The Intrusion Depth of Density Currents Flowing into Stratified Water Bodies

Mathew Wells University of Toronto, Toronto, Ontario, Canada

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Parthiban Nadarajah University of Toronto, Toronto, Ontario, Canada

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Abstract

Theory and laboratory experiments are presented describing the depth at which a density current intrudes into a linearly stratified water column, as a function of the entrainment ratio E, the buoyancy flux in the dense current B, and the magnitude of the stratification N. The main result is that ZE−1/3B1/3/N. It is shown that the depth of the intrusion scales as Z ∼ (3 ± 1)B1/3/N for laboratory experiments, and as for oceanic density currents. The velocity of a large-scale density current is controlled by a geostrophic balance defined as Ugeo = 0.25gs/f, where s is the slope and f is the Coriolis parameter. The geostrophic buoyancy flux is then defined by Bgeo = gUgeoh, with g′ the reduced gravity and h the thickness of the current. The scaling herein implies that the depth of an oceanic intrusion is relatively insensitive to changes in source water properties but is very sensitive to changes in the stratification of the water column, consistent with the previous scaling of Price and Baringer. For example, if the buoyancy flux of a dense current were to double while the stratification remained constant, then there would only be a 25% increase in the intrusion depth, whereas doubling the stratification would result in a 50% decrease of the intrusion depth.

Corresponding author address: Mathew Wells, 1265 Military Trail, Department of Physical and Environmental Sciences, University of Toronto, Toronto, ON M1C 1A4, Canada. Email: wells@utsc.utoronto.ca

Abstract

Theory and laboratory experiments are presented describing the depth at which a density current intrudes into a linearly stratified water column, as a function of the entrainment ratio E, the buoyancy flux in the dense current B, and the magnitude of the stratification N. The main result is that ZE−1/3B1/3/N. It is shown that the depth of the intrusion scales as Z ∼ (3 ± 1)B1/3/N for laboratory experiments, and as for oceanic density currents. The velocity of a large-scale density current is controlled by a geostrophic balance defined as Ugeo = 0.25gs/f, where s is the slope and f is the Coriolis parameter. The geostrophic buoyancy flux is then defined by Bgeo = gUgeoh, with g′ the reduced gravity and h the thickness of the current. The scaling herein implies that the depth of an oceanic intrusion is relatively insensitive to changes in source water properties but is very sensitive to changes in the stratification of the water column, consistent with the previous scaling of Price and Baringer. For example, if the buoyancy flux of a dense current were to double while the stratification remained constant, then there would only be a 25% increase in the intrusion depth, whereas doubling the stratification would result in a 50% decrease of the intrusion depth.

Corresponding author address: Mathew Wells, 1265 Military Trail, Department of Physical and Environmental Sciences, University of Toronto, Toronto, ON M1C 1A4, Canada. Email: wells@utsc.utoronto.ca

1. Introduction

In many parts of the World Ocean, the renewal of deep or intermediate waters is due to density currents that flow down the continental margin. For example, the combination of brine rejection and cooling in the Weddell and Ross Seas produces very cold and dense water that flows as a density current down the Antarctic continental slope to form the deep Antarctic Bottom Water. In addition to regions of the World Ocean where the major water masses are formed, there are many other smaller density currents that have been documented by Ivanov et al. (2004). The water in these density currents is formed from either a combination of cooling and brine rejection at high latitudes, or because of excessive evaporation of saltwater in marginal seas at low latitudes. In almost all of the 61 density currents documented by Ivanov et al. (2004) the dilution caused by turbulent entrainment meant that the density current intruded into the water column at a shallower depth than the level of neutral buoyancy of the initial source waters. For instance, the outflow of the Mediterranean Sea is initially so dense at the Strait of Gibraltar that it could sink to the bottom of the North Atlantic, but because of entrainment of lighter overlying waters the plume intrudes at a depth of around 1 km (Price and Baringer 1994). The depth at which the Mediterranean plume intrudes has important implications for the stability of the present climate because of its important role in the global thermohaline circulation (Price and Yang 1998; Rahmstorf 1998; Wu et al. 2007), so it is important to understand how the intrusion depth might change in response to an increase or decrease in the salinity of the Mediterranean Sea. In this paper we will investigate the underlying scaling laws of the intrusion depth as a function of the buoyancy flux, the entrainment ratio, and the background stratification. Large-scale density currents in the ocean are influenced by Coriolis forces, and so their buoyancy flux will be determined by their geostrophic velocities (Nof 1983; Price and Baringer 1994). The development of an idealized scaling law is intended to help understand the intrusion dynamics of density currents in more complex stream-tube models, or in low-resolution global circulation models, such as those of Papadakis et al. (2003), Legg et al. (2006), or Wu et al. (2007).

In addition to oceanic density currents, there are dynamically similar density currents that occur in many lakes. These can occur when a cold river flows into a warmer stratified reservoir and forms an “underflow” (Hebbert et al. 1979; Fischer et al. 1979). Underflows can also develop when density currents form due to differential cooling in the shallow regions of lakes (Monismith et al. 1990; Wells and Sherman 2001; Fer et al. 2001). In a density-stratified lake or reservoir an “interflow” occurs after the underflow separates from the boundary and intrudes into the layer where the flow is neutrally buoyant. The depth at which a river intrudes into the stratified water column can determine where pathogens are placed (Antenucci et al. 2005), the locations of biological hot spots (MacIntyre et al. 2006), and whether nutrients from runoff will be emplaced in the euphotic or the benthic zones (McCullough et al. 2007).

The depth at which either an oceanic or a limnological density current will intrude into a stratified water body is controlled by the total amount of mixing of overlying lighter water. One common approach to model the trajectory and the ultimate level of intrusion of oceanic density currents has been through the development of stream-tube models. In these models the Coriolis force, buoyancy force, bottom drag, shear-driven entrainment, and increasingly complex topography can be incorporated to predict the behavior of density currents (Smith 1975; Killworth 1977; Speer and Tziperman 1990; Price and Baringer 1994; MacCready et al. 1999). One of the central assumptions in these models is the entrainment hypothesis of Taylor (1945), whereby the rate at which the fluid in a density current turbulently entrains the surrounding ambient fluid is assumed to be linearly proportional to the mean downstream velocity U. This idea is often expressed in terms of an entrainment ratio, defined as
i1520-0485-39-8-1935-e1
where we is the entrainment velocity normal to the current. For a vertical plume, the entrainment ratio is constant and equal to 0.1, but for a density current the entrainment ratio decreases as the stability of the current increases (Ellison and Turner 1959; Turner 1986).

