Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue8.2000:

Beach water table ¯uctuations due to spring±neap tides: moving

boundary e€ects

L. Li

a,*

, D.A. Barry

a

, F. Stagnitti

b

, J.-Y. Parlange

c

, D.-S. Jeng

d a

School of Civil and Environmental Engineering and Contaminated Land Assessment and Remediation Research Centre, The University of Edinburgh, Edinburgh EH9 3JN, UK

b

School of Ecology and Environment, Deakin University, Warrnambool, Vic. 3280, Australia

c

Department of Agricultural and Biological Engineering, Cornell University, Ithaca, NY 14853-5701, USA

d

School of Engineering, Grith University, Gold Coast, Qld 9726, Australia Received 7 March 2000; accepted 8 March 2000

Abstract

Tidal water table ¯uctuations in a coastal aquifer are driven by tides on a moving boundary that varies with the beach slope. One-dimensional models based on the Boussinesq equation are often used to analyse tidal signals in coastal aquifers. The moving boundary condition hinders analytical solutions to even the linearised Boussinesq equation. This paper presents a new perturbation approach to the problem that maintains the simplicity of the linearised one-dimensional Boussinesq model. Our method involves transforming the Boussinesq equation to an ADE (advection±di€usion equation) with an oscillating velocity. The perturbation method is applied to the propagation of spring±neap tides (a bichromatic tidal system with the fundamental frequenciesx1andx2) in the aquifer. The results demonstrate analytically, for the ®rst time, that the moving boundary induces interactions between the two primary tidal oscillations, generating a slowly damped water table ¯uctuation of frequencyx1ÿx2, i.e., the spring±neap tidal water table ¯uctuation. The analytical predictions are found to be consistent with recently published ®eld observations. Ó ₂₀₀₀ Elsevier Science Ltd. All rights reserved.

Keywords:Spring±neap tides; Water table ¯uctuations; Moving boundary; Coastal aquifer

1. Introduction

Groundwater tables in a coastal aquifer ¯uctuate with oceanic tides. These ¯uctuations have been found to a€ect coastal processes such as beach sediment transport [6] and chemical transfer from the aquifer to the ocean [9]. One-dimensional models based on the Boussinesq equation are often used to help understand and analyse coastal aquifers_Õ behaviour, e.g., to predict the water table ¯uctuations [1,4,10,11]. The tidal forcing occurs over the beach slope, creating a moving bound-ary condition at the shore (Fig. 1). This hinders ana-lytical solutions to even the linearised Boussinesq equation for small amplitude tides to the extent that most existing solutions assume a ®xed location of the boundary condition (i.e., a vertical beach face).

Nielsen [10] presents perhaps the only analytical in-vestigation where the assumption of a ®xed location of

the shoreline boundary condition is relaxed. He derived a perturbation solution for small amplitude water table ¯uctuations based on the linearised Boussinesq equation by matching a prescribed series solution with the moving boundary condition, whence he obtained

hx_;t_{† ˆ}A_{exp… ÿj}x_†cos…xt_ÿjx_†

‡eA

2 1

h

‡p2exp_ÿp2jx

cos 2 xt_‡p

4ÿ



2

p

jxi_‡Oÿ_e2

; …1_†

wherehis the (¯uctuating) water table elevation relative to the still water table height, H.Aandxare the tidal amplitude and angular frequency, respectively; j… 

n_ex=2KH p

, wheren_{e andKare the e€ective porosity}

and hydraulic conductivity of the beach sand, re-spectively) is the amplitude damping rate and wave number. The perturbation variable, e, is de®ned as

A_{jcot…b†, wherebis the beach angle (Fig. 1). The O(e)}

term in (1), including an overheight (eA_{/2) and a}

har-monic wave at angular frequency 2x, is due to the www.elsevier.com/locate/advwatres

*

Correspoding author. Tel.: 44-131-650-5814; fax: 44-131-650-5814. E-mail address:Ling.Li@ed.ac.uk (L. Li).

moving shoreline. Such e€ects are qualitatively similar to those caused by the nonlinearity of ®nite amplitude tides (i.e., as can be derived from the nonlinear Bous-sinesq equation [11]).

