JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,

.

An edited version of this paper was published by AGU. Copyright (2011) American Geophysical Union.Toffolon, M., and Savenije, H. H. G. (2011), Revisiting the linear solution for estuarinehydrodynamics, J. Geophys. Res., doi:10.1029/2010JC006616.

Revisiting linearized one-dimensional tidal propagation

Marco ToffolonDipartimento di Ingegneria Civile e Ambientale, University of Trento, Italy.

Hubert H. G. SavenijeDepartment of Water Management, Delft University of Technology, Delft, Netherlands.UNESCO-IHE Institute for Water Education, Delft, Netherlands.

Abstract.In this paper we extend the validity of the classical linear solution for tidal hydrody-

namics including the effects of width and depth convergence. Reworking such a solutionin the light of externally defined, dimensionless parameters, we are able to provide sim-ple relationships to predict the most relevant features of the tidal wave at the estuarymouth (velocity amplitude, phase lag, wavelength, damping), and to reproduce the maindynamics of tidal wave propagation along finite- and infinite-length channels. We alsohighlight the need for an accurate treatment of the linearized bed shear stress by ex-ploiting an iterative procedure, and we show the improvement that can be reached bysubdividing the entire estuary in shorter reaches. Different versions of the analytical so-lution are compared with numerical results, highlighting the strengths and weaknessesof the linear model.

1. Introduction

Analytical solutions for tidal hydrodynamics have beenthe subject of extensive study exploiting different assump-tions for different classes of estuaries [Lanzoni and Semi-nara, 1998, e.g.]. The influence of along-channel geometri-cal variations (primarily funneling) has been the object ofseveral papers, which studied in particular infinitely long es-tuaries for which elegant solutions can be found [Jay, 1991;Friedrichs and Aubrey, 1994; Lanzoni and Seminara, 1998;Kukulka and Jay, 2003; Savenije et al., 2008, e.g.]. In thecase of a landward barrier or when riverine discharge hasto be considered, the presence of a reflected wave tendsto make the problem more complicated. A way to over-come this difficulty is to linearize the governing equations,an approach that has been applied even recently [Souza andHill, 2006, e.g.]. In fact, classical linear solutions [Dronkers,1964], based on the assumption that the tidal amplitude issmall with respect to the flow depth, have been the refer-ence test for a long time, until the appearance of personalcomputers. From that time onward, numerical simulationsgained ground because of their capability to describe com-plex dynamics, but the interest for simplified, explicit solu-tions for estuarine hydrodynamics remained.

One of the main issues in deriving a linearized solution isthe treatment of the friction term. In his seminal work,Lorentz [1926] proposed a linearization of the bed shearstress which has been the basis for simple solutions for manydecades. This approach requires a suitable estimate of thelinearization constant through a reference velocity, which is

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unknown. Although iterative approaches for its determina-tion have been proposed, this is often assumed as a calibra-tion parameter. A problem of a different nature comes fromthe need to follow along-channel variations of the estuarinesections, and has been tackled by subdividing the estuary inmultiple reaches. These solution techniques (iteratively de-termined friction and multi-section models) have been usedby many researchers, for instance, Giese and Jay [1989] andJay and Flinchem [1997].

In this paper we propose a modified derivation of the lin-ear model, taking into account the effect of the variationof channel width and depth explicitly. Working in the con-text of a harmonic analysis, we investigate the behavior ofthe first harmonic (usually the most important componentfor water level and velocity oscillations) of the tidal signal,assumed to be the sum of sine waves characterized by sta-tionary parameters. In order to derive simple yet generalrelationships, we analyze the problem using a dimensionlessformulation that separates external and internal parameters[Toffolon et al., 2006]. We examine three different models,which differ from each other because of the iterative refine-ment of the friction term and the subdivision of the estuaryin multiple reaches. Therefore, given the tidal forcing andthe geometry of the channel, we provide simple relationshipsfor velocity amplitude, phase lag, wavelength, and damp-ing, whose knowledge is especially important at the estuarymouth. We compare such results with a recent analyticalsolution [Savenije et al., 2008] and with the solution of afully non-linear numerical model.

2. Formulation of the problem

We describe the tidal wave propagation by means of theusual one-dimensional continuity and momentum equations

σ∂D

∂t+ U

∂D

∂x+D

∂U

∂x+UD

B

dB

dx= 0 , (1)

1

X - 2 TOFFOLON AND SAVENIJE: LINEARIZED TIDAL PROPAGATION

Figure 1. Sketch of water levels in the tidal channel.

∂U

∂t+ U

∂U

∂x+ g

∂H

∂x+

τ

ρD= 0 , (2)

where D is the depth, U the velocity, H the free surfaceelevation, ρ the water density, g the gravity acceleration, tis time, x is the longitudinal coordinate directed landward,

τ

ρ=

|U |UC2

h

(3)

is the bed shear stress (Ch is the dimensionless Chezy pa-rameter). We assume that the flow is concentrated in amain rectangular cross section with area BD, with a possiblepresence of lateral storage areas [Speer and Aubrey, 1985],whose effect is described by the storage ratio σ [Savenije etal., 2008, e.g.], and mainly determines an increase of flowvelocity [Seminara et al., 2010]. The possible dynamic ef-fect of the storage areas on the momentum equation (2) isneglected as a first approximation (see also Toffolon andLanzoni [2010]).

We also assume that the hydrodynamics is tide-dominated and that the semidiurnal M2 tidal component isdominant, with frequency ω = 2π/T and period T ≃ 12.4h.Then, introducing the mean depth Y = ⟨D⟩ (where anglebrackets represent tidal average), the water level oscillationsare described by the tidal amplitude A with respect to themean depth Y , and the bed elevation (fixed in time, butvarying along the channel) is given by (−Z) = H − D =⟨H⟩ − Y (see Figure 1). The amplitude of the semidiur-nal component of velocity oscillations is represented by thevariable V .

In order to recast the problem in dimensionless form, weuse the values of average depth Ds (= Ys), width Bs, andwave amplitude As at the seaward end of the channel asreference scales (identified by a subscript s). The resultingscale for the (a priori unknown) velocity amplitude Vs isgiven by σϵCs [Savenije et al., 2008], where Cs is the fric-tionless wave celerity

Cs =

√gDs

σ(4)

and ϵ is the dimensionless tidal amplitude (see also Table2). The natural choice for the length scale is Ls = Cs/ω.As usual, we describe the width of funnel-shaped estu-aries by means of an exponential convergence law B =Bs exp(−x/Lb). In this case B−1dB/dx = L−1

b is a con-stant, and the convergence term in (1) is null for Lb → ∞.In the case of an estuary of finite length, then its length isindicated by Le.

In the case of negligible river discharge, the estuarine hy-drodynamics are controlled by a few dimensionless parame-ters [Toffolon et al., 2006; Savenije et al., 2008]. The param-eters that depend only on the geometry and on the externalforcing are listed in Table 2, with reference to the scales de-fined at the seaward end of the channel. Other parameters,

which depend on the resulting tidal motion in the channel(mainly because they are concerned with the velocity), arelisted in Table 3, where ϕA and ϕV are the phases of the wa-ter level and velocity oscillations, respectively, and LA andLV are the actual wavelength of water level and velocity,respectively. It is also worth noting that λ can be seen asthe ratio between the frictionless wave celerity Cs and theactual wave speed [Savenije et al., 2008], i.e. λA = Cs/CA,λV = Cs/CV , CA and CV being the celerities of the waterlevel and velocity waves, respectively.

