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1566 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 43 Estuary Classification Revisited ANIRBAN GUHA AND GREGORY A. LAWRENCE Civil Engineering Department, and Institute of Applied Mathematics, The University of British Columbia, Vancouver, British Columbia, Canada (Manuscript received 19 July 2012, in final form 31 March 2013) ABSTRACT Studies over a period of several decades have resulted in a relatively simple set of equations describing the tidally and width-averaged balances of momentum and salt in a rectangular estuary. The authors rewrite these equations in a fully nondimensional form that yields two nondimensional variables: (i) the estuarine Froude number and (ii) a modified tidal Froude number. The latter is the product of the tidal Froude number and the square root of the estuarine aspect ratio. These two variables are used to define a prognostic estuary classification scheme, which compares favorably with published estuarine data. 1. Introduction Since the introduction of the stratification–circulation diagram by Hansen and Rattray (1966), numerous estuarine classification schemes have been proposed. The reader might ask—why revisit this topic? Our motivation for pursuing a new classification scheme stems from notable recent advances in estuarine physics, many of which are reviewed in MacCready and Geyer (2010). These advances led us to hypothesize that there might be a simple means to determine the conditions under which a sufficiently well behaved estuary will be well mixed, partially mixed, or highly stratified. We start by outlining the classical tidally averaged model as presented by MacCready and Geyer (2010). We then rewrite the equations of this model in nondimensional form. Using this new set of equations we develop our classification scheme, and then compare its predictions with field observations. exchange flow, and the tides determines the estuarine velocity and salinity structure. We consider an idealized rectangular estuary of depth H and width B. The origin of the coordinate system is at the free surface at the mouth of the estuary with the horizontal (i.e., x) axis pointing seawards and the vertical (i.e., z) axis pointing upward. Therefore, both the horizontal and vertical distances within the estuary are negative quantities. To obtain the width- and tidally averaged horizontal velocity u, and salinity s distribution in the estuary, these quantities are first decomposed into depthaveraged (overbar) and depth-varying (prime) components: u 5 u(x, t) 1 u0 (x, z, t) and s 5 s(x, t) 1 s0 (x, z, t). The quantity u 5 QR /A is the cross-section-averaged river velocity, where QR is the mean river flow rate, and A 5 BH. The solution for both partially and well-mixed estuaries was given by Hansen and Rattray (1965) [for a recent review, see MacCready and Geyer (2010)]: u 5 u 1 u0 5 uP1 1 uE P2 2. Classical tidally averaged model The physics of estuarine circulation is governed by the competing influences of river and oceanic flows. While the former adds freshwater, the latter adds denser saltwater, which moves landward because of the combined effect of tides and gravitational circulation (or exchange flow). The complicated balance between the river, the Corresponding author address: Anirban Guha, Civil Engineering Department, The University of British Columbia, 2002–6250 Applied Science Lane, Vancouver, BC V6T 1Z4, Canada. E-mail: [email protected] DOI: 10.1175/JPO-D-12-0129.1 Ó 2013 American Meteorological Society s 5 s 1 s0 5 s 1 where and H2 s (uP3 1 uE P4 ) , KS x (1) (2) 3 3 P1 5 2 j2 , 2 2 P2 5 1 2 9j2 2 8j3 , 7 1 1 1 j2 2 j 4 , and P3 5 2 120 4 8 1 1 2 3 4 2 5 P4 5 2 1 j 2 j 2 j . 12 2 4 5 (3) AUGUST 2013 In Eq. (3), j 5 z/H 2 [21, 0] is the normalized vertical coordinate. The subscript x implies ›/›x, where x is dimensional. The vertical eddy diffusivity is KS . For exchange-dominated estuaries, an important parameter is the exchange velocity scale: uE 5 c2 H 2 Sx /(48KM ) . ð d S dx 5 2u0 S0 1 KH S x 2uS , dt |ﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ} |{z} |ﬄﬄﬄ{zﬄﬄﬄ} |ﬄ{zﬄ} accumulation exchange tidal where E2 E1 T LE 5 0:019uH 2 /KS , 1 3 The different terms in Eq. (6) are as follows: R is the river term; T is the tidal term; and E1, E2, and E3 are the different components of the exchange term. Hansen and Rattray (1965) presented Eq. (6) in a slightly different form, and MacCready (2004, 2007) introduced the length scales in Eq. (7). The length scales in Eq. (7) depend upon the mixing coefficients: KH , KS , and KM . Making use of an extensive study of Willapa Bay, Banas et al. (2004) proposed the following: (9) E3 (10) T Chatwin (1976) further reduced Eq. (10) to two simple cases with analytical solutions: the exchange-dominated case (T / 0) and the tidally dominated case (E3 / 0). While these approximations have been widely used, there does not appear to have been any serious attempt to determine the conditions under which they are applicable. 