Residual Currents Induced by Asymmetric Tidal Mixing in Weakly Stratified Narrow Estuaries

Transcription

1 SEPTEMBER 21 C H E N G E T A L Residal Crrents Indced by Asymmetric Tidal Mixing in Weakly Stratified Narrow Estaries PENG CHENG* AND ARNOLDO VALLE-LEVINSON Department of Civil Coastal Engineering, University of Florida, Gainesville, Florida HUIB E. DE SWART Institte for Marine Atmospheric Research Utrecht, Utrecht University, Utrecht, Netherls (Manscript received 14 Jly 29, in final form 1 May 21) ABSTRACT Residal crrents indced by asymmetric tidal mixing were examined for weakly stratified, narrow estaries sing analytical nmerical models. The analytical model is an extension of the work of R. K. McCarthy, with the addition of tidal variations of the vertical eddy viscosity in the longitdinal momentm eqation. The longitdinal distribtion of residal flows driven by asymmetric tidal mixing is determined by the tidal crrent amplitde by asymmetries in tidal mixing between flood ebb. In a long channel, the magnitde of the residal flow indced by asymmetric tidal mixing is maximm at the estary moth decreases pstream following the longitdinal distribtion of tidal crrent amplitde. Larger asymmetry in tidal mixing between flood ebb prodces stronger residal crrents. For typical tidal asymmetry, mixing is stronger dring flood than dring ebb reslts in two-layer residal crrents with seaward flow near the srface lward flow near the bottom. For reverse tidal asymmetry, mixing is weaker dring flood than dring ebb the reslting residal flow is lward near the srface seaward near the bottom. Also, the residal flow indced by tidal asymmetry has the same order of magnitde as the density-driven flow therefore is important to estarine dynamics. Nmerical experiments with a primitive-eqation nmerical model [the Regional Ocean Modeling System (ROMS)] generally spport the pattern of residal crrents driven by tidal asymmetry sggested by the analytical model. 1. Introdction Pioneering stdies of estarine exchange flow focsed on the mean effects after a tidal time scale sggested that residal estarine crrents reslted mainly from the longitdinal baroclinic pressre gradient (Pritchard 1956; Hansen Rattray 1965; Chatwin 1976; MacCready 24). The inflence of tidal processes was inclded in those theories sing a constant vertical eddy viscosity that represented mixing over tidal time scales. However, it has been recognized that tides can contribte to the creation of residal crrents in many estaries throgh * Crrent affiliation: Large Lakes Observatory, University of Minnesota, Dlth, Minnesota. Corresponding athor address: Peng Cheng, Department of Civil Coastal Engineering, University of Florida, Gainesville, FL chengp@d.mn.ed nonlinearities throgh the temporal variation of tidal mixing. Ianniello (1977) showed that, nder relatively strong trblent mixing, the Elerian residal crrents indced by tidal nonlinearities are seaward at all depths the Lagrangian residal crrents indced by tidal nonlinearities oppose the two-layer density-driven flow. Sbseqent stdies have concentrated on temporal variations of tidal mixing. Jay Smith (199) introdced the concept of tidal asymmetry, sggesting stronger mixing dring flood tides enhanced stratification dring ebb tides. They demonstrated the importance of tidal asymmetry with observations at the Colmbia River. Asymmetric tidal mixing can arise from strain-indced periodic stratification (SIPS) (Simpson et al. 199) or extra freshwater inpts from a side embayment (Lacy et al. 23; Fram et al. 27). Stacey et al. (28) sed a one-dimensional nmerical model to examine the tidal asymmetry cased by extra freshwater inpts its role in the creation of estarine sbtidal circlation. They demonstrated that the residal flow generated by this DOI: /21JPO Ó 21 American Meteorological Society

2 2136 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 4 tidal asymmetry is of the same vertical strctre magnitde as the expected estarine gravitational circlation. They also fond that the residal flow is strongly dependent on the timing of stratification beginning within the tidal cycle developed a blk Richardson nmber topredicttheonsetofstratification. Tidal straining is one of the most common mechanisms for creating tidal asymmetries. The role of tidal straining in creating asymmetries in trblent mixing at tidal time scales has been well docmented throgh observations (Nepf Geyer 1996; Simpson et al. 25; Stacey Ralston 25; Sclly Friedrichs 27) nmerical simlations (Simpson et al. 22; Li et al. 28). The strain-indced tidal asymmetry creates asymmetries in the shear of the tidal crrent profile, reslting in a nonzero net flow profile. Dring ebb tides, tidal crrents stratify the water colmn throgh the straining of the density field by the interaction between a longitdinal density gradient a vertically sheared velocity profile. Dring flood tides, this straining is reversed the water colmn tends to be well mixed, intensifying crrents near the bottom. This asymmetric mixing velocity profile can lead to a residal flow with the same strctre as the density-driven circlation: seaward flow near the srface lward flow at the bottom (Jay Msiak 1996; Stacey et al. 21). However, the vertical strctre of this reslting residal flow has been obtained on the basis of conceptal analysis bt has not been evalated in detail with analytical or nmerical models. The prpose of