In order to find a simple analytical solution, we adopt theusual Lorentz’s linearization of the bed shear stress [Lorentz,1926]

τ

ρ= rU , r = κ

U

C2h

, κ =8

3π, (5)

where U is a reference maximum velocity, which shouldstrictly vary with x, but it is usually assumed as a constant.A natural choice is to select U = Vs, which is however un-known a priori. Then, a dimensionless friction parametercan be defined for the linear stress as follows

χ =r

ωDs= κ

Vs

C2hωDs

. (6)

It is clear that χ is not an external parameter, since it de-pends on the resulting velocity Vs. Nevertheless, a simplerelationship exists with χ in the form

χ = κµχ . (7)

2.1. Linear solution

In order to linearize the differential system, we assumethat the tidal oscillations are small with respect to the depth(ϵ ≪ 1) and introduce the following structure for the un-knowns (a superscript star denotes dimensionless variables),

H = ϵDs [A∗ exp(iωt) + c.c.] /2 + h.o.t. ,

D = Ds Y∗ + ϵDs [A

∗ exp(iωt) + c.c.] /2 + h.o.t. ,

U = σϵCs [V ∗ exp(iωt) + c.c.] /2 + h.o.t. , (8)

where A∗ and V ∗ are unknown complex functions (c.c. rep-resents the complex conjugate of the preceding term), andh.o.t. represent higher order terms (residual head and veloc-ity, overtides), not considered in the present analysis becausetheir order of magnitude is assumed to be much smaller thanthe tidal amplitudes (A∗, V ∗). Thus the average dimension-less depth is simply ⟨Y ∗⟩ = Z∗. The function A∗, Y ∗ andV ∗ vary along the dimensionless coordinate x∗ = x/Ls. Thestructure of (8) implies that the hydrodynamics are domi-nated by a single sinusoidal tidal constituent (M2), and theactual dimensional oscillations of water level and velocityare given by

A cos(ωt+ ϕA) = ϵDs [A∗ exp(iωt) + c.c.] /2 , (9)

V cos(ωt+ ϕV ) = σϵCs [V∗ exp(iωt) + c.c.] /2 , (10)

where also ϕA and ϕV are functions of x∗.The differential system (1)-(2) can be easily linearized.

The equations governing the variation of the tidal ampli-tudes (which are complex functions) are obtained by select-ing the terms proportional to exp(iωt):

TOFFOLON AND SAVENIJE: LINEARIZED TIDAL PROPAGATION X - 3

iA∗ + Z∗ ∂V∗

∂x∗− γ V ∗Z∗ = 0 , (11)

iV ∗ +∂A∗

∂x∗+ χ

V ∗

Z∗ = 0 , (12)

where γ and χ may vary with x∗ in principle.The two equations (11) and (12) can be combined into a

single, second order differential equation and solved for oneof the two unknowns if the coefficients are constant. Thismeans that we have to neglect any variation of the param-eters χ and γ, and of the average depth Z∗ (except thatimplicitly included in γZ) along x∗. This is a key pointof the analysis, since the above assumptions are in generalnot strictly valid. Nevertheless, we will proceed with an ap-proximate solution, which is indeed correct for short estuaryreaches, whereby we can assume Z∗ = 1 and assign the well-known structure of the constant-coefficient solution to bothvariables as follows

A∗ = a∗1 exp (w∗1x

∗) + a∗2 exp (w2x) , (13)

V ∗ = v∗1 exp (w∗1x

∗) + v∗2 exp (w∗2x

∗) , (14)

where w∗l = m∗

l +ik∗l (l = 1, 2) is a complex number, withm∗l

representing the amplification factor and k∗l the wavenum-ber. In this case the continuity equation gives the relation-ship

a∗l = i (w∗l − γ) Z∗ v∗l , (15)

which is valid locally and hence we retain the variable depthZ∗ (it will be used when applying the landward boundarycondition). Substituting the information from (15) into themomentum equation, the second order equation

w∗l2 − γw∗

l + 1− iχ = 0 (16)

can be obtained, which gives the solution

w∗l =

γ

2±∆ , ∆ =

√−Γ + iχ , (17)

where the parameter

Γ = 1−(γ

2

)2

(18)

measures the distance from frictionless critical convergenceγ = 2 [Jay, 1991; Savenije et al., 2008].

In more detail, (17) gives two solutions: the first with thepositive root, and the second with the negative root,

w∗1 =

γ

2+ ∆ , w∗

2 =γ

2−∆ . (19)

Consequently, equation (15) gives the following relationships

v∗1 =i a∗1

(γ/2−∆) Z∗ , v∗2 =i a∗2

(γ/2 + ∆) Z∗ . (20)

The parameter ∆ can be separated into its real and imagi-nary parts:

ℜ(∆) = K , ℑ(∆) =χ

2ℜ(∆)=

√Ω+ Γ

2, (21)

where

Ω =√

Γ2 + χ2 , K =

√Ω− Γ

2. (22)

The relationships (21) imply that m∗l = γ/2 ± ℜ(∆) and

k∗l = ±ℑ(∆) (with plus sign for l = 1 and minus for l = 2).We note that Γ2 + χ2 ≥ 0 always, so Ω is a real number,and that Ω ≥ |Γ| always, so the right hand sides of (21) areindeed real functions.

2.2. Boundary conditions

The amplitudes a∗l and v∗l are determined exactly bymeans of two boundary conditions applied at the reach ends.We can impose the boundary condition in terms of a knownfree surface elevation in a generic section xh as follows:

H = ϵDs h∗ [exp i(ωt+ θh) + c.c.] /2 (x∗ = x∗h) , (23)

where h∗ is the dimensionless amplitude of the tide, and θhis the phase. The relationship between a∗1 and a∗2 is thenreadily determined as

a∗1 exp (∆x∗h) + a∗2 exp (−∆x∗h) = h∗ exp(iθh − γ

2x∗h

).

(24)

The other relevant case is where a flux is imposed as aboundary condition in the form

UD = σϵCsDsZ∗q∗ [exp i(ωt+ θq) + c.c.] /2 (x∗ = x∗q) ,

(25)

where q∗ is the amplitude of the imposed dimension-less flux and θq its phase. After linearization UD =σϵCsDsZ

∗[V ∗ exp(iωt) + c.c.]/2, and hence

v∗1 exp(∆x∗q

)+ v∗2 exp

(−∆x∗q

)=

q∗

Z∗|x∗q

exp(iθq −

γ

2x∗q

),

(26)

or exploiting (20), which allows one to cancel Z∗,

exp(∆x∗q

)(∆− γ/2)

a∗1 −exp(−∆x∗q

)(∆ + γ/2)

a∗2 = iq∗ exp(iθq −

γ

2

).(27)

The actual solution of the problem is determined by aphysically reasonable combination of (24) and (27) for depthand velocity oscillations. Once the complex functions A∗

and V ∗ have been determined by means of suitable bound-ary conditions, the physical waves are defined by (9)-(10),and the amplitudes and phases can be calculated as follows

A = ϵDs|A∗| , V = σϵCs|V ∗| , (28)

tan(ϕA) =ℑ(A∗)

ℜ(A∗), tan(ϕV ) =

ℑ(V ∗)

ℜ(V ∗). (29)

The dimensionless parameters defined in Table 3 are cal-culated using the functions A and V at the seaward end of

X - 4 TOFFOLON AND SAVENIJE: LINEARIZED TIDAL PROPAGATION

the estuary reach. Then, the dimensionless scale of velocityis µ = |V ∗|, the phase lag is ϕ = ϕV − ϕA, the dampingcoefficients and the wavenumbers of the two waves can becalculated as

δA = ℜ(

1

A∗dA∗

dx∗

), δV = ℜ

(1

V ∗dV ∗

dx∗

), (30)

λA = ℑ(

1

A∗dA∗

dx∗

), λV = ℑ

(1

V ∗dV ∗

dx∗

), (31)

respectively. The derivation of (30)-(31) is given in Ap-pendix A.

