The effect of a seasonal stratification variation on the load exerted by internal solitary waves on a cylindrical pile

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1 Acta Oceanol. Sin., 4, Vol. 33, No. 7, P. 6 DOI:.7/s E-ail: hyxbe@63.net The effect of a seasonal stratification variation on the load exerted by internal solitary waves on a cylindrical pile CAI Shuqun *, XU Jiexin,, CHEN Zhiwu, XIE Jieshuo,, DENG Xiaodong,, LV Haibin State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Acadey of Sciences, Guangzhou 53, China University of Chinese Acadey of Sciences, Beijing 49, China Received Septeber 3; accepted 8 October 3 The Chinese Society of Oceanography and Springer-Verlag Berlin Heidelberg 4 Abstract On the basis of Morison s epirical forula and odal separation ethod in estiating the force and torque exerted by internal solitary waves (ISWs) on a cylindrical pile, it is found that the loads exerted by the ISWs change largely in different seasons at the sae site of the continental shelf in the South China Sea (SCS) even under the condition that the aplitudes of ISWs are the sae. Thus, the effect of a seasonal water stratification variation on the force and torque exerted by the ISWs is investigated, and a three-paraeter stratification odel is eployed. It is shown that the loads exerted by the ISWs depend largely on the water stratification. The stronger the water stratification, the larger the force and the torque; when the depth where the axiu therocline appears is deepened, the force decreases but the torque increases; when the width of the therocline is narrowed, the force increases but the torque decreases. The seasonal variation of the force and the torque exerted by the ISWs in four seasons in the SCS is thus explained. Key words: internal solitary waves, force, torque, water stratification, South China Sea Citation: Cai Shuqun, Xu Jiexin, Chen Zhiwu, Xie Jieshuo, Deng Xiaodong, Lv Haibin. 4. The effect of a seasonal stratification variation on the load exerted by internal solitary waves on a cylindrical pile. Acta Oceanologica Sinica, 33(7): 6, doi:.7/ s Introduction The internal solitary waves (ISWs) are very active in the South China Sea. Induced strong underwater currents during the propagation of these internal solitons ay cause severe threats to the ocean engineering structures, such as deep-sea oil drillings or pipelines (e.g., Ebbeseyer et al., 99). On the basis of the Korteweg-de Vries (KdV) odel, Cai et al. (3) and Cai et al, (6) introduced Morison et al. s (95) epirical forula and a odal separation ethod to estiate the forces and torques exerted by the ISWs on a cylindrical pile. Furtherore, the effect of shear flows on the coputed wave speed, wave for, environental paraeters of a nonlinear internal wave, and the corresponding force and torque on the cylindrical pile was also discussed (Cai et al., 8). Siilarly, Zha et al. () also proposed a ethod for estiating the load exerted by the ISWs on the cylindrical pile using nautical X-band radar observations and in-situ buoyancy frequency data. Using the generalized KdV odel, Si et al. () studied the shear forces and torques exerted by the ISWs on a cylindrical pile, and found that the shear force was the largest at the turning point of the horizontal velocity induced by the ISWs, while the torque was the largest at the pile foot. On the basis of a continuously stratified nonlinear odel, Xie et al. () siulated the generation of ISWs over a sill by tidal flows and calculated the force and the torque on the cylindrical pile exerted by the siulated ISW packet; they also obtained an epirical forula on the force with the wave aplitude. Recently, the action of ISWs on other arine structures was also investigated (Song et al., ; Guo et al., 3). Most of the above studies suggested that the force exerted by the ISWs increased with wave aplitudes. In fact, the ISW wave strength and aplitude are also related to the water stratification, e.g., in the northern SCS, the ISWs can occur in a whole year, but there exists a distinct seasonal variation, which is closely related to the seasonal variation of therocline (Zheng et al., 7). However, the effect of seasonal stratification change on the load exerted by the ISWs on the cylindrical pile has received no concern. It is also interesting to question the effect of stratification on the force and torque exerted by the ISWs on the cylindrical pile if the ISW aplitude were fixed. In this paper, based on the Korteweg-de Vries (KdV) odel, Morison et al. s epirical forula and odal separation ethod, we try to answer the above questions. Since it is shown that 94.75% of the force exerted by the ISWs on the cylindrical pile is contributed by the first ode wave (Cai et al., 6), to facilitate