Density-driven exchange flow in terms of the Kelvin and Ekman numbers

Transcription

1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2007jc004144, 2008 Density-driven exchange flow in terms of the Kelvin and Ekman numbers Arnoldo Valle-Levinson 1 Received 5 February 2007; revised 4 October 2007; accepted 13 December 2007; published 3 April [1] The pattern of density-induced flow influenced by basin s width, friction, and Earth s rotation is investigated as a function of the Ekman (E k ) and Kelvin (K e ) numbers. A semianalytical solution is used to determine the conditions under which the densityinduced exchange flow is vertically sheared or horizontally sheared. Solutions are obtained over diverse laterally varying bathymetries. It is found that the exchange flow is horizontally sheared under high frictional conditions (E k > 1) independently of the width of the basin (K e ). The horizontally sheared pattern describes inflow in the channel and outflow over shoals, with the inflow occupying the entire water column. The exchange flow pattern is also horizontally sheared under weak friction (E k! 0) and in wide (K e > 2) basins. In that case, however, the outflow is concentrated on the left (looking into the basin in the Northern Hemisphere) portion of the cross section and inflow appears on the right. Also, under weak friction, the exchange pattern becomes more vertically sheared, with outflow at surface and inflow underneath as the width of the basin becomes small (K e < 1). Bathymetry is not very influential in the weak friction exchange patterns. Finally, under moderate friction (0.01 < E k < 0.1), the exchange pattern is both horizontally and vertically sheared for all widths. The horizontally sheared pattern is best defined in wide basins (high K e ), whereas the vertically sheared pattern practically dominates in narrow basins (low K e ). These findings allow classification of various estuaries in the E k -K e parameter space. Citation: Valle-Levinson, A. (2008), Density-driven exchange flow in terms of the Kelvin and Ekman numbers, J. Geophys. Res., 113,, doi: /2007jc Introduction [2] It has been traditionally recognized that a basin s width determines whether Earth s rotation effects on density-induced or wind-induced water exchange are appreciable or not [e.g., Pritchard, 1952]. The common view is that the basin should be wider than the internal Rossby radius R i for rotation to be important [e.g., Gill, 1982]. In a densityinduced flow, R i is given by (g 0 h) 1 = 2 /f, where g 0 is the reduced gravity, h is the depth of the buoyant part of the densityinduced flow, and f is the Coriolis parameter. In turn, g 0 equals g Dr/r o, where g is the gravity acceleration, r o is a reference water density, and Dr is the contrast between the buoyant water density and the density underneath. The importance of R i in containing the buoyant flow may be characterized by the nondimensional Kelvin number K e, which compares the basin s width B to R i, i.e., K e = B/R i [Garvine, 1995]. Earth s rotation effects are supposed to be most prominent when K e >1. [3] Kasai et al. [2000] and Winant [2004] pointed out that water column depth, rather than basin s width, should determine whether Earth s rotation (or Coriolis) effects are 1 Civil and Coastal Engineering Department, University of Florida, Gainesville, Florida, USA. Copyright 2008 by the American Geophysical Union /08/2007JC004144$09.00 important. Their argument was that over depths greater than several Ekman layers D E (e.g., >4 D E ), Coriolis effects were important regardless of the width. The value of D E is given by (2 A z /f ) 1 = 2, where A z is the flow s eddy viscosity. Earth s rotation effects on exchange flows may then be cast in terms of the Ekman number E k (E k = A z /[fh 2 ], where H is water depth), which compares frictional to Coriolis effects. Coriolis effects become negligible at high E k (>1). The objective of this paper is to reconcile these ideas with the help of semianalytical results that portray density-induced exchange flows in terms of the Ekman and Kelvin numbers. This study extends that of Valle-Levinson et al. [2003] by considering the effects of basin width, i.e., the K e dependence, on density-induced exchange flows. Results show that the density-induced exchange pattern is independent of the basin s width (or K e ) at high E k and depends on width (or K e ) at low and moderate E k. 2. Approach [4] Density-induced exchange flow patterns are obtained with a semianalytical solution [see Valle-Levinson et al., 2003] that compares very favorably with observations. The model solves for the nontidal or mean along-basin u and transverse v flows at one basin cross section. The flows are produced by pressure gradients and assumed to be modified only by Coriolis and frictional influences. Advective effects 1of10

