188 JOURNAL OF PHYSICAL VOLUME 34

A Dense Current Flowing down a Sloping Bottom in a Rotating Fluid

C. CENEDESE AND J. A. WHITEHEAD Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

T. A . A SCARELLI Universita' di Roma ``La Sapienza,'' Rome, Italy

M. OHIWA Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

(Manuscript received 21 August 2002, in ®nal form 15 July 2003)

ABSTRACT A density-driven current was generated in the laboratory by releasing dense ¯uid over a sloping bottom in a rotating freshwater system. The behavior of the dense ¯uid descending the slope has been investigated by systematically varying four parameters: the rotation rate, the bottom slope, the ¯ow rate of the dense ¯uid, and the density of the dense ¯uid. Over a wide range of parameter values, the following three ¯ow types were found: a laminar regime in which the dense current had a constant thickness behind the head, a wave regime in which wavelike disturbances appeared on the interface between the dense and fresh ¯uids, and an regime in which periodic formation of cyclonic eddies in the fresh overlying ambient ¯uid was observed. All of the experiments revealed that increasing the slope angle and the density of the bottom ¯uid allowed the ¯ow to evolve from the laminar to the wave regime. Furthermore, increasing rotation rate induced the formation of eddies. A theoretical solution for the downslope velocity ®eld has been found using a steady-state model. Comparison between the theoretical and experimental downslope velocities gave good agreement. The wave regime was observed to occur for values of the Froude number greater than 1. The laminar regime was found for values of the Froude number less than 1. The amount of mixing between the dense and the ambient ¯uids was measured. Mixing increased signi®cantly when passing from the laminar to the wave regime, that is, with increasing Froude number. Good agreement between the amount of mixing observed in the and in the laboratory experiments is encouraging and makes the waves observed in the present experiments a possible candidate for the mixing observed during oceanic dense current over¯ows.

1. Introduction out¯ow (Price et al. 1993). Consequently they descend The descent of dense water from the continental the continental slope and enter the open ocean. shelves into the deep ocean over the continental slope Furthermore, dense water descends downstream of is an important component of the thermohaline circu- gaps separating deep abyssal basins. Gaps with docu- lation. Sources of deep-water formation are found main- mented unidirectional ¯ows of the densest water include ly in the polar regions (Warren 1981), and out¯ows to the Ceara , the Vema Channel, Vema Gap, the main are found in the Denmark Strait (Dick- Discovery Gap, and the Romansch son and Brown 1994), in the Faroe Bank Channel (Saun- (Whitehead 1998). In all cases, the dense water descends ders 1990), on the Arctic continental shelves (Aagaard into the downstream basin and mixes with the surround- et al. 1981), and at various sites on the ing water during its descent. of Antarctica in winter (Foster and Carmack 1976). The dynamics of such dense currents on slopes have There are also dense water currents from enclosed seas been modeled in the past both theoretically and exper- where evaporation contributes to increasing the density imentally starting with Ellison and Turner (1959) and of the enclosed waters. These currents ¯ow into the open Britter and Linden (1980), and a review on gravity cur- seas through a strait, as in the case of the Mediterranean rents can be found in Grif®ths (1986). A review of ex- isting observations of downslope ¯ows around Antarc- tica together with possible dynamical mechanisms is Corresponding author address: Dr. Claudia Cenedese, Department of , Woods Hole Oceanographic Institution, presented by Baines and Condie (1998). In a simpli®ed MS #21, Woods Hole, MA 02543. ``streamtube'' model (Smith 1975; Killworth 1977; E-mail: [email protected] Price and Barringer 1994), the dense plume ¯owed

᭧ 2004 American Meteorological Society JANUARY 2004 CENEDESE ET AL. 189 downslope balancing buoyancy, Coriolis and friction signet (2002) investigated the mechanism of mixing in forces. This model considered the overlying ambient a nonrotating gravity current descending a slope using ocean at rest and averaged or ignored any three-dimen- a two-dimensional, nonhydrostatic, numerical model sional effects so that the plume was not subject to any with a high horizontal and vertical resolution that per- nonlinear instabilities. Price and Barringer (1994) in- mitted the capture of the Kelvin±Helmhotz instability cluded a Froude number parameterization of the en- in the head and trailing ¯uid of the gravity current. The trainment and an Ekman number parameterization of turbulent entrainment of ambient ¯uid associated with the broadening. Strong mixing was found to be localized this instability is found to be stronger than the laminar to regions with large bottom slope, typically the shelf± entrainment related to the head growth caused by the slope break, and a substantial entrainment of oceanic drag exerted by the fresh ¯uid in front. Entrainment into water was observed for the four major over¯ows ana- a turbulent jet descending into a rotating strati®ed res- lyzed in this study: the Mediterranean, the Denmark ervoir has been investigated also in the laboratory by Strait, the Faroe Bank Channel out¯ows, and the Filch- Davies et al. (2002) and resulted in the formation of an ner ice shelf out¯ow into the Weddell Sea. This model intermediate density . considered a laterally integrated streamtube, while Jung- The purpose of this paper is to investigate, over a claus and Backhaus (1994) resolved the plume hori- wide range of parameters, the behavior of a dense cur- zontally using a hydrostatic, reduced gravity, two-di- rent ¯owing down a slope and to quantitatively identify mensional primitive equation numerical model. This lat- the transition between the three main regimes observed ter study investigated the transient character of dense separately in the past: the laminar, the wave, and the currents ¯owing down a slope and the effects of to- eddy regime. Although each of these regimes have been pographic disturbances. A model lying between the previously discussed in different studies, a comprehen- streamtube and the full three-dimensional models was sive study of all of the regimes showing the fundamental developed by Shapiro and Hill (1997). Their 1½-layer parameters that de®ne each regime is still lacking. In theoretical model included bottom friction and entrain- section 2, we develop a steady-state model similar to ment and brought to light that a typical thickness of the Nagata et al. (1993), but valid for any slope angles. Our plume is twice the bottom Ekman-layer scale, that ver- model is a simpli®ed version of the Shapiro and Hill tical mixing enhances downslope propagation of the (1997) model but it allows us to ®nd analytical solutions dense current, and that an interior current can either for the downslope velocity of the dense current while enhance or oppose the dense current descent. Recently, the advection±diffusion type of governing equation of numerical models have included the dynamic interaction the Shapiro and Hill (1997) model allows estimates of of the dense ¯ow with the ambient ¯uid above (Chap- the velocity, but the full solution can only be found man and Gawarkiewicz 1995; Jiang and Garwood 1995, numerically. The predicted velocities found with our 1996, 1998; Spall and Price 1998; Swaters 1998) and model are in good agreement with the experimental ve- have shown that a strong coupling between the two locities measured in both the laminar and wave regime. layers can lead to the formation of cyclonic vortices. The experimental apparatus is described in section 3. In Similar vortex formation was observed near the Den- section 4 we describe the behavior of the dense current mark Strait (Bruce 1995; Krauss 1996) and in earlier for the nonrotating experiments, while the rotating ex- laboratory experiments (Whitehead et al. 1990). The periments are discussed in sections 5 and 6. In section eddies can be generated by either a process in which 7, we investigate the in¯uence of the observed growing the upper-layer columns get stretched, conserving po- waves on the mixing between the two layers and com- tential vorticity (Lane-Serff and Baines 1998, 2000; pare the mixing to the values measured in the ocean and Etling et al. 2000), or by baroclinic instability (Smith in previous laboratory experiments in section 8. The 1977; Etling et al. 2000; Morel and McWilliams 2001). conclusions of the work are also discussed in section 8. Once formed, these eddies move across slope as de- scribed by Mory et al. (1987). However, the descent of 2. Theoretical model dense water down a slope is not always associated with eddy formation. Shapiro and Zatsepin (1996) and Zat- The purpose of this model is to provide a solution sepin et al. (1996) observed the interface above a dense for a laminar ¯ow descending a sloping bottom. bottom current to become unstable to growing waves, and a linear stability analysis showed that the stability a. Formulation of the model threshold depended on the Froude number. Nagata et al. (1993) compared the downslope velocity component of We seek a solution for steady ¯ow of a dense layer a dense current with a theoretical estimate based on a of constant thickness over a sloping bottom. We con- steady-state model that is valid for small slope angles. sider two ¯uids of constant uniform densities ␳1 and ␳ 2, The agreement was good for low values of the Coriolis with ␳ 2 Ͼ ␳1 as shown in Fig. 1. Both ¯uids have ki- parameter, while for large values, the magnitude of the nematic viscosity ␯. The bottom layer has constant experimental speeds was larger than the theoretical es- thickness so the interface is parallel to the sloping bot- timates. Last, a recent study by Ozgokmen and Chas- tom with slope s ϭ tan␪, where ␪ is the angle between 190 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34

