Internal wave simulation for different angles and shapes of continental shelf

Transcription

1 Internal wave simulatin fr different s and shapes f cntinental Himansu K. Pradhan*, A.D. Ra, K.K.G. Reddy and S. Mhanty Centre fr Atmspheric Sciences, Indian Institute f Technlgy Delhi, New Delhi, India ceancalling@gmail.cm Abstract- The average slpe f the cntinental in the wrld cean is.5 and its width varies cnsiderably. This paper illustrates experimental studies describing the internal wave run-up n different gradients f cntinental varying frm. t.5. MIT general circulatin mdel is cnfigured with a variable grid, tidal infrmatin in the mmentum equatins and backgrund stratificatin f density as initial fields t simulate internal waves. The mdel simulated density and temperature time-series is subjected t Fast Furier Transfrm t cmpute the energy spectra f internal waves. The results reveal that the peak f internal wave activity varies spatially fr different s f the cntinental. The experiments are further cntinued fr cncave castline gemetry t lk at the internal wave energy distributin ver the. The results shw that in a cncave castline the energy is large cmpared t a straight castline inferring cnvergence f internal wave energy. Keywrds Cntinental, Internal Waves; MITgcm; Energy spectra. I. INTRODUCTION Internal waves (IWs) are cmmn phenmena ver the cntinental shelves and internal tides are IWs at a tidal frequency that are generated when bartrpic tides/surface tides interact with shallw tpgraphic features [1]. Occurring beneath the free surface f a density stratified water bdy; they have utmst imprtance in submarine acustics, underwater navigatin, ffshre structures, cean mixing, bigechemical prcesses, etc. ver the -slpe regin. The gemetry f the -slpe regin and lcal hydrgraphic cnditins determine the effect f this interactin. Wave trains frmed after the interactin prpagate bth inward ver the cntinental and utward int the pen cean []. The cnversin f bartrpic tidal energy int barclinic energy in the frm f internal tides takes place at tpgraphic bundaries. Earlier studies f Legg and Adcrft [] examined mixing generated by internal wave reflectin frm variable tpgraphic slpes f cnvex and cncave shapes. The upper part f the cntinental slpe plays an imprtant rle in nn-linear effects and mixing as fund by Lien and Gregg [4] in their in-situ bservatins. Wallace and Wilkinsn [5] described the internal wave run up and its dissipative phase n unifrm slpes f. and.54 radian frm a series f labratry experiments. Peridic wave trains prpagate nt the slpe, steepens frms slitary wave as they travel shreward int the cntinental. Numerical study cnducted by Gerkema et al [6] fcused n the nn-linear evlutin f the internal tide and shwed that the main regin f generatin f IWs in upper part f the slpe using tw- dimensinal MITgcm. Studies by Munre and Lamb [7] and Hllway and Merrifield [8] have shwn that there are changes in cnversin rates in threedimensinal studies as cmpared t tw-dimensinal. Earlier studies cnsidered simulatin f IWs ver Gaussian tpgraphy, labratry experiments, etc. with tw-layered and multi-layered stratificatins. Present numerical study cnsiders the real aspect rati f the -slpe tpgraphy fund in the wrld cean with a stratified water clumn f a typical trpical cean. The cntinental acrss wrld cean varies and the average gradient f the cntinental is.5. Similarly the average gradient f cntinental slpe is -4. Here, we at first reprduce the generatin f IWs n different gradients f cntinental and cmpute the spectral energy distributin. Secndly, a cmparisn is made fr tw different stratificatin f May and February. Finally, we demnstrate the crucial rle f the castal gemetry. Fr this a cmparisn f spectral energy distributin is made between a straight and a cncave castline in a particular stratificatin. II. NUMERICAL MODEL CONFIGURATION MITgcm [9],[1] is an pen surce cde available t the cmmunity, (see designed t study bth ceanic and atmspheric phenmena. This finite vlume, z-crdinate mdel slves the incmpressible Navier-Strke equatins with Bussineq apprximatin n an Arakawa-C grid. The lpped cell representing the tpgraphy [11] is essential fr accurate representatin f the interactin f bartrpic tide with tpgraphy. Fr the current study the mdel uses plynmial equatin f state [1] fr the cmputatin f density field. The main physical parameters gverning the tidal flw are /14/$1. 14 IEEE

