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Convectively driven exchange in a shallow coastal embayment
Continental Shelf Research 19 (1999) 1599}1616
Convectively driven exchange in a shallow coastal embayment Murray C. Burling, Gregory N. Ivey, Charitha B. Pattiaratchi* Department of Environmental Engineering, Centre for Water Research, The University of Western Australia, Nedlands, 6907, Western Australia, Australia Received 13 December 1996; received in revised form 29 September 1997; accepted 30 March 1999
Abstract Shark Bay is a large coastal embayment (length &250 km, width &100 km) located on the central west coast of Australia. The Bay is comprised of two major reaches, which are characterised by average depths of 10 m and salinities which increase with longitudinal distance away from the Bay entrance. Maximum salinities in the Bay exceed 60 (Practical Salinity Scale), and occur in Hamelin Pool at the southern end of Hopeless Reach, the eastern region of the Bay. Exchange between Hamelin Pool and Hopeless Reach is severely restricted by the presence of a sill, and occurs predominantly through a single 2 km wide, 6 m deep channel (Herald Loop). CTD measurements taken in Hopeless Reach show variable strati"cation: vertically well-mixed in summer and strongly strati"ed during the winter survey. A shallow cavity natural convection model was applied to the Herald Loop channel to determine the contribution of the saline discharge from Hamelin Pool to the observed variable strati"cation. The model shows that the discharge from Hamelin Pool is a di!usive process. Thus while the discharge is an important salinity source, it will not directly contribute to the variability observed in Hopeless Reach. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Natural convection; Saline exchange; Inverse estuary
1. Introduction Many shallow coastal water bodies exhibit longitudinal gradients in density. In estuaries, such gradients arise as a result of the mixing of seawater with low salinity
* Corresponding author. Tel.: #0061-8-9380-3179; fax: #0061-8-9380-1015. E-mail address: [email protected] (C.B. Pattiaratchi) 0278-4343/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 7 8 - 4 3 4 3 ( 9 9 ) 0 0 0 3 4 - 5
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tributary in#ows. Longitudinal density gradients may also be a feature of coastal regions with little freshwater in#ow, due to a dominance of evaporation over rainfall and run-o!. In this case, however, the density gradient is reversed (or negative), and maximum salinities occur in those regions with minimal direct oceanic in#uence. Examples of such `inverse estuariesa include the Spencer Gulf in South Australia (Nunes and Lennon, 1986) and Shark Bay in Western Australia (Logan and Cebulski, 1970). Often, the degree of wind and tidal mixing is su$cient to overcome lateral and vertical buoyancy inputs (e.g. stream#ow, solar heating), resulting in vertically wellmixed conditions. In the absence of strong mixing, a longitudinally strati"ed water body would naturally tend to adjust to the internal pressure gradient, and given time, become vertically strati"ed. Mixing retards this adjustment by increasing the internal friction to the baroclinic #ow; the stronger the mixing, the larger the time-scale for exchange between each end of the system. The characteristic circulation of the system is thus governed by a balance between the buoyancy forces due to the longitudinal gradient in density and the viscous forces due to turbulent mixing (e.g., O$cer, 1976). In the limit of very strong vertical mixing, the system becomes di!usive, and the characteristic longitudinal density gradient is linear. The circulation that results from an assumed buoyancy-viscous balance was "rst investigated by Hansen and Rattray (1965). In their study of estuarine #ow, they applied a similarity technique to solve the coupled equations for momentum and density conservation, assuming a priori a linear longitudinal salinity distribution. O$cer (1976) decoupled the momentum and density equations, solving instead for the velocity "eld generated by the special case of a constant longitudinal density gradient. The e!ect of the velocity "eld on the salinity distribution was then calculated via a simple advection-di!usion balance. Both solution sets included the surface wind stress contribution to the circulation, derived earlier by Baines and Knapp (1965), and can be shown to be equivalent (O$cer, 1976). A perturbation solution for the linearised equations of the di!usive regime of estuarine #ow was obtained by Van De Kreeke and Zimmerman (1990). However, in order to solve for the salinity distribution, the form of the longitudinal di!usivity must be known. Common to the arguments of Hansen and Rattray (1965) and O$cer (1976) is the assumption that the longitudinal density gradient is linear. Thus, it is not possible to de"ne the conditions under which the assumption of a di!usive transport regime (characterised by a linear longitudinal density gradient) is valid. The problem of buoyancy driven estuarine circulation has also motivated a number of studies into steady state laminar natural convection in cavities of low aspect ratio (e.g., Cormack et al., 1974; Imberger, 1974). While shallow coastal water bodies are turbulent systems, the large time-scale of the gravitational circulation allows the use of mean, or e!ective, eddy exchange coe$cients and the turbulent natural system (e.g. an estuary) may then be treated as analogous to a laminar system (Imberger, 1974). Buoyancy driven #ow in a shallow cavity is dependent upon three non-dimensional parameters (e.g., Cormack et al., 1974; Imberger, 1974). The geometry of the cavity is described by the aspect ratio, A"h/¸, which is typically very much smaller than 1.
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The buoyancy-viscous force balance is described by the Grashof number, gb *S h Gr" l
(1)
where h is the cavity height, ¸ is the cavity length, l is the kinematic viscosity, g is the acceleration due to gravity, b is the saline expansion coe$cient, *S is the end-to-end salinity di!erence. The "nal parameter is the Prandtl number, Pr"l/i, where i is the molecular di!usion coe$cient (for heat or salt). An asymptotic solution to the governing equations can be found by making use of the fact that the aspect ratio of the cavity, A, is very small. Cormack et al. (1974) "rst introduced this method to the problem of shallow cavity convection with di!erentially heated endwalls and no-slip surfaces, as a compliment to the experimental work of Imberger (1974). Bejan and Tien (1978), Bejan (1984) and Ivey and Hamblin (1989) have also investigated the problem using similar methods. The e!ect of a free surface was investigated by Cormack et al. (1975), including the cases of free-slip and an imposed shear stress. These studies have shown that for a restricted range of Gr and A, transport within the cavity is dominated by longitudinal di!usion. It will be shown in Section 4.1 that this range is de"ned by GrA(10. A bene"t of this approach is therefore the ability to de"ne the limits of the solution. An additional advantage is that only one assumption is required, that A;1. In this study we utilise the asymptotic model to investigate the saline driven exchange within Shark Bay, Western Australia. Much of Shark Bay is hypersaline: salinities in excess of 60 have been recorded. Discharge from the largest `salt reservoira in the Bay, Hamelin Pool, is severely restricted by the presence of a sill. In this region, estimates of the mean tidally generated vertical eddy viscosity suggest that the value of GrA will generally be below the critical value of 10. Thus, the transport regime can be characterised as di!usive, and an appropriate transport coe$cient can be found. It will also be shown that the analytical model is sensitive to both wind direction and magnitude, in particular when the prevailing wind is in the direction of decreasing salinity. Furthermore, it is shown that the asymptotic model solution is equivalent to the classical solutions of Hansen and Rattray (1965) and O$cer (1976). We begin with a description of the "eld site, followed by a review of the asymptotic theory.