this stdy is to investigate the spatial pattern strength of residal crrents indced by tidal-straining-indced tidal asymmetry in a two-dimensional estary channel. Becase of the difficlty in solving the nonlinear momentm eqations analytically, this stdy is limited to weakly stratified narrow estaries. These can be treated as weakly nonlinear systems whose governing eqations can be solved sing a pertrbation method. The approach is also restricted to narrow estaries where lateral variability of flow the earth s rotation can be neglected. The weakly stratified estary is considered as an approximately well-mixed estary. With these assmptions, the analytical model developed by McCarthy (1993) is extended to inclde an additional term representing tidal variations in vertical mixing. To obtain the soltion, a relatively weak asymmetry in trblent mixing between flood ebb tides mst be considered. Also, the analytical model ses an artificially designed eddy viscosity, which decoples trblent mixing from tidal crrents stratification. Frthermore, a primitive-eqation nmerical model, with a two-eqation trblent closre, is also sed to validate the analytical soltions. The remainder of this paper is strctred as follows: Section 2 presents a two-dimensional analytical model FIG. 1. Schematic of an idealized estary channel (length not to scale): a vertical longitdinal section is along the centerline of the channel. The dashed lines represent tidal elevations. the soltions of residal crrents indced by tidal asymmetry. Section 3 presents a nmerical experiment in an estary channel to evalate the generality of the analytical model show the inflence of stratification on tidal mean mixing. Finally, conclsions are presented in section Analytical model This analytical model is an extension of the work of McCarthy (1993) by adding tidal variations of the vertical eddy viscosity in the longitdinal momentm eqation. Most of the model assmptions eqation-solving techniqes follow those of Ianniello (1977) McCarthy (1993). The novel featre of this model is that asymmetric tidal mixing is inclded tidal-asymmetry-indced residal crrents can be explicitly compted. The model domain is a straight narrow estary channel that has constant depth width. Lateral variations of bathymetry are neglected so that the estary can be simplified as a two-dimensional along-channel section. The coordinate system is shown in Fig. 1, where x is positive from the moth toward the head z is positive pward with the bottom at z 52h. Here, narrow means that the along-channel residal dynamics are not significantly affected by lateral circlations; namely, both lateral advection the earth s rotation can be neglected. The conditions nder which these assmptions apply are given in Hijts et al. (29) Cheng Valle-Levinson (29). The first is that the Rossby nmber is large, which indicates that Coriolis forcing is negligible. In the case of a long estary this implies that its width shold be small compared to the Rossby radis of deformation. On the other h, a large Rossby nmber sggests that lateral advection cold be important in narrow estaries, as in the Hdson River (e.g., Sclly et al. 29). Therefore, the second condition is that vertical trblent mixing is relatively

3 SEPTEMBER 21 C H E N G E T A L large, nder which lateral advection is negligible. Accordingly, a narrow estary, like the one stdied here, can be considered as an estary with large Rossby Ekman nmbers. This condition is satisfied for wellmixed weakly stratified estaries with small width. a. Governing eqations The governing eqations for momentm, continity, salt transport the eqation of state are ^ ^ ^ ^h 1 ^ 1 ^w 5 g ^t ^x ^z ^x g^r ^h ^r ^z ^x d^z 1 ^ ^K, ^z ^z (1a) ^h ^t 1 ^x ^ ^x 1 ^w ^z 5, ^h ^h (1b) ^d^z 5, (1c) ^s ^s ^s 1 ^ 1 ^w ^t ^x ^z 5 ^K 2^s x ^x 2 1 ^z ^s ^K ^z, (1d) ^r 5 ^r (1 1 b^s), b (1e) Here, a caret denotes a dimensional qantity; ^t is time; ^ ^w are the longitdinal vertical velocity components, respectively; ^h is the free srface elevation; ^h is the water depth; ^s is salinity; ^r is water density; ^r is freshwater density; ^b is the haline contraction coefficient; ^Kx is the horizontal eddy diffsivity; ^K represents vertical eddy viscosity diffsivity, assming the Schmidt nmber (ratio of eddy viscosity to diffsivity) is 1. The above eqations are the same as those sed by McCarthy (1993) except that the salt balance is sed here instead of density. To inclde asymmetric tidal mixing in the dimensionless eqations, we assme that the vertical trblent mixing coefficient can be decomposed in two parts: ^K 5 ^K* 1 ^K9. (2) Here ^K* measres the degree of mixing that wold occr nder well-mixed conditions, whereas ^K9 represents a tidal variation part cased by tidal straining. Ianniello (1977) arges that in a tidally dominated, well-mixed estary ^K* 5 C d ^U ^h^f (^z), (3) where C d is a drag coefficient, ^U the leading order depth-averaged tidal velocity, ^f (^z) a fnction of the vertical coordinate ^z; we assme ^U is a constant. A similar pertrbation techniqe to Ianniello s (1977) McCarthy s (1993) is followed. The sbseqent scalings are introdced: t 5 ^tv, x 5 ffiffiffiffiffi ^x ^z q g ^h, z 5 p ffiffiffiffiffiffiffiffiffiffiffiffiffi, /v 2K /v ^h d 5 p ffiffiffiffiffiffiffiffiffiffiffiffiffi, h 5 ^h 2K /v h*, ^ ^w 5 q ffiffiffiffiffiffi h* g/ ^h, w 5 p h* ffiffiffiffiffiffiffiffiffiffiffiffi 2K v/ ^h, s 5 ^s, K 5 ^K, s c K K