the question, in this study, we only pay attention to the effect of the water stratification on the force and torque exerted by the first ode ISWs. In this paper, data and coputational ethod are presented in Section. Results and discussion are described in Section 3, and conclusions are presented in Section 4. Foundation ite: The Strategic Priority Research Progra of the Chinese Acadey of Sciences under contract No. XDA; the National Basic Research Progra of China under contract Nos CB37 and 3CB956; the National Natural Science Foundation of China (NSFC) under contract No. 459; the Knowledge Innovation Progra of the Chinese Acadey of Sciences under contract No. SQ3; LTO Independent Research Progra under contract No. LTOZZ34. *Corresponding author, E-ail: caisq@scsio.ac.cn

2 CAI Shuqun et al. Acta Oceanol. Sin., 4, Vol. 33, No. 7, P. 6 Data and coputational ethod The cliatologic onthly ean teperature and salinity data at a station (.5 N, 6.5 E) near the Dongsha Islands in the northern SCS where the ISWs are active are retrieved fro the resolution data set by NODC ( gov/oc5/woa9/pubwoa9.htl), and the onthly buoyancy frequencies are coputed. Under the conditions of linear approxiation, no background current and Boussinesq approxiation, the vertical velocity W(z) satisfies the following boundary value proble (Gill, 98), d W W N z + ( ) =, () c W =, z =, () W =, z =, (3) g dρ where N( z) = is the buoyancy frequency, ρ is the ρ water density, g the gravity acceleration, and z the vertical coordinate; and H the depth of the ocean. Fro Eqs () (3), the linear first ode eigenvalue c (i.e., the ode- phase speed) and eigenfunction W can be found. By the Thopson-Haskell ethod, Eqs () (3) can be solved (Cai et al., 3). For the shallow-water ISWs, the aplitude η satisfies the KdV equation (Apel et al., 995), According to Cai et al. (3), under the conditions of Boussinesq approxiation and no background current, and supposing the diaeter of a pile, D, is less than the wave length, L, i.e., D/L.5, the force F exerted on the cylindrical pile by the first ode ISWs can be reduced to a drag coponent in ter of the hydrodynaic coefficient C D, while the torque M by the first ode ISWs can be calculated in ter of two hydrodynaic coefficients, C D and C M, i.e., H F ρ C DDu u () M = MI + MD πd u ρ z C M + ρ z C DDu u d, z 4 () where M I and M D are the inertial and drag coponents, respectively, and since, we have, dw u= ηv sech ϕ, () η V dw u = ϕ ϕ sech tanh, (3) 4 H dw dw F = ρcddηv sech ϕ d z, (4) η 3 + c η + αη η + β η =, 3 x x x (4) πd dw I Mρη ϕ ϕ M = C V sech tanh z d z, (5) where x is the horizontal coordinate; the nonlinear paraeter for the first ode wave is, 3 3 c (d W/ d z) d z = ; (d W / d z) α and the dispersion paraeter is c W = (d / d ) d β. W z z The steady solution for the first ode wave to Eq. (4) has the for, (5) (6) η = η W ( z)sech ϕ, (7) where, ϕ = ( x Vt )/, and the nonlinear ISW speed is and the characteristic half-width is V = c + αη / 3, (8) = β / αη. (9) 4 H dw dw MD = CDρη DV sech ϕ z d z. (6) Finally, by solving the following equations: F =, M =, (7) (8) the tie, t ax, when the force and the torque are at their axiu can be obtained. Substituting t ax into Eqs (4) (6), we can get the axiu force F ax and torque M ax. 3 Results and discussion The coputed onthly buoyancy frequency N based on the cliatologic teperature and salinity data at (.5 N, 6.5 E) is shown in Fig.. The depth at the site, H, is 3. It should be noted that the water stratification changes largely with seasons, e.g., in June, the axiu buoyancy frequency with a value of.3 3 Hz appears at a depth of 3, whilst in Deceber, the axiu buoyancy frequency with a value of only. 4 Hz appears at a depth of 5. Figure shows the distribution of the noralized aplitude W (z) versus the depth in March, June, Septeber and Deceber. The depths of the axiu aplitudes are also very different, e.g., in March, Septeber and Deceber, although all of the axiu aplitudes appear at a depth of 5, the aplitudes at other depths in these three onths are different, whilst in June, the axiu apli-

