bottom: the role of friction

Transcription

1 Published in J. Phys. Oceanogr.2003, 33, No2., The alternative density structures of cold/salt water pools on a sloping bottom: the role of friction G.I. Shapiro 1,3 and A.E. Hill 2 1 Institute of Marine Studies, University of Plymouth, UK. 2 Proudman Oceanographic Laboratory, Bidston, UK 3 PP Shirshov Institute of Oceanology, Moscow,Russia Revised Version 8 July 2002 EDITORS: PLEASE ADDRESS ALL CORRESPONDENCE TO Dr. A.E. Hill, Proudman Oceanographic Laboratory, Bidston Observatory, Prenton, Merseyside CH43 7RA. United Kingdom. ehill@pol.ac.uk

2 ABSTRACT Observed density sections through dense water pools or lenses on sloping topography typically have an asymmetric structure. Usually one side of the dense lens is bounded by isopycnals that slope steeply down to the sea bed whilst, on the other side, the slope of isopycnals is more gentle. A common situation is for the steepest sloping isopycnals to be on the up-slope side of a lens (which we term the head-up state), but occasionally the reverse is true (which we call the head-down state). Here we use a one-and-a-half layer reduced gravity model which resolves the bottom boundary layer to provide physical insight into the three-dimensional evolution of these alternative forms. It is found that the head-up state arises when the thickness of the central core of a lens exceeds about two Ekman depth scales, whilst the head-down state arises when the converse is true. The speed of along- and cross-slope motion of the central, thick core of a dense lens is also investigated and the results from an ensemble of runs with the 3-dimensional reduced gravity model are found to accord suprisingly well with some approximations derived from bulk dynamics. From a practical point of view, the results concerning the shape of isopycnals bounding dense lenses on slopes can provide valuable information from which to infer important aspects of the underlying dynamics. 1

3 1. Introduction In many parts of the world dense water formed on continental shelves propagates along a sloping sea bed, either on the continental shelf itself or after it has spilled over the shelf edge onto the steeper continental slope (Whitehead, 1987). When the sources of dense water are variable in space and time, the resulting dense water masses are likely to take the form of either isolated lenses (cold or salt pools) or dense plumes with significant 3-dimensional structure. Even when the source of dense water is continuous there is evidence that downstream lenses will form (Zatsepin et al, 1998; Jiang and Garwood, 1998). There are few, if any, field measurements of the detailed 3-dimensional structure of such features. Nevertheless, two-dimensional slices through dense bottom water masses on slopes characteristically show an asymmetric structure. For example, in Spencer Gulf (South Australia) dense (salty) water is formed at the head of the gulf by evaporative salinization and flows out of the gulf periodically in a series of blobs at neap tides when tidal mixing is minimal (Bowers and Lennon, 1987). Fig. 1a shows a section through such a dense lens. The lens propagates along-slope with the shallow water to its left (southern hemisphere) and also moves down-slope under the influence of friction (Bowers and Lennon, 1987). From the point of view of this paper, the significant aspect of the structure of the lens is that isopycnals slope steeply down to the bed on its up-slope side whereas there is a longer tail of gently sloping isopycnals on the down-slope side. Similar density structure can be seen in other systems such as the outflow of cold bottom water in the Adriatic Sea (Artegiani and Salusti, 1987; Fig 1b) and the Denmark Strait outflow (eg: Bruce, 1995; Fig. 1c). Not all dense outflows exhibit this structure, however. In dense water cascades such as in the Bass Strait (Tomczak, 1985; Fig. 1d), the most steeply sloping isopycnals are on the downslope side. Flows such as that depicted in Fig. 1d are often thought of as classical gravity currents in which 2

4 the head of the current penetrates downslope. We shall borrow the terminology for this paper from gravity currents. We shall refer to as 'head-down', a structure in which the steepest sloping isopynals (normally associated with the head of a gravity current) are on its down-slope side. Similarly, when the steepest sloping isopycnals are on the up-slope side, we shall use the term 'head-up'. A number of authors have considered the propagation of isolated lenses or eddies (e.g. Nof, 1983; Swaters and Flierl, 1991; Swaters, 1998; Swaters, 1999; Whitehead et al, 1990). Some of these consider in detail the dynamical coupling between active lower (lens) and upper layers. However, the common feature of all of these previous studies is the inviscid dynamical treatment. Some frictional effects are considered in studies of steady near-bottom currents on a sloping seabed, e.g. Smith (1975), Chapman and Lentz (1997). These models simulate the bulk effect of friction using a 'drag law' formulation, however they ignore Ekman velocity veering and differential drag within the bottom boundary layer. Several full-physics primitive equation numerical modelling treatments (discussed below) include dense lenses and frictional effects are incorporated within the model dynamics. However, results from such models suffer from the lack of a dynamical theory with which to interpret the role of frictional effects. In this paper we shall focus on frictional effects. We therefore simplify the dynamics using the fact that the main body of upper-layer water is often much thicker than the lens itself. This simplification was made along the lines of the conventional reduced gravity approach with one active (bottom) layer but with the improvement that the upper layer may also move. However, motion in the upper layer is allowed to impact on the lower layer but not vice versa (one-and-ahalf layer dynamics). As a result of this simplification we are able to complement previous 3

