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Density-driven exchange flow in terms of the Kelvin and Ekman numbers
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Arnoldo Valle-Levinson
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Civil and Coastal Engineering Department
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University of Florida
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Gainesville, FL 32611
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Tel. 352-392-9537 ext. 1479
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arnoldo@ufl.edu
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Abstract
The pattern of density-induced flow influenced by basin’s width, friction and Earth’s
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rotation is investigated as a function of the Ekman (Ek ) and Kelvin (Ke) numbers. A semi-
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analytical solution is used to determine the conditions under which the density-induced exchange
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flow is vertically sheared or horizontally sheared. Solutions are obtained over diverse laterally
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varying bathymetries. It is found that the exchange flow is horizontally sheared under high
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frictional conditions (Ek > 1) independently of the width of the basin (Ke). The horizontally
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sheared pattern describes inflow in the channel and outflow over shoals, with the inflow
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occupying the entire water column. The exchange flow pattern is also horizontally sheared
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under weak friction (Ek →0) and in wide (Ke > 2) basins. In that case, however, the outflow is
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concentrated on the left (looking into the basin in the northern hemisphere) portion of the cross-
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section and inflow appears on the right. Also under weak friction, the exchange pattern becomes
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more vertically sheared, with outflow at surface and inflow underneath, as the width of the basin
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becomes small (Ke < 1). Bathymetry is not very influential in the weak friction exchange
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patterns. Finally, under moderate friction (0.01< Ek < 0.1) the exchange pattern is both
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horizontally and vertically sheared for all widths. The horizontally sheared pattern is best
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defined in wide basins (high Ke) whereas the vertically sheared pattern practically dominates in
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narrow basins (low Ke). These findings allow classification of various estuaries in the Ek - Ke
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parameter space.
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Introduction
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It has been traditionally recognized that a basin’s width determines whether Earth’s
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rotation effects on density-induced or wind-induced water exchange are appreciable or not (e.g.
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Pritchard, 1952). The common view is that the basin should be wider than the internal Rossby
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radius Ri for rotation to be important (e.g. Gill, 1982). In a density-induced flow, Ri is given by
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(g’h)½/f, where g’ is the reduced gravity, h is the depth of the buoyant part of the density-induced
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flow and f is the Coriolis parameter. In turn, g’ equals g Δρ/o, where g is the gravity
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acceleration, o is a reference water density and Δρ is the contrast between the buoyant water
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density and the density underneath. The importance of Ri in containing the buoyant flow may be
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characterized by the non-dimensional Kelvin number Ke, which compares the basin’s width B to
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Ri, i.e., Ke = B/Ri (Garvine, 1995). Earth’s rotation effects are supposed to be most prominent
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when Ke > 1.
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Kasai et al. (2000) and Winant (2004) pointed out that water column depth, rather than
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basin’s width, should determine whether Earth’s rotation (or Coriolis) effects are important.
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Their argument was that over depths greater than several Ekman layers DE (e.g. > 4 DE), Coriolis
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effects were important regardless of the width. The value of DE is given by (2 Az / f ) ½, where Az
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is the flow’s eddy viscosity. Earth’s rotation effects on exchange flows may then be cast in
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terms of the Ekman number Ek (= Az / [f H 2], where H is water depth), which compares frictional
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to Coriolis effects. Coriolis effects become negligible at high Ek (>1). The objective of this
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paper is to reconcile these ideas with the help of semi-analytical results that portray density-
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induced exchange flows in terms of the Ekman and Kelvin numbers. This study extends that of
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Valle-Levinson et al. (2003) by considering the effects of basin width, i.e. the Ke dependence, on
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density-induced exchange flows. Results show that the density-induced exchange pattern is
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independent of the basin’s width (or Ke) at high Ek and depends on width (or Ke) at low and
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moderate Ek.
