ARTICLE IN PRESS. Continental Shelf Research

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1 Continental Shelf Research 28 (28) Contents lists available at ScienceDirect Continental Shelf Research journal homepage: Numerical studies of internal waves at a sill: Sensitivity to horizontal grid size and subgrid scale closure Jarle Berntsen a,, Jiuxing Xing b, Alan M. Davies b a Department of Mathematics, University of Bergen, Johannes Brunsgate 12, N-8 Bergen, Norway b Proudman Oceanographic Laboratory, Joseph Proudman Building, 6 Brownlow Street, Liverpool L DA, UK article info Article history: Received December 27 Received in revised form March 28 Accepted 1 March 28 Available online 8 April 28 Keywords: Non-hydrostatic Ocean modelling Tides Sills Internal waves Grid resolution Mixing abstract A non-hydrostatic terrain following model in cross sectional form is applied to study the generation and propagation of tidally forced internal waves near a sill in an idealized loch. On inflow internal waves are generated behind the sill, and they propagate from the sill. There is a transfer of energy from the barotropic tide to internal waves and then to irreversible mixing. The range of length scales involved goes from the scale of the forced tide to scales associated with wave breaking. This wide range of scales makes it very difficult to resolve all relevant length scales with a numerical model using a uniform finite difference grid. The sensitivity of the numerical results to the grid resolution and the parameterisation of subgrid scale mixing is investigated. Initial calculations are performed with constant values of viscosity and diffusivity, and subsequently they are repeated using large eddy simulations and including Richardson number dependant vertical mixing. In the studies with sub-critical flow over the sill, the results are robust to changes in the grid size. With stronger forcing, the flow over the sill becomes super-critical during maximum inflow, and the frequency and the nature of the internal waves generated in the lee of the sill becomes grid size dependant. For the super-critical case, the solution is sensitive to the subgrid scale closure scheme. Calculations show that with constant and small values of vertical viscosity and diffusivity, strong internal waves are generated on inflow in the lee of the sill. When using Richardson number dependant vertical viscosity and diffusivity, more of the tidal energy goes into vertical irreversible mixing, and less into internal waves. & 28 Elsevier Ltd. All rights reserved. 1. Introduction The role of turbulent mixing on the ocean circulation has been discussed in many recent papers, see for instance Munk and Wunsch (1998), Samelson (1998), Spall (21), Wunsch and Ferrari (2). How energy due to the major forcing mechanisms namely wind and tide is converted into small scale mixing and how this feeds back into the large scale circulation is a major problem in oceanography. It has been suggested that internal waves interacting with topography play an important role in this mixing process, see Polzin et al. (1997), Kunze and Llewellyn Smith (2). Many studies focus on the processes along the continental slopes, see Wunsch (197), Armi (1978), Legg and Adcroft (2) and the discussion in Thorpe (2). The process by which large scale energy of wind and tidal origin is converted into mixing involves the cascade of energy through a spectrum of time and space scales. This makes it very challenging to measure Corresponding author. Tel.: +7 88; fax: addresses: jarle.berntsen@math.uib.no, jarleb@mi.uib.no (J. Berntsen), jxx@pol.ac.uk (J. Xing), amd@pol.ac.uk (A.M. Davies). or model all details of internal wave generation, propagation, breaking, and mixing. Many notable studies of these processes are summarised in Thorpe (2) and Vlasenko et al. (2). Internal waves are also found in fjords. Fjords are smaller and shallower than the open ocean, and many of the important physical phenomena of the deep ocean are also found in fjord systems. Therefore, fjords may be used as laboratories to investigate baroclinic processes, and in particular processes near sills where energy is transferred by non-linear effects from large to small scales and eventually mixing, see for instance Stigebrandt (1976), Stigebrandt and Aure (1989), Parsmar and Stigebrandt (1997), Stigebrandt (1999), Cushman-Roisin et al. (1989), Tverberg et al. (1991), Farmer and Freeland (198), Stacey and Gratton (21), Skogseth et al. (2, 27), Fer et al. (2), Fer (26), Inall et al. (2, 2), Eliassen et al. (21), Vlasenko et al. (22), Vlasenko and Stashchuk (26), Stashchuk and Vlasenko (27). Laboratory experiments are also very useful in studies of the interactions between stratified flow and topography, see Sveen et al. (22), Guo and Davies (2), Guo et al. (2). To gain insight into how barotropic tidal energy is converted into internal waves and mixing Xing and Davies (26a) 278-/$ - see front matter & 28 Elsevier Ltd. All rights reserved. doi:1.116/j.csr