3. Results

The solution obtained in the previous section represents alinear approximation of the fully non-linear hydrodynamics.However, it is not expressed in terms of external parametersbecause χ contains the dimensionless scale of velocity µ atthe seaward end of the channel, as described by (7).

In general, an explicit solution for µ cannot be derived,thus an iterative refinement is needed to obtain the correctwave behavior. The following procedure usually convergesin a few steps: (i) assume χ = χ as a first attempt, andcalculate µ = |V ∗| using the analytical solution; (ii) modifyχ according to (7) and calculate a new tentative value for µ;(iii) when the process converges, calculate the other parame-ters δ, λ, and ϕ. Such an iterative refinement allows for thereintroduction of one of the aspects of the quadratic non-linearity of the friction term in the solution, which thereforeshould not be considered strictly linear, although it is ob-viously not able to reproduce non-linear effects such as thegeneration of overtides. The procedure is simple and doesnot require specific numerical codes1: since the convergenceis usually fast, a spreadsheet can be a suitable tool for themajority of practical problems. Moreover, the problem withthe constant-coefficient assumption (in particular Z∗ = 1)can be overcome by subdividing the estuary in short reachesand then solving the overall problem.

In the following sections we compare the analytical solu-tion with fully non-linear numerical results using the numeri-cal model described in Toffolon et al. [2006]. The numericalresults are interpreted as follows: (i) a Fourier analysis isperformed on water levels and velocities along the estuaries;(ii) the amplitude and phase of the first tidal constituent(M2) are calculated; (iii) the dimensionless parameters µ, δ,λ, ϕ are estimated following the definitions of Table 3. Theresidual water level and velocity and the overtides are notconsidered in the analysis. Moreover, for sake of simplicitywe set σ = 1 (no storage areas).

In the next section we show how an accurate solutioncan be found with the linear model. Then, in the followingsections we will discuss the problem of characterizing theestuarine hydrodynamics with simple relationships and wewill derive the dimensionless parameters at the mouth of thechannel in some particular cases.

3.1. Sequence of reaches

In this section we set up the general problem of an es-tuary subdivided into N reaches . In this way the lon-gitudinal variability can be fully taken into account, bothfor the variation of the external parameters (i.e. geome-try) and for internal parameters (e.g. µ, assumed constantwithin each part). This also justifies the use of a constant Z∗

in the derivation of the analytical solution. Therefore, thewhole estuary is subdivided into a sequence of linear prob-lems determined by suitable boundary conditions: external(seaward and landward) and internal (at the nodes between

reaches). We enumerate the reaches starting from the sea(j = 1) to the landward end (j = N). In order to find sim-pler relationships, we use a moving origin of the axis x∗ forevery reach, whose length is L∗

j . For simplicity, we assumethe same value of ϵ (calculated at the estuary mouth) for allreaches. This means that the scale of water level variationsis ϵDs,j , with Ds,j the actual depth of the seaward end ofreach j, and hence h∗ = 1 in the boundary condition (23)for the generic reach.

Let us first consider the internal boundary conditions be-tween a reach j and its upstream neighbour j+1. Assumingthat the water level cannot be discontinuous implies that

a∗1,j exp(w∗

1,j L∗j

)+ a∗2,j exp

(w∗

2,j L∗j

)=

= αj

(a∗1,j+1 + a∗2,j+1

), (32)

where αj = Ds,j+1/Ds,j is the ratio assuring the dimen-sional correspondence. The same considerations hold for thedischarge: assuming that both depth and width are contin-uous at the node yields the condition

exp(w∗

1,j L∗j

)w∗

2,j

a∗1,j +exp(w∗

2,j L∗j

)w∗

1,j

a∗2,j =

= α1/2j

[a∗1,j+1

w∗2,j+1

+a∗2,j+1

w∗1,j+1

], (33)

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

|V* |

lin0linRlinNnum

0 0.2 0.4 0.6 0.8 1 1.2

0.8

1

1.2

1.4

|A* |

0 0.2 0.4 0.6 0.8 1 1.20.8

1

1.2

1.4

1.6

φ [r

ad]

0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

x*

χ<

Figure 2. Dimensionless amplitudes |A∗|, |V ∗|, phaselag ϕ, and linearized friction parameter χ along x∗: com-parison among three versions of the linear model (‘lin0’non-refined, ‘linR’ refined with single reach, ‘linN’ refinedwith multiple reaches) and the numerical results (‘num’)for a channel with horizontal bed (γZ = 0). See alsoSection 3.1.

TOFFOLON AND SAVENIJE: LINEARIZED TIDAL PROPAGATION X - 5

for the velocity, where the exponent of αj is 1/2 because thescale of velocity ϵCs ∝

√Ds. The set of equations (32)-(33)

is defined for the internal nodes j ∈ [1, N − 1] and provides2(N − 1) relationships for the 2N unknowns. The two miss-ing equations are given by the external boundary conditions:the forcing tide (24)

a∗1,1 + a∗2,1 = 1 , (34)

and the landward condition (27), which can be easily im-posed either for a closed channel,

exp(w∗

1,N L∗N

)w∗

2,N

a∗1,N +exp(w∗

2,N L∗N

)w∗

1,N

a∗2,N = 0 , (35)

or for an ideally infinite reach, a∗1,N = 0 (see also Section3.2). The linear system can be solved for the unknownsa1,j , a2,j by means of a suitable numerical scheme. We notethat a standard Gaussian elimination method can be used tosolve the linear system [Giese and Jay, 1989]. In AppendixB we show an explicit recursive derivation, which howevermay give unreliable numerical results.

It is worth looking at the performances of the differentanalytical solutions against fully non-linear numerical re-sults. In Figures 2 and 3 we compare the dimensionlessamplitudes of water level and velocity oscillations and the

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

|V* |

lin0linRlinNnum

0 0.2 0.4 0.6 0.8 1 1.2

0.8

1

1.2

1.4

|A* |

0 0.2 0.4 0.6 0.8 1 1.20.8

1

1.2

1.4

1.6

φ [r

ad]

0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

x*

χ<

Figure 3. Dimensionless amplitudes |A∗|, |V ∗|, phaselag ϕ, and linearized friction parameter χ along x∗: com-parison for a channel with decreasing depth (γZ > 0).See also Figure 2 and Section 3.1.

phase lag in four cases: i) non-refined linear model (‘lin0’,assuming χ = χ, i.e. µ = 1/κ); ii) iteratively refined linearmodel with a single reach (‘linR’); iii) iteratively refined lin-ear model with multiple reaches (‘linN’, with N = 20); iv)numerical simulations (‘num’). The example is chosen totest the linear model in a complex case: ϵ = 0.2 is not small(As = 2m, Ds = 10m); the channel is closed landward,so the reflected wave is important, but it is long enough(Le = 90 km) to exhibit a non trivial behavior; friction isrelevant (χ = 3.17), and width convergence (γB = 1.17,given by Lb = 60 km) plays a role. The only difference be-tween the two figures is the inclusion of depth convergencein Figure 3 (γZ = 0.78 from an exponential variation ofthe bed level characterized by Lz = 90 km), compared withthe horizontal bed of Figure 2. Assuming the numerical re-sults as a benchmark, the comparison shows that in bothcases the ‘linN’ model performs almost perfectly, while the‘lin0’ model is not satisfactory. A specific remark concernsthe single-reach refined model (‘linR’): with horizontal bed(γZ = 0, Figure 2) it works quite well, whereas when thedepth convergence is significant (Figure 3) it strongly devi-ates from the correct solution while moving landward. It isworth noting however that this model reproduces the veloc-ity amplitude at the mouth very satisfactorily in both cases:this fact, which implies that the total convergence γ has astrong significance in determining the wave behavior at themouth, will be exploited in the following analysis.