2. Physical setting Shark Bay is a large coastal embayment (length&250 km, width&100 km, depth &10 m) located on the central west coast of Australia, in a region bound by latitudes 24330 and 26345 , and is semi-enclosed by three large islands (Fig. 1). The majority of the Bay is comprised of two reaches (Hopeless Reach to the east and Freycinet Reach to the west) aligned approximately north}south, separated by the Peron Peninsula. The climate is semi-arid, with most of the average annual precipitation of around 200 mm falling sporadically in winter and during summer cyclone events. For most of
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Fig. 1. Shark Bay, Western Australia. Bathymetric contours of 4, 10, 20 m are marked. Note particularly the Faure Sill (1 m deep region shaded) and the Herald Loop channel in the eastern reach of the Bay.
the year, the region is in#uenced by strong southerly winds with a seasonally variant mean magnitude (e.g. Fig. 2). Winds during the spring and summer periods are generally the strongest, with mean speeds of 8 ms\ at both Denham and Carnarvon. The mean wind direction during summer is approximately southerly (1903) with a standard deviation of 503, demonstrating that winds are predominantly con"ned to a small arc ranging from southwesterly to southeasterly. Wind speed reduces during the winter and autumn periods, although the average remains approximately 5 ms\. Along with a reduction in wind speed during these months, the wind direction moves around toward the south-east (150}1603) and is accompanied by an increased variation (standard deviation&803). The maximum variance occurs during June, July and August, owing to the periodic passage of winter weather systems.
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Fig. 2. Three-hourly wind vectors at Denham, 1994: (a) summer and (b) winter. Positive vectors represent a southerly component, i.e. toward the north.
The combination of strong, persistent winds and high temperatures generate an annual evaporation (&2 m) which is ten times greater than the annual rainfall (&200 mm). Consequently, large sections of the Bay are hypersaline (salinity'50), and both Freycinet Reach and Hopeless Reach (Fig. 1) exhibit a salinity distribution analogous to an inverse estuary (Logan and Cebulski 1970). During a 13 yr study period, Logan and Cebulski (1970) observed little seasonal or inter-annual variation in the salinity structure of the entire Bay, leading them to conclude that the system is at steady state. They also noted that most of the Bay was vertically well-mixed. The single anomalous data set (winter, 1965) collected by Logan and Cebulski (1970) revealed that, although the salinity throughout the Bay was reduced by approximately 10%, the salinity gradients were preserved. Recent intensive "eld experiments in February (summer) and August (winter) of 1995 have shown that the salinity structure in the Bay, particularly in the eastern reach, can in fact be seasonally variant. Figs. 4 and 5, for example, show a salinity transect taken in an east}west line cutting Cape Peron (Fig. 3), for the summer and winter surveys, respectively. Although the range of salinities is unchanged, it is immediately apparent that the strong two-layer structure observed to the east of Cape Peron during the winter survey (Fig. 5) represents a dramatically di!erent #ow regime
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Fig. 3. CTD Transects, Shark Bay, February and August 1995. (A) Cape Peron and (B) Herald Loop. Average distance between stations is 2 nautical miles.
to that observed in summer (Fig. 4). Variability in the discharge from the hypersaline regions at the southern end of the Bay may contribute to the variation apparent in Figs. 4 and 5. Alternatively, a reduction in the energy available for turbulent vertical mixing may lead to an inertia-buoyancy balance and the formation of clearly de"ned gravity currents (Linden and Simpson, 1986; Simpson and Linden, 1989). A similar phenomenon has been observed in Spencer Gulf (Nunes and Lennon, 1987). Hamelin Pool, situated at the southern end of the Hopeless Reach, is the major hypersaline region within the Bay. Exchange between Hamelin Pool and the remainder of the Bay is restricted by the presence of the Faure Sill (Fig. 1). The average depth in the sill region is 1 m. Most of the exchange #ow is con"ned to a single narrow
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Fig. 4. Salinity contours at Cape Peron, February (summer) 1995. Transect location as per Fig. 3. Note that to the east of Cape Peron (proj. dist."18,000 m) the water-column is well-mixed.
channel, Herald Loop, which is approximately 2 km wide, 20 km long and has an average depth of 6 m. Salinity pro"les were taken through Herald Loop during winter 1995, with the resultant longitudinal transect of the vertical salinity structure in the channel shown in Fig. 6. This transect was taken during a #ooding tidal phase in a north}south direction, beginning in the lower region of Hopeless Reach and extending into Hamelin Pool (Fig. 3), where salinities approaching 60 are observed. The corresponding horizontal density di!erence over this region is 8 kgm\. Despite the strong density gradient, the system is well-mixed vertically and has a linear gradient in salinity (&10 over 20 km), both characteristic of a di!usively dominated system. The data collected by Logan and Cebulski (1970) is supportive, showing the persistence of this salinity gradient over the entire sill region, including the Herald Loop channel.
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Fig. 5. Salinity contours at Cape Peron, August (winter) 1995. Transect location as per Fig. 3. Note the two-layer strati"cation now visible to the east of Cape Peron (proj. dist."18,000 m).
3. Model formulation Our purpose here is to model the transport through the Herald Loop channel in order to assess the contribution of this saline #ux to the dynamics of Shark Bay as a whole. The channel between Hamelin Pool and the remainder of Shark Bay can be represented by a two-dimensional cavity (Fig. 7). The end regions of the cavity are dynamically passive in terms of any contribution to the #ow "eld in the interior of the cavity (e.g., Cormack et al., 1974) allowing the `endwallsa to be chosen as the ends of the Herald Loop channel. This simply means that we have chosen the relevant longitudinal length scale to be equal to the channel length. Each boundary represents a transition between the channel waters and a large reservoir, whose mean properties are assumed to be "xed and independent of mixing in the channel. Consequently the boundary conditions for the channel circulation remain independent of the internal #ow. The bottom is assumed to be a no-slip boundary and the surface is subject to an applied shear stress. We will neglect tidal #ow in the channel, except for
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Fig. 6. Salinity contours through Herald Loop, #ooding tide, August (winter) 1995. The transect stations are shown in Fig. 3. Hamelin Pool is located to the right. The Herald Loop channel is located approximately in the region of linear longitudinal gradient (proj. dist. "3000}23,000 m). The salinity gradient is approximately 0.5 km\.
the contribution to the mean turbulent exchange coe$cients, assuming that tidal motion simply advects the entire `cavitya back and forth (note that the tidal excursion length is approximately 5 km). Further, the system is assumed to be at steady state. The governing equations may be non-dimensionalised using the following variables (e.g. Cormack et al., 1974; Bejan, 1984; Ivey and Hamblin, 1989) x( z( ¸u( ¸w( S!S x" , z" , u" , w" , p" ¸ h Grl GrlA S !S
(2)
where x( and z( are the longitudinal distance (increasing in the direction of increasing salinity) and depth and u( and w( are the velocities in the x( and z( directions, respectively. The length, depth and width scales of the channel are ¸"20 km, h"6 m and ="2 km and the aspect ratio is A"3;10\. The salinity, S, is scaled with the `endwalla salinities, S &50 (Hopeless Reach) and S &60 (Hamelin Pool), i.e. *S"10.
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Fig. 7. Schematic diagram of the Herald Loop channel, where h"6 m, ¸"20 km, S "50 and S "60. The coordinate system is as shown.