The above solution considers only one tidal constit-uent. In reality, tides are more complicated and often bichromatic, containing oscillations of two slightly dif-ferent frequencies. For example, a semi-diurnal solar tide has periodT_{1ˆ12 h and frequencyx1ˆ0:5236 rad h}ÿ1

whileT_{2ˆ12:42 h andx2ˆ0:5059 rad h}ÿ1_{for a}

semi-diurnal lunar tide. As a result, the spring±neap cycle (i.e., the tidal envelope) is formed with a longer period [5],

T_{sn ˆ2}p=…x₁ÿx₂† ˆ14:78 d. Recently, Raubenheimer et al. [12] observed water table ¯uctuations of periodT_sn.

These ¯uctuations (called spring±neap tidal water table ¯uctuations hereafter, abbreviated as SNWTF) occurred much further inland than the primary tidal signals (i.e., diurnal and semi-diurnal tides). While one may relate this long period ¯uctuation to the spring±neap cycle, and take its occurrence and the slow damping feature for granted, the cause of such a phenomenon is not readily apparent, i.e., no models exist in the literature that pre-dict water table ¯uctuations of periodT_sn.

Spring±neap tides are bichromatic signals, i.e., they can be described asA_1cos…x1t_{† ‡}A_2cos…x2t_{ÿd) where}

A1 and A2 are the amplitude of the semi-diurnal solar

and lunar tide, respectively, anddis the phase di€erence

between them. In other words, there are only two pri-mary forcing signals at the boundary. If they propagate in the aquifer independently (as would occur in a lin-earised model assuming a vertical beach face), then the water table response will also be bichromatic and simply described by A_1exp…ÿj1x_{†cos …x1}t_ÿj1x_{† ‡}A_2exp …ÿj2x†cos…x2tÿdj2x†. Both j1…

 n_ex1=2KH p

† and

j2…

 n_ex2=2KH p

†are relatively high damping rates that quickly extinguish the tidal signal in the coastal aquifer. Thus, a slowly damped spring±neap tidal water table ¯uctuation cannot be predicted using a vertical beach face in a linearised model. However, the beach face is sloping and hence the seaward aquifer boundary moves with the tidal oscillations. The moving boundary in-duces interactions between the two primary tidal signals as they propagate inland. Such interactions underlie the generation of the SNWTF as shown by Raubenheimer et al. [12] using numerical simulations. The purpose of this paper is to develop an analytical approach for quantifying the moving boundary e€ects and the re-sulting SNWTF. Clearly, an analytical solution will provide direct insight into the physics of the SNWTF. Since the damping length of the SNWTF is much larger than those of the primary tidal water table ¯uctuations, its in¯uence on coastal groundwater dynamics can be very important. To determine these e€ects, analytical predictions of the SNWTF are also desirable.

Nomenclature

A tidal amplitude [L]

Asn amplitude of the spring±neap tidal water table

¯uctuations [L]

H still water table height [L] HT high-tide sea level

h ¯uctuating water table elevation [L] K hydraulic conductivity [LTÿ1_]

LT low-tide sea level MT mid-tide sea level ne e€ective porosity

T tidal period [T]

Tsn period of spring±neap tides [T] t time [T]

X x-coordinate of the moving shoreline [L] x cross-shore distance from the shore [L] z transformed coordinate,x_ÿXt_{†, [L]}

b beach angle [rad] e perturbation variable

g tidal shoreline oscillations [L] j wave number [Lÿ1_]

x tidal angular frequency [rad Tÿ1_]

The paper is organised as follows: ®rst a new perturbation approach is described (Section 2). The approach is used to revisit the propagation of mono-chromatic tidal signals and then to derive a perturbation solution of the SNWTF. In Section 3, a discussion of the accuracy and applicability of the SNWTF solution will be presented. Finally, conclusions are drawn in Section 4.