5 inviscid lens studies by exploring the impacts of bed and interfacial friction on the dynamics of bottom layer lenses. This is particularly relevant for shallow coastal seas or over continental slopes where, for instance, tidally generated bed turbulence is likely to mean that bottom friction cannot be ignored. The propagation of isolated, inviscid, dense lenses on sloping topography was discussed by Nof (1983) although he considered only the inviscid (frictionless) case. His central result was that dense lenses translate along-slope at the speed gs/f where g is the reduced gravity of the lens compared to overlying (static) ambient fluid, S is the bottom slope and, f,the Coriolis parameter. Moreover, he found that this speed (which we call the Nof speed) was an invariant because all lenses would propagate along-slope with this speed regardless of their shape and size. The seminal attempt to describe the internal structure of dense lenses propagating on bottom slopes under the influence of friction was due to Shaw and Csanady (1983). They only considered the case where density is uniform in the vertical (from surface to bottom) and found that the density perturbation could be represented by Burger s equation so that a shock-like density front was formed at the leading edge of a lens as it propagated along-slope. Chapman and Lentz (1997) considered a steady along-isobath flow, which was assumed to consist of an inviscid interior with horizontal isopycnals above a well mixed bottom boundary layer with vertical density isolines. The vertical shear in the horizontal velocities was attributed exclusively to the horizontal density gradients. Shapiro and Hill (1997) looked at the problem in terms of a one-and-a-half layer model including bottom friction and resolving the structure of velocity veering in the Ekman layer. That model (used in this paper also) uses a simplified two-layer density stratification, although it provides the three-dimensional structure of both horizontal and vertical velocity. They found the thickness of the lower layer to be governed by a non-linear advection-diffusion 4

6 type equation (which also admitted shock-wave type solutions in some cases) and gave rise to complex three-dimensional evolution of dense water on a slope. On the basis of the latter three papers, at least, we should not be surprised that dense lenses influenced by friction have an asymmetric density structure, particularly in the along-slope direction. In addition to the direct measurements of dense lenses on slopes, several numerical simulations (some with full primitive equation models) have been applied to the problem and exhibit similar asymmetric structure of lenses. Jiang and Garwood (1998) used a primitive equation model to examine the behaviour of dense water on sloping bottom topography. Their simulations showed the evolution of complex three-dimensional structure and, for their parameter sets, the development of a head-up density structure (Fig. 2a). Krauss and Käse (1998) also used a primitive equation model to examine eddy formation in the Denmark Strait overflow. The lenses of cold (dense) bottom water that form in this model also have a 'head-up' structure (Fig. 2b). The descent of sediment-laden gravity plumes was examined numerically by Fohrmann et al (1998) by numerical solution of a reduced gravity plume model. For the situation they describe, the density front associated with the plume is on the downslope side, ie: the 'head-down' state (Fig 2c). In summary, asymmetry of the density structure of lenses is to be expected on sloping bottom topography under the influence the non-linear dynamics of friction. In the along-slope direction sharper density gradients are formed at the leading edge of a lens as it propagates along slope whilst a tail of weaker density gradient is trailed out behind. Although the detailed density structure is likely to be very complex in particular cases, observations and numerical model simulations alike point to two broad tendencies for the density structure in the cross-slope 5

7 direction. Either there is (a) a tendency for the steepest sloping isopycnals (eg: density fronts) to form on the up-slope side of a lens (the 'head up' state) or, (b) the steepest sloping isopycnals occur on the down-slope side (the 'head-down' state). Despite the several modelling studies referred to above, we are not aware of any attempt to examine whether it is, in fact, the case that lenses will tend to evolve into one or other of these states and, if so, to provide a physical explanation for it. This purpose of the this paper is to provide, by application of a reduced-physics model, the necessary physical insight into this problem in order to permit interpretation of measurements and the behaviour of simulations with more complex numerical models. 2. The Model The model that we shall apply to this problem is based on that described by Shapiro and Hill (1997) and provides the basis for clear physical interpretation. It occupies a niche between the full primitive equation approach and the classical stream-tube methods (both of which include frictional effects). The model describes the variable thickness of the bottom layer as well as the vertical structure of current velocities in the bottom Ekman layer and in a frictional sub-layer of the upper-layer near the density interface. It is the capacity to resolve the vertical structure of the Ekman layer that distinguishes this model from other layer models such as that of Jungclaus and Backhaus (1994) and the classical stream-tube approach of Smith (1975). In both of the latter, frictional stress is parameterised by bottom drag laws, thereby effectively distributing friction over the entire lower dense layer. The implications of this will become apparent later. A right handed coordinate set (x,y,z) is used with the z axis pointing vertically upwards. A 6

8 7 uniform bottom slope is assumed and the heights above a horizontal datum line of the (timevarying) density interface and the sea bed are respectively ξ(x,y,t) and b(x,y), where t is time. Consequently h(x,y,t)= ξ(x,y,t) - b(x,y) is the thickness of the bottom layer. Following Shapiro and Hill (1997), the equation is derived from the shallow water mass conservation equation, namely = 0 + F t h (1) where the volume flux, F, is given by = ),, ( ), ( ),,, ( t y x y x b dz t z y x F ξ u (2) The velocity u, is obtained from the linearised two-layer equations of motion subject to boundary conditions of no-slip at the sea bed, continuity of stress and velocity at the interface (possibly with prescribed upper layer flow, u 0 ) and negligible stress in the upper layer interior. When u is substituted into (2) above, which in turn is substituted into (1), the following non-linear governing equation for the lower-layer interface thickness is obtained (see Appendix) (3) Here k is the unit vector vertically upwards f is the Coriolis parameter and g the reduced gravity ) x ( + ) ( = ) u 0 k u u u u h R h R f h g h R R R R +( t h E 5 E B S S B