2 from tidal currents are assumed to be at least one order of magnitude smaller than other influences [e.g., Geyer et al., where B is the basin s width. The value of D that satisfies a prescribed N and R is Rfa 2 iag D ¼ Z B 0 h i NðyÞ e ahy þ ah y tanh ahy 1 e ah y þ a 2 Hy 2 =2 dy i Z B 0 tanh ah y ahy dy ð5þ 2000], and their influence on the pattern of density-induced exchange flows is insignificant [Huijts et al., 2006]. In a right-handed coordinate system (x, y, z), where x points seaward, y points across the basin, and z points upward, the nontidal (or steady) momentum balance is a set of two differential equations, f þ z þ 2 u 2 þ g z þ 2 ð1þ v 2 where f, g, r, h, and A z are the Coriolis parameter, gravity acceleration (9.8 m s 2 ), water density (kg m 3 ), surface elevation (m), and vertical eddy viscosity homogeneous in z and y (m 2 s 1 ), respectively. Equation (1) may be solved for a complex velocity w = u + iv, where i 2 = 1 is the imaginary number, wz ðþ¼gnf 1 ðþþ z F 2 ðþ: z In (2), N represents the sea level slope from the barotropic pressure gradient (@h/@x + i@h/@y). The functions F 1 and F 2 depict the vertical structure of the barotropic (from sea level slope) and baroclinic (from density gradient) contributions to the flow, respectively, " # F 1 ¼ i cosh az 1 ð Þ f cosh ah y ð2þ " # : ð3þ F 2 ¼ id coshðazþ ð f a eaz azþ e ahy ah y cosh ah y In (3), D equals g/r(@r/@x + i@r/@y) and is independent of depth; the parameter a equals (1 + i)/d E, where D E is the Ekman layer depth [2A z /f] 1 = 2. Equation (3) is obtained by assuming no stress at the surface (@F 1 /@z 2 /@z =0at z = 0) and no slip at the bottom (F 1 = 0 and F 2 =0atz = H y ). Solutions (2) and (3) require prescription of H y (as any function of y), a sea level slope N, an eddy viscosity A z, and a density gradient D that is dynamically consistent with N. The dynamically consistent value of D may be obtained by assuming a net volume flux R (m 3 s 1 ) along or across a cross section [Kasai et al., 2000], i.e., Z B 0 Z 0 H y wdzdy¼ R; ð4þ and is a constant independent of y and z. As explained by Kasai et al. [2000], the solution consists of a unidirectional outflow, represented by the barotropic contribution gnf 1 in (2) and (3), and bidirectional exchange flows given by the baroclinic contribution F 2 in (2) and (3). The key to the solution is the way in which N(y) is prescribed [Valle- Levinson et al., 2003]. On the basis of observations and numerical model results, they prescribed a slope with a value N 0 at the coast that decayed exponentially across the basin: N = N 0 {1 + i exp[ (y/b) 2 ]} (Figure 1). [5] In all of Valle-Levinson et al. s [2003] solutions, the Kelvin number K e was 1 because the exponential decay of the transverse slope of sea level spanned the width B of the basin. Those results are extended here by allowing a more general range of K e values. This is done by normalizing the across-basin distance with the internal radius of deformation R i rather than with B, n h io N ¼ N 0 1 þ i exp ðy=r i Þ 2 : ð6þ The real part of (6) could also be prescribed as a function of y, but the results are practically the same [Valle-Levinson et al., 2003]. Solutions (2) and (3) are obtained for a value N 0 of , R of zero, and different values of E k (function of A z ) and K e (function of R i ) over various bathymetric profiles. The bathymetric variation across the domain H y is given by 2=b 2 H y ¼ H 0 exp y y p 1 ; ð7þ where y p is the across-basin location of the deepest part of the channel (H 0 ) and b 1 determines the lateral slope of the channel. Furthermore, assuming along-basin uniformity, the vertical component of motion v w may be w /@z. [6] As discussed by Valle-Levinson et