2 J. Berntsen et al. / Continental Shelf Research 28 (28) (hereafter XD6a) applied an idealised two-dimensional (2D) cross sectional model of Loch Etive, a region of recent intensive measurements, see Inall et al. (2, 2). Subsequently Xing and Davies have studied the influence of stratification and topography upon internal wave spectra (Xing and Davies (26b) hereafter XD6b), and the influence of stratification and tidal forcing upon in particular how stratification in the region near the upper part of the sill influenced mixing (Davies and Xing, 27). The importance of non-hydrostatic pressure effects is demonstrated in Xing and Davies (27). The tidal flow over the sill in Loch Etive is also recently discussed in Stashchuk et al. (27). The results reported in XD6a,b were produced with the MITgcm, see Marshall et al. (1997a, b) and Adcroft et al. (1999), using a horizontal grid size of 1 m near the sill and gradually coarser resolution away from it. A grid size of 1 m is very small if we relate it to grid sizes often used in numerical ocean modelling, see for instance Haidvogel and Beckmann (1999). However, both observations and numerical model results for a similar system namely Knight Inlet show processes around this inlet that may not be well resolved with a grid size of 1 m, see for instance Farmer and Armi (1999), and Cummins et al. (2). Therefore, we want to investigate the sensitivity of the numerical results for the model area in XD6a to the grid resolution. Using the XD6a model domain it will be very computationally demanding to refine the grid further. The approach chosen in the present paper is, therefore, to apply a sequence of equidistant grid sizes ranging from 1 to 12. m to examine convergence. In 2D cross sectional model studies, important three-dimensional (D) effects are ignored and one may therefore expect gradually more D numerical studies. Given limited computer resources, the horizontal grid size often has to be increased to facilitate D studies. With larger grid sizes, important physical process may be lost, and waves may become misrepresented or aliased on the wrong wave lengths. Before moving towards D modelling and coarser grids, it will be useful with more insight into these possible consequences of increased grid sizes. The model applied in the present studies is a non-hydrostatic cross sectional s-coordinate model, see Berntsen et al. (26). By using a s-coordinate model, we hope to achieve a better representation of the processes in the bottom boundary layer, see for instance Haidvogel and Beckmann (1999), than XD6a,b who used a z-coordinate model with shaved cells in the bed region. The experiments in XD6a,b were performed with constant values of viscosity and diffusivity, both horizontally and vertically. In other studies of flow over sills, a range of subgrid scale closure schemes are applied, and the sensitivity of the results to the eddy viscosity/diffusivity is discussed in Lamb (2) for the Knight Inlet. The sill depth at Loch Etive is approximately 1 m, whereas the corresponding depth at Knight Inlet is approximately 6 m. The processes and energy transfers near the sill are affected by the sill depth, and Loch Etive may therefore be a suitable study area for fjords or lochs with sills that are shallower than the sill in Knight Inlet. In this paper we want to follow up by investigating this sensitivity for the present case with a shallower sill. The sensitivity of the numerical results to the eddy viscosity/ diffusivity is in the present study investigated for all grid sizes applied. In Section 2 the numerical experiments are described. Some results from experiments where the maximum velocities over the sill are less than the internal wave speed are given in Section. In Section corresponding results for the case with super-critical flow over the sill are given. The sensitivity of the results to the parameterisation of subgrid scale effects is discussed in Section. A summary and a general discussion is given in Section The numerical experiments 2.1. The model and experimental setup The s-coordinate ocean model applied in the present studies is a 2D, ðx; zþ, version of the model described in Berntsen (2) where x and z are the horizontal and vertical Cartesian coordinates, respectively. The model is available from www. math.uib.no/bom/. The variables are discretized on a C-grid. In the vertical, the standard s-transformation, s ¼ðz ZÞ=ðH þ ZÞ, where Z is the surface elevation, and H the bottom depth, is applied. For advection of momentum and density a total variance diminishing (TVD)-scheme with a superbee limiter described in Yang and Przekwas (1992) is applied in the present studies. The standard second order Princeton ocean model (POM) method is applied to estimate the internal pressure gradients (Blumberg and Mellor, 1987; Mellor, 1996). The model is mode split with a method similar to the splitting described in Berntsen et al. (1981) and Kowalik and Murty (199). In the 2D model, all variations in the cross channel direction, y, are neglected, and all q=qy terms in the governing equations are accordingly set to zero. There will still be cross channel velocities due to the Coriolis effect since the Coriolis frequency f is equal to 1:2 1 s 1. Using a splitting technique, the non-hydrostatic pressure may be computed by inserting the expressions for the non-hydrostatic velocity corrections into the equation of continuity (Kanarska and Maderich, 2; Heggelund et al., 2). Due to the s-transformation, additional terms appear in the pressure equations, which complicates the computations. An alternative method, that has been adopted in the present study, is to view the non-hydrostatic pressure directly as Pðx; s; tþ, where t is time, or a pressure due to convergence or divergence in the s-coordinate system (Berntsen and Furnes, 2). The time steps are performed with a predictor corrector method where the leapfrog method is used as predictor and the y-method, where y is the degree of implicitness, is used as corrector. For stability y must be in the range :pyp1, and in the present experiments y ¼ 1: (i.e. fully implicit) (Haidvogel and Beckmann, 1999; Casulli, 1999). The model has recently been applied to study lock release gravity currents and the propagation of solitary waves up an incline (Berntsen et al., 26) and the results are related to measurements from laboratory experiments and to corresponding results from the MITgcm (Marshall et al., 1997a,b; Adcroft et al., 1999). In Berntsen et al. (26) more details about the estimation of the non-hydrostatic pressure are given. The set up of the numerical experiments is very similar to that described in XD6a. The topography and the initial stratification for the first set of experiments are given in Fig. 1. The initial buoyancy frequency N is :1 s 1 in all experiments. The depth profile HðxÞ in meters is specified according to 8 >< þ 1 þðx=w sill Þ ; xo; HðxÞ ¼ 8 >: 1 þ 1 þðx=w sill Þ ; x assuming x ¼ m at the top of the sill and W sill is the half width of the sill. In the first set of experiments W sill is chosen to be 1 m, and in the second set it is set to m. There is no flow through the sea bed or the closed right end of the loch. Initially the water elevation is Z, and there is no flow. The M 2 tide is forced into the loch by specifying a velocity in the x- direction u as u ¼ u tide cðz Z tide Þ=H I ; 8z

