A weakly dispersive edge wave model

Transcription

1 Coastal Engineering wwwelseviercomrlocatercoastaleng Technical Note A weakl dispersive edge wave model A Sheremet ), RT Guza Center for Coastal Studies, Scripps Institution of Oceanograph, UniÕersit of California, La Jolla, San Diego, CA USA Received 19 Januar 1999; accepted 18 March 1999 Abstract We derive a general linear, weakl dispersive, Boussinesq-tpe equation that can be used to stud edge waves on beaches with slow cross-shore variation of the depth and the alongshore current The equation is more accurate than the non-dispersive shallow water equations and simpler than the full dispersive elliptic mild slope equation Žespeciall for a non-zero alongshore current The improved performance of the new Boussinesq-tpe model is demonstrated using analtic solutions for edge waves on a plane beach with zero alongshore current q 1999 Elsevier Science BV All rights reserved Kewords: Edge waves; Weak dispersion; Boussinesq model; Mild slope model 1 Introduction Edge waves refractivel trapped near the shoreline of a sloping beach b cross-shore variations in water depth and mean alongshore currents often have been studied using the linear shallow water equations Ž eg, Bran et al, 1998 and references therein These equations accuratel describe linear low mode edge waves that have limited cross-shore extent Ž and are thus confined to shallow water on beaches with mild topographic variations and weakl sheared alongshore currents The limitations of the shallow water model are well known On a plane beach with vanishing mean currents, the shallow water dispersion relation Ž Eckart, 195 is v s< k < Ž nq1 s Ž 1 g ) Corresponding author Fax: q ; alex@coastucsdedu r99r$ - see front matter q 1999 Elsevier Science BV All rights reserved PII: S

2 48 A Sheremet, RT GuzarCoastal Engineering whereas the dispersion relation from the full linear theor Ursell, 195 is v s< k < sin Ž nq1 u with Ž nq1 u-pr Ž g where v is the radian frequenc, k is the alongshore wave number, sstan u with u the beach slope in radians, and n is an integer mode number In the full theor Ž, there Ž < < are a finite number of modes eg, n q 1 u- pr and all modes satisf k ) v rg In the shallow water approximation Ž 1, there is no limit on the mode number and edge waves are possible for all < k < ) 0 The accurac of the shallow water edge wave dispersion relation degrades as Ž n q 1 s increases and the turning point location moves offshore into deeper water Ž Fig 1a Full theor analtic solutions are known onl for the special case of a plane beach with no mean current Ž Ursell, 195 The non-dispersive shallow water theor is relativel straightforward to solve numericall with arbitrar cross-shore variation of the bathmetr and mean alongshore current Ž Bran et al, 1998 but has a restricted range of validit The elliptic mild slope Ž EMS equation Ž Eckart, 195 is a full dispersive alternative to the shallow water equation and describes wave evolution from deep water to the shoreline On a planar beach with small slope, edge wave solutions given b the EMS equations agree closel with the exact Ursell solutions for all mode numbers However, solving EMS equations on irregular cross-shore bathmetr is more complicated numericall than solving the shallow water equations, and the complexit of the EMS equations has apparentl precluded their application to edge waves in the presence of mean alongshore currents The EMS and the shallow water models are briefl reviewed in Section In Section 3 we derive a Boussinesq-tpe model for edge waves in the presence of alongshore currents that is simpler than the EMS model and has improved dispersive properties relative to the shallow water equations In Section 4, analtic edge wave solutions to Fig 1 Relative error in the edge wave alongshore wave number k versus the planar beach slope Ž a shallow water ŽEq Ž 1 and Ž b classical Boussinesq ŽEq Ž 1 approximations are compared to the full linear theor ŽEq Ž Mode numbers are indicated on the figure

3 A Sheremet, RT GuzarCoastal Engineering shallow water and Boussinesq-tpe equations are compared to full theor Ursell solutions for the case of a plane beach with no alongshore currents Edge wave models The full dispersive EMS equation for the sea surface elevation h over bathmetr with slow cross-shore variation of the depth h x and the mean alongshore current V x is and x Ihx Ih s kh q Ž k k s0 with Is 1q Ž 3 s s k sinhž kh s s vkv sgk tanh kh, 4 where k is the wave number, s is the intrinsic angular wave frequenc observed in a reference frame moving with alongshore velocit V, and v is the absolute angular frequenc observed in a fixed reference frame The alongshore wave number k and absolute frequenc v are constants The fixed coordinate frame origin is at the shoreline, and x, and z are the cross-shore, alongshore, and vertical coordinates, respectivel The shallow water equations are retrieved as a limiting case for h 0 With the substitution ssav, asž 1VrC where Ž C svrk, and defining an equivalent depth hsa h that completel accounts for the mean current Ž Howd et al, 199, Eq Ž 3 becomes gh gh h q 1 k x hs0 Ž 5 v v x A similar equivalent depth cannot be defined for the general mild-slope Eq Ž 3 and Ž the coefficients Irs and k of Eq Ž 3 are complicated non-linear functions of both v and k ŽEq Ž 4 3 Weakl dispersive Boussinesq-tpe models The complexit of the EMS equation is reduced b replacing the full dispersion relation Ž 4 b a truncated expansion in powers of the non-dimensional shallowness parameter b sž kh The phase and group velocities are written as s gh ds s s ghf Ž b s, and s F Ž b, Ž 6 1 k 1qbGŽ b dk k

