ARTICLE IN PRESS
Deep-Sea Research II 56 (2009) 874–883
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Deep-Sea Research II
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Tidal effect on the dense water discharge, Part 1: Analytical model$
Hsien-Wang Ou a,Ã, Xiaorui Guan a, Dake Chen b a Division of Ocean and Climate Physics, Lamont-Doherty Earth Observatory, Columbia University, 61 Route 9W, Palisades, NY 10964, USA b The State Key Lab of Satellite Ocean Environment Dynamics, SIO, Hangzhou, China article info abstract
Article history: It is hypothesized that tidally induced shear dispersion may significantly enhance the spread and Accepted 26 October 2008 descent of the dense water. Here, we present an analytical model to elucidate the basic physics, to be Available online 18 November 2008 followed (in Part 2) by a numerical study using a primitive-equation model. Keywords: While tides enhance the vertical mixing and thicken the Ekman layer, the tidal dispersion is seen to Shear dispersion produce a disparate benthic layer several times the Ekman depths. With its top boundary lying above Benthic layer the frictional shear, the diapycnal mixing with the overlying water would be curtailed on account of its Dense outflow Richardson-number dependence. Over the shelf, this reduced erosion of the dense water and the generation of the Ekman flow by the thermal wind within the layer would act in concert to propel the dense water much beyond the deformation radius, which thus may reach the shelf break with less densification by the air-sea fluxes. Over the slope, even with the strong Ekman flow induced by the sloping interface, the dense water remains confined to the Ekman layer in the absence of tides, hence subjected to strong diapycnal mixing that would curb its descent. With the tidal augmentation of the benthic layer beyond the Ekman depth, on the other hand, the much-reduced diapycnal mixing would allow the Ekman advection to operate more effectively, which may propel the dense water to the lower slope, as observed in the western Ross Sea. & 2008 Elsevier Ltd. All rights reserved.
1. Introduction flow induced by the along-isobath current can advect the dense water offshore (Nagata et al., 1993; Condie, 1995; Cenedese et al., Through air–sea fluxes, dense water may form in marginal seas 2004), the frictional torque acting on a detached benthic plume or on polar shelves, which then descends the continental slope to can propel it to the concave lower slope (Ou, 2005), and enter the deep ocean. Prominent examples include the Mediter- entrainment of the overlying water would augment the transport ranean Outflow—a significant salt provider to the North Atlantic, to extend its outer edge (Smith, 1975; Price and Baringer, 1994; the Denmark Strait Overflow—a major source of the North Condie, 1995; Baines and Condie, 1998). On the other hand, the Atlantic Deep Water (NADW), and the High-Salinity Shelf Water Coriolis constraint can be overcome if the dense water is (HSSW) from Antarctic shelves, which seeds the Antarctic Bottom channeled through canyons, blocked by ridges, or incorporated Water (AABW) to ventilate the deepest of the world oceans. Partly into frontal eddies (Jungclaus and Backhaus, 1994; Gawarkiewicz for the dynamic intrigue and partly for its obvious climatic and Chapman, 1995; Chapman and Gawarkiewicz, 1995; Jiang and importance, how the dense water may descend the slope and Garwood, 1995). And in polar waters, the thermobaric effect may enter the deep ocean has generated considerable research interest exert additional buoyancy force to lessen the constraint of a in the past, as attested by an extensive body of literature and neutral density level (Killworth, 1977). many recent review articles on the subject (Griffiths, 1986; Price Clearly, different processes dominate over different regions or and Baringer, 1994; Baines and Condie, 1998; Ivanov et al., 2004). stages of the phenomenon; but despite the rather exhaustive list, Given the spatio-temporal scales involved, the Earth’s rotation one dynamical aspect of the phenomenon has received scant would impose a strong constraint on the down-slope motion, attention—not even mentioned in the above reviews, which deflecting it instead to move along isobaths within about the concerns the tidally rectified effect. This is somewhat surprising deformation radius. As such, frictional and mixing processes are since tides are typically strong along some major dense- often needed in simulating the observed descent (Ezer, 2005; Legg water routes and tidal shear dispersion is a well-known and et al., 2006). Among the more prominent mechanisms, the Ekman effective transport mechanism of passive tracers (Okubo, 1967; Zimmerman, 1986).
$ As the western Ross Sea is a major formation region of the Lamont-Doherty Earth Observatory Contribution Number 7218. Ã Corresponding author. AABW (Jacobs et al., 1970; Orsi and Wiederwohl, 2009), a field E-mail address: [email protected] (H.-W. Ou). program (AnSlope) was carried out in recent years to examine the
0967-0645/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.dsr2.2008.10.031 ARTICLE IN PRESS
H.-W. Ou et al. / Deep-Sea Research II 56 (2009) 874–883 875 dense water formation and discharge. A hydrographic section with speed up to a half knot; and since the instruments are from AnSlope is shown in Fig. 1, which shows a benthic layer located well within the benthic layer, the temperature fluctuation about 200 m thick (on average) extending all the way to the lower is due mainly to the tidal advection of the lateral temperature slope. The topography in the experimental site is relatively gradient. While there are short bursts of energetic cascades during smooth—devoid of major canyons, and there is no evidence of the spring tide—hence likely tidally triggered (Gordon et al., prevalent meso-scale eddies given the relatively stable descent 2004)—their contribution to the net flux is nonetheless minor angle of the dense water (Gordon et al., 2009, their Fig. 17), so the compared with that due to the mean motion. mean down-slope motion is due primarily to the friction-induced The observation raises some obvious questions: If the Ekman Ekman flow. We show in Fig. 2 moored time series 20 m above the flow dominates the down-slope transport, shouldn’t the benthic bottom from a mid-slope site, which exhibit strong diurnal tides layer defined by the anomalous density be confined to the much
Fig. 1. Property distributions in the western Ross Sea. Data are from the AnSlope (courtesy of Arnold Gordon and Martin Visbeck). Northward is approximately offshore.
