is expected. The currents are again intensified near the seabed but to a lesser extent than for the critical slope. Maximum currents are about 1.5 times the surface currents. These properties are also seen in the amplitude functions of elevation and current (Fig. 19). Bottom currents lead the surface flow by approximately 240°. This is the reverse of the critical slope case where the surface currents lead. The elevations, although strongly intensified near the seabed, are nearly constant in phase with depth. The elevation amplitude

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Fig. 16. Time sequence of vertical profiles of onshore currents, K and temperature at 120 m water depth . for a critical seabed slope, a/s"1.0.

increases almost linearly with depth from zero at the surface to a maximum at the top of the boundary layer. The maximum in elevation over a steeply sloping seabed is primarily determined by the vertical component of velocity parallel to the seabed. Near the seabed, the vertical velocity is given as w "u dh/dx where u is the horizontal velocity near the seabed " " " and, for sinusoidal motion, the vertical displacement near the seabed will be

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53

Fig. 17. Time series, over 2 wave periods, of vertical eddy viscosity (K ), velocity (º) and temperature (¹ ) . computed at 1.4, 5.1 and 11 m above the seabed. Results are for the critical slope, a/s"1.0.

f "(u /u) dh/dx. The maximum in velocity of 0.26 m s~1 for the supercritical slope " " then gives f "30 m, in agreement with the model value (Fig. 19). Similar agreement is " found for the maxima in u and f for the critical slope. Asymmetry in the upslope to downslope flow near the seabed is seen in the time sequence of velocity profiles (Fig. 20). Maximum downslope flow near the seabed is approximately 1.5 times maximum upslope flow. In addition, the shape of the upslope and downslope velocity profiles are quite different. During maximum downslope flow near the seabed, the boundary layer is very thin, approximately 5 m, and there is a near linear gradient in the velocity in the lower half of the water column and above the boundary layer. During maximum upslope flow near the seabed, the velocity profile is more rounded with the maximum about 15 m above the seabed. However, it

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Fig. 18. Section of vertical displacement amplitude and phase and onshore current amplitude and phase for internal wave forcing and a super-critical seabed slope, a/s"2.0.

is during upslope flow that eddy viscosity (K ) is a maximum (Fig. 20). Temperature . profiles show that, during upslope flow, a deeper mixed-layer forms at the seabed, allowing K to become larger than during downslope flow when there is virtually no . bottom mixed layer and the strong stratification suppresses the turbulence. Time series of temperature, velocity and K over 2 tidal cycles and at 1.4, 5.3 and 9.5 m . above the seabed are plotted in Fig. 21. Distortion of the velocity and temperature time series from sinusoidal is stronger than for subcritical and critical slopes. Again, vertical mixing is strongest 5 m above the seabed, reaching values of 40]10~3 m2 s~1 at times of maximum upslope flow. The mixing comes in short 2 h pulses. Close to the seabed (1.4 m), temporal variability in K resembles that for the other seabed slopes. . At 9.5 m, K is zero during most of the wave period. . 5. Internal wave observations Moored current meter observations are analysed to resolve the vertical currentstructure of semi-diurnal internal waves. Observations are from 19 current meters on 3 moorings around the shelf break on the NWS during February to April 1992

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Fig. 19. Vertical profiles of vertical displacement and onshore current amplitude and phase at 120 m depth for internal wave forcing and a super-critical seabed slope, a/s"2.0.