The introduction of the stream function t, de"ned by *t *t u" , w"! *z *x
(3)
allows the non-dimensional steady equations of motion for the central core region of the cavity (far removed from the e!ect of the endwalls) to be written as GrA A
*u *u *p *(u, t) "A # ! *x *z *x *(x, z)
*t *t # "!u *x *z
GrPrA
*(p, t) *p *p "A # *x *x *(x, z)
(4) (5) (6)
where A is the aspect ratio h/¸, Pr is the Prandtl number l/i , where i is the di!usion Q Q coe$cient of salt, and Gr is the Grashof number de"ned by Eq. (1). For the purpose of this analysis, l, the laminar momentum exchange coe$cient, and i are inter-changeQ able with the mean (spatial and temporal) vertical turbulent eddy viscosity (e ) and X vertical turbulent di!usivity (N ) appropriate to the large scale physical system. We X also assume that Pr is 1 for the turbulent system. The appropriate boundary conditions are (de"ning the vertical coordinate such that z"1 at the surface) vertical: z"0 : z"1 :
p "0; t"0; t "0 X X p "0; t"0; t "B X XX
(7) (8)
horizontal: x"0 : p"0; t"0; t "0 V x"1 : p"1; t"0; t "0 V
(9) (10)
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Following Cormack et al. (1975), the wind induced surface stress has been nondimensionalised in (8) in the following manner: q "u "u " H H B" kGrl/h¸ gb*ShA
(11)
where the surface shear velocity scale, u , can be obtained from H o (12) u " C ; H " o where o is the density of air, o the reference density, C is a drag coe$cient " (1.4;10\, e.g., Friehe and Schmitt, 1976) and ; is the wind speed at 10 m. We can now seek a solution in terms of an asymptotic expansion in the small parameter A, i.e.,
p"p #Ap #Ap #2 t"t #At #At #2 u"u #Au #Au #2 (13) Substitution of the above asymptotic expansions into Eqs. (4)}(6), and neglecting all terms O(A) and above, we obtain the following equations at O(A), O(A) and O(A): O(A): p #t "0 V XXXX p "0 XX O(A):
(14)
p #t "0 V XXXX p "0 XX O(A):
(15)
Gr(t t !t t )"!p !2t !t , V XXX X XXV V VVXX XXXX GrPr(p t !p t )"p #p , (16) V X X V VV XX where the numerical subscripts denote the order (with respect to A) of the function and the alphanumeric subscripts denote partial di!erentiation with respect to the variable. The boundary conditions at each order of A are all homogeneous, with the exception of the surface stress condition, t "B, and the right hand boundary condition, p"1, XX which are applied only to the O(A) Eq. (14). The above equations may then be progressively solved, enabling u and to be determined to within two constants, which may be evaluated by asymptotically matching the solution for the core region #ow to the solution for each end region (e.g. Cormack et al., 1974), or by an integral matching technique (e.g. Bejan and Tien,
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1978). The "nal solution, to O(A) accuracy, in terms of the non-dimensional variables is (Cormack et al., 1975),
u"!
z 5z z B ! # # (3z!2z) 6 16 8 4
p"x!GrA
(17)
5z z 1 B z z z ! # !B ! ! ! 120 192 48 16 12 720 120
(18)
The important feature of the solution is that di!usive transport of salt, characterised by the leading term in (18), is dominant. The convective transport enters the solution only at O(A), and thus the resulting distribution is linear in x and only weakly strati"ed in z. The solutions are shown graphically in Fig. 8 for a number of surface stress scenarios using parameters (Gr"10, A"10\) approximate to those expected in the Herald Loop channel.
Fig. 8. Solution pro"les for u and p at x"0.5 (cavity mid-point) as given by Eqs. (17) and (18) for Gr"10, A"3;10\ and B"0 (a), 0.1 (b), !1/12 (c) and !0.3 (d). The vertical axis is z. Note that although a two-layer #ow structure is apparent under each condition shown, the degree of vertical strati"cation is almost negligible. In this case the vertical strati"cation is less than 0.1% of the end-to-end salinity di!erence.
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By comparison, O$cer (1976) found the velocity "eld was given by (using his notation) gjh h¹ u( " (8g!9g#1)# (1!4g#3g) 48o N 4o N X X
(19)
where g"z( /h, with the vertical coordinate axis de"ned such that g"1 at the bottom (i.e. in the opposite sense to that used in the derivation of Eqs. (17) and (18)), N is X a constant vertical eddy viscosity interchangeable with l, j is the constant longitudinal density gradient and ¹ is the surface wind stress. As O$cer (1976) assumed the gradient is linear, then *o b *S o j" " ¸ *x
(20)
and noting that ¹/o "u and g"(1!z) then H gb *S h hu u( " (8(1!z)!9(1!z)#1)# H (1!4(1!z)#3(1!z)) 48l¸ 4l
(21)
Substituting now for u in terms of B gives H Grl 1 Grl B u( " (!8z#15z!6z)# (3z!2z) ¸ 4 ¸ 48
(22)
which is identical to the dimensionalised form of Eq. (17). The corresponding solutions for the salinity pro"le can be shown to be identical in similar fashion. Thus the earlier models of Hansen and Rattray (1965) and O$cer (1976) are perfectly consistent with the shallow enclosure limit (i.e. A;1). The advantage of the method leading to Eqs. (17) and (18) is that the formal limits of the solution can be established (see below), and in particular, the conditions under which the assumption of a linear gradient is justi"ed.
4. Application 4.1. Salt transport The di!usive transport regime characterised by the the shallow cavity convection solution, Eqs. (17) and (18), is limited to the range of Gr and A that remains consistent with the buoyancy-viscous balance assumed in Eqs. (4)}(6). Imberger (1974) showed that the numerical and experimental heat transfer results converged to the asymptotic theory (in the case of the closed cavity) providing GrA(10. For the same case, Bejan and Tien (1978) suggested that the #ow was di!usion dominated if GrA&10. Ivey and Hamblin (1989) suggested GrA(10 for the free-slip surface case when using a non-linear equation of state.
