On breaking internal waves over the sill in Knight Inlet

 

Ya.D.Afanasyev* and W.R.Peltier

 

Department of Physics, University of Toronto, Toronto, Ontario Canada M5S 1A7

 

Submitted to Proceedings of the Royal Society: Series A

 

October 1, 1999


Corresponding author: W.R.Peltier

Department of Physics

University of Toronto

Toronto, Ontario

Canada M5S 1A7

Fax:(416) 978 8905

e-mail: peltier@atmosp.physics.utoronto.ca

-------

*Present address: Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John's, NF, Canada, A1B 3X7, e-mail: yakov@physics.mun.ca

Abstract

 

A new series of numerical simulations of stratified flow over localized topography is herein described which has been designed to address issues arising from a recently published sequence of detailed observations from a coastal oceanographic setting. Results demonstrate that the numerically simulated flow is very similar to that which develops in Knight Inlet, British Columbia, a fjord which is subject to periodic tidal forcing, and that the detailed dynamical characteristics of this flow are also strikingly similar to those of severe downslope windstorms that often occur in the atmosphere. A typical sequence of events observed in such flows includes the ``breaking'' of a forced stationary internal wave induced by the topography which results in irreversible mixing and the formation, through wave-mean flow interaction, of a decelerated mixed layer that extends downstream from the level of breaking. The formation of this mixed layer is a necessary precondition for transition of the flow into a supercritical hydraulic regime in which a low-level high velocity jet develops in the lee of the topographic maximum. Simulations with both fixed inflow velocity and harmonically varying inflow velocity are performed and intercomparison of the results clearly demonstrates that flow evolution in the unsteady forcing case can be described, to reasonable approximation, by the results of the corresponding quasi-steady simulations, at least during the accelerating stage when inflow velocity is slowly increasing. At later times of flow evolution, however, the well mixed fluid accumulates and the flow enters a statistically steady hydraulic-like regime which is characterized by a constant mean drag exerted by the topography on the flow even while the inflow velocity slowly decreases.

1  Introduction

The severe downslope atmospheric windstorms that are often observed to occur in the lee of major mountain ranges constitute a phenomenon that has turned out to be extremely rich in the array of fundamental hydrodynamic interactions that support their occurrence. Although such flows have been intensively investigated for a period of at least 50 years, they remain the subject of serious hydrodynamic analysis. That they may occur, in principle, in any density stratified fluid which flows over topography is well understood even though observations of the phenomenon in nature have been exclusively restricted, until recently, to atmospheric examples. Very recently, however, Farmer and Armi (1999a) have presented an oceanographic example in the form of detailed measurements of the velocity and density fields that were observed to accompany tidally driven flow over a sill. Their acoustic Doppler observations were acquired in Knight Inlet (British Columbia, Canada), a long narrow fjord subject to periodic tidal forcing, and reveal a strong visual similarity to typical downslope windstorms in the atmosphere. As first demonstrated by Peltier and Clark (1979), the primary event which leads to the formation of downslope windstorms in the atmosphere is the ``breaking'' of the topographically forced internal wave above its source. This process leads to intense irreversible mixing of the initially stable density stratification that was required to support the upward propagation of the internal wave, as well as the eventual development of a secondary shear instability which develops on the upper surface of the low-level jet that subsequently forms beneath the overlying mixed layer and which extends downstream as the flow develops under steady upstream forcing. This sequence of events which attends the formation of downslope windstorms in the atmosphere is now well understood and has been carefully analyzed using both theoretical and numerical methods. When orographically forced internal waves break over the obstacle by inverting the local density stratification they induce a local critical level (Peltier and Clark 1979, 1983) for upward propagating waves at which the reflection coefficient may equal (or exceed) unity. This supports a resonant growth of the wave field in the cavity formed by this critical layer and the surface, and to ``transition'' of the flow into a ``stratified hydraulic'' regime. In its fully developed form the flow comes to include a hydraulic jump of the kind that may be generated in one-layer unstratified flow over an isolated obstacle (Long 1953).

Our concern in the present paper will be to address a mild controversy that has arisen with respect to the interpretation of the above mentioned oceanographic example of such flows whose properties have been thoroughly documented using acoustic Doppler profiler and CTD measurements. This concerns primarily the issue as to the dynamical origin of the intermediate layer of well mixed fluid. Farmer and Armi (1999 a) suggest that this fluid is created by diapycnal mixing associated with continuing action of small scale shear instability over the long time scale of tidal evolution rather than by the ``breaking'' of the parent internal wave as has clearly been demonstrated to be the case with respect to the atmospheric phenomenon. A further discussion of the same Knight Inlet data set has also been presented very recently by Klymak and Gregg (1999) who suggest that the flow is significantly three dimensional, thus undermining the basis of the Farmer and Armi interpretation. Their budget analysis suggests that the source of the intermediate water that is the focus of the Farmer and Armi interpretation is most probably lateral isopycnal entrainment. In the present study our goal will be to demonstrate, by performing an appropriate sequences of high resolution numerical simulations, that breaking of the primary forced internal wave remains the main source of mixing in two-dimensional geometry and the dynamical properties of the flows in this regard are strikingly similar to their atmospheric counterparts, and to the Knight Inlet observations, in essentially all respects.

Under appropriate simplifying conditions, steady-state stratified flow over finite amplitude topography can be described using theoretical ideas developed by Long (1953). When the upstream inflow velocity and buoyancy frequency are height independent and the flow is assumed to be two dimensional and time independent, the nonlinear advection terms in the momentum and internal energy equations vanish identically and the field equation which determines the stream function becomes identical to the form that is obtained on the basis of linear perturbation theory. Although one is still obliged to solve this equation subject to the full nonlinear lower boundary condition (e.g. see Laprise and Peltier 1989 c) it is possible to obtain analytic solutions for certain topographic shapes. In general, however, Long's steady-state ideas are inappropriate even when the upstream inflow conditions are fixed. This has to do with the fact that for sufficiently strong forcing these flows become unstable (Peltier and Clark 1979, 1983; Durran 1986; Laprise and Peltier 1989 a, b, c; Scinocca and Peltier 1989, 1991, 1994 a, b; Peltier 1993). This instability develops when streamlines overturn so as to induce a region of convective instability in the developing primary wave field and matures into the finite amplitude fully developed ``storm''.

