The effect of a background shear current on large amplitude internal solitary waves

Transcription

1 PHYSICS OF FLUIDS 8, The effect of a background shear current on large ampltude nternal soltary waves Wooyoung Cho a Department of Mathematcal Scences, Center for Appled Mathematcs and Statstcs, New Jersey Insttute of Technology, Newark, New Jersey Receved 8 October 005; accepted 4 January 006; publshed onlne 4 March 006 Large ampltude nternal waves nteractng wth a lnear shear current n a system of two layers of dfferent denstes are studed usng a set of nonlnear evoluton equatons derved under the long wave approxmaton wthout the smallness assumpton on the wave ampltude. For the case of unform vortcty, soltary wave solutons are obtaned under the Boussnesq assumpton for a small densty jump, and the explct relatonshp between the wave speed and the wave ampltude s found. It s shown that a lnear shear current modfes not only the wave speed, but also the wave profle drastcally. For the case of negatve vortcty, when compared wth the rrotatonal case, a soltary wave of depresson travelng n the postve x drecton s found to be smaller, wder, and slower, whle the opposte s true when travelng n the negatve x drecton. In partcular, when the ampltude of the soltary wave propagatng n the negatve x drecton s greater than the crtcal value, a statonary recrculatng eddy appears at the wave crest. 006 Amercan Insttute of Physcs. DOI: 0.063/.809 I. INTRODUCTION Large ampltude nternal waves are no longer rare phenomena evdenced by an ncreasng number of feld observatons and the study of ther physcal propertes has been an actve research area n recent years; for example, see Lynch and Dahl, Ostrovsky and Stepayants, and Helfrch and Melvlle. 3 Classcal weakly nonlnear theores ncludng the well-known Korteweg de Vres KdV equaton have been found to fal n ths strongly nonlnear regme, and t has been suggested that new theoretcal approaches are necessary to better descrbe large ampltude nternal waves. These strongly nonlnear waves have often been studed numercally for both contnuously stratfed and multlayer fluds Lamb; 4 Grue et al. 5. Alternatvely, wth the long wave approxmaton but no assumpton on the wave ampltude, the strongly nonlnear asymptotc models were proposed Myata; 6 Cho and Camassa; 7 Cho and Camassa 8, and t was found that these smplfed models are ndeed vald for large ampltude waves. In partcular, for travelng waves, the system of coupled nonlnear evoluton equatons can be reduced to a sngle ordnary dfferental equaton, and ts soltary wave soluton shows excellent agreement wth laboratory experments and numercal solutons of the Euler equatons Camassa et al. 9. The success of ths asymptotc model even for large ampltude waves can be attrbuted to the fact that the nternal soltary wave becomes wder as the wave ampltude ncreases and, therefore, the underlyng long wave assumpton s vald even for large ampltude waves. Many of these prevous studes have neglected the background shear current effect, whch s mportant n real ocean envronments but s not well understood. In ths paper, the strongly nonlnear model s further generalzed to nclude the a Electronc mal: wycho@njt.edu shear current effect, and ts soltary wave solutons and ther wave characterstcs are studed. For contnuously stratfed flows, Maslowe and Redekopp 0 consdered weakly nonlnear nternal waves n shear flows and showed that the KdV and Benjamn-Ono equatons can be derved for long waves n the shallow and deep confguratons, respectvely, both wth and wthout crtcal layers. Stastna and Lamb nvestgated numercally the effect of shear currents on fully nonlnear nternal waves. For a lnear shear current of negatve constant vortcty, they found the nternal soltary wave of elevaton travelng n the postve x drecton s taller and narrower, whle the wave propagatng n the opposte drecton becomes shorter and wder. Ther numercal solutons showed that the maxmum wave ampltude soluton approaches the conjugate flow lmt or a front soluton, although the so-called wave breakng and shear nstablty lmts are also possble for some parameter ranges. For two-layer system, Breyanns et al. consdered nternal waves n a lnear shear current numercally usng a boundary ntegral method. Snce