2. A new perturbation approach

To focus on the e€ects of the moving boundary, we consider only small amplitude tides, modelled using the linearised Boussinesq equation [2]

oh

The e€ect of capillarity is assumed to be negligible for tidal water table ¯uctuations [1,8]. As shown in Fig. 1, tidal oscillations on a sloping beach create a moving boundary

h X t_{‰ … †;}t_{Š ˆg… †}t _andX t_{† ˆ cot… †b g… †}t _{; …3†}

whereX(t) is thex-coordinate of the moving boundary (the origin of the x-coordinate is located at the inter-section between the mid-tidal sea level and the beach face), b is the beach angle, and g(t) represents tide-in-duced oscillations of the mean sea level (i.e., very high frequency signals such as wave run-up/run-down [8] are not included in this analysis). Note that the seepage face has been assumed negligible, an assumption that is reasonable for slow (e.g., tidal) sea level oscillations, relatively small tidal amplitudes and steep beaches (further discussion of seepage face e€ects is presented in Section 3). By introducing the new variablez_ˆx_ÿXt_†,

Eqs. (2) and (3) can, respectively, be transformed to [7],

oh

Eq. (2) is mapped to a ®xed boundary problem. The governing equation now, however, includes an advec-tion term with an oscillating velocity. A perturbaadvec-tion approach can be used to obtain approximate solutions to Eqs. (4)±(6).

2.1. Perturbation solution: revisiting the case of a monochromatic tide

We consider a monochromatic tide,g…t_{† ˆ}A_cos…xt_†,

and thus Eqs. (4) and (5) become

oh

respectively. We seek a solution of the form

h_ˆh_0‡eh_1‡Oÿ_e2

; …9†

where eˆA_{jcot…b† and jˆ}p_x=2D_{. Physically, e}

represents the ratio of the horizontal tidal excursion to the wavelength of primary mode tidal water table ¯uc-tuations. From Eqs. (7) and (8), the following pertur-bation equations can then be derived

oh₀

ot ˆD o2_h

0

oz2 ; …10a†

with the following boundary condition,

h_0…0;t_{† ˆ}A_{cos… †x}t _{; …10b†}

with the boundary condition,

h_1…0;t_{† ˆ0: …11b†}

The solution to Eqs. (10a) and (10b) is simply [4]

h_0ˆA_{exp… ÿj}z_†cos…xt_ÿjz_{†: …12†}

Using Eq. (12), we rewrite Eq. (11a) as,

oh₁

The solution to Eq. (13) is

To obtain the solution in the x-coordinate, one only needs to substitute z_ˆx_ÿA_{cot…b†cos…x}t_{† into Eq.}

(15). The resulting solution containsin the exponential and cosine functions. Expanding these functions as Taylor series inxand collecting terms of equal order in

, one can deduce Eq. (1) from Eq. (15). While both Eqs. (15) and (1) predict the same overheight, A_{e/2, a}

dis-tinction between the two is that Eq. (15) satis®es exactly the moving boundary condition, Eq. (8), while Eq. (1) only meets this condition approximately.

Obviously, the above approach can be extended to obtain Oe2

†or higher terms. These solutions, including harmonic waves of higher frequencies, are complicated and lengthy, and thus will not be presented here.

2.2. Application to the propagation of bichromatic tides

As discussed in the introduction, tidal signals are, in reality, more complicated than the case presented in Section 2.1. A more realistic model is to take the tidal signals as being bichromatic, containing both semi-di-urnal solar (T_{1ˆ12 h and x1ˆ0:5236 rad h}ÿ1_{) and}

semi-diurnal lunar (T_{2ˆ12:42 h and x2ˆ0:5059 rad}

hÿ1_{) constituents [5]. These di€ering frequencies}

com-bine to produce spring and neap tides. In this section, we consider the propagation of spring±neap tides in the aquifer. The boundary condition is then