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Approach
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Density-induced exchange flow patterns are obtained with a semi-analytical solution (see
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Valle-Levinson et al., 2003) that compares very favorably with observations. The model solves
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for the non-tidal or mean along-basin u and transverse v flows at one basin cross-section. The
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flows are produced by pressure gradients and assumed to be modified only by Coriolis and
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frictional influences. Advective effects from tidal currents are assumed to be at least one order
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of magnitude smaller than other influences (e.g. Geyer et al., 2001) and their influence on the
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pattern of density-induced exchange flows is insignificant (Huijts et al., 2006). In a right-handed
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coordinate system (x, y, z), where x points seaward, y across the basin and z upward, the non-tidal
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(or steady) momentum balance is a set of two differential equations:
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 g 
 2u
 f v  g

z  Az 2
 x  x
z
 g 
 2v
f u  g

z  Az 2
y  y
z
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where f, g, , , Az are the Coriolis parameter, the gravity acceleration (9.8 m/s2), water density
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(kg/m3), surface elevation (m), and vertical eddy viscosity homogeneous in z and y (m2/s),
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respectively. Equations (1) may be solved for a complex velocity w = u + iv, where i 2 = 1 is the
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imaginary number:
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w(z) = gNF1 (z) + F2 (z) .
(1)
(2)
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In (2), N represents the sea level slope from the barotropic pressure gradient (∂η/∂x+ i∂η/∂y).
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The functions F1 and F2 depict the vertical structure of the barotropic (from sea level slope) and
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baroclinic (from density gradient) contributions to the flow, respectively:
F1 
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iD
F2 
f
i
f

cosh  z 
1 
 cosh  H y

 





 z
cosh  z 
H y
 H y
 e  z  e
cosh  H y






.
(3)

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In (3), D equals g/ρ(∂ρ/∂x + i∂ρ/∂y) and is independent of depth; the parameter  equals (1 +
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i)/DE, where DE is the Ekman layer depth [2Az / f ]½. Equations (3) are obtained by assuming no
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stress at the surface (F1 /z = F2 /z = 0 at z = 0) and no-slip at the bottom (F1 and F2 = 0 at
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z = Hy). Solutions (2) and (3) require prescription of Hy (as any function of y), a sea level slope
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N, an eddy viscosity Az, and a density gradient D that is dynamically consistent with N. The
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dynamically consistent value of D may be obtained by assuming a net volume flux R (m3/s)
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along or across a cross-section (Kasai et al., 2000), i.e.,
B 0
  w dz dy  R
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(4)
0 H y
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where B is the basin’s width. The value of D that satisfies a prescribed N and R is
B

R f 2  i g  N ( y ) (e
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D
H y
 H y ) tanh( H y )  (1  e
H y

  2 H y2 2) dy
0
B


(5)
i  tanh( H y )  H y dy
0
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and is a constant independent of y and z. As explained by Kasai et al. (2000), the solution
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consists of a unidirectional outflow, represented by the barotropic contribution gNF1 in (2 and 3),
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and bidirectional exchange flows given by the baroclinic contribution F2 in (2 and 3). The key to
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the solution is the way in which N(y) is prescribed (Valle-Levinson et al., 2003). On the basis of
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observations and numerical model results, they prescribed a slope with a value N0 at the coast
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that decayed exponentially across the basin: N = -N0 {1+ i exp[-(y/B)2]} (Fig. 1).
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In all of Valle-Levinson et al.’s (2003) solutions, the Kelvin number Ke was 1 because the
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exponential decay of the transverse slope of sea level spanned the width B of the basin. Those
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results are extended here by allowing a more general range of Ke values. This is done by
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normalizing the across-basin distance with the internal radius of deformation Ri rather than with
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B:
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N = - N0 {1+ i exp[-(y/Ri)2]}.
(6)
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The real part of (6) could also be prescribed as a function of y but the results are practically the
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same (Valle-Levinson et al., 2003). Solutions (2) and (3) are obtained for a value N0 of 1×10-6, R
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of zero and different values of Ek (function of Az) and Ke (function of Ri) over various
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bathymetric profiles. The bathymetric variation across the domain Hy is given by:
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Hy = H0 exp(-(y-yp)2/b12),
(7)
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where yp is the across-basin location of the deepest part of the channel (H0) and b1 determines the
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lateral slope of the channel. Furthermore, assuming along-basin uniformity, the vertical
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component of motion vw may be calculated from
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v
v
 w.
y
z
As discussed by Valle-Levinson et al. (2003), the analytical solution ignores advective
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accelerations and also assumes uniform Az. Despite these simplifications, the patterns produced
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by solutions (2) and (3) emulate the essence of those observed. Discrepancies exist only in
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details such as exact area of outflows/inflows and exact slope of isotachs (lines of equal speed).