3 CAI Shuqun et al. Acta Oceanol. Sin., 4, Vol. 33, No. 7, P March June Septeber Deceber N/ 3 Hz 5 3 March June Septeber Deceber W Fig.. Monthly variation of the buoyancy frequency with depth at (.5 N, 6.5 E) in March, June, Septeber and Deceber. Fig.. Distribution of noralized aplitude W (z) versus depth at (.5 N, 6.5 E) in March, June, Septeber and Deceber. tude appears at a depth of. The phase speed, the nonlinear paraeter, the nonlinear phase speed and the characteristic half-width of the first odal ISWs are also calculated and shown in Table, respectively. E.g., the phase speed, the nonlinear paraeter, the nonlinear speed and the characteristic width in Septeber are.93 s,.88 s,.355 s and 36., respectively. Liu et al. (4) suggested that the aplitudes of the depression ISWs near the above site ranged fro 95 to 63 and less in the shallow water. For convenience, we set the ISW aplitude η = 6 in the following calculations. Meanwhile, we set ρ = 5 kg/ 3, and D=5.. There are any studies on the choice of the epirical coefficients C M and C D used in the Morison equation (e.g., Lighthill, 986). These coefficients depend on Reynolds nuber (Re=V D/v, and its value is around fro and in the present study, where V is the nonlinear speed in four onths, and ν is the kineatic viscosity), roughness ratio of the cylinder and Keulegan-Carpenter nuber ( Kc = πη / D in the present study). The highest Reynolds nuber suggests the value of C D is around fro.6 and.7, and CM is about.8. Thus we choose the epirical coefficients C M =.8 and C D =.6 in the following estiations, and calculate the forces and the torques. Details of resulted forces and torques on the pile by the first ode ISW in different seasons are also listed in Table. It is shown that, the value of F ax is around fro 3.89 to 38.8 kn, with a axiu in June, and M ax is around fro 8. and.5 N, with a axiu in Septeber. Note that the forces and the torques change largely, and the axia of the force and the torque do not appear in the sae season. Based on the Eqs () and (3), the horizontal velocity u and its acceleration u / t of the first ode ISW versus depth and tie could be coputed. Figure 3 shows the results at x= in June. It should be noted that, for the first ode ISW, the ISW induced axiu horizontal velocity appears at t= on the surface with a value larger than 3 c/s (Fig. 3a), then the horizontal velocity reduces with depth and tie, and it reduces to at a depth of about ; below this zero- horizontal velocity depth, the horizontal velocity changes to an opposite direction and increases with depth, and a negative axiu horizontal velocity appears at t= on the botto with a value of about 55 c/s. The first ode ISW induced horizontal velocity acceleration field is anti-syetric before and after the arriving tie of the ISW (Fig. 3b), i.e., before (after) t=, the horizontal velocity acceleration varies with depth in the sae variation pattern as that of the horizontal velocity. It is noted that the horizontal velocity in the upper layer is opposite to that in the lower layer, so does the load on the pile whose suberged depth h p is equal to the whole water depth H. Since the pile is soeties suspended in the water and anchored Table. Details of the characteristic paraeters and the resulted loads in four onths Month h / N /Hz c / s α / s V / s / F ax /kn M ax /MN March June Septeber Deceber

4 4 CAI Shuqun et al. Acta Oceanol. Sin., 4, Vol. 33, No. 7, P a 8 5 b u/c s 5. /c s u t/s t/s. Fig.3. Distribution of horizontal velocity u (a) and its acceleration u / t (b) of the first ode ISW versus depth and tie at x= in June. by an anchor line in the Spar platfor, we can expect that the load exerted by the ISWs ight reach its axiu when the pile suberged depth h p is just equal to the zero-horizontal velocity depth. Table shows the related estiated results in four onths. It deonstrates that the axiu force F ax in this case is uch larger than when the pile suberged depth h p is equal to the whole water depth H; however, it is the opposite case for the axiu torque M ax, e.g., in June, F ax is kn in Table, which is about.5 ties larger than that in Table, whilst M ax is only.7 MN in Table, which is only about /4 of that in Table. Since the force depends on the horizontal velocity, and now the horizontal velocity in the upper layer is in the sae direction and larger than that in the lower layer, it is apparent that the force reaches its axiu when the pile suberged depth h p is just equal to the zero-horizontal velocity depth. However, the torque depends ore on the product of horizontal velocity and pile suberged depth. Although the horizontal velocity in the upper layer is larger than that in the lower layer, the lower layer thickness is uch larger than the upper layer one, thus the associate torque only reaches its axiu when the pile suberged depth h p is equal to the whole water depth H. How does stratification affect the load exerted by the ISWs? To elucidate the relationship between the water stratification and the load, here a three-paraeter stratification odel as eployed by Pan et al. (7) is used for further analyses, i.e., N z N ( ) =, C( z/ H + C) + (9) where N is the axiu of N i, C = (H / δh), C = h / H, δh is the width of N at half N and is regarded as the pycnocline thickness; and h is the therocline depth. The three-paraeter buoyancy frequency odel can provide a realistic odel of the oceanic pycnocline in June at (.5 N, 6.5 E), and it is