2 frequency f the bartrpic tide, ω amplitude f the bartrpic tide, U (c) crilis frequency f. The mdel is frced with real-time tides by adding tidal cmpnents in the mmentum equatins [1],[14] f the mdel. An scillating bartrpic flw is impsed unifrmly thrughut the dmain as a surface frcing. Fur tidal cmpnents M, S, K1 and O1 were cnsidered fr tidal frcing. The respective time perids are 1.4hr, 1hr,.9hr and 5.8hr and crrespnding velcity amplitudes are 8.98cm/s, cm/s, 15.41cm/s and 1.94 cm/sec fr the cmpnents. At pen bundaries, Orlanski radiatin bundary cnditins [15] are prescribed which allw any disturbance generated in the dmain t pass thrugh withut any significant distrtin. N-slip bundary cnditin is applied at the bttm and a free-slip at the lateral bundary. It uses implicit free surface and n rigid lid fr surface pressure. Fur experiments with gradients f cntinental varying frm. t.5 and gradient f cntinental slpe is as shwn in Fig. 1. The / edge pint is smthened t avid abrupt fall frm cntinental t cntinental slpe. An rthgnal curvilinear grid bearing 9*18* grid pints in x, y and z directin respectively. Frm the vertical levels, the first 1 levels are frm the sea surface with 1m interval each and next 4 levels are kept at m interval. Hwever, the maximum depth at this crss-sectin f the dmain is 5m (nt shwn in figure). cast pen cean - cntinental -4 ( varies between. t.5 ) RED GREEN cntinental slpe.4 - PINK (avg.~ ) BLUE Fig. 1: Schematic representatin f bttm tpgraphy at a particular crss-sectin. The mnthly climatlgical temperature and salinity fields are derived frm Wrld Ocean Atlas (9) f Natinal Oceangraphic Data Centre that are taken as initial density fields representing the backgrund density [16],[17]. The prfiles are taken fr the mnth f May and February bearing different stratificatins fund in the trpical waters f the Bay f Bengal in the Nrth Indian Ocean. The simulatin is carried ut fr 18 days f which the first 1 days are cnsidered fr spin-up and the next 8 days fr analysis. A.5days time span is chsen cntaining a spring tide fr cmputing the spectral estimates. This is btained by direct methd using the Fast Furier Transfrm algrithm f Cley and Tukey [18]. Spectral estimates f IWs is cmputed frm the time series f density and temperature that describes the distributin f signal pwer ver wave frequencies. III. RESULTS AND DISCUSSION The first set f experiments are carried ut fr different cntinental s.,.,.4 and.5 fr a width f 17km. Crrespndingly, the break falls at a water depth f 7m, 11m, 145m and 175m respectively. The crss-sectin at 9th grid pint that falls at the centre f the dmain alng the y-axis is taken int accunt fr study. The area averaged temperature and salinity fr the mnth f May were given as initial cnditins as shwn in the Fig. a,b,c. This shws that the water is stratified cntinuusly frm the surface bearing a small mixed layer f depth -5m. The Crilis frequency is fixed at 4.E-5 and the beta plane apprximatin is at.18e-11. spectral estimate ( C ) thermcline cntinental salinity ( psu ) density ( kg/m ) Fig. : Vertical prfile fr May Temperature Salinity (c) Density. The experiment details are tabulated belw: width cntinental slpe.- RED.- GREEN.4 - PINK.5 - BLUE pen cean 7 m m impact pints - at -5 fr different s f -4 cntinental Depth at halcline pycncline (c) Slpe Stratificatin. 17km 7m May. 17km 11m May (c).4 17km 145m May (d).5 17km 175m May Fig. : Spectral estimate f temperature fr all cntinental s. Bttm tpgraphy nrmal t the fr all s (left) and vertical temperature prfile (right).

