Analytical description of tidal dynamics in convergent estuaries

Transcription

1 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 1/7 An edited version of this paper was published by AGU. Copyright (8) Amerian Geophysial Union. Savenije, H. H. G., M. Toffolon, J. Haas, and E. J. M. Veling (8), Analytial desription of tidal dynamis in onvergent estuaries, J. Geophys. Res., 113, C15, doi:1.19/7jc448. To view the published open abstrat, go to Analytial desription of tidal dynamis in onvergent estuaries Hubert H. G. Savenije 1,, Maro Toffolon 3, Jennifer Haas 1, Ed J. M. Veling 1 1 Department of Water Management, Delft University of Tehnology, Delft, The Netherlands Uneso-IHE, Institute for Water Eduation, Delft, The Netherlands 3 Department of Civil and Environmental Engineering, University of Trento, Italy Abstrat Analytial solutions of the one-dimensional hydrodynami equations for tidal wave propagation are now available and, in this paper, presented in expliit equations. For given topography, frition and tidal amplitude at the downstream boundary, the veloity amplitude, the wave elerity, the tidal damping and the phase lag an be omputed. The solution is based on the full non-linearised St. Venant equations applied to an exponentially onverging hannel, whih may have a bottom slope. Two families of solutions exist. The first family onsists of mixed tidal waves, whih have a phase lag between zero and π/, whih our in alluvial oastal-plane estuaries with almost no bottom slope; the seond family onsists of "apparent standing" waves, whih develop in short estuaries with a steep topography. Asymptoti solutions are presented for progressive waves, fritionless waves, waves in hannels with onstant ross-setion, and waves in ideal estuaries where there is no damping or amplifiation. The analytial method is aurate in the downstream, marine, part of estuaries and partiularly useful in ombination with eologial or salt intrusion models. The solutions are ompared with observations in the Shelde, Elbe and Mekong estuaries. 1. Introdution Ever sine Barré de Saint-Venant published his equations for visous flow in 1843, efforts have been made to solve the set of equations analytially. At first, out of need. Analytial solutions were the only option to turn the non-linear partial differential equations into pratial appliations. In river systems, the equations ould in general be simplified more easily than in tidal water bodies. Rivers tend to have more or less onstant ross-setions and in large rivers temporal gradients are slow. As a result quasi-steady state solutions or strongly linearized solutions appeared to work rather well in rivers, leading to analytial expressions for bakwater urves, and the kinemati wave solution. In tidal water bodies, however, quasisteady state ould not be assumed.

2 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE /7 A fruitful attempt was made by Lorentz, the 19 Nobel prize winner for physis, who linearized the equations, introduing his linearized frition fator, and solved them in 1919 to ompute the effets of a losure dam on the Duth Zuyderzee, now IJssel lake. Ippen [1966] and Harleman [1966] solved the St. Venant equations for estuaries with onstant ross-setion and estuaries with an exponentially varying ross-setions (whih they alled "real estuaries"), respetively. However, they used the linearized equations, with linearized frition and a horizontal bottom. As a result their solutions were not very aurate. With the appearane on the sene of the digital omputer, in the 197s, the need for analytial solutions subsided. The quest for analytial solutions ontinued, but at a slower pae. The trigger seemed to be uriosity rather than neessity. Still, there were a number of good reasons for finding analytial solutions. Firstly there is the insight that analytial equations provide. It is possible to see diretly what the impat of an intervention, suh as dredging, is on the tidal dynamis and one an see how a ertain parameter affets another. Analytial equations are not only good for eduational purposes, but also to assess the outome of numerial models or to design a model set-up or field researh. The analytial solutions developed to date, onerning one-dimensional models, all require a ertain number of assumptions on topography and flow harateristis. In tidal hydraulis, the ross-setion is usually assumed as onstant (retangular, trapezoidal or triangular) or gradually varying, with an exponential (or power-law) longitudinal variation. Moreover, the tidal range to depth ratio is assumed muh less than unity, the Froude number small, and the fresh water disharge small ompared to tidal flows. Suh ommon assumptions are shared also by reent approahes, suh as by Jay [1991], Friedrihs and Aubrey [1994], Lanzoni and Seminara [1998], Prandle [3] and Savenije [5]. The differene is that most authors use perturbation analysis, where the saled equations are simplified by negleting higher order terms, whereas Savenije [5] uses a simple harmoni solution without simplifying the equations. Both methods give good results as long as the tidal range to depth ratio is small. In this paper we use an exponentially varying ross-setion, allowing for bottom slope, in ombination with the omplete non-linearized St. Venant equations. The methodology makes use of the set of impliit analytial equations presented in Savenije [5] whih is solved expliitly and ompared to observed tidal parameters in three different estuaries: the Shelde in The Netherlands, the Elbe in Germany, and the Tien and Hau (the two main branhes of the Mekong) in Vietnam. The relatively simple expliit equations obtained are potentially powerful in ombination with eologial or salt intrusion models, or to obtain first order estimates of the onsequenes of interventions in estuary topography (e.g. by dredging).. Formulation of the problem The oneptual sketh of the one-dimensional model of tidal wave propagation presented in this ontribution is shown in Figure 1, where x is the longitudinal oordinate measured in landward diretion from the mouth of the estuary, h is the flow depth, z is the water level flutuation, z b is bottom elevation, A is the ross-setional area of the stream profile. The ratio between the storage width and the stream width is r S. Hereafter, an over-bar denotes tidally averaged variables: thus h is the tidal average depth of flow and A is the tidal average rosssetional area. Moreover, we define Q as the tidal flow disharge, U as the tidal flow veloity, ρ as the water density.