ther focus was on surface waves at the ar-water nterface, they added wnd of constant velocty n the upper layer, resultng n a velocty dscontnuty at the nterface. Therefore, along wth the large densty jump that they consdered, ther results are not drectly applcable to nternal waves of our nterest. Here, we consder a system of two layers, each of whch has constant densty and constant vortcty by assumng the detaled vortcty dstrbuton s not mportant for long nternal waves. Soltary wave solutons are computed and compared wth those of the rrotatonal model, and partcular attenton s pad to wave profles, wave speeds, and streamlnes. Ths paper s organzed as follows. Wth the governng equatons n Sec. II, the strongly nonlnear model s presented n Sec. III and s reduced to a sngle equaton for /006/83/03660/7/$3.00 8, Amercan Insttute of Physcs Downloaded 6 Mar 006 to Redstrbuton subject to AIP lcense or copyrght, see

2 Wooyoung Cho Phys. Fluds 8, where u * and w * are the total horzontal and vertcal veloctes, respectvely, whle u and w represent the perturbaton veloctes. From conservaton of vortcty, any twodmensonal perturbaton to unform shear flows must be rrotatonal and, therefore, the perturbaton veloctes can be wrtten as u = x and w = z, where the velocty potental satsfes the Laplace equaton =0. By substtutng nto the lnearzed system of. and.4, = A coshkz h e kx t, = A coshkz + h e kx t,.7 the lnear dsperson relaton between wave speed c and wave number k can be obtaned from FIG.. Internal waves n two-layer system nteractng wth lnear shear currents. soltary waves. In Sec. IV, usng the Boussnesq assumpton and assumng vortcty s unform for smplcty, the propertes of large ampltude nternal soltary waves propagatng n both horzontal drectons are descrbed n detal wth concludng remarks n Sec. V. II. GOVERNING EQUATIONS For an nvscd and ncompressble flud of densty, the velocty components n Cartesan coordnates u *,w * and the pressure p * satsfy the contnuty equaton and the Euler equatons, u * x + w * z =0,. u * t + u * u * x + w * u * z = p * x /,. w * t + u * w * x + w * w * z = p * z / g,.3 where g s the gravtatonal acceleraton and subscrpts wth respect to space and tme represent partal dfferentaton. Here, = = stands for the upper lower flud see Fg. and s assumed for a stable stratfcaton. The boundary condtons at the nterface are the contnuty of normal velocty and pressure, t + u * x = w *, t + u * x = w *, p * = p * at z = x,t,.4 where s the nterface dsplacement. At the upper and lower rgd surfaces, the knematc boundary condtons are gven by w * x,h,t =0, w * x, h,t =0,.5 where h h s the undsturbed thckness of the upper lower flud layer. We assume that the total velocty can be decomposed nto a background unform shear of constant vortcty and a perturbaton so that u * x,y,z = z + u x,z,t, w * = w x,z,t,.6 c coth kh + coth kh c /k g /k =0..8 For long waves kh 0, the lnear long wave speed s found to be c = h h h + h ± h h 4 h + h + gh h,.9 h + h whch, for =0, can be reduced to c =± gh h h + h..0 III. A STRONGLY NONLINEAR MODEL For long surface gravty waves n a sngle layer wth constant vortcty, a system of nonlnear evoluton equatons was derved by Cho 3 by adoptng the smlar asymptotc expanson method used n Cho and Camassa. 8 Snce the detaled dervaton of the model can be found n Cho, 3 t wll be omtted here. Assumng the rato of water depth to the characterstc wavelength s small for long waves, the pressure n.3 s expanded to the second order n ths small parameter to nclude the nonhydrostatc pressure effect, whch, when combned wth. and., results n the nonlnear evoluton equatons for the lower layer, t h x + x + u x =0, = h +, 3. u t h u x + u u x + g x = P x / G x, 3. where P= px,,t s the pressure at the nterface provdng couplng wth the upper layer, and the depth-averaged velocty u and the nonhydrostatc pressure effect G are defned by u x,t = u x,z,tdz, h 3.3 Downloaded 6 Mar 006 to Redstrbuton subject to AIP lcense or copyrght, see

