Denmark Strait overflow: Comparing model results and hydraulic transport estimates

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109,, doi: /2004jc002297, 2004 Denmark Strait overflow: Comparing model results and hydraulic transport estimates F. Kösters Department of Geosciences, University of Kiel, Kiel, Germany Received 27 January 2004; revised 3 June 2004; accepted 30 July 2004; published 20 October [1] The transport of dense water through Denmark Strait is investigated with a highresolution numerical model. The results are compared with different transport estimates from rotating hydraulic theory in order to evaluate such theories with respect to the interpretation of hydrographic data. Generally, estimates far upstream result in upper transport limitations that can be appropriately scaled to give time mean transport estimates, whereas estimates from the sill entrance region can provide a good approximation of the variable transport. INDEX TERMS: 4508 Oceanography: Physical: Coriolis effects; 4255 Oceanography: General: Numerical modeling; 4203 Oceanography: General: Analytical modeling; 4263 Oceanography: General: Ocean prediction; 4520 Oceanography: Physical: Eddies and mesoscale processes; KEYWORDS: rotating hydraulics, Denmark Strait, numerical modeling Citation: Kösters, F. (2004), Denmark Strait overflow: Comparing model results and hydraulic transport estimates, J. Geophys. Res., 109,, doi: /2004jc Introduction [2] For present-day climate conditions, the formation of North Atlantic Deep Water (NADW) is an important part of the thermohaline circulation [Schmitz, 1995]. NADW is mainly originating from the cold and dense water masses formed in the Nordic Seas and subsequently crossing the Greenland-Scotland-Ridge (GSR) to enter the North Atlantic basin [Dickson and Brown, 1994; Hansen and Østerhus, 2000]. One important deep-water gateway across the GSR is the Denmark Strait, situated between Greenland and Iceland. Owing to the importance of deep-water formation in the Nordic Seas for the meridional overturning circulation in the North Atlantic [Döscher and Redler, 1997; Redler and Böning, 1997] the Denmark Strait can be regarded as a key region for assessing climate change. Various field programmes have tried to measure the overflow strength [e.g., Dickson and Brown, 1994], its pathway downstream [Girton et al., 2001] and its sources [Rudels et al., 1999; Jónsson, 1999; Rudels et al., 2002]. Recently the Northern Overflows have obtained renewed attention due to an observed freshening of the overflowing water mass [Dickson et al., 1999] as part of an overall change in the Atlantic freshwater balance [Curry et al., 2003]. [3] The apparent lack of a seasonal signal indicates that the flow is hydraulically controlled and Whitehead et al. [1974] have first applied the theory of rotating hydraulics to describe the overflow. Subsequently, the field of rotating hydraulics evolved with various theories of increasing complexity (see Whitehead [1998] for a review) and with numerical modeling studies [Käse and Oschlies, 2000] supporting the theory of Whitehead et al. [1974]. However, Copyright 2004 by the American Geophysical Union /04/2004JC002297$09.00 analytical models are always idealized descriptions of the overflow processes with associated intrinsic difficulties for quantitative transport estimates using hydrographic data. Helfrich and Pratt [2003] point out that the possible reduction of the Faroe Banks Channel overflow and its climatic implications [Hansen et al., 2001] may be due to changes in the upstream circulation instead of a decrease in the actual dense water transport. This example emphasizes the need for a better understanding, testing and modeling of hydraulic laws. [4] In the following results from a numerical ocean general circulation model will be compared with the recent theoretical approaches of Nikolopoulos et al. [2003] and Helfrich and Pratt [2003] in addition to a modified version of the original theory of Whitehead et al. [1974]. Thus the fidelity of hydraulic theory is tested for the interpretation of oceanographic data and for applications concerning paleoceanographic questions of the Denmark Strait. 2. Model [5] The Regional Ocean Modeling System (ROMS) [Haidvogel et al., 2000] is employed with 30 equally spaced sigma layers in the vertical and an eddy resolving horizontal resolution of Dx = Dy =5km 1/20. Sigma coordinates have been chosen because they can resolve the bottom layer far better than z models [Willebrand et al., 2001], which makes them more suitable for overflow modeling. In order to reduce the pressure gradient error inherent in sigma coordinate models, the bathymetry has been smoothed after interpolating it to the model grid following suggestions from Haidvogel and Beckmann [1999] keeping the realistic values for the depth (580 m) and width (350 km) of the Denmark Strait sill. The model domain (Figure 1) extends from 60.5 N to 70.5 N and from 45 W to16 Whaving closed walls and land areas masked out. 1of10