Oct 2003 Nov 2003
0.5 0.5
0 0
−0.5 −0.5
−1 −1 potential temperature −1.5 potential temperature −1.5 0 102030 0102030
60 60
40 40
20 20
0 0
−20 −20 tidal velocity(cm/s) tidal velocity(cm/s) −40 −40 0 102030 0102030 days days
Fig. 2. Time series of potential temperature and cross-shore velocity measured over the slope in the western Ross Sea (20 m above the bottom in a water depth of 1750 m). Data are from the AnSlope (courtesy of Alejandro Orsi). ARTICLE IN PRESS
876 H.-W. Ou et al. / Deep-Sea Research II 56 (2009) 874–883 thinner Ekman Layer (no more than a few tens of meters, see local tidal flux may be counter-gradient, the vertically integrated Section 6)? And in the presence of diapycnal mixing, can the tidal diffusivity remains positive. That is, the passive tracer in the Ekman flow really propel the dense water to the lower slope? benthic layer, being short-circuited by vertical mixing, would Generally, how important is the tidal flux in transporting the spread in bulk from its source, the essence of the shear dispersion. dense water? It is to address these questions and to assess the To specify the tidal motion, we consider a tidal velocity tidal effect on the spread and descent of the dense water that u^ ð u0 þ iv0Þ governed by the equation motivates the study. ^ ^ ^ Our study consists of two parts pertaining to analytical and iðf sÞðu u1Þ¼uuzz, (2.4) numerical modeling. This part 1 of the analytical study is where f is the Coriolis parameter, assumed positive, and the organized as follows: We first derive in Section 2 the tidal vertical viscosity is taken to be the same as the vertical diffusivity diffusivity of a passive tracer, and discuss in Section 3 the genesis since they stem from the same turbulent mixing. For simplicity, of the benthic layer. The spread of the benthic layer on the shelf we assume a non-slip bottom, although a more general stress and its decent down the slope are discussed in Sections 4 and 5 condition can be easily applied, which however would introduce respectively. In Section 6, the model is applied to the western Ross an additional parameter to complicate the problem; and with the Sea to compare with the observed phenomenon. We summarize tidal shear being greater with the non-slip condition, the derived the main findings and provide further discussion in Section 7. In tidal diffusivity represents an upper bound. Given the tidal forcing part 2, a process study using a primitive-equation numerical u^ 1, we solve (2.4) and (2.1) for the tidal fields, which are then model will be carried out, both to evaluate the analytical model combined in (2.2) to obtain the tidal diffusivity. and to provide a closer simulation of the observed phenomenon. For simplicity, we non-dimensionalize the variables by the 1 scaling rules (the brackets) of [t] ¼ f ,[s] ¼ f,[z] ¼ hE (the 1/2 0 0 0 0 1 ‘‘Ekman’’ depth) (2u/f) , ½u ¼ju1j, ½C ¼½u C¯ xf and 2. Tidal diffusivity 1 ½k ¼½u0 2f , so the tidal fields and diffusivity are governed by a single dimensionless parameter, the tidal frequency. The solution The basic physics of the tidal shear dispersion is well known is given in the Appendix and plotted in Fig. 3 for three non- (for example, Okubo, 1967), and derivation of the rectified flux in dimensionalized frequencies 2, 0.5 and 0.1—the high value finite water depth is available from Ou et al. (2000, referred corresponds to the semi-diurnal tide in mid-latitudes and the hereafter as Ou00). The main extension here lies in its application middle value is representative of the diurnal tide in polar to the dense benthic layer to address its spread and descent. For latitudes—the dominant forcing in the western Ross Sea. Since the logical progression, we shall first consider in this section the the tidal shear decreases exponentially away from the bottom tidal diffusivity in an unstratified ocean of constant vertical (A.1), one may use its e-folding scale to define the ‘‘shear’’ depth diffusivity, and some mathematic steps of Ou00 need to be (hence the subscript ‘‘s’’) or (from (A.3)) repeated for self-containment. 1=2 Although the derivation is for a general frequency hence not hs ¼j1 sj , (2.5) limited to tides, the pejorative descriptor ‘‘tidal’’ will nonetheless be used for convenience. Since without the density stratification, which is plotted in Fig. 4 in the thin solid line. As expected, the the tidal diffusivity should be a function only of the tidal motion, shear depth approaches the Ekman depth in the low-frequency one may use any hypothetical passive property field (denoted by limit, but can be considerably thinner at higher frequencies; and C) to aid its derivation, which we shall take to be one of vertical at the inertial latitude it extends through the whole depth since uniformity. When subjected to cross-shore tidal current u0,a no shear can be sustained. As for the phase, it propagates upward perturbation in the property field would be generated and, apart for the super-inertial tides since the cross-shore pressure gradient from the time-factor e ist (s being the forcing frequency), is countered by the frictional drag acting on the bottom flow but governed by the equation accelerates the interior flow, resulting a lag of the latter. For the sub-inertial tides, the phase propagation is reversed since the 0 0 ¯ 0 isC þ u Cx ¼ uCzz, (2.1) acceleration in the interior is now dominated by geostrophic where overbars and primes denote the time-mean and tidal balance, resulting in a tidal flow directed at right angle to the anti- perturbation respectively, and subscripts in coordinates indicate cyclonic side of—hence leading—the near-bottom flow. This also derivatives. Without the vertical mixing, the property perturba- can be understood by the steady-state limit of a downward tion would be in quadrature with the tidal current, effecting no cyclonic (Ekman) spiral. tidal flux, but with the vertical diffusivity shifting their phases, For the property signal, it contains two e-folding scales (A.2): non-zero tidal flux could ensue, which then