3 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 3/7 In the following, we onsider a sinusoidal tidal wave having a tidal period T, a frequeny ω=π/t, and a wave length L=T, being the wave elerity. As shematially shown in Figure 1, υ is the amplitude of the tidal veloity U, and η is the amplitude of the tidal water level variation z. Another important parameter desribing the tidal wave is the phase lag ε between high water (HW) and high water slak (HWS) (or between low water (LW) and low water slak (LWS)). For a simple harmoni wave, ε=π/-(φ z -φ U ), where φ z is the phase of water level and φ U the phase of veloity..1 Basi equations The tidal dynamis in an alluvial estuary an be desribed by the following set of onedimensional equations [e.g. Savenije, 5]: U U h zb h ρ U U +U + g + g g + g =, (1) t ρ C h A Q r S + =, () t where g is the aeleration due to gravity, C is Chezy's frition fator; the longitudinal gradient of the water density ρ drives a residual water surfae slope. In the derivation of these equations two assumptions have been used, the first being that the Froude number is O(.1) or smaller. Subsequently a seond neessary assumption is that the hannel is well-defined, and that r S <. If not, then the seond term in equation (1) needs to be modified. For the purpose of saling, we rewrite (1)-() onsidering z as the water level flutuation in relation to the tidal average water level: h = z + h. (3) Assuming that the depth onvergene is small ompared to the width onvergene and the tidal amplitude to depth ratio is small, equations (1) and () an be modified into: U U z U U +U + g + gσ + g =, (4) t C h z z U hu A r S +U + h + =, (5) t A where σ = ( zb + h ) h ( ρ ) ρ inludes the free surfae residual slope and the density term. In alluvial estuaries, the tidal average ross-setional area A an be well desribed by an exponentially funtion x A = A exp -, (6) a where A is the ross-setional area at the origin (the mouth of the estuary), and a is the onvergene length. Equation (5) then modifies into: z z U hu r S +U + h =. (7) t a. Saling the equations In this setion we aim at identifying the dimensionless parameters that an be used to desribe the tidal wave harateristis in a given reah of the estuary, where suitable referene values of

4 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 4/7 the main quantities are adopted as onstant sales. Thus we introdue a saling on (4) and (7), similar to that used in Toffolon et al. [6], to derive dimensionless equations, the asterisk supersript denoting dimensionless variables: L U = υu, h = hh, z = ηz, x = π x T, t = π t g h, C = C, σ = σ, (8) f L where the sales have been already introdued and f is the dimensionless frition fator. The residual slope and density term σ, although already dimensionless, is resaled with the depth to length ratio. Assuming that the adopted sales are onstant in the domain of solution, we obtain: U υt U gηt z gth υtf U U + U + + = + t L x L x L σ, (9) h υ υ π C h z υt z hυt U hυt + U + h h U = t Lr S x Lr S x ar, (1) η πη S There is the problem to determine the real sales of veloity (υ) and wave length (L), so we use the values for a fritionless tidal wave in a hannel with zero onvergene (U, L ) as a referene: υ = U µ, (11) L = L / λ, (1) where we introdue the unknown veloity number µ and elerity number λ. Moreover, the dimensionless tidal amplitude ζ is defined: ζ = η / h. (13) Taking into aount the storage effet, the following relationships are valid for the lassial wave elerity (zero onvergene and no frition), the fritionless veloity amplitude U and the fritionless wave length for zero onvergene L : gh =, (14) r s U ζ r s L T =, (15) =. (16) Thus the dimensionless equations read U U λ z λ U U + rsζµλ U + + σ + µχ =, t µ ζµ C h (17) z z U + ζµλu + µλh µγh U =, t (18) where the dimensionless parameters χ and γ have been introdued as the frition number and estuary shape number, respetively: L χ = rs f ζ = rs f ζ, ωh π h (19) γ L = ωa π a () The estuary shape number γ depends on the ratio of the square root of the depth to the onvergene length and is the main indiator for estuary shape. Note that the (14), (15) and (19) are slightly different from the definitions used in Toffolon et al. [6] beause they inlude the effet of the parameter r s. As a onsequene, the orresponding definitions of the veloity number and the elerity number are:

5 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 5/7 1 υ 1 υh µ = =, (1) r ζ r η S L = S λ =, () L where = L T is the atual elerity of propagation of the tidal wave. Moreover, the definition of γ used in Toffolon et al. [6] only referred to the width onvergene, assuming a horizontal bottom, an unneessary assumption here. It is interesting to note that the dimensionless form of the governing equations (17)-(18) does not ontain r S expliitly, apart from the seond term of (17), whih is a small term ompared to the first term (as long as the Froude number r Sζµ is small), and hene the effet of r S, whih is lose to unity, may generally be disregarded in this term. If the fator r S is disregarded in this term, then these equations are idential to the equations used in Toffolon et al. [6], apart from the residual slope and density term..3 Analytial solutions Equations (1) and () an be written in the form of four impliit analytial equations, as desribed by Savenije [5; hapter 3]. These derivations are based on earlier papers by Savenije [199, 1993, 1998, 1] and Savenije and Veling [5]. Table 1 lists the assumptions made in the derivations of these equations. The basi assumption is that the ross-setional area of the estuary an be desribed by an exponential funtion, following equation (6), as is the ase in alluvial estuaries (assumption 1). A seond fundamental assumption for all equations is that the ratio of the tidal amplitude to depth is less than unity (assumption ). The third assumption is that the fresh water disharge is small ompared to the amplitude of the tidal disharge (assumption 3). Although during a flood situation or in the most upstream part of an estuary this may not always be the ase, in exponentially shaped estuaries this is not a restritive assumption. Horrevoets et al. [4] looked into the validity of this assumption, and onluded that it was aeptable in the downstream, tide dominated, part of alluvial estuaries. The three basi assumptions (1, and 3), indiated in bold in Table 1, are quite aeptable for small amplitude tidal waves in the downstream end of alluvial estuaries. The other assumptions listed in Table 1 are derivatives of these three basi assumptions. The requirement that the Froude number is small (assumption 4) is essentially the same as assumption, but less restritive; the Froude number is generally smaller than the amplitude to depth ratio (see equation (4)). Assumption 5, stating that the tidal wave an be desribed by a simple harmoni funtion, follows from assumptions and 3. If the tidal amplitude to depth ratio and the fresh water to tidal disharge ratio are small, then the tidal wave is not muh deformed by non-linear effets, but it is unavoidable that as the wave travels further inland, the wave deforms, resulting in a longer ebb and shorter flood duration. A non-restritive assumption is that the width to depth ratio should be large and that the storage width ratio should be modest (assumption 6). This assumption is not really important sine it only affets the seond term of equation (1), whih sales at the Froude number, and moreover alluvial estuaries have a large width to depth ratio. The requirement that the salt intrusion is partially or well mixed (assumption 7) relates to assumption 3, while it is not a restritive assumption. Assumption 8 requires that tidal damping is modest. It implies that the length sale of the damping/amplifiation proess is muh longer than the distane traveled by a water partile (the tidal exursion E). This assumption is also not restritive in alluvial estuaries, as is shown empirially (e.g. in Savenije [5]). Some methods require the absene of bottom slope, but this is not a neessary ondition in this approah, as long as the bottom slope effet is