2 Rutgers University Ocean Model SCRUM, the previous model version of ROMS, as used in the work of Käse and Oschlies [2000] where the source terms are added in the vertical momentum equations. However, since the model is run for 60 days only the reservoir height is not strongly changing and a minor influence of the sources on the transport can be assumed. Figure 1. Map of the bathymetry in the area of the Denmark Strait located between Greenland and Iceland. The box outlined shows the model area for the numerical experiments. The lines labeled A D within this box define the section locations used for analyzing the model results. Data sampled at line E provide information on the model state in along-strait direction. [6] The horizontal advection scheme is of third order with implicit diffusion [Shchepetkin and McWilliams, 1998] and therefore no explicit diffusivities or viscosities are needed to maintain stable solutions. Vertical advection is fourth-order centered and vertical mixing is implemented as the Large- McWilliams-Doney closure [Large et al., 1994]. The bottom friction is set as quadratic bottom drag, with a coefficient of c D =10 3 as in the work of Käse and Oschlies [2000]. The equation of state is simplified assuming that density r depends only on potential temperature q T, according to r = kg m 0.08 kg 3 Cm q 3 T, which was derived from a linear regression of density as a function of temperature. Thus the salt content is not considered explicitly. For more accurate transport calculations, especially to take into account for entrainment at the sill, an additional passive tracer is added. [7] The model is initialized with two water masses having a potential temperature of q T = 1 C (r = kg m ) north of the sill and q 3 T =+5 C (r = kg m ) south of the sill, resembling the average two respective densities. 3 A dam-break scenario at the sill is used as experimental setup with buoyancy forcing only. The dam initially separating the two water masses is removed at time zero and dense water starts to descend the sill. In all runs, the model is integrated for 60 days, and days 20 to 60 are analyzed. After 20 days the model is at steady state and after 60 days the overflow plume hits the southern boundary where reflections occur. [8] The model is forced by buoyancy only with neither wind stress nor surface fluxes, therefore the reservoir height has to be kept artificially constant. The warm water advected from the south is replaced with cold water, using sinks and sources at the northern boundary. These source and sink terms are included in the horizontal tracer and momentum equation. A northward flow is prescribed in the top five layers whereas the flow is set to be southward in the bottom five layers. It thus resembles the advection of newly formed deep water. This is different from the S coordinate 3. Revision of Theory [9] Sidewall and bottom constrictions of Denmark Strait (Figure 1) show only small depth variations in alongchannel direction, which allows to describe the transport using rotating hydraulic laws as in the work of Whitehead et al. [1974] (WLK in the following). The hydrography may be simplified since the observed density structure [Conkright et al., 2003] can be approximated as a two-layer system Criticality of Flow [10] The lack of seasonal variations [Saunders, 2001], strong influence from rotation and the topographic setting suggests that the flow through Denmark Strait is hydraulically controlled. However, frictional effects could lead to a similar appearance without hydraulic control being active [Pratt, 1986]. Therefore it is essential to determine the criticality of the flow. [11] In a nonrotating channel information propagates at p ffiffiffiffiffiffiffi gravity wave speed c = g 0 D, with g 0 = gdr/r as reduced gravity. This can be expressed in terms of a composite Froude number G, which is defined for a two-layer system and small density contrasts [Armi, 1986] as G 2 ¼ F 2 1 þ F2 2 ; with F 2 i = v 2 i /g 0 h i as the Froude number of a single layer with thickness h i. A Froude number smaller than unity (G < 1) indicates that the flow is subcritical, if bigger than unity (G > 1) it is supercritical and if equal unity (G = 1) it is critical. Control points are such critical points where long-wave disturbances (gravity waves) travel neither upstream nor downstream. Unfortunately rotation is important for Denmark Strait and the simple nonrotating assessment of flow criticality might not be applicable. For critical conditions in rotating channel flow [Pratt et al., 2000] defined a generalized Froude number F S based on the critical condition of Stern [1974]. A different approach is to define a semigeostrophic Froude number F d as in the work of Gill [1977] and Helfrich and Pratt [2003]. [12] The location of the control section for different Froude numbers (Figure 2) shows that the in nonrotating systems commonly used composite Froude number G is still meaningful even in the case of strong rotation. The Froude number analysis is shown for an experiment with H eff = 480 m but the position of the control point does not dependent on the effective height. Even though the downstream behaviour is different, the flow reaches a critical state at roughly the same distance from the sill, whereas is slightly shifts for an increasing density contrast toward the sill (not shown). As the Denmark Strait is wide compared to the internal Rossby radius R d information will be ð1þ 2of10