9 based on the density difference, ρ, between the lower and upper layers. The Ekman depth scale is h E = (2K/f) 1/2 where K is the (assumed constant) vertical eddy viscosity coefficient. The effect of entrainment has been excluded. The velocities that appear in equation (3) are u B g g = k x b = - S u b f B f (4a) These velocities have the same absolute value, u B which is equal to the speed of inviscid along-slope translation of a dense lenses (g S/f) that we call the Nof speed (Nof, 1983; Shapiro and Hill, 1997). The Nof speed is a result of a balance between down-slope reduced gravity and Coriolis forces. The interior velocity in the upper layer is u 0 and the terms in equation 3 related to this quantity are given by 1 u = k x p ρ f 1 = - ρf S 0 0 u p (4b) where ρ is water density and p is the pressure in the upper layer. The upper layer interior flow is thus an imposed quantity which is able to influence the dynamics of the lower layer. On the other hand the lower layer dynamics does not feed back into the upper layer flow so the governing dynamics is one-and-a-half layer. 8

10 The magnitude of local and advective accelerations is assumed small in comparison with the dominant balance of forces involving the pressure-gradient; friction and Earth-rotation. In particular, small local acceleration means that effects such as interfacial gravity waves and topographic Rossby waves are also likely to be less important than the dominant influence of friction. Consequently the horizontal momentum equation may be assumed steady and linear (i.e. small Rossby number). The advantage of this simplification is that wave effects are filtered out of the dynamics enabling us to focus on the role of friction in influencing the structure of dense water pools. In any event, because the dense water pools considered in this paper are closed (i.e. the density interface bounding these structures intersects the bottom around the edges of these structures), then even if gravity or topographic waves were admitted to the dynamics, they could not radiate energy away from a dense lense into the far field. Even though the momentum balance can be assumed linear, the governing equation for h (equation 3) has no linear limit because the interface between layers is allowed to intersect the sea-bed. Consequently, there will always be places (i.e. around the margins of a lens) where interface perturbations will be comparable with, or exceed, the mean lower-layer thickness. The non-linearity of the dynamics in (3) is formally expressed by the functions R 1 to R 6, which are deterministic functions of the non-dimensional layer thickness, h/h E and are given (Shapiro and Hill, 1997) by R 1 (η) = 2Q(η) - Q(2η), R 2 (η) = Q(η), R 3 (η) = 2P(η) - P(2η), 9

11 R 4 (η) = P(η), R 5 (η) = 1/2(P(η) - Q(η) ), R 6 (η) = P(η) - Q(η) +1/4 (Q(2η) - P(2η)), P(η) = 1 - cos(η) e -η, Q(η) = sin(η) e -η (5) It should be noted that the model does not include free parameters, the only tunable parameter being the vertical eddy viscosity coefficient K (or equivalently the Ekman depth scale h E ). Equation 3 has been solved numerically using methods described by Shapiro and Hill (1997). The numerical method is based on the operator splitting method according to which the spatial operator in (1) is split into several advection and diffusion terms. An explicit scheme is used for diffusion with small time steps and a Godunov scheme with larger time steps for advective terms. The problem is solved over a rectangular domain with h = 0 on all boundaries. Horizontal grid resolution was typically 1 km x 1 km. 3. The shape of a lens In the first instance we shall be concerned only with the question of how the internal dynamics of a lens affects its shape on a bottom slope. For the present we shall set the upper-layer flow to zero (u 0 = 0) and return later to the influence of upper-layer flows. Firstly, to illustrate the contrasting dynamical behaviour of lenses on slopes the model has been run under a range of contrasting conditions. In order that the results can be more readily appreciated from a physical point of view, all parameters used are quoted in dimensional terms 10

12 and their conversion to non-dimensional forms is set out by Shapiro and Hill (1997). In what follows the Coriolis parameter is taken as f = 10-4 s -1 and the density contrast between layers is ρ = 0.1 kg m -3. In the following model runs, the vertical eddy viscosity ranges from K = 10-6 m 2 s -1 (representing nearly inviscid cases) up to K = 10-2 m 2 s -1 corresponding to an Ekman depth scale, h E = 14 m. In the model a circular dense bottom water lens is assumed to be placed on the sea bed and the initial shape of the lower-layer (lens) interface is given by r h= h0 exp - r 0 4 (6) where r is the radial distance from the centre of the lens, r 0 is the radius of the lens and h 0 is the initial thickness of the lens at its centre. This shape has been chosen because it gives the lens an initially uniform thickness (flat top) prior to its collapse after release. The centrifugal force is not included in the linearised dynamical equations that give rise to (3). This is justified by the fact that the ratio of the centrifugal to Coriolis forces for a circular lens of radius, r 0, and thickness, h 0, is gh 0 /f 2 r 2 0 which is small and typically has a value of It does not exceed 0.2 even in the extreme cases of the smallest lenses considered below. (a) A lens on a steep bottom slope In the first set of examples we consider a constant bottom slope equivalent to an increase in bottom depth of 1000 m in a distance of 100 km (S =10-2 ), a gradient that is characteristic of continental slopes. The radius of the lens is taken to be r 0 = 10 km and its initial thickness is h 0 11