(Fig. 22). Current meters were set at 2.5, 5.0 and 10 m above the seabed as well as through the rest of the water column. Most instruments were Steedman Acoustic current meters sampling 2 min vector averages of currents with 4 InterOcean S4 electromagnetic current meters sampling 6 min vector averages. All instruments measured temperature. The observations are obtained from the shelf break where the steepest topography is at Slope with a gradient of a"0.0035. Stratification during the mooring deployment varies slightly with time but is characterised by a buoyancy frequency of 0.013 s~1 giving a characteristic slope of s"0.010. Then a/s"0.34, a value well below critical. However critical slopes are found further down the continental slope at depths of around 125 and 350 m and possibly deeper (Holloway, 1996). There are several problems in resolving the internal tide in the measurements. As a first approximation, the barotropic flow can be defined as the depth averaged current and the baroclinic current the residual, assuming the two components are linearly superimposed. However, within the bottom boundary layer, it is not possible to resolve barotropic and baroclinic contributions without knowing how the flows interact. In order to analyse the observations, depth average flows are calculated neglecting measurements at 2.5 and 5.0 m above seabed so as not to bias the averages

56

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Fig. 20. Time sequence of vertical profiles of onshore currents, K and temperature at 120 m water depth . for super-critical seabed slope, a/s"2.0.

toward bottom values. With the close proximity of the Break and Shelf moorings (only 1500 m apart) and the expectation that the barotropic tide will be nearly constant over this region, a common depth average is calculated using the upper 4 current meters from both locations. Baroclinic currents are then computed at all measurement locations, including 2.5 and 5 m above seabed, as the difference between total and depth averaged currents. These calculations are performed on the

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Fig. 21. Time series, over 2 wave periods, of vertical eddy viscosity (K ), velocity (º) and temperature (¹ ) . computed at 1.4, 5.3 and 9.5 m above the seabed. Results are for the super-critical slope, a/s"2.0.

cross-shelf velocity component with data reduced to hourly sampling using a running mean filter. The baroclinic time series are complex demodulated with a modulation period of 12.42 h. This identifies the amplitude and relative phase of the internal tides over a broad semi-diurnal period as a function of time and at each measurement depth. This is an alternative to using tidal harmonic analysis where the signal is resolved at specific tidal frequencies. Vertical profiles of the resulting amplitude and phase from each location at two alternating Spring and Neap tides, approximately 7 d apart, are plotted in Fig. 23. In general, velocity profiles show maxima near the surface and seabed with a change in phase of approximately 180° from top to bottom, as would be expected. Maximum currents are around 0.3 m s~1. A number of profiles show strong

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Fig. 22. Bathymetric cross-section showing mooring and instrument locations for the observations. Current meters were Steedman Acoustic current meters (d) or InterOcean S4 electromagnetic current meters (j).

bottom intensification of the baroclinic currents in the bottom two or three current meters (2.5, 5.0 and 10.0 m above the seabed). When the intensification is observed, it is seen at all three moorings. The most pronounced example, on 7 March, shows a maxima at 5.0 m above the seabed at the central Break mooring with maxima at 2.5 m at the other moorings. This suggests a very thin boundary layer, strong shear and turbulence. Similar examples are seen on 20 March and 5 April. A sequence of measured velocity and temperature time series at 2.5, 5 and 10 m above the seabed are shown for the Slope location in Fig. 24 where records have been low-pass filtered to remove oscillations with periods shorter than 2 h. Time series show strong asymmetry between upslope and downslope flow with near uniform flow speeds at each depth around times of maximum upslope flow and strong shear during downslope flow. Maximum upslope current is weaker than maximum downslope current. Temperature in the bottom 10 m is close to homogeneous during falling temperatures while the layer stratifies during rising temperatures. These features are consistent with the model predictions, e.g. Fig. 17. A feature of these observations not seen in the model is that, during downslope flow, the flow at 5 m is weaker than at 2.5 and 10 m.

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Fig. 23. Vertical profiles of semi-diurnal internal tide amplitude (DºD) and phase (º ) at locations Slope, ' Break and Shelf. Profiles are plotted approximately every 7 d at alternating Spring and Neap tides. Results are from the complex demodulation of the baroclinic current time series.

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Fig. 24. Time series of onshore currents and temperatures at 2.5, 5 and 10 m above the sea bed from the Slope location over a 2 d period in 1992. Records have been low pass filtered to remove oscillations with periods shorter than 2 h.