A density current flowing down a slope in a stratified environment entrains surrounding fluid until the density difference with the environment is eliminated and the fluid intrudes laterally. For a point source, the total buoyancy flux B has units of m4 s−3 and by dimensional analysis the intrusion depth Z scales as ZE−1/2B1/4/N3/4, where N is the buoyancy frequency (Morton et al. 1956). By examining the observed intrusion heights from smokestacks into stratified atmospheres, Briggs (1969) showed that there is excellent agreement with the scaling of Morton et al. (1956) from laboratory scales (cm) up to atmospheric scales (km). A density current flowing into a stratified water column will behave in a similar fashion, but is usually better represented by a line source of buoyancy than by a point source of buoyancy, because many currents are constrained to channels and do not make significant changes to their width. In this case the dimensions of the buoyancy flux per unit width are m3 s−3. By using a similar dimensional analysis to Morton et al. (1956), we predict that the intrusion depth Z will scale as ZE−1/3B1/3/N. A similar scaling has been previously discussed in the context of vertical line plumes by Fischer et al. [1979, their Eq. (10.3)] and in Turner [1986, his Eq. (24)]. Previous laboratory experiments with a line plume and a linear stratification found intrusion depths in the range from Z = (2 ± 1)B1/3/N (Wright and Wallace 1979) to Z = (3 ± 1)B1/3/N (Fischer et al. 1979; Bush and Woods 1999). However, in these previous studies the scaling laws were neither rigorously derived nor extended to study the case of an intruding density currents.

To extend the scaling ZE−1/3B1/3/N to a density current on a slope we need to know how the entrainment ratio of Eq. (1) varies with the slope. To solve this problem we will use the experimental observation of Wells and Wettlaufer (2005, 2007) that the total entrainment ratio of fluid per unit depth is constant for a given buoyancy flux. This observed behavior of nonrotating laboratory-scale density currents occurs because the decreasing entrainment ratio on low-angle slopes appears to be balanced out by the greater distance that density current must travel to achieve the same vertical drop. This result was also predicted theoretically by Hughes and Griffiths (2006), and has also been seen in numerical simulations of density currents by Ozgokmen et al. (2006). If the total entrainment ratio of fluid per unit depth Eeq is constant, then it follows that the intrusion depth of a plume should scale as Z = CB1/3/N, with a constant that only depends on the value of Eeq and is independent of slope for a nonrotating system. This scaling law will be the main result of this paper, which we will test against laboratory experiments. We also compare the scaling of to observations from the oceanographic literature, where rotation is important and the buoyancy flux Bgeo is described in terms of the geostrophic velocity of the density current.

This paper is arranged in the following fashion. First, in section 2 we review the experimental observations of Wells and Wettlaufer (2005) that indicate that the vertically integrated entrainment ratio appears to be a constant value that is independent of slope, if the buoyancy flux is constant. Then, in the subsection, we extend the previous theory of Morton et al. (1956) to the case of a line source of buoyancy. The resulting scalings are then tested against laboratory experiments on slopes between 15° and 90° in section 3, and in section 4 they are tested against a large dataset of intrusion depths from the oceanographic literature. In section 4 we also compare our scaling with a simplified version of the Price and Baringer (1994) stream-tube model. Finally, we conclude with a discussion of how these simple results might be used to test the fidelity of complex parameterizations of density currents in global circulation models.

2. Theory

The mechanism by which dense fluid mixes with overlying waters is of central importance to understanding the dynamics of an intruding density current. The situation we are considering is sketched in Fig. 1, where a body of dense water is flowing down a slope. Initially there is a large density difference and the velocity of the current is high. As the density current progresses entrainment of overlying fluid will reduce the density contrast, which reduces the buoyancy force that drives the flow. At some point there will be no density contrast between the density current and the surroundings, so that the current will form a subsurface intrusion. The momentum of the current may lead to an initial overshooting of the level of neutral buoyancy, but this intruding water will quickly rise back to the level of neutral buoyancy.

To determine the depth at which a density current reaches its level of neutral buoyancy in a stratified water column, it is essential to know how much entrainment has occurred per unit depth, rather than per unit length of slope. In the Ellison and Turner (1959) model the entrainment ratio is defined in terms of the rate of the increase in plume depth as a function of distance along slope, whereas we are interested in the rate of increase in density current thickness as a function of depth, which we will denote as Eeq. When the slope angle Θ is decreased the inferred entrainment for the density current along a slope E is related to the depth-integrated entrainment value by
i1520-0485-39-8-1935-e2

Based upon experimentally measured values of Eeq, Wells and Wettlaufer (2005) found that Eeq then appears to asymptotically approach a constant value of around Eeq = 0.08 for slopes of between 10° and 90° if the buoyancy flux is constant. Experimental observations of Cenedese and Adduce (2008) found that if the buoyancy flux of a density current was varied, then the entrainment ratio will decrease as either the Froude number or Reynolds number of the flow decrease. The result that the total entrained volume of fluid does not appear to depend on the slope for a constant buoyancy flux has also been seen in the high-resolution numerical simulations of density currents of Ozgokmen et al. (2006) for slopes of 3°, 5°, and 7°. The result that Eeq is constant was also theoretically predicted by Hughes and Griffiths (2006), based upon extrapolation of the experiments of Ellison and Turner (1959), who had found that entrainment ratio E of their density current decreased from 0.1 to 0.02 as the slope decreased from 90° to 10°.