also shown in Fig. 4 by Curve S, which is very siilar to the ean cliatologic buoyancy frequency in June (denoted by Curve S ) and in which paraeters H, h, δ h, and N are taken as 3, 3,, and.3 3 Hz, respectively. Based on Curve S, three ore curves S (in which the axiu buoyancy frequency N is reduced), (in which the pycnocline depth h is deepened) and (in which the pycnocline thickness δ h is thinned) as shown in Table 3 and Fig. 4 are constructed, i.e., when copared with that in Curve S, in Curve S, only the paraeter N is changed into. 4 Hz; in Curve, only the paraeter h is changed into ; while in Curve, only the paraeter δ h is changed into. Figure 5 shows the distribution of the noralized aplitude W (z) versus depth in the four types of water stratifications. Since in Types S and S, the variation of pycnocline with depth is siilar except for the different axiu buoyancy frequency, the distribution of the noralized aplitude W (z) versus depth is the sae and the curves in two types of water stratifications S and S overlap. The phase speed, the nonlinear paraeter, the nonlinear phase speed and the characteristic half-width of the first odal ISWs in four types of the water stratifications are calculated and shown in Table 4, respectively. It is shown that the inertial torque coponent M is uch less than the drag torque I Table. Loads exerted by the ISWs when the pile suberged depth h p is just equal to the zero- horizontal velocity depth in four onths Month h p / F ax /kn M ax /MN March June Septeber Deceber Table 3. Values of the paraeters characterizing four types of water stratifications given by Eq. (9) Stratification type h/ N /Hz δh/ S S

5 CAI Shuqun et al. Acta Oceanol. Sin., 4, Vol. 33, No. 7, P S 5 S S 5 S, S N/ 3 Hz W Fig.4. Variation of the buoyancy frequency with depth at (.5 N, 6.5 E) in June (denoted by Curve S ) and in four constructed types of water stratifications (denoted by Curves S, S, and ). Fig.5. Distribution of noralized aplitude W (z) versus depth for the four types of water stratifications (note that the curves in Types S and S overlap). coponent M D, i.e., the ain contribution to torque M ax is M D. For Type S, the phase speed, the nonlinear paraeter, the nonlinear speed, the characteristic width, the force F ax and the torque M ax are.3 /s,.394 s,.4 /s, 5.3, 76. kn and.4 MN, respectively. For Type S, when the axiu buoyancy frequency N is reduced, the phase speed and the nonlinear speed also reduce to values of.8 4 and.999 /s, respectively, and as we can see fro Eqs () and (4) (6), the force and the torque increase with the nonlinear speed. Thus, in this case, they also reduce to values of 38. kn and 5.4MN, respectively. For Type, when the pycnocline depth h is deepened when copared with that in Type S, although the phase speed and the nonlinear speed also increase to values of and.744 /s, respectively, according to dw dw Eqs (4) (6), the force increases with d z, and dw dw dw the torque increases with z and d z z. In this case, when copared with that in Type S, according dw dw to the coputational results, since d z decreases largely, the force F ax decreases all the sae with a value of 8.88 kn; although M I also decreases to a value of.3 MN with an increasing characteristic width, it contributes less to dw dw the torque M ax than does M D, and since z increases largely, M D (.57 MN ) and the torque M ax (.8 MN ) increase all the sae. For Type, when the pycnocline thickness δ h is thinned when copared with that in type S, although the nonlinear speed reduces to a value of.5 /s, dw dw since d z increases largely, the force F ax increases all the sae with a value of 9.3 kn; although M I increases to a value of.6 MN with a decreasing characteristic width, it contributes less to the torque M ax than does M D, and dw dw since z decreases largely, M D (3.78 MN ) and the torque M ax (5.38 MN ) decrease all the sae. This deonstrates that, under the condition that the aplitudes of ISWs are the sae, when the water stratification is strengthened (as Type S versus Type S ), the force and the torque increase; when the depth where the axiu therocline appears is deepened (as Type S versus Type ), the force decreases but the torque increases; when the width of the therocline is narrowed (as Type S versus Type ), the force increases but the torque decreases. Table 4. Details of the characteristic paraeters and the resulted loads in four types of water stratifications Stratification type c / s α / s V / s / F ax /kn M I /MN M D /MN M ax /MN S S