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In the analytical solution, the effects of tidal forcing on mixing may be approximated by
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prescribing different values of Az to emulate the observations. Obviously, the complete approach
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to this problem would be to analyze numerical model results that allow Az to change in space and
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in time. Nonetheless, the results presented below and those in Valle-Levinson et al. (2003)
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exhibit essentially the same across-basin distribution as non-linear, turbulence closure numerical
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results for high Ek (e.g.Valle-Levinson and O’Donnell, 1996) and for low to moderate Ek (e.g.
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Weisberg and Zheng, 2006). Still, a comprehensive study based on numerical experiments
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should follow the approach proposed here.
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Results
Results are presented for the across-basin distribution of the density-induced along-basin
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flow u and cross-basin flow v, i.e., for the real and imaginary parts of (2), respectively. All
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representations show non-dimensional flows looking into a northern hemisphere basin where f =
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1×10-4 s-1. The along-basin flows are normalized by the maximum inflow. First, five
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bathymetric configurations are chosen to portray the lateral structure of flows for 3 selected
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values of Ek and Ke. Second, the strength of the exchange flow over the deepest part of the
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channel (H0) is examined for a wide range of Ek and Ke values. The strength of exchange flow is
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characterized by the difference between maximum inflow and outflow and is explored for the
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same five bathymetric distributions plus a sixth one. Third, the strength of the exchange flow
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allows characterization of a basin as vertically sheared or horizontally sheared, or between these
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two extremes, according to its locations in the Ek vs. Ke parameter space. Examples of several
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estuaries are placed in such parameter space.
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Lateral structure of flows
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In the depiction of the lateral structure of flows, three bathymetric distributions feature
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the deepest part H0 in the middle of the section. A fourth one shows H0 close to the left edge of
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the section and a fifth one exhibits H0 close to the right edge (looking into the basin). The three
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bathymetric distributions with H0 in the middle of the section had different lateral slopes to
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explore the shape of the flows over different morphologies. The other two bathymetric
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distributions examined the influence of the channel’s position in the section.
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Flows over weak lateral slope
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The exchange pattern (along-basin flows) over a nearly flat cross-section (contours in
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Fig. 2) shows a strong dependence on Ke at low Ek (very weak friction). In contrast, the
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exchange pattern remains nearly unaffected by Ke at high Ek. Under large Ke (wide basin) and
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weak to moderate friction (Ke ≥ 1 and Ek < 0.1) the isotachs are steeply sloped (Fig. 2a, 2b, 2d,
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2e). Most of the outflow in Figures 2a and 2b appears constrained by the internal radius of
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deformation and the inflow occupies most of the cross-sectional area. As Ke decreases from 4 to
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1, the outflow occupies up to one half of the cross section and the isotachs tilt less steeply (Fig.
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2d and 2e) until they become almost flat at Ke of 0.25 (Fig. 2g and 2h). When friction becomes
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much more dominant than rotation (Fig. 2c, 2f, 2i) then the exchange becomes that of typical
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estuarine circulation with outflow at the surface and inflow underneath across the entire section.
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This pattern is independent of Ke as Coriolis is not relevant anymore and the dynamics are
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determined by the balance between pressure gradient and friction (e.g. Pritchard, 1952; Hansen
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and Rattray, 1965). Still, slight bathymetric effects favor the development of two branches of
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outflow over the shallower portions of the section. This becomes more evident as the
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bathymetric slopes increase (e.g. Wong, 1994).