Finally, the fourth subplot of Figures 2 and 3 shows howthe linearized friction parameter χ changes along the chan-nel for the ‘linN’ model in comparison with the constantvalues assumed by the ‘linR’ (where χ is iteratively refinedat the mouth only) and ‘lin0’ (χ = χ). The variation isparticularly interesting for the case with decreasing depth(Figure 3) since a maximum for χ exists due to the oppos-ing effects of depth reduction (increasing χ) and velocitydecay (decreasing µ).

3.2. Infinitely long channel

Now we examine some particular cases. A relevant one isthat of long channels, where there is not a landward bound-ary condition. This simple yet general case can be obtainedby setting h∗ = 1 (tidal amplitude at the mouth equal tothe forcing amplitude scale) and θh = 0 (which can be al-ways obtained by shifting the origin of time), and choosingx∗h = 0. Since the landward boundary condition is ideallyimposed at ∞, and using (20), the solution is determined by

a∗1 = 0 , a∗2 = 1 , v∗1 = 0 , v∗2 =i

(∆ + γ/2), (36)

which suggest that there is no reflected wave (a∗1 = v∗1 = 0),and give

A∗ = exp (m∗2 x

∗ − ik∗2 x∗) , (37)

V ∗ =

(k∗2 + im∗

2

m∗22 + k∗2

2

)exp (m∗

2 x∗ − ik∗2 x

∗) . (38)

where m∗2 = ℜ(∆) + γ/2 and k∗2 = ℑ(∆).

Focusing attention on the dependent dimensionless pa-rameters at the mouth, where the single-reach solution(‘linR’) is reasonably accurate, it is straightforward to derive

µ =1√

m∗22 + k∗2

2=

(Ω+

γ2

4+ γ

√Ω− Γ

2

)−1/2

.(39)

The phase lag is given by ϕ = ϕV − ϕA, where ϕA = 0, andhence

X - 6 TOFFOLON AND SAVENIJE: LINEARIZED TIDAL PROPAGATION

tan(ϕ) =m∗

2

k∗2=

Ω− Γ

χ+γ

χ

√Ω− Γ

2. (40)

The damping δ = m∗2 = γ/2 − K and the wavenumber

λ2 = k∗22 = (Ω+Γ)/2 are the same for both water level and

velocity. The complete set of the dependent dimensionlessparameters for the general case is reported in Table 1.

Now we can test the performance of some analytical solu-tions (the different versions of the present linear model, andSavenije et al. [2008] model, which will be examined alsolater) against numerical results. In particular, we focus onthe dimensionless parameters µ, δ, λ, and ϕ, evaluated atthe mouth of the estuary. As for the numerical model, weadopt a reference depth Ds = 10m at the mouth, and letthe other quantities vary (tidal amplitude, friction, widthand depth convergence lengths); both the width and depthconvergence are assumed exponential. The behavior of thedimensionless parameters is shown in Figure 4, 6, 7, wherenumerical results obtained with different values of ϵ (from0.1 to 0.5) are denoted by different markers. An example ofthe increasing divergence of the linear solution from numer-ical results when ϵ becomes larger is given in Figure 5.

0 2 4 6 8 10

0.4

0.5

0.6

0.7

0.8

χ

φ

0 2 4 6 8 10

0.4

0.6

0.8

1

µ

0 2 4 6 8 10

−1.5

−1

−0.5

0

δ

0 2 4 6 8 10

1

1.5

2

λ

Sav.lin0linRlinNnum:ε=0.1

ε=0.3

ε=0.5

Figure 4. Infinitely long estuary: main dimension-less parameters (velocity µ, damping δ, wavenumber λ,phase lag ϕ) obtained with various analytical relation-ships as a function of friction parameter χ, for constantγ = 0.78 (with γZ = 0). Legend: ‘Sav.’ Savenije et al.[2008] solution; ‘lin0’ non-refined linear model (χ = χ),‘linR’ refined with single reach, ‘linN’ refined with mul-tiple reaches (L∗

N = 1, N = 20), ‘num’ numerical withdifferent values of ϵ.

The variation of the dependent dimensionless parame-ters with χ is shown in Figure 4 for a given value of γ(=0.78). For this case the numerical values of the dimen-sionless scale of velocity µ are perfectly approximated by allanalytical models expect the usual ‘lin0’ (possible deviationsof Savenije et al. [2008] model around the critical conver-gence are discussed below), almost independently of ϵ. Thisis surprisingly true also for larger ϵ (Figure 5), at least un-til ϵ reaches excessive values (∼ 0.8), which are far beyondthe limits of a linear model assuming small values of thisparameter. The same behavior occurs for the dimensionlesswavenumber λ, with just a more sensible dependence on ϵ.The damping δ is well fit by the analytical solutions only forsufficiently small ϵ, but it clearly feels the increase of thisparameter (see also Toffolon et al. [2006]). The phase lag ϕ,apart from the dependence on ϵ, is well reproduced (in thelimits of applicability of the linear model) only by the ‘linN’model, although the deviation of the ‘linR’ model is not toolarge. This suggests that the solution at the mouth is af-fected by the behavior along the channel, and in particularby the damping of the velocity for stronger friction (see theplot of δ), which changes the effective x-dependent frictionparameter χ.

Figure 6 and Figure 7 describe the variation of the de-pendent dimensionless parameters with γ when χ (=3.56) isfixed: the former consider the case of horizontal bed (γZ = 0,γ = γB), while the latter the other limit, i.e. constant width(γB = 0, γ = γZ). It is evident from Figure 6 that the ‘linR’and ‘linN’ models work perfectly for µ, λ, and ϕ (where asmall deviation of ‘linR’ appears for small γ). The damping

0 10 20 30 40 50 60

0.4

0.5

0.6

0.7

0.8

χ

φ

0 10 20 30 40 50 60

0.4

0.6

0.8

1

µ

0 10 20 30 40 50 60

−6

−4

−2

0

δ

0 10 20 30 40 50 60

0.5

1

1.5

2

2.5

λ

linNnum:

ε=0.2

ε=0.4

ε=0.6

ε=0.8

Figure 5. Infinitely long estuary: the ‘linN’ model asin Figure 4 compared with numerical results with largevalues of ϵ.