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Following Ivey and Hamblin (1989), it is possible to obtain a validity limit based on the Nusselt number, Nu. The Nusselt number is de"ned as the ratio of the total saline transport to the purely di!usive transport, and is given (in terms of the nondimensional variables) by
*p !Grup dz (23) *x In order to remain within the di!usive regime, we require that the convective contribution to Nu be small compared to the dominating di!usive term (e.g. )10%). Using Eqs. (17) and (18), Nu for the general applied shear case can be evaluated to give Nu"
Nu"1#(1.31;10\#1.74;10\B#5.95;10\B)(GrA)
(24)
It is clear from the above expression for Nu that the convective contribution to the salt transport is strongly dependent on B, the non-dimensional wind-stress. The mean value of B in Shark Bay, based on 2 yrs of 3-hourly data from Denham and Carnarvon (Fig. 1), is !0.4. In general, the range is such that "B")1. Applying this range to Eq. (24), we "nd GrA(10 is the general requirement for the validity of this solution, in agreement with the results of Ivey and Hamblin (1989) and Bejan and Tien (1978). In the "eld, turbulent di!usivities (i.e. e ) are applicable, rather than laminar X transport coe$cients. The above result now allows us, in conjunction with the de"nition in Eq. (1), to determine a lower bound for the mean turbulent vertical eddy viscosity, e . As we have no in situ measurements of e , evaluation of a lower bound is X X useful in determining the applicability of the model. Using the previously de"ned values of A and *S we "nd that in order to satisfy the condition GrA(10, then we require that e '4;10\ ms\. X An estimate of the mean value of e can be made using a simple open-channel X expression (e.g. Fischer et al., 1979). Tidally generated currents in the channel have an amplitude of approximately 1 ms\, and a mean of approximately 0.3 ms \, giving e "0.067 hu , h&6 m, u &0.1u Ne &10\ ms\ X H H X Thus the mean tidal currents are capable of generating values of e in excess of those X required by the validity criteria. The strong wind "eld would also contribute to the turbulence, and clearly e of O(10\) ms\ is likely to prevail to maintain the X vertically well-mixed conditions as observed in Fig. 6, for example. 4.2. Wind stress ewects The magnitude and direction of the surface wind-stress must also be considered before applying the solution given by Eqs. (17) and (18). If the wind-drift velocity scale, u &10u (e.g. Baines and Knapp, 1965; Wu, 1975) exceeds the convective velocity H scale, Grl/¸ (from Eq. (2)), then the convective scaling applied to the governing equations may be inappropriate. This is equivalent to requiring that 10B(GrA for the scaling in Eq. (2) to be valid. The observed range of B, discussed in Section 4.1, satis"es this requirement. We can now combine this result with that of Section 4.1 to
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give the general conditions under which the solution, and hence the assumption of a linear gradient made by O$cer (1976) and Hansen and Rattray (1965), holds, i.e. 10"B"(GrA)10, A;1
(25)
A system satisfying these conditions is unlikely to exhibit signi"cant vertical salinity strati"cation. Vertical strati"cation is possible if the longitudinal advective #ux of salt (&10u h *S) is greater than the vertical di!usive #ux of salt (&e (*S/h)¸). ComparH X ing these scales and applying the conditions in Eq. (25) shows that for A(0.1, the vertical di!usive #ux is always greater than the longitudinal advective #ux. The direction of the advective (wind-driven) salt #ux is also of importance. It is easy to show that the solutions for the cases of no-slip and free-slip surfaces can be obtained from Eqs. (17) and (18) by choosing an appropriate value for B, i.e. !1/12 and 0 respectively. When B(!1/12, advection against the salinity gradient will lead to rapid vertical mixing due to overturning, and the salinity distribution given by Eq. (18) is invalid (e.g. Fig. 9). Under these conditionsthe system will become vertically homogeneous, retaining a mean velocity pro"le given by Eq. (17).
Fig. 9. Analytical solutions of (a) u and (b) p for a wind stress in the direction of decreasing salinity, where Gr"10, A"10\ and B"!0.3. The vertical axis is z. Note that the predicted salinity pro"le is unstable.
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5. Discussion The above analysis has shown that the applicability of the model is sensitive to the level of turbulence in the channel, characterised by e . The value of e required by the X X buoyancy-viscous force balance inherent in the solution set Eqs. (17) and (18), is consistent with the empirically predicted range for the Herald Loop channel. We have also shown that the directional component of the wind stress can be important, as the form of salinity pro"le Eq. (18) is no longer appropriate under certain conditions. In the present case however, the observations show that tidal mixing in the channel appears to be su$ciently strong to preclude any vertical strati"cation. Hence, we can conclude that the velocity pro"le given by Eq. (17) is applicable, allowing the salt discharge to be calculated using a depth independent salinity for all wind scenarios. The steady-state (over an annual time-scale) salinity within Hamelin Pool requires that the rate of salt production due to evaporation is balanced by the rate of saline discharge. As the discharge through Herald Loop is also di!usive, we can expect that the discharge across the shallow Faure Sill (width&25 km) as a whole is di!usive, and must be incorporated into the salt balance. The total di!usive discharge is thus given by Q +K *S(= A #= A )+3K *S = A (26) Q V A A Q Q V A A where = , A and = , A are the width and aspect ratio of the channel and sill, A A Q Q respectively. K is the longitudinal di!usion, or dispersion, coe$cient representative V of the net e!ect of tidal, wind driven and convective currents on the exchange process, and is assumed constant for both the sill and the channel. The rate of salt production due to the evaporative #ux is equal to EA S , where E is the evaporation rate (in &. ms\), A is the surface area of Hamelin Pool and S is the salinity. Balancing this &. with Eq. (26) yields the result EA S &. K" (27) V (1.1)3= *SA A A where Nu is assumed to be the maximum allowable value for the di!usive regime (1.1). Taking typical values of E"2 myr\, A "10 m, S "60, = "2000 m, *S" &. A 10 and A "3;10\ results in K &200 ms\, which probably has a range of A V $50% owing to the seasonal variation of the rate of evaporation. The magnitude of the di!usive discharge therefore remains at the same order over an annual period. The estimated value of K lies within the literature range quoted in Rutherford (1994) for V natural river and channel environments, and is also similar to those calculated by Nunes and Lennon (1986) for Spencer Gulf.
6. Conclusion In general, shallow systems in which rotation is unimportant and whose dynamics satisfy the expression 10"B"(GrA)10, are di!usive. These systems are
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characterised by linear longitudinal density gradients and little vertical strati"cation. The application of the model to the Herald Loop channel showed that the discharge of salt from Hamelin Pool into the remainder of Shark Bay is a steady, di!usive process (i.e., NuP1). Further, the annual average value of the transport coe$cient K was estimated to be 200 ms\. Thus it appears that the seasonal switching in V salinity structure east of Cape Peron is independent of the discharge from hypersaline regions at the southern end of the Bay, and is therefore driven by local processes.
Acknowledgements We are grateful to Terry Smith for his e!orts during the "eld program, and also to the crew of the =oomerangee. Thanks also to the anonymous reviewers, Prabhath de Silva, Kraig Winters and Roshanka Ranasinghe for providing invaluable comments on the manuscript, and to Eleanor Bruce for the bathymetric data. The "eld program was funded in part by the Gascoyne Development Commision and P. McGowan.
References Baines, W.D., Knapp, D.J., 1965. Wind driven water currents. Journal of the Hydraulics Division. Proceedings of the American Society of Civil Engineers 91 (HY2), 205}221. Bejan, A., 1984. Convection Heat Transfer. Wiley, New York, pp. 176}185 (Chapter 5). Bejan, A., Tien, C.L., 1978. Laminar natural convection heat transfer in a horizontal cavity with di!erent end temperatures. Journal of Heat Transfer 100, 641}647. Cormack, D.E., Leal, L.G., Imberger, J., 1974. Natural convection in a shallow cavity with di!erentially heated end walls. Part 1. Asymptotic theory. Journal of Fluid Mechanics 65, 209}229. Cormack, D.E., Stone, G.P., Leal, L.G., 1975. The e!ect of upper surface conditions on convection in a shallow cavity with di!erentially heated end-walls. International Journal of Heat and Mass Transfer 18, 635}648. Fischer, H.B., List, E.J., Koh, R.C.Y., Imberger, J., Brooks, N.H., 1979. In: Mixing in Inland and Coastal Waters. Academic Press, San Diego. Friehe, C.A., Schmitt, K.F., 1976. Parameterization of air-sea interface #uxes of sensible heat and moisture by the bulk aerodynamic formulas. Journal of Physical Oceanography 6, 801}809. Hansen, D.V., Rattray Jr., M., 1965. Gravitational circulation in straits and estuaries. Journal of Marine Research 23, 104}122. Imberger, J., 1974. Natural convection in a shallow cavity with di!erentially heated end walls. Part 3. Experimental results. Journal of Fluid Mechanics 65, 247}260. Ivey, G., Hamblin, P., 1989. Convection near the temperature of maximum density for high Rayleigh number, low aspect ratio, rectangular cavities. Journal of Heat Transfer 111, 100}105. Linden, P.F., Simpson, J.E., 1986. Gravity-driven #ows in a turbulent #uid. Journal of Fluid Mechanics 172, 481}497. Logan, B.W., Cebulski, D.E., 1970. Sedimentary Environments of Shark Bay, Western Australia. American Association of Petroleum Geologists.. Tulsa, Oklahoma, pp. 1}37. Nunes, R.A., Lennon, G.W., 1986. Physical property distributions and seasonal trends in Spencer Gulf, South Australia: an inverse estuary. Australian Journal of Marine and Freshwater Research 37, 39}53. Nunes, R.A., Lennon, G.W., 1987. Episodic strati"cation and gravity currents in a marine environment of modulated turbulence. Journal of Geophysical Research 92 (C5), 5465}5480. O$cer, C.B., 1976. Physical Oceanography of Estuaries (and Associated Coastal Waters). Wiley, New York, pp. 117}129, 191}201.
