Laboratory study and analysis of the instability of alongshore currents

Acta Oceanologica Sinica 2009, Vol.28, No.4, p.96-106 .http://www.hyxb.org.cnE-mail: hyxbe@263.netLaboratory study and analysis of the instability ofalongshore currentsREN Chunpingl*, ZOU Zhili1State Key Laboratory of Coastal and Ofshore Engineering, Dalian University of Technology,Dalian 116024, ChinaReceived 6 April 2008; accepted 27 November 2008AbstractA laboratory experiment on the instability of alongshore currents was conducted on a plane beachwith slope 1:40. Low-frequency fuctuations of alongshore currents with the period of approxi-mately 100 s were observed. The dominant frequency and amplitudes of the oscillations of along-shore currents were determined using the maximum entropy method and the regression method oftrigonometric function. The variations across the beach cross-section of the oscillation amplitudesof the alongshore current were given. The linear shear instability theory was used to analyze themechanism of the oscillation, and the calculated results agreed with measurements. This confirmsthat the observed fuctuation of alongshore currents is due to the shear instability of alongshoreKey words: alongshore currents, shear instability, water waves, coast1 Introductionby resonant triad interaction. Noyes et al. (2004)examined the shear wave properties with field obser-Oltman-Shay et al. (1989) observed a new type ofvations in surf zone. Their results suggest that totaloscillations associated with the presence of alongshoreroot mean square shear wave fluctuations are betweencurrents in field experiment and these oscillations are10%-40% of local observed mean alongshore currentof shorter wavelength and much less phase speed thanand shear waves are generated primarily in the highlygravity waves for the same frequency. Their frequen-sheared region immediately seaward of the location ofcies are approximately 0.001~0.01 Hz. The relatedthe maximum mean alongshore current.theoretical studies show that this kind of oscillationsIn the present study, a laboratory experiment wasis caused by shear istabilit in the alongshore cu- conducted on a plane beach with slope 1:40 to studyrents, and termed as shear waves. The studies of thethe shear instabilities of longshore currents. The oscilla-shear istabilitieis of longshore currents have great sig- tions with the period of ppoximately 100 s were observed.nificance for the researches of the sediment and con-The maximum entropy method for spectral analysis andtamination transport in coastal water.Bowen and Holman (1989) presented a theoreticalto analyze the dominant frequency and the variations ofmodel to describe these motions, which is based on theoscillation amplitudes in cross-shore direction. The linearlinearized, inviscid, 2D shallow-water equations withinstability model was applied to analyzing the shear insta-a rigid-lid assumption.Dodd and Thornton (1990)bilities of longshore currents observed in the present exper-made the analysis of the more realistic velocity andiment. And a comparison was made between the dominantbottom profile, and got the result that backshear isperiod obtained from the spectral analysis and period ob-very important to the shear intability of longshore tained from the numerical calculation.currents. Putrevu and Svendsen (1992) employed anumerical techmique to study the characteristics of 2 Experiment and data analysisshear instabilities of longshore currents on four bottomprofiles (horizontal,“equilibrium” profile, plane slopeThe experiment was conducted in the wave basinand barred beach). Baquerizo et al. (2001) provedof the State Key Laboratory of Coastal and Ofshorethe existence of instabilities due to the presence of a Engineering in Dalian University of Technology, whichsecond extremum of background vorticity on the front is 55 m long, 34 m wide and 1.0 m deep (Fig. 1). Theside of longshore currents. Dodd et al. (2004) inves-plane slope of 1:40 is adopted. The water depth in thetigated the possibility that the observed instability in horizontal bottom part is 0.45 m. The beach makes anfield and laboratory can grow in linearly stable flows angle of 30°中国煤化Inerator in orderFoundation item: The National Natural Science Foundation of China under contraMHCNMHG* Corresponding author, E-mail: Chunpingren@ 163.com.REN Chunping et al. Acta Oceanologica Sinica 2009, Vol. 28, No. 4, P.96-10697_5Sm .n wave guide 主三.wwwwwwwwnetArry 2怎"esu,30ensm_x wave gauge。