TOFFOLON AND SAVENIJE: LINEARIZED TIDAL PROPAGATION X - 7

δ, well reproduced by both models for small ϵ, tends to de-viate for larger tidal amplitudes. It is interesting to discussthe deviations of the Savenije et al. [2008] model aroundγ = 2.3 (the critical convergence for the assigned χ): thediscontinuous behavior and the transition towards a stand-ing wave do not seem to be physically reasonable (at leastin the numerical approximation).

Figure 7 shows the limitation of assuming a constant Z∗

to find the solution of the differential system (11)-(12). Nowthe comparison between numerical and analytical results isworse, especially for large γZ (very steep bed) and spe-cific parameters (in particular the wavenumber λ). How-ever, although the case of a strong exponential variation ofthe depth is mainly academic, the analytical model retainssome of the effects of the variable bed level on hydrodynam-ics, which would have been totally disregarded adopting aconstant-depth model.

3.3. Particular cases for infinite channel

It is worth examining some particular cases, for whichanalytical solutions have already been derived. In partic-ular, we refer to the solution discussed in Savenije et al.[2008], where these cases have already been identified andcompared with previous results. Incidentally, we note thatthe assumption δA = δV introduced to derive Savenije etal. [2008] solution is not an independent requirement, but

0 0.5 1 1.5 2 2.5 3 3.5

0.5

1

1.5

γ

φ

0 0.5 1 1.5 2 2.5 3 3.5

0.3

0.4

0.5

0.6

0.7

µ

0 0.5 1 1.5 2 2.5 3 3.5

−1

−0.5

0

0.5

δ

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

λ Sav.lin0linRlinNnum:ε=0.1

ε=0.3

ε=0.5

Figure 6. Infinitely long estuary: main dimensionlessparameters (velocity µ, damping δ, wavenumber λ, phaselag ϕ) obtained with various analytical relationships asa function of convergence γ (with γZ = 0), for constantχ = 3.56, compared with numerical results. For the leg-end see Figure 4.

just a consequence of having considered the case of infinitelength estuaries.

We identify the following specific conditions: 1) the fric-tionless case (χ = 0), where we can distinguish the caseof 1a) subcritical convergence (γ < 2) and the case of 1b)supercritical convergence (γ > 2); 2) the case of constantcross section (γ = 0); and 3) the case of ideal estuary (i.e.without damping, δ = 0). All these particular solutions arereported in Table 1.

1a) Frictionless, subcritical convergence. The solutioncan be easily obtained by setting χ = 0 (hence Ω = Γ forΓ > 0, and K = 0). In the same case, Savenije et al. [2008]find exactly the same relationships as an asymptotic case ofthe mixed wave solution.

1b) Frictionless, supercritical convergence. The solu-tion is derived by setting Γ < 0 and χ = 0, which yieldΩ = −Γ = γ2/4 − 1 and K =

√−Γ, and hence the wave-

length tends to be infinite (λ = 0, see Table 1). Differently,the relationships proposed by Savenije et al. [2008] for thecase of apparent standing wave are always characterized byλ = 0 when the threshold condition χ < −2γΓ+(γ2−2)

√−Γ

is satisfied, even with friction. Actually, in the present modela clear separation between the subcritical and the supercrit-ical cases exists only for vanishing friction (χ = 0). In fact,when Γ < 0 but friction is present,

λ2 =−Γ

2

[−1 +

√1 +

χ2

Γ2

], (41)

0 0.5 1 1.5 2 2.5 3 3.5

0.5

1

1.5

γ

φ

0 0.5 1 1.5 2 2.5 3 3.5

0.3

0.4

0.5

0.6

0.7

µ

0 0.5 1 1.5 2 2.5 3 3.5

−1

−0.5

0

0.5

δ

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

λ Sav.lin0linRlinNnum:ε=0.1

ε=0.3

ε=0.5

Figure 7. Infinitely long estuary: main dimensionlessparameters as in Figure 6 but with γ = γZ (γB = 0;χ = 3.56).

X - 8 TOFFOLON AND SAVENIJE: LINEARIZED TIDAL PROPAGATION

which can attain very small values (i.e. very long wave-length) as far as χ2 ≪ Γ2, but it is always λ2 > 0. There-fore, Savenije et al. [2008] apparent standing wave is verifiedexactly only for the idealized case of complete lack of friction(χ = 0, or in approximate form χ2 ≪ Γ2).

2) Constant cross section. Since the present model con-siders the effect of width and depth variations together, we

can set γ = 0 (hence Γ = 1, Ω =√

1 + χ2) to find the solu-tion for this case. The analogous case treated in Savenije etal. [2008] suggests a different set of relationships, which canbe rewritten as

µ0 =

√−2δ0χ

, λ20 = 1 + δ20 , tan(ϕ0) = − δ0√

1 + δ20,

δ0 =3√R0 − χ− 3

√R0 + χ

2, R0 =

√χ2 +

8

27. (42)

The two sets of solutions are different, but they tend to co-incide for weak friction since (42) give

µ = 1− χ2

4+O(χ4) , δ = −χ

2+O(χ3) ,

λ = 1 +χ2

8+O(χ4) , ϕ =

χ

2+O(χ3) , (43)

and the solution in Table 1 gives the same expansions interms of χ. When χ cannot be approximated by χ, the so-lution reported in Table 1 should be refined recalling (7).Instead of using iteration, in this case the solution can beobtained in closed form by solving

κ2χ2µ6 + µ4 − 1 = 0 , (44)

which is comparable with the relationship used in Savenijeet al. [2008], χ2µ6 +2µ2 − 2 = 0. The real root of (44) gives

µ =1

χκ

√1

3

(m1

2+

2

m1− 1), δ = − 1

µ

√1− µ2

2,

λ2 =1 + µ2

2µ2, tan(ϕ) =

1− µ2

χκ µ3, χκ = κχ ,

m1 = 3

[4χ4

κ − 8

27+ 4χ2

κ

√χ4κ − 4

27

]1/3

. (45)

The comparison between Savenije et al. [2008] relationshipsand the linearized ones for the case of constant cross sec-tion suggests that the iteratively refined solution (45) isvery similar to Savenije et al. [2008] results. Finally, whenχ → 0 (frictionless limit) the well-known result µ = λ = 1,δ = ϕ = 0 is obtained.

3) Ideal estuary. The condition of no damping is easily setby posing δ = 0, which implies K = γ/2 and χ2 = γ2. Using(7), and noting that γ must be positive in order to balancethe damping due to friction, the relationship becomes

χ =γ

κ

√γ2 + 1 . (46)

The results of the linear model are given again in Table 1.The relationships found by Savenije et al. [2008] are

µI =1√γ2 + 1

, δI = 0 , λ2I = 1 , tan(ϕI) = γ ,(47)

with

χI = γ(γ2 + 1) , (48)

which is different from (46). For small friction (small χ andhence small γ), (46) can be expanded as

χ =γ

κ+γ3

2κ+O(γ5) , (49)

which tends to (48) for small γ if κ ≃ 1. As a whole, the re-lationships (47) are verified by the present solution exactly,apart from the last expression.