A fundamental property of this fully developed ``storm'' is the eventual emergence of the previously mentioned Kelvin-Helmholtz instability in the shear layer that forms between the low-level high velocity jet and the overlying almost stagnant region created by wave breaking. The detailed properties of this secondary instability have been explored in Scinocca and Peltier (1989, 1991, 1993, 1994 a, b), Peltier and Scinocca (1990), and Peltier (1993). Our own most recent study (Afanasyev and Peltier 1998, 1999) also demonstrated that the downstream propagating Kelvin-Helmholtz vortices are themselves subject to a further three-dimensional instability which takes the form of streamwise oriented vortices of opposite sign, the same instability which controls the mixing transition in free shear layers (Peltier et al., 1978; Klaassen and Peltier 1984, 1989, 1990; Caulfield and Peltier 1994). Thus a complex picture of the developing storm as comprising a nested sequence of distinct hydrodynamic interactions and different forms of instability has now been developed.

Long's theory, however, may still be applied to mimic some of the time averaged properties of the developed flow, as clearly demonstrated with the model suggested by Smith (1985). In this simple model based on Long's equations the existence of an intermediate mixed layer is taken as given rather than being a feature of the flow to be explained. The streamlines are assumed to split around this internal ``block'' thus creating an effectively pliant upper boundary for the lower layer. A second approximate approach that has been invoked to describe flow over the obstacle is of course that based upon a simple hydraulic analysis. The application of hydraulic analysis requires that the flow under consideration be divided into layers with uniform density and velocity. The principles of mass and momentum conservation are then applied to each layer. Hydraulic theory may be employed to describe the steady-state characteristics of a flow subject to certain constraints. In particular an asymmetric flow with different conditions upstream and downstream of the obstacle may exist only if an appropriately defined Froude number for the flow achieves the value of unity at the obstacle crest (e.g. Armi 1986, Baines 1995). These asymmetric flows are referred to as being ``controlled'' in the language of hydraulics. Such hydraulic analysis has been employed by both Farmer and Armi (1999 a) and by Klymak and Gregg (1999) primarily as a consistency check on the flow observed in Knight Inlet. In the course of the present work we will also provide an estimate of the appropriate nondimensional parameter (Froude number) for our numerically simulated flows in order to demonstrate that the flows do indeed undergo transition at the sill crest from a subcritical into a supercritical regime.

For incident flows characterized by uniform velocity (U) and constant buoyancy frequency (N), dimensional analysis requires the entire flow to be governed by three primary nondimensional parameters (these parameters were collectively referred to as Froude numbers in some previous publications), namely

Fv =

hN


U

,  Fh =

aN


U

,  Fd =

DN


U

.

These parameters represent the ratios of different length scales (namely height h, and width a of the topography and depth of the fluid layer D when this is finite as in the oceanographic case of interest to us here) to the intrinsic vertical length scale of the flow U/N. The parameter Fv is a measure of the degree of nonlinearity (due to the finite amplitude lower boundary condition) to be expected of the flow. When Fv is large enough the topographic wave can achieve critical steepness and ultimately overturn (break). The second parameter Fh provides a measure of the importance of nonhydrostatic effects. The third parameter, Fd = DN/U is an additional parameter that does not arise in the infinite depth atmospheric case. A fourth parameter Fr = HN/U, in which H is the density scale height also requires mention. This latter nondimensional parameter measures the importance of non-Boussinesq effects. It was shown in a recent study (Afanasyev and Peltier, 1998) that these effects can be a significant factor in the dynamics of severe downslope windstorms in the atmosphere, in fact they are determinant of the intensity of the Kelvin-Helmholtz instability induced pulsations which themselves control the strength of the transience that is characteristic of the fully developed flow. For the purpose of the analyses to be discussed herein this additional parameter may be safely disregarded. In what is to follow we will vary the first of these parameters, Fv, by varying the amplitude of the topography. On the other hand the width, a, of the topography will be kept constant in all simulations so that the second parameter Fh will vary only with the inflow velocity, U. Since the typical width of the topography is quite large, one may assume the flows to be essentially hydrostatic on the largest scales. The third parameter, Fd, determines the number of vertical wavelengths of the primary hydrostatic forced internal wave that will be contained over the finite depth of the oceanic layer. This number will also be varied by varying the inflow velocity while the depth, D, of the layer will be kept constant in all simulations.

Our main purpose in this paper will be to more fully explore a number of basic dynamical features of the observed Knight Inlet flow in a specially designed series of two-dimensional numerical simulations. Such oceanic flows have not previously been explored using modern methods of computational fluid dynamics. In the following sections of this paper, the equations of motion and boundary conditions that define the numerical model that we employ are fully discussed. The new sequence of simulations which document the evolution of the flow for different values of inflow velocity and amplitude of topography are described and discussed in section 3. Conclusions are summarized in section 4.

2  Model equations and boundary and initial conditions

The basic computational structure of the numerical model to be employed to perform the simulations discussed herein is essentially that described in Clark (1977) and first employed in the context of analyses of topographically forced atmospheric internal waves in Clark and Peltier (1977). For the purpose of the present application in an oceanographic context the original anelastic model was modified to allow the simulation of essentially Boussinesq flows of thermally stratified water, the differing contributions of temperature and salinity to the determination of the density field being irrelevant in the present context (see Bush, McWilliams and Peltier 1994 for further discussion). The governing equations of the dynamical system on which the ensuing analyses are based are therefore:

 

 

 

 


r
 

 

dui


dt

= -i p-di3gr+j


r
 

KM Dij

(1)

 

 

i(ui) = 0

(2)

 

 

 


r
 

 

dq


dt

= i (


r
 

KH i q)

(3)

 

in which ui (i = 1,2,3) are the components of velocity in (x,y,z) Cartesian coordinates , and q is potential temperature. p and r are pressure and density fluctuations respectively, such that:

 

 

 

p =

_
p
 

(z)+p(x,y,z,t)

(4)

 

 

r =


r
 

+r(x,y,z,t)

(5)

 

 

 

d

_
p
 


dz

= -


r
 

g

(6)

 

The overbar denotes a hydrostatic background state variable, the density in which is assumed to be constant, and the deformation tensor Dij is given by

Dij = j ui +i uj -

2


3

dijk uk

(7)

The Boussinesq approximation filters sound waves and removes the impact of the background variation of density on internal wave amplitude. The connection between the thermodynamic variables r, q will be herein assumed to be given by a linearized equation of state for density, namely:

r =


r
 

(1-aq),

(8)

in which [`(r)] is the background density and a = 2 10-4 K-1 is the thermal expansion coefficient. The values of the mixing coefficients for momentum, KM, and for heat, KH, are taken to be equal to zero so that the the flow domain is assumed to be effectively inviscid.