al. [2003], the analytical solution ignores advective accelerations and also assumes uniform A z. Despite these simplifications, the patterns produced by solutions (2) and (3) emulate the essence of those observed. Discrepancies exist only in details such as exact area of outflows/inflows and exact slope of isotachs (lines of equal speed). In the analytical solution, the effects of tidal forcing on mixing may be approximated by prescribing different values of A z to emulate the observations. Obviously, the complete approach to this problem would be to analyze numerical model results that allow A z to change in space and in time. Nonetheless, the results presented in section 3 and those presented by Valle-Levinson et al. [2003] exhibit essentially the same across-basin distribution as nonlinear, turbulence closure 2of10

3 Figure 1. Cross-channel distribution of the lateral slope prescribed in equation (5) as compared to numerical and observational results (as in the work by Valle-Levinson et al. [2003]). numerical results for high E k [e.g., Valle-Levinson and O Donnell, 1996] and for low to moderate E k [e.g., Weisberg and Zheng, 2006]. Still, a comprehensive study based on numerical experiments should follow the approach proposed here. 3. Results [7] Results are presented for the across-basin distribution of the density-induced along-basin flow u and cross-basin flow v, i.e., for the real and imaginary parts of equation (2), respectively. All representations show nondimensional flows looking into a Northern Hemisphere basin where f = s 1. The along-basin flows are normalized by the maximum inflow. First, five bathymetric configurations are chosen to portray the lateral structure of flows for three selected values of E k and K e. Second, the strength of the exchange flow over the deepest part of the channel (H 0 )is examined for a wide range of E k and K e values. The strength of exchange flow is characterized by the difference between maximum inflow and outflow and is explored for the same five bathymetric distributions plus a sixth one. Third, the strength of the exchange flow allows characterization of a basin as vertically sheared or horizontally sheared, or between these two extremes, according to its locations in the E k versus K e parameter space. Examples of several estuaries are placed in such parameter space Lateral Structure of Flows [8] In the depiction of the lateral structure of flows, three bathymetric distributions feature the deepest part H 0 in the middle of the section. A fourth one shows H 0 close to the left edge of the section, and a fifth one exhibits H 0 close to the right edge (looking into the basin). The three bathymetric distributions with H 0 in the middle of the section had different lateral slopes to explore the shape of the flows over different morphologies. The other two bathymetric distributions examined the influence of the channel s position in the section Flows Over Weak Lateral Slope [9] The exchange pattern (along-basin flows) over a nearly flat cross section (contours in Figure 2) shows a strong dependence on K e at low E k (very weak friction). In contrast, the exchange pattern remains nearly unaffected by K e at high E k. Under large K e (wide basin) and weak to moderate friction (K e 1 and E k < 0.1), the isotachs are steeply sloped (Figures 2a, 2b, 2d, and 2e). Most of the outflow in Figures 2a and 2b appears constrained by the internal radius of deformation, and the inflow occupies most of the cross-sectional area. As K e decreases from 4 to 1, the outflow occupies up to one half of the cross section and the isotachs tilt less steeply (Figures 2d and 2e) until they become almost flat at K e of 0.25 (Figures 2g and 2h). When friction becomes much more dominant than rotation (Figures 2c, 2f, and 2i), then the exchange becomes that of typical estuarine circulation with outflow at the surface and inflow underneath across the entire section. This pattern is independent of K e as Coriolis is not relevant anymore and the dynamics are determined by the balance between pressure gradient and friction [e.g., Pritchard, 1952; Hansen and Rattray, 1965]. Still, slight bathymetric effects favor the development of two branches of outflow over the shallower portions of the section. This becomes more evident