6 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 6/7 inorporated in the gradient of the ross-setional area (i.e. in the onvergene length a). Finally, the derivation of the elerity equation through the methods of harateristis [Savenije and Veling, 5] uses the assumptions that the wave elerity and the phase lag between HW and HWS are onstant over an estuary reah (assumption 9) and that the tidal wave travels as a simple harmoni funtion. The latter orresponds with assumption 5, the former is an additional assumption. This assumption orresponds with an ideal estuary where there is no tidal damping or amplifiation and holds true if assumption 8 is stritly adhered to. Finally, we note that previous authors have proposed relationships whih may look similar to the ones summarized in this paper. However, in most ases they are valid only in some limiting ases (see for instane the omments below about Friedrihs and Aubrey s [1994] relationships). The four impliit analytial equations are summarized as follows: The phase lag equation ωa a dη tan ε = / 1 η d x (3) where is the elerity of the tidal wave for average depth, not ompensating for HW or LW. The phase lag ε is a ruial parameter defining the type of wave ourring in an estuary. If ε=, the tidal wave mimis a standing wave; if ε=π/, the tidal wave is a progressive wave. In onvergent estuaries the value of ε is always between and π/. A wave with a phase lag between and π/ is alled a "mixed tidal wave" [Savenije, 5]. We an see in equation (3) that if there is modest tidal damping or amplifiation, the phase lag depends solely on the ratio of the onvergene length to the wave length of the tidal wave (L=T ). This equation was derived by Savenije [199, 1993] from the onservation of mass equation (), using a simple harmoni wave in a Lagrangean referene frame. A similar equation was derived by Prandle [3] for strongly onvergent estuaries with a triangular ross-setion. We also note that, in the limit of standing waves (ε ) and vanishing amplifiation, equation (3) beomes ε=aω /, whih orresponds to equation 37 in Friedrihs and Aubrey [1994]. The saling equation r η υ 1 h = S sin ε (4) whih introdues the amplitude of the tidal flow veloity, υ. This equation, first published in Savenije and Veling [5], makes use of the analytial solution of the onservation of mass equation published in Savenije [1993] and equation (3). For a progressive wave, where ε=π/, this equation orresponds with the equation used by other authors (e.g. Friedrihs and Aubrey [1994], and Jay [1991]) to sale the St. Venant equations. Also for a stranding wave (ε ) with modest amplifiation, using the simplified version of (3) disussed above, equation (4) beomes υ=gω ηa/, whih orresponds to equation 11 in Friedrihs and Aubrey [1994]. The damping/amplifiation equation 1d η 1+ α 1 sin = f υ ε (5) η d x α a h where α = υ sinε ( gη) is a tidal Froude number, and f is the frition fator defined in equation (8). This equation follows from a ombination of equations (1), () and (6), and was derived by Savenije [1]. This equation followed from subtration of the two asymptoti solutions for high water (HW) and low water (LW). The two terms on the right hand side represent the balane between onvergene and frition. If the estuary is strongly onvergent