3 178 J. Berntsen et al. / Continental Shelf Research 28 (28) T (ci=.2 C) Fig. 1. Initial temperature and topography for half sill width equal to 1 m. at the left boundary. p In the above equation H I is the depth at the inflow point, c ¼ ffiffiffiffiffiffiffi gh I, Z is the water elevation just inside the left open boundary, Z tide is the forced tidal water elevation, and u tide is the corresponding forced tidal velocity. The tidal forcing is computed from a standing wave solution given in (Gill, 1982, p. 11) Z tide ¼ Z cosðkl x Þ cosðo M2 tþ, and u tide ¼ cz sinðkl x Þ sinðo M2 tþ, H I where o M2 ¼ :126 rad s 1, k ¼ o M2 =c, and L x the length of the channel ðl x ¼ 21 mþ. The value of Z is. m in the first set of experiments giving sub-critical flow over the sill at maximum inflow. In the second set of experiments Z ¼ :8m giving super-critical flow over the sill at maximum inflow as in XD6a. The standing wave solution is strictly for a channel with constant depth. However, by forcing with the above open boundary condition, the tidal signal propagate with very little disturbances into and out of the domain. For the temperature a Neumann boundary condition is applied at the open boundary. In the vertical 1 equidistant s-layers are used in all experiments. Horizontally the grids applied are equidistant and the experiments are run with horizontal grid sizes Dx equal to 1,, 2, and 12. m. In the experiments with Dx ¼ 12: m the internal time step used is.6 s, and external time steps are used for each internal step. The time steps are increased proportionally with Dx in the experiments with coarser grid resolutions. The experiments are performed with constant values of viscosity and diffusivity as in XD6a, that is: horizontal viscosity A h ¼ 1 1 m 2 s 1, vertical viscosity A V ¼ 1 m 2 s 1, horizontal diffusivity K h ¼ 1 7 m 2 s 1, and vertical diffusivity K V ¼ 1 7 m 2 s 1. By using very small values of the diffusivities, the mixing becomes primarily controlled by convective overturning and instabilities, see XD6a and XD6b. The much larger values of the viscosities are used to avoid grid scale noise due to the cascade of energy towards the grid scale in non-linear models. With the present grid sizes all relevant length scales are not resolved and the effects to be parameterised by eddy diffusivity/ viscosity will depend on the grid resolution. Therefore, the experiments are also performed with values of eddy viscosity and eddy diffusivity that depend on the grid size and that vary in time and space. For the horizontal viscosity a Smagorinsky (196) type formulation is applied, and the viscosities are computed from " A h ¼ C M Dx 2 qu 2 þ 1 # qv 2 1=2, (1) qx 2 qx where C M is a non-dimensionless viscosity parameter and v is the velocity component in the cross loch direction. Statement (1) is a 2D version of the formulation applied in the Princeton Ocean Model (Blumberg and Mellor, 1987; Mellor, 1996). Values of C M in the range..2 are suggested in Mellor (1996) and in Haidvogel and Beckmann (1999). In the present studies C M is chosen to be.2. To compute the vertical eddy viscosities and diffusivities, the Mellor and Yamada (1982) level model with the modifications due to Galperin et al. (1988) are used. The minimum allowed value of A V is set to 1 m 2 s 1 and the minimum allowed value of K V is 1 7 m 2 s 1. The experiments with the Smagorinsky model and the Mellor and Yamada model are denoted as large eddy simulations (LES) Model diagnostics In the numerical experiments the kinetic energy E kin computed in each time step according to E kin ðtþ ¼ Z 1 m Z Zðx;tÞ m HðxÞ :rðx; z; tþðuðx; z; tþ 2 þ vðx; z; tþ 2 þ wðx; z; tþ 2 Þ dz dx, (2) where r is the density computed from the temperature and a linear equation of state and u, v, and w are the velocity components in x, y, and z direction, respectively. Defining ðuðx; tþ; vðx; tþþ ¼ 1 Z Zðx;tÞ ðuðx; z; tþ; vðx; z; tþþ dz, () Dðx; tþ HðxÞ where Dðx; tþ ¼HðxÞ + Zðx; tþ, the energy from the depth averaged velocity component E kin2d is computed from E kin2d ðtþ ¼ Z 1 m Z Zðx;tÞ m HðxÞ :rðx; z; tþðuðx; tþ 2 þ vðx; tþ 2 Þ dz dx. () is

4 J. Berntsen et al. / Continental Shelf Research 28 (28) The kinetic energy due to deviations of the velocity components from the depth averaged values E kin-bclin is computed from E kin-bclinðtþ ¼E kin ðtþ E kin2d ðtþ. () At each time step the maximum velocity component U max in x- direction is recorded according to U max ðtþ ¼max uðx; z; tþ. (6) 8x;z To investigate the sensitivity of the stability of the water masses in the area around the sill to the parameters involved in the various calculations, the area of the water masses with Richardson numbers (Ri) less than the critical value 1 (A Ri crit) is computed at each time step, namely, A Ri crit ¼ Z 2 m Z Zðx;tÞ 1 m HðxÞ where ( 1 if Riðx; zþo:2; d Ri ðx; zþ ¼ otherwise; d Ri ðx; zþ dz dx, (7) and Ri ¼ g qr=qz r ðqu=qzþ 2 þðqv=qzþ. (8) 2 The area (7) will be time dependent. The time averaged value of A Ri crit Z 2T A Ri crit ¼ 1 A 2T Ri crit dt (9) where T is a M 2 tidal period, is used to indicate the potential for mixing in different calculations. The 1 criterion for the Richardson number used above is not a sufficient condition for instabilities, see Kundu and Beardsley (1991), Helfrich and Melville (26) and Kundu and Cohen (28). However, there is also evidence that it still may be regarded as a useful guide for the prediction of instabilities, see Kundu and Beardsley (1991), Kundu and Cohen (28) and references therein. The time and space integrated values of the energy losses due to bottom friction E t are computed according to E t ¼ Z 2T Z 1 m m ~u b ðx; tþ~t b ðx; tþ dx dt, where ~u b ðx; tþ ¼ðu b ðx; tþ; v b ðx; tþþ is the velocity vector in the nearest bottom grid cell, and u b ðx; tþ and v b ðx; tþ are the velocity components in the x-direction and the cross channel direction, respectively. The bottom stress vector ~t b ðx; tþ is specified by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~t b ðx; tþ ¼r C D u 2 b þ v2 ~u b b ðx; tþ, (1) where r is the reference density. The drag coefficient C D is given by " # k 2 C D ¼ max :2; ðlnðz b =z ÞÞ 2 (11) and z b is the distance of the nearest grid point to the bottom. The von Karman constant k is. and the bottom roughness parameter is chosen to be z ¼ :1 m, see Blumberg and Mellor (1987). Similarly integrated values of the losses due to horizontal viscosity E H and vertical viscosity E V are computed Z 2T Z 1 m Z Zðx;tÞ E H ¼ ra h m HðxÞ " qu 2 þ qv 2 þ qw # 2 dz dx dt, qx qx qx and Z 2T Z 1 m Z Zðx;tÞ E V ¼ ra V m HðxÞ " qu 2 þ qv 2 þ qw # 2 dz dx dt. qz qz qz Errors in the internal pressure gradient estimation may create artificial flows in s-coordinate models, see for instance Haney (1991), Mellor et al. (1998), Shchepetkin and McWilliams (2) and Thiem and Berntsen (26). However, in the present experiments with relatively small grid sizes this is not a serious problem. For instance in experiments with Dx ¼ 12: m and W sill ¼ m where the flow is driven only by the artificial internal pressure gradients, the kinetic energy E kin remains less than 1 1 Jm 1, and U max less than 1 1 ms 1.. Results from the sub-critical experiments The maximum tidal velocity across the sill u in the first set of calculations is approximately :6 m s 1. This means that the maximum Froude number, Fr ¼ u=nh, defined as in XD6a, at the sill crest, where H is the sill depth, is approximately.. The half width of the sill is 1 m. In general the results for this case are independent of the horizontal grid size for the range of grids examined here. This may for instance be seen from the temperature fields at maximum inflow in the second tidal cycle, see Figs. 2 and. In Fig. the values of the various diagnostics are given. From Figs. a and b we notice that the kinetic energy is approximately three times larger at the time of maximum inflow ðt ¼ =TÞ when both barotropic and baroclinic motions are strong than at t ¼ 2T, the time when the tide turns and the system essentially contains baroclinic signals. The results for this low Froude number case are in general less sensitive to the grid size than to the choice of viscosity and diffusivity for the range of grid sizes applied. When using the present values of constant viscosity and diffusivity, the results are fairly robust to the grid size. However, from Figs. a and b we notice that the solution becomes less energetic as the grid size is reduced. This is due to slightly larger viscous losses and losses due to bottom friction when Dx is reduced. In a mathematical sense we can produce a converged solution by reducing Dx for this low Froude number case keeping the diffusivity constant and the viscosity large and constant. There is a different message from the large eddy simulations. For this case, there is a clear increase in energy as the grid size is reduced. This is due to the grid dependent viscosities (1) allowing more small scale processes to enter the solution as Dx is reduced. This may for instance be seen in the time series of vertical velocities for the case with Dx ¼ 12: m, see Fig.. It is also seen in the increase in the estimates of the areas of the water masses with Ri less than the critical value, see Fig. e. These low Ri water masses are found near the sea bed over the sill, but also higher up in the water column near the sill, see Fig. 6. For this case with grid dependent viscosity, the turbulent part of the water column will probably grow in extent if the grid size was reduced even further. In essence the non-linear nature of the problem (even at a subcritical Froude number) in this region is such that there is a rapid cascade of energy from long waves to short waves and associated mixing. This cascade is balanced by the viscous terms, which in the case of constant values may lead to a rapidly converged solution, as Dx is reduced (see in particular Fig. e). However, in the LES calculations, reducing Dx decreases the viscous term and hence the balance in the energy cascade to short waves is affected.