as the bathymetric slopes increase [e.g., Wong, 1994]. [10] The cross-basin flows, depicted by arrows in Figures 2 to 6, show three features that are consistent among all the bathymetric configurations explored. The first feature is a clockwise circulation pattern, in the vertical plain, that remains qualitatively the same at low E k regardless of K e (top row of Figures 2 through 6). This clockwise gyre is composed of currents with similar magnitude as the along-basin flows and is tied to those along-basin flows through Coriolis effects. Thus the lateral flow at surface moves to the right (in the Northern Hemisphere) of the outflow, and the lateral flow in the layer underneath moves to the right of the inflow. Such linkage between along- and cross-basin flows is best illustrated in Figures 2d, 2g, 3d, 3g, 4d, 4g, 5d, 5g, 6d, and 6g. The second general feature is that the cross-basin circulation has the opposite direction at high E k (bottom row of Figures 2 to 6) than at low E k (top row of Figures 2 to 6). In general, cross-basin flows are from left to right at surface (0 < y < 0.3 in Figures 2c, 3c, 4c, 5c, and 6c) and from right to left in the deepest part of the section. This pattern contrasts that of low E k and illustrates a region of flow convergence over the channel. The lateral flow at such high E k is rather weak (consistently <1 cm s 1 ) and is caused by the lateral pressure gradient being balanced by friction rather than Coriolis. The dynamic balance between pressure gradient force and friction results in a sideways estuarine circulation [Valle-Levinson et al., 2003], even though it is weak. The third general feature has to do with intermediate E k (middle row in Figures 2 to 6). The crossbasin flow pattern reverses from that at high K e (e.g., Figure 2b), which also resembles the flow pattern at low E k (Figure 2a), to that at intermediate and low K e (e.g., Figures 2e and 2h). This means that at high K e and intermediate E k (Figure 2b), the lateral flow is to the right in the deep part of the channel, but it becomes toward the left at lower K e (Figures 2e and 2h). The lateral flow at such 3of10

4 Figure 2. Along-estuary (normalized by maximum inflow) and cross-estuary (scale appearing above top right corner in cm s 1 ) flows in a cross section with very weak lateral slopes. Bathymetry is drawn from equation (7) for y p in the middle (y/b = 0.5) and b 1 of 1.9. Darker areas denote regions of inflows. Contours are drawn at 0.2 intervals. Views are looking into the estuary. For the top row, A z =110 4 m 2 s 1 ; for the middle row, A z =110 2 m 2 s 1 ; and for the bottom row A z =110 1 m 2 s 1. 4of10

5 Figure 3. Same as Figure 2 but with b 1 in equation (7) of 0.5. intermediate E k values is 2 3 times smaller than the alongbasin flow, so it is relevant to the transport of solutes Flows Over Moderate Lateral Slope [11] The along-basin patterns over a moderately sloped cross section are consistent with those over weak slope but only for the low-friction cases (Figure 3). As K e decreases, the isotachs tilt less and the exchange flow changes from horizontally sheared to vertically sheared (Figures 3a, 3d, and 3g). The influence of friction allows the development of two branches of outflow. Under moderate friction, the exchange pattern becomes vertically sheared for all K e examined (Figures 3b, 3e, and 3h). Furthermore, the left branch of outflow (looking into the basin) is more prominent because of the influence of Coriolis accelerations. Such prominence of the left branch, however, decreases with decreasing K e, i.e., the asymmetry becomes less evident in narrow channels. The asymmetry also diminishes under strong forcing (Figures 3c, 3f, and 3i). At high E k, the exchange pattern is (1) symmetric about H 0, (2) independent of K e, and (3) horizontally sheared. This exchange pattern at high E k contrasts that over weak lateral slope, which is vertically sheared. [12] The cross-basin flow exhibits the same three general features described for the bathymetry with weak lateral slope. The magnitude of the lateral flows is greater over this bathymetry than over a gentler slope. This is the result of the same prescribed pressure gradient acting over a smaller cross section. The ratio of cross-basin to along-basin flows