7 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 7/7 and smooth, the tidal wave is amplified; if it is moderately onvergent and rough, the tidal wave is damped. In an "ideal" estuary these terms anel out and there is no tidal damping. In the latter ase, the energy gained per unit volume of water by the onvergene of the banks is equal to the energy dissipated by frition. In the derivation of this equation the upstream boundary ondition used is that hannel is infinitely long (no weir or obstrution) and that the fresh water disharge is negligible. Horrevoets et al. [4] solved the equation for a nonnegligible disharge, whih yielded additional terms in the frition term (see Savenije [5], setion 3.3.3). The elerity equation 1 sinε osε υsinε = gh / 1 f = (6) rs ( 1+ α) ωa ωh 1 D where D is the damping term. If the damping term equals zero, then we have an "ideal" estuary where the wave elerity equals the lassial wave elerity. As already reognized by previous authors [e.g. Friedrihs and Aubrey, 1994], if the damping term is negative, then the wave is damped and the wave elerity is less than the lassial wave elerity; if the damping term is positive, then the wave is amplified and the wave elerity exeeds the lassial wave elerity. The upper boundary of the damping term is D=1. This situation orresponds with an "apparently standing wave". This is not the same as a standing wave due to full refletion. In fat, as in the ase of a strongly onvergent estuary, the inident wave alone results in an infinite wave elerity and, in so doing, mimis a standing wave [Jay, 1991]. The elerity equation was derived by Savenije and Veling [5] using the method of harateristis for an amplified or damped tidal wave. Rearranging (5)-(6) with (3) and (4), after some algebrai manipulations one an find (/ ) -1=dη/dx(1- a/η dη/dx)(/ω) /(aη); in the ase of small amplifiation, this orresponds to a manipulation of equation 9 in Friedrihs and Aubrey [1994]. Hene, a ombination of methods was uses to derive the analytial equations. The phase lag equation and the saling equation are based on a Lagrangean approah, where the integration is made following a water partile. The damping equation followed from substitution of these equations in the momentum balane equation and solving for asymptoti solutions. The fourth equation, however, was derived by the method of harateristis. In ombining these approahes inonsistenies may arise, whih may limit the appliability of the method. However, sine any analytial (linear) solution of the non-linear equations is an approximation with limited appliability, this may not be a problem, as long as the result is as aurate, or more aurate than the methods used thus far..4 The governing dimensionless equations These four equations an be saled making use of the dimensionless parameters derived in Setion.: the estuary shape number γ given by (), the veloity number µ given by (1), the elerity number λ given by (), the frition number χ given by (19), and a damping number δ, defined as: 1dη δ = η d x ω. (7) Substitution of these dimensionless parameters in equations (3)-(6) yields the following four dimensionless equations.

8 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 8/7 The phase lag equation λ tan ε = (8) γ δ The saling equation sinε osε µ = = λ γ δ (9) The damping/amplifiation equation µ δ = ( γ χµ λ ) (3) µ + 1 The elerity equation osε λ = 1 D = 1 δ = 1 δ ( γ δ ) (31) µ This is an impliit set of four equations that an ompute: the elerity of tidal wave propagation, the tidal damping/amplifiation, the phase lag and the veloity amplitude for a given geometry, frition oeffiient, and tidal amplitude to depth ratio. We see that the St. Venant's equations (1) and () are two equations with two unknowns (U and h ) for given topography and frition, as a funtion of x. Equations (8)-(31) form a set of four equations with four unknowns (ε, λ, δ and µ ) that an be solved for a given topography γ, and frition χ. It was reognised by Jay [1991], building on previous work by Green [1837] and Prandle and Rahman [198], that the two main fators governing tidal propagation are frition and onvergene. In this paper these fators are desribed by γ and χ, similar to Jay's [1991] parameters - l and R'/ω. The effet of the storage ratio r S has already been inluded in the dimensional saling. We also note that the frition parameter χ linearly depends on the dimensionless tidal amplitude ζ. It is, in fat, through the frition parameter that the tidal range to depth ratio omes into the equations. Hene, we onsider γ and χ as the independent variables and ε, λ, δ, and µ as the dependent variables (see Toffolon et al. [6] for an estuarine haraterization in terms of external variables). The equations (8)-(31) represent a loal solution, beause they relate the veloity amplitude, the phase and length of the tidal wave and the amplitude longitudinal variation (amplifiation or damping) to the loal referene value of the tidal amplitude η itself (along with the geometrial harateristis of the hannel). 3. Solution of the set of equations The set of equations (8)-(31) an be solved iteratively, but it is also amenable to an analytial solution, as shown by Toffolon et al. [6]. Given the non-linear harater of the system, different families of solutions an be derived. Making use of the trigonometri equation ( osε ) = 1+ ( tan ε ) be ombined to eliminate the variable ε to give:, equations (8) and (9) an

9 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 9/7 ( γ δ ) = λ 1. (3) µ Equations (3) and (31) an be rewritten as: δ γ δ = χµ λ +, (33) µ 1 λ γ δ =. (34) δ Isolating λ from (34) and substituting into (3) and (33), we an write two equations in the unknowns δ and µ. After some algebra, it is possible to obtain a single tenth order equation for µ: 6 4 ( χ µ + γχµ + µ )( µ γµ + 1)( µ + γµ + 1) =. (35) 4 µ ( γχµ + µ + ) The denominator of (35) is always stritly positive in the physially meaningful ases. Conerning the numerator, the relationships delimited by the first and seond parentheses give rise to two different families of solutions (the first family representing the mixed tidal wave and the seond family the "apparent standing" wave), whih will be onsidered separately below. The equation delimited by the third parenthesis gives no positive roots for µ. 3.1 Solution for the mixed wave (the first family of solutions) We an now derive simpler relationships onsidering only a single family of solutions. Introduing (33) and (34) into (3), we end up with 1 λ δ 1 + ( 1 ) = χµ λ, (36) µ whih an be simplified for λ, negleting the seond family of solution. In this ase, (36) along with (33) gives a simple relation between δ and µ γ χµ m δ m =. (37) where the subsript m denotes the mixed tidal wave. Substituting (37) into (34) it is also possible to find 4 χ µ m γ λm = + 1. (38) 4 These two equations for δ m and λ m an be used to eliminate δ and λ from (3). This leads to a single equation for µ m : χ µ = m γχµ m µ m, (39) whih orresponds to the first family of solution of (35) and represents the propagation of a mixed wave with <ε<π/. It an be solved as a third-order equation in µ. The real solution is µ m = 1 γ 6 m γ + 3χ m (4a) with: [ 7χ + ( 9 γ ) γ χ + ( 9 γ ) γχ + 8 γ ] 1 3 m =, (4b) With this solution for µ m, expliit solutions an be obtained for λ m, δ m, and ε m, by substitution in (38), (37) and (8), respetively.