6. Discussion In this paper a finite-difference, primitive equation, numerical model has been used to investigate the vertical structure of the velocity and vertical displacements of tidal period internal waves under the influence of bottom friction and turbulence. The dependence of internal wave structure on the bottom slope, vertical stratification and background barotropic flow have been considered. The production and dissipation of turbulence is modelled using a Mellor Yamada level 2.5 turbulence closure scheme which determines a temporally and spatially varying vertical eddy viscosity depending on the current shear and buoyancy forces. While this model has been widely used for modelling barotropic tidal flows in homogeneous oceans, few applications have considered the effects of stratification. In most parts of the water column, the buoyancy forces are too strong and the current shear too weak for the model to produce non-zero vertical eddy viscosities. The value of K is only non-zero in the . bottom boundary layer and then only during part of the internal wave period. The turbulence model does not predict mixing of momentum throughout the water column and this may be a weakness in the model. At the top of the boundary layer where K becomes zero, a sharp gradient . or discontinuity is produced in the velocity profile. This feature does not seem physically reasonable and appears to be a failure in the turbulent closure scheme. In order to overcome this problem, a weak background value is added to the computed K . This tends to slightly smooth the velocity profile in the region at the . top of the bottom boundary layer without significantly affecting the rest of the velocity profile. Internal waves on a sloping seabed are characterised by bottom intensification of currents, the production of strong current shears and strong vertical mixing. This is

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Fig. 25. Sections of temperature for a critical seabed slope (a/s"1.0) plotted at times corresponding to the end of upslope and downslope flows.

most pronounced for critical bottom slopes where the seabed slope equals that of the internal wave characteristics. In this case, maximum currents (about double the surface currents) are found at 5 m above the seabed. There is strong asymmetry in flows over a tidal period with downslope flow near the seabed much stronger than upslope flow. During downslope flow, the boundary layer is thinner than during upslope and the velocity profile more elongated. During upslope flow, the velocity profile is rounded. The density stratification is advected upslope and downslope by the currents. This results in a thick near-homogeneous bottom boundary layer of cold water during upslope flow and a strongly stratified layer all the way to the seabed during

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downslope flow. As a result, production of turbulent energy and vertical mixing of momentum is strongest during upslope flow when the shear is moderate but stratification is weak. During downslope flow, shear is strongest but stratification resulting from the advection of warm water down slope is also strong and little turbulent energy is generated by the turbulence closure model. The structure of the stratification in the boundary layer is seen in the temperature sections (Fig. 25) plotted at times corresponding to the end of upslope and downslope flows. At the end of upslope flow a deep temperature-homogeneous boundary layer is seen. However, at the end of downslope flow, a thin strongly stratified layer is seen near the seabed beneath a thick homogeneous layer. This appears to result from the fluid close to the seabed being advected downslope more slowly than the fluid above. For subcritical and supercritical slopes, there is only weak vertical mixing during downslope flow and this is confined within 1 m of the seabed. At times around peak upslope flow K reaches a maximum. For critical slopes, shear is sufficiently strong . during upslope and downslope flows to create significant turbulent energy for vertical momentum mixing. Non-zero eddy viscosity (K ) comes in short intense bursts rather . than being continuous over a wave period. Momentum mixing is the most intense for supercritical slopes but only persists for a small fraction of a wave period during upslope flow. For critical and subcritical seabed slopes, momentum mixing is less intense but persists for a larger fraction of a wave period. The results have interesting implications for the re-suspension and transport of sediment by internal waves. The predominance of strongest momentum mixing during upslope flow near the seabed suggests that maximum re-suspension of sediment would occur at this time and that net transport of sediment would be upslope, although this would depend on sediment settling rates. It is only for critical slopes that strong mixing is likely to occur during both upslope and downslope flows. Note that there will always be a frequency that corresponds to internal wave characteristics at the critical slope.

Acknowledgements The data described in this paper was collected by Steedman Science and Engineering Pty. Ltd. as part of an Australian Research Council funded project with Ray Steedman and Chris Fandry. Steve Buchan was responsible for much of the data collection and organisation. Support for the work described in this paper was provided by an Australian Research Council Small Grant. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.