3. Nondimensional tidally averaged model In this section, we rewrite the governing Eqs. (1), (2), and (6) in nondimensional form in anticipation of: (i) revealing the important nondimensional parameters governing the problem and (ii) facilitating comparison of the relative magnitude of each of the terms in Eq. (6). Defining X 5 x/LE3 , Eq. (6) can be rewritten as and (7) KS 5 KM /Sc, S 5 (LE S x )3 1 LH S x . 3 |{z} |ﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ ﬄ} |ﬄﬄ{zﬄﬄ} 2 2 1/3 LE 5 0:024(c/u)1/3 cH 2 /(KS KM ) . and where a0 5 0.028, CD 5 0.0026, and Sc 5 2.2 is a Schmidt number. We will use Eqs. (8) and (9) in the development of a nondimensional set of equations. While the governing Eqs. (1), (2), and (6) are elegant representations of the problem of estuarine circulation, they are sufficiently complicated that simplifications have been sought after. Numerous investigators, including Hansen and Rattray (1965), Chatwin (1976), Monismith et al. (2002), MacCready (2004), and MacCready and Geyer (2010) have assumed u uE , which yields R LH 5 KH /u, LE 5 0:031cH 2 /(KS KM )1/2 , KM 5 a0 CD uT H river S 5 (LE S x)3 1 (LE S x)2 1 LE S x 1 LH S x , (6) 3 2 1 |{z} |ﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ ﬄ} |ﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ ﬄ} |ﬄﬄﬄ{zﬄﬄ ﬄ} |ﬄﬄ{zﬄﬄ} (8) where a1 5 0.035, and uT is the amplitude of the depthaveraged tidal velocity. Based on field studies and modeling of the Hudson River estuary, Ralston et al. (2008) obtained (5) where KH is the horizontal diffusivity. This equation physically implies that the temporal salt accumulation in an estuary is due to the competition between salt addition and removal processes. While exchange (note that u0 S0 is negative) and tidal processes add salt, river inflow removes it. At steady state, Eq. (5) can be rewritten as E3 KH 5 a1 uT B, (4) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Here, c 5 gbsocn H is twice the speed of the fastest internal wave that can be supported in an estuary (MacCready and Geyer 2010). The vertical eddy viscosity is KM and b ﬃ 7.7 3 10 24 psu21 . The nondimensional salinity is defined as S 5 s/socn, where socn is the ocean salinity. Equations (1) and (2) were derived under the assumption that the density field is governed by the linear equation of state: r 5 r0(1 1 bs), where r0 is the density of freshwater. The details of the derivation are well documented in MacCready (1999, 2004). The salt balance is given by R 1567 GUHA AND LAWRENCE S 3 5SX where 1 !2 LE 2 LE 3 2 SX 1 ! LE 1 LE 3 ! LH SX 1 SX, LE (11) 3 !2 0:031 2 1/3 2/3 Sc FR 5 2:17FR2/3 , 0:024 LE 3 LE 0:019 1 5 Sc2/3 FR4/3 5 1:34FR4/3 , and 0:024 LE 3 LH a 0 a 1 CD Sc21/3 (B/H)FT2 FR22/3 . 5 LE 0:024 LE 5 2 3 (12) 1568 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 43 The velocity u and uT have been nondimensionalized by c to obtain the densimetric estuarine Froude number FR 5 u/c and the tidal Froude number FT 5 uT/c. Substituting Eq. (12) into (11) yields 3 2 f 2 F 22/3 S , S 5 S X 1 C1 FR2/3 S X 1 C2 FR4/3 S X 1 C3 F R T X |{z} |{z} |ﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} R E3 E2 E1 T (13) 3ﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1025 where C1 5 2.17, C2 5 1.34, C3 5 8.16 p ﬃ , and the f modified tidal Froude number FT 5 FT B/H . Like the salt balance equation, the momentum and salinity equations [Eqs. (1) and (2)] can also be expressed in nondimensional form as follows: U 5 C4 FR1/3 S X P2 1 FR P1 and 2 S 5 S 1 C5 FR2/3 S X P4 1 C6 FR4/3 S X P3 . (14) (15) The constant C4 5 0.667, C5 5 47.0, and C6 5 70.5. In Eq. (14), the quantity U 5 u/c is the nondimensional horizontal velocity (not to be confused with FR, which is u/c). Equations (13)–(15) are the nondimensional governing equations for our idealized estuary. Equation (13) poses a nonlinear initial value problem, which can only be solved numerically. For that, the conditions at the estuary mouth have to be determined. One such condition is S(0, 21) 5 1; meaning that the salinity at the bed of the estuary at its mouth has to be the same as the ocean salinity. Substituting Eq. (15) into (13) and making use of this condition, we obtain (S X j0 )3 1 C7 FR2/3 (S X j0 )2 f2 F 22/3 )S j 5 1, 1 (C8 FR4/3 1 C3 F T R X 0 (16) where C7 5 5.31 and C8 5 6.04. Equation (16) is actually the nondimensional version of Eq. (19) of MacCready (2004). Being a cubic equation, it can be solved analytically to evaluate the salinity gradient at the estuary mouth S X j0 . Additionally, Eq. (16) indicates that S X j0 is fT . The variation of S X j with only a function of FR and F 0 these two Froude numbers is depicted in Fig. 1. The figure shows that 0 , S X j0 , 1 over the entire parameter space. 