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The cross-basin flows, depicted by arrows in Figures 2 to 6, show three features that are
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consistent among all the bathymetric configurations explored. The first feature is a clockwise
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circulation pattern, in the vertical plain, that remains qualitatively the same at low Ek regardless
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of Ke (top row of Figures 2 through 6). This clockwise gyre is comprised of currents with similar
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magnitude as the along-basin flows and is tied to those along-basin flows through Coriolis
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effects. Thus, the lateral flow at surface moves to the right (in the northern hemisphere) of the
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outflow and the lateral flow in the layer underneath moves to the right of the inflow. Such
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linkage between along and cross-basin flows is best illustrated in panels d) and g). The second
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general feature is that the cross-basin circulation has the opposite direction at high Ek (bottom
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row of Figures 2 to 6) than at low Ek (top row of Figures 2 to 6). In general, cross-basin flows
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are from left to right at surface (0 < y < 0.3 in the c) panels) and from right to left in the deepest
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part of the section. This pattern contrasts that of low Ek and illustrates a region of flow
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convergence over the channel. The lateral flow at such high Ek is rather weak (consistently < 1
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cm/s) and is caused by the lateral pressure gradient being balanced by friction, rather than
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Coriolis. The dynamic balance between pressure gradient force and friction results in a
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‘sideways’ estuarine circulation (Valle-Levinson et al., 2003) even though it is weak. The third
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general feature has to do with intermediate Ek (middle row in Figures 2 to 6). The cross-basin
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flow pattern reverses from that at high Ke (e.g. Fig. 2b), which also resembles the flow pattern at
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low Ek (Fig. 2a), to that at intermediate and low Ke (e.g. Fig 2e and 2h). This means that at high
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Ke and intermediate Ek (Fig. 2b) the lateral flow is to the right in the deep part of the channel but
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it becomes toward the left at lower Ke (Fig 2e, 2h). The lateral flow at such intermediate Ek
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values is 2-3 times smaller than the along-basin flow so it is relevant to the transport of solutes.
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Flows over moderate lateral slope
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The along-basin patterns over a moderately sloped cross-section are consistent with
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those over weak slope but only for the low friction cases (Fig. 3). As Ke decreases, the isotachs
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tilt less and the exchange flow changes from horizontally sheared to vertically sheared (Fig. 3a,
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3d, 3g). The influence of friction allows the development of two branches of outflow. Under
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moderate friction the exchange pattern becomes vertically sheared for all Ke examined (Fig. 3b,
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3e, 3h). Furthermore, the left branch of outflow (looking into the basin) is more prominent
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because of the influence of Coriolis accelerations. Such prominence of the left branch, however,
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decreases with decreasing Ke, i.e., the asymmetry becomes less evident in narrow channels. The
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asymmetry also diminishes under strong forcing (Fig. 3c, 3f, 3i). At high Ek the exchange
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pattern is a) symmetric about H0, b) independent of Ke and c) horizontally sheared. This
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exchange pattern at high Ek contrasts that over weak lateral slope, which is vertically sheared.
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The cross-basin flow exhibits the same three general features described for the
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bathymetry with weak lateral slope. The magnitude of the lateral flows is greater over this
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bathymetry than over a gentler slope. This is the result of the same prescribed pressure gradient
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acting over a smaller cross-section. The ratio of cross-basin to along-basin flows remains similar
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to that over the gentler sloped bathymetry. This means that lateral flows are of similar
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magnitude as the along-channel flows for low Ek, 2-3 times smaller for intermediate Ek, and
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around 5 times smaller for large Ek.
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Flows over steep slope
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The exchange flow patterns resulting over a steep slope are very similar, qualitatively, to
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those obtained over a moderate slope (Fig. 4). Once again, the exchange pattern shows a strong
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dependence on Ke under very weak friction (top row of Fig. 4). For a wide channel (Fig. 4a) the
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flows are greatly influenced by rotation and the exchange pattern is horizontally sheared,
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whereas for a narrow channel (Fig. 4g) the exchange is vertically sheared. The high friction
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cases (Fig. 3c, 3f and 3i) show symmetric exchange patterns, independent of Ke, with net inflow
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appearing throughout the water column. The cross-basin flows again show the same three
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features discussed above and an increase in magnitude because of the reduced cross-section.
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Also, the ratio of the cross-basin to along-basin flow magnitudes remains similar to the other
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bathymetries explored. The generalities of the along-basin and cross-basin flow patterns stay the
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same regardless of the position of H0. Only a few details change with H0 located toward the left
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or right.
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Flows over moderate slope, H0 on the left and right
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The exchange patterns arising over a channel located toward the left of the cross-section
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(Fig. 5) are analogous to those of a channel in the middle (Fig. 3). There is a difference that
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appears in the high Ke and moderate friction case (Fig 5b). In this case, there is a region of net
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inflow that develops from bottom to surface. In turn, the patterns related to a channel on the
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right of the cross-section (Fig. 6) are very similar to those of Figure 3. Therefore, the position of
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the channel seems to have a minor effect on determining the shape of the exchange patterns.