The extent of the model domain to be employed for all of the analyses to be reported in what follows will be 1000Dx * 200Dz where Dx = 5m and Dz = 0.5m so the numerical resolution will be exceptionally high in these simulations. At the inflow and outflow boundaries, open boundary conditions are applied while a rigid lid condition is imposed on the upper horizontal boundary of the model. Since the model is intended to simulate the actual flow in Knight Inlet, the topography (h) to be employed in all of the simulations is given by the following piecewise linear approximation to the actual topography of the Knight Inlet sill (see Farmer and Armi 1999a):

h(x) = h0(a+bx),  









a = b = 0,

if x < 1000m;

a = -1.74, b = 1.74 10-3,

if 1000m x < 1420m;

a = 0.18, b = 3.86 10-4,

if 1420m x < 2340m;

a = 18.68, b = -7.52 10-3,

if 2340m x < 2490m;

a = b = 0,

if x 2490m;

 

(9)

in which h0 is the amplitude of the topography. In most of the experiments the parameter h0 will be fixed to the constant value h0 = 40 m since this is the value for which the above representation fits the observed topography most accurately. However, we will also perform a series of simulations for topography of variable height in which h0 will be varied through the range of values h0 = 4-40 m in different experiments.

In all of the experiments to be discussed in the present paper, the upstream profile of u will be assumed to be independent of depth. In the experiments where time dependent tidal flow is simulated, the inflow velocity will be specified to be a harmonic function with a period of 7 h which provides a good fit to the local tidal oscillations at Knight Inlet. In the experiments where quasi-steady upstream flow is imposed the velocity is kept constant in time in a given experiment, while its' amplitude is varied through the range U0 = 0.1 - 0.5 ms-1 between different experiments, again spanning the range that characterizes the oceanographic observations. In most of the experiments to be described the background profile of density will be assumed to be representable by a piecewise linear approximation to the density profile given in Farmer and Armi (1999 a), namely:

r(z) = r0(a+bz),  



a = 1.0, b = 1.22 10-5,

if 0m z < 92.6m;

a = 0.779, b = 2.05 10-3,

if 92.6m z < 100m;

 

(10)

in which r0 = 1025 kgm-3 is the value of density at z=0. The values of the parameters employed in the different experiments are summarized in Table 1. In all our simulations the flows were initialized impulsively at t = 0. This gives rise to potential (irrotational) flow at this time, which is described by a Laplace equation for the stream function with appropriate boundary conditions. This formulation of the physical problem implies that one can use the potential flow over a given topography for ``smooth'' start-up of numerical calculations in order to avoid nonphysical transients which otherwise occur in ``shock'' start-up (when the flow is initialized without topography and then the topography is suddenly introduced). All of the analyses that we will report herein have been performed on the CRAY J916 computer system in our laboratory in Toronto.

As a primary target for these analyses we show in Figure 1 a typical acoustic Doppler image of the density and velocity vector fields for a typical image of the flow derived from the Farmer and Armi observations. Our goal will be to understand as much of the scale dependent structure as possible on the basis of very high resolution direct numerical simulations of tidally induced flow evolution.

3  Results and interpretation of the simulations

Several numerical solutions of the initial value problem have been constructed for different values of inflow velocity and amplitude of the topography (Table 1). The first series of simulations, in which the inflow velocity was varied from experiment to experiment while the amplitude of topography was fixed at h0 = 40 m, was performed for the purpose of investigating different stages of tidal flow evolution characterized by different tidal velocities. Since a typical half cycle of the tide is 6 - 7 h one would expect a quasi-steady approximation to be reasonably representative of the actual flow at each time in the course of its evolution as long as the typical time to achieve a fully developed statistically steady state is significantly less then a half period of the tidal cycle.

3.1 Time independent upstream conditions: the influence of inflow velocity

Two representative sequences of images illustrating flow evolution for two experiments (2, 4 in Table 1) are presented in Figures 2 and 3. Note that in all of our images the flow is directed from left to right. The typical structure of the stationary (zero horizontal phase velocity) wave induced by the sill can be observed in both sequences of images (Figure 2 a,b and Figure 3 a,b). A fluid particle moving vertically with initial velocity U in the stratified fluid with buoyancy frequency N can ascend or descend to a maximum height U/N. In our case the buoyancy frequency in the lower layer is approximately N = 0.01 s-1 which gives U/N = 22, 44 m for U = 0.25, 0.5 ms-1 respectively. It is clear that particles from the lower levels do not have sufficient kinetic energy to reach the crest of the sill at the height of 40 m for smaller values of inflow velocity and upstream blocking effects therefore occur in the first example. The obvious difference between the two cases is the vertical wavelength of the wave induced by the sill. The vertical wavelength lv for a steady wave can be easily estimated as lv = 2pU/N which gives lv = 237, 475 m for U = 0.25, 0.5 ms-1 respectively. Since there is a maximum negative deflection of isotherms at the topography (the topography can be considered to constitute a lower streamline so long as boundary layer separation does not occur: see below) and there must be vanishing deflection at the upper boundary of the fluid layer, it is clear that one should observe an odd number of quarter wavelengths. This estimate gives 1/4lv = 34, 69 m respectively, and it is not therefore surprising that one can observe only one quarter wavelength in the case of higher inflow velocity (Figure 3) while an additional positive maximum in the deflection of isotherms will be observed in the lower velocity case(Figure 2) which indicates that the depth is sufficient for three quarter wavelengths to develop in this case. Note that the wave appears to be contained to the vicinity of the horizontal location of the crest of the sill and is therefore essentially nondispersive, as is expected in the hydrostatic regime. The horizontal wavelength lh of the wave is quite short compared to the horizontal extent of the topography since the sill is asymmetric to significant degree. The horizontal wavelength can best be approximated as lh = gU/N, where g = 3.75 is the ratio of horizontal extent (150 m) of the region of steep slope on the lee side of the sill to its vertical extent (40 m). One then obtains an estimate lh = 82, 165 m for U = 0.25, 0.5 ms-1 respectively, which agrees well with the observations.