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Rutherford, J.C., 1994. In: River Mixing. Wiley, Chichester. Simpson, J.E., Linden, P.F., 1989. Frontogenesis in a #uid with horizontal density gradients. Journal of Fluid Mechanics 202, 1}16. Van De Kreeke, J., Zimmerman, J.T.F., 1990. Gravitational circulation in well- and partially-mixed estuaries. In: Le MeH hauteH , B., Hanes, D.M. (Eds.), Ocean Engineering Science, The Sea, Vol. 9. Wiley, New York, pp. 495}522. Wu, J., 1975. Wind-induced drift currents. Journal of Fluid Mechanics 68, 49}70.
Convectively driven exchange in a shallow coastal embayment Murray C. Burling, Gregory N. Ivey, Charitha B. Pattiaratchi* Department of Environmental Engineering, Centre for Water Research, The University of Western Australia, Nedlands, 6907, Western Australia, Australia Received 13 December 1996; received in revised form 29 September 1997; accepted 30 March 1999
Abstract Shark Bay is a large coastal embayment (length &250 km, width &100 km) located on the central west coast of Australia. The Bay is comprised of two major reaches, which are characterised by average depths of 10 m and salinities which increase with longitudinal distance away from the Bay entrance. Maximum salinities in the Bay exceed 60 (Practical Salinity Scale), and occur in Hamelin Pool at the southern end of Hopeless Reach, the eastern region of the Bay. Exchange between Hamelin Pool and Hopeless Reach is severely restricted by the presence of a sill, and occurs predominantly through a single 2 km wide, 6 m deep channel (Herald Loop). CTD measurements taken in Hopeless Reach show variable strati"cation: vertically well-mixed in summer and strongly strati"ed during the winter survey. A shallow cavity natural convection model was applied to the Herald Loop channel to determine the contribution of the saline discharge from Hamelin Pool to the observed variable strati"cation. The model shows that the discharge from Hamelin Pool is a di!usive process. Thus while the discharge is an important salinity source, it will not directly contribute to the variability observed in Hopeless Reach. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Natural convection; Saline exchange; Inverse estuary
1. Introduction Many shallow coastal water bodies exhibit longitudinal gradients in density. In estuaries, such gradients arise as a result of the mixing of seawater with low salinity
* Corresponding author. Tel.: #0061-8-9380-3179; fax: #0061-8-9380-1015. E-mail address: [email protected] (C.B. Pattiaratchi) 0278-4343/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 7 8 - 4 3 4 3 ( 9 9 ) 0 0 0 3 4 - 5
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tributary in#ows. Longitudinal density gradients may also be a feature of coastal regions with little freshwater in#ow, due to a dominance of evaporation over rainfall and run-o!. In this case, however, the density gradient is reversed (or negative), and maximum salinities occur in those regions with minimal direct oceanic in#uence. Examples of such `inverse estuariesa include the Spencer Gulf in South Australia (Nunes and Lennon, 1986) and Shark Bay in Western Australia (Logan and Cebulski, 1970). Often, the degree of wind and tidal mixing is su$cient to overcome lateral and vertical buoyancy inputs (e.g. stream#ow, solar heating), resulting in vertically wellmixed conditions. In the absence of strong mixing, a longitudinally strati"ed water body would naturally tend to adjust to the internal pressure gradient, and given time, become vertically strati"ed. Mixing retards this adjustment by increasing the internal friction to the baroclinic #ow; the stronger the mixing, the larger the time-scale for exchange between each end of the system. The characteristic circulation of the system is thus governed by a balance between the buoyancy forces due to the longitudinal gradient in density and the viscous forces due to turbulent mixing (e.g., O$cer, 1976). In the limit of very strong vertical mixing, the system becomes di!usive, and the characteristic longitudinal density gradient is linear. The circulation that results from an assumed buoyancy-viscous balance was "rst investigated by Hansen and Rattray (1965). In their study of estuarine #ow, they applied a similarity technique to solve the coupled equations for momentum and density conservation, assuming a priori a linear longitudinal salinity distribution. O$cer (1976) decoupled the momentum and density equations, solving instead for the velocity "eld generated by the special case of a constant longitudinal density gradient. The e!ect of the velocity "eld on the salinity distribution was then calculated via a simple advection-di!usion balance. Both solution sets included the surface wind stress contribution to the circulation, derived earlier by Baines and Knapp (1965), and can be shown to be equivalent (O$cer, 1976). A perturbation solution for the linearised equations of the di!usive regime of estuarine #ow was obtained by Van De Kreeke and Zimmerman (1990). However, in order to solve for the salinity distribution, the form of the longitudinal di!usivity must be known. Common to the arguments of Hansen and Rattray (1965) and O$cer (1976) is the assumption that the longitudinal density gradient is linear. Thus, it is not possible to de"ne the conditions under which the assumption of a di!usive transport regime (characterised by a linear longitudinal density gradient) is valid. The problem of buoyancy driven estuarine circulation has also motivated a number of studies into steady state laminar natural convection in cavities of low aspect ratio (e.g., Cormack et al., 1974; Imberger, 1974). While shallow coastal water bodies are turbulent systems, the large time-scale of the gravitational circulation allows the use of mean, or e!ective, eddy exchange coe$cients and the turbulent natural system (e.g. an estuary) may then be treated as analogous to a laminar system (Imberger, 1974). Buoyancy driven #ow in a shallow cavity is dependent upon three non-dimensional parameters (e.g., Cormack et al., 1974; Imberger, 1974). The geometry of the cavity is described by the aspect ratio, A"h/¸, which is typically very much smaller than 1.
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The buoyancy-viscous force balance is described by the Grashof number, gb *S h Gr" l
(1)
where h is the cavity height, ¸ is the cavity length, l is the kinematic viscosity, g is the acceleration due to gravity, b is the saline expansion coe$cient, *S is the end-to-end salinity di!erence. The "nal parameter is the Prandtl number, Pr"l/i, where i is the molecular di!usion coe$cient (for heat or salt). An asymptotic solution to the governing equations can be found by making use of the fact that the aspect ratio of the cavity, A, is very small. Cormack et al. (1974) "rst introduced this method to the problem of shallow cavity convection with di!erentially heated endwalls and no-slip surfaces, as a compliment to the experimental work of Imberger (1974). Bejan and Tien (1978), Bejan (1984) and Ivey and Hamblin (1989) have also investigated the problem using similar methods. The e!ect of a free surface was investigated by Cormack et al. (1975), including the cases of free-slip and an imposed shear stress. These studies have shown that for a restricted range of Gr and A, transport within the cavity is dominated by longitudinal di!usion. It will be shown in Section 4.1 that this range is de"ned by GrA(10. A bene"t of this approach is therefore the ability to de"ne the limits of the solution. An additional advantage is that only one assumption is required, that A;1. In this study we utilise the asymptotic model to investigate the saline driven exchange within Shark Bay, Western Australia. Much of Shark Bay is hypersaline: salinities in excess of 60 have been recorded. Discharge from the largest `salt reservoira in the Bay, Hamelin Pool, is severely restricted by the presence of a sill. In this region, estimates of the mean tidally generated vertical eddy viscosity suggest that the value of GrA will generally be below the critical value of 10. Thus, the transport regime can be characterised as di!usive, and an appropriate transport coe$cient can be found. It will also be shown that the analytical model is sensitive to both wind direction and magnitude, in particular when the prevailing wind is in the direction of decreasing salinity. Furthermore, it is shown that the asymptotic model solution is equivalent to the classical solutions of Hansen and Rattray (1965) and O$cer (1976). We begin with a description of the "eld site, followed by a review of the asymptotic theory.