The experimental results that we will present in section 3, and those of previous experimental studies of Wells and Wettlaufer (2005) and Ellison and Turner (1959), only apply to the slopes > 10°. While oceanic density currents on low-angle slopes may not necessarily have Eeq = 0.08, the use of Eq. (2) implies that typical oceanic entrainment ratios, which are in the range of E = O(10−3–10−4) and on low-angle slopes are between 0.5° and 2° (Cenedese et al. 2004), are actually consistent with Eeq = O(0.1–0.01). In the next section we will show that there is only a weak dependence of the intrusion depth upon the value of the entrainment ratio, so that the intrusion depth of oceanic density currents can be expected to scale similarly to the results from laboratory density currents.

a. Predicting the depth of intrusion

The intrusion height of a point source of buoyancy into a linear stratification was described by Morton et al. (1956) using a set of conservation equations for the plume’s volume, buoyancy, and momentum, as a function of height. As the plume increased the density deficit decreased by the entrainment of ambient fluid, and they were able to calculate the height at which the density difference between the plume and the environment disappeared, and the height at which momentum disappeared. Briggs (1969) subsequently showed that their scaling was in good agreement with the measured intrusion heights from the centimeter scales of their original laboratory experiments up to kilometer scales of the plume rising in the atmosphere, as shown in Fig. 2. The Morton et al. (1956) model has also been extended by Caulfield and Woods (1998) to consider the intrusion height of a point source in a nonlinear stratification. A further extension was made by Cardoso and Woods (1993) to describe the time-dependent influence of entrainment upon the stratification in a confined domain, and they found excellent experimental agreement with their predictions. In all of these models the only parameter that needs to be empirically fitted is the entrainment ratio of Eq. (1). We now aim to develop an analogous result for the intrusion depth of a 2D vertical line source of buoyancy in a linear stratification, which we will subsequently test against laboratory density currents flowing down slopes between 10° and 90°.

Following Morton et al. (1956), the conservation equations for volume, momentum, and density deficiency in the 2D vertical line plume when integrated over the cross section are
i1520-0485-39-8-1935-e3
i1520-0485-39-8-1935-e4
i1520-0485-39-8-1935-e5
where b is the plume width, w is the plume velocity, Δ = g(ρρ1)/ρ1 is the gravity anomaly of the plume, and Δo = g(ρoρ1)/ρ1 is the gravity anomaly of the environment, with g as gravity, ρ and ρo as the densities inside and outside the plume, and ρ1 as the reference density.
We introduce the following variables for the volume, momentum, and buoyancy flux per unit length:
i1520-0485-39-8-1935-e6
to simplify Eqs. (3)(5). These variables are then nondimensionalized by
i1520-0485-39-8-1935-e7
i1520-0485-39-8-1935-e8
i1520-0485-39-8-1935-e9
i1520-0485-39-8-1935-e10
where the asterisk denotes a dimensionless variable. The last equation is of central importance to this paper, because once we know the dimensionless depth z* at which either M or F = 0, then we have determined where the intrusion will occur. It is worth noting that this scaling of the intrusion depth (10) is consistent with Price and Baringer (1994), who found that there was a greater sensitivity to the product water density (and hence the depth of intrusion) upon the stratification N than upon the entrainment ratio E. In the process of making the above variables nondimensional we have used the implicit hypothesis that the depth of intrusion depends on source buoyancy flux and stratification, so that F cannot depend upon the entrainment ratio E. The scaling of Q, M, and z with E then follows from dimensional analysis. Using these nondimensionalized variables, we transform Eqs. (3)(5) to
i1520-0485-39-8-1935-e11
i1520-0485-39-8-1935-e12
i1520-0485-39-8-1935-e13
with boundary conditions of Q* = M* = 0 and F* = 1 at z = 0.
Using a similar approach to Morton et al. (1956), these equations are integrated numerically with a fourth–order Runge–Kutta scheme, with the calculations initialized at small Δz with the first terms in a series of solutions to (11)(13),
i1520-0485-39-8-1935-e14
The solution to these equations is plotted in Fig. 3 in terms of the dimensionless velocity U* = M*/Q*, the dimensionless plume width b* = Q*2/M*, and the dimensionless buoyancy anomaly ½Δ* = ½F*/Q*. It is seen that the dimensionless buoyancy is zero when z* = 2.0391 and that velocity disappears at z* = 2.9621. These values are similar to the values where buoyancy and momentum disappears (z* = 2.1315 and z* = 2.8245, respectively) for the point source of buoyancy considered by Morton et al. (1956).
The intrusion depth in Eq. (10) can be seen to be only weakly dependent upon the entrainment ratio. Additionally, the previous experimental of Wells and Wettlaufer (2005) found that if the buoyancy flux was constant then the value of Eeq = 0.08 was nearly constant, so that we expect that in our laboratory experiments there will be almost no dependence of the intrusion depth upon the slope. The intrusion depth is then given by z = CB1/3/Nz*, where the constant C is determined either by using the dimensionless depth z*, where the momentum is zero, so that , or the depth where the buoyancy anomaly is zero, so that . Hence, we expect the intrusion depth to be in the range of
i1520-0485-39-8-1935-e15
This range of values is similar to previous laboratory experiments with a line plume (Fischer et al. 1979; Wright and Wallace 1979; Bush and Woods 1999) that found intrusion depths in the range Z = (2 ± 1)B1/3/N to Z = (3 ± 1)B1/3/N. We note that the experiments by Bush and Woods (1999) were conducted in a rotating and stratified system, and that their intrusion depths were not noticeably influenced by the system rotation. We will now use laboratory experiments to determine if the intrusion depth obeys the same scaling when the slope of the density current is varied.