remains similar to that over the gentler sloped bathymetry. This means that lateral flows are of similar magnitude as the along-channel flows for low E k, 2 3 times smaller for intermediate E k, and around 5 times smaller for large E k Flows Over Steep Slope [13] The exchange flow patterns resulting over a steep slope are very similar qualitatively to those obtained over a moderate slope (Figure 4). Once again, the exchange pattern shows a strong dependence on K e under very weak friction (top row of Figure 4). For a wide channel (Figure 4a), the flows are greatly influenced by rotation and the exchange pattern is horizontally sheared, whereas for a narrow channel (Figure 4g), the exchange is vertically sheared. The high friction cases (Figures 3c, 3f, and 3i) show symmetric exchange patterns, independent of K e, with net inflow appearing throughout the water column. The cross-basin flows again show the same three features discussed in section and an increase in magnitude because of the reduced cross section. Also, the ratio of the cross-basin to along-basin flow magnitudes remains similar to the other bathymetries explored. The generalities of the along-basin and cross-basin flow patterns stay the same regardless of the position of H 0. Only a few details change with H 0 located toward the left or right. 5of10

6 Figure 4. Same as Figure 2 but with b 1 of Flows Over Moderate Slope, H 0 on the Left and Right [14] The exchange patterns arising over a channel located toward the left of the cross section (Figure 5) are analogous to those of a channel in the middle (Figure 3). There is a difference that appears in the high-k e and moderate-friction case (Figure 5b). In this case, there is a region of net inflow that develops from bottom to surface. In turn, the patterns related to a channel on the right of the cross section (Figure 6) are very similar to those of Figure 3. Therefore the position of the channel seems to have a minor effect on determining the shape of the exchange patterns. [15] In summary, under high frictional conditions (E k >1), the exchange pattern is practically invariant to K e. In contrast, under very weak frictional conditions (E k! 0), the exchange pattern depends on K e. The exchange flow is horizontally sheared under high K e (dynamically a wide basin) and vertically sheared under low K e (narrow basin). These findings are explored further by examining the strength of exchange flows in the deepest part of the channel H Strength of Exchange Flows Over H 0 [16] The strength of the exchange flows over H 0 is determined by the difference (Du) between maximum outflow (positive normalized values) and maximum inflow (negative normalized values). The values of Du are always positive. When the maximum inflow develops over H 0 and there is inflow from bottom to surface, then Du =1,asin Figures 3c, 3f, and 3i. If 0 < Du < 1, then there is only inflow over H 0, but the maximum inflow in the section is found outside the channel, as in Figures 4a and 5a. Those cases of Du between 0 and 1 represent horizontally, rather than vertically, sheared flows. Values of Du > 1 indicate vertically sheared exchange flows. The greater Du, the stronger the vertical shear in the exchange flows is. A total of 3131 solutions of equations (2) and (3) plus corresponding values of Du were obtained for a combination of 31 E k values and 101 K e values for each of six different bathymetric sections. These results are portrayed in Figure 7, where the darkest shades indicate the parameter space for which unidirectional inflows occupy practically the entire water column at H 0 (Du 1.1). The unshaded areas represent values of E k and K e for which maximum exchange develops over H 0 (Du > 1.5). The lightly shaded areas denote the parameter space of two-layer exchange in which net inflows occupy a greater portion of the water column than outflows over H 0 (1 < Du < 1.5). The results portrayed in Figures 2 to 6 are also placed in the context of Figure 7 in terms of where each of those solutions lies in the E k -K e parameter space. [17] Noteworthy of these results are three main features that should be expected in basins where Coriolis accelerations and frictional effects are relevant to the exchange hydrodynamics. First, for every bathymetric section and K e considered, the largest values of Du develop at moderate frictional influences, i.e., at E k between 0.01 and 0.1 6of10