velocity meter_ 1wave guide士'Fig.1. Experimental set-up.to create a large incident angle of waves and a longer records was filtered before making spectral analysis bybeach and thus more room for instabilities of along-low passing (the threshold frequency is 0.02 Hz) (Hu,shore curents to develop. At the lateral and back 2003). Figure 3 shows the maximum entropy spec-ends of the beach a channel with the width of 3.0 m trums corresponding to velocity time series in Fig. 2.was made to maintain the water remains circulation The maximum peak frequencies obtained from thesecaused by the longshore currents. The horizontal ve- spectrums at three locations of VM (Table 2) weregiven in Table 2, also given were the average of threetwo-dimensional velocity meters (VMs), and were set frequencies and the period corresponding to the av-at one third of the water depth from the bottom, whereerage frequency. The results show that the dominantis approximately the location of depth-average along- period of longshore current oscillations in cross-shoreshore currents. These flow meters were deployed in and longshore direction is approximately 100 s.two arrays normal to the beach, each has 16 velocityThe reason for using the MEM is because themeters with the interval 0.5 m or 1.0 m. Random， time series of longshore currents only contain threeunidirectional incident waves were generated, and the or four periods of instability oscillation, and for thesewave conditions are given in Table 1short time series the MEM is better than the FourierFigure 2 gives the recorded time series of croSs-analysis for the spectral analysis, especially for theshore (left column) and alongshore current (right col-peak frequency of the spectrums. However, the MEMumn) velocities for Case 7. The distances of the also has its weakness that the estimated amplitudeflow meters from shoreline are also given in the fig-by the MEM is not very reliable (Xu and Bai, 1990).ure (x=2.5 m, denotes that the distance of VM to Hence, the MEM was only used in the present studyshoreline is 2.5 m). It can be seen in the figure that for detecting the possible peak frequencies f(l=1 tothe long period oscillations (approximately 100 s) ocm) while the regression method of trigonometric func-cur in both the cross shore and alongshore direction. tion (Xu and Bai, 1990) was used to determine theSimilar results were obtained for other cases.amplitude Al and θq phase of the oscillations in long-The maximum entropy method (MEM) was used shore currents. This method gives the expressionto make the spectral analysis for the time series of for the time series of the oscillation longshore currentlongshore current fuctuation. Each of the velocityvelocity:中国煤化工Table 1. Test conditionsCase1 Case2Case3 Case4 (Case 5MYHCNM H GgCase 9Wave period/s1.01..01.5.52.12.0Wave height/cm3.65..84.06.33.">.78.7.98REN Chunping et al. Acta Oceanologica Sinica 2009, Vol. 28, No. 4, P.96-10630日-E二-10-15E-20WuMyhwMwx=2.5mx=2.-3030E-5Eg-10=-15r=3.0 mx=3.0 m-25sE-sEM县8-15x=3.5 m-30 l的I-10wwww=4.0 m-4.0 m- 20-1wwwww10000 3004005000300t/:1/sFig.2. Recorded time series of cross-shore velocity u (the left) and alongshore velocity v (the right) forCase 7.have from Eq.(1)x()= > Alcos(2πfit +0).