3 balance. The governing equations are the momentum equation and the equation of continuity. For semigeostrophic flow the horizontal momentum equation þ fu ; ð2þ Fv ¼ ; ð3þ Figure 2. Comparison of the position of the control section based on the composite Froude number G, critical conditions for rotating flow F S and the semigeostrophic Froude number F d. transmitted via Kelvin waves along the boundary of the sill. Baroclinic Kelvin wavesp travel ffiffiffiffiffiffiffi with a phase speed of internal gravity waves at c = g 0 D. With typical values of the Denmark pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Strait one yields for the gravity wave speed of c = 0:0033 m s m = 0.72 m/s. Comparing this with the observed velocities at the sill of up to 1.3 m/s and a mean velocity of 0.56 m/s [Girton and Sanford, 2003] the fluid advection speed exceeds the gravity wave speed. From Figure 2 one can conclude that the fluid advection speed is small when approaching the sill (negative distance) and the Froude number is subcritical (G < 1), after passing through the sill the fluid accelerates when descending and reaches a critical Froude number (G = 1) at 80 km downstream. When descending further the fluid can gain even more speed and gets supercritical. The position of the critical section 80 km downstream of the sill is in excellent agreement with observations [Girton, 2001] and expected from hydraulic theory [Pratt, 1986; Garrett and Gerdes, 2003]. Besides the effects of rotation and friction, stratification will affect the overflow [Killworth, 1992] and alter the criterion for critical flow with no commonly agreed solution so far. Therefore the following Froude number analysis is restricted to the definition given by equation (1) Rotating Hydraulics [13] The reference system for investigating rotating hydraulics is commonly chosen as a rotating channel aligned in the y direction with cross-channel direction x and bottom topography T(x, y) positive upward. Then a 1 1 = 2 -layer model is used, which consists of an upper, stagnant layer of density r and a lower, moving layer of density r + Dr (Figure 3a). The height of the interface above the sill is denoted as h(x), the depth of the lower layer is D(x), and the topography is described by T(x, y), hence D(x) =h(x) T(x) (Figure 3). [14] In Figure 3a the parabolic shape represents the overflow with the intersection points a and b of the parabola with the topography and with h(a) and h(b) as the corresponding height above sill level. Owing to the slow variation in topography along the y axis the along-channel velocity component of the overflow is in geostrophic where u and v are the velocities in x and y direction, respectively and f is the Coriolis parameter. The equation of continuity is vd Þ ¼ 0: ð4þ From this, the conservation of potential vorticity G can be shown and yields z þ f D ¼ GðyÞ; where is the relative vorticity and y is the transport stream function, ¼ u: This is the basis for the following approaches which then differ in the treatment of potential vorticity conservation Figure 3. Schematic representation of the Denmark Strait Overflow. (a) An idealized cross-channel section as viewed from the North with Iceland to the left and Greenland to the right (adopted from Nikolopoulos et al. [2003]) and (b) the along-channel structure. ð5þ ð6þ 3of10

4 (equation (5)). At first the classical WLK model is outlined followed by a review of two recently published studies. Borenäs and Nikolopoulos [2000] extended the WLK approach to take into account realistic bathymetry which has been applied to the Denmark Strait by Nikolopoulos et al. [2003]. The study of Helfrich and Pratt [2003] presents a revised version of the Gill model [Gill, 1977] and puts emphasis on the upstream conditions Deep Upstream Basin [15] First, the case in which the upstream basin is deep and upstream velocities are negligible is considered. The Bernoulli potential B(y) is that of the upstream basin B(y)= g 0 h ups, which is here assumed to be constant and hence db dy = G(y) = 0. With little curvature in y = 0 this simplifies equation (2) ¼ f : Now the transport is calculated by integrating v h from the right hand wall to the intersection point a. The maximum value of h is the undisturbed upstream value h ups, which results in a maximum transport Q WLK Q WLK ¼ 1 2 ð7þ g 0 h 2 ups : ð8þ f Note that assuming G(y) is small is often referred to as zero-potential