Ϫz/␦ Ϫz/␦ ␷ 2 ϭ C sin(z/␦)e ϩ D cos(z/␦)e

z/␦ z/␦ ϩ E sin(z/␦)e ϩ F cos(z/␦)e ϩ ␷ p and (7)

Ϫz/␦ Ϫz/␦ u2 ϭϪC cos(z/␦)e ϩ D sin(z/␦)e ϩ E cos(z/␦)ez/␦ Ϫ F sin(z/␦)ez/␦, (8) where

1(␳21Ϫ ␳ ) sin␪ ␷ p ϭ g (9) 2⍀ ␳1 cos␪ is the particular solution of (3) and C, D, E, F are con-

FIG. 1. Sketch of the model con®guration. stants. Equation (9) is the across-slope velocity for a geostrophic ¯ow (neglecting friction) and it is also the long speed (Nof 1983). the slope and the horizontal. The upper ¯uid is consid- ered to be in®nitely deep. The ¯uids are in a frame b. Solutions rotating with angular velocity ⍀, parallel to gravity. The The values of the constants in these solutions depend Coriolis parameter is f ϭ 2⍀. The centrifugal force is on the boundary conditions. Let us consider a nonslip neglected, and we subtract the hydrostatic pressure dis- bottom boundary condition, ␷ ϭ 0 and u ϭ 0 for z ϭ tribution within the upper ¯uid. In the rotating frame of 2 2 ϪH, and set the velocity and the stress to be continuous reference, we adopt the tilted coordinates i (directed up on the interface between the two layers, ␷ ϭ ␷ , u ϭ the slope), j (directed across slope, the left looking up 1 2 1 .z at z ϭ 0ץ/ uץz ϭץ/ uץ z, andץ/ ␷ץz ϭץ/ ␷ץ , u the slope being positive), and k (directed upward normal 2 1 2 1 2 These conditions produce to the bottom) shown in Fig. 1. The rigid sloping bottom is de®ned by the boundary z ϭϪH, where H is the A ϭ C ϭ ␷ pJ, B ϭ D Ϫ F, depth of the lower layer, while the interface is de®ned 1 by the surface z ϭ 0. The velocity has components un, D ϭ ␷ ppK, E ϭ 0, and F ϭϪ ␷ , (10) ␷n, wn, n ϭ 1, 2, parallel to the tilted x, y, z axes, 2 respectively. where We seek fully developed ¯ow parallel to the sloping bottom and thus set wn ϭ 0. The linearized, steady Bous- 2HeϪ2H/␦ sinesq momentum equations in the upper ¯uid obey J ϭϪsin ϩ sin(H/␦)eϪH/␦ and ΂΃␦ 2 2 u1 ץ Ϫ2H/␦ Ϫ(2⍀ cos␪)␷ 1 ϭ ␯ and (1) 2He .␦/z2 K ϭ cos Ϫ cos(H/␦)eϪHץ ΂΃␦ 2 2 ␷ 1 ץ (2⍀ cos␪)u1 ϭ ␯ . (2) The full solution is z2ץ zz1 z The lower layer obeys u ϭ ␷ ϪJ cos eϪz/␦ ϩ K sin eϪz/␦ ϩ sin eϪz/␦ , 1 p ␦␦2 ␦ 2u (␳ Ϫ ␳ ) ΂΃ץ Ϫ(2⍀ cos␪)␷ ϭ ␯ 221Ϫ g sin␪ and (3) 2 2 (11) z ␳1ץ

2 zz1 z ␦/␷ 2 Ϫz/␦ Ϫz/␦ Ϫz ץ (2⍀ cos␪)u ϭ ␯ . (4) ␷ 1 ϭ ␷ p J sin e ϩ K cos e ϩ cos e , ␦ z2 ΂΃␦␦2ץ 2 The last term on the right-hand side of (3) represents (12) the buoyancy forcing of the system due to the dense zz1 z Ϫz/␦ Ϫz/␦ z/␦ ¯uid layer on the slope. Equations (1) and (2) are the u2 ϭ ␷ p ϪJ cos e ϩ K sin e ϩ sin e , standard equations having solution ΂΃␦␦2 ␦ ␷ ϭ A sin(z/␦)eϪz/␦ ϩ B cos(z/␦)eϪz/␦ and (5) (13) 1 and Ϫz/␦ Ϫz/␦ u1 ϭϪA cos(z/␦)e ϩ B sin(z/␦)e , (6) zz1 z Ϫz/␦ Ϫz/␦ z/␦ where ␦ ϭ 2␯/ f is the Ekman layer depth, f ϭ 2⍀ ␷ 2 ϭ ␷ p J sin e ϩ K cos e Ϫ cos e ϩ 1. ͙ * * ␦␦2 ␦ cos␪, and A and B are constants. Similarly in the lower ΂΃ layer the solution takes the form (14) JANUARY 2004 CENEDESE ET AL. 191

The ¯ux in the top ¯uid in the across-slope direction is

ϱ ␦␷ 1 u ϭ udzϭϪp J ϩ K ϩ , (15) 1F ͵ 1 22 0 ΂΃ and the across-slope bottom layer ¯ux is

0 u ϭ udz 2F ͵ 2 ϪH ␦␷ HH ϭ p (ϪJ ϩ K) cos eH/␦ Ϫ (J ϩ K) sin eH/␦ 2 [ ␦␦ 1 HH 1 ϩ sin ϩ cos eϪH/␦ ϩ J Ϫ K Ϫ . 2΂΃␦␦ 2] (16) The ¯ux in the top ¯uid in the downslope direction is

ϱ ␦␷ 1 ␷ ϭ ␷ dz ϭ p J ϩ K ϩ , (17) 1F ͵ 1 22 0 ΂΃ and in the bottom layer it is

0 ␷ ϭ ␷ dz 2F ͵ 2 ϪH ␦␷ HH ϭ p (J ϩ K) cos eH/␦ Ϫ (J Ϫ K) sin eH/␦ 2 [ ␦␦ 1 HH 1 ϩ cos Ϫ sin eϪH/␦ Ϫ J Ϫ K Ϫ 2΂΃␦␦ 2]