defines the tidal one associated with the tidal shear and the other measures the diffusivity k via reach of the vertical diffusion in one tidal cycle. This latter scale will be referred as the ‘‘diffusive’’ depth or (from (A.5)) 0 0 u C kC¯ x. (2.2) 1=2 hd ¼ s , (2.6) Since the tidal perturbation is linear in the mean lateral gradient (2.1), the latter would cancel from both sides of (2.2) to which is plotted in Fig. 4 in the thin dashed line. Since vertical render a tidal diffusivity indeed a function only of the tidal diffusion is not subjected to the Coriolis force, the diffusive depth motion, as alluded to earlier. This diffusivity however can be grows with decreasing frequency and exceeds the shear depth for highly heterogeneous in the vertical, possibly even changing sign the low-frequency forcing. Since the property signal is originated (Ou00). Nonetheless, when substituted from (2.1), (2.2) can be in the interior by the tidal advection, it diffuses toward the bottom integrated vertically to yield (of homogeneous boundary condition), resulting in downward Z Z phase propagation near the bottom. For an easier comparison with 1 1 1 K k dz ¼ ðC¯ Þ 2 uC02 dz40, (2.3) that of the motion field (solid curve), the phase of the property 2 x z 0 0 signal (dashed curve) has been advanced 901, so its departure in which the symbol ‘‘N’’ indicates the interior above the benthic from that of the motion field represents the effect of the vertical layer and the conditions of vanishing vertical flux at the boundary diffusion, which gives rise to the non-zero rectified flux and hence and in the far field have been applied. It is seen that, although the the tidal diffusivity (see (2.2)). ARTICLE IN PRESS
H.-W. Ou et al. / Deep-Sea Research II 56 (2009) 874–883 877
6 k |C'| |u'| h's
5 0.1 8
4 6
3 k 0.01 4 k θ + 90° 2 k hd 2 C hb hs θ u 1 0.001 0.1 0.2 0.5 1 2 5 10 σ 0 0.5 1 -0.05 0 0.05 θ's Fig. 4. The depths of the shear (hs), diffusive (hd) and benthic (hb) layers, all scaled -45° 0 45° by the Ekman depth, plotted against the frequency in units of the inertial frequency. The thick dashed line is the non-dimensionalized mean tidal diffusivity in the benthic layer. The shaded vertical columns mark the frequencies whose 6 solutions are shown in Fig. 3.
5 k feature. For s ¼ 0.1, on the other hand, the growing diffusive θ |u'| depth and hence the gentler slope in the property phase-curve 4 u k causes it to lag that of the tidal current near the bottom, reversing the above sign of the tidal diffusivity near the bottom. Despite 3 θ + 90° |C'| C these variations, it is the vertically integrated diffusivity that is germane to the mean property distribution, which would be 2 short-circuited by vertical mixing; and for this total diffusivity, we have already seen that it is positive (2.3), so the overall effect of 1 the tidal rectification is diffusive.
1 2 0 0 0.1 3. Genesis of the benthic layer 0 45° One particularly distinct feature seen in Fig. 3 is that the tidal diffusivity retains significant magnitude well beyond the Ekman 12 depth (the unit of the vertical scale). For a benthic layer defined by the anomalous property concentration, its genesis requires lateral 10 transport from the source region, so if such tidal flux is the progenitor of the benthic layer, the latter depth should be much k 8 thicker than the Ekman depth. Given the undulating profile of the |C'| tidal diffusivity, there is however no obvious marker for the 6 |u'| benthic layer depth; but with its genesis being a lateral process, one may argue that this depth is the one with the maximum tidal θ + 90° k 4 c dispersion—as it would overtake the less effective ones to emerge from the source region. Specifically, we seek therefore a layer 2 depth for which the vertically averaged tidal diffusivity of the θ underlying column (marked by ‘‘tilde’’) has a maximum. With this u |u'| diffusivity defined by 0 0.5 1 -0.5 0 0.5 1 Z 1 z |C'| k~ðzÞ k dz (3.1) 0 5 10 z 0 o θ's 0 45 its maximization (dk~=dz ¼ 0) yields ~ ~ ~ Fig. 3. The tidal fields (left panel) and diffusivity (right panel) for selected k ¼ kmax when k ¼ k (3.2) frequencies (in units of the inertial frequency) plotted against the vertical distance above the bottom (scaled by the Ekman depth). Also plotted in the right panel is or the benthic-layer depth is where the averaged tidal diffusivity the tidal diffusivity averaged over the underlain column (dashed), with the solid of the underlain column equals its local value, as indeed seen in squares marking its maximum. (A) s ¼ 2, (B) s ¼ 0.5 and (C) s ¼ 0.1. Fig. 3 (the solid squares). As such, the upper portion of the benthic layer contains the maximum tidal diffusivity, which in fact Since the tidal diffusivity involves the product of the motion provides an additional argument for our choice of the benthic and property fields, it naturally contains more structure than layer if it is characterized by a positive density anomaly: that is, either field and, depending on the subtler alignment of the phase with the dense water more effectively advancing near the top of curves, it may assume either sign with varying patterns for the benthic layer by the tidal flux, it would convect downward to different frequencies (Fig. 3). For s ¼ 2, for example, the opposite further homogenize the layer and sharpen its definition—despite trends in the phase curves toward the bottom together with their the rather convoluted appearance of the tidal diffusivity. It is alignment in the interior lead to a negative tidal diffusivity near recognized however that the tidal diffusivity is derived based on a the bottom. For s ¼ 0.5, the reversal of the phase propagation in constant vertical diffusivity, which obviously does not hold the motion field apparently is not sufficient to alter the above crossing the density stratification, but to the degree that the ARTICLE IN PRESS