5 18 J. Berntsen et al. / Continental Shelf Research 28 (28) T (ci=.2 C) T (ci=.2 C) T (ci=.2 C) T (ci=.2 C) Fig. 2. The temperature fields for the sub-critical case after t ¼ =T (where T is the M 2 tidal period) for the experiments with constant viscosity and diffusivity. The results are for (a) Dx ¼ 1m, (b) Dx ¼ m, (c) Dx ¼ 2m, and (d) Dx ¼ 12:m T (ci=.2 C) T (ci=.2 C) T (ci=.2 C) T (ci=.2 C) Fig.. The temperature fields for the sub-critical case after t ¼ =T for the large eddy simulations. The results are for (a) Dx ¼ 1m, (b) Dx ¼ m, (c) Dx ¼ 2m, and (d) Dx ¼ 12:m.

6 J. Berntsen et al. / Continental Shelf Research 28 (28) x 1 x 1 1. Kinetic energy [J/m] Kinetic energy [J/m] x 1.9 Kinetic energy [J/m] U [m/s] Area [m 2] Energy loss [J/m] x Energy loss [J/m] Energy loss [J/m] x Fig.. Sensitivity of (a) E kin ð=tþ, (b) E kin ð2tþ, (c) E kin-bclinð2tþ, (d) U maxð=tþ, (e) A Ri crit, (f) E t, (g) E H, and (h) E V to the grid resolution for the sub-critical case. The results for the constant viscosity and diffusivity experiments are given with solid lines and squares. The results for the large eddy simulations are given with dashed lines and circles.

7 182 J. Berntsen et al. / Continental Shelf Research 28 (28) Depth [m] 6 W (ci=.2 mm/s) Fig.. Vertical velocities for the sub-critical case at x ¼ 1 m during a part of the second tidal cycle for the large eddy simulations with Dx ¼ 12: m. (Time is scaled with T.) Time Fig. 6. Richardson number distribution for the sub-critical case at t ¼ 1=8T for the large eddy simulations with Dx ¼ 12: m. The Ri ¼ : and.2 contours are given with thick solid lines, and the Ri ¼ : contours with thin dashed lines. In a mathematical sense the problem changes with Dx, and this is reflected in the rate of convergence.. Results from the super-critical experiments The maximum tidal velocity across the sill in the second set of calculations is approximately :6 m s 1 giving a maximum value of Fr of approximately. The sill width is reduced to m in these experiments. The increased inflow velocity and the reduced sill width have the effect of substantially increasing the non-linear nature of the problem compared to the previous set of experiments. The calculations show that for this case the numerical solutions are much more sensitive to the grid size, see Figs. 7 and 8. Note in particular the increase in amplitude of the internal waves at depth as Dx decreases from 2 to 12. m in the constant viscosity case (Fig. 7). However, in the LES simulations these waves are smoothed by the viscosity and diffusivity terms (Fig. 8). In essence the solutions qualitatively change nature when the subgrid scale closure is changed. In the constant viscosity and diffusivity experiments, internal waves appear in the lee of the sill as reported in XD6a,b. The period of these waves depend on the grid size, see the time series of vertical velocities in the lee of the sill given in Fig. 9a. For the highest resolution, Dx ¼ 12:m, the period is approximately 1 12 min consistent with the results produced with the MITgcm and Dx ¼ 1 m near the sill, see

8 J. Berntsen et al. / Continental Shelf Research 28 (28) T (ci=.2 C) T (ci=.2 C) T (ci=.2 C) T (ci=.2 C) Fig. 7. The temperature fields for the super-critical case after t ¼ =T for the experiments with constant viscosity and diffusivity. The results are for (a) Dx ¼ 1m, (b) Dx ¼ m, (c) Dx ¼ 2m, and (d) Dx ¼ 12:m T (ci=.2 C) T (ci=.2 C) T (ci=.2 C) T (ci=.2 C) Fig. 8. The temperature fields for the super-critical case after t ¼ =T for the large eddy simulations. The results are for (a) Dx ¼ 1m, (b) Dx ¼ m, (c) Dx ¼ 2m, and (d) Dx ¼ 12:m.