13 =25 m. The eddy viscosity, K, varies from K = 10-6 m 2 s -1 to K = 10-2 m 2 s -1 in order to examine the effect of varying friction. In Fig. 3 the lens is shown at time, t = 0 and at time t = 4 days later. In each case the contours show the thickness of the lower layer. Cross sections (perpendicular to the slope and through the peak of the lens at both t = 0 and t = 4 days) are also shown. For the nearly inviscid case K = 10-6 m 2 s -1 (Fig.3a). The lens moves along-slope more or less preserving its symmetry. The along-slope translation speed is precisely the Nof speed. However, even very small friction results in some reshaping of the lens periphery (e.g. Fig. 3a cross section). Over time the lens core also collapses (as seen in the cross section) because it spreads under the influence of friction into a thin but extensive tail behind the lens friction (i.e. due to the diffusive term in equation (3)). The effect of slightly increased friction (K = 10-4 m 2 s -1 ) which would correspond to an Ekman depth of just h E =1.4 m is shown in Fig 3b. At t=0 the lens has precisely the same shape as in Fig. 3a and so only the starting position of the centre of the lens is depicted (by the solid circle). The trajectory of the thickest part of the lens is along-slope (Fig. 3b). An important result from the work of Shapiro and Hill (1997) is that significant frictional influence on the lens propagation is only felt directly within about two Ekman depth scales (2 h E ) off the bottom. Layers much thicker than this will behave like the inviscid state and thus propagate along-slope like inviscid lenses (Nof, 1983). This explains why the thick centre of the dense lens in Fig. 3b moves along-slope at a speed of approximately 0.2 m s -1 (the Nof speed for inviscid lenses). This thick central core moves along-slope more quickly than the thinner, frictionally influenced outer edges of the lens. Consequently along-slope asymmetry begins to develop as discussed by 12

14 Shaw and Csanady (1983) and a density front forms ahead of the lens core with a longer, thinner tail trailing behind it. The inviscid lens core is influenced indirectly by friction and collapses as water is pumped out into the tail by the secondary circulation in the Ekman layer. Recent measurements by Sherwin et al (1999) provide evidence for such asymmetry in the along-slope direction. The relative sharpening of contours of interface thickness on the up-slope side of the lens with respect to the down-slope side indicate the development of a head-up configuration. The cross section of interface shape in Fig 3b confirms this and the formation of a thin tail of fluid (about two Ekman depth scales thick) is clearly visible draining out of the central core of the lens on the down-slope side. The peak of the lens itself has, however, not moved perceptibly down-slope. In the bottom Ekman layer there is net down-slope transport so there is significant downslope motion of water over the entire depth of the thinner down-slope parts of the lens and in the thin tail behind. Similarly, down-slope Ekman transport on the thin up-slope periphery of the lens steepens the slope of isopycnals on the up-slope side creating a density front on the up-slope side (as seen be the tighter contour spacing). A section through the dense lens shows a resulting density structure which has a 'head up' form similar to that illustrated in Figs. 1a-c and Fig. 2a. For even higher friction K = 6 x 10-4 m 2 s -1 which corresponds to an Ekman depth of h E =3.5 m (Fig. 3c). An increased proportion of the lens is now within the influence of the bottom Ekman layer and thus takes the form of a thickened down-slope draining tail (in comparison with the thin draining tail in Fig. 3b). The layer thickness contours show some sharpening on both upslope and down-slope sides and also an indication of some down-slope movement of the central 13

15 core of the lens also. The cross section through the lens illustrates the situation more clearly. The thickest point of the lens has shifted down-slope and the interface slopes downwards steeply both up- and down-slope of the central core. Finally, Fig. 3d shows a high friction case corresponding to K = 10-2 m 2 s -1 or an Ekman depth of h E =14 m. The core of the lens with peak thickness has been displaced 20 km down-slope. Much of the lens is now within the Ekman layer and fluid drains down-slope forming a steep interface slope on the down-slope side of the lens with only a gently-sloping interface on the up-slope side of the lens. Consequently this results in a clear head-down configuration. However in cases with strong friction most of the fluid in the lens is trailed out in a tail behind the central peak of the lens. When the entire lens (including the central core) lies within the Ekman layer, then even the along-slope translation speed is slowed in comparison with the inviscid Nof speed as seen by the along-slope translation distance of the core in Fig 3d as compared with Figs 3a-c. The fluid mass of the lens was conserved in all cases. The above sequence thus illustrates neatly how the head-up configuration is the consequence of a situation in which the lens is thick compared to the Ekman depth (e.g compare Fig 3d with Figs 3b,c). In the head-up case most of the lens fluid remains up slope and only a thin tail drains away in the down-slope direction. The steepest interface slope thus occurs on the up-slope side of the lens. On the other hand, when the lens thickness is small compared to the Ekman depth (e.g. Fig. 3d), fluid drains down-slope and consequently the lens thins out on the up-slope side. The non-linearity of equation 3 (i.e. the dependence of the functions R 1 to R 6, on the local lens thickness) means that the thicker parts of the lens within the Ekman layer experience lower frictional forces and will propagate down-slope more quickly than the thinner parts. The 14