10 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 1/7 Equations analogous to (39) an be written also for the other unknowns. For instane, the equation for λ reads [ ] λm χ ( γ 4)( + 1) 4 ( γ 4) λ + ( γ 4) + 4( γχ + 1) [ ] = 6 16λm + 8 m γχ, (41) whih an be seen as a third-order equation in λ m. The solution is physial only if real roots exist, i.e. if λ. If λ = the threshold ondition, aording to (41), reads χ ( γ 4)( γ χ + 1) =, (4) where γ is the limit for ritial onvergene, defined as the threshold ondition for the transition from the mixed tidal wave (first family of solutions) to the "apparent standing" wave (seond family). Below we show that the seond family of solution is ompletely determined by the onvergene alone. Hene, ritial onvergene an also be defined as the limit for whih the solution is influened by frition. For weak frition, this boundary for ritial onvergene is similar to the one defined by Jay [1991] as the onvergene rate at whih onvergene and aeleration effets are equal and opposed. For a fritionless tidal wave γ =, whih is the same value as obtained by Jay [1991]. At this point the phase lag approahes zero, whih is similar to the impedane phase φ of 9 o, obtained by Jay [1991]. On the other hand, when frition beomes important, the two definitions of ritial onvergene lead to different results. The relationship (4) defines the region of existene of ritial onvergene in the χ-γ plane and an be solved for one of the two parameters as a funtion of the other one. The expression for χ where γ=γ reads: 1 ( ) ( γ ) χ ( γ ) = γ γ 4 + γ 4, (43) where the ondition γ (always satisfied, beause γ as shown below) has been used to simplify the solution. The solution for γ is a more omplex expression: 1 m1 (1χ + 1) γ = 1+ (44a) 3χ m1 with ( 3χ + 8) 8 + 1χ 3 ( χ ) ( 7 ) m 1 = 36χ χ 4. (44b) The threshold ondition is shown in Figure : the solution for the mixed wave exists for γ < γ ( χ ), where γ is the threshold for ritial onvergene, whih inreases with χ starting from γ = for the fritionless ase Solution for an estuary with onstant ross-setion A speial ase of the mixed wave solution is the situation where there is no onvergene, i.e. when γ=. In that ase equations (37), (9) and (31) yield: χ δ = µ, (45) χ 3 os ε = µ δ = µ, (46a)

11 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 11/7 tan ε 1+ δ λ = = (46b) δ δ 1 χµ λ = +, (47) where the subsript denotes the zero onvergene situation. Using (4), and the ondition that γ=, we an derive expliit expressions for µ, δ, ε and λ : m 6 µ =, (48) 3m χ m 6 δ =, (49) 6m 3 1 m 6 osε = 3, (5a) χ m 6m = + m 6 tanε 1 (5b) m 6 λ = 1+ 6 m, (51) with: m = 3 χ + χ +. (5) 7 With these equations the damping and phase lag of a wave in an estuary with a onstant rosssetion an be omputed. We see that the damping, wave propagation and phase lag purely depend on the frition ratio, whih is in agreement with the literature (e.g. Lorentz [196], Lamb [193], Dronkers [1964], Ippen [1966], Van Rijn [199]). In these solutions the tidal amplitude is damped exponentially and the tidal wave elerity is redued aordingly. All these authors used linearized St Venant equations and a onstant damping number (exponential damping). As a result, their equations for the tidal wave propagation in a hannel of onstant ross-setion under the influene of frition are less aurate and less general than the solution presented here. In the speial ase of no frition we obtain m = 6 and hene, as expeted, that λ =1 (the lassial wave elerity) and δ = (no damping). One an also demonstrate that as χ approahes zero, µ approahes unity and ε approahes π/: the ase of the undamped progressive wave. 3.3 Solution for the apparent standing wave (the seond family of solutions) It is also worthwhile to onsider the solution in the ase of an apparent standing wave (the wave is not a standing wave in the formal sense; rather it is an inident wave that mimis a standing wave). For an apparent standing wave λ = applies, whih is the solution we have exluded in the simplifiation of equation (36) used to derive the first ase. Imposing λ = in the set of equations, we find that (3) is redundant sine it is exatly the produt of (33) and (34). Then, equating the latter two equations, one easily obtains:

12 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 1/7 µ s γµ s + 1 =, (53) µ =, (54) s δ s ε s =, (55) where the subsript s stands for the standing wave solution, whih pertains to the ase where γ > γ χ. The last equation follows diretly from (8). ( ) This ase orresponds to the seond family of solutions of equation (35). Equation (53) gives real solutions only if γ, whih means beyond ritial onvergene; the physial solution hene reads: 1 µ ( 4) s = γ γ. (56) If γ goes to infinity, this equation will approah µ s = 1/ γ. Similarly we an derive for the damping ratio: 1 δ ( 4) s = γ γ, (57) whih again gives the limit δ = 1/ γ when γ goes to infinity. s The threshold ondition (4) ensures that the transition between the two families of solution is smooth from the mixed tidal wave, given by (41), to the apparent standing wave where λ =. It is possible to demonstrate that also the transition between the solutions (4) and (56) is ontinuous. 3.4 Ideal estuary An interesting speial ase of the first family of solutions (the mixed tidal wave) is that of the ideal estuary (subsript I), where there is neither tidal damping nor amplifiation. Hene, where: δ I =. (58) In this ase, the system (3)-(34) yields: λ I = 1, (59) 1 γ µ I = =, γ + 1 χ I (6) whih is the ondition of solubility (representing the relationship between frition and onvergene parameters neessary to obtain a balane between damping and amplifiation of the wave amplitude). This yields: χ ( I = γ γ +1), (61) whih has been tested against numerial results in Toffolon et al. [6]. The phase lag an be estimated from equation (8): 1 tan ε I =. γ (6) In addition, it follows from substitution of these values in (9) that: sinε I = γ 1 = χ γ I + 1 (63) and:

13 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 13/7 3 γ 1 osε I = = γ (64) χ γ + 1 I These relations for χ I and ε I are also presented in Figure. 3.5 Fritionless estuary Finally, a speial ase of the first family of solutions is the fritionless estuary. The seond family of solutions, already is independent of frition. This is beause in the damping equation (5) the frition is neutralised if ε=. In fat, an apparent standing wave has zero veloity at HW and LW and, as a result, the frition term falls out of the equation in the derivation of (5) [Savenije, 1]. If a mixed wave has an infinitely small tidal amplitude, or if the estuary is fritionless, we also obtain χ=. Under that ondition, the expression for the veloity number (39) simplifies substantially, leading to: µ f = 1 (65) where the subsript f denotes the fritionless situation. In addition we find: δ f = γ / (66) ( / ) λ f = 1 γ (67) and osε f = γ /. (68) The resulting expression for ε f is shown in Figure. The speial ase of a fritionless estuary with onstant ross-setion (γ=) results in a purely progressive wave, with ε=π/, δ= and λ=1. In the next setion, the urves representing the fritionless situation will be learly visible in the graphial representations. The equations for δ f and os(ε f ) are simply straight lines; the equation for λ f is a parabola. Surprisingly, the equations desribing os(ε) and δ appear to behave as near-straight lines for the ases with frition as well. 4. Results and disussion 4.1 Graphial representation Figures 3-7 present the solution of the phase lag, the veloity number, the damping number, the elerity number and the damping number as a funtion of γ and χ. What we see very learly in these figures is that there exist two distint types of estuaries that show very different behaviour. For large values of γ (the strongly onvergent estuaries) we see a behaviour that no longer depends on the frition parameter χ and whih orresponds to an apparent standing wave (the seond family of solutions, where ε=). In this solution, λ=, implying an infinite elerity or, equivalently, an infinite wave length; thus HW and LW our instantaneously along the estuary. For smaller values of γ (the weakly onvergent estuaries) we see a pattern that is strongly dependent on χ and whih orresponds to the lass of estuaries with a mixed tidal wave (the first family of solutions, where <ε<π/). The transition zone in the middle is the area where ritial onvergene ours (with values of γ >). The set of points where ritial onvergene ours is desribed by (4), with as a speial point the fritionless ritial onvergene, whereγ =. This fritionless ritial onvergene oinides with the orresponding ritial onvergene desribed by Jay [1991]

14 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 14/7 and defines the point where the tidal wave approahes a standing wave due to the onvergene of the banks. Tidal waves are of the apparent standing wave type in short estuaries that are fored by a steep topography (as desribed e.g. by Wright et al., 1973); estuaries with a mixed tidal wave our in river valley estuaries with a long oastal plain, as desribed by Savenije [5]. In the former ase the length of the estuary is typially one quarter of a wave length (with a node at the upstream end of the estuary), whereas the latter is muh longer. We see that for a very small tidal amplitude (χ ), or fritionless wave, this happens at γ=, or at a wave length L =4πa. Aording to Wright et al., [1973], short estuaries with a standing wave have a length of L /4. Apparently the ritial onvergene for an estuary to develop an apparent standing wave (under near fritionless onditions) is L /(4π), and the length of the estuary is πa. Substitution of this length in (6) shows that this orresponds to a point where the ross-setional area has redued to 4% of the area at the mouth: A / A = exp( π ).4. In Figure 3 we see how the phase lag develops as a funtion of γ and χ. Figure 3a shows the variation of os(ε) with χ and Figure 3b of sin(ε). It is astonishing that the urves for os(ε) seem to be straight lines. The fritionless ase is indeed a straight line, aording to (68), but the urves for χ> also approah straight lines. The interept of the line is given by (46) and the line reahes os(ε)=1 at γ = γ, aording to (44). In Figure 4 we see that the maximum value of the veloity number, µ = 1, is reahed for χ=. The urves learly have two branhes, representing the two families of solutions. At ritial onvergene the urves hange from a near linear relation to a hyperbole-like urve, earlier desribed by Toffolon et al. [6]. The urve obeys (56) and for high values of γ tends to µ 1/γ. Figure 5 shows a similar piture for the damping number δ. For lower values of the estuary shape number, almost straight lines are found, whih hange into a hyperbole after the transition zone. In the fritionless ase (66) applies as an exat straight line. The interepts of these lines orrespond to (49). The transition of the lines to the hyperbole ours at ritial onvergene where γ = γ. Beyond ritial onvergene, (57) applies, whih approahes δ=1/γ for large values of γ. In this region there is always amplifiation, albeit modest. The line where δ= orresponds with the set of ideal estuaries. The point where both δ= and γ= orresponds with the fritionless progressive wave. Figure 6 shows a similar piture for the elerity number λ. Ideal estuaries our where λ=1. Tidal damping orresponds with a wave elerity below the lassial wave elerity (λ>1); tidal amplifiation orresponds with a wave propagation that exeeds the lassial wave elerity (λ<1). Figure 7, whih is losely related to Figure 6, shows that the damping term D=1 for large values of γ. What we annot see from these figures is how in a given estuary, with a given topography and frition, the parameters hange as a funtion of x, beause ζ, and hene χ, is not neessarily onstant with x. Only in an ideal estuary, where ζ is onstant and hene all parameters are fixed for a given value of γ, estuaries an be haraterised by a single dot in these graphs. If these variables are not onstant, then an estuary is represented by a line segment. For a onstant value of γ this is a vertial line segment. If γ varies along the estuary axis (e.g. beause of shallowing), then the line segments an go aross the graphs. In the following