9 18 J. Berntsen et al. / Continental Shelf Research 28 (28) W [m/s] W [m/s] t t Fig. 9. Time series of vertical velocities in m s 1 at ðx; zþ ¼ð m; 2 mþ for the super-critical case. The results for Dx ¼ 1 m are given with dashed thin lines, the results for Dx ¼ m with dashed thick lines, the results for Dx ¼ 2 m with solid thin lines, and the results for Dx ¼ 12: m with solid thick lines. The time is given in minutes. The results from the constant viscosity and diffusivity experiments are given in (a), and the corresponding results from the large eddy simulations are given in (b). Notice the change in scale on the axis for the vertical velocities. XD6b. The vertical velocities at ðx; zþ ¼ð m; 2 mþ are generally much smaller in the LES calculations, see Fig. 9b, and the time intervals during inflow with vertical excursions are much shorter. The vertical motions are also less harmonic, see Fig. 9b. However, in the time series for Dx ¼ 12: m a few oscillations with periods of approximately 12 min are still seen, even if they are much weaker than in the corresponding results produced with constant viscosity and diffusivity. Since the internal waves produced in the LES calculations generally are much weaker, the solutions away from the sill, see Figs. 1 and 11, are also affected by the choice of subgrid scale closure. With constant viscosity and diffusivity the strong internal waves produced in the lee propagate away from the sill. In the corresponding results from the large eddy simulations this signal is much weaker. The internal waves leaving the sill may subsequently break at topographic features away from the sill. In the constant viscosity/diffusivity calculations there may thus be a substantial turbulent dissipation away from the sill. In the LES calculations more mixing will take place near the sill. The choice of subgrid scale closure scheme thus affects the spatial distribution of the mixing, and the background stratification and the residual flow will be affected in long time integrations. The numerical studies presented here are motivated by the measurements in Loch Etive (Inall et al., 2, 2; Stashchuk et al., 27). The present studies are for a more idealised system than the corresponding physical counterpart. The initial buoyancy frequency is for instance constant in the numerical simulations. The numerical results may therefore not be directly related to corresponding measurements from Loch Etive. However, some of the processes found in our numerical results, are also described in Inall et al. (2). For instance, Inall et al. (2) report on internal waves propagating away from the sill with a phase speed of approximately :2ms 1, see their Fig. 7. They state that the shallow pycnocline oscillations... take on the appearance of soli-bores released from the tidal jet. In our numerical studies, especially the studies with constant viscosity and constant diffusivity, internal waves are also generated during inflow, and the generated sequence of waves leaves the sill as the tide weakens and turns and the waves propagate away from the sill with a phase speed of approximately :22 m s 1, see our Figs. 1a and 11a. This internal wave speed has been reproduced also with the MITgcm with an equidistant horizontal grid size of 12. m. With a coarser resolution away from the sill, the wave propagation in the coarse grid part of the domain will not be captured, and this may also affect the spatial distribution of mixing in the calculations. In Stashchuk et al. (27) phase speeds in the range from.22 to : m s 1 are suggested. These estimates were based on modal analysis and a more realistic density profile. In their discussion they also state that the waves can propagate freely and leave the slope when the Froude number becomes less than unity, which is consistent with our findings. Boundary layer separation is discussed in the papers focusing on the Knight Inlet sill, see Farmer and Armi (1999) and Cummins (2). Afanasyev and Peltier (21a) applied a slip bottom boundary condition that did not allow bottom boundary layer separation, whereas Lamb (2) used a no-slip condition at the bottom boundary giving separation from near the top of the sill. In the present studies, a no-slip bottom boundary condition is also applied, and in all super-critical experiments bottom boundary layer separation occurs in the lee of the sill approximately at m depth between and m from the top of the sill, see Figs. 7, 8, and 1. This agrees with the position of the measured separation point reported in Inall et al. (2), see their Fig.. The numerical separation point is fairly robust to the parameters investigated. A horizontal length scale of the horizontal movements of the separation point may be estimated from l ¼ L s =H s U=N, see Afanasyev and Peltier (21a). Here L s is the horizontal extent of the region of the slope on the lee side where the separation occur (1 m), H s is the corresponding vertical extent (2 m), and U is a typical current speed near the separation point ð:2 m s 1 Þ giving a length scale l of approximately 12 m consistent with the range found in our numerical studies. Closer to the top of the sill, between 1 and 2 m depth, flow separation is also found in the numerical results from some of our experiments with high resolution, see Fig. 7d. In our studies with a larger computational domain and larger grid sizes than the values used in Stashchuk et al. (27), this upper separation point is more sensitive to both the grid size and the choice of subgrid scale closure scheme.