16 consequence is a shock-wave like steepening of the interface within the Ekman layer on the down-slope side of the lens. The overall effect, therefore is that the interface slopes most steeply to the bottom on the down-slope side with only a gently sloping interface on the up-slope side where the lens has been thinned, i.e. a head-down configuration (similar to Fig. 1d, and Fig. 2c). The fundamental physics behind the effects described above is a differential drag, i.e. the differential gravitational effect due to unequal drag on various portions of the lens. Differential frictional effects within the lens distort the lens shape. We represent in the model the differential drag by resolving in detail the three-dimensional structure of both bottom and interfacial Ekman layers (unlike slab models). The frictional dynamics is then captured by a single equation (3), for the interface thickness, which has properties giving rise to phenomena such as shock-wave formation. An important consequence of the Ekman-layer resolving approach used in the present model is that, not only do different parts of the dense lens travel at different speeds, but also different parts of the lens can move in different directions (greater along-slope component for thick parts of the lens with greater downslope motion for the thin parts as in Fig. 3). The resulting distortion of the lens shape thus generates frontal structures in different parts of the lens periphery. This aspect of the dynamics is not well reproduced in slab models that use drag laws for friction. In the latter cases, the drag formulation causes friction to be distributed over whole layer (like in Figs. 3b,c) and hence down-slope motion is induced throughout the lens. Consequently, slab-models exaggerate the tendency for down-slope motion in all parts of a lens. Moreover, they will inevitably tend to produce a 'head-down' density structure by producing fronts on the down-slope side. From our model, head down states are produced when the lower layer is largely within the Ekman layer. Consequently when friction is distributed over a lower 15

17 layer (even one much thicker than the Ekman layer) by drag-law formulations in slab models, the effect is to produce head down states. We now turn to the question of the rate of spreading of lenses under the influence of friction, firstly on a purely flat bottom and then over a slope. (b) A lens on a flat bottom Fig 4 illustrates the spreading of a lens under the influence of friction. To isolate the influence of topography we consider first only a flat sea-bed. The initially circular lens has radius r 0 = 20 km and the Ekman depth-scale is h E =14 m. There is no net translation of the lens because, in the absence of a bottom slope, the Nof speed is zero. The lens merely spreads symmetrically as fluid drains away from the central core within two Ekman depth scales of the bottom. The spreading tendency of dense lenses can be quantified from equation (3). Elimination of the advective terms on the left hand side (because the upper layer is assumed to be at rest and the Nof speed is zero) gives an equation which is mathematically equivalent to frictionally-induced diffusion of the density interface on a flat bottom. From Shapiro and Hill (1997) the instantaneous position of the front that bounds the thin frictionally influenced periphery of the lens is given by X fr = C η 3/2 F g h f E f t (7) 16

18 where C is a constant which is approximately unity and η F is the non-dimensional thickness of the frictional periphery of the lens, also approximately unity. In the derivation of equation (7) it is assumed that initially the bounding fronts were infinitely steep. (c) A lens on a gently sloping bottom We now turn to the situation on a gentle bottom slope characteristic of continental shelves. The slope is taken to be an order of magnitude less than that used previously, equivalent to an increase in bottom depth of 200 m in 100 km (S = 2 x 10-3 ). Since we are now dealing with a potentially wider shelf region compared to the narrower continental slope regions examined in sub-section a above, larger lenses are considered with r 0 = 20 km. The eddy viscosity is set as K = 10-2 m 2 s -1 equivalent to an Ekman depth scale of h E =14 m. In the first case, the initial thickness of the lens is h 0 = 100 m and exceeds significantly the Ekman scale, and thus allows the central core to behave in an inviscid manner (Fig. 5a). However, when the bottom slope is gentle, the along-slope translation of the lens is only small (0.02 m s -1 ) because the Nof speed is proportional to the bottom slope. In these circumstances the frictional spreading (i.e. gravitational collapse) of the lens is more important than its alongslope self-advection (Fig.5a) as evidenced by the expansion in size of the lens. Nevertheless, the thickest part of the lens still moves along-slope more quickly than the thinner periphery and a thinner tail is drawn out behind the lens giving rise to along-slope asymmetry in the density structure. In the cross slope direction, however, the dense lens remains fairly symmetrical (Fig. 5a). When the initial thickness of the lens is smaller (h 0 = 25 m), it moves down-slope under frictional influence and steeper isopycnal slopes are formed on the down-slope side of the lens ('head down' state) as shown in the cross section (Fig. 5b) 17

19 Assmuing that the result (equation 7 above) for the rate of frictional spreading on a flat bottom also gives a reasonable indication of the frictional spreading rate on a slope, then the expected time, τ, after which the speed of frictional spreading of the front matches the along-slope Nof speed, g S/f, is given by τ C 4 he f g S 2 3 = η F 2 (8) This time-scale shows strong dependence on the bottom slope. For small slopes, it takes a long time for the frictional spreading to slow down and become comparable with the along-slope translation speed. Equation (7) exactly relates to the spreading of an idealized dense water pool with extremely abrupt fronts. In the first instance, the fronts produce pressure gradients, which exceed the gravitational force due to sloping seabed and hence the speed of spreading due to pressure gradients exceeds the Nof speed. As the lens spreads out, however, the bounding fronts become more gentle and the speed of spreading due to the pressure gradient force matches the Nof speed on the time-scale given by equation (8). This equation gives an approximate estimate in cases when the Nof speed is small and a lens first spreads due to frontal pressure jump. 4. Trajectory of a lens The previous section has demonstrated how the shape of a lens evolves under the influence of friction. However, the lens as a whole also moves both along- and down-slope at rates which depend upon the bottom slope and the thickness of the lens in comparison with the Ekman depth 18