ϩ ␷ pH. (18) FIG. 2. Nondimensional velocity section in (top left) the x±z plane We will discuss two limits obtained for large and small and (top right) the y±z plane and (bottom) 3D velocity vectors (a) Ekman numbers, Ek ϭ (␦/H)2. It is useful to inspect the for values of the Ekman number Ek ϭ 0.043 and the Ekman layer depth ␦ ϭ 0.10 cm ( f ϭ 2sϪ1, H ϭ 0.5 cm, gЈϭ2.0 cm sϪ2, and solution for Ek K 1 by setting H/␦ k 1. In that case s ϭ 0.4). (b) As in (a) but for values of the Ekman number Ek ϭ H/ there are Ekman layers. We set J ϭ K ϭ O(eϪ ␦), which 0.01 and the Ekman layer depth ␦ ϭ 0.05 cm ( f ϭ 8.0 sϪ1, H ϭ 0.5 is small; consequently, the ¯uxes to ®rst order are cm, gЈϭ2.0 cm sϪ2, and s ϭ 0.4). The x, y, and z axes correspond to the tilted coordinate system shown in Fig. 1. ␦␷ u ϭ ␷ ϭ p and (19) 1F 1F 4

u2F ϭϪ3u1F, ␷ 2Fpϭ ␷ H ϩ u2F. (20) The other limit is for large Ek for which we expand The effective volume ¯ux thickness is ␦/4. Fluid moves the variables as a power series in z/␦ for H/␦ K 1. The up the slope by Ekman pumping in the top ¯uid and velocity in the top ¯uid is three times more ¯uid moves down the slope in the HzHz bottom layer. Fluid in the bottom layer moves across u ϭϪ␷ 1 Ϫ cos Ϫ sin eϪz/␦, and (21) the slope in thermal wind balance with an additional 1 ␷ []΂΃␦␦␦␦ ¯ux in the boundary layers, and the top ¯uid ¯ux is found in a boundary layer moving across the slope in HzHz ␷ ϭ ␷ 1 Ϫ sin ϩ cos eϪz/␦, (22) the same direction as the thermal wind ¯ow. Since H 1 ␷ []΂΃␦␦␦␦ k (␦/4) the thermal wind ¯ux ␷pH across the slope is 2 larger than the ¯ux in the boundary layers as also found where ␷␷ ϭ ␷p(H/␦) is the Stokes ¯ow velocity for a by Shapiro and Hill (1997). Typical ¯ows are shown layer of viscous ¯uid traveling down the slope [see Eq. for H/␦ large but close to 1 (Ek ϭ 0.043) in Fig. 2a (6±18) in Schubert and Turcotte (2002)]. The ¯ux in the and for larger H/␦ (Ek ϭ 0.01) in Fig. 2b. top ¯uid is 192 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34

␦␷ ␷ ϭϪu ϭ ␷ . (23) 1F 1F 2 A large volume ¯ux is dragged toward a deeper region, and this ¯uid also moves across the slope in the direction of a current driven by thermal wind. The velocity in the bottom layer is

z2 u ϭϪ␷ 1 Ϫ and (24) 2 ␷ ΂΃H 2 1 ␷ ϭ ␷ (z ϩ H). (25) 2 ␦ ␷ The former is the simple viscous ¯ow down a slope. The latter is smaller than the former, as also found in Shapiro and Hill (1997), and is not usually included in studies of geophysical gravity currents where planetary rotation is completely neglected. However, since ␦ k H in this limit, the ¯ux of the top ¯uid (23) is greatest of all and rotation produces an important effect that is usually neglected. Figure 3 shows such ¯ows for two cases with H/␦ ϭ 1.5 (Ek ϭ 0.43) and H/␦ ϭ 0.3 (Ek ϭ 10.8). The ¯ux in the bottom layer is 21H 2 u ϭϪ ␷ H and ␷ ϭ ␷ . 2F 32␷ 2F ␷ ␦ These solutions will be used to determine some of the properties of the experimental runs.

3. Experimental apparatus The experiments were conducted in a glass tank of depth 60 cm and length and width of 61 cm. This was mounted concentrically on a 1-m-diameter rotating turn- FIG. 3. Nondimensional velocity section in (top left) the x±z plane table with a vertical axis of rotation. We used a square and (top right) the y±z plane and (bottom) 3D velocity vectors (a) tank to avoid optical distortion from side views asso- for values of the Ekman number Ek ϭ 0.43 and the Ekman layer depth ␦ ϭ 0.33 cm ( f ϭ 0.2 sϪ1, H ϭ 0.5 cm, gЈϭ2.0 cm sϪ2, and ciated with a circular tank. A sketch of the apparatus is s ϭ 0.4). (b) As in (a) but for values of the Ekman number Ek ϭ shown in Fig. 4. The tank had a bottom slope, s ϭ tan␪, 10.8 and the Ekman layer depth ␦ ϭ 1.6 cm ( f ϭ 0.008 sϪ1, H ϭ where ␪ is the angle between the slope and the hori- 0.5 cm, gЈϭ2.0 cm sϪ2, and s ϭ 0.4). The x, y, and z axes correspond to the tilted coordinate system shown in Fig. 1. zontal, and it was ®lled with freshwater of density ␳1, which was initially in solid body rotation. A reservoir

(B) of salted and dyed water of density ␳ 2 Ͼ ␳1 was placed on the rotating table and connected to a nozzle the water in the tank was mixed and a new ambient via a submersible pump and a plastic tube. A ¯owmeter density was measured for the following experiment. (C) was positioned between the reservoir and the nozzle Two modi®cations were made to the apparatus described together with a valve (D) for ¯ow rate regulation. The above when we conducted a set of experiments aimed nozzle was covered with sponge in order to reduce mix- at measuring the mixing between the dense current and ing near the source between the dense ¯uid and the fresh the overlying fresh ambient water (see section 7). The ambient water, and it was positioned on the right side ®rst modi®cation was made to reduce the mixing at the (looking upslope) of the shallowest part of the tank nozzle. We eliminated the nozzle, and roughly in the about 1±2 cm from the bottom. same location we positioned, on the sloping bottom, a Over 300 runs were conducted. For each rotating ex- plastic square box (10 ϫ 10 cm) 5 mm in height, with periment, the ambient ¯uid was spun up for as long as the wall looking downslope removed. The tube from 5 times the Ekman spinup time. The experiment com- the pump was connected to an opening made on the top menced by switching on the submersible pump; at this of this plastic box, and the dense water ®lled the box time the video camera (E) was turned on. A current was thickness and then exited the box on the downslope side established during each experiment, at the end of which (see Fig. 4, insert ii). The second modi®cation was made JANUARY 2004 CENEDESE ET AL. 193

FIG. 4. Sketch of the experimental apparatus. (left) Side view and (right) plan view of the tank: (A) freshwater tank, (B) dense water reservoir, (C) ¯owmeter, (D) regulation valve for the dense