3. Experimental results

To test the scaling for the intrusion depth developed in the last section, we conducted a series of 41 laboratory experiments where a dense plume of saline water was pumped at a controlled rate onto a slope and allowed to flow down into a water sample with linear density stratification. The experimental setup is shown in Fig. 4. The Plexiglas tank that we used measured 280 cm × 45 cm × 36 cm. Inside this tank there was a movable channel that had an internal width of 32 cm, which allowed the slope to be changed from 15° to 90°. The density current was produced by pumping a dense saline solution through a 32-cm-long permeable hose, which was located 1 cm above the sloping channel in order to initiate turbulent mixing in the density current. This may lead to a slight increase in total mixing but does not change the buoyancy flux. The flow rate was varied from 0.5 < Q < 30 cm3 s−1 and the reduced gravity of the inflow varied from 30 < g′ < 190 cm−2. The buoyancy flux per unit length then varied from 0.5 < B < 350 cm3 s−3. A linear stratification was made using the standard double-bucket technique, so that the buoyancy frequency was varied from 0.3 < N < 1.7 Hz. In these experiments both B and N were independently varied, and with the range of B and N used, the expected intrusion depth Z = 3 B1/3/N varied between 3 and 40 cm. A small amount of dye was added to the dense saline solution and the background was matte white so that high-contrast images could be taken. The depth of the intrusion is defined as the vertical distance from the source to the center of the initial intrusion. Because the initial thickness of the intrusion shown in Fig. 5 was of the order of 5–8 cm, this leads to the relatively large error bars of the resulting intrusion depths shown in Fig. 6. The actual experiments only lasted for 1 min, because the intruding density current would quickly start to modify the stratification of this confined tank in a manner very similar to that in Cardoso and Woods (1993). After each experiment the tank was drained and a new linearly stratified system was made.

The depths of intrusion shown in Fig. 6a are plotted as a function of the length scale B1/3/N. The straight line corresponds to Z = 3B1/3/N and is in fair agreement with experimental observations. In Fig. 6b the observed intrusion depth is divided by the length scale B1/3/N and plotted as a function of slope angle, to again show the result that Z = (3 ± 1)B1/3/N. This result is consistent with the previous observations for vertical plumes (Wright and Wallace 1979; Bush and Woods 1999), and we have now shown that it also applies to density currents on slopes. The collapse of the data for the different slopes is due to the weak dependence of the intrusion depth with the entrainment ratio expressed by ZE−1/3, as well as the previous experimental observation that the entrainment per unit depth [Eq. (2)] was almost constant for slopes between 10° and 90° (Wells and Wettlaufer 2005), if the buoyancy flux is constant. It seems likely that our result may hold for slopes less than 10° as well, because the scaling for the intrusion depth depends only weakly upon the actual entrainment ratio. While our experiments are similar to those of Baines (2001, 2005), we used larger slopes and consequently did not observe the same detrainment dynamics as reported by these previous studies.

4. Scaling of the intrusion depth of oceanographic density currents

We now test the scaling of the intrusion depth (15) against observations of oceanic intrusion depth reported in the literature. Of the 61 observations described in Ivanov et al. (2004), there are 40 independent locations distributed around the world with intrusion depths between 24 and 2700 m. From these 40 locations, there is sufficient information in their paper to determine the buoyancy flux, the background stratification, and the depth of intrusion for 33 cases. The velocity and downward trajectory of these large-scale density currents are controlled by a balance between Coriolis forces, buoyancy forces, and drag. The small-scale interfacial entrainment dynamics, however, are not directly influenced by Coriolis forces, but rather can change through the influence of the Coriolis forces upon the bulk velocity of the density current. The depth Z at which the density current reaches neutral density and intrudes is still set by the total integrated entrainment into the sinking density currents, as in the stream-tube model of Price and Baringer (1994) or the experimental observations of Bush and Woods (1999). When we compare the scaling of (15) against oceanic data, the buoyancy flux per unit width of these density currents was calculated based upon the reported geostrophic velocity Ugeo, the thickness h, and the reduced gravity g′, so that Bgeo = gUgeoh. The geostrophic velocity reported by Ivanov et al. (2004) is defined as Ugeo = 0.25 × gs/f, so that the velocity of the density current is a function of the slope s, the reduced gravity g′, and the Coriolis parameter f (Nof 1983; Price and Baringer 1994). Using the expression for the geostrophic velocity of the density currents we can explicitly describe the dependence of the intrusion depth upon the Coriolis forces and the slope as
i1520-0485-39-8-1935-e16
The buoyancy frequency N was estimated based upon the ambient density difference between the initial and final depths of the plume. In our estimate of N we have implicitly assumed a linear stratification, which will probably underestimate the maximum buoyancy frequency experienced by the plume.

In Fig. 7a we plot data of the observed intrusion depths of the oceanographic density currents and note that there is good agreement with the intrusion depth scaling as . In this graph we are taking into account the combined influences of the slope and the Coriolis forces upon large-scale rotating density currents by using the geostrophic buoyancy flux. In Fig. 7 additional data points are taken from the literature for intrusions produced by overflows from the Weddell Sea, the Ross Sea, the Mediterranean overflow, the Denmark Strait overflow, and the Faroe Banks overflow. In the Weddell Sea, Foldvik et al. (2004) report a density current that descends from a sill at 500-m depth to intrude at 2000 m. The horizontal line represents the range in buoyancy fluxes resulting from the range in reported velocities. In the Ross Sea, Gordon et al. (2004) report a density current that descends from a sill at 600-m depth to intrude at 2000 m. Their data suggest a higher estimate for the ratio B1/3/N than the data from Ivanov et al. (2004), and a slightly lower estimate of the intrusion depth compared to the 2700-m depth reported in Ivanov et al. (2004). To show the range of values we have connected all of the reported values of the Ross Sea overflow in Fig. 7. The Mediterranean outflow is reported by Baringer and Price (1999) to descend from a sill at 300 m to form an intrusion at 850-m depth in the Atlantic. The water from the Faroe Bank descends from 800 to 3000 m, where it forms an intrusion above the very slightly less dense water from the Denmark Straight (Swift 1984; Price and Baringer 1994).

It is helpful to compare our scaling of Z = (3 ± 1)(g2hs/f )1/3/N with the scaling shown in Fig. 36 of Ivanov et al. (2004). Here the authors show that there is good agreement between a nondimensional dynamic parameter of “cascade activity,” defined as Π = g′/f2Hc, and the combination of structural parameters 18s−1/2r−2h, where s is the slope, Hc is the initial thickness of the dense water layer before descending the slope, f is the Coriolis parameter, h is the dimensionless thickness of the dense water cascade, and r is the “mixedness” defined in terms of the change between the source and product water densities as r ≡ (ρsourceρproduct)/(ρsourceρocean). The correlation between their log(Π) and log(18s−1/2r−2h) was R2 = 0.87, indicating a good fit to the data. Their combination of structural parameters was chosen based upon a statistical analysis. In our Fig. 7 the correlation between log(Z) and log[(3g2hs/f )1/3/N] also has a good fit with R2 = 0.98, and is now based upon a physical description of the flow rather than a statistical fit.