3 Fig. a shws the cmputed spectral estimate f mdel simulated temperature fr different s (.,.,.4 and.5 in Fig. b) f the represented in different clurs. The activity is predminant ver the -slpe regin as the wave excitement tends t increase with the height and the steepness f the -slpe. As the increases, the internal wave activity is seen mre twards inner. Particularly fr., the activity is just ver the and the estimate is less since the water clumn is limited t the upper part f the thermcline at the, whereas depth at the break increases fr all ther s [see Fig. b ]. In general, the spectral estimate in the pen cean is less due t less r n interactin with the bttm tpgraphy cntinental cntinental slpe. - RED. - GREEN.4 - PINK.5 - BLUE pen cean 7 m m impact pints - at -5 fr different s f -4 cntinental density (kg/m ) Fig. 4: Spectral estimate f density acrss all crss-sectins. Bttm tpgraphy nrmal t the fr all s (left) and vertical density prfile (right). The spectral estimate f density representing spectral energy distributin is als shwn in Fig. 4. On cmparisn f Fig. and 4, the behaviur f the prfile fr spectral estimate f bth temperature and density is similar except its magnitude. Since the variables are different, it is expected the magnitudes f the spectral estimate t be different. The abve simulatins are cmpared with an experiment using a different stratificatin fr February using February temperature while retaining May salinity t see the effect f temperature alne. The vertical prfile f temperature, salinity and density calculated frm bth temperature and salinity are shwn in Fig thermcline salinity ( psu ) density ( kg/m ) Fig. 5: Vertical prfile fr February Temperature Salinity (c) Density. The detail f the experiments are tabulated belw: width Depth at Slpe. 17km 7m Feb. 17km 11m Feb (c).4 17km 145m Feb (d).5 17km 175m Feb spectral estimate ( C ) cntinental Stratificati n Fig. 6: Spectral estimate f temperature acrss all crsssectins. Bttm tpgraphy nrmal t the (left) fr all s and vertical temperature prfile (right). cntinental slpe halcline pen cean pycncline (c). - RED. - GREEN.4 - PINK.5 - BLUE 7 m m impact pints - at -5 fr different s f -4 cntinental

4 cntinental cntinental slpe. - RED. - GREEN.4 - PINK.5 - BLUE pen cean 7 m m impact pints - at -5 fr different s f -4 cntinental density (kg/m ) Fig. 7: Spectral estimate f density acrss all crss-sectins. Bttm tpgraphy nrmal t the (left) fr all s and vertical density prfile (right). Fig. 6 shws spectral estimate f temperature fr all cntinental s fr February stratificatin. It als shws the prpagatin f IWs twards the cast enhances with the larger gradient f cntinental as in the case f May. Hwever, the verall spectral estimate are less cmpared t that f simulated fr May. This fact is attributed t prevailing deeper mixed layer (extending up t 6m) in Feb. The peak spectral estimate at. lies utside the cntinental regin as bserved in May due t small prtin f stratified water clumn interacts at the (the depth f the is 7m as shwn in figure). The spectral estimate f energies fr May and Feb at.5 is apprximately the same, which may be attributed t less effect f the mixed layer at higher depth (175m) f. In bth the mnths, the at 175m interacts with lwer prtin f the pycncline. Als in these cases, the extent f spectral estimate f energy prpagatin ver the is nearly the same hwever their peak varies. In all ther cntinental s (.,. and.4 ) the spectral estimate in Feb is less than May, as a deep mixed layer desn't supprt grwth f IWs prpagatin/generatin. As discussed previusly, the spectral estimate f density shwn in Fig.7 describes the energy spectra have the same behaviur as spectral estimate f temperature. The experiments are further cntinued t see the effect f the gemetry f the castline. Fr this, an experiment with May stratificatin is carried ut with cncave castline fr a cntinental f.4 and cntinental slpe f [see Fig. 8a]. Spectral energy distributin at the crss-sectin f interest i.e. at, 8, 9, 1 and 15 in slid lines shwn in Fig. 8b(iv) is analysed. The 8th, 9th and 1th grid pint falls in the cncave prtin with the 9th pint falling at the centre f curvature. In this experiment, rtatin is nt taken int accunt i.e. crilis frce is kept as zer. latitude pints & (145m) c(i) grid pints 15 lngitude b(i) pints & 1 depth (m) b(ii) c(ii) grid pints b(v) b(iv) b(iii) c(iii) grid pints Fig. 8: The dmain f the cncave castline. b(i-v) The spectral energy distributin at different crss sectins. c(i-iii) The bttm tpgraphy at the respective crss sectins. latitude pints & (145m) c(i) grid pints lngitude b(i) pints & 1 depth (m) grid pints Fig. 9: The dmain f a straight castline. b(i-v) The spectral energy distributin at different crss sectins. c(i-iii) The bttm tpgraphy at the respective crss sectins. 8 b(ii) c(ii) b(v) b(iv) b(iii) c(iii) grid pints