¯ow, and (E) the video camera; ␳1 and ␳2 are the density of the freshwater and dense water, respectively. The dashed circle and arrowhead indicate the rotation direction. Inserts (i) and (ii)

show details of the modi®cations made for the experiments described in section 7; ␳b is the density of the current after it descended the slope. to collect the dense water at the bottom of the slope 1.2, 1.5, 1.7, 2.0, and 2.2 sϪ1. For most of the experi- without introducing further mixing. A 5-cm-high wall ments, the bottom slope s ϭ tan␪ took the values of: was attached orthogonal to the bottom and sides (dashed 0.1, 0.25, 0.3, 0.4, 0.5, 0.6, and 0.7. However, in the line in Fig. 4, insert i) of the slope, creating a pocket experiments that measured mixing (see section 7), 18 at the bottom of the slope where we collected samples values of the slope s within the same range (0.1 to 0.7) of the dense water after it descended downslope (see were used. The dense ¯uid was pumped in at a ¯ow Fig. 4, insert i). Samples were collected while the ex- rate Q that ranged between 0.45 and 41.60 cm3 sϪ1. periment was still rotating by means of a small diameter Typically, about six values of Q were used with each tube placed in the bottom of the pocket. As discussed values of f, s, and gЈ. The nozzle inner radius was kept in section 6, we presume that the mixing was caused constant at 0.3 cm. The buoyancy forces are described by highly nonlinear waves with such high amplitudes by the reduced gravity gЈϭg(␳ 2 Ϫ ␳1)/␳1, where g is that the thickness of the dense ¯uid in the wave trough the gravitational acceleration, and gЈ took the values: is less than a millimeter. Hence, the mixing induced by 2, 4, 10, and 20 cm sϪ2. The densities were determined the waves is believed to affect the entire ¯uid within with a model DMA58 Anton Paar densitometer with an the density current. Therefore, the ¯uid in the pocket at accuracy of 10Ϫ5 gcmϪ3. The thickness of the dense the bottom of the slope is presumed to be of uniform current, H, was measured by eye, with an error of ap- density; that is, the lower dense layer is well mixed. proximately Ϯ2 mm, looking at the side view of the The video camera (E) was mounted above the tank tank where a ruler, attached to the side wall, was placed and ®xed to the turntable so that measurements could orthogonal to the sloping bottom. be obtained in the rotating frame. The dense current was The velocity of the dense current, de®ned as the ve- made visible by dyeing the ¯uid with food coloring and locity of the current head or of the wave crests (see was observed both from the top and side view. The ¯ow below), was measured by digitizing images of the ex- at the free surface was observed by adding buoyant periment from the video tape at ®xed time intervals and paper pellets. The depth of the freshwater on the shal- by selecting, on those images, a rectangular area where lowest side of the tank was kept constant at 10 cm for the dense ¯uid was present. The dye in the dense ¯uid every slope inclination and the following values of the made regions with higher thickness (like the head of the Coriolis parameter f were used: 0.0, 0.2, 0.5, 0.7, 1.0, dense current or the crests of wavy interfacial pertur- 194 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 bations) have darker intensities than regions with lower two ¯uids. From the side view, we observed that the thickness (like the trough of wavy interfacial pertur- raised head was a zone of ``backward breaking'' waves, bations). We then calculated the values of the cross- but the viscous forces dominated over the buoyancy correlation function, Cn, between an image at an initial forces and the head was very small. The head was not time t 0 and an image at time t1, t2,...,tn, respectively. observed to grow in size, and very little mixing occurred The distance between the origin and the maximum of behind it (see section 7). the cross-correlation function, Cn, was the average dis- placement of the current head or of the wave crests in b. Waves the time interval (tn Ϫ t 0), and from the displacement and the time interval we obtained the velocity and the A wavelike disturbance on the interface between the angle of propagation of either the current head or of the dense and fresh ¯uids was observed in the plane or- wave crests in each time interval. We de®ned the ve- thogonal to the bottom slope. These disturbances, anal- locity and the angle of propagation in each experiment ogous to those previously observed by Alavian and Asce by the mean of these values. Furthermore, we obtained (1986), were parabolic in plane view and traveled down- the value of the wavelength and, again, the direction of slope slightly faster than the head of the current. A propagation of the waves by computing the autocorre- detailed description of the waves is presented for the lation function, A 0, of one image. The distance between rotation experiments in the next section. the origin and the maximum of the autocorrelation func- In these experiments the bottom slope was held con- tion was the average wavelength and indicated the angle stant at s ϭ 0.30 and only the ¯ow rate Q and the of propagation of the wave crests in the image consid- reduced gravity gЈ were varied. For values of gЈϭ2 ered. cm sϪ2 waves were not observed and the dense current descended the slope in a laminar fashion. The waves 4. Nonrotating experiments appeared to grow on the interface between the two ¯uids for all values of Q Ն 4cm3 sϪ1 and gЈϭ10 cm sϪ2. All the experiments described hereafter are constant For gЈϭ20 cm sϪ2, waves appeared to grow for all ¯ux experiments in which ¯uid of density ␳ 2 was added values of Q. We expect that increasing the slope with continuously from a small source positioned near the a ®xed value of gЈϭ2cmsϪ2 would have given similar bottom. After the dense water was released at the nozzle results because, as the slope increases, the waves would on the right side (looking upslope) of the shallowest have been generated within the dense current. As ex- part of the tank, it started ¯owing downslope. The lead- pected, eddies were never observed in the nonrotating ing part of the front formed a ``head'' that was thicker experiments. than the ¯uid behind and had the characteristics de- scribed by Simpson (1997) with a region, behind the head, where some mixing occurred. The amount of mix- 5. Rotating experiments: Qualitative results ing related to the head of the current depends on the When f 0sϪ1, the current was in¯uenced by ro- Reynolds number (Simpson 1997; Schmidt 1911). For Ͼ tation and its ¯ow had an across-slope component. The low Reynolds number, viscous forces predominate over dense current initially moved downslope for a length buoyancy forces and very little mixing is apparent. As scale approximately equal to the Rossby radius of de- the Reynolds number increases, the buoyancy forces prevail on the viscous forces and eventually, for Re Ͼ formation, RD ϭ ͙gЈH/ f. This motion was de¯ected 1000, eddies can be seen streaming back from the head. leftward (looking upslope) by the effect of rotation until In the experiments described in this paper Re ϭ UH/␯ the current reached a steady state in which the Coriolis Ͻ 300, hence the mixing associated with the head of force associated with the across-slope velocity balanced the current was very small. The width of the current the viscous force and the downslope pressure force as- grew as it ¯owed down the slope and, for f ϭ 0sϪ1, sociated with the dense current. This transient regime it reached an approximately constant value some dis- took place over a time scale on the order of one inertial tance downstream. For the experiment with f ϭ 0sϪ1, period. We systematically varied the slope s, the Coriolis the dense current descended exclusively downslope parameter f, the reduced gravity gЈ, and the ¯ow rate (with no across-slope ¯ow). Over the range of parameter Q in order to observe their in¯uence on the ¯ow be- values investigated, two ¯ow types were found: laminar havior. Over the range of parameter values investigated, ¯ow and waves. three ¯ow types were found: laminar ¯ow, waves, and eddies. a. Laminar ¯ow a. Laminar ¯ow In some cases, after leaving the source region the dense ¯uid behind the head had a constant thickness This regime has been described in the previous sec- and a very sharp interface with the freshwater layer, tion and the effect of rotation caused the ¯ow to veer indicating that no mixing was occurring between the to the left when looking upslope as shown in Fig. 5. JANUARY 2004 CENEDESE ET AL. 195

FIG. 7. Sketch of the waves moving downslope.