g_{† ˆ}t A_1cos…x1t_{† ‡}A_2cos…x2t_{ÿd†: …16†}

We denote r_{1ˆx2=x1ˆ0:9662 and} A₂₌A_1ˆr_{2. The}

transformed governing equation is

oh

Again, we look for a perturbation solution to Eq. (17), i.e., assuming Eq. (9) with e_ˆA_{1j1cot…b† and}

will not change the solution fundamen-tally (i.e., a symmetric solution). The perturbation equations can be derived as before

oh₀

The solution to Eq. (18) is

h_0ˆA_{1exp… ÿj1}z_†cos…x1t_ÿj1z_†

‡A_{2exp… ÿj2}z_†cos…x2t_ÿj2z_{ÿd†: …20†}

Substituting Eq. (20) into Eq. (19) results in

oh₁

These four functions represent four additional forcing terms as a result of interactions between the two primary tides, caused by the moving boundary. Each forcing term will induce water table responses correspondingly as shown below. The solution to Eqs. (21a)±(21e) is

w₂ˆr2A2

Again, the solution in the x-coordinate can be ob-tained by substituting z_ˆx_{ÿ ‰}A_{1cot…b†cos…x1}t_† ‡A_{2cot…b†cos…x2}t_{ÿd†Šinto Eqs. (20) and (22a)±(22e).}

Expansions of these equations inxlead to

h_ˆh_0‡h_10‡h_11‡h_12‡h_13‡h_14†e‡Oÿ_e2_{; …23a†}

landward of the inter-tidal zone. It is clear that, in a bichromatic tidal system, the moving boundary con-dition generates an overheight [h_{10e, Eq. (23c)], and}

additional harmonic waves of frequency 2x1 [h11e, Eq.

(23d)], 2x2[h12e, Eq. (23e)],x1‡x2[h13e, Eq. (23f)] and x1ÿx2 [h14e, Eq. (23g)]. The wave of x1ÿx2

repre-sents the spring±neap tidal water table ¯uctuations due to spring±neap tides. Since the damping rate,j4, is much

smaller than j1, j2 and j3, the SNWTF propagates

much further inland than the primary forcing oscilla-tions and their sub-harmonics (i.e., those of frequencies x1,x2, 2x1, 2x2andx1‡x2). The damping distance (1/ j4) for the SNWTF is ®ve times larger than those for the

primary mode water table ¯uctuations. In Fig. 2, we plot the simulated water table ¯uctuations atx_{ˆ20; 50; 100}

and 200 m (parameter values used in the calculation are listed in Table 1). The results clearly show that the SNWTF occurs much further inland than the semi-di-urnal tides, consistent with recent experimental ®ndings on spring±neap tidal water table ¯uctuations [12].

3. Spring±neap tidal water table ¯uctuations

To examine the SNWTF, we will focus on h_{14e as}

given by Eq. (23g), since this term predicts the spring± neap tidal signal in the aquifer. In the following, we discuss the accuracy and applicability of this solution.

3.1. Solution's accuracy and comparison with numerical solution

It can be demonstrated (details not shown here) that

h_{14eis actually accurate to the second order ofe, i.e., with}

a truncation error of Oe3

numerically simulated water table ¯uctuations were av-eraged over 25 h to obtain the SNWTF, which were then compared with the predictions by h_{14e (Fig. 3). Again,}

good agreement between numerical and analytical results is evident. From the averaged water table ¯uctuations, the amplitude for the SNWTF (A_{sn) can be calculated.}

The variation of A_{sn with x, as predicted numerical}

simulations and analytical solutions, is compared in Fig. 4. The overall di€erence between the two predictions is extremely small. We quantify such a di€erence as

Diff _ˆ

RL A_1‡A₂

… †cot… †b …Asn nÿAsna† 2

dx

RL A_1‡A₂

… †cot… †b …Asn a† 2

dx ; …24†

where L is the simulated cross-shore distance, much larger than the damping distance of the SNWTF. We observed that, if Diff<0:01, the di€erence between the numerical and analytical predictions is trivial. Di€ was

Fig. 3. 25-h averaged water table ¯uctuations: circles are from nu-merical simulations and solid lines are from analytical solutions.