2. Physical setting Shark Bay is a large coastal embayment (length&250 km, width&100 km, depth &10 m) located on the central west coast of Australia, in a region bound by latitudes 24330 and 26345 , and is semi-enclosed by three large islands (Fig. 1). The majority of the Bay is comprised of two reaches (Hopeless Reach to the east and Freycinet Reach to the west) aligned approximately north}south, separated by the Peron Peninsula. The climate is semi-arid, with most of the average annual precipitation of around 200 mm falling sporadically in winter and during summer cyclone events. For most of
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Fig. 1. Shark Bay, Western Australia. Bathymetric contours of 4, 10, 20 m are marked. Note particularly the Faure Sill (1 m deep region shaded) and the Herald Loop channel in the eastern reach of the Bay.
the year, the region is in#uenced by strong southerly winds with a seasonally variant mean magnitude (e.g. Fig. 2). Winds during the spring and summer periods are generally the strongest, with mean speeds of 8 ms\ at both Denham and Carnarvon. The mean wind direction during summer is approximately southerly (1903) with a standard deviation of 503, demonstrating that winds are predominantly con"ned to a small arc ranging from southwesterly to southeasterly. Wind speed reduces during the winter and autumn periods, although the average remains approximately 5 ms\. Along with a reduction in wind speed during these months, the wind direction moves around toward the south-east (150}1603) and is accompanied by an increased variation (standard deviation&803). The maximum variance occurs during June, July and August, owing to the periodic passage of winter weather systems.
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Fig. 2. Three-hourly wind vectors at Denham, 1994: (a) summer and (b) winter. Positive vectors represent a southerly component, i.e. toward the north.
The combination of strong, persistent winds and high temperatures generate an annual evaporation (&2 m) which is ten times greater than the annual rainfall (&200 mm). Consequently, large sections of the Bay are hypersaline (salinity'50), and both Freycinet Reach and Hopeless Reach (Fig. 1) exhibit a salinity distribution analogous to an inverse estuary (Logan and Cebulski 1970). During a 13 yr study period, Logan and Cebulski (1970) observed little seasonal or inter-annual variation in the salinity structure of the entire Bay, leading them to conclude that the system is at steady state. They also noted that most of the Bay was vertically well-mixed. The single anomalous data set (winter, 1965) collected by Logan and Cebulski (1970) revealed that, although the salinity throughout the Bay was reduced by approximately 10%, the salinity gradients were preserved. Recent intensive "eld experiments in February (summer) and August (winter) of 1995 have shown that the salinity structure in the Bay, particularly in the eastern reach, can in fact be seasonally variant. Figs. 4 and 5, for example, show a salinity transect taken in an east}west line cutting Cape Peron (Fig. 3), for the summer and winter surveys, respectively. Although the range of salinities is unchanged, it is immediately apparent that the strong two-layer structure observed to the east of Cape Peron during the winter survey (Fig. 5) represents a dramatically di!erent #ow regime
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Fig. 3. CTD Transects, Shark Bay, February and August 1995. (A) Cape Peron and (B) Herald Loop. Average distance between stations is 2 nautical miles.
to that observed in summer (Fig. 4). Variability in the discharge from the hypersaline regions at the southern end of the Bay may contribute to the variation apparent in Figs. 4 and 5. Alternatively, a reduction in the energy available for turbulent vertical mixing may lead to an inertia-buoyancy balance and the formation of clearly de"ned gravity currents (Linden and Simpson, 1986; Simpson and Linden, 1989). A similar phenomenon has been observed in Spencer Gulf (Nunes and Lennon, 1987). Hamelin Pool, situated at the southern end of the Hopeless Reach, is the major hypersaline region within the Bay. Exchange between Hamelin Pool and the remainder of the Bay is restricted by the presence of the Faure Sill (Fig. 1). The average depth in the sill region is 1 m. Most of the exchange #ow is con"ned to a single narrow
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Fig. 4. Salinity contours at Cape Peron, February (summer) 1995. Transect location as per Fig. 3. Note that to the east of Cape Peron (proj. dist."18,000 m) the water-column is well-mixed.
channel, Herald Loop, which is approximately 2 km wide, 20 km long and has an average depth of 6 m. Salinity pro"les were taken through Herald Loop during winter 1995, with the resultant longitudinal transect of the vertical salinity structure in the channel shown in Fig. 6. This transect was taken during a #ooding tidal phase in a north}south direction, beginning in the lower region of Hopeless Reach and extending into Hamelin Pool (Fig. 3), where salinities approaching 60 are observed. The corresponding horizontal density di!erence over this region is 8 kgm\. Despite the strong density gradient, the system is well-mixed vertically and has a linear gradient in salinity (&10 over 20 km), both characteristic of a di!usively dominated system. The data collected by Logan and Cebulski (1970) is supportive, showing the persistence of this salinity gradient over the entire sill region, including the Herald Loop channel.
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Fig. 5. Salinity contours at Cape Peron, August (winter) 1995. Transect location as per Fig. 3. Note the two-layer strati"cation now visible to the east of Cape Peron (proj. dist."18,000 m).
3. Model formulation Our purpose here is to model the transport through the Herald Loop channel in order to assess the contribution of this saline #ux to the dynamics of Shark Bay as a whole. The channel between Hamelin Pool and the remainder of Shark Bay can be represented by a two-dimensional cavity (Fig. 7). The end regions of the cavity are dynamically passive in terms of any contribution to the #ow "eld in the interior of the cavity (e.g., Cormack et al., 1974) allowing the `endwallsa to be chosen as the ends of the Herald Loop channel. This simply means that we have chosen the relevant longitudinal length scale to be equal to the channel length. Each boundary represents a transition between the channel waters and a large reservoir, whose mean properties are assumed to be "xed and independent of mixing in the channel. Consequently the boundary conditions for the channel circulation remain independent of the internal #ow. The bottom is assumed to be a no-slip boundary and the surface is subject to an applied shear stress. We will neglect tidal #ow in the channel, except for
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Fig. 6. Salinity contours through Herald Loop, #ooding tide, August (winter) 1995. The transect stations are shown in Fig. 3. Hamelin Pool is located to the right. The Herald Loop channel is located approximately in the region of linear longitudinal gradient (proj. dist. "3000}23,000 m). The salinity gradient is approximately 0.5 km\.
the contribution to the mean turbulent exchange coe$cients, assuming that tidal motion simply advects the entire `cavitya back and forth (note that the tidal excursion length is approximately 5 km). Further, the system is assumed to be at steady state. The governing equations may be non-dimensionalised using the following variables (e.g. Cormack et al., 1974; Bejan, 1984; Ivey and Hamblin, 1989) x( z( ¸u( ¸w( S!S x" , z" , u" , w" , p" ¸ h Grl GrlA S !S
(2)
where x( and z( are the longitudinal distance (increasing in the direction of increasing salinity) and depth and u( and w( are the velocities in the x( and z( directions, respectively. The length, depth and width scales of the channel are ¸"20 km, h"6 m and ="2 km and the aspect ratio is A"3;10\. The salinity, S, is scaled with the `endwalla salinities, S &50 (Hopeless Reach) and S &60 (Hamelin Pool), i.e. *S"10.