5 The energy is maximum at the 9th crss-sectin and decreases n either side indicating the cnvergence f energy due t the cncave shape f the castline. T understand further this energy accumulatin due t the cncave shape f the castline, an experiment with straight castline is carried ut keeping the same cngfiguratin (cntinental f.4 and cntinental slpe f ) shwn in Fig. 9. The energy distributin shws n change at all the crsssectins [see Fig. 9b(i-v)]. The spectral energy in the cncave castline is almst duble at 9th crss-sectin as cmpared t the straight castline. Thus, the energy maximum at the curved prtin is the due t the gemetry f the castline. IV. CONCLUSION The edge remains the regin f maximum internal wave activity. With the increasing depth f the the stratified water clumn allws mre IW activity ver the cntinental. The peak f spectral estimate f energy fr. lies utside the regin in bth the cases (May & Feb) as the interactin with the cntinental regin is limited nly t the upper part f the thermcline. Because f the presence f a deeper mixed layer (6-7m) in Feb, the assciated spectral energy estimates are relatively less cmpared t that f cntinuusly stratified waters f May. The peak spectral estimate f energy fr.5 fr bth the mnths (May & Feb) is nearly the same as the effect f mixed layer depth f 175m is negligible. The gemetry f the castline als plays a majr rle in IW energy distributin with an increased magnitude seen in the cncave part f the cast. [8] P. E. Hllway and M. A. Merrifield, "Internal tide generatin by seamunts, ridges, and islands," J. Gephys. Res., vl. 14, n. C11, pp , Nv [9] J. Marshall, A. Adcrft, C. Hill, L. Perelman, and C. Heisey, "A finite-vlume, incmpressible navier stkes mdel fr studies f the cean n parallel cmputers," J. Gephys. Res., vl. 1, n. C, pp , Mar [1] J. Marshall, C. Hill, L. Perelman, and A. Adcrft, "Hydrstatic, quasi-hydrstatic, and nnhydrstatic cean mdeling," J. Gephys. Res., vl. 1, n. C, pp , Mar [11] A. Adcrft, C. Hill, and J. Marshall, "Representatin f tpgraphy by shaved cells in a height crdinate cean mdel," Mn. Wea. Rev., vl. 15, n. 9, pp. 9-15, Sep [1] T. J. McDugall, D. R. Jackett, D. G. Wright, and R. Feistel, "Accurate and cmputatinally efficient algrithms fr ptential temperature and density f seawater," J. Atms. Oceanic Technl., vl., n. 5, pp , May. [1] S. Khatiwala, "Generatin f internal tides in an cean f finite depth: analytical and numerical calculatins," Deep Sea Research Part I: Oceangraphic Research Papers, vl. 5, n. 1, pp. -1, Jan.. [14] C. Gu, X. Chen, V. Vlasenk, and N. Stashchuk, "Numerical investigatin f internal slitary waves frm the luzn strait: Generatin prcess, mechanism and three-dimensinal effects," Ocean Mdelling, vl. 8, n. -4, pp. -16, Jan. 11. [15] I. Orlanski, "A simple bundary cnditin fr unbunded hyperblic flws," Jurnal f Cmputatinal Physics, vl. 1, n., pp , Jul [16] R. A. Lcarnini, A. V. Mishnv, J. I. Antnv, T. P. Byer, H. E. Garcia, O. K. Baranva, M. M. Zweng, D. R. Jhnsn, "Wrld Ocean Atlas 9," vl. 1, Temperature, NOAA Atlas NESDIS, vl. 68, edited by S. Levitus, pp. 184, NOAA, Silver Spring, Md, 1. [17] J.I. Antnv, D. Seidv, T. P. Byer, R. A. Lcarnini, A. V. Mishnv, H. E. Garcia, O. K. Baranva, M. M. Zweng, D. R. Jhnsn, Wrld Ocean Atlas 9, vl., Salinity, NOAA Atlas NESDIS, vl. 69, edited by S. Levitus, pp. 184, NOAA, Silver Spring, Md, 1. [18] J. W. Cley and J. W. Tukey, "An algrithm fr the machine calculatin f cmplex furier series," Mathematics f Cmputatin, vl. 19, n. 9, pp. 97-1, REFERENCES [1] C. Wunsch, "On the prpagatin f internal waves up a slpe," Deep Sea Research and Oceangraphic Abstracts, vl. 15, n., pp , Jun [] O. M. Phillips, "Dynamics f the upper cean",cambridge University Press, pp. 61, [] S. Legg and A. Adcrft, "Internal wave breaking at cncave and cnvex cntinental slpes*," J. Phys. Oceangr., vl., n. 11, pp. 4-46, Nv.. [4] R. C. Lien and M. C. Gregg, "Observatins f turbulence in a tidal beam and acrss a castal ridge," J. Gephys. Res., vl. 16, n. C, pp , Mar.1. [5] B. C. Wallace and D. L. Wilkinsn, "Run-up f internal waves n a gentle slpe in a tw-layered system," Jurnal f Fluid Mechanics, vl. 191, pp , Jun [6] T. Gerkema, C. Staquet, and P. Buruet-Aubertt, "Nn-linear effects in internal-tide beams, and mixing," Ocean Mdelling, vl. 1, n. -4, pp. -18, Jan. 6. [7] J. R. Munre and K. G. Lamb, "Tpgraphic amplitude dependence f internal wave generatin by tidal frcing ver idealized three-dimensinal tpgraphy," J. Gephys. Res., vl. 11, n. C, pp. C 1+, Feb. 5.