10 J. Berntsen et al. / Continental Shelf Research 28 (28) T (ci=.2 C) T (ci=.2 C) Fig. 1. The temperature fields for the super-critical case after t ¼ =2T for (a) the experiment with constant viscosity and diffusivity and (b) the large eddy simulation. The results are for Dx ¼ 12:m. In this study, the focus has been mainly on the response in the lee of the sill in order to relate our findings to corresponding results reported in XD6a,b. However, topographic features like sills create a form drag and a blocking effect that will affect also the upstream response, see for instance Pierrehumbert and Wyman (198), Stigebrandt (1999) and McCabe et al. (26). This upstream numerical response will also depend on the grid size and the subgrid closure scheme. This sensitivity of the upstream response may to some extent be seen in the plots of the temperature fields at the top of the sill (Figs. 7 and 8). Cummins et al. (2) describe for the Knight Inlet case upstream propagating internal waves. For our case with a much shallower sill, internal waves generated in the lee below the sill depth may be blocked by the sill from propagating upstream. By comparing the energy levels in Figs. a and b for the subcritical case, with the corresponding Figs. 12a and b for the supercritical case, we notice approximately to times higher energy levels in the more strongly forced case. As for the subcritical case, the energy levels for the super-critical case are highest at the time of maximum inflow and for this case the maximum energy levels are more than four times higher than the corresponding levels at the time when the tide turns. From the diagnostics for the super-critical case given in Fig. 12 we also notice that they vary both with the grid size and the subgrid scale closure. However, contrary to the findings for the weakly forced case, the results from the LES calculations seem to be less sensitive to the grid size, see for instance the maximum velocities reported in Fig. 12d. The reason for this, can in part be understood in terms of the much larger short wave components in the constant viscosity/diffusivity solutions compared to the LES solutions. As in any numerical calculation, if the solution is dominated by short waves, a finer grid must be used than when the solution is composed of mainly long waves. Thus, for the constant viscosity case, the energy losses due to horizontal viscosity increase very much when the grid size is reduced, see Fig. 12g. These losses are due to the term ðq=qxþa h ðqu=qxþ in the governing equations, and since A h is constant, it must mean that qu=qx changes substantially with the grid size since the solution contains large short wave components. For the LES calculations it is noticeable that the corresponding losses are approximately

11 186 J. Berntsen et al. / Continental Shelf Research 28 (28) T (ci=.2 C) T (ci=.2 C) Fig. 11. The temperature fields after t ¼ 2T for (a) the experiment with constant viscosity and diffusivity and (b) the large eddy simulation. The results are for Dx ¼ 12:m. equal for Dx ¼ 2 m and Dx ¼ 12: m since the short wave components of the solution are small. Focusing on the areas of the water masses with Ri less than the critical value, we notice by comparing Figs. e and 12e that these areas are more than 1 times larger in this more strongly forced case. Furthermore, in the higher Froude number case the areas seem to increase as the grid size is reduced. However, the areas for the LES calculations using Dx ¼ 2 m and Dx ¼ 12: m are very similar, and almost equal to the value for Dx ¼ 12: m using constant viscosity and diffusivity. Even if the total areas are almost the same, the spatial distributions of these regions with low Ri are different, see Figs. 1 and 1. Comparing Figs. 1 and 1 we notice a good correspondence between the temperature fields and the Ri number distributions.. Sensitivity to the subgrid scale closures In the previous sections the results were found to be very sensitive to the subgrid scale closure. In order to investigate this sensitivity further, four additional experiments for the supercritical flow case with Dx ¼ 12: m were performed. The horizontal diffusivity K h is equal to 1 7 m 2 s 1 in all studies. The values, or methods for computing, A h, A V, and K V in all six experiments with Dx ¼ 12: m are given below Experiment 1: A h ¼ 1 1 m 2 s 1, and A V ¼ 1 m 2 s 1, and K V ¼ 1 7 m 2 s 1 as in XD6a,b. Experiment 2: A h is computed using Eq. (1) with C M ¼ :2, A V and K V are estimated using the Mellor Yamada scheme as described in Section 2. Experiment : A h is computed using Eq. (1) with C M ¼ :2, A V ¼ 1 m 2 s 1, and K V ¼ 1 7 m 2 s 1. Experiment : A h is computed using Eq. (1) with C M ¼ :, A V ¼ 1 m 2 s 1, and K V ¼ 1 7 m 2 s 1. Experiment : A h is computed using Eq. (1) with C M ¼ :, and A V and K V are estimated using using the Mellor Yamada scheme. Experiment 6: A h ¼ 1 1 m 2 s 1, and A V and K V are computed with the method given in Pacanowski and Philander (1981) (hereafter PP81). The equations are A A V ¼ ð1 þ ariþ þ A b; k A V K V ¼ ð1 þ ariþ þ K b, (12)

12 J. Berntsen et al. / Continental Shelf Research 28 (28) x 1 7 x 1 6 Kinetic energy [J/m] Kinetic energy [J/m] x 1 6 Kinetic energy [J/m] U [m/s] x 1 6 x 1 6 Area [m 2].. Energy loss [J/m] x x 1 6 Energy loss [J/m] 2 1 Energy loss [J/m] Fig. 12. Sensitivity of (a) E kin ð=tþ, (b) E kin ð2tþ, (c) E kin-bclinð2tþ, (d) U maxð=tþ, (e) A Ri crit, (f) E t, (g) E H, and (h) E V to the grid resolution for the super-critical case. The results for the constant viscosity and diffusivity experiments are given with solid lines and squares. The results for the large eddy simulations are given with dashed lines and circles.