(1)=1x(iO)= >中国煤化工The temporal discretization is made by lettingl=MYHCNMHGt = i△where i = 0,1,2,... ,N (N is the number2 C[Ycos(2πfiO) + Yt+msin(2πfi0)], (2)of time step) and △is a constant time step. Then we.REN Chunping et al. Acta Oceanologica Sinica 2009, Vol. 28, No. 4, P.96-10699where Yi = A[cos0;Yi+m = -Aisin0(l = 1,2,... ,m); ple, there are three peak frequencies (0.006 1, 0.009 8,A =√32+1+1,01 = arctg( - htm)which can 0.015 3 Hz) in Fig. 3 for x=2.5 m and two peak fre-be obtained by solving the over determined equationsquencies (0.010 4, 0.019 5 Hz) for x=4.5 m, and there .formed from Eq. (2). The over determined equationsis only one peak frequency for other locations of VM.can be solved easily using QR decomposition (Xu andHence, m=3 for x = 2.5m,m= 2 for x=2.5 m, m=1Bai, 1990). The number and value of f, can be deter-and for other locations of VM were adopted. Figure 4mined from the figures of the spectrums. For exam-gives the calculated amplitudes and phases correspo-2000px=2.5 m300x=2.5m1500|"自 1000502 5004000|x=3.0m3 000100025 00012 000 rx=3.5mx3.5m0 00015 000分100004 0005 0001 200x=4.0m6 000=4.0 m80030002 000=4.5mx=4.5m100-自100中国煤化工0.010.020.030.04f/HzMHCNMHGFig.3. Maximum entropy spectrums of velocity time series in Fig. 2 (the left: u, the right: v)..100REN Chunping et al. Acta Oceanologica Sinica 2009, Vol. 28, No. 4, P.96-106Table 2. Experiment resultsCasef1/Hzf2/Hzf3/Hz .于=(f1+f2+ f3)/3]/HzT(=1/F)/s1x=2.5 mx=3 m0.009 80.009 8.102.04x=2.5 m .x=3.0 mx= =3.5mx=3.5 mx=4.0 mx=4.5 m0.010 40.010 0100.00:=2.5 mx=3mx=3.5m0.010298.03 9x=5 m0.010496.15x= =3.5 m0.0098x=4 m0.009r=5 mx=5.5 m98.03nding to the alongshore velocity spectrums in Fig. 3 acceleration due to gravity; and t is time. Subscriptsand the comparison between the calculated and mea-denote partial differentiation with respect to the indi-sured time series. Figure 5 gives the cross. -shore varia- cated variable.tions of oscillation amplitude (o') of longshore currentThe flow field is assumed to consist of a steadytogether with the measured mean longshore currents longshore current V(x) and a perturbation velocityfor Cases 1 to 9. It is shown that the variations ofo' u'(x,y) = [u'(x,y),v'(x,y)], such that u(x,y) =have the same trend as the mean longshore curents [u'(x,y),V(x) + v'(x, y)], substituting it into Eqs (3)and the maximum of v' is approximately one sixth of and (4) and omitting nonlinear term yieldsthe maximum of the mean alongshore current and is .located near the location of the mean alongshore cur-u+Vu',= -gne,(5rent maximum.v +u'Vx +Vu;= -gMy.(63 Theoretical analysis for the longshore cur-A stream function ψ given by the following equationsis introduced due to the continuity Eq. (3):rent instability of the experimentψx= ho',(7a)The shear-wave instability theory is used in thepresent study to discuss if the observed oscillationsψy= -hul'.(7b)of longshore current are caused by shear instabilities.Eliminating η in Eqs (5) and (6) yieldsThe theoretical model is based on the 2D shallow-water equations. The rigid lid assumption is adopted,uly + Vu -咖-u%Vx-u'Vxx-which means that temporal variation of the surface el-V0- Voyx= 0.evation in the continuity equation is neglected and thefollowing equations are yielded:Substituting Eqs (7a) and (7b) in Eq. (8) leads toV.(hu)=0,(3)(号)+(号)0w+(号)心-(号)2V-Ut +(u. V)u+ gVη= 0.(4)()xx+V;(号)v+V(")wx=0.(9where x and y are the horizontal coordinates (x in-Equation (9)i中国煤化工for the instacreases seaward and x = 0, is the shoreline, y is thealongshore coordinate, see Fig. 1); V = (0/8x, 8/8y);bility of longsYHCNMH(:bation streamU is the depth-averaged velocity field; η is the surfacefunction mayelevation; h is the undisturbed water depth; g is theψ(x, y,t) = 4p(x)expli(kxy - wt)],.REN Chunping et al. Acta Oceanologica Simica 2009, Vol. 28, No. 4, P.96-106101where k: is wavenumber;w = Wr + iwi, wr is angular fre- difference. With the boundary conditions applied, thequency of shear instability and w; is the growth rate of following matrix is formedthe shear instability; so(x) is the amplitude of streamfunction. Substituting Eq. (10) into Eq. (9) yieldsAψ= cBF,(13)(V -c)(ψx_k2φ- 4正工h，(11)whereAandBareNbyNmatrices;ψisavectorofsize N. For given water depth h(x) and mean veloc-where c=w/k=Cr +ic, Cr is the phase speed of theity profile V(x), and wavenumber k is given, solvingshear instabilities of longshore currents, Cr = wr/k .Eq. (13) will give N eigenvalues of c and eigenfunction4(x). The case Wi > 0 shows the existence of the shearThe boundary conditions areinstabilities of longshore currents, otherwise longshoreφ=0 (x=0，x→∞).