In Fig. 7b we plot data from our laboratory experiments along with the oceanographic observations and find good agreement with Z = (3 ± 1)B1/3/N. This graph covers five orders of magnitude of depth and a range of slopes between 2° and 90°. The collapse of such a range of data to our simple scaling analysis is the main result of this paper, and this shows that the variation in intrusion depth is primarily a function of the ambient stratification and the buoyancy flux of the plume. In the large-scale oceanic density currents the buoyancy flux will be determined by the geostrophic velocity of the current, so the intrusion depth is implicitly dependant upon the slope and the Coriolis forces. As noted earlier the expected range of the entrainment ratio in the oceanic density currents will vary by at least an order of magnitude between E = 10−4 and 10−3. This implies that will vary by only a factor of 2, and may account for some of the scatter in Fig. 7a. It is also apparent that with a steeper slope there would be better agreement between the prediction and oceanic observations in Fig. 7a. A steeper slope would be consistent with the value of Eeq being lower in oceanic density currents.

Our laboratory experiments were not conducted in a rotating frame, so it is initially surprising that the same scaling holds for the intrusion depth of oceanic density currents that display a strong deflection resulting from Coriolis forces. The action of rotation upon the entrainment in density currents has been studied in laboratory experiments of Cenedese et al. (2004), Wells (2007), and Cenedese and Adduce (2008), who found that the action of Coriolis forces made a strong impact on the trajectory and velocity of density currents flowing down a slope. If the buoyancy flux was kept constant, they found that the effect of increasing the Coriolis forces was to deflect the current to the right and reduce the velocity of the current. The reduction in velocity meant that the flow had a lower Froude number and hence became more stable and entrained less ambient fluid. It seems possible that the increased length of the trajectory may balance the reduction in entrainment ratio (Wells 2007), so that the density of the product water may not be greatly influenced by the action of rotation, and that only the buoyancy flux and the total vertical fall are the important parameters in determining the product water densities. The assumption that there is in fact such a balance between increasing the plume pathlength and a decreased entrainment ratio, has previously been made by Wahlin and Cenedese (2006) and Hughes and Griffiths (2006), and may partially account for the collapse of the oceanographic data to the nonrotating scaling observed in Fig. 7b. However, the existence of such a balance has not yet been clearly demonstrated in laboratory experiments and would be a fruitful area of further research. An additional complicating factor in understanding these dynamics of oceanic density currents is that most of the continental slopes are not smooth, as in the laboratory models of Cenedese et al. (2004) and Cenedese and Adduce (2008), but rather are incised by deep canyons that steer the flow (Wahlin 2002; Darelius and Wahlin 2007; Wahlin et al. 2008) and may actually result in faster downslope velocities than would be estimated based upon the mean gradient of the continental slope.

There is a gap visible in the data shown in Fig. 7 for scales from 5 to 50 m, which is typical of the depth of intrusion of many river plumes into stratified lakes. However, no data have been included because most thermally stratified lakes are usually best described as having two-layer stratification, where a well-mixed epilimnion is separated from a cold and uniform hypolimnion by a very sharp metalimnion (thermocline). In such two-layer systems there are only two possibilities for the intrusion depth of a dense underflow: either the currents penetrate through the thermocline to form a deep underflow or an intrusion will form at the level of the thermocline (Fischer et al. 1979; Monaghan et al. 1999; Monaghan 2007; Wells and Wettlaufer 2007).

a. Applications of scaling to numerical stream-tube models

One approach to modeling oceanic density currents is to use a stream-tube model to calculate the trajectory and dilution of the density current (Smith 1975; Price and Baringer 1994). The advantage of such models over the relatively simplified approach we have taken is that they can determine the trajectory of a dense current flowing through complex bathymetry with a nonlinear density stratification, and they can explicitly include the effect of Coriolis forces. We will now look at a simplified version of the Price and Baringer (1994) model in order to see if our simple scaling ZE−1/3B1/3/N emerges when the Coriolis forces is turned off and the density current is allowed to flow down a uniform slope into a linear stratification. For this purpose we have used a version of the Price and Baringer (1994) model that has been programmed into Matlab by Bower et al. (2000) and has been previously used to study laboratory scale flows by Cenedese and Adduce (2007).

The results of a simulation for laboratory-scale flows are shown in Fig. 8, where we show that our simple scaling for the intrusion depth emerges as a robust feature of the more complex stream-tube model. For the numerical results shown in Fig. 8 we used conditions relevant to the laboratory experiment, that is, a density current with a constant width of 30 cm and an initial depth of 2 cm enters a tank of depth of 50 cm that has a buoyancy frequency between 1.5 and 2.5 Hz and a slope of 45°. The Coriolis term is set to zero and a drag coefficient of CD = 0.25 is used, comparable to the drag coefficient that was used by Cenedese and Adduce (2007). In Fig. 8a the entrainment ratio is kept at a constant value of E = 0.05 while both B and N are varied, while in Fig. 8b the entrainment ratio E is varied with B and N kept constant. In Fig. 8a the modeled intrusion depth shows good agreement with our scaling of ZB1/3/N. The variation of the intrusion depth with a varying entrainment ratio can be seen to scale very well with ZE−1/3 in Fig. 8b. The scalings of ZB1/3/N and ZE−1/3 now quantify the intuitive prediction that the intrusion depth should increase as the value of the entrainment ratio decreases, as the buoyancy flux increases, or as the stratification decreases. We hope that these two simple scaling laws will be used to test the fidelity of different entrainment parameterizations used in numerical models of intrusions in reservoirs or in the ocean.

A major difference between the numerical scheme used by Price and Baringer (1994) and the theory of Morton et al. (1956) is that the initial source condition of the density currents is not infinitely small but rather has a finite thickness. This means that the effective “virtual source” (Baines and Turner 1969) is located slightly above the finite size source. This is the reason for the offset in the intrusion depth observed in Fig. 8a. Typically the entrainment ratio E used in many stream-tube models is not constant, and it is described as a decreasing function of the local flow Richardson number (Turner 1986). However, in order to make a meaningful comparison with the set of Eqs. (3)(5) we did not take account of the variation of E with the Richardson number.