878 H.-W. Ou et al. / Deep-Sea Research II 56 (2009) 874–883 approximation holds within the benthic layer, the mean tidal where all variables represent their non-dimensionalized counter- diffusivity of the layer derived above (referred henceforth simply parts, which can be taken from Fig. 4. For the diurnal tide in polar as the tidal diffusivity) should remain essentially valid. oceans (that is, s ¼ 0.5), for example, l can be estimated to be In Fig. 4, we have plotted the benthic-layer depth (thick solid) 0.32, so the spreading distance is about one third of the tidal and the tidal diffusivity (thick dashed) based on calculations excursion. It is seen that since changing the value of the vertical from selected frequencies, which exhibit some robust features. diffusivity, say, by tides, would be countered by the corresponding As the inertial frequency is approached, the benthic layer grows change in the layer depth, the vertical diffusivity does not enter indefinitely—mirroring that of the shear depth, but the mean tidal the spreading distance. diffusivity also vanishes, rendering ineffective tidal dispersion. In For a benthic layer containing a positive density anomaly, there fact, at the inertial frequency, there is no tidally induced benthic is buoyancy-driven flow u, which is simply the Ekman flow linked layer, so its depth cannot be defined. Besides this singular feature, to the geostrophic along-isobath flow via (e.g., Condie, 1995) the benthic depth generally trends upward with decreasing uE vg=2, (4.4) frequency, which reflects the variation in the diffusive depth hd as the latter marks the depth of the phase shift that gives rise to where the subscripts ‘‘E’’ and ‘‘g’’ stand for ‘‘Ekman’’ and the non-zero tidal flux. ‘‘geostrophic’’, respectively. To underscore the tidal effect, let us For the tidal diffusivity, other than its dip at the inertial first consider the non-tidal case so the dense water is propelled frequency as explained above, it trends upward with decreasing solely by the Ekman flow. As such, the benthic layer should be frequency and asymptotes to a value of about 0.05. Noting that the aligned with the Ekman layer, and the balance of the Ekman tidal diffusivity is plotted in the log scale, so there is a 5-fold advection and the vertical diffusion in (4.1) yields a spreading increase between s ¼ 2 and 0.5, underscoring a much more distance of effective shear dispersion for sub-inertial tides; and for s ¼ 2, 2 which could be representative of the diurnal tides in polar oceans, lE uEhE =u 2uE=f vg=f , (4.5) the tidal diffusivity is already near its upper limit. which is thus proportional to the geostrophic along-isobath flow, ~ It is seen from Fig. 3 that while k has a distinct peak in the the latter an outcome of the governing frontal dynamics, as vertical for major tidal constituents, this peak is barely discernible discussed next. for low-frequency forcing and there is in fact an additional peak For the benthic layer on the shelf, one may invoke the emerging in the lower water column. Since this lower peak lies geostrophic adjustment of a vertical front in setting the along- well within the diffusivity depth, its effect on the mean property isobath flow. Although the process is often posed as an initial- field would be short-circuited by the vertical mixing. Because of value problem, it is relevant in the oceanic setting if the initial this, we retain the upper peak as more pertinent in defining the imbalance is more broadly interpreted as a tendency continually benthic boundary, with the cautionary note that the benthic layer forced by convective overturning events (Ou, 1984). In the is less sharply defined for low frequency forcing. adjusted state, the front is tilted with its foot extending a
To recap, the most robust feature of the tidally induced deformation radius Rc beyond the mean frontal position, where benthic layer, as can be gleaned from Fig. 4, is that it is several the along-isobath flow is (Csanady, 1982) times the Ekman depth, so its top boundary lies well above the frictional shear—the basis for our hypothesized tidal effect, as vg fRc. (4.6) seen later. Substituting (4.6) into (4.5), the spreading distance is
lE Rc (4.7) 4. Spread of the benthic layer or the Ekman flow would propel the dense water another deformation radius before it is eroded by the diapycnal mixing. To examine the spread of the benthic layer defined by the In fact, with the benthic layer aligned with the Ekman layer, no anomalous property concentration C, we consider the tidal-mean geostrophic alongshore flow can develop beyond the front, so the equation of the form (dropping henceforth the overbars) Ekman flow itself would cease, just as the density anomaly. The situation however is quite different in the tidal regime ðuC kCxÞx ¼ðuCzÞz (4.1) since the tidal flux, as seen in Section 3, would produce a benthic- after the property field has attained quasi-equilibrium. The layer depth several times the Ekman depth. Over a flat shelf, the bracketed term on the left-hand side is the property flux layer top is level, so the sustained shear is confined to the Ekman consisting of both advective and tidally rectified components, depth (we neglect the internal tides, which are episodic hence of whose divergence is balanced on the right-hand side by the lesser importance); and with the layer top lying well above the vertical mixing. In the above equation, u is the buoyancy-driven frictional shear, it is subjected to much reduced diapycnal mixing flow, which would be zero if the benthic layer were defined by on account of its Richardson-number dependence. Together with passive tracers and contained no density signal. In such cases, the free convection that produces a more sharply defined benthic the lateral transport is due solely to the tidal dispersion layer, one may integrate (4.1) through this layer to yield a bulk and, setting u ¼ 0, the spreading distance would be ‘‘scaled by’’ balance of (the symbol ‘‘ ’’) 0 0 0 ður krxÞx ¼ ar =hb, (4.8) 1=2 l hbðk=nÞ (4.2) where we have replaced the concentration C by the positive density anomaly r0 of the benthic layer (against the overlying and since the tidal diffusivity is quadratic in the tidal amplitude, water of uniform density r0), and all variables refer hereafter to the spreading distance is proportional to the tidal excursion le the vertical means in the benthic layer. The transfer coefficient is 0 2ju1j=s (the subscript ‘‘e’’ for ‘‘excursion’’). Using this excursion as defined as a ui/hi (‘‘i’’ for ‘‘interfacial’’) and since both interfacial the distance scale, the non-dimensionalized version of (4.2) diffusivity and thickness depend on small-scale processes not becomes (see the scaling rules defined in Section 2) addressed here, we shall take the transfer coefficient to be qffiffiffiffiffiffiffiffi external whose value can be estimated from observation or l shb k=2, (4.3) diagnosed from numerical solutions. ARTICLE IN PRESS