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In summary, under high frictional conditions (Ek >1) the exchange pattern is practically
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invariant to Ke. In contrast, under very weak frictional conditions (Ek →0) the exchange pattern
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depends on Ke. The exchange flow is horizontally sheared under high Ke (dynamically a wide
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basin) and vertically sheared under low Ke (narrow basin). These findings are explored further
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by examining the strength of exchange flows in the deepest part of the channel H0.
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Strength of exchange flows over H0
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The strength of the exchange flows over H0 is determined by the difference (Δu) between
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maximum outflow (positive normalized values) and maximum inflow (negative normalized
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values). The values of Δu are always positive. When the maximum inflow develops over H0 and
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there is inflow from bottom to surface then Δu = 1, as in Figure 3c, 3f, 3i. If 0 < Δu < 1, then
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there is only inflow over H0 but the maximum inflow in the section is found outside the channel
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as in Figures 4a, 5a. Those cases of Δu between 0 and 1 represent horizontally, rather than
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vertically, sheared flows. Values of Δu > 1 indicate vertically sheared exchange flows. The
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greater Δu, the stronger the vertical shear in the exchange flows. A total of 3131 solutions of
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equations (2) and (3) plus corresponding values of Δu were obtained for a combination of 31 Ek
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values and 101 Ke values for each of 6 different bathymetric sections. These results are
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portrayed in Figure 7 where the darkest shades indicate the parameter space for which
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unidirectional inflows occupy practically the entire water column at H0 (Δu ≤ 1.1). The
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unshaded areas represent values of Ek and Ke for which maximum exchange develops over H0
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(Δu > 1.5). The lightly shaded areas denote the parameter space of two-layer exchange in which
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net inflows occupy a greater portion of the water column than outflows over H0 (1 < Δu < 1.5).
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The results portrayed in Figures 2 through 6 are also placed in the context of Figure 7 in terms of
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where each of those solutions lies in the Ek - Ke parameter space.
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Noteworthy of these results are three main features that should be expected in basins
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where Coriolis accelerations and frictional effects are relevant to the exchange hydrodynamics.
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First, for every bathymetric section and Ke considered, the largest values of Δu develop at
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moderate frictional influences, i.e. at Ek between 0.01 and 0.1 (between -2 and -1 in the abscissa
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of Fig. 7). Second, under any particular frictional influence and for all bathymetries, the greatest
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values of Δu appear mostly at low Ke (< 1.6, i.e., 0.2 in the ordinate of Fig. 7) and in general Δu
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tends to decrease with increasing Ke. Third, in cases with appreciable bathymetric lateral
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variability (Fig. 7c through f), net inflow develops from surface to bottom at Ek > 0.3 (equivalent
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to -0.5 in the abscissa of Fig. 7). A few more comments are pertinent for each of these three
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features.
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The development of largest Δu under moderate friction indicates that some friction is
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required to generate vertically sheared exchange flows. Otherwise, for too little or too much
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friction the inflow may occupy most of the water column over H0. Values of Δu > 1 will
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develop, however, under low Ke (< 1.6, i.e., 0.2 in the ordinate of Fig. 7) and low Ek (<0.01, i.e.,
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-2 in the abscissa of Fig. 7). This implies that under very weak friction, the exchange will be
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vertically sheared in a narrow basin (low Ke) and horizontally sheared in a wide basin (high Ke).
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The Δu dependence on Ke is related to the second feature, mentioned above, that for any given Ek
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the largest Δu develops mainly at low Ke. For large Ke (wide basins), Coriolis accelerations limit
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the outflow to the left portion (looking into the basin) of the cross-section. However, vertically
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sheared exchange flows (Δu > 1.6) may develop under moderate friction (Ek between 0.03 and 1,
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equivalent to -1.5 and -1 in the abscissa of Fig. 7). With too little friction the inflow occupies the
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whole water column and the maximum inflow appears to the right of H0. The third feature is
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associated with horizontally sheared exchange flows developing over relatively steep lateral
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bathymetric slopes. For less steep cross-sections, tending toward a flat bottom, bathymetric
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effects obviously do not play a role and the flow is vertically sheared even under strong frictional
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effects. In the cross-section as a whole (figure not shown), the difference between maximum
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outflow and inflow at the entire section (not only over H0) is greatest at intermediate Ek (same
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values as above) and high Ke. The results depicted in all figures suggest that the combined
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influence of Coriolis and friction is crucial for the development of vigorous exchange flows in a
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basin.