The further time evolution of the internal wave over the sill demonstrates that the wave gradually achieves a vertical orientation of its streamlines and then overturns convectively. In Figure 4 a succession of images is presented to demonstrate in detail the evolution of instability in the overturned wave. These images were obtained from experiment 3 (Table 1) with the inflow velocity equal to U = 0.35 ms-1 and is representative of the typical picture of wave breaking that is observed in all of the simulations. It will be observed on the basis of Figure 4 a, b that vertically oriented density interfaces become unstable and small scale perturbations (horizontal intrusions) develop. These are characteristic of the typical development of the gravitational (convective) instability that occurs in a stationary topographically forced internal wave (see e.g. the recent atmospheric simulations reported in Afanasyev and Peltier 1998, 1999) or the instability of a vertically oriented density interface where the formation of horizontal intrusions has been observed in laboratory experiments (e.g. Voropayev et al., 1993). These small scale instabilities develop on the background of large scale shear due to the primary wave field which acts to further overturn the density stratification. These processes result in intense mixing and ultimately in the creation of an almost stagnant mixed layer of fluid situated over the high velocity jet which is simultaneously forming adjacent to the lower boundary (see Figure 5). Since our model is essentially nondiffusive, by mixing we here imply the process whereby significant small scale granulation develops (on the scale of the mesh size of the computational grid) in the density field due to small-scale turbulence rather than truly irreversible mixing which requires the explicit action of molecular diffusion. Mixed fluid can therefore re-stratify to some degree in our simulations. This only occurs over an extremely long time scale, however, and we therefore expect will have only a minor effect on the main features of the flow. It is interesting to compare the simulated wave breaking event with results of the field studies of Farmer and Armi (1999 a). Figure 6 shows their data for the velocity and density distribution acquired during the fully developed tidally forced flow when acoustic sounding images (see Figure 1) indicate the existence of a strong low level jet on the lee slope of the sill. It is clear from the analysis of our numerical simulations that the picture of density interfaces in Figure 6 represents a breaking wave. It is also interesting to note the existence of small scale features (indicated by arrows in Figures 4 and 6) at the tip of the wave and at vertical interfaces. These features are typical of the oceanographic observations as well as of all our numerical simulations. The large scale picture of the density field is also very similar between simulated and observed flows.

An important question in the present oceanographic context is whether the mixing effected by wave breaking is the primary mechanism responsible for the accumulation of fluid in the intermediate (mixed) layer over the slope of the obstacle as opposed to isopycnal fluxes into the layer. Consider for example the volume balance for the layer of density between 24.2 and 24.3 (green in the diagram in Figure 4). The total volume of the layer in the computational domain is conserved since there is no mass exchange between the layers due to diffusion inside the domain and the flux into the layer at the inflow boundary is very precisely equal to the flux at the outflow boundary (otherwise mass would accumulate within the domain). However the volume of the layer may increase locally (consider e.g. the region over the slope of the obstacle) due to thickening of the layer which would be associated with a nonzero local isopycnal flux into the layer. Since the total volume is conserved, such local thickening would require that the layer become narrower upstream (or downstream) of the locally thickened region. Such a flow adjustment can only take place before the steady flow establishes. Thus the ``pumping'' of water from one location along the layer into another cannot be a permanent source of mixed layer water. The estimates of the volume (labelled ``partial volume''in Figure 7) of the ``green'' layer in the region over the slope demonstrate that it increases only during the initial phase of flow evolution. After the onset of wave breaking and the subsequent establishement of the quasy-steady hydraulic-like flow, the volume ceases to grow further and is only subject to oscillations due to episodic wave activity. The most important mechanism for the creation of intermediate layer water is clearly wave breaking and the ``mixing'' thereby induces. Here we might better use the term ``entrainment'' rather than mixing because, as previously commented, there is no diffusion in the model and, hence, there is no irreversible mixing. Although the patches of less dense (sand-yellow layer) and more dense (blue-green) water (Figure 4d) entrained into the green layer do not loose their identity, the process of entrainment is dynamically equivalent to mixing in the present circumstance because restratification is a much slower process compared with typical time of flow evolution. Thus the process of entrainment represents diapycnal exchange and results in intermediate water formation. The volume (labelled ``total volume'' in Figure 7) of intermediate water including the entrained water has also been estimated. This volume rapidly exeeds the volume initially accumulated due to local isopycnal advection. In experiment 3 (Figure 4) the entrained volume at the end of the simulation is four times larger than the advected volume while in experiment 4 (Figure 5), in which mixing is more intense, the advected volume can be entirely neglected in comparison to that entrained. Note that in Figure 5 the downstream advance of the front of the mixed layer is clearly seen. Persistent creation of intermediate layer water is clearly required to support the downstream extension of this layer. Such continuous creation clearly can not be explained by any adjustment of the flow (narrowing of the layer upstream). Thus the process of entrainment due to wave breaking is indeed the primary cause of accumulation of intermediate layer water.

A further interesting feature of the oceanographic flow described by Farmer and Armi (1999 a) is the existence of an apparent Kelvin-Helmholtz-like instability in the shear layer between the low level high velocity jet and the overlying fluid. The finite amplitude form of this instability was characterized by Farmer and Armi (1999 a) as comprising entrained eddies (see Figure 13 a-c in Farmer and Armi, 1999 a) which cause diapycnal mixing. This feature of such flows is well known on the basis of past analyses of atmospheric examples (e.g. see Scinocca and Peltier 1989 and Peltier and Scinocca 1990) wherein its' importance is found to be very sensitive to non-Boussinesq effects (Afanasyev and Peltier 1998). In our present simulations the Kelvin-Helmholtz vortices are not well formed, although some indication of the existence of such vortices is clearly evident in the example shown in Figure 4 d.