13 188 J. Berntsen et al. / Continental Shelf Research 28 (28) Fig. 1. The Richardson number distribution for the super-critical case after t ¼ =2T for (a) the experiment with constant viscosity and diffusivity and (b) the large eddy simulation. The results are for Dx ¼ 12: m. The Ri ¼ : and.2 contours are given with thick solid lines, and the Ri ¼ : contours with thin dashed lines. where we have chosen A ¼ :1 m 2 s 1, a ¼, k ¼ 2, A b ¼ 1 m 2 s 1, K b ¼ 1 m 2 s 1, and Ri is computed from Eq. (8). The temperature fields after =T for experiments 6 are given in Fig. 1, and the diagnostics are given in Fig. 16, including also the diagnostics from the previous studies with Dx ¼ 12:m. By comparing Fig. 1 with Figs. 7d and 8d, we notice that the internal waves in the lee of the sill appear clearly in all the results produced using both A V and K V constant. If the vertical viscosity and diffusivity are computed with the Mellor Yamada scheme or Eq. (12), the internal waves are far less evident. The results are also sensitive to the horizontal viscosity A h, see Fig. 16. However, the solutions do not change qualitatively when replacing A h ¼ 1 1 m 2 s 1 with the values computed using Eq. (1), or by adjusting the coefficient C M. A noticeable result from Fig. 16 is that the energy losses due to horizontal viscosity, E H, are substantially reduced when replacing the constant values of A V and K V with values given by the Mellor Yamada scheme or PP81. The explanation is that when using the Mellor Yamada scheme or PP81 (Eq. (12)), more of the tidal energy goes into irreversible mixing, and the internal wave generation is reduced. The horizontal gradients qu=qx consequently become smaller, and the viscosity terms ðq=qxþa h qu=qx are accordingly reduced. From Fig. 16e we notice that the areas with less than critical Richardson numbers are fairly consistent for the first five experiment. The area is considerably reduced when using PP81, and generally larger values of A V and K V when using PP81 may be the explanation. This is also consistent with larger energy losses due to bottom friction and vertical viscosity in Experiment 6, see Figs. 16f and h. 6. Discussion A cross sectional s-coordinate model has been applied to study the dynamics near a sill in an idealised loch. The buoyancy frequency is assumed to be constant in all studies. For studies with sub-critical flow over a wide sill, the results are very robust to changes in the horizontal grid size in the range from 1 to

14 J. Berntsen et al. / Continental Shelf Research 28 (28) Fig. 1. The Richardson number distribution for the super-critical case after t ¼ 2T for (a) the experiment with constant viscosity and diffusivity and (b) the large eddy simulation. The results are for Dx ¼ 12: m. The Ri ¼ : and.2 contours are given with thick solid lines, and the Ri ¼ : contours with thin dashed lines. 12. m. With stronger tidal forcing, a narrower sill, and supercritical flow over the sill during inflow, we get flow separation and generation of high amplitude internal waves in the lee of the sill. The frequency and the amplitude of these waves changes noticeably as the horizontal grid size is reduced from 1 to 12. m. With a grid size of 12. m and constant values of viscosity and diffusivity, the results are very similar to those presented in XD6a,b. The present study is partly motivated by measurements in Loch Etive (Inall et al., 2, 2; Stashchuk et al., 27) which may be regarded as a suitable study area for fjords or lochs with shallow sills. Even if the calculations are for a more idealised system than the physical counterpart, it is found that the boundary layer separation during flood tide in the numerical results agree very well with the measured boundary layer separation. Inall et al. (2) report on a sequence of internal waves propagating away from the sill, and such waves are also reproduced in the numerical experiments, especially in the experiments with constant viscosity and constant diffusivity. The tidally driven processes near Knight Inlet, British Columbia, have also been well investigated using both measurements (Farmer and Armi, 1999; Armi and Farmer, 22; Klymak and Gregg, 21, 2, 2), and numerical model experiments (Stacey, 198; Cummins, 2; Afanasyev and Peltier, 21a; Klymak and Gregg, 2; Cummins et al., 2; Lamb, 2; Vlasenko and Stashchuk, 26; Stashchuk and Vlasenko, 27). Hydrostatic, non-hydrostatic, rigid-lid, and free surface models are used for this system. The numerical studies are performed with both slip and no-slip bottom boundary conditions. A wide range of grid sizes, horizontally and vertically, and subgrid scale closure schemes is applied in these studies. The sensitivity of the flow to the stratification is also investigated. The generation mechanisms behind the formation of a wedge of partly mixed fluid downstream of the sill are discussed in several recent papers, see Afanasyev and Peltier (21a), Farmer and Armi (21) and Afanasyev and Peltier (21b). In Farmer and Armi (1999) and Farmer and Armi (21) it is argued that the small scale instabilities are essential, whereas Afanasyev and Peltier (21a)

15 19 J. Berntsen et al. / Continental Shelf Research 28 (28) T (ci=.2 C) T (ci=.2 C) T (ci=.2 C) T (ci=.2 C) Fig. 1. The temperature fields for the super-critical case after t ¼ =T for (a) Experiment, (b) Experiment, (c) Experiment, and (d) Experiment 6. point at irreversible mixing due to breaking internal waves. To facilitate high resolution numerical studies, 2D versions of the numerical models are typically used. Important D effects (Klymak and Gregg, 21, 2) that also may play an important role in the creation of the downstream wedge are thus neglected. The controversy over the mechanisms behind the formation of the partly mixed body of fluid downstream of the sill, may motivate systematic studies of the sensitivity of numerical model results to the physical and numerical parameters involved. The studies in Lamb (2) may be regarded as contributions in this direction. Correspondingly, the present work may regarded as an investigation of the role of the parameters affecting the small scale processes near the sill in Loch Etive. Three-dimensional effects are ignored in the present study. In a real loch or fjord, vortices will be created behind headlands and they will influence the wave generation and mixing, see Klymak and Gregg (21), McCabe et al. (26) and Zhao et al. (26). However, with limiting computer resources, three-dimensional studies will require the use of larger grid sizes. Our results indicate that if the grid size is increased, compared to that used in XD6a,b, the generation, propagation, and breaking of the internal waves and the associated mixing may be misrepresented. Even if internal waves are found also in the results produced with coarser grid sizes, the wave lengths, periods, and amplitudes may be wrong. On a much finer scale there will also be three-dimensional effects associated with shear instabilities, see Dörnbrack (1998) and Smyth et al. (2). However, numerical studies of these three-dimensional effects require grid sizes far smaller than those considered here. In numerical studies with grid sizes that are too large to represent important physical processes, as they will be in threedimensional numerical studies of super-critical flow in fjords, the quality of the computed mean fields depends strongly on the quality of the chosen subgrid scale closure scheme, see Griffies (2). Having this in mind, it may be worrying that the results from the present calculations are very sensitive to the choice of subgrid scale closure. In the low Froude number studies, the results are more sensitive to the subgrid scale closure scheme than to changes in the grid size, even if the results are generally consistent for the sub-critical case. In this case the short wave components are less significant than in the high Froude number calculations. For the high Froude number case, the solution may change nature when changing the subgrid scale closure scheme. With constant and low values of vertical diffusivity as in XD6a, strong internal waves are generated on inflow in the lee of the sill. These waves subsequently propagate away from the sill with almost constant wave speed. In this case processes at the scale of the grid resolution and subgrid scale processes are more important than in the low Froude number calculations. By using a Richardson number dependant vertical diffusivity (Mellor and Yamada, 1982 or Pacanowski and Philander, 1981) more of the tidal energy goes into vertical irreversible mixing on inflow, and far less into the generation of internal waves propagating away from the sill. There is a large literature on subgrid scale closures in ocean modelling, see Haidvogel and Beckmann (1999), Kantha and Clayson (2) and Burchard (22). One could, however, hope that as the grid sizes are gradually reduced, there will be less unresolved important processes, and that the sensitivity to the subgrid closure scheme therefore should be reduced. The results from the present studies indicate that with a grid size of 12. m, there are still important unresolved processes, particularly in