3 The effect of a background shear current Phys. Fluds 8, G x,t = u xt h u xx + u xx + u u xx u x. 3.4 The knematc equaton 3. s exact, whle the dynamc equaton 3. contans an error of O 4, where s defned by the rato of water depth to the characterstc wavelength. The second term of the rght-hand sde of 3. represents the nonlnear dspersve effect of large ampltude nternal waves. For the upper layer, by smply replacng,g, by, g, and the subscrpt by, we can obtan an approprate set of equatons. Then, the complete set of equatons for the four unknowns,u,u, P s, n dmensonal form, gven by di dt dtu d + 6 u xx dx =0 =,, 3. dp dt dt d = u U h dx =0, 3. whch physcally correspond to conservaton laws for mass, rrotatonalty, and horzontal momentum of a perturbaton to the background lnear shear current. In order to fnd soltary wave solutons travelng wth constant speed c, we assume that x,t = X, u x,t = u X, X = x ct. 3.3 t + U x U h x + u x =0, 3.5 Substtutng 3.3 nto 3.5 and ntegratng once wth respect to X yelds u t + U u x + u u x + g x = P x + 3 u xt + U u xx 3 U u xx + u u xx u h x, x 3.6 where U s the velocty at the rgd boundary gven by U = h, U = h. 3.7 For =0, the system gven by 3.5 and 3.6 can be reduced to the system of equatons of Cho and Camassa. 8 When computng travelng wave solutons, t s useful to wrte the horzontal momentum equaton 3.6 n the followng conserved form: u + 6 u +U u + xxt u + gx = P x + u xt + 3 U u xx U u xx 3 h + 3 u u xx u x, 3.8 x or, after multplyng 3.6 by, u U h + U u + u t U h U h u + 3 U h = P x + + g x 3x 3 u xt + U u xx U u xx 3 h + u u xx u x. 3.9 x From 3.5, 3.8, and 3.9, t s easy to see that the followng three conservaton laws hold: dm d dx =0, dt dt 3.0 u = c U h + U h h, 3.4 where we have used the boundary condtons at nfnty: h and u 0 as X. After substtutng 3.3 nto 3.8 and ntegratng wth respect to X once, Eq. 3.8 can be wrtten as 3 c + U + u U h u XX 3 u X = c + U u + u + g + P, 3.5 where we have used u X, u XX 0asX. When we subtract 3.5 for = and to elmnate P, we have = 3 c + U + u U u XX u h X = c + U u + = u + g. 3.6 On the other hand, when we add 3.9 for =, and ntegrate once wth respect to X, we have = 3 c + U + u U u XX u h X = = c + U + u u c + U U h U + 3 h 3 g h + h P, 3.7 where we have used + =h +h to obtan the last term n the rght-hand sde. After usng 3.5 wth = for P and addng 3.6 and 3.7, the fnal equaton for X can be found, after a lengthy manpulaton, as where X = 3 A 3 + A + A 3 + A 4 B 5 + B 4 + B B 4 + B 5 + B 6, 3.8 Downloaded 6 Mar 006 to Redstrbuton subject to AIP lcense or copyrght, see

4 Wooyoung Cho Phys. Fluds 8, A = U h + U h, A = U h h 4h U h h 4h + g, A 3 = U 3 h U h h h c U U + c + gh h, A 4 = c U h + U h + c h + h gh h, B = U 4 h U 4 h, B = U 4 h h 4h + U 4 h h 4h, U B 3 = h h U h h h h h h + c U U, U B 4 = h h U h + h h h + cu h h + cu h h, B 5 = ch h U U + c h h, B 6 = c h h h + h. When U =0, 3.8 becomes the equaton for rrotatonal nternal waves Myata; 4 Cho and Camassa 8, X = 3 g + gh h c + c h + h gh h c h h h + h h h For the one-layer case =03.8 can be reduced to the model for surface waves n a unform shear flow, 3 X = + 4+gh /U + h /U c cu gh +h +c/u h, 3.0 where, from 3.7, U = h. IV. BOUSSINESQ ASSUMPTION WITH UNIFORM VORTICITY To further smplfy Eq. 3.8, we assume a small densty change across the nterface true for most oceanc applcatons and use the Boussnesq assumpton, so that we can replace = 0 / and = 0 +/ by 0 except for terms proportonal to the gravtatonal acceleraton g. Also, for smplcty, unform vortcty s assumed and, thus, =. Then, the equaton for X gven by 3.8 can be smplfed to 3 A + A 3 + A 4 X = B 4 + B B 4, + B 5 + B 6 where the coeffcents are gven by A = 0 h + h /4 + g, A 3 = 0 ch + h + gh h, A 4 = 0 c h + h gh h, B = 3 0 h + h /4, B 3 = 0 h h + 0 ch + h, B 4 = 0 h h h + h 0 ch h, 4. B 5 = 0 ch h h + h + 0 c h h, B 6 = 0 c h h h + h. From the fact that the numerator has to vansh at the wave crest where =a wth a beng the wave ampltude, the wave speed c can be found, n terms of a, as c = a ±g/ 0h ah + a h + h. 4. Notce that, compared wth the rrotatonal =0 case, the nonlnear wave speed s ncreased by a/, whle the lnear wave speed c 0 gven, from 4. wth a=0, by c 0 =± g/ 0h h h + h, 4.3 s unaffected by constant vortcty under the Boussnesq assumpton. As n the rrotatonal case, when A a +A 3 a+a 4 =0 n the numerator of 4. has double roots, we have the maxmum wave ampltude, at whch, nstead of a soltary wave, a front soluton appears and beyond whch no soltary wave soluton exsts. Ths condton can be wrtten as A 3 =4A A 4 to yeld the followng expresson for the maxmum wave speed c m : Downloaded 6 Mar 006 to Redstrbuton subject to AIP lcense or copyrght, see