growth in size of the head of the current, possibly be- cause the width of the waves (almost the same as the width of the current) increased while ¯owing downslope (see Fig. 6), hence redistributing the dense ¯uid over a larger area. FIG. 5. Top view of the laminar ¯ow observed after 3.8 revolutions We allowed some experiments to run suf®ciently long for f ϭ 0.7 sϪ1, Q ϭ 2.5 cm3 sϪ1, s ϭ 0.3, and gЈϭ2cmsϪ2.A correction factor of 1.5 must be applied to the scale on the left side for the head of the current to reach the bottom of the of the ®gure because the ruler was positioned on the top of the tank tank. This did not affect the nature of the upstream instead of at the level of the dense current. gravity current. Hence, we do not focus on resolving the details at the gravity current head, which we consider Other than this aspect, the behavior of the current was to be part of a transition stage that lasts until the current similar to the nonrotating current. head reaches the bottom of the slope and the waves are well established along the gravity current covering the whole length of the slope. In the plane orthogonal to b. Waves the bottom slope, the dense current had a wave pro®le For some experiments, the dense ¯uid ¯owed down- with alternating regions in which the bottom layer was slope, exhibited a ``laminar ¯ow'' region (see top part thick (crests) and thin (troughs) as shown schematically of Fig. 6), and then started developing wavelike dis- on Fig. 7. The wave amplitude grew as each crest trav- turbances on the interface between the dense and fresh eled downslope and the classic ``backward breaking'' ¯uids. In the plane parallel to the sloping bottom the crests were observed in most of the experiments (Fig. waves had a parabolic front. The wave crests are visible 8). We presume that mixing between the dense salty in the middle and bottom part of Fig. 6 as the regions water and the lighter freshwater occurred. Furthermore, of darker intensity. The crests traveled downslope in the for some parameter values, the waves grew to such large same direction as, and slightly faster than, the head of amplitudes that they were observed to break in a three- the current. However, it was not possible to discern a dimensional fashion, giving the dense current a more

FIG. 8. Picture showing the three-dimensional waves. The ``back- FIG. 6. Top view of the wave regime observed after 3.1 revolutions ward breaking'' of the wave crests is also visible, as indicated by the for Q ϭ 5cm3 sϪ1, f ϭ 0.7 sϪ1, s ϭ 0.3, and gЈϭ10 cm sϪ2. lighter gray regions. 196 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34

FIG. 9. Top view of the eddies observed after 23.3 revolutions for Q ϭ 1.45 cm3 sϪ1, f ϭ 1.8 sϪ1, s ϭ 0.25, and gЈϭ2cmsϪ2. turbulent appearance (Fig. 8). The in¯uence of wave breaking on the amount of mixing between the two wa- ter masses has been investigated for some of these ex- periments as discussed in section 7. c. Eddies In some experiments, the downslope movement of the dense ¯uid caused the periodic formation of cyclonic eddies in the freshwater above the dense ¯uid (Fig. 9). A possible mechanism for eddy generation is as follows: when dense water was introduced by the source near the bottom, the upper-layer thickness decreased from a depth h 0 to a depth h 0 Ϫ hc, where h 0 is the ambient ¯uid depth at the source before the dense ¯uid is intro- FIG. 10. Diagram showing the distribution of the three regimes for two sets of experiments with gЈϭ2cmsϪ2 and (a) s ϭ 0.3 and (b) duced in the tank and hc is the dense current depth at the souce. In order to conserve potential vorticity, the s ϭ 0.4. The other four diagrams discussed in section 6 are not shown. upper layers acquired a vorticity ␨ given by 6. Rotating experiments: Quantitative results ffϩ ␨ ϭ . (26) hhϪ h The three regimes described in section 5 occurred for 00c different values of the parameters s, gЈ, f, and Q. Figure

Hence ␨ ϭϪfhc/h 0 and the upper layer started spinning 10 illustrates the experiments for a ®xed value of gЈϭ anticyclonically capturing the dense ¯uid below. The 2cmsϪ2 and two different slopes, s ϭ 0.3 and s ϭ 0.4. dense ¯uid then coupled with the layer above and carried For the low value of the bottom slope s ϭ 0.25 and the the anticyclonic column of freshwater above it down- lowest value of gЈϭ2cmsϪ2 (not shown), the wave slope; consequently, the vorticity in the upper layer regime was not observed and the passage from the lam- changed sign due to vortex stretching and potential vor- inar regime to the eddy regime occurred while increas- ticity conservation, and the freshwater column started ing f. For ¯ow rates in the range 0.45 Յ Q Ͻ 6.00 cm3 spinning cyclonically. Similar behavior has been found sϪ1, the transition between the two regimes was found previously, as discussed by Whitehead et al. (1990), for 0.7 Յ f Յ 1.5 sϪ1, while for 6.00 Յ Q Յ 18 cm3 Lane-Serff and Baines (1998, 2000), Spall and Price sϪ1, the transition occurred for 0.5 Յ f Յ 0.7 sϪ1. (1998), and Etling et al. (2000). The cyclonic eddies in For all other parameter values, the wave regime was the freshwater column were observed to move zonally observed. While keeping the reduced gravity ®xed at gЈ leftward (looking upslope) until they reached the wall ϭ 2cmsϪ1 (Fig. 10) and increasing the bottom slope of the tank (Fig. 9) and they caused the lower dense from s ϭ 0.30 to s ϭ 0.50, the wave regime ®rst oc- layer to acquire a dome shape under each eddy with a curred for the lowest values of f (approximately 0 Ͻ height larger than the dense current thickness. f Յ 0.3 sϪ1) and Q Ն 4cm3 sϪ1; while for the highest JANUARY 2004 CENEDESE ET AL. 197 value of s ϭ 0.5 (not shown) it was observed for ap- proximately f Յ 1.3 sϪ1 for Q Յ 10 cm3 sϪ1 and for f Յ 1.0 sϪ1 for Q Յ 15 cm3 sϪ1. For a slope s ϭ 0.30, the transition between the laminar regime and the eddy regime (Fig. 10a) was almost unmodi®ed from the s ϭ 0.25 case (not shown). With larger s, the laminar regime disappeared for values of Q larger than 6 cm3 sϪ1. The transition to the eddy regime occurred for 1.3 Յ f Յ 1.8 sϪ1 and for 1.8 Յ f Յ 2.0 sϪ1, for s ϭ 0.4 (Fig. 10b) and s ϭ 0.5 (not shown), respectively. A similar trend was observed while increasing the reduced gravity gЈ from 2 to 10 cm sϪ2 and keeping the slope constant at s ϭ 0.30 (not shown). The wave regime was present for values of Q Ն 2cm3 sϪ1 and f Յ 0.6 sϪ1 for gЈϭ4cmsϪ2 and for almost all values of Q for f Յ 1.2 sϪ1 and gЈϭ10 cm sϪ2. Again, the laminar regime disappeared for the highest values of Q, and for gЈϭ10 cm sϪ2 it was only present for Q Յ 3cm3 sϪ1 FIG. 11. Value of the Froude number Fr ϭ U/͙gЈH cos␪ , where U ϭ u22ϩ ␷ vs values of the Ekman number Ek ϭ (␦/H)2. The and the transition to eddies occurred for f between 1.5 ͙ 2a␷ 2a␷ Ϫ1 wave regime occurred primarily for values of Fr Ն 1, while the and 2.2 s . laminar and eddy regimes occurred for Fr Ͻ 1 and Ek Ͼ 0.1 and Ek As expected, the eddies were present for the larger Ͻ 0.1, respectively. values of the Coriolis parameter. The wave regime was observed for larger values of f when increasing the bottom slope s or the reduced gravity gЈ. From (13) and we assumed that in the wave regime the dense ¯uid form- (14) the velocities in the lower layer are directly pro- ing the waves moved with the velocities u2 and ␷2 given by (13) and (14) (i.e., in the wave regime the entire portional to ␷p ϭ gЈs/ f; therefore, increasing f will de- crease the velocity of the dense current, while increasing transport is associated with the waves). As shown in Fig. s or gЈ will increase its value. 7, the perturbations to the mean ¯ow in the wave regime In some cases, increasing the ¯ow rate Q while the were highly nonlinear and had such high amplitudes (a other parameters were kept constant (Fig. 10) caused the few mm) that the dense ¯uid in the current was redis- ¯ow to be in a different regime. It was observed that tributed in ``blobs'' corresponding to the wave crests, increasing the value of Q caused an increase of the dense while in the wave troughs the thickness of the dense ¯uid lower layer depth H. Hence, we expect a decrease of the decreased to less than a millimeter (almost a dry region), Ekman number and an increase of the velocity U while thinner than the Ekman layer thickness, which varied increasing Q and keeping all other parameters ®xed. The between 0.1 Ͻ ␦ Ͻ 0.7 cm. Hence, we assumed that a experiments in the laminar and wave regimes had a ten- wave of amplitude H moved downslope with a velocity dency to increase their Froude number with increasing U equal to the velocity of a dense layer of thickness H. Q, hence explaining the transition from laminar to wave The theoretical velocities obtained using this assumption regime observed. Also, the increase in H observed for have been compared with the measured velocities Uexp increasing values of Q induced the column of freshwater of the wave crests, giving good agreement (see Fig. 13). above the dense current to reduce its initial height. Hence, The magnitude of the velocity of a dense layer of depth it experienced a larger ``stretching'' while it was carried H has been de®ned as downslope and eddies formed for suf®ciently large f, 22 U ϭ ͙u2a␷ ϩ ␷ 2a␷ , (27) explaining the transition to the eddy regime observed while increasing Q for large values of f. where In order to determine whether the speed 1100 gЈH cos␪ was the critical velocity required by the u2a␷ ϭ udz22and ␷ a␷ ϭ ␷ 2 dz, (28) ͙ HH͵ ͵ dense current to generate waves in the bottom dense layer, ϪH ϪH we computed the value of the Froude number for a dense and it has been calculated for each experiment as well current over a sloping bottom (Turner 1973), Fr ϭ U/ as the value of the Froude number and the Ekman num- ͙gЈH cos␪. Here, U is the magnitude of the average ber as shown in Fig. 11. From Figs. 11 and 12 it is clear velocity in the lower dense layer [as de®ned later in (27)]. that the wave regime occurred for Fr Ն 1. Assuming We had observed that two successive wave crests never that the waves initially behaved as long waves, this merged while moving downslope, and that the wave ve- indicates that the gravity wave speed was indeed the locity measured for different time intervals (section 3) critical velocity required for the dense current in order was approximately the same. Hence we assumed that, for the waves to be produced. The error in evaluating for each experiment, all the waves had the same speed the theoretical velocity U was mainly associated with and that this speed was constant in time. Furthermore, the error in measuring the current depth H; however, 198 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34