3.4. Closed end channels

A well-known case is that of a channel forced by the tideseaward and closed landward. Assuming h∗ = 1, θh = 0in x∗h = 0, and q∗ = 0 in x∗q = L∗

e , with L∗e = Le/Ls the

dimensionless length, the boundary conditions (24) and (27)yield

a∗1 =

[1 + exp(2∆L∗)

(∆ + γ/2)

(∆− γ/2)

]−1

,

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

1

1.2

1.4

1.6

Le*

φ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

µ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−4

−3

−2

−1

0

δ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.5

1

1.5

λ

Sav.lin0linRlinNnum:ε=0.1

ε=0.3

ε=0.5

Figure 8. Finite-length estuary: main dimensionless pa-rameters (velocity µ, damping δ, wavenumber λ, phaselag ϕ) as a function of the dimensionless length L∗

e . Theparameter χ = 3.56 and γ = 1.60 (γZ = 0) are chosen tohave no damping for the infinite length channel. The so-lution of Savenije et al. [2008] for infinite channel is alsoshown for a comparison.

TOFFOLON AND SAVENIJE: LINEARIZED TIDAL PROPAGATION X - 9

a∗2 = 1− a∗1 . (50)

The integration constants for velocity, v∗1 and v∗2 , are deter-mined through (20) at the seaward end, and read

v∗1 =−i a∗1

(∆− γ/2), v∗2 =

i (1− a∗1)

(∆ + γ/2). (51)

It is easy to show that a∗1 = 0 and a∗2 = 1 for L∗ → ∞ (re-minding that ℜ(∆) = K ≥ 0), thus recovering the solutionof Section 3.2.

Finding simple explicit relationships for the dependentparameters µ, δ, λ and ϕ from (50)-(51) is a complex task.Nevertheless, we can derive that

V ∗|x∗=0 = v∗1 + v∗2 =−γ/2 + ∆(1− 2 a∗1)

χ+ i, (52)

δA + iλA =1

A∗dA∗

dx∗

∣∣∣∣x∗=0

=γ

2−∆(1− 2 a∗1) , (53)

δV + iλV =1

V ∗dV ∗

dx∗

∣∣∣∣x∗=0

= γ +1− iχ

−γ/2 + ∆(1− 2 a∗1).(54)

Although it is not evident, δV + iλV = δA + iλA for a∗1 = 0,as in the case of an infinite estuary (Section 3.2). Finally,µ = |V ∗| and the phase lag can be obtained as in (29).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.5

1

1.5

λ V

Le*

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−1

−0.5

0

0.5

δ A

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−4

−3

−2

−1

0

δ V

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.5

1

1.5

λ A

Sav.lin0linRlinNnum:ε=0.1

ε=0.3

ε=0.5

Figure 9. Finite-length estuary: dimensionless damp-ing, δA (water level) and δV (velocity), and dimension-less wavenumber, λA (water level) and λV (velocity), asa function of the dimensionless length L∗

e , for the samecondition as in Figure 8.

An example of the variation of the main parameters atthe mouth with the dimensionless length of the estuary L∗

e isgiven in Figure 8, where ideal conditions (i.e. no damping)are chosen for the infinite length case, according to (46),and the bed is kept horizontal. Both the ‘linR’ and ‘linN’models reproduce almost perfectly the numerical results, forwhich no visible effect of ϵ can be noted. It is interesting tonote that, for this set of parameters, the effect of the finitelength of the channel can be neglected for L∗

e > 2. It is alsoworth noting that, for the same reason, Savenije et al. [2008]solution is a reasonable approximation when L∗

e >1-1.5 (forthis case).

It is important to note that damping and wavenumber arein general not the same for water level and velocity waves.The differences between δA and δV , and between λA and λV ,are shown in Figure 9. When the channel become shorter,the closed end condition (no flux) causes the velocity damp-ing δV to reach large negative values (actually, the decreaseof the velocity with x becomes linear along the channel).

3.5. Along-channel variation of parameters and wavereflection

The results discussed in previous sections show that therefined linear model with a single reach (‘linR’) predicts thehydrodynamic behavior at the estuary mouth satisfactorilyfor the velocity scale µ and the wavenumber λ, whereas thetotal damping δ is significantly dependent on ϵ, and thephase lag ϕ is not so perfectly reproduced even in the limitof small ϵ. On the contrary, the results for ϕ are muchbetter when using multiple reaches (‘linN’). There are twomain differences between the ‘linR’ (single reach) and ‘linN’(multiple reaches) models: i) the ‘linN’ considers the pos-sible variation of the parameter χ and γ along the estuary;

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

x*

|V* |

total

reflected

direct

linR linN (γZ=0) linN (γ

B=0)

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

|A* |

total

reflected

direct

Figure 10. Wave reflection in a closed estuary (γ =1.17, χ = 3.17): dimensionless amplitudes |A∗| and |V ∗|.Different colors indicate the total solution (thick blackline), the direct wave (blue), and the reflected wave (red).The comparison is made for the same value of γ using:‘linR’ model (continuous lines); width-convergent ‘linN’model (dashed lines with ’+’, horizontal bed, γZ = 0);depth-convergent ‘linN’ model (dashed lines with ’x’, con-stant width, γB = 0).

X - 10 TOFFOLON AND SAVENIJE: LINEARIZED TIDAL PROPAGATION

ii) the ‘linN’ includes the possible effect of wave reflectionalong the estuary (see also Jay and Flinchem [1997]). In thecase of an infinite estuary, the second aspect might seem lessrelevant, since no wave reflection is expected.

In order to illustrate the behavior of the two models werefer to the case studied in Figure 2, where a convergent,finite length channel (Le = 90 km, Lb = 60 km, Lz = ∞) isconsidered. Figure 10 shows the total solution (|A∗|, |V ∗|,black lines), together with the direct wave (i.e. that travel-ling landward: |a∗2 exp (w∗

2x∗) |, |v∗2 exp (w∗

2x∗) |, blue lines),

and the reflected wave (|a∗1 exp (w∗1x

∗) |, |v∗1 exp (w∗1x

∗) |, redlines). The ‘linR’ model (continuous lines) is compared withthe ‘linN’ model (dashed lines with ’+’, corresponding tohorizontal bed, γZ = 0), and with a further case havingthe same value of γ, but generated only by bed exponentialvariation (dashed lines with ’x’, for constant width, γB = 0).When γZ = 0, the main differences regard the velocity: the‘linR’ model underestimates both direct and reflected wavesin the landward part with respect to ‘linN’, although thetotal solution is very similar. Stronger deviations appearwhen considering only depth convergence (γB = 0): in thiscase, according to the ‘linN’ model (the ‘linR’ solution doesnot change, since the value of γ is the same), the reflectedwave is less intense, the total solution for |A∗| is dampedand the velocity is lower.

It is reasonable that wave reflection is primarily causedby the closed channel end. Therefore, it is worth having alook at the corresponding infinite channel case (i.e. withexactly the same parameters but with a different boundarycondition). This is shown in Figure 11: the reflected wave isalmost absent. Both water level and velocity waves are moredamped in the case of depth convergence, but the solution inthe seaward part is the same. This suggests that, when thereflected wave is vanishing, the solution at the mouth is lo-cally controlled by the outer forcing (sea tide), and the effectof depth variation is not different from that of width varia-tion. Moreover, the reflected wave is negligible when γZ = 0(on the contrary being small but visible in the case of depthreduction), implying that width convergence does not causeany reflection (see also Friedrichs and Aubrey [1994]). Thepartial reflection in the case with γZ > 0 is likely due to thereduction of the wave celerity with decreasing depth implicitin (4).

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

x*

|V* |

total

reflected

direct

linR linN (γZ=0) linN (γ

B=0)

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

|A* |

total

reflected

direct

Figure 11. Wave reflection in an open estuary (infinitelength, γ = 1.17, χ = 3.17): see the caption of Figure 10.