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One notes from (4.8) that the tidal rectification may enhance before being eroded by the diapycnal mixing. In the tidal regime, the spread of the dense water in multiple ways. Besides the tidal rectification would produce a benthic layer several times thickening the benthic layer and, hence, reducing the diapycnal the Ekman depths to curtail the diapycnal mixing and induce mixing, the tidal dispersion itself may transport the property as additional Ekman flow, both acting in concert to propel the dense entailed in the tidal diffusivity. In addition, with bulk of the water beyond the tidal excursion, which thus may reach the shelf benthic layer lying above the Ekman layer, the thermal wind break with less densification by the air–sea fluxes. associated with the cross-shore density gradient would generate geostrophic along-isobath flow within the benthic layer, which in turn would induce an offshore Ekman flow to propel the dense 5. Descent of the dense water water. Since these flows stem from thickening of the benthic layer by the tidal flux, they must be regarded as rectified flows (that is, Once the dense water reaches beyond the shelf break, there is non-existent without tides)—a nonlinear generation mechanism additional buoyancy force associated with the sloping interface not previously recognized. Applying the thermal wind, the mean that would further propel its spread. To illustrate the essential cross-shore flow is approximately tidal effects, it suffices to assume this interface to be parallel to 0 the topographic slope, which moreover is taken to be a constant hE vg hE 1 grx hE u uE ðhb hEÞ . (4.9) for simplicity. Although the observed slope is not constant and hb 2 hb 2 f r0 hb indeed its curvature is demonstrably important for a certain With the density anomaly scaled by its source value and mechanism (Ou, 2005; see also Section 1), such variation does not offshore distance by the tidal excursion le, and recognizing that change fundamentally the proposed tidal effects and hence is the tidal diffusivity is a constant over a flat bottom, (4.8) is non- neglected. In the absence of tides, the balance leading to (4.5) dimensionalized to remains unchanged, only that the geostrophic flow would be linked to the interfacial slope s via (Nof, 1983) ð u^rxr rxÞx ¼ a^ 1r, (4.10) 0 where u^ and a^ 1 are the (dimensionless) parameters measuring the vg gr s=ðf r0Þ, (5.1) importance of the Ekman advection and diapycnal mixing, which when substituted into (4.5), yields a spreading distance of respectively, with the expressions: 0 2 0 lE gr s=ðf r Þ. (5.2) 1 hE g½r 0 u^ 1 hE (4.11) 2 hb r0fk Since this spreading distance is linear in the bottom slope, the and vertical reach of the dense water would increase as square of the bottom slope. Nonetheless, as estimated in Section 6, even over a 2 a^ 1 ale =ðkhbÞ. (4.12) steep slope of the western Ross Sea, the density anomaly is largely eroded over the upper slope. This deduction underscores the The subscript ‘‘1’’ in the diapycnal mixing parameter distin- effectiveness of the diapycnal mixing in neutralizing the dense guishes it from that of the slope case considered later. Eq. (4.10) water in the non-tidal regime because of the relatively thin can be solved numerically subjected to the boundary conditions benthic layer. that r ¼ 1atx ¼ 0 and that the solution decays exponentially in With tides, their rectified effects in the flat-bottom case remain the far field. Examples of the solution for the case of a^ ¼ 1, u^ ¼ 1 operative: the benthic layer would be several times the Ekman ð0; 1Þ are plotted in Fig. 5. The far-field condition is set at x ¼ 5 depth to curtail the diapycnal mixing and a weak Ekman flow which seems adequate, and the choices of the parameter values would be induced by cross-shore density gradient within the are based on that of the western Ross Sea (see Section 6). In the benthic layer; both however would be strongly modified over the absence of the Ekman advection, the density anomaly is seen from slope by the geostrophic shear associated with the sloping (4.10) to decay exponentially with an e-folding distance of interface (5.1). Some obvious effects of this shear can be a^ 1=2 ¼ 1, which is moderately extended when the Ekman 1 immediately discerned: (a) it would augment the diapycnal advection is comparable with the tidal flux. mixing of the flat-bottom case; (b) it may trigger entrainment if To recap, in the absence of tides, the dense water is confined to the internal Froude number exceeds unity (Baringer and Price, the Ekman layer and can advance only one deformation radius 1997); and (c) it would induce a much stronger offshore Ekman flow. 1 To include the entrainment in the property balance (4.8), we simply add a term of the same form as the diapycnal mixing but 0.8 with the transfer coefficient replaced by the entrainment rate we. Based on laboratory and field studies (Turner, 1986; Price and Baringer, 1994), we assume this entrainment rate to be propor- 0.6 tional to the geostrophic shear α1 = 1
ρ we ¼ gvg (5.3) 0.4 with the entrainment parameter a function of the bulk Froude u = 0 g 1 number defined by