x 5 ^K x gh* 2 / ^hv (4) in which the variables withot a caret are dimensionless qantities. Here K 5 C d ^U ^h^f (^z) is a constant vertical eddy viscosity, h* is the tidal amplitde at the moth, s c is the ocean salinity, v is the anglar freqency of tidal forcing, d is a measrement pffiffiffiffiffiffiffiffiffiffiffiffiffi of the depth in terms of the frictional length scale 2K /v (note that 1/d is analogos to the Ekman nmber); pffiffiffiffiffi the horizontal length scale is the tidal wavelength g ^h /v. Sbstitting these scalings into Eq. (1) reslts in the dimensionless eqations: 1 «1 «w x h 1 1 d x 5 h x «hd «g z s x dz K 2 2, (5a) x 1 w 5, (5b) «hd dz 5, (5c) s s s 1 «1 «w x 5 2 s «2 K x x K s, (5d) where «5 h*/ ^h g 5 bs c /d. Of particlar interest is the nonlinear effect represented by the tidal amplitde to depth ratio «. The assmption is that the parameter «is not negligible bt is mch smaller than 1. Since «1, the nonlinear advection terms do not affect the tidal soltion «becomes the expansion parameter for the dependent variables. The parameter g measres the relative importance of the baroclinic pressre gradient

4 2138 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 4 is assmed O(1). As McCarthy (1993) discssed, if g ; O(1), the estary is dominated by barotropic tides, stratification is weak, the baroclinic forcing term mainly contribtes to residal crrents. The dimensionless set of eqations is the same as those of McCarthy. The dimensionless vertical mixing coefficient is written as Herein, K 5 K* 1 «K9; K* 5 ^K* K ; K95 ^K9 «K. (6) K* 5 f (z) (7) with f (z) 5 ^f (^z), K9(z, t) depends on the vertical coordinate z time t in a ser-specified manner. Variations in trblent vertical mixing owing to tidal straining are assmed to be small compared to trblent mixing indced by frictional drag at the bottom. Moreover, it is assmed that f(z) 5 1; that is, the trblent mixing coefficient K* does not depend on the vertical coordinate. In many previos analytical stdies, K is assmed constant over a tidal cycle. In this stdy, we se two constants that represent eddy viscosities dring flood ebb tides (Fig. 2). The difference between the two constants translates into asymmetric tidal mixing. This assmption ses the flood-averaged mixing to represent the entire flood ses the ebb-averaged mixing to represent the entire ebb. This cold be the simplest case of asymmetric tidal mixing. More complicated tidal variations of mixing mst await ftre stdies. It is noteworthy that K* can be chosen as the tidally averaged eddy viscosity or a vale smaller than the eddy viscosity of ebb tides, as shown in Fig. 2. Althogh the two choices prodce similar reslts, the latter is preferred becase the mean of the tidal variation component is not zero, consistent with the idea that tidal asymmetry prodces extra eddy viscosity over a tidal cycle. The ratio between the variable the steady components of K is assmed to be O(«). This indicates a weakly stratified estary where variations of vertical mixing indced by stratification are comparatively small. With this assmption, asymmetric tidal mixing only inflences residal crrents. Following the pertrbation method, the dependent variables h,, w, s are exped in an asymptotic series in powers of «, yielding the lowest- [O(«)] the first- [O(«1 )] order pertrbation eqations. The lowest order eqations represent tidal motion are the same as those of Ianniello (1977), the governing eqations of salt transport are the same as those of McCarthy (see appendix). Soltions to tidal motion salinity have FIG. 2. Tidal evoltion of dimensionless eddy viscosity K; K* is the steady component, while «K9 represents a tidal variation component of K. been provided by Ianniello (1977) McCarthy (1993), respectively, are omitted here. This stdy concentrates on the residal crrents only. Time averaging the first-order momentm continity eqations after a tidal cycle gives x 1 w 5 h 1 x 1 g ds K9 dx z d 1 dz 1 j z5 h 5 R, (8a) (8b) where sbscripts 1 represent lowest first order, respectively. The overbar denotes tidal averages, R is the dimensionless river flow velocity, which is the ratio of the dimensional river flow velocity ^ r to the scale of theq nonlinear ffiffiffiffiffi flow (McCarthy 1993); namely, R 5 ^ r /(«2 g ^h ). The residal crrents 1 are considered to have for components: (i) the tidally rectified flow 1N, (ii) the river-indced flow 1R, (iii) the density-driven flow 1D, (iv) the flow indced by asymmetric tidal mixing 1A. The present model setp implies that the for components are linearly independent. Ths, 1 the residal elevation h 1 can be written as 1 5 1N 1 1R 1 1D 1 1A h 1 5 h 1N 1 h 1R 1 h 1D 1 h 1A. (9a) (9b) Sbstitting Eqs. (11) into (1), we obtain five eqations corresponding to the for sorces of residal crrents:

5 SEPTEMBER 21 C H E N G E T A L x 1 w 5 h 1N x N 2 2, (1a) 5 h 1R x R 2, (1b) 2 5 h 1D x 1 g ds dx z D 2 2, (1c) 5 h 1A x A K9, (1d) 1 d 1N dz 1 1 d 1 1 d 1R dz 1 1 d 1A dz 1 j z5 h 5 R. 1D dz (1e) b. Soltion of residal crrents indced by asymmetric tidal mixing The for components of residal crrents can be obtained by solving Eqs. (1). The tidally rectified flow [Eq. (1a)] has been solved by Ianniello (1977, 1981). The soltions of river-indced flow [Eq. (1b)] density-driven flow [Eq. (1c)] have been provided by McCarthy (1993). The aim of this stdy is to obtain the soltion of the flow indced by asymmetric tidal mixing. The bondary conditions sed to solve the momentm eqation [Eq. (1d)] inclde no slip at the bottom no shear at the srface. The continity eqation for flow indced by asymmetric tidal mixing can be obtained from Eq. (1e). According to Ianniello (1977), 5. Also, it is known that d 1Ð 1R dz 5 R d 1Ð 1D dz 5 (McCarthy 1993). Ths, the continity eqation [Eq. (1e)] can be redced to d 1Ð 1A 5. On the basis of those conditions, the soltion of the flow indced by tidal asymmetry [integrating Eq. (1d) twice sing continity] is d 1Ð 1N dz 1 j z5 h 1A 5 h z 1A x (z2 d 2 ) h 1A x 5 3 z 2d 3 K9 dz9 (11a) 9 K9 dz9 dz (11b) 9 In which z9 is a dmmy variable. The soltion shows that the residal flow indced by tidally asymmetric mixing is determined by the lowest order tidal crrent the FIG. 3. Dimensionless analytical soltions for a long channel: p (a) amplitde of tidal crrent scaled by U that eqates «ffiffiffiffiffi gh ; (b) scaled lowest order salinity s,wheres c is the ocean salinity; (c) residal tidally rectified crrents; (d) density-driven crrents. Darker area denotes negative vales (otflow), L d is a tidal dissipation length. Scaled parameters are «5.1, d 5 1, R 5.1, L d , K x 5.9. The soltions follow Ianniello (1977, 1981) McCarthy (1993). tidal variation of vertical mixing K9. The soltion for has been given by Ianniello (1977). A proper form of vertical eddy viscosity needs to be designed so as to examine the pattern strength of 1A. As assmed, K* is a steady part has no vertical distribtion; K9 has both temporal vertical variations. It is constant throghot the flood or ebb bt has a different magnitde for each phase of the tidal cycle (Fig. 2). With stratification, the vertical profile of eddy viscosity tends to show an asymmetric parabolic distribtion with the maximm vale moving toward the bottom (Bsinger et al. 1971). Therefore, the vertical profile of K9 is designed as an exponential fnction K9(z) 5 (d 1 z)/ «exp[2(d 1 z)/a ], which is controlled by the parameter a. Once K9 is determined, 1A is obtained by integrating Eq. (11) nmerically. Taking advantage of the tidal soltion from Ianniello (1977), the residal crrents indced by tidally asymmetric mixing are obtained for an infinitely long channel. The soltion for a channel with a reflecting wall has a similar pattern to the infinitely long channel will not be presented here. Figre 3 shows longitdinal distribtions of tidal crrent amplitde,depth- time-mean salinity s, tidal-nonlinearity-indced residal crrents 1N, density-driven crrents 1D for a long

6 214 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 4 FIG. 4. Dimensional soltions of residal crrents indced by asymmetric tidal mixing: (left) vertical profiles of eddy viscosity for flood ebb tides (right) an axial section of residal crrents indced by asymmetric tidal mixing; (top) case 1, a typical tidal asymmetry; (middle) case 2, a typical tidal asymmetry with larger asymmetry of eddy viscosity between flood ebb tides, relative to case 1; (bottom) reverse tidal asymmetry. The scaled parameters are as in Fig. 3. Negative vales (shaded) denote seaward flow. channel. The vales of the dimensional variables were chosen as h* 5 1m,^h 5 1 m, ^ r 5.1 m s 21,^v s 21 (M 2 tide). The scaled parameters are «5.1, d 5 1, R 5.1, L d , K x 5.9, where L d is a tidal dissipation length scale defined by Ianniello (1977). The parameter d indicates mixing effects a vale of 1. represents a vertical eddy viscosity K 5.7 m 2 s 21 (relatively strong mixing). The scaled tidal crrent amplitde decreases from abot 1. at the moth to at the head of the estary becase of friction (Fig. 3a). The salinity is scaled by seawater salinity sch that a vale of one indicates seawater a scaled salinity of zero represents freshwater. The salinity is one at the moth approaches zero pstream (Fig. 3b). The maximm horizontal salinity gradient occrs near x 5 L d.tidalnonlinearity-indced residal crrents are seaward at all depths decrease from the moth toward the head from srface to bottom (Fig. 3c). The maximm tidalnonlinearity-indced crrents are concentrated in the pper water colmn at the moth. The density-driven flow shows a two-layer strctre, that is, otflow near the srface inflow near the bottom (Fig. 3d). The maximm flow is located at the region of maximm horizontal salinity gradient is consistent with conventional estarine dynamics theory (e.g., Hansen Rattray 1965; Chatwin 1976; Officer 1976). The strength of the density-driven flow is smaller than the tidally indced residal crrents so that density-driven flow is relatively less important in well-mixed or weakly stratified tidal estaries. On the basis of the soltion for, three cases of K9 1A are examined (Fig. 4). The first case tests typical asymmetric tidal mixing, that is, stronger vertical mixing dring flood than on ebb tides (Figs. 4a,b). The depth mean of K9 is 1.25 dring flood.75 dring ebb. The tidal average of the depth mean of K9 is 1 so that it flfills the reqirement that K9 has a magnitde of O(1), jst as K*. The residal crrents indced by asymmetric tidal mixing show a two-layer strctre with seaward flow near the srface lward flow near the bottom. This is similar to the density-driven flow, bt the maximm magnitde of 1A appears at the moth of the estary. Clearly 1A is partly determined by tidal crrents, the strength of 1A follows the longitdinal distribtion of tidal crrent amplitde, which decreases lward.