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Fig. 7. Schematic diagram of the Herald Loop channel, where h"6 m, ¸"20 km, S "50 and S "60. The coordinate system is as shown.
The introduction of the stream function t, de"ned by *t *t u" , w"! *z *x
(3)
allows the non-dimensional steady equations of motion for the central core region of the cavity (far removed from the e!ect of the endwalls) to be written as GrA A
*u *u *p *(u, t) "A # ! *x *z *x *(x, z)
*t *t # "!u *x *z
GrPrA
*(p, t) *p *p "A # *x *x *(x, z)
(4) (5) (6)
where A is the aspect ratio h/¸, Pr is the Prandtl number l/i , where i is the di!usion Q Q coe$cient of salt, and Gr is the Grashof number de"ned by Eq. (1). For the purpose of this analysis, l, the laminar momentum exchange coe$cient, and i are inter-changeQ able with the mean (spatial and temporal) vertical turbulent eddy viscosity (e ) and X vertical turbulent di!usivity (N ) appropriate to the large scale physical system. We X also assume that Pr is 1 for the turbulent system. The appropriate boundary conditions are (de"ning the vertical coordinate such that z"1 at the surface) vertical: z"0 : z"1 :
p "0; t"0; t "0 X X p "0; t"0; t "B X XX
(7) (8)
horizontal: x"0 : p"0; t"0; t "0 V x"1 : p"1; t"0; t "0 V
(9) (10)
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Following Cormack et al. (1975), the wind induced surface stress has been nondimensionalised in (8) in the following manner: q "u "u " H H B" kGrl/h¸ gb*ShA
(11)
where the surface shear velocity scale, u , can be obtained from H o (12) u " C ; H " o where o is the density of air, o the reference density, C is a drag coe$cient " (1.4;10\, e.g., Friehe and Schmitt, 1976) and ; is the wind speed at 10 m. We can now seek a solution in terms of an asymptotic expansion in the small parameter A, i.e.,
p"p #Ap #Ap #2 t"t #At #At #2 u"u #Au #Au #2 (13) Substitution of the above asymptotic expansions into Eqs. (4)}(6), and neglecting all terms O(A) and above, we obtain the following equations at O(A), O(A) and O(A): O(A): p #t "0 V XXXX p "0 XX O(A):
(14)
p #t "0 V XXXX p "0 XX O(A):
(15)
Gr(t t !t t )"!p !2t !t , V XXX X XXV V VVXX XXXX GrPr(p t !p t )"p #p , (16) V X X V VV XX where the numerical subscripts denote the order (with respect to A) of the function and the alphanumeric subscripts denote partial di!erentiation with respect to the variable. The boundary conditions at each order of A are all homogeneous, with the exception of the surface stress condition, t "B, and the right hand boundary condition, p"1, XX which are applied only to the O(A) Eq. (14). The above equations may then be progressively solved, enabling u and to be determined to within two constants, which may be evaluated by asymptotically matching the solution for the core region #ow to the solution for each end region (e.g. Cormack et al., 1974), or by an integral matching technique (e.g. Bejan and Tien,
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1978). The "nal solution, to O(A) accuracy, in terms of the non-dimensional variables is (Cormack et al., 1975),
u"!
z 5z z B ! # # (3z!2z) 6 16 8 4
p"x!GrA
(17)
5z z 1 B z z z ! # !B ! ! ! 120 192 48 16 12 720 120
(18)
The important feature of the solution is that di!usive transport of salt, characterised by the leading term in (18), is dominant. The convective transport enters the solution only at O(A), and thus the resulting distribution is linear in x and only weakly strati"ed in z. The solutions are shown graphically in Fig. 8 for a number of surface stress scenarios using parameters (Gr"10, A"10\) approximate to those expected in the Herald Loop channel.
Fig. 8. Solution pro"les for u and p at x"0.5 (cavity mid-point) as given by Eqs. (17) and (18) for Gr"10, A"3;10\ and B"0 (a), 0.1 (b), !1/12 (c) and !0.3 (d). The vertical axis is z. Note that although a two-layer #ow structure is apparent under each condition shown, the degree of vertical strati"cation is almost negligible. In this case the vertical strati"cation is less than 0.1% of the end-to-end salinity di!erence.
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By comparison, O$cer (1976) found the velocity "eld was given by (using his notation) gjh h¹ u( " (8g!9g#1)# (1!4g#3g) 48o N 4o N X X
(19)
where g"z( /h, with the vertical coordinate axis de"ned such that g"1 at the bottom (i.e. in the opposite sense to that used in the derivation of Eqs. (17) and (18)), N is X a constant vertical eddy viscosity interchangeable with l, j is the constant longitudinal density gradient and ¹ is the surface wind stress. As O$cer (1976) assumed the gradient is linear, then *o b *S o j" " ¸ *x
(20)
and noting that ¹/o "u and g"(1!z) then H gb *S h hu u( " (8(1!z)!9(1!z)#1)# H (1!4(1!z)#3(1!z)) 48l¸ 4l
(21)
Substituting now for u in terms of B gives H Grl 1 Grl B u( " (!8z#15z!6z)# (3z!2z) ¸ 4 ¸ 48
(22)
which is identical to the dimensionalised form of Eq. (17). The corresponding solutions for the salinity pro"le can be shown to be identical in similar fashion. Thus the earlier models of Hansen and Rattray (1965) and O$cer (1976) are perfectly consistent with the shallow enclosure limit (i.e. A;1). The advantage of the method leading to Eqs. (17) and (18) is that the formal limits of the solution can be established (see below), and in particular, the conditions under which the assumption of a linear gradient is justi"ed.
4. Application 4.1. Salt transport The di!usive transport regime characterised by the the shallow cavity convection solution, Eqs. (17) and (18), is limited to the range of Gr and A that remains consistent with the buoyancy-viscous balance assumed in Eqs. (4)}(6). Imberger (1974) showed that the numerical and experimental heat transfer results converged to the asymptotic theory (in the case of the closed cavity) providing GrA(10. For the same case, Bejan and Tien (1978) suggested that the #ow was di!usion dominated if GrA&10. Ivey and Hamblin (1989) suggested GrA(10 for the free-slip surface case when using a non-linear equation of state.