4. Estuary classification Our goal is to develop a simple classification scheme that distinguishes between well-mixed, partially mixed, and highly stratified estuaries. A relevant parameter for classifying estuaries is the nondimensional salinity FIG. 1. Variation of the horizontal gradient of depth-averaged salinity at the estuary mouth (i.e., S X j0 ) with the estuary control fT . The solid lines represent isocontours of S X j . variables—FR and F 0 stratification at the estuary mouth F0 . It is defined as follows: F0 5 1 2 S(0, 0). (17) This parameter ranges between 0 and 1. While the lower limit implies a very well-mixed estuary, the upper limit indicates the transition to salt wedge. Substituting Eq. (15) into (17) yields: F0 5 C9 FR2/3 (S X j0 )2 1 C10 FR4/3 S X j0 , (18) fT are known, where C9 5 7.06 and C10 5 8.82. If FR and F then S X j0 can be directly obtained by solving Eq. (16). Consequently, F0 can be evaluated from Eq. (18), yielding Fig. 2. We follow Hansen and Rattray (1966) and use the condition F0 5 0.1 to define the transition between wellmixed and partially mixed estuaries. To distinguish between partially mixed and highly stratified estuaries, we use the condition F0 5 1.0 which corresponds to fresh surface water extending to the mouth of the estuary. Our classification scheme is obtained by plotting these transifT 5 0, the transition betional criteria on Fig. 2. When F tween well-mixed and partially-mixed estuaries is predicted to occur at FR 5 0.0018, and from partiallymixed to highly stratified at FR 5 0.113. The value of FR fT increases, the infor both transitions increases as F crease being more rapid for the transition from partially mixed to highly stratified estuaries. These results are in qualitative agreement with Fig. 2.7 of Geyer (2010). 5. Discussion Together, Eqs. (16) and (18) provide new insight into estuarine physics. Apart from broadly classifying AUGUST 2013 1569 GUHA AND LAWRENCE parameters (MacCready 2007). For example, many estuaries are too sluggish to respond to fortnightly variability in tidal forcing. To test the applicability of our classification scheme we made use of the field data presented in Prandle (1985). Using these data we have computed FR , FT , B/H, and fT directly, and F0 from Eqs. (16) and (18); see Table 1. F We have compared the computed values of F0 with the measured values in Fig. 3. The comparison is good confT can be desidering the accuracy to which FR and F termined from field data. fT are It is interesting to note that if both FR and F small then Eq. (16) reduces to S X j0 5 1 and Eq. (18) reduces to FIG. 2. Estuary classification diagram with lines representing isocontours of F0 . The three regions are three types of estuaries: (a) light gray is well mixed, (b) white is partially mixed, and (c) dark gray is highly stratified or salt wedge. For data and expansion of abbreviations, see Table 1. estuaries into three categories, namely highly stratified, partially mixed, and well mixed, these equations reveal that under the given assumptions just two parameters, fT , are needed to determine the stratification FR and F at the estuary mouth pﬃﬃﬃﬃﬃﬃﬃﬃﬃF ﬃ 0 . The new nondimensional pafT 5 FT B/H reveals that the ‘‘tidal effect’’ is rameter F not simply represented by the tidal Froude number FT, but the latter combined with the square root of the estuarine aspect ratio B/H. Moreover the equation set fT . If the estuarine condition predicts F0 , given FR and F changes (for example, the river flow or the estuary fT will change accorddepth), the parameters FR and F ingly. These newly obtained Froude numbers will produce a new F0 , which reflects the response of estuarine circulation and mixing to these changes. However, we should be mindful of the fact that estuaries can be ‘‘sluggish’’ in their response to changes in the governing F0 ’ 7FR2/3 . (19) The same result was obtained by MacCready and Geyer (2010) by combining Knudsen’s relations (Knudsen 1900) with Eq. (9). Since Knudsen’s relations are derived from mass and salt balances and do not consider momentum balance, Eq. (19) is an approximation of Eq. (18). For a given FR , Eq. (19) always yields a higher value for F0 than Eq. (18). The difference is small when fT are small, but increases