A further interesting characteristic of the oceanographic flow that we have simulated concerns the existence of small scale instability at the density interface between the upper and lower layers at approximately 10 m depth (90 m height above the floor) upstream of the crest of the sill. This feature is indicated by the arrows in Figures 3 d and 5 d. Similar small scale instability is also evident in the oceanographic observations (see the acoustic images in Figure 1 in this paper and 7 a in Farmer and Armi, 1999 a). Farmer and Armi have suggested that this instability might itself be responsible for the onset of irreversible mixing which they suggest might in turn be responsible for gradually creating the intermediate mixed layer which, in turn, they suggest to have been necessary for the deepening and acceleration of the lower layer. They imagine therefore that it is this small scale feature which controls the development of the ultimately hydraulic response. The very high resolution DNS results reported herein, however, provide no support at all for the notion that this instability plays any significant role in the development of the mixed layer that develops downstream of the crest of the sill. Note that a typical scale of Kelvin-Helmholtz billows observed by Farmer and Armi (1999 a) was approximately 30 m in the horizontal and the same in the vertical. The spatial resolution of the model we used in our simulations is 5 m in the horizontal by 0.5 m in the vertical which provides 6 by 60 grid points across structures of the scale in question. Thus such structures are expected to be well resolved in our simulations. The prominence of the well developed Kelvin-Helmholtz billows in the Knight Inlet flow, which are clearly revealed in the Farmer and Armi (1999 a) observations nevertheless requires comment. The most probable influence involved in their somewhat more prominent appearance in the Knight Inlet flow is that associated with separation of the boundary layer in the flow over the sill that has been documented to occur only in the ebb phase of the tidal cycle. It is clear that Kelvin-Helmholtz instability is most noticeable in those acoustic images provided by Farmer and Armi (1999 a) in which the boundary layer has clearly separated from the crest of the sill, rather than in those for which boundary layer separation has been suppressed by hydraulic effects (as in Figure 1 in spite of the fact that the flow is very intense at this moment). Since our numerical model does not permit boundary layer separation in the lee of the topography, this is a most probable cause of the remaining discrepancies between the simulated and observed flows. These discrepancies include not only the prominence of the upstream Kelvin-Helmholtz billows but also the relatively long timescale on which observations suggest the downslope storm to develop. In our simulations of this oceanographic process, however, as in the corresponding atmospheric phenomenon, it is the breaking of the primary large scale internal wave over the crest of the sill that drives the downstream development of the mixed layer rather than the diapycnal mixing engendered by Kelvin-Helmholtz billows. Once the supercritically steepened wave ``breaks'' through the onset of instability of the slowly evolving Long's model solution (explicit analyses of Long's model derived steady-state solutions have been presented by Laprise and Peltier 1989 a) the continuously stratified flow enters the hydraulic regime. Continuous forcing of the supercritically steepened wave by flow over the topography continuously reinforces the occurrence of convective mixing in the region over the crest of the sill, the mixed fluid being thereafter advected downstream behind the advancing ``chinook front''. It is important to note that, once the mixed region is thereby created, the flow may once again be described by an appropriate solution of Long's equation if one assumes the elevated region of mixed fluid maintained by wave breaking to represent an internal ``block'' around which streamlines are split (Smith, 1985). Since the solutions of Long's equation are in a sense an analogue of potential flow for stratified fluid (Baines, 1995), these flows are established rapidly.

Although the flows under consideration herein obviously take on an appearance that is similar to simple one or two-layer hydraulic flows over an obstacle, there is no straightforward procedure whereby hydraulic analysis may be applied in such circumstances because of the continuous density stratification. As previously noted, the application of conventional hydraulic analysis requires that we may divide the flow into layers with uniform density and velocity. In the present circumstance one is obliged to divide a flow having a complex profile of density into many layers and then to apply these hydraulic principles. An iterative procedure whereby this may be attempted has been described by Baines (1995) and this allows one to construct a hydraulic flow in such a many layer system. However, to the extent that one is obliged to employ a large number of layers one inevitably looses any semblance of the simplicity which is the primary motivation for applying the hydraulic analysis. It would be attractive of course, if it were possible to justify, to employ a model comprised of only one or two layers. However, in addition to the complex background density profile of the flows under consideration the individual layers are observed to lose their identity due to mixing. Hence the traditional hydraulics approach is not applicable.

A somewhat different approach was proposed by Farmer and Armi (1999 a) in their analysis of the Knight Inlet flow in which they considered the flow to consist of two layers, the upper layer being almost stagnant and dynamically passive but experiencing intense mixing such as to change its density. The lower layer of constant density was then assumed to move beneath a wedge of fluid with horizontally varying density and thickness. The usual hydraulic principles, namely those of mass and momentum conservation, could then be applied to the lower layer, the only difference being that a horizontally varying reduced gravitational acceleration, defined as g(x) = g[(Dr(x))/(r)], was assumed to act upon it. Here Dr is the density difference between the two layers, g is acceleration due to gravity and r is the density of the lower layer. The dynamics of one-layer flow is described then by the equations of motion and continuity in the form (see e.g. Baines 1995):

 

 

 

 

u


t

+u

u


x

= -g

h


x

 

(11)

 

 

 

y


t

+


x

(yu) = 0

(12)

 

where h is the displacement of the interface and y = y0+h-h is the thickness of the layer (y0 being the initial thickness). Equations (11) and (12) are based upon the assumption of hydrostaticity. For stationary flow one can easily obtain, on the basis of (11) and (12), the following constraint:

(

u2


gy

-1)

dy


dx

=

dh


dx

 

(13)

which demonstrates that at the crest of the obstacle, where [dh/dx] = 0, either [dy/dx] = 0 or the local Froude number, defined by

Fr2 =

u2


gy

 

(14)

is unity. The first condition ( [dy/dx] = 0 at the crest of the obstacle) implies that the flow remains either supercritical or subcritical everywhere in the domain if the Froude number upstream of the crest of the obstacle is respectively greater or less than unity. The second option (Fr2 = 1 at the crest of the obstacle) allows the flow to undergo a transition from a subcritical regime upstream to a supercritical regime downstream of the crest. For stationary flow equations (11) and (12) may be integrated, yielding:

 

 

 

 

u2


2g

=

u2c


2gc

-


x

xc 

g

d


dx

(y+h)dx

(15)

 

 

uy = ucyc = const

(16)

 

where the subscript c denotes the values of variables at the crest of the sill. Integrating the right hand side of (15) by parts one readily obtains the result:

 

u2


2g

+y+h =

u2c


2gc

+yc+hc+

1


g

 


x

xc 

 

dg


dx

(y+h)dx.