Many engineering studies of river inflow into reservoirs also use a stream-tube approach of which Antenucci et al. (2005) is a good example. The equations used in this model have been programmed in Java as a Web-based tool that (available online at http://www.cwr.uwa.edu.au/webtools/inflow/index.html). This model is based upon the theory of Dallimore et al. (2004) and is similar to the stream-tube models of Smith (1975) and Price and Baringer (1994). One can quickly use this Web-based tool to generate similar results as those in Fig. 8, when a linear stratification is used and the total depth is much greater than the thickness of the inflow.

b. Comparison with the marginal sea boundary condition of Price and Yang

In addition to determining the depth of intrusion of an oceanic density current, it is also important to know the water mass properties of the intruding density current. In many global circulation models there is insufficient spatial resolution to attempt to resolve the physics of a dense current. One widely used model is the marginal sea boundary condition of Price and Yang (1998). Part of their model describes the normalized change in water mass properties as a function of the buoyancy flux and flow velocity of the density current. They define an “entrainment parameter” as
i1520-0485-39-8-1935-e17
which is essentially the same as the mixedness in Cenedese et al. (2004) and Ivanov et al. (2004). When Φ = 0 there is no entrainment and ρsourceρproduct, while Φ = 1 indicates that entrainment is extremely large, so that ρproductρocean. Price and Baringer (1994) show that the mixedness depends upon the properties of the density current as
i1520-0485-39-8-1935-e18
where Bgeo is the buoyancy flux per unit width of the geostrophic density current, and Ugeo is the velocity. Price and Baringer (1994) show that this group can also be interpreted in terms of the Froude number Fr of the flow, so that . With this model the parameter Φ will increase as Bgeo is decreased or as Ugeo is increased.
Based upon our empirical observation that , we can also make a prediction of how the mixedness parameter Φ depends upon the density current properties. In a linear stratification the difference between the source and product water is determined by the depth of intrusion as ρsourceρproduct = ρoceanN2Z/g. The density difference between the source water and the ocean water is defined by the buoyancy flux as ρsourceρocean = ρoceanBgeo/gUgeoh. Using these definitions and (15), the mixedness is written as
i1520-0485-39-8-1935-e19
Our new equation for mixedness has a different functional form to (18), but has the same behavior in that Φ increases as Bgeo decreases or as Ugeo increases. We tested both (18) and (19) against the dataset from Ivanov et al. (2004) that was used in Fig. 7, but neither equation actually collapse this dataset to a single curve relating Φ to Bgeo and Ugeo. This is despite the apparent success in Price and Baringer (1994) of Eq. (18) explaining the observation that the density ordering of the source waters is the reverse of the density ordering of product waters of the major oceanic overflows from the Filchner Ice Shelf, the Denmark Strait, the Faroe Bank channel, and the Mediterranean Sea. The use of the marginal sea boundary condition of Price and Yang (1998) has been shown by Wu et al. (2007) to lead to improvements in representing the overflow physics of a coarse-scale ocean circulation model of the North Atlantic, and enabled the model to attain the correct depth for the intrusion of the Mediterranean salt tongue in the North Atlantic. We suggest that our Eq. (15) could also be used in such coarse-resolution models in a manner similar to the marginal sea boundary condition of Price and Yang (1998) to describe the intrusion depth of a density current.

5. Conclusions

The main conclusions of this paper is that the intrusion depth of a density current in a linear stratification scales with the buoyancy flux and the buoyancy frequency as Z = (3 ± 1)B1/3/N. For the oceanic density currents the buoyancy flux is calculated from the geostrophic velocity and we found good agreement with oceanographic observations where density currents intrude at a depth given by . This scaling for oceanic intrusion depths can also be written as Z = (3 ± 1)(g2hs/f )1/3/N. In both cases the scaling is consistent with previous experiments for vertical plumes (Wright and Wallace 1979; Bush and Woods 1999). The empirical observation in itself is a useful scaling tool to predict the changing depth that a density current may intrude into the ocean when there is a change in density of the source waters. While most situations of interest in the ocean do not have a linear stratification, the scaling we developed should give guidance to testing the fidelity of more complicated numerical schemes. Of particular interest to such models would be to know the effect of changing the entrainment parameterization upon the intrusion depth, or how the intrusion depth of a density current may change in possible future climates where there might be changes to the buoyancy forcing and the stratification. Our theoretical prediction that ZE−1/3B1/3/N shows that the depth of intrusion is most sensitive to changes in the ocean stratification, and that changes to the entrainment ratio or the buoyancy forcing make less of a difference to the intrusion depth. This dependence of intrusion depth upon the buoyancy forcing, the entrainment ratio, and the ocean stratification should then also be robust features of any model that hopes to capture the physics of density currents in large global circulation models, such as those of Papadakis et al. (2003), Legg et al. (2006), and Wu et al. (2007). We hope that (15) can serve as a benchmark to test the performance of future representations of density current physics in coarse-resolution ocean models.

Acknowledgments

Support for this work was provided by NSERC, the Canadian Foundation for Innovation, the Ontario Research Fund, and the Connaught Committee of the University of Toronto. PN was supported by an NSERC USRA while conducting the experiments for this paper. Comments from Colm Caulfield, Claudia Cenedese, and an anonymous reviewer are gratefully acknowledged.

REFERENCES

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Fig. 1.
Fig. 1.

A sketch of a subsurface intrusion from a density current. The initial density current has a velocity U and thickness h and flows down a slope of angle Θ before intruding into the stratified waters at a depth Z below the surface.

Citation: Journal of Physical Oceanography 39, 8; 10.1175/2009JPO4022.1

Fig. 2.
Fig. 2.

The scaling of the intrusion height of 3D plumes in a linear stratification was predicted by Morton et al. (1956) to be Z = 5B1/4N−3/4. In this figure (adapted from Briggs 1969) there is very good agreement between this scaling and the observed height of the plume rise in calm, stratified surroundings over a range of five orders of magnitude. We will show that similar results can be derived for the intrusion depth of density currents.