(12) currents are stable. Of N eigenvalues ofc, it is as-sumed that the eigenvalue with the largest imaginaryThe Eq. (11) is solved numerically by dividing the component dominates the instability of that wavenum-cross-shore length into N grids the partial derivatives ber, and we take this largest imaginary component asare approximated by the second-order centered finite the final solution for c of the Eq. (13).2.5 m°0 0.005 0.01 0.015 0.0200.0050.01 0.015 0.02100150 200 250300 350”400450 .0Ex=3.0mx=3.0 m-3.0moEL0.005 0.010.015 一0.020.005 .150200250300350400 4x=3.5 mx=3.5moE l0.0050.01 0.015 0.020.0050.00.015 0.02100 150 200 250 30035( ) 400 450x=4.0mx=4.0 m0 0.005 0.01 0.015 0.020.005 0.00.015 0.0100 150 200 250 300 350 400 450目x=4.5 mx=4.5m00.0050.010.0150.0200.0050.010.015一 020350400450f/Hz中国煤化工CNM HGFig.4. Calculated amplitude (the left), phases (the middle), the calculated (dotted lIne) and measured(solid line) time series of oscillation longshore currents (the right)..102REN Chunping et al. Acta Oceanologica Sinica 2009, Vol. 28, No. 4, P.96-1060.200.300. 30pCase 1Case 2Case 3●:*:0.15-0.000.00>04110980088.8...o.00t.08889g..:i.0.30 r0.40。Case 4Case 5Case 6g0.20:i:i,“,;e..i80.00 .Case 7Case 8Case9g0.10f“。is:g0.20-*':，0g81884。,1012/mFig.5. The variations of measured oscillation amplitude v' (o) and mean alongshore current (●) for Case9.一- represents the calculated perturbation velocity v' using Eq.(14b).1.02.0 5b0.8目0.65 1.0. plane slope .----Eql. pofile0.2--●barred profile-.-horizontal0.00t).5.50.x/x。0.3. plane slope-●Eql. profile-. - barred profile0.2-- plane slope香--- Eql. profile-.- barred profile100中国煤化工Fig.6. Theoretical velocity profiles (a), four topographies (b), the coMYHCNMHGPutrevuand Svendsen's and present results (o) (c) and the comparison for wr (d)..REN Chunping et al. Acta Oceanologica Simica 2009, Vol. 28, No. 4, P.96-106103Matrix Eq. (13) was solved using the routine The increase of propagation speed can be explainedbased on the algorithm of Garbow (1978) and Molerby the factor that propagation speed is proportional toand Stewart (1973). Figure 6 shows the calculated re- the maximum mean longshore current, which increasessults of the variations of wi and wr versus k for four to- with the increase of wave height. The decreases ofpographies (plane slope,“equilibrium” profile, barred wavenumber may be caused by the decrease of the vari-profile and horizontal, the width of surf zone xo=1,ation rate of shoreward and seaward mean longshore△x=0.01) using the present scheme. The numerical currents (Fig. 7) due to the increase of wave height,solution used 200 nodes and△k=0.01. Putrevu and which leads to a wide surf zone. However, the productSvendsen (1992) who used a fourth order finite dif- of them, that is, the frequency Wr = kCr, changes lit-ference scheme to discrete equation. The comparison tle and these results in very near frequency or periodis also made in the figure with the numerical result of shear instability for different cases shown in Tableof Putrevu and Svendsen (1992), who used a fourth- 3: the period is approximately 100 s for all the cases.order finite difference scheme to discrete equation, the This table gives the wavenumber ko corresponding toagreement is good and this second-order scheme used the maximum growth rate, the propagation speed Crby the present