IMPACT OF BOTTOM STRESS AND CURRENTS ON WAVE-CURRENT INTERACTIONS

Transcription

1 IMPACT OF BOTTOM STRESS AND CURRENTS ON WAVE-CURRENT INTERACTIONS Yin Baoshu 1, 3, Will Perrie 3, HouYijun 1, Lin Xiang 1,, Cheng Minghua 1 1 Inst.of Oeanology, Chinese Aademy of Sienes, Qingdao, PR China Graduate shool, Chinese Aademy of Sienes, Beijing, PR China 3 Fisheries & Oeans Canada, Bedford Institute of Oeanography, Dartmouth, Nova Sotia, Canada 1. INTRODUCTION Wave-urrent interations are important for alulation of impats on bed stress due to wind wave ativity in the oastal region. This is important for a robust theoretial basis for sediment transport estimates, beah evolution alulations and onomitant proesses related to the study of land-sea interations in the oastal area. In reent years, there has been inreased interest in developing models that an resolve the near-bed region, taking aount of impats due to near-bed turbulene related to wind-wave ativity. However, how to alulate the bottom stress has beome an important problem in understanding the proesses that ontrol and define waveurrent interations. At present, there are two widespread methods to alulate the bottom stress. One is alled the integral method, given by Grant and Madsen (1979). The other is alled the separation method, given by Christoffersen and Jonsson (1985). Signell et al.(1990) inluded wave-urrent interation effets in an idealized estuarine model and showed that they ould influene the flow field. Davies et al.(1994) used a similar approah to examine the influene of enhanements in bed stress due to wave-urrent interations upon the wind-indued irulation of the eastern Irish Sea. In this paper, a two-dimensional urrent model is oupled with a third generation shallow water wave model to examine the hanges of bottom stress and urrent veloity due to wave-urrent interations in the oastal region of the Bohai Sea.. HYDRODYNAMIC MODEL.1 Two-Dimensional Current Model This model is motivated by a similar development by Fang and Cao (1990). A Cartesian oordinate system is assumed, with x and y oordinates direted to the east and north, respetively. The ontinuity and momentum equations are ζ + [( h + ζ ) u] + [( h + ζ ) v] = 0 (1) t x y du dt dv dt ζ 1 τ fv = g + x ρ ζ 1 τ + fu = g + y ρ ax τ h + ζ ay bx τ h + ζ by + A x + A u u + x y y v v + x y where ( u, v) are the eastward and northward omponents of the depth-averaged veloity vetor, respetively; t is time, f is the Coriolis parameter, A x and A y are the eddy visosities; g is the gravitational aeleration, v ρ is the density of sea water, ( h + ζ ) is the total depth (mean water + surfae elevation ), τ s = ( τ x, τ y ) is v surfae wind stress, and τ = τ, τ ) is the bottom stress. b ( bx by () (3)

2 . Initial and Boundary Conditions We assume the initial onditions, ζ = u = v = 0 (4) and lateral boundary onditions, that the flow is zero normal to the solid boundary and along the open boundary Pb Po ζ = + fi H i os[ ωit + ( v + u) i gi ]. (5) ρg Here, Po and P b are the atmospheri pressures outside a storm and at the open boundary, ρ is the density of sea water, g is the gravitational aeleration, the ten tidal onstituents are taken as K 1, O 1, P 1, Q 1, M, S, N, S sa. The radian frequeny is ω i ; harmoni onstants; f is the nodal fator of eah onstituent; ( K, S a, of eah onstituent; i orretion angle. H i and v + u is the initial phase and i ) i g i are the amplitude and phase angle u is the nodal Surfae wind stress is generally assumed to take the form v v v τ s = ρ a C d w 10 w 10 (6) where ρ a is the air density; C d, is the surfae aerodynami drag oeffiient and w v 10 denotes wind veloity vetor at 10m referene height. A wave-age dependent C d formulation would assume the funtional dependene of the HEXOS relation of Smith et al. (199). However, in this study, we take a onventional approah, following Hsu (1986), and we assume that C takes the form d C d 0.4 = λ v (7) ln w10 where λ is an adjustable oeffiient depending on differing weather onditions (~1.1 for typhoons, ~1.0 for extra-tropial systems). When we don t onsider wave-urrent interations, bottom stress is assumed to be, v v v ng τ = ρ γ u u, γ = (8) b w z where z is the Chezy-Manning oeffiient and u v is the urrent veloity vetor. Aording to the wave-urrent interation model of Grant and Madsen (1979), in the form published by Signell et al (1990), a ollinear flow implies that the total bed shear stress is given by: τ = τ + τ (9) T w where τ is an instantaneous urrent shear stress and τ w is the maximum wave bed stress, as given by 1 τ w = ρf w U 0 (10) where U 0 is the maximum near-bed wave orbital veloity, and f w is the wave frition fator. The near-bed