4 50 A Sheremet, RT GuzarCoastal Engineering with lim F s 1, where F, G and F are rational functions of b Ž b 0 1, 1 given for several Boussinesq variants in Schaeffer and Madsen, 1995 Substituting Eq Ž 6 into Eq Ž 3 ields gh Ž F 1 F b gh gh h q b h q 1 qbgž b k hs0 Ž 7 x x x s FF s s x 1 the general weak-dispersion version of the mild slope Eq Ž 3 The correction term proportional to b G introduces the usual weak dispersion in constant depth With constant depth and no mean current shear, the bx term vanishes and substitution of h x s cos H k k d x ields the classical Boussinesq dispersion equation ( ghk v s Ž 8 1qv hr3g The term proportional to bx incorporates the effects of the bottom slope and current shear in the weakl dispersive sstem Ž 7 We consider in detail onl the classical Boussinesq version of Eq Ž 7, correspond- ing to F sf s1r 1qbr3 and Gs1r3 Using s sv a from Eq Ž 1 4 and bss hrgqož b from Eq Ž 8, Eq Ž 7 simplifies to gh gh va4 h 4 h x Ž a h x hh x q 1 k q hs0 Ž 9 v 3 v 3g x An equivalent depth Žin the sense of the shallow water Eq Ž 5 cannot be defined for Eq Ž 9 and h is retained onl for convenience Eq Ž 9, referred to below as the classical Boussinesq equation, ma also be derived directl from the original Boussinesq sstem of equations written in terms of depth-averaged velocit and accounting explicitl for currents The explicit dispersion relation Ž 8 makes both analtical and numerical treatment of Eq Ž 9 simpler than for the more complicated improved dispersion versions Ž 6 and Ž 7 or the full dispersive EMS ŽEqs Ž 3 and Ž 4 Numerical solutions to Eq Ž 9 can be found using the method and shoreline boundar conditions often used to solve the shallow water equations Ž Howd et al, 199 The shoreline boundar conditions are unaltered because Eqs Ž 9 and Ž 5 are equivalent at the shoreline 4 Plane beach solutions The improved accurac of edge-wave solutions to the classical Boussinesq Eq Ž 9 relative to the shallow water Eq Ž 5 is illustrated for a plane beach in the absence of currents In this case, analtic solutions are known for both the shallow water theor Ž Eckart, 195 and the full linear theor Ž Ursell, 195 The dispersion relations are given in Eqs Ž 1 and Ž, respectivel The Boussinesq Eq Ž 9 becomes ž / 4 v s v v Ž xhx x xhxq x k hs0 Ž 10 3 g gs 3g

5 Exact solutions to Eq 10 are hsae j r LnŽ gj with A Sheremet, RT GuzarCoastal Engineering ž / n / 1 v s v js Knx, gs 1q and K ns ž 1 ns Ž 11 3 gk gsž nq1 3 where Ln is the LaGuerre polnomial of integer order n and a is the shoreline amplitude The corresponding Boussinseq dispersion relation is 4 v s Ž nq1 4 4 k ns 1q s q s nž nq1 Ž 1 g s Ž nq For small Ž nq1 s, the first term dominates Eq Ž 1 and shallow water dispersion Ž 1 is recovered Žthe eigenfunctions Ž 11 also collapse to their shallow water counterparts The second and third terms in Eq Ž 1, of Os Ž relative to the first term, correspond identicall to terms of this order in the small slope expansion of the Ursell dispersion relation Ž The third term Žoriginating from the b G term in Eq Ž 7 is significant when Ž nq1 ssož 1 but the second term Ž originating from the b term x is never significant if the bottom slope is small The phsical significance of the last Os Ž 4 term is doubtful since the original mild slope equation was itself truncated at this order, but this term is negligibl small if s is small Note that for small s, Eq Ž 1 ields < < k )v r63g This is consistent with the requirement that < k < )k` slim h `k for a turning point to exist, since, from the dispersion relationship Ž 8, k sv r' ` 3 g Recall that the EMS and full theories support a finite number of edge modes satisfing < < k ) k` s v rg whereas the shallow water equations support an infinite number of modes with < k < ) k s 0 The Boussinesq model ields improved agreement with the ` Fig Analtic full linear Ž (((, Ursell, 195, Boussinesq Ž, Eq Ž 11, and shallow water Ž===, Eckart, 195 solutions for edge waves with 16 s period, unit shoreline amplitude and mode number ns8 on Ž a plane beach of 7% slope The non-dimensional cross-shore coordinate v hr g is approximatel equal to b sž kh, the nondimensional shallowness parameter

6 5 A Sheremet, RT GuzarCoastal Engineering full theor dispersion relationship The improvement introduced b the new Boussinesq dispersion relation Ž 1 is shown in Fig 1 For example, over a relativel steep plane beach of 7% slope the full dispersive Ursell sstem has onl 10 eigenmodes For mode ns8 the shallow water and Ursell dispersion relations differ b about % Ž Fig 1a compared to 5% error in the Boussinesq dispersion relation Ž Fig 1b The cross-shore structure of the n s 8 edge wave on a 7% slope is shown in Fig and further demonstrates the improved agreement of the Boussinesq solution with the exact Ursell solutions Acknowledgements This research was supported b the Office of Naval Research Ž Coastal Dnamics, the National Science Foundation Ž CooP program, and the Mellon Foundation References Bran, KR, Howd, PA, Bowen, AJ, 1998 Field observations of bar-trapped edge waves J Geophs Res 103, Eckart, C, 195 The propagation of gravit waves from deep to shallow water National Bureau of Standards, Circular 0, Howd, PA, Bowen, AJ, Holman, RA, 199 Edge waves in the presence of strong longshore currents J Geophs Res 97, Schaeffer, HA, Madsen, PA, 1995 Further enhancements of Boussinesq-tpe equations Coastal Eng 6, 1 14 Ursell, F, 195 Edge waves on a sloping beach Proc R Soc London 14, 79 97