vorticity assumption but as Pratt and Lundberg [1991] have pointed out it should be referred to as deep-upstream-basin case, since G(y) is dimensional and can be quite large even though the ratio of h sill /h ups Borenäs and Nikolopoulos [2000] [16] One basic simplification of the WLK model is that it assumes a rectangular cross section. This restriction was lifted by Borenäs and Nikolopoulos [2000] who developed a method for calculating the maximum deep-water flow for real topography. From geostrophy (equation (2)) and potential vorticity (equation (5)) an equation for the interface height h can be obtained h ¼ f 2 x 2 2g 0 þ fv 0 g 0 x þ h 0 : This equation contains two constants, the interface height h 0 and the velocity V 0 at x = 0, which can be determined from the intersection points a and b and their according depth T(a) and T(b) (Figure 3a). To determine combinations of a and b consider the case of zero potential vorticity where the Bernoulli function is a constant B(y) = g 0 h 1 for all streamlines and can be written as BðyÞ ¼ 1 2 v2 þ g 0 h ¼ g 0 h ups : Now h 0 and V 0 can be combined using the Bernoulli equation. From this one obtains a defining function for the constants a and b and all combinations of intersection points a and b can be found. Owing to the approximately parabolic cross-sectional topography of Denmark Strait a and b ð9þ correspond to an inverse parabola shifted along the y axis. Nikolopoulos et al. [2003] suggest two different transport limitations. The shape of h implies return flow where the slope of h is < 0) and as Killworth [1994] has pointed out this area of reversed flow might be replaced with a stagnant water mass to evaluate maximum transport bounds. Therefore two different transports are calculated, for Q NB the integral from a to b and for Q NB2 the integral from a to the maximum of h, which yields: Q NB ¼ Z b a vddx ¼ g0 2f Z hmax Q NB2 ¼ vddx ¼ g0 a 2f Z h 2 ðþ h b 2 b ðþ a vtdx; Z h 2 ðh max Þ h 2 hmax ðaþ vtdx: a a ð10þ ð11þ Nikolopoulos et al. [2003] applied this approach to the Denmark Strait to obtain transport estimates from hydrographic sections. Moreover, the authors extended the zero potential vorticity assumption to the case of constant potential vorticity but found that changes due to a finite value of potential vorticity are small and therefore the zero potential vorticity case will be discussed in the following only Helfrich and Pratt [2003] [17] The coupling of upstream circulation with the hydraulic flow within the strait has been examined by Helfrich and Pratt [2003]. They extended the classical Gill model [Gill, 1977] to a finite upstream basin and compared it with results from a 1 1 = 2 layer reduced gravity model. Important here is that different source types (inflow and downwelling) and their influence on the transport over the sill have been considered. Using the basin potential vorticity budget to investigate the basin-strait coupling [Pratt and Llewellyn-Smith, 1997] found that for interior downwelling an anticyclonic boundary current will develop, whereas a boundary inflow will split into two boundary currents. They showed that their model correctly predicts the transport independent of the upstream circulation. The transport as suggested in their model can be calculated at the strait entrance region as: Q ¼ g0 h 2 R 2f h2 L ; ð12þ where h R is the maximum and h L is the minimum value of h. Q WLK is from Whitehead et al. [1974]; Q WLK2 is the same as Q WLK but taken at the sill entrance; Q NB is from Nikolopoulos et al. [2003]; Q NB2 is the same as Q NB but no recirculation; Q HPG is from Helfrich and Pratt [2003], extended Gill model; Q p is modeled transport (from passive tracer); Q t is modeled transport (from temperature); and Q td is modeled transport (depth restricted, from temperature). 4. Results [18] Even though the numerical model is simplified, the main features from observations [Dickson and Brown, 1994; 4of10

5 Figure 4. (a) Cross section of the density field along the strait (section E in Figure 1) for the model initial conditions, (b) a snapshot of the modeled steady state (day 35), and (c) observations from Käse et al. [2003]. The thick contour line of the 2 C isothermal depicts the overflow. Girton et al., 2001; Girton and Sanford, 2003] are reproduced as in previous modeling experiments [Käse et al., 2003]. After dam-break at day zero (Figure 4) a dense water plume descends from the sill and follows the topographic slope along the Greenland shelf break. The model is in steady state after 20 days and the mean plume position stays relatively constant in time. Downstream of the sill baroclinic instabilities occur and eddies develop consistent with modeling experiments of Jungclaus et al. [2001]. The initial conditions and a snapshot of the modeled steady state for day 35 are shown in Figures 4a and 4b in along-strait direction (following section E from Figure 1). [19] With increasing distance from the sill, the overflow thickness