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Following Ivey and Hamblin (1989), it is possible to obtain a validity limit based on the Nusselt number, Nu. The Nusselt number is de"ned as the ratio of the total saline transport to the purely di!usive transport, and is given (in terms of the nondimensional variables) by
*p !Grup dz (23) *x In order to remain within the di!usive regime, we require that the convective contribution to Nu be small compared to the dominating di!usive term (e.g. )10%). Using Eqs. (17) and (18), Nu for the general applied shear case can be evaluated to give Nu"
Nu"1#(1.31;10\#1.74;10\B#5.95;10\B)(GrA)
(24)
It is clear from the above expression for Nu that the convective contribution to the salt transport is strongly dependent on B, the non-dimensional wind-stress. The mean value of B in Shark Bay, based on 2 yrs of 3-hourly data from Denham and Carnarvon (Fig. 1), is !0.4. In general, the range is such that "B")1. Applying this range to Eq. (24), we "nd GrA(10 is the general requirement for the validity of this solution, in agreement with the results of Ivey and Hamblin (1989) and Bejan and Tien (1978). In the "eld, turbulent di!usivities (i.e. e ) are applicable, rather than laminar X transport coe$cients. The above result now allows us, in conjunction with the de"nition in Eq. (1), to determine a lower bound for the mean turbulent vertical eddy viscosity, e . As we have no in situ measurements of e , evaluation of a lower bound is X X useful in determining the applicability of the model. Using the previously de"ned values of A and *S we "nd that in order to satisfy the condition GrA(10, then we require that e '4;10\ ms\. X An estimate of the mean value of e can be made using a simple open-channel X expression (e.g. Fischer et al., 1979). Tidally generated currents in the channel have an amplitude of approximately 1 ms\, and a mean of approximately 0.3 ms \, giving e "0.067 hu , h&6 m, u &0.1u Ne &10\ ms\ X H H X Thus the mean tidal currents are capable of generating values of e in excess of those X required by the validity criteria. The strong wind "eld would also contribute to the turbulence, and clearly e of O(10\) ms\ is likely to prevail to maintain the X vertically well-mixed conditions as observed in Fig. 6, for example. 4.2. Wind stress ewects The magnitude and direction of the surface wind-stress must also be considered before applying the solution given by Eqs. (17) and (18). If the wind-drift velocity scale, u &10u (e.g. Baines and Knapp, 1965; Wu, 1975) exceeds the convective velocity H scale, Grl/¸ (from Eq. (2)), then the convective scaling applied to the governing equations may be inappropriate. This is equivalent to requiring that 10B(GrA for the scaling in Eq. (2) to be valid. The observed range of B, discussed in Section 4.1, satis"es this requirement. We can now combine this result with that of Section 4.1 to
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give the general conditions under which the solution, and hence the assumption of a linear gradient made by O$cer (1976) and Hansen and Rattray (1965), holds, i.e. 10"B"(GrA)10, A;1
(25)
A system satisfying these conditions is unlikely to exhibit signi"cant vertical salinity strati"cation. Vertical strati"cation is possible if the longitudinal advective #ux of salt (&10u h *S) is greater than the vertical di!usive #ux of salt (&e (*S/h)¸). ComparH X ing these scales and applying the conditions in Eq. (25) shows that for A(0.1, the vertical di!usive #ux is always greater than the longitudinal advective #ux. The direction of the advective (wind-driven) salt #ux is also of importance. It is easy to show that the solutions for the cases of no-slip and free-slip surfaces can be obtained from Eqs. (17) and (18) by choosing an appropriate value for B, i.e. !1/12 and 0 respectively. When B(!1/12, advection against the salinity gradient will lead to rapid vertical mixing due to overturning, and the salinity distribution given by Eq. (18) is invalid (e.g. Fig. 9). Under these conditionsthe system will become vertically homogeneous, retaining a mean velocity pro"le given by Eq. (17).
Fig. 9. Analytical solutions of (a) u and (b) p for a wind stress in the direction of decreasing salinity, where Gr"10, A"10\ and B"!0.3. The vertical axis is z. Note that the predicted salinity pro"le is unstable.
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5. Discussion The above analysis has shown that the applicability of the model is sensitive to the level of turbulence in the channel, characterised by e . The value of e required by the X X buoyancy-viscous force balance inherent in the solution set Eqs. (17) and (18), is consistent with the empirically predicted range for the Herald Loop channel. We have also shown that the directional component of the wind stress can be important, as the form of salinity pro"le Eq. (18) is no longer appropriate under certain conditions. In the present case however, the observations show that tidal mixing in the channel appears to be su$ciently strong to preclude any vertical strati"cation. Hence, we can conclude that the velocity pro"le given by Eq. (17) is applicable, allowing the salt discharge to be calculated using a depth independent salinity for all wind scenarios. The steady-state (over an annual time-scale) salinity within Hamelin Pool requires that the rate of salt production due to evaporation is balanced by the rate of saline discharge. As the discharge through Herald Loop is also di!usive, we can expect that the discharge across the shallow Faure Sill (width&25 km) as a whole is di!usive, and must be incorporated into the salt balance. The total di!usive discharge is thus given by Q +K *S(= A #= A )+3K *S = A (26) Q V A A Q Q V A A where = , A and = , A are the width and aspect ratio of the channel and sill, A A Q Q respectively. K is the longitudinal di!usion, or dispersion, coe$cient representative V of the net e!ect of tidal, wind driven and convective currents on the exchange process, and is assumed constant for both the sill and the channel. The rate of salt production due to the evaporative #ux is equal to EA S , where E is the evaporation rate (in &. ms\), A is the surface area of Hamelin Pool and S is the salinity. Balancing this &. with Eq. (26) yields the result EA S &. K" (27) V (1.1)3= *SA A A where Nu is assumed to be the maximum allowable value for the di!usive regime (1.1). Taking typical values of E"2 myr\, A "10 m, S "60, = "2000 m, *S" &. A 10 and A "3;10\ results in K &200 ms\, which probably has a range of A V $50% owing to the seasonal variation of the rate of evaporation. The magnitude of the di!usive discharge therefore remains at the same order over an annual period. The estimated value of K lies within the literature range quoted in Rutherford (1994) for V natural river and channel environments, and is also similar to those calculated by Nunes and Lennon (1986) for Spencer Gulf.
6. Conclusion In general, shallow systems in which rotation is unimportant and whose dynamics satisfy the expression 10"B"(GrA)10, are di!usive. These systems are
M.C. Burling et al. / Continental Shelf Research 19 (1999) 1599}1616
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characterised by linear longitudinal density gradients and little vertical strati"cation. The application of the model to the Herald Loop channel showed that the discharge of salt from Hamelin Pool into the remainder of Shark Bay is a steady, di!usive process (i.e., NuP1). Further, the annual average value of the transport coe$cient K was estimated to be 200 ms\. Thus it appears that the seasonal switching in V salinity structure east of Cape Peron is independent of the discharge from hypersaline regions at the southern end of the Bay, and is therefore driven by local processes.
Acknowledgements We are grateful to Terry Smith for his e!orts during the "eld program, and also to the crew of the =oomerangee. Thanks also to the anonymous reviewers, Prabhath de Silva, Kraig Winters and Roshanka Ranasinghe for providing invaluable comments on the manuscript, and to Eleanor Bruce for the bathymetric data. The "eld program was funded in part by the Gascoyne Development Commision and P. McGowan.
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Rutherford, J.C., 1994. In: River Mixing. Wiley, Chichester. Simpson, J.E., Linden, P.F., 1989. Frontogenesis in a #uid with horizontal density gradients. Journal of Fluid Mechanics 202, 1}16. Van De Kreeke, J., Zimmerman, J.T.F., 1990. Gravitational circulation in well- and partially-mixed estuaries. In: Le MeH hauteH , B., Hanes, D.M. (Eds.), Ocean Engineering Science, The Sea, Vol. 9. Wiley, New York, pp. 495}522. Wu, J., 1975. Wind-induced drift currents. Journal of Fluid Mechanics 68, 49}70.