0.2 0 1=2 Fr vg ðgr hb=r0Þ . (5.4) There is considerable uncertainty in this functional dependence x (for example, Cenedese et al., 2004), but as its most robust feature, 123 the entrainment rises sharply when the Froude number exceeds unity, a near-singular increase that in fact strongly caps the Fig. 5. Density anomaly in the benthic layer over a flat bottom plotted against the offshore distance in units of the tidal excursion. The two cases are for without observed Froude number (Baringer and Price, 1997, their Fig. 7). (solid) and with (dashed) the Ekman advection, respectively. To include this essential behavior, it suffices to assume a step ARTICLE IN PRESS
880 H.-W. Ou et al. / Deep-Sea Research II 56 (2009) 874–883 function of which then specifies the solution (5.12). Beyond this point of vanishing entrainment, the solution to (5.6) is g 3 10 3 for Fr41, g ¼ 0 for Frp1 (5.5) r ¼ rc a^ 2ðx lcÞ=2 for lcoxoW, (5.15) in which the supercritical value is gleaned from Baringer and Price where W is the width of the slope region (recall that it is scaled by (1997, their Fig. 7, Section C, and their Table 2) by setting the the deformation radius). It is seen that because the Ekman flow 3 1 1 entrainment rate of 1.2 10 ms and a mean flow of 0.4 m s . weakens with decreasing density stratification, the density One may fashion a more complicated functional form, which anomaly actually decays linearly offshore, rather than at the however does not alter the essential solution and in any event more rapid exponential rate had the Ekman flow been constant. If may not be justified for our extremely crude model. the density anomaly remains finite beyond the slope, it needs to In contrast to the flat-bottom case when the tidal rectification be matched to the flat-bottom solution (Section 4), which would (and its attendant Ekman flow) provides the only cross-shore decay rapidly given the weak tides in the deep ocean. transport mechanism, there is an inherent offshore Ekman Using the parameter values pertinent for the western Ross Sea flow due to the sloping interface—with or without tides. In (Section 6), we plot the solution in Fig. 6 (thick solid line) for the anticipation also of the dominance of the Ekman advection over case of w^ ¼ 1:39, a^ ¼ 0:22, Fr0 ¼ 1.58, H ¼ 4 and W ¼ 3.4. It is seen the tidal flux (see Section 6), we shall non-dimensionalize the that density anomaly is eroded rapidly by entrainment and then property balance based on the Ekman flow. Denoting the values at more slowly by the diapycnal mixing, with the transition 0 0 the shelf break by the subscript ‘‘0’’, we set ½r ¼r0,[h] ¼ H0 occurring at rc 0:4 and lc 1:05. Since r ¼ 0.14 at x ¼ W, the (shelf depth), [x] ¼ Rc (deformation radius based on the shelf dense water retains significant density signature as it reaches the ¼ ¼ ½ 0 ð Þ 1 depth), [u] uE,0hE/hb with uE;0 g r s 2f r0 (from (4.4) and upper rise, as indeed the observed situation. (5.1)). Including the entrainment and non-dimensionalized, (4.8) Although the tidal flux is small compared with the Ekman becomes advection, the tidal effect is essential in its augmentation of the benthic layer, thus reducing the entrainment and diapycnal ðr2 k^r =h2Þ ¼ ðw^ r þ a^ Þr (5.6) x x 2 mixing. We have seen from scaling argument (5.2) that the dense with the dimensionless parameters: water remains on the upper slope in the non-tidal case, a deduction that can be checked by the analytical solution with k k^ ¼ 0 , (5.7) properly adjusted parameter values. Specifically, equating the ½u R c benthic depth with the Ekman depth, we have Fr0 ¼ 3.16 (a 2-fold increase), and replacing the interfacial diffusivity by that due to 2gR the tidal shear and the interface thickness by the Ekman depth, we ^ c w ¼ , (5.8) 3 1 hE have a 2:63 10 ms , yielding a^ 2 1:16 (5 times greater). Using these values, we plot the non-tidal solution in the thick and dashed line, which contrasts sharply the tidal solution. It is seen aR that the entrainment alone would have largely removed the ^ ¼ c a2 . (5.9) : ½u hb density anomaly (since rc 0 1), which is wholly eliminated just beyond the deformation radius (that is, x ¼ 1.11). The subscript ‘‘2’’ in the diapycnal mixing parameter is to distinguish it from that of the flat-bottom case (4.12). Given the 1 jump of the entrainment on the Froude number (5.5), we need to ENT keep tag on the latter, which when expressed in the non- TIDAL dimensionalized density anomaly, is given by 0.8 NO-ENT
1=2 Fr ¼ Fr0r , (5.10) 0.6 where ENT
ρ NO-TIDE NO-ENT s g½r0 1=2 0.4 Fr0 ¼ (5.11) f r0hb is the Froude number at the shelf break. Eq. (5.6) can be solved 0.2 numerically, but since the tidal flux is seen later (Section 6) to be typically small compared with the Ekman advection, we shall x neglect this term, which allows an analytical solution. 1 2 3 Since generally Fr041, there is entrainment as the dense water begins the descent, with the solution 1 r ¼ð1 þ a^ 2=w^ Þ expð wx^ =2Þ a^ 2=w^ for xolc, (5.12) H = 4 W = 3.4 Η in which lc (‘‘c’’ for ‘‘critical’’) marks the distance beyond which Fr falls below unity and the entrainment ceases. Let the density anomaly at this juncture be denoted by rc, we have then W r ¼ Fr 2 (5.13) c 0 Fig. 6. Density anomaly over a slope that is representative of the western Ross Sea (distance scaled by the deformation radius). Tidal (solid) and non-tidal (dashed) and setting r to rc in (5.12), we derive that cases are specified by a^ 2 ¼ 0:22 and 1.16; the cases with (thick) and without (thin) entrainment correspond to w^ ¼ 1:39 and w^ ¼ 0, respectively. It is seen that the 2 rc þ a^ 2=w^ descent of the dense water is severely constrained in the non-tidal regime whether lc ¼ ln , (5.14) w^ 1 þ a^ 2=w^ or not there is entrainment. ARTICLE IN PRESS