15 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 15/7 setion the methodology to draw suh a line segment is presented and subsequently examples of real estuaries are given. 4. Longitudinal solution for tidal wave propagation in real estuaries To represent the estuary parameters of tidal wave propagation along the estuary axis, a longitudinal solution is required. The water level variation along the estuary axis annot be read diretly from the derived equations, but it is desribed by the integration of the damping number δ. The omputation depends on the two independent variables γ and χ. For a given topography, storage width ratio and frition fator, the value of γ an be omputed, but χ still depends on the tidal amplitude, whih an only be obtained from integration of the damping number. This is done by simple expliit integration of the linear differential equation. Hene, the set of equations an be solved by seleting a tidal amplitude at the seaward boundary η, and by integration of the damping number over a distane x, leading to a tidal amplitude η 1 at a distane x upstream, whih is repeated for the entire length of the estuary. Figure 8 shows the longitudinal omputation applied to the Shelde and Elbe estuaries at different dates, with different tidal amplitudes. Table presents data of the Shelde, Elbe, Tien (Mekong) and Hau (Mekong) on whih these omputation and the ones in the next setion are based. Chezy's oeffiient has been determined with the Strikler-Manning 1/6 relation C = Kh with K=45 m 1/3 s -1. In both estuaries, the orrespondene with observations is good. 4.3 Comparing different estuaries In Figures 9-1 we see the graphs for the phase lag, the elerity number, the damping number and the veloity number. In these graphs the Shelde, Elbe, Tien and Hau estuaries are represented by line segments of red, blue, dark green and light green olor, respetively. Next to the segments the distane in km is written indiating the length over whih a segment is representative. We an see that in the Shelde the downstream parts of -11 km forms a vertial line segment with a onstant estuary shape number. Further upstream the pattern beomes more irregular due to shallowing. The Hau has a onstant reah from km and the Tien from km. The Elbe has a less regular depth profile and hene does not have a onstant estuary shape number. For the data used, referene is made to Table. The tidal amplitude represents a normal spring tide and the river disharge an indiation of dry season flow. Of the four estuaries, the Tien and the Hau have the smallest estuary shape number. They are learly under the influene of river disharge, whih auses these estuaries to experiene more damping than an ideal estuary would. They have a riverine harater with a long onvergene length and a high phase lag. The Shelde is a more marine estuary with the largest estuary shape number, the smallest phase lag (losest to the apparent standing wave) and it is amplified. The Elbe lies in between; the tidal wave is not muh damped nor amplified and the estuary behaves almost as an ideal estuary.

16 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 16/7 In Figure 9 we see that there is a transition range near γ=1.5 where the lines for different values of χ ross. For lower estuary shape numbers, the phase lag dereases with frition; for higher values, the phase lag inreases with frition. In the transition zone, the range of phase lags is very small and not very sensitive to frition (around 7 minutes for an M tide). The Elbe lies in this transition zone. We see that the natural estuaries lie lose to the line of the ideal estuaries. Marine estuaries experiene amplifiation if they lie below the line of the ideal estuary; riverine estuaries, however, experiene damping if they lie below the line. 5. Conlusions The expliit analytial solutions and the graphs presented in this paper offer the opportunity to look into the funtioning of the St. Venant equations, without the need for a hydrodynami model. In fat they allow us to determine the tidal veloity amplitude, the rate of tidal damping and the wave elerity diretly on the basis of a geometry indiator, frition and tidal foring. The method is aurate ompared to earlier analytial methods and even very aurate in the marine part of an estuary where the tide dominates over the river disharge. This happens to be the area where salinity intrusion and estuary eology is relevant, whih makes this method very powerful to be ombined with salt intrusion models, eologial models or deision support systems. In addition, the method an be used diretly for the variable of interest, e.g.: the tidal amplitude, the veloity amplitude, the wave elerity, or the phase lag, without the need to look at the entire proess. One only requires the loal geometry, the tidal amplitude and the frition. Espeially the phase lag between HW and HWS is not a parameter easily determined in other methods. The method shows a satisfatory omparison with field data, partiularly in view of the simpliity of the geometri shematisation. This indiates that the exponential funtion used is a natural shape of alluvial estuaries and that the funtion obeys natural laws of selforganisation, in line with the "ideal estuary" [Savenije, 5]. This paper shows that lassifiation of estuaries should be based on two parameters, the estuary shape number γ and the frition sale χ (see also Toffolon et al. [6]). The phase lag is a key parameter to lassify an estuary. Sine natural estuaries appear to follow the trend of ideal estuaries, the position that an estuary has in Figure 9, identifies it as, e.g.: a riverine estuary with a low estuary shape number and a high phase lag whih is mainly damped (Mekong), a marine estuary with a high estuary shape number and a small phase lag whih is mainly amplified (Shelde), or a transitional estuary with an estuary shape number in the order of 1.5, a phase lag lose to.π (7 minutes for an M tide) and almost zero damping (Elbe). Aknowledgement The authors would like to thank the two anonymous referees for their valuable omments and suggestions, whih have greatly improved this paper.