Citation: Journal of Physical Oceanography 39, 8; 10.1175/2009JPO4022.1

Fig. 3.
Fig. 3.

A numeric solution of Eqs. (11)(13) showing U* = M*/Q*, b* = Q*2/M*, and ½Δ* = ½F*/Q*. The dimensionless buoyancy is zero at z* = 2.0391 and vertical velocity is zero at z* = 2.9621.

Citation: Journal of Physical Oceanography 39, 8; 10.1175/2009JPO4022.1

Fig. 4.
Fig. 4.

Plan and side views of the tank used in these experiments. Photographs of intrusion depth were taken by viewing from the side. A linear stratification was made by the standard double bucket method. All measurements are in centimeters.

Citation: Journal of Physical Oceanography 39, 8; 10.1175/2009JPO4022.1

Fig. 5.
Fig. 5.

A laboratory picture of an intruding density current. The depth of intrusion Z is measured from the surface to the middle of the density current.

Citation: Journal of Physical Oceanography 39, 8; 10.1175/2009JPO4022.1

Fig. 6.
Fig. 6.

(a) Experimental results for slopes of 15°, 33°, 45°, 60°, 75°, and 90°. For these 41 laboratory experiments, there is fair agreement with the scaling of the intrusion depth scales as Z = (3 ± 1)B1/3/N. The linear salt stratification was varied between N = 0.3 and 1.7 s−1. (b) Same data as in (a), but the ratio of observed depth of intrusion to the length scale B1/3/N, as a function of slope, is plotted. The mean value this ratio has a value close to 3 (as shown by the dashed line) so intrusion depth scales as Z = (3 ± 1)B1/3/N for all of the slopes studied in this experiment.

Citation: Journal of Physical Oceanography 39, 8; 10.1175/2009JPO4022.1

Fig. 7.
Fig. 7.

(a) The depth of intrusion of oceanic density currents is plotted in along with lines indicating . (b) The oceanic observations along with laboratory results are plotted. There is good agreement with the scaling of the intrusion depth scales as Z = (3 ± 1)B1/3/N over six orders of magnitude, similar to the scaling shown in Fig. 2.

Citation: Journal of Physical Oceanography 39, 8; 10.1175/2009JPO4022.1

Fig. 8.
Fig. 8.

The depth of intrusion is determined using the simplified version of the Price and Baringer (1994) stream-tube model. The numerical results (circles) and the predicted scaling (dashed line) are shown. (a) The entrainment ratio E is kept constant and the depth of the intrusion is seen to scale well with ZE−1/3B1/3/N as both B and N are varied. There is a constant offset from the predictions resulting from the finite size of the initial density current, but otherwise the agreement is good. (b) The buoyancy flux B and the buoyancy frequency N are kept constant and the intrusion depth is seen to scale as ZE−1/3.

Citation: Journal of Physical Oceanography 39, 8; 10.1175/2009JPO4022.1

Save
  • Antenucci, J. P., J. D. Brookes, and M. R. Hipsey, 2005: A simple model for quantifying cryptosporidium transport, dilution, and potential risk in reservoirs. J. Amer. Water Works Assoc., 97 , 8693.

    • Search Google Scholar
    • Export Citation
  • Baines, P. G., 2001: Mixing in flows down gentle slopes into stratified environments. J. Fluid Mech., 443 , 237270.

  • Baines, P. G., 2005: Mixing regimes for the flow of dense fluid down slopes into stratified environments. J. Fluid Mech., 538 , 245267.

    • Search Google Scholar
    • Export Citation
  • Baines, W. D., and J. S. Turner, 1969: Turbulent buoyant convection from a source in a confined region. J. Fluid. Mech., 37 , 5180.

  • Baringer, M. O., and J. F. Price, 1999: A review of the physical oceanography of the Mediterranean outflow. Mar. Geol., 155 , (1–2). 6382.

    • Search Google Scholar
    • Export Citation
  • Bower, A. S., H. D. Hunt, and J. F. Price, 2000: Character and dynamics of the Red Sea and Persian Gulf outflows. J. Geophys. Res., 105 , (C3). 63876414.

    • Search Google Scholar
    • Export Citation
  • Briggs, G. A., 1969: Optimum formulas for buoyant plume rise. Proc. Roy. Soc. London, 265A , 197203.

  • Bush, J. W. M., and A. W. Woods, 1999: Vortex generation by line plumes in a rotating stratified fluid. J. Fluid Mech., 388 , 289313.

  • Cardoso, S. S. S., and A. W. Woods, 1993: Mixing by a turbulent plume in a confined stratified region. J. Fluid Mech., 250 , 277305.

  • Caulfield, C. P., and A. W. Woods, 1998: Turbulent gravitational convection from a point source in a non-uniformly stratified environment. J. Fluid Mech., 360 , 229248.

    • Search Google Scholar
    • Export Citation
  • Cenedese, C., and C. Adduce, 2007: Mixing induced in a dense plume flowing down a sloping bottom in a rotating fluid: A new entrainment parameterization. Proc. Fifth Int. Symp. on Environmental Hydraulics Conf., Tempe, AZ, IAHR.

    • Search Google Scholar
    • Export Citation
  • Cenedese, C., and C. Adduce, 2008: Mixing in a density-driven current flowing down a slope in a rotating fluid. J. Fluid Mech., 604 , 369388.

    • Search Google Scholar
    • Export Citation
  • Cenedese, C., J. A. Whitehead, T. A. Ascarelli, and M. Ohiwa, 2004: A dense current flowing down a sloping bottom in a rotating fluid. J. Phys. Oceanogr., 34 , 188203.

    • Search Google Scholar
    • Export Citation
  • Dallimore, C. J., J. Imberger, and B. R. Hodges, 2004: Modeling a plunging underflow. J. Hydraul. Eng., 130 , 10681076.

  • Darelius, E., and A. Wahlin, 2007: Downward flow of dense water leaning on a submarine ridge. Deep-Sea Res. I, 54 , 11731188.