7 SEPTEMBER 21 C H E N G E T A L The second case examines larger asymmetric mixing between flood ebb tides: K9 is the same as that in the first case dring flood bt is weaker dring ebb (Fig. 4c). The distribtion of 1A is still a two-layer strctre similar to the first case, bt the magnitde is stronger (Fig. 4d). This shows that a larger asymmetry of mixing between flood ebb prodces stronger residal crrents. The third case tests reverse asymmetric tidal mixing; namely, mixing dring flood is weaker than dring ebb. The magnitde of K9 in the third case is opposite to that in the first case; that is, K9 at flood is the same as that dring ebb in the first case (Fig. 4e). As in the first case, 1A has similar magnitde bt exhibits the opposite pattern of lward flow near the srface seaward crrents near the bottom layer (Fig. 4f). This indicates that reverse asymmetric tidal mixing prodces an opposite pattern of residal crrents to the typical tidal asymmetry. It is noteworthy that the magnitde of 1A is of the same order as that of the density-driven flow, so they will compete to determine the residal flows. Three types of reverse tidal asymmetry can be identified on the basis of previos stdies. First, reverse tidal asymmetry can be created by a freshwater sorce from a side embayment of the estary (Lacy et al. 23; Fram et al. 27). When less dense waters from a side embayment move into the estary channel dring flood tides, stratification can be stronger dring the flood than the ebb, reslting in reverse tidal asymmetry. Second, most estaries have lateral bathymetric variations. The deep channel shoals may ndergo opposite tidal evoltion of stratification. At the thalweg, the water colmn exhibits a typical tidal asymmetry de to tidal straining; while over the shoals the water colmn shows a reverse tidal asymmetry. Lateral straining enhances stratification dring flood tides, whereas vertical mixing redces stratification dring ebb tides (Cheng et al. 29). Third, reverse tidal asymmetry can appear in the region of the axis of varying stratification (Fgate et al. 27). This sally happens at the pper reaches of the estarine salt intrsion. Dring flood tides, stratified water is advected pstream the water colmn becomes stratified. Dring ebb tides, the water colmn is well mixed as freshwater that extends from srface to bottom moves downstream. The phasing of stratification has been demonstrated to be important in the creation of tidal-asymmetry-indced residal crrents (Stacey et al. 28). To examine the effects of phasing of stratification on residal crrents, we carried ot a series of experiments following case 1. We assme that half of the tidal cycle is well mixed, represented by a relatively large eddy viscosity (solid line in Fig. 4a), that the other half of the tidal cycle is weakly stratified, represented by a small eddy FIG. 5. Tidal-asymmetry-indced residal crrents as a fnction of tidal phase at which stratification starts. The horizontal axis is the timing of the onset of stratification (phase of the tidal cycle): (p) represents the beginning of the flood (ebb). It is assmed that half of the tidal cycle is stratified. Vertical axis is dimensionless depth. Profiles are near the estary moth. Contors show dimensionless magnitde of the residal crrents. Negative vales (shaded) denote seaward flow. viscosity (dashed line in Fig. 4a). The onset of stratification can occr at any stage of the tidal cycle. Stratification starting at the beginning of flood represents stratified flood well-mixed ebb, that is, reverse tidal asymmetry. When stratification starts at the beginning of ebb, it represents stratified ebb well-mixed flood, that is, typical tidal asymmetry. A station near the estary moth is selected to show the vertical strctre of tidal-asymmetry-indced residal crrents for each tidal phase of stratification onset (Fig. 5). When the entire flood is well mixed (at phase of p) or stratified (at phase of or 2p), the residal crrents are strongest have opposite sign. When half of the flood is stratified (at phases of p/2 3p/2), the tidal-asymmetry-indced residal crrents are essentially zero. When the onset of stratification occrs from p/2 to 3p/2, the residal crrents show a similar pattern to that of density-driven circlation. When the onset of stratification occrs in the rest of the tidal cycle (at phases of p/2 or 3p/2 2p), the residal crrents show an opposite pattern to the density-driven flow. These reslts are consistent with those of Stacey et al. (28). 3. Nmerical experiment The analytical model demonstrated that residal flow can be prodced from asymmetric tidal mixing revealed its vertical strctre. However, the generality of the analytical reslts may be jeopardized by its

8 2142 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 4 FIG. 6. Model domain of the nmerical model. Contors denote water depth on the shelf. assmptions. The analytical model is bilt for wellmixed estaries (McCarthy 1993) for which the tidal variation of vertical mixing has to be assmed small. The eddy viscosity is artificially prescribed is not copled to tidal crrents stratification. To overcome these limitations of the analytical model, a primitive-eqation nmerical model with a more realistic trblence model is reqired. The main prpose of the nmerical experiment is to examine the validity of the analytical reslts instead of carrying ot a comprehensive nmerical stdy for all types of estaries. The nmerical experiment focses on weakly stratified, narrow estaries: partially mixed highly stratified estaries will be addressed in a ftre stdy. a. Model configration The Regional Ocean Modeling System (ROMS) is sed to