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Besides the essential role played by tides in propelling the non-tidal case (basically the tidal excursion versus the deforma- descent, the solutions also underscores the importance of tion radius), which thus may reach the shelf break with less entrainment as a countering agent to the Ekman advection. To densification by the air-sea fluxes. One should note however that see this, we have removed the entrainment phase from the two this deduction does not imply that the discharged HSSW would be solutions, as indicated by the corresponding thin lines. It is seen less dense in the tidal regime, only that—everything else being that the descent of the dense water is significantly enhanced, but equal—it would reach the shelf break earlier in its westward nonetheless it remains severely constrained in the non-tidal progression on the shelf and hence the discharge would span a regime, reaching only the mid-slope (x ¼ 1.72). greater longitudinal stretch. To assess the dense water descent on the slope, we set the density anomaly at the shelf break to be 0.3 kg m 3, which would 6. Application yield a geostrophic alongshore velocity (5.1) of 1 m s 1 and an Ekman flow of 0.5 m s 1, not unlike that observed in AnSlope Although the model is extremely crude, the relevant para- (Gordon et al., 2009). In the absence of tides, the spreading meters will nonetheless be estimated to facilitate the model distance (5.2) based on scaling argument would be 8.3 km, so the application to the western Ross Sea. Since the diurnal tides dense water remains sequestered to the upper slope in a thin dominate the oscillatory forcing (Fig. 2), the parameter estimates Ekman layer of 38 m. With tides, on the other hand, the benthic are based on this particular frequency. In any event, since the layer would be 152 m thick, so the cross-shore flow averaged over semidiurnal tides are near the inertial frequency, they are the benthic layer (same as the velocity scale [u] in Section 5) 1 ineffective in generating the shear dispersion according to Fig. 4. would be 0.13 m s . In addition, we estimate Fr0 1:58 and 3 1 To assess the vertical diffusivity, we use the empirical formulae of weE3 10 ms near the shelf break, both are plausible based Csanady (1982) on Muench et al. (2008). With the above values, we estimate the ^ ^ 1 0 2 dimensionless parameters for the slope case to be k 0:25, w u ð200f Þ Cdju1j (6.1) ^ 1:39 and a^ 2 0:22. To the degree that k is quite smaller than 4 1 3 0 and assign values of f 1:4 10 s , Cd 3 10 , and ju1j unity, it is justifiably neglected in the analytical solution, so the 1ms 1 (hence tidal excursion of 28 km) to arrive at thick solid curve in Fig. 6, which is based on above parameter u 10 1 m2 s 1, a value indeed observed in strong tidal regime estimates, could be representative of the observed distribution. It (Sanford and Lien, 1999). Although tides weaken offshore with is seen therefore that the density anomaly would be reduced by increasing depth, there are other perturbations that may con- entrainment to about 0.12 kg m 3 (from its shelf-break value of tribute to the vertical mixing, so as an order of magnitude 0.3 kg m 3) at a distance of about 10 km, after which it decreases estimate we shall retain this same value throughout the model more gradually and retains a signature of about 0.04 kg m 3 when domain. With this vertical diffusivity, the Ekman depth is 38 m, it reaches the continental rise. Without tides (the thick dashed both shear and diffusive depths can be seen from Fig. 4 to be 53 m, line), on the other hand, the dense water would be largely eroded and the benthic layer depth is 152 m. Since the latter is over the upper slope. commensurate with that observed (Fig. 1), the tidal rectification To the degree that the tidal flux is typically small compared provides a plausible explanation as to why it is so much thicker with the Ekman advection, it should not significantly affect the than the Ekman layer. mean density anomaly of the discharged HSSW, which is governed With the above tidal amplitude, the scale of the tidal diffusivity by thermohaline balances of the frontal system not considered (that is, [k] in Section 2) would be 7.1 103 m2 s 1, so the mean here (see for example Ou, 2007). In any event, such modification is tidal diffusivity is about 2.9 102 m2 s 1 according to Fig. 4. Since likely minor compared with the sharp contrast between the tidal this is due to the diurnal tides alone, it may represent a significant and non-tidal solutions depicted in Fig. 6; the model thus provides underestimate of the effective diffusivity (see Section 7). To assess a plausible account of the observed descent in the western Ross the diapycnal mixing, one recognizes that it is highly uncertain, Sea, in support of the importance of the tidal rectified effect. but based on Muench et al. (2008, their Table 2), we shall take—as representative—values of 10 3 m2 s 1 for a level interface and an order of magnitude greater for a sloping interface due to its 7. Summary and discussion accompanying geostrophic shear. Assuming an interfacial layer 20 m thick, the transfer coefficient a would be 5 10 5 ms 1 for a It is recognized that myriad processes may curtail the spread level interface and 5 10 4 ms 1 for a sloping interface. For the and descent of the dense water, including the Earth’s rotation, the flat-bottom case, the diapycnal mixing parameter (4.11) then has diapycnal mixing and the ambient stratification, so the challenge the value a^ 1 ¼ 0:9. in explaining the observed phenomenon lies in uncovering the As representative of the western Ross Sea, we take the shelf mechanisms that may counter these constraints. Through the depth to be 500 m, the abyssal depth, 2 km, and width of the slope present study, we propose a robust mechanism pertaining to the region, 30 km, which yield a topographic slope of 5 10 2. tidal shear dispersion, which would produce a benthic layer much Assuming a density anomaly of 0.3 kg m 3 for the dense shelf thicker than the Ekman depth; and with the layer top lying water (see e.g., Orsi and Wiederwohl, 2009), the deformation outside the frictional shear, the diapycnal mixing with the radius based on the shelf depth is 8.8 km. Without tides, this is the overlying water would be greatly reduced on account of its distance the dense