study also has enough accuracy.corresponding to ko, the numerical results of period ofshear waves (T = 2π/(koc)) and the period of shear4 Numerical resultwave obtained from experiment. Table 3 also gives theFigure 7 shows the measured mean longshore cur-corresponding experimental results T by the analysisrent for Cases 1- 9 (Zou et al, 2003) and the fittedin Section 2. We can see from Table 3 that the numer-curves used for the instability analysis mentioned inical results of period of shear waves agree well withthe period obtained from the experiment.the previous section.Figure 11 gives the calculated perturbation ve-Figures 8-10 show the variations of the growthlocity field (u',v') for Cases 4, 5, 6. u' and v' arerate wi and propagation speed Cr versus k for Cases 1-calculated using the following equations:9. For each figure the wave period is the same but thewave height has different values so the difference in w;(Cr) is due to the different wave heights. It is seen iniρkexp(w;t)expl[(ky - wrt)]，(14a)these figures that with the increase of wave height thewavenumber corresponding to the maximum growth虹_ 4品=∞iex(w(t)expli(ky -urt)]， (14b)rate decreases and the propagation speed increases.0.20.30Case 1Case 2Case 3。0.20:β0.10 1二0.10H≥0.10 t0.050.00L0.000.40Case 4Case 5 .Case 60.15叶g0. 10-g0.20 |>0.10-0. 10-0.Case 7Case 8.Case 90.15-。0.20-了g0. 100.20-0.05--0.107中国煤化工x:/m/mMHCNMHGFig.7. Measured mean longshore currents.●Data andthe fitted cures..104REN Chunping et al. Acta Oceanologica Sinica 2009, Vol. 28, No. 4, P.96-1062-0.4一 Case 1Case 1-----Case 2).3------- Case 32日0.2.1-0.50.02.1.k/rad. m-'k/rad :m"'Fig.8. The variations of the growth rate wi versus k for Cases 1-3 (a) and propagation speed Cr versus kifor Case 1-3 (wave period 1.0s) (b).0ECase 48E---- Case 50.3|----- Case.....-.. Case 6----- Case 6百4.1k/rad . m-'k/rad " m-'Fig.9. The variations of the growth rate wi versus k for Cases 4 6 (a) and propagation speed Cr versus kfor Cases 4-6 (wave period 1.5 s) (b).).4 rCase 7-Case 7-----. Case 8-----Case8Case g----ase.9e 0.2F0.1-0.50 1.k/rad.m1Fig.10. The variations of the growth rate wi versus k: for Cases 7-9 (a) and propagation speed Cr versus k:for Cases 7-9 (wave period 2.0s) (b).Table 3. Calculated resultsCaseko/rad.m 1_cr/m-s-1 .T/s0.5200.120100.64102.0410.3600.167104.460.17497.55100.0000.5700.115 .95.810.4200.15497.0998.0390.3400.19296.2096.1540.6600.10392.380.3700.18193.770.2800.215104.32Eqs (14a) and (14b) are obtained by Eqs (7a) and However, th中国煤化工inate with the(7b). The perturbation velocity given by the present developmentMHCNMHG resut inafi-study grows exponently with time, and this is because nite amplitude oI perturbation velocity, instead of infi-the linear instability theory is used for the analysis.nite amplitude as given by the linear instability theory..REN Chunping et al. Acta Oceanologica Simica 2009, Vol. 28, No. 4, P.96-106105This agrees with the present experiment result of the Thus, the maximum of perturbation velocity calcu-oscillation amplitude as shown in Fig. 5. It is seen lated using Eq. (14b) is equal to the one sixth of thethat the maximum of v' is approximately one sixth maximum of mean longshore currents. The amplitudeof the maximum of the mean alongshore current fromof v' obtained in this way is also given in Fig.5, andFig. 5. In order to account for this finite amplitude its trend agrees with the measured result also showneffect, a constant M is introduced in drawing Fig. 11 in the figure.to replace the growth factor [exp(w;t)] in Eqs (14a)The calculated total velocity fields (u',V +o') areand (14b). This will lead to a finite amplitude instead given for Cases4 6 in Fig.12. It is seen that the oS-of the infinite amplitude in Fig. 11. The value of Mcillations due to the shear instability of longshore cur-is determined by the relation M(4x/h)max = Vmax/6. rents occur and agree with the recorded velocity timeThis