5 The effect of a background shear current Phys. Fluds 8, FIG.. Maxmum wave ampltude a m gven by 4.5 for varyng U /c 0, where U =h and c 0 s the lnear long nternal wave speed defned by 4.3: ---,h /h =.5;, h /h =5. The densty rato s / =.00, and the superscrpts and ndcate the waves propagatng n the postve and negatve x drectons, respectvely. FIG. 3. Wave speed c vs wave ampltude a gven by 4. for the case of h /h =5 and / =.00:, U /c 0 =0; - - -, U /c 0 = or / g/h =0.0, where U =h and c 0 s the lnear long nternal wave speed defned by 4.3. Notce that c + =c for =0. The densty rato s / =.00, and the superscrpts and ndcate the waves propagatng n the postve and negatve x drectons, respectvely. c m ± = 4 h h ± 4g/0 h + h + h + h. 4.4 Then, the maxmum wave ampltude a m can be found, from 4., as a ± m = g/ ± 0h h +h + h c m 4g/ h + h For =0, the maxmum wave speed and the maxmum wave ampltude can be reduced to c m ± =± g/ 0 h + h, a m ± = h h, 4.6 whch are the results of Cho and Camassa 8 under the Boussnesq assumpton. In ths paper, wthout loss of generalty, we assume 0 and consder waves travelng n both postve and negatve x drectons. The densty rato / =.00 s chosen for the results presented here, and h h s assumed so that a soltary wave of depresson s expected, unless s large. As shown n Fg., when propagatng n the negatve x drecton, the maxmum wave ampltude a m ncreases consderably as or, equvalently, U ncreases, approachng h. On the other hand, when propagatng n the postve x drecton, the maxmum wave ampltude a m + decreases and ts polarty changes as ncreases. Even though the thckness of the lower layer s greater than that of the upper layer, t s possble for a soltary wave of elevaton to exst, for example, when h /c for h /h =5. Fgure also shows that, for 0, the maxmum wave ampltude can be zero, mplyng that no soltary wave soluton exsts, even at a depth rato dfferent from the crtcal rato h /h = for =0. From Fg. 3, for a gven depth rato and h /c 0,tcan be seen that the wave speed c s greater for the wave travelng n the negatve x drecton compared wth that of the rrotatonal soltary wave of same ampltude, whle the opposte s true for the wave travelng n the postve x drecton. As shown n Fg. 4a, soltary wave profles of a/h = 0.5 are obtaned by solvng 4. numercally usng MATH- EMATICA, and t s found that the soltary wave travelng n the postve x drecton s wder than that of the soltary wave FIG. 4. Soltary wave profles for U /c 0 = / g/h =0.0, dashed lne, where U =h, compared wth those for =0 sold lne for / =.00 and h /h =5. a a/h = 0.5, b a=0.99a m, where a m /h and for waves travelng to the rght and left, respectvely. Downloaded 6 Mar 006 to Redstrbuton subject to AIP lcense or copyrght, see

6 Wooyoung Cho Phys. Fluds 8, FIG. 5. Streamlnes for a soltary wave propagatng n the negatve x drecton, wthout and wth a recrculatng eddy, for U /c 0 = / g/h =0.0, dashed lne, / =.00, and h /h =5: a a=0.7a m and b a=0.99a m, where a m /h Notce that the frame of reference movng wth the soltary wave s used and the vertcal scale s exaggerated. travelng n the negatve x drecton. Also, as can seen n Fg. 4b, the wave profles near the maxmum wave ampltudes are very dfferent for two waves travelng n opposte drectons. For example, the wave propagatng n the negatve x drecton s much taller and narrower, whch s consstent wth the numercal soluton of Stastna and Lamb for contnuously stratfed shear flows. It s nterestng to notce that, when a large ampltude wave s travelng n the negatve x drecton, t s possble for a recrculatng eddy to appear at the wave crest. Fgure 5 shows the streamlnes for soltary waves of two dfferent wave ampltudes. A well-defned statonary recrculatng eddy can be observed for a/h =0.99a m, where a m /h = , as shown n Fg. 5b, whle t dsappears for a/h =0.7a m. In the absence of shear, the presence of a statonary recrculatng eddy was observed prevously n twolayer system wth a constant Brunt-Väsälä frequency along wth a densty jump across the nterface by Voronovch. 