7 Figure 5. Same as Figure 2 but with b 1 of 0.3 and y p at y/b = 0.3. (between 2 and 1 in the abscissa of Figure 7). Second, under any particular frictional influence and for all bathymetries, the greatest values of Du appear mostly at low K e (<1.6, i.e., 0.2 in the ordinate of Figure 7), and in general, Du tends to decrease with increasing K e. Third, in cases with appreciable bathymetric lateral variability (Figures 7c through 7f), net inflow develops from surface to bottom at E k > 0.3 (equivalent to 0.5 in the abscissa of Figure 7). A few more comments are pertinent for each of these three features. [18] The development of largest Du under moderate friction indicates that some friction is required to generate vertically sheared exchange flows. Otherwise, for too little or too much friction, the inflow may occupy most of the water column over H 0. Values of Du > 1 will develop, however, under low K e (<1.6, i.e., 0.2 in the ordinate of Figure 7) and low E k (<0.01, i.e., 2 in the abscissa of Figure 7). This implies that under very weak friction, the exchange will be vertically sheared in a narrow basin (low K e ) and horizontally sheared in a wide basin (high K e ). The Du dependence on K e is related to the second feature, mentioned above, that for any given E k the largest Du develops mainly at low K e. For large K e (wide basins), Coriolis accelerations limit the outflow to the left portion (looking into the basin) of the cross section. However, vertically sheared exchange flows (Du > 1.6) may develop under moderate friction (E k between 0.03 and 1, equivalent to 1.5 and 1 in the abscissa of Figure 7). With too little friction, the inflow occupies the whole water column and the maximum inflow appears to the right of H 0. The third feature is associated with horizontally sheared exchange flows developing over relatively steep lateral bathymetric slopes. For less steep cross sections, tending toward a flat bottom, bathymetric effects obviously do not play a role and the flow is vertically sheared even under strong frictional effects. In the cross section as a whole (figure not shown), the difference between maximum outflow and inflow at the entire section (not only over H 0 ) is greatest at intermediate E k (same values as above) and high K e. The results depicted in all figures suggest that the combined influence of Coriolis and friction is crucial for the development of vigorous exchange flows in a basin. 4. Implications on Natural Systems [19] In an attempt to bring these results into a real context, Figure 7d has been recast with the inclusion of several estuaries for which values of E k and K e are known from observations. The bathymetry of Figure 7d has been chosen as a generic bathymetry for estuaries as it consists of a deep channel flanked by shoals that extend to the shores. The systems chosen are only illustrative of where various estuaries would lie in the E k versus K e parameter space and the type of subtidal exchange expected. Detailed observations of the lateral structure of subtidal exchange are available for each system portrayed in Figure 8. Most of 7of10

8 Figure 6. Same as Figure 2 but with b 1 of 0.3 and y p at y/b = 0.7. those systems represented in the E k versus K e diagram of Figure 8 were compared to analytical results depicted by equations (2) and (3) in work by Valle-Levinson et al. [2003]. These include a wide estuary with relatively weak frictional influences, the Gulf of Fonseca, on the Pacific side of Central America; a wide system with moderate frictional influences, the lower Chesapeake Bay; a moderately wide system with moderate friction, the James River; and a narrow system with moderate friction, Guaymas Bay, on the mainland side of the Gulf of California. Three other systems have been included in the diagram: a narrow estuary with moderately weak friction, Saint Andrew Bay on northern Florida s coast in the Gulf of Mexico [Murphy and Valle-Levinson, 2007]; a narrow fjord with weak friction, Reloncavi Fjord [Valle-Levinson et al., 2007]; and a narrow estuary with moderate to strong friction, the thoroughly studied Hudson River [e.g., Lerczak et al., 