(17)

Using the critical condition for the Froude number at the crest of the sill and the continuity equation (16), then gives

Fr2c =

u2c


2gcyc

=

u2y2


gcyc3

= 1.

(18)

Eliminating u from (17) and normalizing y and h by the thickness yc of the layer at the crest and normalizing g by gc, furthermore yields the result:

 

1


2

_
g
 

_
y
 

2
 

+

_
y
 

+

_
h
 

=

5


2

+

1


_
g
 

 


x

xc 

 

d

_
g
 


dx

(

_
y
 

+

_
h
 

)dx

(19)

in which the overbar denotes a nondimensional variable. The equation obtained (which differs from that obtained by Farmer and Armi (1999) by the integral term on the right hand side) allows one to predict the thickness of the lower layer if the density step Dr is given a priori, the analysis then serving essentially as a consistency check upon global conservation properties. Farmer and Armi (1999) showed that the depth of the interface estimated using their method agreed satisfactorily with the observed depth of the interface in the vicinity of the crest of the sill. Based upon our own detailed DNS analyses we may attempt to make contact with this work by estimating the squared Froude number based on the reduced gravitational acceleration and on the depth averaged total velocity of the layer for the numerically simulated flows. The first density isoline that plunges down the lee slope is chosen as the interface between the mixed upper layer and accelerated stratified lower layer in these calculations. The graph in Figure 8 presents the horizontal variation of Fr2 for the flow obtained in experiment 2 (Table 1). This graph demonstrates that Fr2 increases monotonically over the obstacle, reaches a value close to unity at the obstacle crest (at x = 2.1 km), and continues to grow over the lee slope until it drops at the leading edge (hydraulic jump) of the high-velocity jet. These results indicate that the flow is indeed ``controlled'' in terms of the hydraulic terminology (Fr2 1 and [dFr/dx] 0 at the crest) and that it undergoes a transition from a subcritical regime upstream to a supercritical regime downstream of the crest, thus providing direct support of the initial impression concerning the hydraulic-like character of the low-level flow.

The typical time required for the fully developed flow to form may be estimated on the basis of the behavior of one of the primary global characteristics of the flow, namely the wave drag, Dw, exerted by the obstacle on the flow (or vice versa). Our computations of surface wave drag in these simulations have been done using the expression

Dw = -


+

- 

p

h


x

dx

in which p is the pressure perturbation and h is the topography as before. The data shown on Figure 9 a (which are taken from experiment 1) demonstrate that the flows achieve a state in which the drag remains statistically constant after approximately 15 min. Qualitatively similar temporal behavior of the wave drag has been noted in a wide range of atmospheric simulations (e.g. see Scinocca and Peltier, 1989) where the flows are shown to develop into a ``high-drag'' state that is the most characteristic feature of a severe downslope windstorm. Thus the hypothesis that the phenomenon under investigation involves the quasi-steady development of a slowly varying flow seems to be valid, at least as a first approximation. Of course, one has to account for the irreversible mixing which takes place when the topographic wave breaks and for the gradual accumulation of mixed fluid that forms the intermediate layer. The degree to which the accumulation of mixed layer water plays a dynamically significant role will be further discussed in what follows in connection with the simulations with slowly varying inflow velocity.

3.2 Time independent upstream conditions: the influence of topographic height

In our second series of simulations in which the amplitude of the topography is varied from experiment to experiment in the range h0 = 4 - 40 m, while the inflow velocity is kept constant at U = 0.25 ms-1, the influence of nonlinearity due to finite amplitude topography on the breaking of internal waves may be assessed. The analysis of analytical solutions of Long's equation obtained for isolated obstacles in the hydrostatic approximation demonstrate that the critical value of the nondimensional parameter Fv which is necessary for the wave to achieve critical steepness, lies in the range Fcv = 0.5 - 1 (for a review of this problem see Baines, 1995). If the obstacle is asymmetric, the smaller values of Fv occur when the downstream face is steeper (which is the case for Knight Inlet flow). For given U and N the critical amplitude of topography can be estimated as h0c = 12.5 m (for Fcv = 0.5). Overturning and subsequent wave breaking is observed, however, in all of our simulations even with the smallest amplitude of topography h0 = 4 m (Figure 10), although breaking occurs only very locally at low level and over an extremely long time scale (approximately 100 min) in this case. This result is in agreement with the results of numerical simulations by Lamb (1994) in which transient motions which contained overturning regions were observed in the flow over smooth topography for Fv < Fcv. A further interesting feature is observed in the behavior of the wave drag for the two flows with subcritical values of the amplitude of topography (h0c = 4, 10 m, experiments 5, 6 in Table 1). For these experiments the wave drag history reveals variations (see the graph in Figure 11) having a characteristic time scale of approximately 40 min, the variations being clearer in the simulations having the lowest value of the amplitude of topography (Figure 11 a). Such variations have been observed by a number of different authors in both numerical simulations and laboratory experiments (see e.g. Baines, 1995) and were believed to be associated with the generation of forced long wavelength waves that propagate upstream.