4. Conclusions

We have shown that a suitably formulated version of thelinear model for tidal wave propagation is able to correctlyreproduce the dynamics of the main harmonic constituentof water level and velocity oscillations for not too largevalues of the dimensionless tidal amplitude (approximatelyϵ < 0.5). Three major points have been recognized by com-paring the linear model results with numerical simulations:i) the effect of width and depth convergence, together withthe possibly finite length of the channel, must be taken intoaccount, at least in an approximate way; ii) the frictionconstant r in Lorentz’s linearization of the bed shear stress(equation 5) can, and has to, be calculated within the modeland iteratively corrected; iii) the description of wave prop-agation can be improved by dividing the estuary in reachesand solving the linear system of the along-channel variablecoefficient of the linear wave (‘linN’ model).

The iteratively refined model provides good approxima-tions of the main features of the tidal wave at the estuarymouth also when the single-reach version (‘linR’ model) isused, in particular in the case of infinitely long channels orfor short closed channels. It is worth noting that the di-mensionless velocity at the mouth µ is almost independentof the dimensionless amplitude ϵ (at least explicitly, sincean indirect effect is included in χ, see also Toffolon et al.[2006]), whereas the most sensitive quantity is the dampingparameter δ, followed by the phase lag ϕ between velocityand water level.

We have also compared the performance of the presentsolutions with Savenije et al. [2008] model, which howeveris strictly valid only for infinite estuaries: results coincideexactly in the frictionless limit, and are similar far from thethreshold between sub- and super-critical conditions. Sincethe discontinuous behavior predicted by Savenije et al. [2008]does not seem to have a counterpart in numerical results(unless in almost frictionless conditions), such a thresholdshould be considered only as an approximate distinction be-tween two qualitatively hydrodynamic configurations.

Finally, we note that focusing on simplified hydrodynam-ics in estuary is not a trivial matter, in particular in thosepractical cases where few measurements are available. Forinstance, the method allows one to estimate a reliable valueof the velocity in a simple way by knowing few geometricaldata and the tidal forcing at the mouth. This can be partic-ularly crucial when studying salt intrusion [Savenije , 2005].Moreover, the possibility to easily apply the linear model tomultiple branching estuaries may give valuable informationabout the behavior of complex systems.

Appendix A: Characteristic parametersfrom complex solution

In this section we show how to derive the main charac-teristic parameters starting from the solution in complexform. Let us consider the water level wave described byA cos(ωt + ϕA), as in (9). The amplitude and phase aregiven by

A(x) = ϵDs

√ℜ(A∗)2 + ℑ(A∗)2 , (A1)

ϕA(x) = k x = arctan

[ℑ(A∗)

ℜ(A∗)

], (A2)

where k = 2π/LwA is the actual dimensional wavenumber(possibly varying along x). Then, following the definitionsof Table 3, the dimensionless damping coefficient is given by

δA =1

A

dA

dx∗=

1

|A∗|2

[ℜ(A∗)

dℜ(A∗)

dx∗+ ℑ(A∗)

dℑ(A∗)

dx∗

].

(A3)

TOFFOLON AND SAVENIJE: LINEARIZED TIDAL PROPAGATION X - 11

and the dimensionless wavenumber can be obtained as

λA =dϕA

dx∗=

1

|A∗|2

[ℜ(A∗)

dℑ(A∗)

dx∗−ℑ(A∗)

dℜ(A∗)

dx∗

].

(A4)

The sum δA + iλA obtained by the right hand sides of (A3)and (A4) can also be cast in the form

1

A∗dA∗

dx∗=

1

|A∗|2

[ℜ(A∗)

dℜ(A∗)

dx∗+ ℑ(A∗)

dℑ(A∗)

dx∗

]+

+i

[ℜ(A∗)

dℑ(A∗)

dx∗−ℑ(A∗)

dℜ(A∗)

dx∗

], (A5)

from which it is straightforward to obtain (30)-(31). Analo-gous considerations hold for the velocity wave.

Appendix B: Solution of linear system

The internal conditions (32)-(33) can be cast in the formof a linear system and solved to give

a∗1,j = F1,j a∗1,j+1 +G1,j a

∗2,j+1 ,

a∗2,j = F2,j a∗1,j+1 +G2,j a

∗2,j+1 , (B1)

where

F1,j =

√αj w

∗2,j(w

∗1,j −

√αj w

∗2,j+1)

(w∗1,j − w∗

2,j)w∗2,j+1 exp(w

∗1,jL

∗j ),

G1,j =

√αj w

∗2,j(w

∗1,j −

√αj w

∗1,j+1)

(w∗1,j − w∗

2,j)w∗1,j+1 exp(w

∗1,jL

∗j ),

F2,j = −√αj w

∗1,j(w

∗2,j −

√αj w

∗2,j+1)

(w∗1,j − w∗

2,j)w∗2,j+1 exp(w

∗2,jL

∗j ),

G2,j = −√αj w

∗1,j(w

∗2,j −

√αj w

∗1,j+1)

(w∗1,j − w∗

2,j)w∗1,j+1 exp(w

∗2,jL

∗j ). (B2)

In general, the interactions between the reaches j andj + n are described by

a∗1,j = F(n)1,j a

∗1,j+n +G

(n)1,j a

∗2,j+n ,

a∗2,j = F(n)2,j a

∗1,j+n +G

(n)2,j a

∗2,j+n , (B3)

where the rules

F(n)1,j = F

(n−1)1,j+1 F1,j + F

(n−1)2,j+1 G1,j ,

G(n)1,j = G

(n−1)1,j+1F1,j +G

(n−1)2,j+1G1,j ,

F(n)2,j = F

(n−1)1,j+1 F2,j + F

(n−1)2,j+1 G2,j ,

G(n)2,j = G

(n−1)1,j+1F2,j +G

(n−1)2,j+1G2,j (B4)

represent an iterative backward construction of the coeffi-cients (n ≥ 2), starting from F

(1)m,j = Fm,j , G

(1)m,j = Gm,j

(m = 1, 2, j = N − 1).Therefore, the linear system can be translated into a cou-

ple of equations linking the seaward reach (j = 1) to thelandward one (j = N). The seaward boundary conditiona∗1,1 + a∗2,1 = 1 yields

[F

(N−1)1,1 + F

(N−1)2,1

]a∗1,N +

[G

(N−1)1,1 +G

(N−1)2,1

]a∗2,N = 1 .

(B5)

Therefore, imposing a landward boundary condition in the

form a∗1,N = ψa∗2,N , where

ψ = −w∗

2,N

w∗1,N

exp[(w∗

2,N − w∗1,N )L∗

j

](B6)

for the closed boundary and ψ = 0 for the infinite channel,

the condition (B5) gives

a∗2,N =[F

(N−1)1,1 + F

(N−1)2,1

]ψ +

[G

(N−1)1,1 +G

(N−1)2,1

]−1

.

(B7)

Once that a∗1,N , a∗2,N are determined, the other unknowns

can be reconstructed using (B1) backwards.

We note that this method, although theoretically usable

to determine the solution of the linear system explicitly, is

affected by accumulation of numerical errors and may give

unreliable results also for relatively small values of N .