as FR and F fT both FR and F increase (Fig. 4). We also compare the theoretical results with the field data of Prandle (1985) in Table 1 and Fig. 2.6 of Geyer (2010) in Fig. 4. We have chosen to plot fT fT 5 0 and 30, because most estuaries have F Eq. (18) for F within this range. Ideally, most of the partially and wellmixed estuaries should cluster within the gray region fT 5 0 and 30, which is indeed the bounded by the lines F case. The most important aspect of this comparison is that the theoretical curves follow the overall trend of the field data. However, F0 tends to be overpredicted at high values of FR , that is, mixing is underpredicted. This discrepancy could arise if the values of F0 were measured at an upstream location, rather than at the mouth, TABLE 1. Estimates of estuarine parameters calculated using the data of Prandle (1985) and values of B obtained from maps. Equation (18) is used to obtain F0 (theory). Estuary Name Abbreviation FR FT B/H fT F F0 F0 (theory) Vellar Columbia James Tees Southampton Waterway Tay Narrows of the Mersey Bristol Channel V C J Te SW Ta NW B 1.27 0.026 0.004 0.014 0.0012 0.014 0.0009 0.006 0.64 0.43 0.25 1.03 0.37 1.38 0.83 1.59 200 150 360 75 200 400 65 300 9.0 5.3 4.7 8.9 5.2 27 6.7 27 1.00 0.40 0.22 0.18 0.10 0.10 0.05 0.02 1.00 0.50 0.17 0.33 0.06 0.17 0.05 0.06 1570 JOURNAL OF PHYSICAL OCEANOGRAPHY FIG. 3. Comparison of stratification at the estuary mouth obtained from theory or from field data. or could be a result of increasing stratification inhibiting vertical mixing. A plot similar to Fig. 4 is presented in Geyer (2010, Fig. 2.6). In this figure, a line [labeled Eq. (2.22)] VOLUME 43 corresponding to F0 ’ 3FR2/3 provides a good fit to the data. However, when Eq. (2.22) is evaluated using the coefficients provided in Geyer (2010), the result F0 5 8:73FR2/3 is obtained. Finally, we refer to the assumptions behind our theoretical analyses and their consequences. We have simplified the problem by assuming a tidally averaged estuary with rectangular geometry. In real estuaries, bathymetry can play a crucial role in determining estuarine circulation. Moreover, the appearance of just two fT ) in our equations is a conseparameters (FR and F quence of the empirical Eqs. (8) and (9). These equations also determine expressions for the coefficients C1, C2, . . . , C10, given in Table 2. All these coefficients depend upon the Schmidt number (i.e., Sc), which is an empirical quantity that could conceivably vary from estuary to estuary, within a given estuary, or with time. Although Eqs. (8) and (9) are simple and elegant, they may not be very realistic. In real estuaries, both KM and KS are variables. Moreover, other empirical parameterizations have shown that KM depends upon the Richardson number (MacCready and Geyer 2010). While the inclusion of the Richardson number, or any other relevant parameter, might improve the predictability of the classification scheme, the value of this improvement would have to be weighed against the added complexity of the resulting classification scheme. FIG. 4. Comparison of the estuary classification scheme with the approximation F0 5 7FR2/3 in Eq. (19). The gray area indicates the region where estuaries should ideally cluster. Field data from Geyer (2010) and Prandle (1985) are plotted for comparison with the theoretical predictions. AUGUST 2013 1571 GUHA AND LAWRENCE TABLE 2. List of coefficients used in different equations. Coefficient Value C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 1/3 1.67Sc 0.792Sc2/3 41.7a0a1CDSc21/3 0.868Sc21/3 36.2Sc2/3 41.7Sc1/3 4.08Sc1/3 3.57Sc2/3 5.43Sc1/3 5.21Sc2/3 6. Conclusions The equations governing the physics of estuarine circulation have been presented in nondimensional form. The two resulting nondimensional parameters are the estuarine Froude number FR and the modified tidal fT . Given these parameters, the nonFroude number F dimensional salinity gradient at the estuary mouth S X j0 and the nondimensional salinity stratification (also at the estuary mouth) F0 can be computed. The latter result forms the basis of a classification scheme that can be used to predict whether an estuary is fully or partially mixed, or highly stratified. The predictions of this classification scheme compare well with estuarine data. REFERENCES Banas, N., B. Hickey, P. MacCready, and J. A. Newton, 2004: Dynamics of Willapa Bay, Washington: A highly unsteady, partially mixed estuary. J. Phys. Oceanogr., 34, 2413–2427. Chatwin, P. C., 1976: Some remarks on maintenance of salinity distribution in estuaries. Estuarine Coastal Mar. Sci., 4, 555–566. 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