20 scale. To investigate this further, the model has been run over a range of parameters (Fig. 6). In the following set of model runs the initial thickness of the lens core is allowed to vary, as is the bottom slope. The eddy viscosity is fixed at K = 10-2 m 2 s -1 (h E =14 m). Note that equation (3) is scalable. This means it can be converted to a non-dimensional form in such a way that it does not include any dimensional or non-dimensional parameters (see Appendix ). This allows us to re-scale the same non-dimensional numerical results to cover other values of the frictional coefficient, if required. The along-slope velocity of the central core increases with both increasing bottom slope and initial lens thickness (Fig. 6a). As lens thickness increases the flow becomes increasingly invisicid in nature (i.e. as the Ekman number h/h E becomes small) and hence the along-slope velocity asymptotes to the (inviscid) Nof speed of along-slope lens propagation. The down-slope (negative direction) velocity is larger for thinner (more frictional) lenses, as the Ekman number h/h E becomes large and with increasing bottom slope (Fig. 6b). For progressively thicker (i.e. more inviscid lenses) the cross-slope velocity tends to zero, consistent with pure along-slope Nof translation. To investigate these dependencies more closely we have plotted from the range of model runs (Fig. 7a) the along-slope velocity of the central core (determined after 4 days) normalised by the along-slope Nof speed, g S/f. This quantity is plotted against the thickness of the central core (after 4 days) normalised by the Ekman depth scale. It is found that the results from all runs collapse onto a single curve (solid circles in Fig 7a) in which thick lenses propagate along-slope at the Nof speed and thinner lenses travel more slowly. Similarly, the angle of down-slope 19

21 movement of the lens core has been plotted against the thickness of the core at the end of the model run (normalised by the Ekman depth scale). The angle is defined as the ratio of cross- to along-slope distances travelled by the lens and has been determined after 4 days of evolution. This too (Fig. 7b) shows that the numerical results all fall on a single curve in which progressively thicker lenses have smaller down-slope motion. The full governing equation (3) is necessary to describe the detailed internal dynamics of a lens and, in particular, how its shape evolves over time. However, given the similarity form of the properties of the trajectories of the lens cores (Fig. 7), it is of interest to examine whether the physics behind this can be understood in terms of a simplified scale-analysis of the lens dynamics. Consequently we consider the dynamical balance of a water column assumed to represent the thick, central region close to the lens core. The momentum balance within the lens is given by 2 u fv g' S + K 2 z 2 v fu + K = 0 2 z = 0 (9) where positive x indicates the up-slope direction and positive y is the along-slope direction (in the direction of Nof translation) and all terms have been defined previously. Integrating this over the entire thickness, h, of the lens (and denoting depth-averaged velocities by U and V) gives 20

22 fvh g ' Sh + τ top, x τ bottom, x = 0 fuh + τ top, y τ bottom, y = 0 (10) The stresses at the top and bottom of the lens are then parameterized in an order of magnitude sense as follows. Firstly, consider the case when the lens is thinner than the Ekman depth scale (h<< h E ). At height h E above the bed, the velocity in the fluid (in the upper layer) is zero and similarly at the sea-bed the velocity is also zero (no-slip condition). The velocity at the top of the lens (height, h, above the bed) is taken to be characterized by the depthaveraged velocity of the lens (U,V). The stresses at the top and bottom of the lens in the y (along-slope direction) are thus approximated by τ τ top, y = bottom, y V K he V = K h τ τ top, x bottom, x = K U h E = K U h (11) The difference between parameterizations of these stresses arises because the near-bottom velocity shear is confined into a scale h (the actual lens thickness) whereas, at the top of the lens, the shear is formed by the Ekman spiral above the lens which extends a distance h E into the upper layer of lighter water. Using these stress parameterizations in (10) above, gives U 1 1 = 1 + V 2h' h' V V = ( h') ( 1 h ) 2 Nof 1+ + ' 4 21

23 (13) where h = h/h E. Secondly, for the case of a lens that is thicker than the Ekman depth, the shear stress is better represented by equations τ τ top, y bottom, y = K V h E = K V h E τ τ = top, x 0 bottom, x = K U h E (14) In this case the thickness over which the bottom stress is calculated is h E because there is now enough room inside the lens for the Ekman spiral to be accommodated. The other significant difference is that in (14) the stress in the x-direction at the top of the layer is zero. This is because, when the lens is thick, down-slope motion is confined to the bottom Ekman layer at the base of the lens. In the interior of the central column of lens all motion is along-slope. Using these parameterizations, the equivalent of equation (13) for the thick lens case is U V V V 1 = h' Nof 1 = ( h') 2 (15) The results of this simplified scale-analysis (13) and (15) are plotted in Fig 7 (denoted by SA for scale-analysis ), together with the results from the ensemble of numerical model runs.. 22