2006]. [20] The systems that lie in the unshaded region of the E k versus K e diagram (Figure 8), namely, the lower Chesapeake Bay, Saint Andrew Bay, Reloncavi Fjord, and the Hudson River, exhibit a well-developed vertically sheared exchange flow over their deepest channel of their cross sections. Note that the Hudson River lies close to the boundary between lightly shaded and unshaded regions. This indicates that an increase in frictional influences will cause the subtidal exchange to be less vertically sheared than under reduced frictional influences. Indeed, this is the transition observed from neap to spring tides [Lerczak et al., 2006]. A similar situation develops in the James River, which also lies close to the boundary between unshaded and lightly shaded regions in the E k versus K e diagram. The James River exhibits a fortnightly transformation from vertically sheared exchange flows in neap tides to less vertically and more horizontally sheared exchange flows in spring tides. It is likely that systems close to the boundaries in Figure 8 will exhibit marked fortnightly transformations in exchange patterns. Guaymas Bay lies in the region influenced by both vertically and horizontally sheared exchange flows, as does the Gulf of Fonseca, as confirmed by observations. However, Guaymas Bay s pattern is the result of weak to moderate tidal forcing, and Fonseca s pattern is the result of Coriolis effects on a wide system. It also should be acknowledged that natural systems will exhibit different scales of temporal variability that will position them at various locations in the E k versus K e diagram. Therefore the position of the symbols representing each estuary on Figure 8 is nominal. A better representation would be to describe each estuary with an ellipse that circumscribes the range of possible E k and K e values observable for the system. This might be the challenge for future investigations of a given estuary. In general, any estuary could be cast in the E k versus K e parameter space, and its position would yield information on its pattern of density-induced exchange flows. [21] Noteworthy in Figure 8 is the absence of examples in the dark-shaded regions. These would represent either extremely strong frictional systems, for E k > 0.3 (or 0.5 8of10

9 Figure 7. Values of Du (maximum normalized outflow (positive) minus maximum normalized inflow (negative)) at H 0 for a range of K e and E k values and bathymetries. The subpanels on top of each panel show the bathymetry related to each set of results and the location of H 0. Contour interval is 0.1. Unshaded areas indicate the parameter space for which vertically sheared along-estuary exchange develops. Dark shades denote the parameter space for horizontally sheared exchange flow. Lightly shaded areas represent horizontally and vertically sheared exchange flows. Numbers accompanied by letters (e.g., 2a ) indicate the K e -E k flows illustrated in the corresponding figure (e.g., Figure 2a). 9of10

10 [24] 2. Under weak frictional conditions (E k! 0), the exchange pattern is horizontally sheared in wide basins (K e > 2) and vertically sheared in narrow basins (K e <1). This response is independent of the bathymetric profile because under weak friction the exchange flows are insensitive to bottom effects. [25] 3. Under moderate friction conditions (0.01 < E k < 0.1), the exchange flow is both horizontally and vertically sheared in most basin widths. Only very narrow systems (K e < 0.25) display preferentially vertically sheared flow. The latter situation of moderate frictional conditions is the one expected in most estuarine systems (e.g., the observational examples in the work by Valle-Levinson et al. [2003]). By locating a particular system in the proposed E k versus K e parameter space, one should be able to infer the pattern of exchange flows to be expected in the system. [26] Acknowledgments. This study was funded by NSF projects and Conversations with R. Garvine yielded the ideas developed in this manuscript, which is dedicated to him. The comments from R. Chant, S. Monismith, R. Weisberg, and an anonymous reviewer are gratefully appreciated. Figure 8. Same as Figure 7d but including some examples of natural systems: F, Gulf of Fonseca, Central America; C, lower Chesapeake