relation is from the result of Fig. 5, that is, the series (Fig. 2). The three cases (4, 5, 6) have the sameamplitude of oscillation of longshore current is Vmax/6. wave period and different wave heights. We can find25 E20 F2020 Hε15g151000.02 m/s0 t1N226工山山“x/m“Case 4Case 5Case 6Fig.11. Calculated perturbation velocity fields for Cases 4 6.2520 h20 C11↑↑F↑[1↑↑0hH↑↑!1111.0.2m/s! 1↑1 10.02468 10中国煤化工.MCHCNMH GFig.12. The calculated total velocity fields for Uases'4-o..106REN Chunping et al. Acta Oceanologica Sinica 2009, Vol. 28, No. 4, P.96-106that the range of the instability occuring becomesReferencesmore and more large with the increase of wave height.Hereinbefore, characteristics observed in experi-Baquerizo A, Caballeria M, Losada M A, et al. 2001.Frontshear and backshear instabilities of the meanment basically recur in the numerical results. Theselongshore current. J Geophys Res, 106: 16997-17011characteristics include the oscillation period, the vari-Bowen A J, Holman R A.1989. Shear instabilities of theations of amplitude in cross-shore direction and themean longshore current: 1. Theory. J Geophys Res,shape of stream line. It is concluded that the oscilla-94 (C12): 18023- 18030tions observed in experiment can be attributed to theDodd N, Iranzo V, Cballeria M.2004. A subcritical insta-shear instabilities. Thus, the observed oscillations arebility of wave-driven alongshore currents. J Geophysshear waves.Res, 109: C02018, doi: 10. 1029/2001JC001106Dodd N, Thornton E B.1990. Growth and energetics of5 Conclusionsshear waves in the nearshore. J Geophys Res, 95(C9): 16075- -16083The laboratory experiments on the instability ofGarbow B S.1978. CALGO Algorithm 535: the QZ al-alongshore currents were performed on a plane beachgorithm to solve the generalized eigenvalue problemwith slope 1:40. Low-frequency fuctuations of along-for complex matrices. ACM Transactions on Math-shore currents with the period of approximately 100 sematical Software, 4: 404- 410were observed.Hu Guangshu. 2003. Digital Signal Processing (in Chi-The dominant frequency and amplitudes of thenese). Bejing: Tsinghua University Pressoscillations of alongshore currents were determined us-Moler C, Stewart G W.1973. An algorithm for general-ing the maximum entropy method and the regres-ized matrix eigenvalue problems.SIAM Journal onsion method of trigonometric function, and the re-Numerical Analysis, 10: 241-256sults show that the variations of the amplitudes ofNoyesJ T, Guza R T, Elgar S, et al. 2004. Field obser-the alongshore current oscillations have similar trendsvations of shear waves in the surf zone. J Geophys .with the variations of the mean alongshore currentsRes, 109: C0103, doi: 10. 1029/2002JC001761in the cross-shore direction. And the maximum of OS-Oltman-Shay J, Howd P A, Birkemeier W A.1989. Shearinstabilities of the mean longshore current: 2. Fieldcillation amplitudes is approximately one sixth of theobservations. J Geophys Res, 94 (C12): 18031-maximum of the mean alongshore current and is lo-8042cated near the maximum of the mean alongshore cur-Putrevu U, Svendsen I A. 1992. Shear instability of long-rent.shore currents: a numerical study. J Geophys Res,The instability modes of longshore currents in97: 7283- 7303the present experiments were calculated using the lin-Xu Boxun, Bai Xubin. 1990. Application of the maxi-ear shear instability model. The comparison betweenmum entropy estimation in oil/ gas detection. Geo-the calculated oscillation periods and periods obtainedphysical Prospecting for Petroleum (in Chinese), 2: .from experiments show a good agreement. These re-1-12sults show that the periodic oscillations observed inZou Zhili, Wang Shuping, Qiu Dahong. 2003. Longshoreexperiment can be attributed to the shear waves duecurrents of random waves on different plane beaches.to the instabilities of longshore currents.China Ocean Engineering, 17(2): 265- -276中国煤化工MHCNMH G.