24 Although the scale-relations (13) and (15) were derived with the aim of providing only an order-of-magnitude indication of lens translation properties, the numerical model results and the scale-relations are surprisingly close. This probably means that the main physics has been captured reasonably well. The conclusion is that the dependence of the translation of the central core of a lens (i.e. the water column in the vicinity of the region of maximum thickness) can be understood from a straightforward dynamical balance which nevertheless takes into account some detailed understanding of the nature of the Ekman layer within a lens. On the other hand, this analysis will not adequately describe the motion of parts of the lens away from the central core. This is because results using the full dynamics (such as illustrated in Fig. 3) reveal the importance of detailed differential motion within a lens in which features such as thin tails are formed in the outer regions of a lens where the influence of friction is strongest. Effects such as interface-steepening by shock-wave-like phenomenon on account of the non-linear (layer-thickness dependent) propagation are also found in the outer margins of the lens. In order to describe the details of these aspects of lens motion, the full governing equation (3) is needed. 5. The impact of upper-layer flows The interaction between upper and lower layers has been considered in a number of papers concerned with isolated lenses or eddies on slopes. These treatments, however, have all appealed to potential vorticity conservation in inviscid cases (e.g. Swaters and Flierl, 1991; Swaters, 1998, 1999). However, potential vorticity conservation is not applicable in our circumstances where the emphasis is on frictional effects in the lower layer. Our primary interest is in the factors that may bring about a change in shape of a dense lens. Consequently we will restrict the involvement of the upper layer to situations where horizontal upper-layer shear can 23

25 cause distortion of the lower layer by differential forced Ekman transport within different parts of a lens. Consequently the dynamics (represented by equation 3) allows the upper layer to influence the lower layer, but full 2-layer coupling in which the thinner lower layer influences the upper layer is not permitted within the scope of the present analysis. We have explained the asymmetry in lens shapes over slopes (i.e. the head-up and head-down configurations) in the simplest possible terms, which involve only the internal dynamics of lenses themselves. Consequently, we have treated the upper layer as deep and static. In some of the field observations presented (eg: Fig. 1a) this may not necessarily be the case and here we consider briefly the likely effects of flow in the upper layer. It is possible that lens asymmetry could arise by external forcing imposed upon a lens. For example, horizontal shear in an interior (geostrophic) flow in the upper layer above a dense lens would give rise to differential Ekman transport rates in the bottom boundary layer which could produce asymmetry. This physical effect is expressed mathematically by the second term on the right hand side of equation 3 which represents the vorticity of the upper layer flow. To isolate the effect of the upper layer flow we take the sea-bed to be flat (S = 0) so there is no Nof translation. The lens radius is taken as r 0 = 20 km and the initial thickness of the lens is h 0 = 25 m. The vertical eddy viscosity is taken to be K = 10-2 m 2 s -1 (h E =14 m). In all cases, contours of the lens thickness, h 0, are shown at t = 0 and at t = 7.3 days. In the first case (Fig. 8a) a uniform (no horizontal shear) upper layer flow with speed u 0 = 0.1 m s -1 is imposed, flowing in the direction indicated in Fig. 8a. The lens moves in the direction of upper-layer flow as expected, but moves to the left under the influence of bottom- layer Ekman transport to the 24

26 left of the upper layer flow. The lens also develops a front at its leading edge as it moves under the influence of upper layer flow. Fig 8b shows the development of the same lens under the influence of upper layer shear in which u 0 varies linearly from 0 to 0.2 m s -1 from left to right over the width of the model domain (100 km). The tail rotates to the left and the interface front steepens on the left as differential Ekman transport produces convergence on the left hand side of the lens. Similarly when the horizontal upper layer shear is in the opposite sense (with the same magnitude) the effect is reversed (Fig. 8c). The lens asymmetry may arise not only due to non-uniformity of flow in the interior but also from other environmental factors such as non-uniform bottom slope or non-uniform density stratification (Chapman and Lentz, 1997). However these external factors may or may not be present in a certain situation, while internal dynamics due to friction within the bottom boundary layer is always present. The relative importance of internal dynamics of the dense lens compared to the external (upper layer) flow is given by the ratio of the Nof speed to the upper layer flow speed (Shapiro and Hill, 1997). On a steep slope the Nof speed is high so external shear flow is likely a relatively minor effect. Moreover, on a steep slope the horizontal scale of a lens will be small (a few kilometres), and, in these circumstances, upper layer shear over this horizontal scale is also likely to be small. On steeper slopes, therefore, internal lens dynamics alone is likely to describe the evolution of lens shapes. On the other hand, on the shelf itself, (small slope or flat bottom), upper layer flows may have relatively more influence. 6. Discussion We have provided a straightforward physical interpretation of the observed structure of dense lenses and an explanation of the behaviour of dense bottom water in numerical model 25