3.3 Harmonically varying upstream conditions: tidal flows

In order to investigate the way in which the temporal variability of the incident tidal flow influences the dynamics of the flow over a sill, two simulations (experiments 8 and 9 in Table 1) in which the inflow boundary conditions are harmonically varied with the time period of 7 h have been performed. The maximum amplitude of the inflow velocity was chosen to be Umax = 0.25 ms-1 in both simulations. In the first simulation the background stratification is piecewise linear as in our previous quasi-steady simulations, while in the second simulation the background stratification is smoothed at the thermocline (between 75 - 90 m) to more closely follow the density distribution for the Knight Inlet flow described in Farmer and Armi (1999 a). Both simulations are initialized from some small value of inflow velocity (U = 0.04,  0.11 ms-1). The evolution of the flow observed in the simulations can be well represented by a single global characteristic, namely the drag exerted by the flow on the obstacle. The temporal variation of the drag for the first simulation is shown on Figure 12 together with the inflow velocity at the upstream boundary. Inspection of the data of Figure 12 shows that the drag monotonically increases and reaches its maximum value when the inflow velocity approaches its maximum, following which the drag falls and then stabilizes in some statistically steady state (Dw = 9  kgm-1s-2) for an extended period of time (approximately 150 min) whereafter it decreases as the flow relaxes. This statistically constant value approximately corresponds to the value of drag in the developed flow observed in the quasi-steady simulation with U = 0.25ms-1 (experiment 2 in Table 1). Thus, a certain asymmetry (one can also characterize this as hysteresis) in the evolution of the flow is evident from the time dependent behavior of the wave drag. This temporal asymmetry can most probably be explained by the fact that the flow enters a hydraulic-like regime which involves strong spatial asymmetry associated with the formation of the intermediate mixed layer due to irreversible mixing caused by breaking of the primary wave. The fluid in the mixed layer is almost stagnant and cannot be advected away very rapidly. The flow over the sill therefore possesses a degree of inertia and remains in the hydraulic regime for some time even after the inflow velocity has relaxed to some degree.

The succession of images presented in Figures 13, 14 and 15 illustrates the evolution of the density field and horizontal velocity (Figure 15) in the flow obtained in experiment 9 at a sequence of times. It is clear that the hypothesis of quasi-steady flow development is justified during the stage of low level acceleration of the flow. On the basis of observations of the general features of the wave field, namely the spatial form of the wave crests and troughs, it is recognized that these are initially similar to those observed in experiment 1 (not shown) in which the inflow velocity is U = 0.1 ms-1, while at later stages the wave field transforms to that appropriate for larger inflow velocity (with corresponding vertical wavelength). As the flow approaches the form that is appropriate to the maximum inflow velocity Umax = 0.25 ms-1, it develops a form that is similar to the flow observed in quasi-steady experiment 2 with the same inflow velocity (compare Figures 13 c, d with Figures 2 c, d). The interesting feature of this simulation is that the leading edge of the breaking wave reaches farther upstream (see Figure 14) from the crest of the sill than it does in a similar experiment 8 or in the quasi-steady simulations. Finite amplitude perturbations (indicated by arrows in Figure 14 c, d) are in this case generated by the breaking wave front and propagate upstream. These perturbations might also be a source of the internal solitary waves which were observed in the flow at Knight Inlet (Farmer and Armi, 1999 b). Farmer and Armi (1999 b) have suggested that these waves are generated as subharmonics of small-scale shear instabilities observed at the descending interface. The stronger upstream influence of the breaking wave in this particular simulation is obviously a consequence of the slightly different background density field in comparison with the remainder of our numerical experiments. In this simulation the transition between the lower and upper layers is smoothed. The increased width of the transition layer clearly allows the wave to penetrate higher into the upper layer and further upstream. The interface deepening over the sill creates an effective sloping bottom so that breaking of the internal wave is visually similar to breaking of surface waves on a beach. This simulation therefore demonstrates that these small but hydrodynamically important details of the flow field are very sensitive to the details of the background density distribution.

4  Discussion

The numerical simulations described herein provide clear evidence that the basic dynamical controls that are exerted on an oceanographic flow over localized topography are very similar in all important respects to the dynamics of those that act in the equivalent atmospheric case in spite of the obvious differences in the upper boundary condition in these two media. This similarity specifically includes the dominant role played by the breaking of a stationary internal wave induced by stratified flow over the topography and which results in intense irreversible mixing. It further includes the formation of the almost stagnant mixed intermediate layer which is a necessary precondition for the transition to a hydraulic-like regime. Once the flow enters this regime it becomes visually similar to one layer flow over an obstacle which includes the formation of a supercritical high-velocity jet in the lee. According to the simulations reported herein (and elsewhere for the atmospheric case) the primary source of irreversible mixing in both atmospheric and oceanographic flows is the breaking of the large scale primary topographically forced internal wave via gravitational (convective) instability.

Our simulations with harmonically varying inflow velocity demonstrate that the topographically forced internal wave field (and the height distribution of crests and throughs) closely corresponds, at each instant of time, to the wave field observed in the corresponding simulations with constant inflow velocity. This result supports our initial hypothesis of quasi-steady behavior of the topographic flow at least at the stage of slowly accelerating tidal forcing. At later times, however, a hydraulic-like regime which is characterized by statistically steady drag, is established in spite of the later variation (decrease) of inflow velocity. This regime has therefore been shown to be stable to variations of the forcing over a range of inflow velocities.

The simulation that we have performed in which the simple piecewise linear approximation to the distribution of density was smoothed at the boundary between the upper and lower layers (at the thermocline), has enabled us to demonstrate that the leading edge of the breaking topographic wave extends farther upstream of the sill crest in this circumstance. Furthermore, in this more realistic circumstance, finite amplitude upstream propagating perturbations develop on the interface, an effect which was not observed in the remainder of the numerical experiments in which the thermocline was assumed to be rather sharp.