Notation

Main variables (∗ denoting dimensionless variables):x, x∗ longitudinal coordinates,t, t∗ temporal coordinates,D instantaneous depth,

Y, Y ∗ tidally averaged depth,H free surface elevation,

Z,Z∗ bed elevation,B width,U velocity,

U reference velocity for Lorentz’s linearation,A, |A∗| amplitude of free surface oscillation (first mode), with A∗ a complex number,V, |V ∗| amplitude of velocity oscillation (first mode), with V ∗ a complex number,

τ bed shear stress,Ch dimensionless Chezy parameter,σ storage ratio,

T, ω tidal period, frequency,ρ water density,g gravity acceleration,

Le, L∗e channel length,r Lorentz’s linearized friction factor,κ Lorentz’s linearization constant.

Variables defined for the estuary reach (denoting with the

subscript s the reference values defined at the seaward end):As tidal wave amplitude,Vs velocity amplitude,Cs frictionless wave celeriy,

Ds(= Ys) reference depth,Bs reference width,Ls intrinsic length scale,

Lb, Lz exponential convergence length (width, depth),LA, LV wave lengths (free surface, velocity),CA, CV wave celerity (free surface, velocity).

Dimensionless parameters (see also Tables 2 and 3):

X - 12 TOFFOLON AND SAVENIJE: LINEARIZED TIDAL PROPAGATION

Table 1. Dimensionless parameters at the mouth of an infinitely long channel: the general solution and some particular cases.

µ δ λ2 tan(ϕ)

general case1√

1 + γK + 2K2

γ

2−K K2 + Γ

γK + 2K2

χK =

√Ω− Γ

2, Ω =√

Γ2 + χ2

frictionless (χ = 0), subcritical(Γ > 0)

1γ

2Γ

γ

2√Γ

Ω = 0, K = 0

frictionless (χ = 0), supercriti-cal (Γ < 0)

γ

2−

√−Γ

γ

2−

√−Γ 0 ∞ Ω = −Γ

constant cross section (Γ = 1) Ω−1/2 −√

Ω− 1

2

Ω + 1

2

Ω− 1

χΩ =

√1 + χ2

ideal estuary (no damping, δ =0)

1√γ2 + 1

0 13

4γ K =

γ

2, χ = γ

ϵ tidal amplitude,γ total convergence,

γB , γZ width, depth convergence,χ friction,χ linearized friction parameter,µ velocity scale,δ damping,

δA, δV free surface, velocity damping,ϕ phase lag,

ϕA, ϕV phase (free surface, velocity).λ wavenumber,

λA, λV wavenumber (free surface, velocity).

Dimensionless variables used in the complex solution:Γ distance from critical convergence,∆ complex parameter,

K,Ω real parameters,a∗1, a

∗2 complex coefficients (for wave amplitude),

v∗1 , v∗2 complex coefficients (for velocity),

w∗1 , w

∗2 complex coefficients (w∗

l = m∗l + ik∗l ),

m∗1,m

∗2 amplification factors,

k∗1 , k∗2 wavenumbers,

Acknowledgments. The work of the first author hasbeen partially co-funded by the Italian Ministry of Educa-tion, University and Research (MIUR) within the project “Eco-morfodinamica di ambienti a marea e cambiamenti climatici”(PRIN 2008).

Notes

1. Two examples of Matlab scripts are provided as on line Aux-iliary Material.

References

Dronkers, J.J. (1964), Tidal Computations in River and CoastalWaters, 518 pp., Elsevier, New Work.

Friedrichs, C., and D. Aubrey (1994), Tidal propagation instrongly convergent channels, J. Geophys. Res., 99, 3321-3336.

Giese, B.S., and D.A. Jay (1989), Modelling tidal energetics ofthe Columbia River Estuary, Estuarine, Coastal and Shelf Sci-ence, 29, 549-571.

Ippen, A.T. (1966), Tidal dynamics in estuaries, part I: Estuariesof rectangular section, in Estuary and Coastline Hydrodynam-ics, edited by A. T. Ippen et al., McGraw-Hill, New York.

Jay, D.A. (1991), Greens Law revisited: Tidal long-wave propa-gation in channels with strong topography, J. Geophys. Res.,96(C11), 20,585-20,598.

Jay, D.A., and E.P. Flimchem (1997), Interaction of fluctuatingriver flow with a barotropic tide: A demonstration of wavelettidal analysis methods, J. Geophys. Res., 102(C3), 5705-5720.

Kukulka, T., and D.A. Jay, Impacts of Columbia River dischargeon salmonid habitat: 1. A nonstationary fluvial tide model,J. Geophys. Res., 108(C9), 3293, doi:10.1029/2002JC001382,2003.

Lanzoni, S., and G. Seminara (1998), On tide propagation in con-vergent estuaries, J. Geophys. Res., 103(C13), 30793-30812.

Lorentz, H.A. (1926), Verslag Staatscommissie Zuiderzee (inDutch), Algemene Landsdrukkerij, The Hague, Netherlands.

Savenije, H.H.G. (2005), Salinity and Tides in Alluvial Estuaries,Elsevier, New York.

Savenije, H.H.G., M. Toffolon, J. Haas, and E.J.M. Vel-ing (2008), Analytical description of tidal dynamics inconvergent estuaries, J. Geophys. Res., 113, C10025,doi:10.1029/2007JC004408.

Seminara, G., S. Lanzoni, N. Tambroni, and M. Toffolon (2010),How long are tidal channels?, Journal of Fluid Mechanics, 643,479-494, doi:10.1017/S0022112009992308.

Souza, A.J., and A. E. Hill (2006), Tidal dynamics in chan-nels: Single channels, J. Geophys. Res., 111, C09037,doi:10.1029/2006JC003469.

Speer, P.E., and D.G. Aubrey (1985), A study of non-lineartidal propagation in shallow inlet/estuarine systems, Estuar-ine, Coastal and Shelf Science, 21, 207-224.

Toffolon, M., G. Vignoli, and M. Tubino (2006), Relevant param-eters and finite amplitude effects in estuarine hydrodynamics,J. Geophys. Res., 111, C10014, doi:10.1029/2005JC003104.

Toffolon, M., and S. Lanzoni, Morphological equilibrium of shortchannels dissecting the tidal flats of coastal lagoons, J. Geo-phys. Res., 115, F04036, doi:10.1029/2010JF001673.

M. Toffolon, Department of Civil and Environmental Engi-neering, University of Trento, Via Mesiano, 77, I-38123 Trento,Italy. (marco.toffolon@ing.unitn.it)

H. H. G. Savenije, Department of Water Management, DelftUniversity of Technology, P.O. Box 5048, NL-2600-GA Delft,Netherlands. (h.h.g.savenije@tudelft.nl)

TOFFOLON AND SAVENIJE: LINEARIZED TIDAL PROPAGATION X - 13

Table 2. Independent dimensionless parameters.

tidal amplitude ϵ =As

Ds

friction χ =σϵCs

C2hωDs

cross section convergence γ = γB + γZ

lateral convergence γB = −Ls

Bs

dB

dx

∣∣∣s=

Ls

Lb

vertical convergence γZ = −Ls

Z

dZ

dx

∣∣∣s

Table 3. Dependent dimensionless parameters.

velocity µ =Vs

σϵCsdamping water level δA =

Ls

As

dA

dx

∣∣∣s

damping δ =δA + δV

2damping velocity δV =

Ls

Vs

dV

dx

∣∣∣s

wavenumber λ =λA + λV

2wavenumber water level λA = 2π

Ls

LA

phase lag ϕ = ϕV − ϕA wavenumber velocity λV = 2πLs

LV