FIG. 12. Bar graph showing the percentages of experiments in the laminar (black), wave (light gray), and eddy (cross hatches) regimes for different Froude number bins. the clear distinction between the laminar and wave re- gimes observed when increasing the Froude number above 1 gives us con®dence that, despite having an error FIG. 13. Theoretical velocity U ϭ u22ϩ ␷ vs the measured in evaluating U, we were still able to determine a critical ͙ 2a␷ 2a␷ experimental velocity Uexp. The solid line is the 1:1 line. The dashed Fr for the transition from the laminar to the wave regime. line is the linear regression of the data. Once formed, the waves grew to large amplitude and became nonlinear and, at this later stage, the long-wave approximation is no longer valid. Both the wave and 7. Mixing induced in the wave regime the laminar regimes occurred primarily for values of the The experiments described in this section were de- Ek Ͼ 0.1 while the eddy regime occurred for Ek Ͻ 0.1. signed to study the mixing between the dense current The Reynolds number did not show any correlation with and the overlying fresh ambient water. In particular, we the different regimes and ranged between 0 Ͻ Re Ͻ wanted to quantify how ef®cient the waves were in mix- 300. ing the two water masses. We believe that mixing be- The value of the experimental velocity Uexp of the tween the dense salty water and the lighter freshwater dense current, de®ned as the velocity of the current head was caused by breaking waves as shown in Fig. 8. For or of the wave crests, was measured for a representative some parameter values the waves grew to such large number of the experiments in the laminar and wave amplitudes that they were observed to break in a three- regimes. We used the technique described in section 3 dimensional fashion, possibly increasing the amount of but, due to the time consuming aspects of the technique, mixing. As discussed in section 3, we measured the did not analyze all of the experiments. The theoretical density of the dense current after it descended the slope values of the velocity U of the current head (in the laminar regime) or of the wave crests (in the wave re- gime) were in good agreement with the measured ex- perimental velocity Uexp for low values U, as shown in Fig. 13. The quality of the agreement decreased for increasing values of U, in which case it is possible that the steady state theoretical velocity U was not attained by the dense current within the ®nite length of the tank. It is possible that experiments on longer slopes would give better agreement for high values of U. We again computed the values of Froude and Ekman numbers using the measured experimental velocity Uexp, and, as shown in Fig. 14, the result was very similar to the result obtained with the theoretical velocity U (Fig. 11) with the wave regime occurring for approximately Fr Ն 1. Furthermore, we measured the angle between the tra- jectory of the dense ¯uid and the horizontal. This angle decreased approximately linearly with increasing Cor- iolis parameter f, as expected. The values of the mea- FIG. 14. Value of the Froude number Fr ϭ Uexp/͙gЈH cos␪ , where sured wavelength (obtained using the technique de- Uexp, de®ned as the velocity of the current head or of the wave crests, is the experimental velocity measured with the technique described scribed in section 3) did not show any dependence on in section 3 vs values of the Ekman number Ek ϭ (␦/H)2. The wave the parameters varied during the experiments and regime occurred mainly for values of Fr Ն 1, while the laminar regime ranged, for the most part, from 4 to 6 cm. occurred for Fr Ͻ 1. JANUARY 2004 CENEDESE ET AL. 199

FIG. 15. Mixing ratio r ϭ (␳b Ϫ ␳1)/(␳2 Ϫ ␳1) dependence on the FIG. 16. Mixing ratio r ϭ (␳b Ϫ ␳1)/(␳2 Ϫ ␳1) dependence on the slope s (Q ϭ 2.5 cm3 s Ϫ1 and gЈϭ4cmsϪ2). Coriolis parameter f (Q ϭ 2.5 cm3 sϪ1, s ϭ 0.6, and gЈϭ4cmsϪ2).