3 wave orbital veloity is given by U 0 awσ = (11) sinh kh where a w is the wave amplitude, σ is wave frequeny, and k is the wave number determined from the linear dispersion relation = ( gk) tanh( kh) σ. (1) The wave frition fator f w an be readily omputed from the semi-empirial expression of Jonsson and Carlsen (1976), based upon laboratory observations, This implies, 4 1 f w + log 1 4 = log 10 f w A k b b (13) where k b = 30z0, and z 0 is the bottom roughness length and A b = / σ. If we assume that the urrent doesn t influene the wave field, the wave frition veloity is given by, U 0 1/ τ * U = w w. (14) ρ At initial time t = 0, an initial urrent frition fator f, exluding wind wave turbulene, is determined from f 0.4 = ln( 30zr / kb ) (15) with kb taken as the Nikuradse roughness kb = kb = 30z0, and zr was taken as 100 m above the bed. One f is determined, U * an be readily omputed from 1/ τ * U = (16) ρ B B where τ = F + G and F B and F G ( u v ) ( u v ) GB are given as 1 = f ρ u (17) 1 = f ρ v. (18) B + B + The ombined frition veloity ( U U ) 1/ * w * * w U * w for waves and urrents is given by U = +. (19)

4 The apparent bottom roughness kb felt by the urrent due to the presene of the waves is given by k b = k b β U * w Ab C1 (0) U w kb with C 1 = 4. 0, from Grant and Madsen (1979), and U* β = 1. (1) U * w The value for k b is then used at the next time step to determine f, and hene the bed stress due to the wave field and the time-evolving visosity field may be estimated, r 1 r r τ = ρ f u u (). Wave Model The third generation shallow water wave model YWE-WAM used in this study is based on the ation balane equation of WAMDI (1988). Soure funtions are taken diretly from the standard WAM model, exept for the dissipation due to depth-indued breaking, whih is expliitly parameterized. The basi equations are N t v v + [( C + u) N] + [ Cσ N] + ( θ N) σ θ g = S σ (3) v v F( σ, θ, x, t) N = N( σ, θ, x, t) = (4) σ C g 1 kd σ = (1 + ) sinh kd k v 1 σ d v u ; Cθ = ( + k ) (5) k d m m v σ d v v u Cσ = ( + u d) C g k (6) d t s. where F ( σ, θ ) is the spetral density, d, k v, u v are water depth, wave number vetor, veloity vetor, s is the spae oordinate in the propagation diretion, θ, the two-dimensional spae gradient is, and m is the spatial oordinate perpendiular to the s diretion,. The formulations for propagation speed C g, C σ and C θ give a representation of the effet of varying depth and urrents on wave propagation. Soure funtions are: S = Sin + Snl + S dis + Sbot + S dbs, inluding wind input, nonlinear interations, white-apping dissipation, bottom frition and depth-limited breaking dissipation. A detailed model desription is in Yin et al. (1996)..3 The Coupling Proedure During the simulation proess, omputed results of interest suh as signifiant wave heights, mean wave periods, diretional wave spetra, water surfae elevations, urrent veloities and bottom stresses, with and without the inlusion of oupled interations an be output by the wave and urrent models. Following Zhang