17 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 17/7 Referenes Dronkers, J.J. (1964), Tidal omputations in rivers and oastal waters. North Holland, Amsterdam. Friedrihs, C.T., and D.G. Aubrey (1994), Tidal propagation in strongly onvergent hannels, J. Geophys. Res., 99(C), Green, G., (1837), On the motion of waves in a variable anal of small depth and width, Trans. Cambridge Philos. So., 6, Harleman, D.R.F. (1966), Tidal dynamis in estuaries part II: Real estuaries. In Ippen et al., Estuary and Coastline Hydrodynamis, MGraw Hill, New York, USA. Horrevoets, A.C., H.H.G. Savenije, J.N. Shuurman, and S. Graas (4), The influene of river disharge on tidal damping in alluvial estuaries. Journal of Hydrology, 94(4): Ippen, A.T. (1966), Tidal dynamis in estuaries Part I: Estuaries of retangular setion. In Ippen et al., Estuary and Coastline Hydrodynamis, MGraw Hill, New York, USA. Jay, D.A. (1991), Green s Law Revisited: Tidal long-wave propagation in hannels with strong topography, J. Geophys. Res., 96(C11),,585-,598. Lanzoni, S., and G. Seminara (1998), On tide propagation in onvergent estuaries, J. Geophys. Res., 13, 3,793-3,81. Lamb, 193, Hydrodynamis. Cambridge University Press. Lorentz, H.A., (196), Verslag Staatsommissie Zuiderzee. Algemene Landsdrukkerij, The Hague (in Duth). Prandle, D. (3), Relationships between tidal dynamis and bathymetry in strongly onvergent estuaries, J. Phys. Oeanogr., 33, Prandle, D., and M. Rahman (198), Tidal response in estuaries, J. Phys. Oeanogr., 1, Savenije, H.H.G. (199), Lagrangean solution of St. Venant's equations for an alluvial estuary, J. Hydraul. Eng., 118(8), Savenije, H.H.G. (1998), Analytial expression for tidal damping in alluvial estuaries, J. Hydraul. Eng., 14(6), Savenije, H.H.G. (1), A simple analytial expression to desribe tidal damping or amplifiation, J. Hydrol., 43, Savenije, H.H.G., and E.J.M. Veling (5), The relation between tidal damping and wave elerity in estuaries, J. Geophys. Res., 11, C47, doi:1.19/4jc78. Savenije, H.H.G. (5), Salinity and Tides in Alluvial Estuaries, Elsevier Publishers.

18 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 18/7 Toffolon, M., G. Vignoli, and M. Tubino (6), Relevant parameters and finite amplitude effets in estuarine hydrodynamis. J. Geophys. Res., 111, C114, doi:1.19/5jc314. Van Rijn, L., 199. Priniples of fluid flow and surfae waves in rivers, estuaries, seas and oeans. Aqua Publ., Amsterdam. Wright, L.D., J.M. Coleman, and B.G. Thom (1973), Proesses of hannel development in a high tide range environment: Cambridge Gulf-Ord river delta, Western Australia, Journal of Geology, 81,

19 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 19/7 List of Symbols symbol name dimension a onvergene length of ross-setional area L A ross-setional area of flow L A ross-setional area at the mouth L wave elerity LT -1 lassial wave elerity LT -1 C Chezy's frition fator L 1/ T -1 D damping term - E tidal exursion L f frition fator - g aeleration due to gravity LT - h ross-setional average depth of flow L K Manning-Strikler frition fator L 1/3 T -1 L wave length L L lassial wave length L L e estuary length L m parameter - Q tidal disharge L 3 T -1 Q f fresh water disharge L 3 T -1 r S storage width ratio - t time T T tidal period T U ross-setional average flow veloity LT -1 x distane L z water level L z b bottom level L γ estuary shape number - δ damping number - ε phase lag - ζ tidal amplitude to depth ratio - η tidal amplitude L η tidal amplitude at the mouth L λ elerity number - µ veloity number - ρ density of water ML -3 υ tidal veloity amplitude - χ frition number - ω tidal frequeny T -1

20 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE /7 Table 1. Equation name phase lag equation saling equation damping equation elerity equation Assumptions made for the derivation of the basi equations equation number referenes assumptions made (3) Savenije [199] 1), ), 3), 4), 5), 6), 7), 8) Savenije [1993] (4) Savenije [1993] 1), ), 3), 4), 5), 6), 7), 8) (5) Savenije [1] 1), ), 3), 4), 5), 6), 7), 8) (6) Savenije and Veling [5] 1), ), 3), 4), 5), 6), 7), 8), 9) legend: 1) exponentially varying ross-setion; a is onstant ) the tidal amplitude to depth ratio is small; ζ<1 3) fresh water disharge is small ompared to the tidal flow 4) small Froude number; this follows diretly from assumption ) 5) simple harmoni funtion to desribe the tide 6) depth to width ratio is small and storage width lose to unity; h/b<<1, r S < 7) the salt intrusion is well-mixed; this is essentially the same as assumption 3) E dη 8) tidal damping is small; << 1 η d x 9) the phase lag ε and the wave elerity are onstant along an estuary reah; this is the ase in an ideal estuary, where damping is small; it is a strit interpretation of assumption 8). Table. Geometri and tidal harateristis of the estuaries studied Estuary name η (m) Elbe 1.5 Depth Estuary Length L e (km) Amplitude Crosssetional area Convergene length Storage width ratio h A a r S (m) (m (km) (-) ) -6km 6-14km , -6km 6-14km -6km 6-14km Shelde km 11-18km 19 6, -11km 11-18km -11km 11-18km Hau km 57-16km 3 6,6-57km 57-16km -57km 57-16km Tien km 45-18km 3 1,6-45km 45-18km -45km 45-18km ) means a depth inrease from 7. to 9. m over the reah -6 km River disharge Q f (m 3 /s)

21 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE 1/7 Figure 1. Sketh and notation. Figure. Phase lag ε and frition parameter χ as a funtion of onvergene parameter γ γ χ from equation (43), under speial onditions: threshold line for ritial onvergene ( ) disriminating between the two families of solution; onditions for an ideal estuary χ I ( γ ) and ε I ( γ ) from equations (61) and (6); phase lag behavior in a fritionless estuary from equation (68).

22 SAVENIJE ET AL.: ANALYTICAL SOLUTIONS OF TIDAL WAVE /7 3a) Figure 3. Relationship between sin ε, os ε and the estuary shape number γ for different values of the frition number χ, indiated by different line types. The blue line with dots represents the ideal estuary (eq. 63 for sin ε and eq. 64 for os ε). 3b)