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Implications on natural systems
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In an attempt to bring these results into a real context, Figure 7d has been recast with the
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inclusion of several estuaries for which values of Ek and Ke are known from observations. The
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bathymetry of Figure 7d has been chosen as a generic bathymetry for estuaries as it consists of a
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deep channel flanked by shoals that extend to the shores. The systems chosen are only
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illustrative of where various estuaries would lie in the Ek vs Ke parameter space and the type of
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subtidal exchange expected. Detailed observations of the lateral structure of subtidal exchange
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are available for each system portrayed in Figure 8. Most of those systems represented in the Ek
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vs Ke diagram of Figure 8 were compared to analytical results depicted by equations 2 and 3 in
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Valle-Levinson et al. (2003). These include a wide estuary with relatively weak frictional
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influences, the Gulf of Fonseca, on the Pacific side of Central America; a wide system with
287
moderate frictional influences, the lower Chesapeake Bay; a moderately wide system with
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moderate friction, the James River; and a narrow system with moderate friction, Guaymas Bay,
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on the mainland side of the Gulf of California. Three other systems have been included in the
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diagram: a narrow estuary with moderately weak friction, Saint Andrew Bay on Northern
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Florida’s coast in the Gulf of Mexico (Valle-Levinson, unpublished data); a narrow fjord with
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weak friction, Reloncavi fjord (Valle-Levinson et al., 2007); and a narrow estuary with moderate
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to strong friction, the thoroughly studied Hudson River (e.g. Lerczak et al., 2006).
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The systems that lie in the unshaded region of the Ek vs Ke diagram (Fig. 8), namely the
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lower Chesapeake Bay, St Andrew Bay, Reloncavi Fjord and the Hudson River, exhibit a well-
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developed vertically sheared exchange flow over their deepest channel of their cross sections.
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Note that the Hudson River lies close to the boundary between lightly shaded and unshaded
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regions. This indicates that an increase in frictional influences will cause the subtidal exchange
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to be less vertically sheared than under reduced frictional influences. Indeed this is the transition
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observed from neap to spring tides (Lerczak et al., 2006). A similar situation develops in the
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James River, which also lies close to the boundary between unshaded and lightly shaded regions
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in the Ek vs Ke diagram. The James River exhibits a fortnightly transformation from vertically
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sheared exchange flows in neap tides to less vertically and more horizontally sheared exchange
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flows in spring tides. It is likely that systems close to the boundaries in Figure 8 will exhibit
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marked fortnightly transformations in exchange patterns. Guaymas Bay lies in the region
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influenced by both vertically and horizontally sheared exchange flows as does the Gulf of
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Fonseca, as confirmed by observations. However, Guaymas Bay’s pattern is the result of weak
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to moderate tidal forcing and Fonseca’s pattern is the result of Coriolis effects on a wide system.
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It also should be acknowledged that natural systems will exhibit different scales of temporal
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variability that will position them at various locations in the Ek vs Ke diagram. Therefore, the
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position of the symbols representing each estuary on Figure 8 is nominal. A better representation
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would be to describe each estuary with an ellipse that circumscribes the range of possible Ek and
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Ke values observable for the system. This might be the challenge for future investigations of a
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given estuary. In general, any estuary could be cast in the Ek vs Ke parameter space and its
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position would yield information on its pattern of density-induced exchange flows.
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Noteworthy in Figure 8 is the absence of examples in the dark shaded regions. These
317
would represent either extremely strong frictional systems, for Ek > 0.3 (or -0.5 on the abscissa of
318
Fig. 8), or nearly frictionless while strongly rotating systems (upper left corner of diagram). The
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former case of strongly frictional systems would be consistent with Wong (1994) results but
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would require extremely large tidal currents (perhaps > 2 m/s) and very large eddy viscosities
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(>0.05 m2/s). In systems influenced by strong tidal currents the subtidal flow will probably be
15
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dominated by tidal rectification rather than by density gradients and the dynamics presented in
323
(1) would not apply. Similarly, the upper left corner of the diagram might have just a few
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examples represented in nature as it denotes very wide and frictionless systems dominated by
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density-induced flows, like the Gulf of California (Castro et al., 2006). Most estuaries in nature
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are expected to lie in the lightly shaded and unshaded regions of the Ek vs. Ke. As seen in Figure
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7, the limits of these regions are fairly consistent among different bathymetries, at least where
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appreciable lateral slopes exist.