16 J. Berntsen et al. / Continental Shelf Research 28 (28) x 1 7 x 1 6 Kinetic energy [J/m] Kinetic energy [J/m] N N x 1 6 Kinetic energy [J/m] U [m/s] Area [m 2] N x N Energy loss [J/m] N x N 7 x x 1 6 Energy loss [J/m] Energy loss [J/m] N N Fig. 16. Sensitivity of (a) E kin ð=tþ, (b) E kin ð2tþ, (c) E kin-bclinð2tþ, (d) U maxð=tþ, (e) A Ri crit, (f) E t, (g) E H, and (h) E V for the super-critical case to the subgrid scale closure scheme for the experiments 1 6. The N on the horizontal axis indicate experiment number. studies with super-critical flow. This may not be surprising, considering the results in Farmer and Armi (1999) and Cummins et al. (2). However, many studies of internal wave generation, propagation, breaking, and mixing have been performed with grid sizes in the range from 1 to 1 m. Especially in the generation and breaking phases there may be important

17 192 J. Berntsen et al. / Continental Shelf Research 28 (28) unresolved processes in such studies, see for instance Klymak and Gregg (21), Vlasenko et al. (22), Legg and Adcroft (2), Bourgault and Kelley (2), Wang (26) and Lamb (27). The sensitivity of the results presented here to the choice of subgrid scale closure naturally leads to the question: What is the best choice of viscosities and diffusivities in the supercritical case? Underlying such a question is the question: How shall we define best or optimal numerical solutions? The state of the art in the field of ocean modelling generally, and in the subgrid scale closure literature specifically is not mature enough to offer clear answers to the questions above. This is also recently discussed in Griffies (2). The results presented here indicate, however, that by using small and constant values of viscosity and diffusivity, rather than a LES approach, the internal wave propagation becomes qualitatively more in agreement with the corresponding wave propagation reported in Inall et al. (2). One may consider to apply numerical fine scale simulations, in essence direct numerical simulations, to get the true solution and to estimate diffusivities and viscosities, and finally to assess or improve subgrid scale parameterisations, see Smyth et al. (2) and Berntsen et al. (26). However, in direct numerical simulations the grid sizes applied are typically smaller than.1 m allowing only studies for tank scale problems. In physical oceanographic problems such as those considered here, inaccuracies in, for example, topographic variations and changes in bed types and forms on scales of the order of 1 m, which are rarely known, will have an appreciable effect and may negate any advantages of direct numerical simulations even if it was computationally possible. Several complementary approaches are therefore necessary to improve our understanding of the processes occurring in the lee of the sill during super-critical inflow. One approach may still be to use numerical models and to focus more on the area near the sill, allowing a horizontal grid size closer to 1 m as in Cummins et al. (2). As this grid size is significantly larger than that used in direct numerical simulations, there will still be unresolved processes. However, shear interface instabilities, and the effects of these instabilities, may still be captured. Another approach may be to seek high resolution measurements. A potential problem with this approach is that it may be difficult to find a loch or a fjord of sufficient limited extent that a synoptic data set of density measurements can be made particularly in the sill region for model investigations, and subsequent measurements for model validation. In the Arctic there may, however, be fjords with almost constant buoyancy frequency at least during parts of the year, see Fer and Widell (27) which would at least provide initial conditions. third approach would be to undertake laboratory experiments of the kind reported in Sveen et al. (22), Guo and Davies (2) and Guo et al. (2), but for conditions similar to those used in the present study. There are strengths and weaknesses with all three approaches. However, a combined use of all three may lead to deeper insight into the processes in the lee of the sill under conditions of strong inflow, and not least into the effects of these processes on the larger scale fields. Finally some comments related to the initial linear stratification that has been applied in the present studies and in XD6a. The measurements from Loch Etive reported in Inall et al. (2) show a stronger stratification near the surface above approximately m depth, and a weaker stratification below this depth. Therefore, a two layer approach could have been regarded as more relevant for the Loch Etive case. This is also followed up in XD6b and in Davies and Xing (27). With stronger stratification in the surface layer, the Froude number at maximum inflow will be reduced, and the intensity of the internal waves and mixing in the lee of the sill may be reduced, see XD6b and Davies and Xing (27) for further discussion. Acknowledgements This work was done during the first author s sabbatical year at Proudman Oceanographic Laboratory. He thanks the people at the laboratory for their great hospitality, and the Faculty of Mathematics and Natural Sciences, University of Bergen, for support during the sabbatical year. References Adcroft, A., Hill, C., Marshall, J., A new treatment of the coriolis terms in C-grid models at both high and low resolutions. Monthly Weather Review 127, Afanasyev, Y., Peltier, W., 21a. On breaking internal waves over the sill in Knight Inlet. Proceedings of the Royal Society of London A 7, Afanasyev, Y., Peltier, W., 21b. Reply to comment on the paper On breaking internal waves over the sill in Knight Inlet. Proceedings of the Royal Society of London A 7, Armi, L., Some evidence for boundary mixing in the deep ocean. Journal of Geophysical Research 8, Armi, L., Farmer, D., 22. Stratified flow over topography: bifurcation fronts and transition to the uncontrolled state. Proceedings of the Royal Society of London A 8, 1 8. 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