6 6 CAI Shuqun et al. Acta Oceanol. Sin., 4, Vol. 33, No. 7, P. 6 Now we can explain the reason of the seasonal variation in the force and torque exerted by the ISWs in the SCS as shown in Table. In June, the water stratification is the strongest, thus the force exerted by the ISWs is the axiu aong the four seasons, in Septeber, although the water stratification is soewhat weaker than that in June, the therocline is deeper and the width of the therocline is widened than that in June, thus the torque exerted by the ISWs is the axiu aong the four seasons. The above conclusions would be of great use for the design of oil platfors, since the annual variation of the loads exerted by the ISWs on cylindrical piles in soe sea area could be estiated if the aplitudes of ISWs and the water stratifications are known. It should be noted that the above study is based on the KdV equation odel, thus the wave aplitude, the induced horizontal velocity and its acceleration, and even the load on the pile are very different fro those obtained fro the odified KdV equation odel (e.g., Grishaw et al., 7). 4 Conclusions In this paper, based on the Morison s epirical forula and the odal separation ethod in estiating the force and torque exerted by ISWs on a cylindrical pile (Cai et al., 3; Cai et al., 6), the effect of the seasonal stratification variation on the load on a cylindrical pile is studied, and soe interesting conclusions are drawn as follows. () It is found that at a sae site of the continental shelf in the SCS, even under the condition that the aplitudes of ISWs are the sae, the loads exerted by the ISWs change largely in different seasons. () The three-paraeter stratification odel is eployed for the study on the effect of the seasonal water stratification variation on the loads exerted by the ISWs. It is shown that under the condition that the aplitudes of ISWs are the sae, the loads exerted by the ISWs depend largely on the water stratification,, i.e., when the water stratification is strengthened, the force and torque increase; when the depth where the axiu therocline appears is deepened, the force decreases but the torque increases; when the width of the therocline is narrowed, the force increases but the torque decreases. Thus, the seasonal variation of the force and torque exerted by ISWs in four seasons in the SCS can be clearly explained. Acknowledgents The authors would like to thank two anonyous reviewers for giving any iportant coents. References Apel J R, Ostrovsky L A, Stepanyants Y A Internal solitons in the ocean. The Journal of the Acoustical Society of Aerica, 98(5): Cai Shuqun, Long Xiaoin, Gan Zijun. 3. A ethod to estiate the forces exerted by internal solitons on cylindrical piles. Ocean Eng, 3: Cai Shuqun, Long Xiaoin, Wang Shengan. 8. Forces and torques exerted by internal solitons in shear flows on cylindrical piles. Appl Ocean Res, 3: 7 77 Cai Shuqun, Wang Shengan, Long Xiaoin. 6. A siple estiation of the force exerted by internal solitons on cylindrical piles. Ocean Eng, 33: Ebbeseyer C, Cooes C A, Hailton R, et al. 99. New observations on internal waves (solitons) in the South China Sea using an acoustic Doppler current profiler. Marine Tech Soc, 9: Gill A E. 98. Atosphere-Ocean Dynaics. New York: Acadeic Press Guo Haiyan, Zhang Li, Li Xiaoin, et al. 3. Dynaic responses of top tensioned riser under cobined excitation of internal solitary wave, surface wave and vessel otion. Journal of Ocean university of China, : 6 Grishaw R, Pelinovsky E, Talipova T. 7. Modeling internal solitary waves in the coastal ocean. Surv Geophys, 8: Lighthill M J Fundaentals concerning wave loadings on offshore structures. J Fluid Mechanics, 73: Liu A K, Rap S R, Zhao Y, et al. 4. A case study of internal solitary wave propagation during ASIAEX. IEEE Journal of Oceanic Engineering, 9: Morison J R, Johnson J W, Schaaf S A, et al. 95. The forces exerted by surface waves on piles. Journal of Petroleu Technology, (5): Pan Jiayi, Jay D A, Orton P M. 7. Analyses of internal solitary waves generated at the Colubia River plue front using SAR iagery. J Geophys Res, : C74 Si Zongshang, Zhang Yuanling, Fan Zhisong.. A nuerical siulation of shear forces and torques exerted by largeaplitude internal solitary waves on a rigid pile in South China Sea. Appl Ocean Res, 37: 7 3 Song Zhijun, Teng Bin, Gou Ying, et al.. Coparisons of internal solitary wave and surface wave actions on arine structures and their responses. Appl Ocean Res, 33(): 9 Xie Jieshuo, Xu Jiexin, Cai Shuqun.. A nuerical study of the load on cylindrical piles exerted by internal solitary waves. J Fluids and Structures, 7: 5 6 Zha Guozhen, He Yijun, Yu Tan, et al.. The force exerted on a cylindrical pile by ocean internal waves derived fro nautical X- band radar observations and in-situ buoyancy frequency data. Ocean Eng, 4: 3 Zheng Quanan, Susanto R D, Ho C R, et al. 7. Statistical and dynaical analyses of generation echaniss of solitary internal waves in the northern South China Sea. J Geophys Res, : C3