5 The crteron of exstence of ths recrculatng eddy can be found from the fact that a stagnaton pont where the velocty vanshes should appear n the upper layer, more specfcally, az0. At the leadng order, the total stream functon can be approxmated by X,z = z cz + u Xz h for X z h, z cz + u Xz + h for h z X. 4.7 Then, the total horzontal velocty n the upper layer at the orgn X=0 can be found, after substtutng 3.4 for u,as z 0 = a h g/ 0h + a h + h h a. 4.9 z = z c + u 0 = z a + h g/ 0h + a h + h h a, 4.8 from whch the vertcal locaton of a stagnaton pont z 0 can be found as For ths stagnaton pont to le n az 0 0, the followng nequalty between a and must be satsfed: h a g/ 0h + a h + h h a, whch s shown n Fg Downloaded 6 Mar 006 to Redstrbuton subject to AIP lcense or copyrght, see

7 The effect of a background shear current Phys. Fluds 8, Here, a smple, unform vortcty dstrbuton s adopted and a smlar approach used here can be appled to a more general vortcty dstrbuton whch mght be approxmated by the stack of constant vortcty layers. ACKNOWLEDGMENT Ths work was supported n part by the U.S. Offce of Naval Research through Grant No. N FIG. 6. Doman of exstence of recrculatng eddy for / =.00 and h /h =5. Sold lne: the maxmum wave ampltude gven by 4.5; dashed lne: Eq V. CONCLUDING REMARKS We have studed large ampltude nternal soltary wave solutons n a lnear shear current usng the strongly nonlnear asymptotc model. It s found that, even for small background vortcty, the wave characterstcs are sgnfcantly dfferent for waves travelng n opposte drectons. The background lnear shear current changes not only the wave speed, but also the wave shape drastcally. Therefore, the knowledge of background currents s essental for the accurate nterpretaton of an ncreasng number of feld measurements. Valdty of the strongly nonlnear model n the absence of shear has been examned by Ostrovsky and Grue 6 and Camassa et al. 9 It was found that soltary wave solutons of the model show excellent agreement wth Euler solutons and laboratory/feld experments n shallow confguraton n whch the depth rato h /h s less than roughly 0. The model proposed here s therefore expected to be vald for the same depth rato range. As the thckness of one layer s much greater than the other, a deep confguraton model smlar to that n Cho and Camassa 8 should be adopted, although ts valdty mght be lmted to ntermedate wave ampltudes for a fxed depth rato. 9 J. F. Lynch and P. Dahl, Specal Issue on scence and engneerng advances n explorng the Asan margnal seas, IEEE J. Ocean. Eng L. A. Ostrovsky and Y. A. Stepanyants, Internal soltons n laboratory experments: Comparson wth theoretcal models, Chaos 5, K. R. Helfrch and W. K. Melvlle, Long nonlnear nternal waves, Annu. Rev. Flud Mech. 38, K. G. Lamb, Numercal experments of nternal wave generaton by strong tdal flow across a fnte ampltude bank edge, J. Geophys. Res. 99, J. Grue, H. A. Frs, E. Palm, and P. O. Rusas, A method for computng unsteady fully nonlnear nterfacal waves, J. Flud Mech. 35, M. Myata, Long nternal waves of large ampltude, n Proceedngs of the IUTAM Symposum on Nonlnear Water Waves, edted by H. Horkawa and H. Maruo Tokyo, Japan, 988, p W. Cho and R. Camassa, Long nternal waves of fnte ampltude, Phys. Rev. Lett. 77, W. Cho and R. Camassa, Fully nonlnear nternal waves n a two-flud system, J. Flud Mech. 396, R. Camassa, W. Cho, H. Mchallet, P.-O. Rusås, and J. K. Sveen, On the realm of valdty of strongly nonlnear asymptotc approxmatons for nternal waves, J. Flud Mech. 549, S. A. Maslowe and L. G. Redekopp, Long nonlnear waves n stratfed shear flows, J. Flud Mech. 0, M. Stastna and K. G. Lamb, Large fully nonlnear nternal soltary waves: The effect of background current, Phys. Fluds 4, G. Breyanns, V. Bontozoglou, D. Valougeorgs, and A. Goulas, Largeampltude nterfacal waves on a lnear shear flow n the presence of a current, J. Flud Mech. 49, W. Cho, Strongly nonlnear long gravty waves n unform shear flows, Phys. Rev. E 68, M. Myata, An nternal soltary wave of large ampltude, La Mer 3, A. G. Voronovch, Strong soltary nternal waves n a.5-layer model, J. Flud Mech. 474, L. A. Ostrovsky and J. Grue, Evoluton equatons for strongly nonlnear nternal waves, Phys. Fluds 5, Downloaded 6 Mar 006 to Redstrbuton subject to AIP lcense or copyrght, see