carry ot idealized experiments. The model is a free-srface, hydrostatic, primitive-eqations ocean model that ses stretched, terrain-following vertical coordinates orthogonal crvilinear horizontal coordinates on an Arakawa C grid (Haidvogel et al. 2). The model domain is designed as an estary shelf system (Fig. 6) following the stdy of Hetl Geyer (24). The part of the domain corresponding to the estary is straight, 3 km long, has no along-channel bottom slope. The cross-channel section has a rectanglar shape with a depth of 1 m. The width is.6 km. A freshwater discharge with fixed section-averaged velocity of.1 m s 21 is specified at the head of the estary. The inflowing river water is prescribed to have zero salinity temperatre 158C, identical to the backgrond temperatre set throghot the entire domain. The continental shelf is 8 km wide has a fixed cross-shelf slope of.5%. A semidirnal tide S 2 with an amplitde of 1.5 m is imposed at the eastern open bondary. The salinity of the coastal ocean is 35 ps, a sothward weak flow (.3 m s 21 ) is specified on the shelf to sppress the blge of freshwater at the estary moth. The coastal ocean is inclded in the domain to avoid specifying bondary conditions at the estary moth, which are sally difficlt to establish. The two-eqation trblence closre k v is sed to calclate vertical mixing. The model grid is 2 (along channel, x direction) by 8 (cross channel, y direction) by 4 (vertical, z direction) cells. The river has 15 grid cells along the channel 3 grid cells across the channel. The along-channel grid size (Dx) increases exponentially from the estary moth (;5 m) to its head (;11 km), providing a highly resolved region near the moth. The cross-channel grid in the estary is niformly distribted the vertical layers are niformly discretized. The model rns from rest, for 7 days ntil reaching steady state. The reslts of the last day are sed for analysis. b. Residal crrents indced by asymmetric tidal mixing Figre 7 shows the along-channel distribtion of the tidally averaged salinity field. The vertical distribtions of salinity indicates that the estary is approximately well mixed at the pper section (x, 27 km) is weakly stratified at the lower section (x. 27 km to the

9 SEPTEMBER 21 C H E N G E T A L FIG. 8. Longitdinal distribtions of (a) depth-mean tidal crrent amplitde (b) depth-mean eddy viscosity from the nmerical model. Solid (dashed) lines represent flood (ebb) averages. FIG. 7. Longitdinal distribtion of tidally averaged salinity from the nmerical model: (a) axial section of salinity in the estarine region (b) depth-mean salinity. moth). The tidal evoltion of the salinity field (not shown) exhibits a well-mixed flood tide a weakly stratified ebb tide. The longitdinal distribtion of tidally averaged depth-mean salinity (Fig. 7b) shows the general form of the hyperbolic tangent fnction observed in many coastal plain estaries (Pritchard 1952; Hansen Rattray 1965). Salinity increases downstream gradally reaches the terminal vale of the ocean. The maximm horizontal salinity gradient occrs arond x 5 29 km. Near the estary moth (29 3 km), the horizontal salinity gradient decreases the crvatre of the salinity crve becomes negative. This may be attribted to the strong tidal dispersion near the estary moth (McCarthy 1993; MacCready 24). According to the salinity distribtion, we consider that the inner regime of the estary is located at km is approximately well mixed. The central regime is located at km, the oter regime is located at 29 3 km. Both central oter regimes are weakly stratified. Longitdinal distribtions of depth-mean along-channel tidal crrent amplitde vertical eddy viscosity, averaged dring flood ebb tides (Fig. 8), show distinct distribtions. For both tidal phases the tidal crrent amplitde, obtained by harmonic analysis, decreases from the moth becomes nearly zero at the head. The flood-averaged velocity is larger than the ebb-averaged velocity near the head ( 1 km) is smaller than the ebb-averaged velocity near the moth (15 3 km). Between 24 km, where river flow dominates (zero salinity), eddy viscosities are proportional to tidal crrent amplitdes show similar longitdinal distribtions to the velocities. In the estary (x. 24 km), however, the ebb-averaged eddy viscosity is mch smaller than the flood-averaged eddy viscosity, showing an opposite trend as the tidal crrent amplitdes. This indicates a typical asymmetric tidal mixing. The tidal-asymmetry-indced flow is calclated sing a dimensional form of Eq. (11) in which variables are taken from the nmerical model otpt. Here ^K* is chosen as the tidally averaged depth-mean eddy viscosity, ^K9 is the tidal flctation component of the eddy viscosity (i.e., ^K9 5 ^K ^K*), is the modeled tidal crrent amplitde. The tidal-asymmetry-indced flow exhibits a two-layer strctre with seaward flow near the srface lward flow near the bottom (Fig. 9). The pattern is consistent with the predicted residal flow indced by typical tidal asymmetry from the analytical model (e.g., Fig. 4b). The maximm residal flow appears in the central regime of the estary instead of at the estary moth as dedced by the analytical model. This discrepancy may partly reslt from the alongestary distribtion of ^K*, which is assmed constant throgh the estary in the analytical model. In contrast, modeled eddy viscosities show that ^K* is redced in the central regime of the estary (Fig. 1a), according to Eq. (11a), small ^K* reslts in stronger residal crrents. The nmerical reslts also sggest that the analytical model has potential tility in real estaries if proper information of the eddy viscosity tidal crrents is available.