Bay, Virginia; J, James River, Virginia; G, Guaymas Bay, Mexico; S, Saint Andrew Bay, Florida; H, Hudson River, New York New Jersey; R, Reloncavi Fjord, Chile. on the abscissa of Figure 8), or nearly frictionless while strongly rotating systems (top left corner of Figure 8). The former case of strongly frictional systems would be consistent with Wong s [1994] results but would require extremely large tidal currents (perhaps >2 m s 1 ) and very large eddy viscosities (>0.05 m 2 s 1 ). In systems influenced by strong tidal currents, the subtidal flow will probably be dominated by tidal rectification rather than by density gradients, and the dynamics presented in (1) would not apply. Similarly, the top left corner of Figure 8 might have just a few examples represented in nature, as it denotes very wide and frictionless systems dominated by density-induced flows, like the Gulf of California [Castro et al., 2006]. Most estuaries in nature are expected to lie in the lightly shaded and unshaded regions of the E k versus K e. As seen in Figure 7, the limits of these regions are fairly consistent among different bathymetries, at least where appreciable lateral slopes exist. 5. Conclusion [22] The main findings of this study, which extends those of Valle-Levinson et al. [2003] related to density-induced exchange flows over laterally varying (channel-shoals) bathymetry, are as follows. [23] 1. The exchange pattern consisting of net inflows in the channel and outflows over shoals develops only under high frictional conditions (E k > 1) independently of the basin s width (K e ). This is a horizontally sheared exchange pattern. References Castro, R., R. Durazo, A. Mascarenhas, C. A. Collins, and A. Trasviña (2006), Thermohaline variability and geostrophic circulation in the southern portion of the Gulf of California, Deep Sea Res., Part I, 53, Garvine, R. W. (1995), A dynamical system for classifying buoyant coastal discharges, Cont. Shelf Res., 15(13), Geyer, W. R., J. Trowbridge, and M. Bowen (2000), The dynamics of a partially mixed estuary, J. Phys. Oceanogr., 30, Gill, A. E. (1982), Atmosphere-Ocean Dynamics, 661 pp., Academic, New York. Hansen, D. V., and M. Rattray Jr. (1965), Gravitational circulation in straits and estuaries, J. Mar. Res., 23, Huijts, K. M. H., H. M. Schuttelaars, H. E. de Swart, and A. Valle-Levinson (2006), Lateral entrapment of sediment in tidal estuaries: An idealized model study, J. Geophys. Res., 111, C12016, doi: /2006jc Kasai, A., A. E. Hill, T. Fujiwara, and J. H. Simpson (2000), Effect of the Earth s rotation on the circulation in regions of freshwater influence, J. Geophys. Res., 105(C7), 16,961 16,969. Lerczak, J., W. R. Geyer, and R. Chant (2006), Mechanisms driving the time-dependent salt flux in a partially-stratified estuary, J. Phys. Oceanogr., 36(12), Murphy, P., and A. Valle-Levinson (2007), Tidal and residual circulation in the St. Andrew Bay system, Florida, submitted to Continental Shelf Research. Pritchard, D. W. (1952), Salinity distribution and circulation in the Chesapeake Bay estuarine system, J. Mar. Res., 11, Valle-Levinson, A., and J. O Donnell (1996), Tidal interaction with buoyancy driven flow in a coastal plain estuary, in Buoyancy Effects on Coastal and Estuarine Dynamics, vol. 53, Coastal Estuarine Stud., edited by D. G. Aubrey, and C. T. Friedrichs, pp , AGU, Washington, D. C. Valle-Levinson, A., C. Reyes, and R. Sanay (2003), Effects of bathymetry, friction and Earth s rotation on estuary/ocean exchange, J. Phys. Oceanogr., 33(11), Valle-Levinson, A., N. Sarkar, R. Sanay, D. Soto, and J. León (2007), Spatial structure of hydrography and flow in a Chilean fjord, Estuario Reloncaví, Estuaries Coasts, 30(1), Weisberg, R. H., and L. Y. Zheng (2006), Circulation of Tampa Bay driven by buoyancy, tides, and winds, as simulated using a finite volume coastal ocean model, J. Geophys. Res., 111, C01005, doi: /2005jc Winant, C. D. (2004), Three dimensional wind-driven flow in an elongated rotating basin, J. Phys. Oceanogr., 34(2), Wong, K.-C. (1994), On the nature of transverse variability in a coastal plain estuary, J. Geophys. Res., 99(C7), 14,209 14,222. A. Valle-Levinson, Civil and Coastal Engineering Department, University of Florida, Gainesville, FL 32611, USA. (arnoldo@ufl.edu) 10 of 10