5 and Li (1996), and Mastenbroek et al. (1993), a oupling of models was implemented, following the proedure: a) The urrent model is run (15 minute time-steps) using the alulated surfae wind stress and bottom stress from the previous wave model yle time step to get elevations and urrent veloity. This gives newly omputed elevations and urrents, whih are passed to the wave model for the next time step omputations. b) The wave model is run (oarse-grid time-steps, 15 minutes: fine-grid, 5 minutes) using the omputed hange of water depth (mean water depth plus tide-surge elevation) and inhomogeneous and unsteady urrents from the two-dimensional urrent model to obtain related the wave parameters and wave spetrum. ) The bottom stress alulated using the wave spetrum and passed bak to the urrent model. d) The alulated bottom stress is alulated and input to the two-dimension urrent model, whih then exeutes the next time step, resulting in newly omputed elevations and urrents, to be input to the wave model to repeat the sequene of the omputations. 3. NUMERICAL STUDY OF THE HUANGHE DELTA COASTAL AREA Huanghe Delta oastal area is loated in the southwest region of the Bohai Sea. This area has great strategi importane beause it is the most important oil prodution area of the Shengli Oilfield. The fine-resolution grid area, as shown in Figure 1, is the Huanghe Delta oastal area, whih is the fous of this study. The purpose of this study is to onsider the impats on bottom stress and urrent veloities that are due to the wave-urrent interations. A seondary objetive is to offer a feasibility analysis of the potential for the adoption of a oupled wave-urrent model to make preditions of wave heights, urrents and sea level in this region of the Bohai Sea.???? Figure 1. The Bohai and Huanghe Delta oastal area, with buoy loation at A. The loation of additional output points B, C,.. and seleted model grid points are also indiated. 3.1 Case Desriptions Two storm ases ourred on -5 April 1998 and 1- April 1999, where measured wave and urrent as well as sea level data were olleted from the buoy site, N, E, shown in Figure 1. The wind fields were prepared by the Oean University of Qingdao. Nested oarse- and fine-resolution grids for the wave model are and, respetively. See the related paper by Yin et al. (00). We first onsider the wave and urrent data from the olleted the first storm, -5 April 1998, at position N, E indiated in Figure 1. Additional output points are also hosen, at different depths: B(.8m), C(5.5m), D(10m), E(15m), F(0m), G(5m), H(30m), and I(35m). This allows estimation of the effets of waves on the bottom stress and urrent veloity. Grids for both wave model and urrent model are, and time steps are 15 minutes. Comparison of results of oupled and unoupled simulations allows analysis of the effets of waves on the bottom stress and urrent veloity in the oastal area, in our ase speifially the Huanghe Delta oastal region. The following Figures -5 give results at point A for wave heights, surfae elevations, urrent veloities and diretions, as given from the numerial model.

6 Figure. Comparisons of simulated and measured wave heights for the 0UTC April UTC 5 April 1998 storm. Measured data, unoupled wave model ---, oupled wave-tide-surge model,. Figure3. Comparisons of simulated and measured sea level for the 0UTC April UTC 5 April 1998 storm. Measured data, unoupled tide-surge model ---, oupled wave-tide-surge model,. Figure 4. As in Figure, for the 00UTC 01 April UTC 0 April 1999 storm.

7 Figure5. As in Figure 3, for the 00UTC 01 April UTC 0 April 1999 storm. Figure 6. As in Figure 3, omparing simulated and measured urrent veloities for the 1998 storm. Figure 7. As in Figure 6 for urrent diretions for the 1998 storm. Units are degrees.

8 Figure 8. As in Figure 5, omparing simulated and measured urrent veloities for the 1999 storm. Figure 9. As in Figure 8 for urrent diretions for the 1999storm. It follows from Figures and 4 that wave height estimates from the oupled wave-urrent model are in (very slightly) better agreement with measured values, than those obtained from the unoupled wave model. From Figures 3, 5, 6-9, we see that estimated sea level and urrent veloity values from the oupled model are smaller than those obtained from the unoupled tide-surge model. Corresponding urrent diretions show little differene between the two simulations. These effets an be understood in terms of the wave-bottom effets. The impats of waves on the urrent bottom boundary layer are notable: bottom stress is signifiantly inreased due to wave-urrent interations. Therefore, a oupled wave-urrent model is an important onsideration beause it represents important physial fators and also beause it is feasibly pratial to implement for operational simulations. Thus it should be the basis for simulating the urrent veloity and sea level in the near shore region. Further results are presented in the Appendix, where in Tables 1-, we give model outputs for bottom stress and urrent veloity for the output points indiated in Figure 1. These values result from both oupled and unoupled model simulations, for the 1998 storm. This shows that wave-urrent interations in shallow water result in inreased bottom stress. Conomitantly, hanges in bottom stress affet the urrent veloity: inreased bottom stress give dereased urrent veloity values. Results from the 1999 storm, are similar. The extent that bottom stress and urrent veloity hanges varies depends on the intensity of the storm.