(␳b) (see Fig. 4, insert i), and we quanti®ed the amount reversed the direction of rotation of the table, creating of mixing by the mixing ratio r ϭ (␳b Ϫ ␳1)/(␳ 2 Ϫ ␳1), a clockwise rotation with a Coriolis parameter f ϭϪ1.0 where r ϭ 1 when there is no mixing and r → 0 in the sϪ1. The dense current had a zonal velocity in the op- completely mixed case. posite direction than for f ϭ 1.0 sϪ1 and it encountered We ®rst investigated the dependence of the mixing the right (looking upslope) wall soon after exiting the ratio on the slope angle for ®xed values of the Coriolis source. Hence, during the clockwise experiments, waves parameter, reduced gravity, and ¯ow rate. As shown in never appeared and all the mixing was observed during Fig. 15, the effect of the waves on the mixing was most the descent along the boundary wall. The results shown evident for f ϭ 1.0 sϪ1. For this value of the Coriolis in Fig. 15 indicate that some mixing occurred due to parameter, the mixing ratio dramatically decreased when the boundary wall, with a reduction in the mixing ratio passing from the laminar to the wave regime. For s from 0.998 to 0.943 as the slope was increased from s between 0.1 and 0.4, we observed a laminar regime with ϭ 0.3 to s ϭ 0.7. However, the mixing observed during very little mixing occurring within the current. How- these experiments accounts for only 16%±27% of the ever, for s Ͼ 0.4 waves began to appear at the interface total mixing measured for f ϭ 1.0 sϪ1. Therefore, we between the dense current and the freshwater above, were con®dent that the mixing along the boundaries is and the mixing ratio decreased from 0.931, for s ϭ 0.45, only a small contribution to the mixing generated by to 0.628, for s ϭ 0.70. During those experiments, be- the waves. cause of the high value of f, the dense current had a We performed further experiments investigating the strong zonal velocity component and it encountered the dependence of the mixing ratio on the Coriolis param- left wall of the tank (when looking upslope) before eter f, varying f between 0 and 2 sϪ1 with the slope reaching the bottom of the slope. After reaching the left set to s ϭ 0.6. The results shown in Fig. 16 suggest that wall, the dense current descended downslope along the the mixing ratio decreased while increasing f until it wall until it reached the bottom pocket (see section 3; reached the minimum value for f ϭ 1sϪ1 and for values Fig. 4, insert i) where samples of dense water were f Ͼ 1sϪ1, it increased again. This behavior can be collected. In order to investigate the possibility that the explained by examining the regimes for different values mixing was taking place at the boundary (left) wall and of f. For f ϭ 0sϪ1, the waves only appeared close to not on the slope where waves were observed, we con- the bottom of the slope. Therefore the mixing produced ducted two different sets of experiments. First, we re- by waves was active for a very short distance. For great- duced the Coriolis parameter to f ϭ 0.25 sϪ1. Hence, er f, the waves were present over a larger portion of the zonal velocity component of the dense current di- the slope and had a larger amplitude. After f Ͼ 1, the minished and the dense ¯ow no longer encountered the behavior reversed and the waves reduced in amplitude left wall during its descent. Instead, it moved downslope and appeared on a smaller portion of the slope until, for until it reached the bottom pocket. The values of the f ϭ 2sϪ1, the ¯ow was laminar everywhere on the mixing ratio r are shown in Fig. 15. Although r de- slope. Hence, this result ensured that the difference in creased from 0.990 to 0.909 after entering the wave mixing between the f ϭ 1sϪ1 and f ϭ 0.25 sϪ1 sets regime at s ϭ 0.3, the reduction was less signi®cant of experiments was due to the difference in rotation rate than in the experiments with f ϭ 1.0 sϪ1. Second, we and not due to the fact that, for f ϭ 1sϪ1, the currents 200 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 encountered the left (looking upslope) wall and more locity was not too high. Large values of the theoretical mixing occurred because of the wall. velocity provided an upper bound to the speed of the We conducted a few experiments investigating the dense ¯uid in the wave regime. effect of a nonuniform sloping bottom by adding some Both the theoretical and measured values of the ve- across-slope topographic perturbations. The perturba- locity gave values of the Froude number less than unity tions were obtained by placing some tubing of diameter for the experiments in the laminar regime, while the 8 mm across slope, following an isobath, so that the wave regime was observed for values of the Froude perturbation depth was higher than the dense current number larger or approximately equal to unity. This thickness (a few millimeters) and the bottom boundary behavior suggests that wave generation requires that the layer depth (␦ ϭ 1.4 mm). Experiments were performed current speed reach or overcome a critical velocity given with one, two, and three topographic perturbations on by the gravity wave speed͙gЈH cos␪ . We believe the the slope. The differences of the mixing ratio r between observed waves are a manifestation of the roll wave the three experiments were negligible and were ap- instability (Baines 1995) in the bottom dense layer. Roll proximately the same values found with a sloping bot- waves are the spatially periodic steady solutions of the tom with no perturbations. Hence, these experimental equations of motion for nonrotational downslope ¯ows results suggested that small perturbations to the bottom with frictional drag for FrD Ͼ 2, where FrD is the Froude topography were not responsible for enhancing the mix- number de®ned using the steady equilibrium solution ing between the bottom dense current and the ambient obtained with a frictional drag represented by a qua- water. dratic drag law

gЈH sin␪ 8. Discussion and conclusions U ϭ (29) D C Density-driven currents in a two-layer rotating system Ί D over a sloping bottom were produced in a laboratory and CD is the drag coef®cient. If we assume that the 2 tank. Dense ¯uid, supplied continuously through a noz- term associated with the frictional drag ϪCD(/U D H)is 2 2 ( zץ/u 2 ץ)zle on the right side (looking upslope) of the shallowest substituted by the viscous term in Eq. (3), ␯ 2 part of the tank, started ¯owing downslope turning right scaled as ␯(UD/H ), we obtain CD ϭ ␯/HUD ϭ 1/Re. (looking downstream) in¯uenced by the This de®nition of the drag coef®cient is found to be a and increasing its width while descending the slope. good approximation at low Reynolds numbers (Tritton