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330
Conclusion
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The main findings of this study, which extends those of Valle-Levinson et al. (2003)
332
related to density-induced exchange flows over laterally varying (channel-shoals) bathymetry,
333
are as follows. 1) The exchange pattern consisting of net inflows in the channel and outflows
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over shoals develops only under high frictional conditions (Ek > 1) independently of the basin’s
335
width (Ke). This is a horizontally sheared exchange pattern. 2) Under weak frictional conditions
336
(Ek → 0) the exchange pattern is horizontally sheared in wide basins (Ke > 2) and vertically
337
sheared in narrow basins (Ke < 1). This response is independent of the bathymetric profile
338
because under weak friction the exchange flows are insensitive to bottom effects. 3) Under
339
moderate friction conditions (0.01< Ek < 0.1) the exchange flow is both horizontally and
340
vertically sheared in most basin widths. Only very narrow systems (Ke < 0.25) display
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preferentially vertically sheared flow. The latter situation of moderate frictional conditions is the
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one expected in most estuarine systems (e.g. the observational examples in Valle-Levinson et al.,
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2003). By locating a particular system in the proposed Ek vs. Ke parameter space, one should be
344
able to infer the pattern of exchange flows to be expected in the system.
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345
346
347
Acknowledgments
This study was funded by NSF projects 9983685 and 0551923. Conversations with R.
348
Garvine yielded the ideas developed in this manuscript, which is dedicated to him. The
349
comments from R. Chant, S. Monismith, R. Weisberg and an anonymous reviewer are gratefully
350
appreciated.
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352
17
353
References
354
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geostrophic circulation in the southern portion of the Gulf of California. Deep Sea Res. I,
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53, 188-200, 2006.
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Garvine, R.W. A dynamical system for classifying buoyant coastal discharges. Cont. Shelf Res.,
15(13), 1585-1596, 1995.
Geyer, W.R., J. Trowbridge, and M. Bowen, The dynamics of a coastal plain estuary, J. Phys.
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Gill, A.E., Atmosphere-Ocean Dynamics. Academic Press, New York, 661 pp. 1982.
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Hansen, D.V. and M. Rattray, Jr., Gravitational circulation in straits and estuaries. J. Mar. Res.,
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23, 104-122, 1965.
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Huijts, K.M.H., H.M. Schuttelaars, H.E. de Swart, and A. Valle-Levinson, Lateral entrapment of
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sediment in tidal estuaries: An idealized model study, J. Geophys. Res., 111, C12016,
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doi:10.1029/2006JC003615, 2006.
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Kasai, A., A.E. Hill, T. Fujiwara, and J.H. Simpson, Effect of the Earth’s rotation on the
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circulation in regions of freshwater influence, J. Geophys. Res., 105(C7), 16,961-16,969,
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2000.
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Lerczak, J., W.R. Geyer, and R. Chant, Mechanisms driving the time-dependent salt flux in a
partially-stratified estuary, J. Phys. Oceanogr., 36(12), 2296-2311, 2006.
Pritchard, D.W. Salinity distribution and circulation in the Chesapeake Bay estuarine system. J.
Marine Res., 11, 106-123, 1952.
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Valle-Levinson, A. and J. O'Donnell, Tidal interaction with buoyancy driven flow in a coastal
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plain estuary, Coastal and Estuarine Studies vol. 53, Buoyancy Effects on Coastal and
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Estuarine Dynamics, edited by D.G. Aubrey and C.T. Friedrichs, Am. Geophys. Union,
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Washington, DC., pp. 265-281, 1996.
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Valle-Levinson, A., C. Reyes and R. Sanay, Effects of bathymetry, friction and Earth’s rotation
on estuary/ocean exchange, J. Phys. Oceanogr., 33(11), 2375-2393, 2003.
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Valle-Levinson, A. N. Sarkar, R. Sanay, D. Soto, and J. León. Spatial structure of hydrography
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and flow in a Chilean Fjord, Estuario Reloncaví, Estuaries and Coasts, 30 (1), 113-126,
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2007.
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Winant, C.D., Three dimensional wind-driven flow in an elongated rotating basin, J. Phys.
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Weisberg, R. H., and L. Y. Zheng, Circulation of Tampa Bay driven by buoyancy, tides, and
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winds, as simulated using a finite volume coastal ocean model, J. Geophys. Res., 111,
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C01005, doi:10.1029/2005JC003067, 2006.