10 2144 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 4 FIG. 9. Longitdinal distribtion of tidal-asymmetry-indced flow of the nmerical model. The nit of the residal flow is cm s 21. Negative vales (shaded) denote lward flow. c. Tidally averaged eddy viscosity in the central regime of the estary The nmerical model shows a pecliar featre of trblent mixing. The eddy viscosities are markedly redced are approximately constant throghot the central regime of the estary (Fig. 1a). This is consistent with the assmption of the analytical theory of Hansen Rattray (1965), bt the case of this spatial distribtion (constant) is nclear. In general, trblent mixing can be parameterized in terms of a velocity scale a length scale. The velocity scale is sally the tidal crrent velocity the length scale can be taken as the water depth of the channel (MacCready 27). This scale, however, does not accont for the inflence of stratification on trblent mixing is only proportional to the tidal crrent amplitde becase of the niform depth of the estary. The amplitde of tidal crrents, obtained by harmonic analysis, decreases lward almost linearly (Fig. 1b). Obviosly, this scale is not able to predict the constant eddy viscosity in the central regime of estary. To inclde the effects of stratification on trblent mixing, Ralston et al. (28) provided another scale of eddy viscosity that depends on the thickness of the bottom bondary layer ^h bl instead of the water depth, ^K v 5 ac d ^UT ^hbl, (12) where ^K v represents the tidally averaged vertical eddy viscosity diffsivity (assme Schmidt nmber of 1), a is a constant parameter, C d is the bottom drag coefficient, ^UT is the tidal crrent amplitde, all variables are dimensional qantities. The bottom bondary layer h bl can be scaled as (Stacey Ralston 25) ^h bl 5 ^h R 1/2 f, (13) Ri x FIG. 1. Longitdinal distribtions of (a) tidally averaged depthmean eddy viscosity, (b) tidal crrent amplitde, (c) horizontal salinity gradient from the nmerical model. Solid lines are nmerical reslts; dashed lines are predictions sing the scaling of eddy viscosity [Eq. (12)]. where R f is the critical flx Richardson nmber (taken as.2 here) Ri x is the horizontal Richardson nmber. The scale for the bondary layer thickness is obtained for stratified ebb tides assming that the boyancy flx de to straining of d^s/d^x is balanced by shear prodction at the top of the bottom bondary layer (Stacey Ralston 25). The assmption involved in its derivation limits the scale to stratified water colmns; Ri x is a scale of the Richardson nmber by representing the vertical density difference as a fnction of the horizontal density gradient. Stacey et al. (21) proposed 2 gb ^h ^s Ri x 5 ^ 2 ^x, (14) * where ^ * is the friction velocity is related to the tidal crrent velocity ^U p T by ^ * 5 ffiffiffiffiffiffi C d ^UT. To apply Eq. (12), the vales of ^U T, ^h, d^s/d^x were taken from the nmerical experiments (Figs. 1b,c); the C d sed was.25; the best fitted a was.31. The vales of ^K v obtained with Eq. (12) generally match the trend of the modeled eddy viscosity in the central oter regimes of the estary (Fig. 1a). Particlarly, ^Kv exhibits approximately a constant vale in the central regime. This indicates that spatially constant vertical mixing arises from the combined effect of tidal mixing stratification. Becase the inflence of stratification on eddy viscosity is represented with an along-channel salinity gradient, the longitdinal distribtion of the scaled eddy viscosity is approximately a mirror image of the salinity

11 SEPTEMBER 21 C H E N G E T A L gradient: that is, ^Kv } (^s/^x) 1/2. As the along-channel salinity gradient tends to zero in the inner regime of the estary, the scaled eddy viscosity becomes very large nreliable. Althogh Eq. (12) can be calibrated to approximate the constant trblent mixing in the central regime of the estary, distinct discrepancies between scaled modeled eddy viscosity appear in the oter inner regimes of the estary. These discrepancies indicate that the parameterization of estarine trblent mixing still needs frther stdy. 4. Conclsions This stdy examines residal crrents indced by asymmetric tidal mixing. Neglecting lateral processes effects of the earth s rotation assming a weakly stratified estary that can be approximated as a weakly nonlinear system, a two-dimensional analytical model is established. Analytical soltions show that the residal flow indced by asymmetric tidal mixing is determined by tidal crrents by the asymmetry of tidal mixing between flood ebb. The magnitde of tidal-asymmetryindced flow is the same order as that of density-driven flow. Typical tidal asymmetry in mixing leads to a twolayer residal flow similar in strctre to the densitydriven flow, as previosly established (Jay Msiak 1996; Stacey et al. 21). However, nlike the densitydriven flow, which has a maximm at the region of greatest horizontal density gradient, the strongest residal crrents indced by tidal asymmetry appear at the estary moth where the maximm tidal crrent amplitdes are fond. Reverse tidal asymmetry prodces a two-layer strctre that is opposite to that of the densitydriven flow. Nmerical experiments spport the longitdinal pattern of the crrents indced by tidal asymmetry as predicted by the analytical model. However, the modeled tidal-asymmetry-indced flow has maximm magnitdes in the central regime of the estary where trblent mixing is redced. This