Appendix A The second-moment turbulence closure equations (Mellor and Yamada, 1982; Blumberg and Mellor, 1987) describe q2 and q2l where q2/2 is the turbulent kinetic

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63

energy and l is a mixing length of the turbulence which represents the large scale eddies associated with the turbulence. These equations, neglecting gradient terms in the along-bathymetry (y) direction, are

C

D

Lq2 Lq2 Lq2 L Lq2 #u #¼ " K #2K q Lz . Lt dx dz Lz

CA B A B D C D Lu 2 Lv 2 # Lz Lz

2g Lo 2q3 L Lq2 # K ! # A o ) Lz B l Lx ) Lx 0 1

(A.1)

and

C

D

Lq2 l Lq2l Lq2l L Lq2l #u #¼ " K #l E K 2 Lz 1 . Lt dx dz Lz

CA B A B D C D Lu 2 Lv 2 # Lz Lz

lE g Lo q3 L Lq2l # 1 K ! ¼# A ) ) o Lz B Lx Lx 0 1 where ¼ is the wall proximity function defined as

G H

l 2 ¼"1#E 2 i¸ where i"0.4 is the von Karman constant, E and E are constants and 1 2 1 1 1 " # . ¸ f!z H#z

(A.2)

(A.3)

(A.4)

In these equations, u and v are the x and y (horizontal) velocity components, f is the free surface, t is time, K is the vertical mixing coefficient for turbulent kinetic energy, 2 g is gravitational acceleration, o the density and o a reference density, B is 0 1 a constant and A is the coefficient of horizontal eddy diffusion. It is possible to define ) stability functions S , S and S from . ) 2 S (6A A G )#S (1!2A B G !12A A G )"A (A.5) . 1 2 . ) 2 2 ) 1 2 ) 2 S (1#6A 2G !9A A G )!S (12A 2G #9A A G )"A (1!3C ) (A.6) . 1 . 1 2 ) ) 1 ) 1 2 ) 1 1 S "0.20 (A.7) 2 where l2 G " . q2

CA B A B D

Lu 2 Lv 2 1@2 # Lz Lz

(A.8)

and l2 g Lo G" ) q2 o Lz 0

(A.9)

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The stability functions have the effect of damping the turbulence and are functions of the Richardson Number. At a critical Richardson Number of 0.19, S "0"S . The . ) mixing coefficients are then defined as K "lqS , K "lqS and K "lqS and the . . ) ) 2 2 various constants were empirically determined by Mellor and Yamada (1982), from observations of unstratified flows, as (A , A , B , B , C , E , E )"(0.92, 0.74, 16.6, 1 2 1 2 2 1 2 10.1, 0.08, 1.8, 1.33). Given the velocities (u, v) and density at a particular time step from the dynamical model, Eqs. (A.1)—(A.9) are solved to define K and K at the new . ) time step.

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Simpson, J. H., Crawford, W. R., Rippeth, T. P., Campbell, A. R., Cheok, J. V. S., 1996. The vertical structure of turbulent dissipation in shelf seas. Journal of Physical Oceanography 26, 1579—1590. Soulsby, R. L., 1983. The bottom boundary layer of shelf seas. In: Johns, B. (Ed.), Physical Oceanography of Shelf Seas. Elsevier, Amsterdam, pp. 189—266. Taylor, J. R., 1993. Turbulence and mixing in the boundary layer generated by shoaling internal waves. Dynamics of Atmosphere and Oceans 19, 233—258. Wunsch, C., 1978. On the propagation of internal waves up a slope. Deep-Sea Research 15, 251—258. Xing, J., Davies, A. M., 1995. Application of three dimensional turbulence energy models to the determination of tidal mixing and currents in a shallow sea. Progression Oceanography 35, 153—205. Xing, J., Davies, A. M., 1996. Application of a range of turbulence energy models to the determination of M tidal current profiles. Continental Shelf Research 16(4), 517—547. 4