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Wong, K.-C., On the nature of transverse variability in a coastal plain estuary, J. Geophys. Res.,
99, 14,209-14,222, 1994.
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List of Figures
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Figure 1. Cross-channel distribution of the lateral slope prescribed in equation (5) as compared
393
394
to numerical and observational results (as in Valle-Levinson et al., 2003).
Figure 2. Along-estuary (normalized by maximum inflow) and cross-estuary (scale appearing
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above upper right corner in cm/s) flows in a cross-section with very weak lateral slopes.
396
Bathymetry is drawn from equation (7) for yp in the middle (y/B = 0.5) and and b1 of 1.9.
397
Darker areas denote regions of inflows. Contours are drawn at 0.2 intervals. Views are
398
looking into the estuary. For the upper row, Az = 1×10-4 m2/s; for the middle row, Az =
399
1×10-2 m2/s; and for the lower row Az = 1×10-1 m2/s.
400
Figure 3. Same as figure 1 but with b1 in equation (7) of 0.5.
401
Figure 4. Same as figure 1 but with b1 of 0.2.
402
Figure 5. Same as figure 1 but with b1 of 0.3 and yp at y/B = 0.3.
403
Figure 6. Same as figure 1 but with b1 of 0.3 and yp at y/B = 0.7.
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Figure 7. Values of Δu (maximum normalized outflow - positive - minus maximum normalized
405
inflow -negative) at H0 for a range of Ke and Ek values and bathymetries. The subpanels on
406
top of each panel show the bathymetry related to each set of results and the location of H0.
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Contour interval is 0.1. Unshaded areas indicate the parameter space for which vertically
408
sheared along-estuary exchange develops. Dark shades denote the parameter space for
409
horizontally sheared exchange flow. Lightly shaded areas represent horizontally and
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vertically sheared exchange flows. Numbers accompanied by letters (e.g. ‘2a’) indicate the
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Ke -Ek flows illustrated in the corresponding figure (e.g. Fig. 2a).
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413
Figure 8. Same as Figure 7d but including some examples of natural systems: F is Gulf of
Fonseca, Central America; C is lower Chesapeake Bay, Virginia; J is James River,
20
414
Virginia; G is Guaymas Bay, Mexico; S is St Andrew Bay, Florida; H is Hudson River,
415
New York-New Jersey; R is Reloncavi Fjord, Chile.
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21
417
418
419
420
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Figure 1. Cross-channel distribution of the lateral slope prescribed in equation (5) as compared
to numerical and observational results (as in Valle-Levinson et al., 2003).
22
422
423
424
425
426
427
Figure 2. Along-estuary (normalized by maximum inflow) and cross-estuary (scale appearing above upper right corner in cm/s)
flows in a cross-section with very weak lateral slopes. Bathymetry is drawn from equation (7) for yp in the middle (y/B = 0.5) and
and b1 of 1.9. Darker areas denote regions of inflows. Contours are drawn at 0.2 intervals. Views are looking into the estuary.
For the upper row, Az = 2×10-4 m2/s; for the middle row, Az = 1×10-2 m2/s; and for the lower row Az = 1×10-1 m2/s.
23
428
429
Figure 3. Same as figure 1 but with b1 in equation (7) of 0.5.
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431
Figure 4. Same as figure 1 but with b1 of 0.2.
25
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Figure 5. Same as figure 1 but with b1 of 0.3 and yp at y/B = 0.3.
26
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435
Figure 6. Same as figure 1 but with b1 of 0.3 and yp at y/B = 0.7.
27
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437
438
439
440
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Figure 7. Values of Δu (maximum normalized outflow - positive - minus maximum normalized inflow -negative) at H0 for a
range of Ke and Ek values and bathymetries. The subpanels on top of each panel show the bathymetry related to each set of
results and the location of H0. 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 Ke -Ek flows illustrated in the corresponding figure (e.g. Fig. 2a).
28
443
444
445
446
447
448
Figure 8. Same as Figure 7d but including some examples of natural systems: F is Gulf of
Fonseca, Central America; C is lower Chesapeake Bay, Virginia; J is James River, Virginia; G is
Guaymas Bay, Mexico; S is St Andrew Bay, Florida; H is Hudson River, New York-New Jersey;
R is Reloncavi Fjord, Chile.
29