reduces and the width increases due to mixing and entrainment with ambient warmer water. The plume structure resembles that of observations at least as far downstream as 63 N [Kösters et al., 2004]. Since the boundaries are closed the plume structure is influenced by reflections after it hits the southern boundary around day 60. However, the downstream dynamics and the descent of the plume are not primary objectives here and it is assumed that the downstream behaviour is not significant for the overflow transport as information cannot travel upstream as previously discussed. [20] The velocity structure has a significant barotropic component but velocities are increasing toward the bottom. The bottom velocity has a mean value of 25 cm/s, reaching up to 60 cm/s at times. This is less than the observed values of up to 1.3 m/s, but the overall velocity pattern resembles the observations. [21] The dense water transport through the Denmark Strait is commonly defined as that of water colder than 2 C or having a density higher than s q = Owing to entrainment at the sill entrance this will not correctly measure the amount of dense water transported over the sill but is affected by recirculation. Therefore a passive tracer was used to calculate the transport in the model. Since the transport Q p derived from the passive tracer does not relay 5of10

6 Figure 5. Modeled transport as a function of squared 2 upstream reservoir height h ups for different transport definitions. Transport of water colder than 2 C (Q t ), transport for water colder than 2 C and a depth exceeding 350 m (Q td ), and calculated from the passive tracer (Q p ). on a temperature definition it results in a more accurate estimate of the mean transport. For comparison the modeled transport is calculated across the sill at section A (Figure 1) using three different transport definitions, Q p, Q t, the transport of water colder than 2 C, Q td using the same temperature limitation but additionally restricted to that part of the sill where the depth exceeds 350 m. All of these three definitions of volume transport show a quadratic relation with the upstream reservoir height h ups. In Figure 5 this is represented as a linear relation of the volume transports and the squared upstream reservoir height h 2 ups. [22] For observations a passive tracer suitable for quantitative transport estimates is not available yet but might be estimated in the future from analyzing chlorofluorocarbon inventories [Smethie and Fine, 2001]. The results from our simulations suggest that the transport estimated with the temperature criterion (Q t ) underestimates the true overflow. Detailed inspection of the results indicate that the recirculation of cold water on the shelf is responsible for this bias. It can be circumvented by restricting the calculations to depth below 350 m. This is shown by Q td which closely approximates the true transport Q p. [23] For hydraulically controlled flow the transport through the sill can be assessed from the upstream reservoir height and the density contrast alone. The three different methods explained above are used to determine the theoretical transports and compare them with the modeled transport Q p. The deep-upstream basin approach is applied at the sill entrance (Q WLK2 ) at section C and upstream (Q WLK ) at section D. The extended Gill theory (Q HPG )is employed at the sill entrance region along section B. The approach from Nikolopoulos is taken at section A for both the full width (Q NB ) and the part with a positive slope of the interface h only (Q NB2 ). [24] The upstream reservoir height is changed systematically from 80 m to 580 m in steps of 100 m to test the theoretical relations. As an example the transport time series for the case of h ups. = 580 m is shown (Figure 6). [25] Q WLK is clearly an upper bound on the modeled transport, with very little variability in time. It is already determined from the initial conditions of reservoir upstream height and density contrast. If the transport is evaluated closer to the sill (Q WLK2 ) initially the values are similar to the upstream estimate but after the adjustment to steady state (20 days) it is close to the modeled transport. The variability is similar to that of the modeled transport, but the time mean is slightly underestimated. Estimate Q HPG produces transports close to that of the numerical model during the first 20 days but gradually increases toward too high values as the model approaches steady state. Q NB2 overestimates the flow most of the time whereas Q NB predicts too low values. Both estimates are only weakly variable and can be regarded as upper and lower bounds on the modeled transport. [26] Next, the time mean flow is investigated for different external conditions like reservoir height. Therefore the different transport time series have been averaged over days 20 to 60. For each experiment the modeled transport is plotted versus the theoretical transport estimates (Figure 