In none of the oceanographic flows simulated herein have we observed the development of the finite amplitude Kelvin-Helmholtz billows in the shear layer that forms between the accelerated low level jet and the almost stagnant intermediate layer above in the region downstream of the crest of the sill, though vestiges of structures that might be considered indicators of shear instability are evident in some images. One might imagine, however, that for certain distributions of background density and inflow velocity, the flows might prove to be favorable for the development of the Kelvin-Helmholtz instability. The onset of Kelvin-Helmholtz instability upstream of the crest of the sill in the Knight Inlet flow observed by Farmer and Armi (1999 a) appears to occur during the episode of boundary layer separation from the crest which initially acts so as to supress wave overturning and to modify the upstream structure of the upper layer flow. Boundary layer separation effectively reduces the height of the obstacle creating a ``liquid boundary'' downstream of the crest. The fluid below this boundary is almost stagnant. As a consequence, the amplitude of the topographic wave is reduced and therefore may not be sufficient for the onset of breaking. In contrast the velocity shear is enhanced due to the fact that the flow above the ``liquid boundary'' is more strongly confined in height than it would be in the absence of the boundary layer separation. These effects therefore provide more favorable dynamical conditions for the onset of the Kelvin-Helmholtz instability. For the simulations we have performed with varying inflow velocity, the boundary layer separation that does occur initially during the ebb phase of the tidal cycle in the actual Knight Inlet flow is absent from the numerical results. The fact that such boundary layer separation is not included in our simulations undoubtedly accounts for the remaining discrepencies between the simulated and observed flows, especially for the delay in the time of onset of the downslope storm with respect to the onset time of the ebb phase. During the fully developed phase of the storm, however, it is clear on the basis of Figures 1 and 6 that the boundary layer has reattached and that the directly forced internal wave is breaking. These effects are certainly of great interest, however, and deserve further attention. Our numerical model, however, will have to be modified to allow incorporation of the required no-slip boundary condition and also to incorporate a more sophisticated representation of the turbulent exchanges of momentum and heat. Results obtained on the basis of analyses of this kind will be reported in due course.

Although the nearly homogeneous stagnant fluid region formed by convective mixing in the unstable breaking wave is the sole significant contributor to mixing in our simulations, the influence of diapycnal mixing through Kelvin-Helmholtz instability in the earliest stages of flow development could be a factor in Knight Inlet during the earlier stage of the flow development when boundary layer separation effects prevail and wave breaking is suppressed. Eventually, however, breaking of the primary internal wave occurs and drives the flow into the hydraulic regime.

A further required step in the analysis of the flows observed in Knight Inlet will clearly involve investigation of the three-dimensional effects in the flow which are expected to arise, either due to the onset through instability of three dimensional motions (even in the circumstance in which the topographic forcing is taken to be two dimensional) or due to the specific variations of topography in the cross stream direction which characterize the Knight Inlet ``flume''.

Acknowledgments

The research reported in this paper has been supported by the Natural Sciences and Engineering Research Council of Canada under Grant A9627. The authors wish to thank Dr William Hall for his helpful inputs and Dr David Farmer for giving us access to the exceptionally fine data set that has been collected during the course of the Knight Inlet experiments. These extraordinary data have stimulated all of the results reported herein. Table 1. Parameters used in numerical experiments.

 

Experiment

U, ms-1

h0, m

Fv

Fh

1

0.1

40

4

10

2

0.25

40

1.6

4

3

0.35

40

1.16

2.9

4

0.5

40

0.8

2

5

0.25

4

0.16

4

6

0.25

10

0.4

4

7

0.25

20

0.8

4

8

0.25sin(2pt/T)

40

1.6-160

4-400

9

0.25sin(2pt/T)

40

1.6-440

4-1100

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Figure 1. Acoustic image of the developed flow over the sill in Knight Inlet

from Farmer and Armi (1999), courtesy of D.Farmer.

Figure 2. Contour plots of density in the form of st (the density r = (1000+st) kgm-3)

for the simulation with U = 0.25 ms-1 (experiment 2 in Table 1), t=4 (a), 8 (b), 12 (c),

28 min (d).
Figure 3. Contour plots of density for the simulation with U = 0.5 ms-1 (experiment 4

in Table 1), t=4 (a), 8 (b), 12 (c), 20 min (d). Arrow in frame (d) indicates

instability in the shear layer upstream of the sill crest.
Figure 4. Zoomed image of the flow over the sill for the simulation with

U = 0.35 ms-1 (experiment 3 in Table 1), t=4 (a), 8 (b), 12 (c), 14 min (d).

Arrows show the typical features of the breaking wave
Figure 5. Contour plots of horizontal velocity (ux) for the experiment 4 (Table 1),

t=4 (a), 8 (b), 12 (c), 20 min (d). Arrow in frame (d) indicates instability in the

shear layer upstream of the sill crest.
Figure 6. Contours of constant density and velocity vectors for the developed flow over

the sill in Knight Inlet from Farmer and Armi (1999), courtesy of D.Farmer.

Arrows show the typical features of the breaking wave (for comparison see Figure 4).
Figure 7. Plot of volume of the intermediate layer versus time for two simulations:

U=0.5 m/s (dashed lines), U=0.35 m/s (solid lines). Volume is given by

the number of cells of the computational grid occupied by the fluid of intermediate density.

Figure 8. Squared Froude number for the lower layer at different horizontal locations

for the experiment 2 (Table 1) at t = 24 min.
Figure 9. Plots of drag versus time for four simulations: experiment 1 (a), experiment 2 (b),

experiment 3 (c), experiment 4 (d).
Figure 10. Contour plots of density for the simulation with the amplitude of

topography h0 = 4 m (experiment 5 in Table 1), t=84 (a), 120 (b), 128 (c), 160 min (d).
Figure 11. Plots of drag versus time for two simulations: experiment 5 (a), experiment 6 (b).
Figure 12. Drag, Dw, versus time, t for the simulation with variable inflow velocity

(experiment 8 in Table 1). Plot of the inflow velocity, U is also shown.
Figure 13. Contour plots of density for the simulation with variable

inflow velocity (experiment 9 in Table 1), t=20 (a), 50 (b), 80 (c), 100 min (d).
Figure 14. The same as in Figure 12 but for t=105 (a), 120 (b), 140 (c), 150 min (d).

Arrows indicate perturbations propagating upstream from the breaking wave front.

Figure 15. Contour plots of horizontal velocity (ux) for the experiment 9 (Table 1),

t=50 (a), 100 (b), 120 (c), 150 min (d).


 

 

 

 

 

 

Figure 2

 

 

 

 

Figure 3

 

 

 

 

Figure 4

 

 

 

Figure 5

 

 

Figure 6

 

 

Figure 7

 

 

Figure 8

 

 

Figure 9

 

 

Figure 10

 

 

 

Figure 11

 

 

 

 

 

 

 

 

 

 

Figure 12

 

 

 

Figure 13

 

 

 

Figure 14

 

 

 

Figure 15