water would spread on the shelf beyond the Richardson-number dependence. foot of the front, which moreover would be confined to a relatively Over a flat shelf, the reduced erosion of the dense water and thin Ekman layer of 38 m. In the tidal regime, on the other hand, the generation of an Ekman flow by the thermal wind within the the benthic layer would be four times thicker, which would induce benthic layer would propel the dense water much beyond the additional Ekman flow, as measured by the advective parameter u^ deformation radius, which thus may reach the shelf break with (4.12) estimated to be u^ 1:1. The solution shown in Fig. 5 is less densification by the air-sea fluxes. Over the slope, there is an based on these parameter values, which thus could be relevant to inherent Ekman flow associated with the sloping interface, but the observed benthic layer on the shelf. Particularly, it suggests with the dense water confined to the Ekman layer in the non-tidal that the tidal rectification with its attendant Ekman flow would regime, it is rapidly neutralized by diapycnal mixing and hence advance the dense water more than three times over that of the may not descend beyond the upper slope. With the tidal ARTICLE IN PRESS
882 H.-W. Ou et al. / Deep-Sea Research II 56 (2009) 874–883 augmentation of the benthic layer, on the other hand, the reduced debouch through an unconstrained stretch of the shelf, and then diapycnal mixing allows the Ekman advection to operate more carried by a dense flow on the slope streaming past the western effectively, which may propel the dense water to the lower slope. terminus of the shelf. As such, one may not equate the dense It should be stressed that the above tidal reduction of the water formation rate with the offshore Ekman transport, which diapycnal mixing is in comparison with the non-tidal case as the would unduly constrain the front and lead to the improbable benthic layer extends beyond the frictional shear—even though scenario that higher production of the dense water actually the overall vertical mixing is enhanced by tides (6.1). Similarly, reduces the down-slope reach of the dense water (Condie, 1995). since a sloping interface would induce a geostrophic shear on its Once free from such constraints, there need not be frontal own, the diapycnal mixing remains substantial even if it is instability or ensuing eddies as required to absorb the dense reduced from the non-tidal case; and with the geostrophic shear water discharge. In other words, some these features—either spatially separated from the frictional shear, it should manifest as deduced or simulated—stem from two-dimensional balance a local maximum in the vertical mixing, as indeed observed in requirements that might not be prevalent in the actual situation. AnSlope (Visbeck and Thurnherr, 2009). While the importance of the Ekman advection is well recognized in the literature, less appreciated however is that the Acknowledgments turbulent mixing that facilitates the Ekman flow also erodes the density anomaly to severely limit its reach. This being not the case This research is supported by the National Science Foundation for laminar flows in the laboratory, one needs to be cautious in through Grants OPP-0125177 and ANT-0538148. We have bene- extrapolating some laboratory results (for example, Tanaka, 2006) fited from input from the AnSlope participants (a project funded to the oceanic phenomenon. By the same token, reduced-gravity by NSF/OPP) and comments from anonymous reviewers. models that externally specify the benthic-layer depth (Cenedese et al., 2004; Wang et al., 2003) or artificially suppress the diapycnal mixing (Shapiro and Hill, 1997) provide only partial Appendix explanations of the observed phenomenon. For primitive-equa- tion models, the diapycnal mixing is naturally encoded and The solution to (2.1), (2.2) are operative, but sometimes it needs to be suppressed to simulate u0 ¼ 1 e mz, (A.1) the observed descent (Ezer, 2005)—an outcome perhaps not unexpected based on our scaling arguments. Taken together, what i m C0 ¼ þ ie mz e nz , (A.2) seems to be missing is an independent generation mechanism for s n the benthic layer other than the Ekman advection, and our central where hypothesis is that the tidal rectification—with its augmentation of the benthic layer much beyond the Ekman depth—may provide m ¼ð1 sÞ1=2ð1 þ iÞ if sp1 (A.3) this mechanism. There are other processes that may raise the layer boundary, m ¼ðs 1Þ1=2ð1 iÞ if s41. (A.4) including the entrainment and the geostrophic adjustment, but since the offshore flow remains confined to the Ekman layer, there n ¼ s1=2ð1 iÞ. (A.5) is still need of a thickened layer at the leading edge of the dense water, for which the tidal rectification remains a viable mechan- In their non-dimensionalized forms, the tidal diffusivity is ism. On the other hand, the tidal diffusivity decreases rapidly 1 k ¼ RefC0 C0 g, (A.6) offshore, which raises the question of its efficacy farther down the 4 zz slope. For this, one may invoke the fact that the introduction of and its vertical average given by even a slight stratification to an un-sheared region would Z z singularly augment the Richardson number to buffer the vertical ¯ 1 0 0 1 0 2 k ¼ RefC Czgþ jCzj dz (A.7) mixing, so the critical factor lies in the slower vertical decay of the 4z 4z 0 tidal flux than the current shear, as indeed suggested by the both algebraic expressions can be derived and calculated. analytical solution (Fig. 3). And with the tidal preconditioning of the stratification, the Ekman advection would then provide a positive feedback to further enhance the tidal effect. References For the western Ross Sea, the diurnal tides dominate the oscillatory signal so the parameter estimates based on a mono- Baines, P.G., Condie, S., 1998. Observations and modeling of Antarctic downslope chromatic forcing may be justified. Otherwise, one may need to flows: a review. In: Jacobs, S.S., Weiss, R.F. 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