7 with error bars indicating the 95% confidence bounds as in subsequent figures) corroborating the results from the transport time series (Figure 6). [27] The transport estimate from Q WLK2 overestimates the modeled transport by far. Q NB2 and Q HPG slightly overestimate it where Q WLK and Q NB underestimate it. However, all of these theoretical estimates show a quadratic relationship of the transport and the dense water interface height and can be appropriately scaled to predict the correct time mean modeled transport (Table 1). [28] Besides the dependence on the upstream height the modeled transport also depends on the density contrast and the Coriolis force. Each of the three theories presented Figure 6. Transport time series of the modeled transport Q p and the theoretical transports (Q WLK2, Q WLK, Q HPG, Q NB and Q NB2 ) for an upstream height of h ups = 580 m. For the smoothed modeled transport a 3 point running mean filter was applied. 6of10

7 Figure 8. Modeled transport (Q p ) as a function of changing density contrast Dr across the sill. The observations are taken from Girton et al. [2001]. Figure 7. Theoretical transport as function of modeled transport Q p. The theoretical transport is calculated comparing various approaches. (a) Derived from the classical WLK theory (Q WLK ) and from the revised version Q WLK2. (b) The method from Nikolopoulos et al. [2003] is used (Q NB and Q NB2 ), and (c) the method from Helfrich and Pratt [2003] is applied(q HPG ). The dash-dotted line represents the perfect theory Q theory = Q model and the error bars indicate the 95% confidence bounds. above assumes a linear relationship with the density contrast. This was tested in the same systematic way as for the changing upstream reservoir height. The density contrast was changed from 0.16 to 0.72 sigma units, which corresponds to a decrease by a factor of three up to an increase by a factor of 1.5. [29] As one can see from Figure 8, the assumed linear relationship between density contrast and transport holds. Here it has to be kept in mind that the density in the real ocean depends on salinity and temperature connected with a nonlinear equation of state. [30] The Coriolis force depends on the lateral position of the sill, which is only changing on geological timescales. However, in order to fully evaluate the rotating hydraulics theories the assumed inverse linear relationship of f with the transport has been tested. The range of useful values of f is limited because the Rossby radius is changing with f and in the derivations above it is assumed that the sill width is large compared to the Rossby radius. Therefore the variation of f was limited to s 1 to s 1, which relates to a meridional position of the sill from 26 N to 63 N. Figure 9 shows that a linear function of f 1 describes the Table 1. Overview of the Ratio of Experimental to Theoretical Transports Theory Experiment/Theory, % Correlation Coefficient WLK WLK NB NB HPG Figure 9. Modeled transport as a function of the inverse Coriolis parameter f 1. 7of10

8 Figure 10. Ratio of modeled Q p to theoretical WLK transport calculated for the sill and upstream area with a depth greater than 350 m. The vectors show the volume flux of cold water in 150 m 2 /s. See color version of this figure at back of this issue. transport sufficiently well, with a correlation coefficient of R = Discussion [31] Hydraulics The difference between the deepupstream estimates taken at different sections (Q WLK versus Q WLK2 ) poses the question on the spatial structure of the upstream estimate. In order to evaluate how the flux predicted from theory depends on the upstream sites where it is evaluated the theoretical estimate calculated for all upstream points deeper than 350 m is represented as the ratio of modeled to theoretical transport in Figure 10. [32] The transport estimate in the upstream area is constant over a broad spatial range with a ratio of modeled to theoretical transport of 46%, which shows only little spatial variability and provides a robust upper transport bound. Along the boundaries and at the sill the ratio is rapidly increasing. Close to the sill it is almost 100% and the variability is close to that of the modeled transport (Figure 6). [33] The overflow position at the sill as determined from interface height and density contrast according to Borenäs and Nikolopoulos [2000] (Figure 11a) resembles the model results (Figure 11b). [34] The overflow position and shape are best described with a mean of the shapes associated with Q NB and Q NB2 as seen for the transport estimates. The mean position of the overflow core in the model coincides with the mean position from the theory (dashed line in Figure 11). The transport estimates from this method differ strongly depending if the reversed flow is taken into account (Q NB ) or taken as stagnant (Q NB2 ). This ambiguity complicates the application to modern hydrographic sections. However, it is potentially useful when the hydrography is not well known as for paleoceanographic questions. [35] Finally, Q Helfrich gave very convincing results when compared to model data but the method depends strongly on the hydrographic input data. As the interface height is Figure 11. Theoretical shape of overflow as determined from geostrophic relation [Borenäs and Nikolopoulos, 2000] (top panel with the left profile corresponding to Q NB and the right profile to Q NB2 ) and model results (average of days 51 61) for section A with shaded temperature and velocity (negative southward) contours (bottom panel). See color version of this figure at back of this issue. 8of10