  • Ellison, T. H., and J. S. Turner, 1959: Turbulent entrainment in stratified flows. J. Fluid Mech., 6 , 423448.

  • Fer, I., U. Lemmin, and S. A. Thorpe, 2001: Cascading of water down the sloping sides of a deep lake in winter. Geophys. Res. Lett., 28 , 20932096.

    • Search Google Scholar
    • Export Citation
  • Fischer, H., E. List, R. Koh, J. Imberger, and N. Brooks, 1979: Mixing in Inland and Coastal Waters. Academic Press, 302 pp.

  • Foldvik, A., and Coauthors, 2004: Ice shelf water overflow and bottom water formation in the southern Weddell Sea. J. Geophys. Res., 109 , C02015. doi:10.1029/2003JC002008.

    • Search Google Scholar
    • Export Citation
  • Gordon, A. L., E. Zambianchi, A. Orsi, M. Visbeck, C. F. Giulivi, T. Whitworth III, and G. Spezie, 2004: Energetic plumes over the western Ross Sea continental slope. Geophys. Res. Lett., 31 , L21302. doi:10.1029/2004GL020785.

    • Search Google Scholar
    • Export Citation
  • Hebbert, B., J. Imberger, I. Loh, and J. Patterson, 1979: Collie River underflow into the Wellington reservoir. J. Hydraul. Div., 105 , 533545.

    • Search Google Scholar
    • Export Citation
  • Hughes, G. O., and R. W. Griffiths, 2006: A simple convective model of the global overturning circulation, including effects of entrainment into sinking regions. Ocean Modell., 12 , 4679.

    • Search Google Scholar
    • Export Citation
  • Ivanov, V. V., G. I. Shapiro, J. M. Huthnance, D. L. Aleynik, and P. N. Golovin, 2004: Cascades of dense water around the world ocean. Prog. Oceanogr., 60 , 4798.

    • Search Google Scholar
    • Export Citation
  • Killworth, P. D., 1977: Mixing on the Weddell Sea continental slope. Deep-Sea Res., 24 , 427448.

  • Legg, S., R. W. Hallberg, and J. B. Girton, 2006: Comparison of entrainment in overflows simulated by z-coordinate, isopycnal and non-hydrostatic models. Ocean Modell., 11 , 6997.

    • Search Google Scholar
    • Export Citation
  • MacCready, P., W. E. Johns, C. G. Rooth, D. M. Fratantoni, and R. A. Watlington, 1999: Overflow into the deep Caribbean: Effects of plume variability. J. Geophys. Res., 104 , (C11). 2591325935.

    • Search Google Scholar
    • Export Citation
  • MacIntyre, S., J. Sickman, S. Goldthwait, and G. Kling, 2006: Physical pathways of nutrient supply in a small, ultraoligotrophic arctic lake during summer stratification. Limnol. Oceanogr., 51 , 11071124.

    • Search Google Scholar
    • Export Citation
  • McCullough, G. K., D. Barber, and P. M. Cooley, 2007: The vertical distribution of runoff and its suspended load in Lake Malawi. J. Great Lakes Res., 33 , 449465.

    • Search Google Scholar
    • Export Citation
  • Monaghan, J. J., 2007: Gravity current interaction with interfaces. Ann. Rev. Fluid. Mech., 39 , 245261.

  • Monaghan, J. J., R. A. F. Cas, A. M. Kos, and M. Hallworth, 1999: Gravity currents descending a ramp in a stratified tank. J. Fluid Mech., 379 , 3970.

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  • Fig. 1.

    A sketch of a subsurface intrusion from a density current. The initial density current has a velocity U and thickness h and flows down a slope of angle Θ before intruding into the stratified waters at a depth Z below the surface.

  • Fig. 2.

    The scaling of the intrusion height of 3D plumes in a linear stratification was predicted by Morton et al. (1956) to be Z = 5B1/4N−3/4. In this figure (adapted from Briggs 1969) there is very good agreement between this scaling and the observed height of the plume rise in calm, stratified surroundings over a range of five orders of magnitude. We will show that similar results can be derived for the intrusion depth of density currents.

  • Fig. 3.

    A numeric solution of Eqs. (11)(13) showing U* = M*/Q*, b* = Q*2/M*, and ½Δ* = ½F*/Q*. The dimensionless buoyancy is zero at z* = 2.0391 and vertical velocity is zero at z* = 2.9621.

  • Fig. 4.

    Plan and side views of the tank used in these experiments. Photographs of intrusion depth were taken by viewing from the side. A linear stratification was made by the standard double bucket method. All measurements are in centimeters.

  • Fig. 5.

    A laboratory picture of an intruding density current. The depth of intrusion Z is measured from the surface to the middle of the density current.

  • Fig. 6.

    (a) Experimental results for slopes of 15°, 33°, 45°, 60°, 75°, and 90°. For these 41 laboratory experiments, there is fair agreement with the scaling of the intrusion depth scales as Z = (3 ± 1)B1/3/N. The linear salt stratification was varied between N = 0.3 and 1.7 s−1. (b) Same data as in (a), but the ratio of observed depth of intrusion to the length scale B1/3/N, as a function of slope, is plotted. The mean value this ratio has a value close to 3 (as shown by the dashed line) so intrusion depth scales as Z = (3 ± 1)B1/3/N for all of the slopes studied in this experiment.

  • Fig. 7.

    (a) The depth of intrusion of oceanic density currents is plotted in along with lines indicating . (b) The oceanic observations along with laboratory results are plotted. There is good agreement with the scaling of the intrusion depth scales as Z = (3 ± 1)B1/3/N over six orders of magnitude, similar to the scaling shown in Fig. 2.

  • Fig. 8.

    The depth of intrusion is determined using the simplified version of the Price and Baringer (1994) stream-tube model. The numerical results (circles) and the predicted scaling (dashed line) are shown. (a) The entrainment ratio E is kept constant and the depth of the intrusion is seen to scale well with ZE−1/3B1/3/N as both B and N are varied. There is a constant offset from the predictions resulting from the finite size of the initial density current, but otherwise the agreement is good. (b) The buoyancy flux B and the buoyancy frequency N are kept constant and the intrusion depth is seen to scale as ZE−1/3.

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