indicates that the strength of tidal-asymmetry-indced flow is also dependent on the tidal mean eddy viscosity. The spatially constant eddy viscosity in the central regime of the estary is fond to be inflenced by stratification, which is represented by the along-channel density gradient. This demonstrates that trblent mixing is highly related to stratification in stratified estaries. Becase this stdy concentrates mainly on weakly stratified estaries, the residal crrents indced by asymmetric tidal mixing in partially mixed highly stratified estaries are yet to be explored. The linear decomposition of residal flow sed in the analytical model cold be extended to partially mixed highly stratified estaries. Sch extension will allow identification of the strength spatial patterns of the estarine residal crrent components: river-indced, densitydriven, tidal-nonlinearities-indced, tidal-asymmetryindced flows. Acknowledgments. This work was spported by NSF Project OCE AVL also acknowledges the completion of this stdy while a Gledden Fellow at the Centre for Water Research of the University of Western Astralia. We thank David Jay an anonymos reviewer for their insightfl comments that helped to improve this work. APPENDIX The Pertrbation Eqations This appendix gives the derivation of Eqs. (5) (8). More detailed information can be fond in Ianniello (1977, 1981) McCarthy (1993). The dimensionless eqations [Eqs. (5)] are analyzed for the sitation that «1, g 5 O(1), R 5 O(1), K* 5 O(1), K95O(1), K x 5 O(1). (A1) These are typical vales for a well-mixed estary (vertical mixing dominates over advection), where residal crrents generated by tidal advection terms, baroclinic pressre gradients, freshwater discharge are small compared to tidal crrents. Asymptotic soltions of the system can now be constrcted as pertrbation series in the small parameter «, (, w, h, s) 5 (, w, h, s ) 1 «( 1, w 1, h 1, s 1 ) 1. (A2) A frther implication of «1 is that, in the along-channel momentm balance, the baroclinic pressre gradient term can be exped as follows: «hd s «g x dz95 «g z z s dz91. (A3) x As will trn ot below, it sffices to analyze the dynamics p to O(«). Likewise, the bondary conditions at the free srface can be transformed to conditions at the fixed srface z 5 by exping variables in a Taylor series. For example, the kinematic condition becomes p to O(«) atz 5 :

12 2146 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 4 w 1 «hd w h h «1. (A4) x The lowest order eqation of the asymptotic expansion of Eq. (A8) is To lowest order this yields the eqations 1 ds d s 1 dz Rs K x dx 5, (A9) 5 h x , (A5a) x 1 w 5, (A5b) h 1 1 d x s dz 5, (A5c) s 2. (A5d) The corresponding bondary conditions are at z 5 : w 5 h, 5, s 5 ; (A6a) at z 5 : w 5, 5, s 5 ; (A6b) at x 5 : h 5 cos(vt), s 5 1; (A6c) at x! : 1 d dz!, 1 d s dz!. (A6d) This system describes a damped linear tidal wave that travels into the estary. The eqation of s is a diffsion eqation. Considering the bondary conditions of s, for any initial conditions variations in s over the vertical will become smaller in the corse of time; ths, the nontransient soltion of this eqation is s 5 s (x). (A7) Ths, to lowest order the salinity is only a fnction of the along-channel coordinate x, as is to be expected for a well-mixed or weakly stratified estary (in reality, s may vary with time depth in weakly stratified estaries, so the assmption needs frther stdy). To determine s, tidally averaging depth integrating the salt transport eqation (5d) gives «hd s s «K x dz 5. x (A8) where R is the dimensionless river flow velocity, O(1). This eqation reveals a balance between tide-indced salt transport, a term owing to freshwater transport longitdinal dispersion. The first-order eqations can be derived in a straightforward manner. The final eqations contain qite a nmber of terms, bt some of them can be removed becase it is known that s only depends on x. The reslting eqations in this case read as 1 h 1 1 x 1 w 5 h 1 x 1 g ds dx z K9, (A1a) 1 1 d x 1 x 1 w 1 5, (A1b) 1 dz 1 x ( j z5 h ) 5, (A1c) s 1 s 1 x s 1 2 2, (A1d) with corresponding bondary conditions at z 5 : w 1 5 h 1 1 h x, (A11a) 1 1 dh K9 5, (A11b) s 1 at z 5 : w 1 5, 1 5, 5 ; (A11c) s 1 5 ; (A11d) at x 5 : h 1 5, s 1 5 ; (A11e) at x! : 1 d 1 dz! R, 1 d s 1 dz!. (A11f) It is noticed that asymmetric tidal mixing in this model only enters in the momentm balance, not in the salt balance. This is de to or assmption of strong vertical

13 SEPTEMBER 21 C H E N G E T A L mixing, sch that s does not depend on the vertical coordinate. The focs of this paper is on the residal components for which the eqations are obtained by averaging all first-order eqations over a tidal cycle. These can sbseqently be solved. In this stdy, the new aspect concerns the soltion of the residal flow indced by time-varying tidal mixing as cased by the eddy viscosity coefficient K9. Owing to adopting a constant ^K*, the factor of shear stress has been changed from 3 /8 in Ianniello (1977) McCarthy (1993) to ½ in this stdy. The soltion to 1N is the same as that of Ianniello (1977, 1981), which neglected the factor 3 /4 for the case with constant eddy viscosity, whereas the soltion to 1D from McCarthy (1993) has been changed to 1D 5 gd3 24 ds dx z2 z d d 3. (A12) The soltion to 1R was not affected by the factor 3 /4. REFERENCES Bsinger, J. A., J. C. Wyngaard, Y. Izmi, E. F. 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