9 determined in the sill entrance region it is variable in time. Owing to the strong variability a snapshot of the density field as obtained from a field survey might be very different from one taken a few days later. This difficulty could be overcome when key regions can be identified and monitored with moorings. Maybe solving problems with moorings directly at the Denmark Strait where high velocities lead to a very difficult environment for the equipment (R. H. Käse, personal communication, 2004). [36] The linear relationship of density contrast and transport emphasizes the importance of observations of a reduced density or freshening of the overflows [Dickson et al., 2002]. The model clearly shows that a decreased density contrast would lead to a reduced overflow strength. That could in turn affect the overturning in the North Atlantic. A freshening of the overflow and the corresponding salinity decrease is not explicitly included in our model but the resulting density change can be translated to an equivalent change in temperature. [Dickson et al., 2002] found a decrease in salinity of the Denmark Strait Overflow of to a density loss of approximately s 0.As one can derive from equation (8) this would cause a relative decrease in dense water transport by almost 7%. Further confidence toward the theoretical approach is obtained from the variation of f. The theory is confirmed by the model even though it seems that there is a higher variability and the linear relationship is less pronounced as for the density contrast. [37] There are some aspects which are not considered in this study. The effect of a wind driven barotropic component could not be realized due to the limited model domain not allowing for flow around Iceland. This will be left to subsequent studies employing an expanded model area. Moreover the downstream behaviour is sensitive to the stratification, which is not realistically included here. 6. Conclusions [38] The classical WLK approach can be used nearly without any scaling factor to predict the transport if it is evaluated at the sill entrance and not upstream. The upstream value imposes an upper bound on the transport with a mean modeled transport of only about 46% of that predicted by hydraulic limitation. Because of its small spatial variability it might be applicable when the reservoir height and density contrast are only approximately known as for paleoceanographic questions. Q NB2 defines an upper transport bound whereas Q NB sets the lower transport bound, both estimates are only weakly variable and seem to define the model transport extrema well. As a drawback, this leaves quite an uncertainty which makes this method less applicable for the interpretation of hydrographic sections. Q HPG might be most useful for the interpretation of hydrographic sections because of its good skill in predicting the mean transport without any scaling factor needed. [39] Acknowledgments. I would like thank Rolf Käse for initiating and supporting this project and Andreas Schmittner and Michael Schulz for providing valuable help on the later and earlier part of this work respectively. This project was carried out within the Deutsche-Forschungs- Gemeinschaft funded project Ocean Gateways at the University of Kiel. The author would like to thank the two anonymous reviewers and the associate editor W. Han for their constructive comments on the manuscript and R. Murtugudde for editing it. References Armi, L. (1986), The hydraulics of two flowing layers with different densities, J. Fluid Mech., 163, Borenäs, K., and A. Nikolopoulos (2000), Theoretical calculations based on real topography of the maximum deep-water flow through the Jungfern Passage, J. Mar. Res., 58, Conkright, M. E., R. 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11 Figure 10. Ratio of modeled Q p to theoretical WLK transport calculated for the sill and upstream area with a depth greater than 350 m. The vectors show the volume flux of cold water in 150 m 2 /s. Figure 11. Theoretical shape of overflow as determined from geostrophic relation [Borenäs and Nikolopoulos, 2000] (top panel with the left profile corresponding to Q NB and the right profile to Q NB2 ) and model results (average of days 51 61) for section A with shaded temperature and velocity (negative southward) contours (bottom panel). 8of10