Four parameters were varied in different experiments: 1988), appropriate to our experiments. Substituting CD the Coriolis parameter f, the reduced gravity gЈ, the into Eq. (29) gives a value for the velocity UD ϭ 2␷␷, slope angle s, and the ¯ow rate Q. Over a wide range where ␷␷ is the Stokes ¯ow velocity de®ned in section of parameters, the following three ¯ow types were ob- (2b) for a nonrotating layer of viscous ¯uid traveling served: the laminar regime, in which the dense current down the slope. Hence we would expect roll waves to had constant thickness behind the nose; the wave re- appear for values of the Froude number Fr␯ Ͼ 1 when gime, in which a locally planar wave disturbance ap- de®ned using the Stokes ¯ow velocity. The Stokes ¯ow peared on the interface between the dense and fresh is the solution of the equations of motion (1)±(4) in the ¯uids as shown in Figs. 6 and 7; and the eddy regime, limit of f ϭ 0 (i.e., large Ek; see section 2b). The wave in which periodic formation of cyclonic eddies in the regime observed for the nonrotating experiments qual- overlying fresh ambient ¯uid was observed. All of the itatively presented no differences with the wave regime experiments revealed that, for increasing slope angle observed in the rotating experiments. Hence, assuming and reduced gravity, the transition between the laminar the mechanism for roll wave formation is not drastically and the wave regime occurred at higher values of f. affected by rotation, we would expect roll waves to Furthermore, increasing the rotation frequency induced appear for Fr Ͼ 1, as observed in our experiments when the formation of the eddies. Fr is de®ned using the velocity of the rotating viscous A linear steady theory including viscous effects and ¯ow given by (27). We discard the possibility that the rotation has been developed for a rotating dense current observed waves are an expression of the Ekman layer over a sloping bottom. For large values of the Ekman instability since the Rossby and Reynolds numbers do number, the velocity of the dense current had the form not collapse on the stability curves given by Tatro and of a simple viscous ¯ow down a slope. The ¯uid had a Mollo-Christensen (1967). Furthermore the Richardson single Ekman layer adjacent to the sloping bottom. For number for the majority of the wave regime experiments small values of the Ekman number, the ¯ow had three was larger than 1/4, with only seven experiments having Ekman layers (Fig. 2). One was next to the sloping a Richardson number less than 1/4 hence, we believe bottom and the other two were above and below the that the waves that we observed are not an expression interface between the dense and fresh ¯uid. The theo- of the Kelvin±Helmhotz instability. retical velocity of the dense current obtained with this The auto- and cross-correlation technique (see sec- model agreed with the speed of the dense ¯uid in the tion 3) were viable tools for measuring the length and laminar and wave regimes, provided the theoretical ve- speed of the waves. For the ®rst time, both across- and JANUARY 2004 CENEDESE ET AL. 201 downslope components of the velocity have been mea- sured for a dense current down a slope in a rotating environment. (In the present paper we presented only the total velocity obtained as a sum of the two com- ponents mentioned above.) This allows for a more ac- curate estimate of the Froude number when compared to previous experiments that measured only the down- slope component of the velocity (Shapiro and Zatsepin 1996; Zatsepin et al. 1996). As discussed above, in the present experiments the roll waves manifested for Fr Ͼ 1, while in Shapiro and Zatsepin (1996) and Zatsepin et al. (1996) waves appeared for values of Fr Ͼ 0.6, where the Froude number was based on the downslope velocity of the front, neglecting the azimuthal com- ponent of the velocity. A ``wavy-eddy'' regime was also observed by Nagata et al. (1993) and this ¯ow pattern was determined by two nondimensional param- eters: the nondimensional density difference ␳* ϭ gЈ/ FIG. 17. Entrainment rate We, normalized by the velocity difference ␦V(ϭU ), plotted against the internal Froude number. Data are from f 2 H and the Taylor number Ta ϭ f 2 H 4 /␯ 2 . The values exp laboratory experiments analyzed by Price (1979) (shaded area) and of these nondimensional parameters for the experi- Turner (1986) (solid line) and from estimates of the Mediterranean ments discussed in the present paper span a much larger out¯ow by Baringer and Price (1997) (solid dots) and of the Denmark range and do not show any consistent pattern when Straight over¯ow by Girton and Sanford (2002) (crosses) and for the compared to the ¯ow regime diagram of Nagata et al. present experiments (stars). Readapted from Price and Yang (1998). (1993) constructed with only 10 experiments. The dense current generation employed in Nagata et al. lowing analysis of the data. From the values of the mix- (1993) was different than in the present study and, ing ratio, r, obtained in the laboratory, we estimated the given the limited number of experiments, it is dif®cult value of the entrainment rate to understand why the Nagata et al. (1993) results col- lapse for different nondimensional numbers than in the 11 present study. The eddy regime was qualitatively sim- We ϭ Q Ϫ 1, Ar ilar to Lane-Serff and Baines (1998, 2000) and Etling ΂΃ et al. (2000); however, the values of the nondimen- de®ned as the averaged vertical velocity at the interface sional parameters indicating the importance of the vis- necessary to obtain the observed mixing between the cous draining and relative stretching used by Lane- dense ¯uid and the fresh overlying ¯uid over an area Serff and Baines (1998) for which eddies were ob- A. (We estimated the area A covered by the dense ¯ow served were slightly different. In the present study, to be approximately one-half of the sloping bottom area eddies were observed for values of the viscous draining and equal to A ϭ 1090.5 cm2.) In order to normalize parameter larger than 9 instead of 3.5, and values of We with the ¯uid velocity, we wanted to measure the the relative stretching larger than 0.02 instead of 0.07. experimental velocity Uexp for the experiments for which The nature of the present experiment is different than mixing was measured. However, the experimental setup Lane-Serff and Baines (1998) since it presents lower and poor quality of the imaging prevents measurement ¯ow rates and density differences, steeper slopes, and of the current velocity from the experiments as described slightly lower Coriolis parameters. The focus of the in section 3. Therefore, we used the data in Fig. 13 to present work was the wave regime and not the eddy ®nd a relationship between the theoretical velocity U regime, hence we did not investigate any further a pos- and Uexp. A linear regression of the data gives the ex- 2 sible explanation for the slightly different values of pression Uexp ϭ 0.5901U ϩ 0.2248, with an R ϭ those nondimensional parameters for which we found 0.6873. The use of a linear regression is meant to ac- eddies when compared with Lane-Serff and Baines count for the discrepancy between U and Uexp at large (1998). values of U as discussed in section 6. Hence, we nor-

Furthermore, we investigated the mixing between the malized We by the estimated experimental velocity Uexp dense ¯uid and the fresh overlying ¯uid in both laminar given by the expression above and calculated the Froude and wave regimes. Mixing was enhanced when passing number using this Uexp. from the laminar to the wave regime and it also in- These results are shown in Fig. 17 and are compared creased with increasing the Froude number. The highest with previous laboratory and observational values in- values of mixing occurred for a value of the Coriolis vestigated by Price and Yang (1998) and some new parameter f ϭ 1sϪ1. observational values (Girton and Sanford 2003). Figure Since one objective of this study is to understand 17 shows that the present laboratory results (stars) had mixing by such density currents, we conducted the fol- a somewhat lower normalized entrainment rate than the 202 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 experiments analyzed by Price (1979) and Turner Grif®ths, R. W., 1986: Gravity currents in rotating systems. Annu. (1986), which presented classical turbulent entrainment Rev. Fluid Mech., 18, 59±89. Jiang, L., and W. J. Garwood, 1995: A numerical study of three- behavior. However, these values were comparable to the dimensional bottom plumes on a Southern Ocean continental estimates of the Mediterranean over¯ow (solid dots) slope. J. Geophys. Res., 100, 18 471±18 488. (Baringer and Price 1997a,b) and to the Denmark Strait ÐÐ, and ÐÐ, 1996: Three-dimensional simulations of over¯ows over¯ow (crosses) (Girton and Sanford 2002). The on continental slopes. J. Phys. Oceanogr., 26, 1224±1233. agreement between the observations and the laboratory ÐÐ, and ÐÐ, 1998: Effects of topographic steering and ambient strati®cation on over¯ows on continental slopes: A model study. experiments was encouraging and indicated that the J. Geophys. Res., 103, 5459±5476. waves observed in the present experiments may be a Jungclaus, J. H., and J. O. Backhaus, 1994: Application of a transient possible candidate for the mixing observed during oce- reduced gravity plume model to the Denmark Strait over¯ow. anic over¯ows. J. Geophys. Res., 99, 12 375±12 396. Killworth, P. D., 1977: Mixing on the Weddell Sea continental slope. Deep-Sea Res., 24, 427±448. Acknowledgments. We thank Jason Hyatt, Paul Lin- Krauss, W., 1996: A note on over¯ow eddies. Deep-Sea Res., 43, den, and Jim Price for providing invaluable discussions, 1661±1667. carefully reading drafts, and substantially improving the Lane-Serff, G. F., and P. G. 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