Fig. 4. Predicted amplitude damping of the SNWTF: circles are from numerical simulations and solid lines are from analytical solutions. Fig. 2. Predicted total water table ¯uctuations at: (a)x_ˆ20 m, (b)

x_ˆ50 m, (c) x_ˆ100 m, and (d) x_ˆ200 m. Solid lines are from analytical solutions and circles are numerical predictions.

Table 1

Parameter values used in the calculations of results in Figs. 2±4 and 6

Fig. no. A_1(m) A_{2(m) d(rad)} K_(m/s) n_{e tan(b)} H_(m)

2, 3, 4 0.25 0.75 0 0.002 0.2 0.15 2 0.1

6 0.25 0.75 0 0.0007 0.215 0.05a _{4.7 0.34}

a_{Beach slope landward ofx}

calculated for all simulations. The results show that the analytical solution (h_{14e) compares well with the}

nu-merical predictions of the SNWTF for e up to 0.2 (Fig. 5). Note that the numerical solution may be taken as being ``exact'' since standard methods were employed such that the numerical errors were negligible in the simulations.

3.2. Applicability of the solution: seepage face e€ects and nonlinearity of ®nite amplitude tides

The analytical solution assumes that on the bound-ary, the seepage face is negligible. Based on the concept of Dracos [3,13], the following condition can be derived to predict situations where the assumption of a negli-gible seepage face is reasonable

n_eA_1‡A_{2†…x1‡x2†}

4K_sin2_†b 61: …25†

The result implies that the analytical solution only ap-plies to relatively small tidal ranges, highly permeable and steep beaches. In reality, large beach slopes are as-sociated with high beach permeability. While the above condition limits the application of the analytical solu-tion, one would expect that this solution might still provide approximate predictions of the SNWTF for beaches where seepage faces are formed. We applied the solution to data from a ®eld experiment [12] where a seepage face existed. The calculated A_{sn is found to be}

reasonably close to the experimental data (Fig. 6). The amplitude data were estimated as half of the di€erence between the maximum and minimum 25 h-averaged water table elevations (in Figs. 2 and 4 of Raubenheimer et al. [12]). The parameter values used in the calculation are the same as observed in the ®eld (Table 1).

The analytical solution also neglects the nonlinearity of ®nite amplitude tides. However, it can be extended to include the nonlinearity without much diculty by combining the present approach and that of Parlange et al. [11]. Since the focus of this paper is on the moving boundary e€ects, the nonlinearity issue will not be dis-cussed further here.

4. Conclusions

A new perturbation approach has been developed for solving the moving boundary problem in tidal propa-gation in a coastal aquifer. The new approach was ap-plied ®rst to the case of a monochromatic tide. The resulting solution matches the seaward boundary con-ditions exactly, rather than approximately as is the case for the previous prediction [10].

Fig. 5. Estimated di€erences between numerical and analytical pre-dictions ofA_snx_†for di€erent.

Perturbation solutions were obtained, using this ap-proach, for spring±neap water table ¯uctuations. The solution demonstrates analytically that the moving boundary induces interactions between primary tidal oscillations as they propagate in the aquifer. Such inter-actions lead to the generation of long-period and slowly damped spring±neap tidal water table ¯uctuations.

Since the SNWTF propagates much further inland than primary tidal oscillations, including diurnal and semi-diurnal tides, it may a€ect the dynamics of coastal groundwater signi®cantly. The present analytical solu-tion, providing estimates of the SNWTF for given tidal and aquifer conditions, will assist in assessing such ef-fects. The solution assumes small tidal amplitude and neglects the formation of seepage faces. While the ap-proach can be extended easily to nonlinear ®nite am-plitude tides, the comparison of the analytical predictions with the ®eld data (a€ected by seepage faces) indicates that the solution may also provide a reason-able approximation of the SNWTF for situations with seepage faces formed.

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