27 simulations. The evolution of lens shape is described by the non-linear equation for the interface thickness (equation 3) for which the thickest parts of a lens propagate fastest. These dynamics focus on the slow evolution of density structure under an assumed dominant friction-rotationpressure gradient balance from which gravity and topographic wave motions have been filtered on account of simplification of the momentum equation by linearisation and removal of local acceleration. The physical mechanism, which forms the lens asymmetry is differential frictional drag, its relative strength being dependent on the thickness of the dense water pool. Comparison with the results by Chapman and Lentz (1997, Fig3.) shows that the observed asymmetry between 'head-up' and 'head-down' cases cannot be explained assuming bulk frictional drag. When the central core of the lens is thicker than about two Ekman depth scales, it propagates along-slope largely unaffected by friction, with the force balance being between down-slope reduced gravity and the Coriolis force (Nof, 1983). However, even in the central core, water is drained out at the base of the lens and the lens thickness slowly decreases. On the other hand, down-slope Ekman drainage in the thin edges of the lens steepens isopycnals bounding the lens on its up-slope side whilst Ekman drainage also generates a thin tail of fluid extending away from the core both behind and in the on the down-slope side of the thick core. The result is steepened isopycnals on the up-slope side of the lens (the 'head-up' case). On the other hand, if the initial lens is thinner than two Ekman scales, all parts of the lens (including the thickest central core) tend to move down-slope under the influence of friction. Consequently, because the thick core of the lens moves down-slope faster than water on the thinner periphery, isopycnals steepen on the down-slope side of the lens whilst a thin tail is produced on the upslope side (and behind the core), giving rise to the 'head down' state. If the bottom slope is relatively gentle (characteristic of continental shelves), these effects are 26

28 seen in much reduced form. In particular, whilst a lens develops along-slope asymmetry with sharp fronts developing ahead of the thickest core, the lens largely preserves its initial symmetry in the cross-slope direction. However, horizontal shear in the upper layer can generate asymmetry in the shape of dense lenses even on gentle sloping or flat seafloor. Analysis of this case requires knowledge of the upper layer shear. If a dense lens moves through regions of rapidly varying horizontal velocity, then quite complex deformations could take place. In order to illustrate the value of the above dynamics in the interpretation of measurements, we apply these considerations to the observations of Bowers and Lennon (1987) reproduced in Fig.1a. This is a "head-up" case, which implies that the dense layer thickness exceeds the double Ekman scale. Comparing Fig.1a with theoretical results in Figs.3 (a-d) we can judge that the observed lens shape is similar to Fig.3c, which represents a medium viscosity case, corresponding to the lens/ekman-scale ratio of about 2-4. In the natural environment, the observed dense water pool was elongated rather than circular so that we modelled this situation by looking at the evolution a band of dense water, originally extended along isobaths and having a cross section given by Eq (6). From the information presented in the paper by Bowers and Lennon (1987), (mostly their Figs.3,4 and Table 1) we derived the following parameters of the dense water pool and the current flow: ρ=0.6 kg m -3, S = 1.2 x The observed flow velocity was typically 0.1 m s -1, but ranging between 0 and 20 cm s -1. Hence we approximated the flow field as having velocity u=10 cm s -1 at the centre of the pool and the horizontal shear across the pool as 10 cm s -1 per 36.5 km, higher velocities being at the deeper part of the bay. Numerical experiments show that the final shape of the dense water pool is not sensitive to small variations of these parameters. 27

29 Computations were carried out in non-dimensional form. The advantage of this is that the nondimensional form of equation (3) does not contain any parameters and the only "free" parameter to tune was the unknown value of frictional coefficient (or Ekman scale, h E ), which appears in the initial condition. This parameter was used to match the dimensional cross-section area (defined by the isohaline 36.2 psu) with that of the observed plume and was found to be h E = 8.8 m. The computed shape of the plume is superimposed on the observational plot in Fig.9. Numerical results reflect the main features of the observation: "head-up" shape of the plume is clearly revealed, maximum thickness of the plume ( 27 m) and its cross-slope extension (36.5 km) match reasonably well with the observed values of 31 m and 43 km respectively. The relative thickness in the centre of the dense water pool was h/h E 3, which corresponds to the qualitative theoretical considerations discussed earlier in this section. Although upper layer shear flows can contribute to the development of the shape of bottom water lenses, the results of our modelling show quite strong and persistent shears are required to do do this. In the natural environment, however, it is more likely that bottom water lense shape is influenced by frictional effects which are always present in the bottom boundary layer. In general, as a lens expands in the horizontal under the influence of friction, its thickness will decrease. All lenses will, therefore, eventually collapse into the Ekman layer. At this point thinlens dynamics will prevail and motion will be strongly down-slope and isopycnals will steepen on their down-slope side and eventually form a 'head down' configuration. Observations of lenses in the 'head up' condition (assuming no upper layer shear), therefore, tell us that the core of the lens is thicker than about two Ekman depth scales and implies that the lens is 'young' in the sense that it has not yet collapsed into the Ekman layer. 28

30 As a result of this study we now have two complementary theories: (a) Inviscid two-layer theories describe the dynamical impact on lenses and eddies of coupling between active upper and lower layers through vortex stretching over sloping topography but ignore frictional effects. (b) The present work establishes the role of bottom and interfacial friction upon the internal dynamics of dense lenses over sloping topography subject to one-and-a-half layer dynamics (i.e. ignoring coupling between active upper and lower layers) An important goal, therefore will be to combine in future the essential dynamics of each approach to obtain a fuller dynamical treatment of lenses and eddies over sloping bathymetry. Acknowledgements. This work was funded by a Royal Society Joint Project (JP872), the Russian Foundation for Basic Research ( ) and INTAS projects and from the European Union. The assistance of Tatania Akivis with the numerical runs is gratefully acknowledged. 29