9 4. CONCLUDING REMARKS This paper presents a oastal high-resolution ( ) oupled wave-urrent interation numerial model with expliit onsideration of the wave-urrent interation bottom stress mehanism. We show that the bottom stress alulated by using a oupled wave-urrent model is inreased, as one would expet, ompared with bottom stress values omputed by an unoupled urrent model. Moreover, the urrent veloity field is also hanged. The extent of hanges to bottom stress and urrent veloities varies with the intensity of the storm. Therefore, it is neessary to develop a oupled wave-urrent model, taking aount of wave-urrent interations, partiularly for strong storms. From this study, it an be inferred that the adoption of a two-dimensional urrent model may aount for some of the wave effets, although it annot larify the hanges resulting in the vertial urrent profile. We have shown that the bottom stress effets are large and need to be inluded. For this reason, it is neessary to develop a three-dimensional urrent model oupled with a wave model. Thus, we an investigate the impats of wave-urrent interations on the near-bed in shallow seas suh as the Bohai Sea. ACNOWLEGEMENTS This paper was partly finished while the first author visited Bedford Institute of Oeanography. We gratefully aknowledge finanial supported from NSFC , Innovation Projet KZCX-0 from the Chinese Aademy of Sienes, as well as the USA Offie of Naval Researh funding to GoMOOS, The Gulf of Maine Oean Observing System. REFERENCES Davies. A. M., and Lawrene, J., 1994: Examining the influene of wind and wind wave turbulene on tidal urrents, using a three-dimensional hydrodynami model inluding wave-urrent interation. J. Phys. Oeanogr., 4, Christoffersen, J. B., and Jonsson, I. G., 1985: Bed frition and dissipation in a ombined urrent and wave motion. Oean Engineering, 1, Fang Guohong and Cao Deming, 1990: A -D oean irulation model in Bohai Sea. Oeanology and Limnology Sinia (in Chinese), 10(4), Grant, W. D., and Madsen, O.S., 1979: Combined theory wave and urrent interation with a rough bottom. J. Geophys. Res., 84, Hsu, S.A., 1986: A mehanism for the inrease of wind stress oeffiient with wind speed over water surfae: A parametri model. J. Phys., Oeanogr. 16, Jin Zhenghua, Wang Tao, and Yin Baoshu, 1998: The effets of ombined wave-tide-surge bottom stress. Oeanology and Limnology Sinia, 6, (in Chinese). Jonsson, J. G., and Carlsen, N. A., 1976: Experimental and theoretial investigation in an osillatory turbulent boundary layer. J. Hydrauli Res., 14, Mastenbroek, C., Burgers, G., Janssen, P.A.E.M., 1993: The dynamial oupling of a wave model and a storm surge model through the atmospheri boundary layer. J. Phys. Oeanogr., 3, Signell, R.P., et al., 1990: Effet of wave-urrent interation on wind-drived irulation in narrow, shallow embayments. J. Geophys. Res., 95, Smith, S. D., R. J. Anderson, W. A. Oost, C. Kraan, N. Maat, J. DeCosmo, K. B. Katsaros, K. L. Davidson, K. Bumke, L. Hasse, and H. M. Chadwik, Sea surfae wind stress and drag oeffiient: The HEXOS results, Boundary-Layer Meteor., 60, , 199. The WAMDI Group, 1988: The WAM model - A third generation oean wave predition model. J. Phys. Oeanogr., 18, Yin Baoshu, Wang Tao, El-Sabh M.I., 1996: A third generation shallow water wave numerial model YE_WAM. Chinese J. Oeanol. Limnol., 14(), Yin, Baoshu, Perrie, W., Hou Yijun, Lin Xiang, Cheng, Minghua, 00: The impat of radiation stress in a oupled wave-tide-surge model. Proeedings of This Workshop. Zhang, M.Y., and Li, Y.S., 1996: The synhronous oupling of a third-generation wave model and a twodimensional storm model. Oean Engineering 6,

10 APPENDIX TABLE 1. Bottom stress at seleted points in Figure 1. Coupled model simulations are represented by Y, unoupled simulations, by N. April, 1998, hour 0 B C D E F G H I N Y N Y N Y N Y N Y N Y N Y N Y April 3 hours 00, 04, 08, 1, 16, April 4 hours 00, 04, 08, 1, 16, April 5 hour TABLE. As in Table 1, for bottom urrent veloity in m/s, at seleted points in Figure 1. Coupled model simulations are represented by Y, unoupled simulations, by N. April, 1998, hour 0 B C D E F G H I N Y N Y N Y N Y N Y N Y N Y N Y April 3 hours 00, 04, 08, 1, 16, April 4 hours 00, 04, 08, 1, 16, April 5 hour