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Aquatic Procedia 4 ( 2015 ) 41 – 48

INTERNATIONAL CONFERENCE ON WATER RESOURCES, COASTAL AND ENGINEERING (ICWRCOE 2015) Tidal Hydrodynamics along Gulf of , West Coast of

Sathish Kumar Sa, Balaji Rb*

a Indian Institute of Technology Bombay, Powai, – 400 076, India b Indian Institute of Technology Bombay, Powai, Mumbai – 400 076, India

Abstract

In coastal hydrodynamics, tides play a major role and it is important to understand the complexity of their characteristics along the difficult coastal areas like creeks, estuaries, bays and gulfs for any kind of engineering activities like harbor, jetty, dams and protection measures. In this study, six tidal constituents (K1, M2, N2, O1, P1 and S2) variations along the , west coast of India were studied with the spatially varied friction coefficient. A two-dimensional finite element-based numerical model (Telemac-2D) was developed and zones of different friction coefficients were identified by comparing numerically estimated levels and currents with that of measured and reported by earlier studies. Based on the comparisons, the gulf was divided into five zones of different bottom friction values. The numerical model was used to predict the tidal constituent’s propagation along the gulf with the spatially varied bottom friction values and the amplitude and currents were analysed along the gulf. From the results, it was observed that the M2 (semi-diurnal) and K1 (diurnal) constituents influence the hydrodynamics of the gulf of Khmabhat significantly, compared to the other constituents. Further, a composite tide, inclusive of all six tidal constituents was applied to the offshore boundary of the numerical model and the tidal levels and currents were analysed along the gulf locations. It was observed that the combined effect of the constituents further increase the tidal amplitude up to 10m and the velocity up to 4ms-1. The validation of the numerical model showed good agreement.

© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © 2015 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of ICWRCOE 2015. Peer-review under responsibility of organizing committee of ICWRCOE 2015

Keywords: Tidal hydrodynamics; Tidal constituents; Telemac-2D; Gulf of Khambhat; Friction coefficient

* Corresponding author. Tel.: +0-22-2576-7321; fax: +0-22-2576-7302. E-mail address: [email protected]

2214-241X © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of ICWRCOE 2015 doi: 10.1016/j.aqpro.2015.02.007 42 S. Sathish Kumar and R. Balaji / Aquatic Procedia 4 ( 2015 ) 41 – 48

1. Introduction

It is essential, for engineers and scientists, to understand the tidal hydrodynamics of particular coastal features like estuaries, gulfs and lagoons for any kind of coastal developments (harbors, ports, protection measures). Tide is composed of longest ocean waves which usually rises and falls every half day (semi-diurnal) or a day (diurnal). The change of water level due to tidal currents occurs everywhere in the ocean. However, the change is the most significant in the coastal zone due to the shallower water depth in the coastal . To describe the strength of the tide, the difference in water elevation between high tide and low tide is called tidal range. Gulf of Khambhat is an inverted funnel shaped indentation on the west coast of India and spread over several hundred square kilometres, is one of the dynamic natural basins and having high tidal range (Siddiquie et al., 1981; Vora et al., 1980). The entire gulf is relatively shallow, compared to Arabian , with the maximum water depth of about 30m. Tidal phase speed, along-channel amplitude growth, and tidal harmonics in strongly convergent channels are all linked by morphodynamics of that particular basin (Friedrichs and Aubrey, 1994). Complex geography of the gulf amplifies the tidal range to about 10m and tidal currents are about 3ms-1 (Giardino et al., 2014; Kumar and Kumar, 2010; Kumar et al., 2006). In geography, bed friction plays a vital role in tidal amplification. Unnikrishan et al., (1999) suggested that use of appropriate bottom friction coefficient will give better prediction. In this present study, study area is divided into five zones with appropriate friction coefficients based on the sensitivity analysis. Various data, measured during different period of time, are collected and systematically used to calibrate the numerical model. The calibrated model is then used to map the contours of amplitudes of selected tidal constituents (K1, M2, N2, O1, P1 and S2).

2. Numerical Model

2.1. Model Description

A finite element based depth averaged numerical model has been developed to study the tidal hydrodynamics, using Telemac – 2D numerical scheme (Hervouet, 2007) which is capable of simulating free-surface flows in the two dimensions of horizontal space and solves the Saint-Venant equations. Basically, the Telemac-2D module solves the following averaged form of hydrodynamic continuous and momentum equations:

wwhhUhV() w ()  0 (1) wwtx wx

ww()hU ( hUU ) w ( hUV ) wwwww Z§· U §·U  gh¨¸ hveex ¨¸hv  hF (2) wwtx w y wwwww xxxyy©¹©¹

ww()hV ( hUV ) w ( hVV ) wwwww Z§· V §·V  gh¨¸ hveey ¨¸ hv  hF (3) wwtx w y wwwww yxxyy©¹©¹

2.2. Applied Forces in Momentum Equations

2.2.1. Bottom Friction Bottom friction, one of the applied forces, offers resistance to momentum of flow. In case of tidal propagation, the bottom friction in association with geometry and bathymetry, dictates the amplification of levels and magnitude of currents at any place of interest. The formula for friction force at the bottom, to be added to the momentum equation in the non-conservative form, takes as; S. Sathish Kumar and R. Balaji / Aquatic Procedia 4 ( 2015 ) 41 – 48 43

1 g 22 FUx  2 UV (4) CosD hC fr

1 g 22 FVy  2 UV (5) CosD hC fr

1/2 -1 The dimensionless friction parameter (Cfr) can be parameterised either in terms of Chezy (C, in m s ), Manning 1/3 -1 (m, in m s ), Strickler (S) or in Nikuradse (Ks,in mm) friction coefficients. Chezy’s friction coefficient is used for this study, which is applied by the following equation

2g C (6) fr C 2

2.2.2. Coriolis Force As the numerical model domain covers the entire gulf of large area, the effect of Coriolis force due to ’s rotation on the momentum equation is also considered in this study. The following equations, defining Coriolis force along x and y directions are considered as source terms in the momentum equation:

FVx 2sin()ZO (7)

FUy 2sin()ZO (8)

2.3. Model Setup

A relatively large model domain, covering entire Gulf of Khambhat and part of , is considered for the numerical model with a size of about 650x750kms (16°N 68°E to 22°N 74°E) as shown in Fig. 1(a). The entire area is discretized into triangular grids, with a largest size of 50km along the offshore boundary and smallest size of 2km near the coastline. Finite element mesh of the area of study along with the bathymetry is shown in Fig. 1(b). The seabed contours for the model domain are extracted from various admiralty charts and the offshore boundary is forced with tidal constituents, obtained from global tidal models (Haigh, et al 2011).

Fig. 1 (a) Study area and (b) bathymetry – Gulf of Khambhat 44 S. Sathish Kumar and R. Balaji / Aquatic Procedia 4 ( 2015 ) 41 – 48

2.4. Sensitivity Analysis

Sensitivity analysis was carried out (Fernandes et al., 2001; Liu et al., 2009) to test the sensitivity to variations in bottom friction parameters in the numerical model (Telemac – 2D). To investigate the model sensitivity to different laws of bottom friction and the different values of friction coefficient were carried out by applying Chezy’s, Manning’s and Strickler’s laws of bottom friction. The effect on tidal currents for various laws of bottom friction and different friction coefficient values at Bombay High location is illustrated in Fig. 2. It is clear that the magnitude of the tidal current is also controlled by magnitude of the bottom friction coefficients.

Fig. 2 Estimated tidal current at Bombay High (a) Chezy, (b) Manning, (c) Strickler laws of bottom friction and (A) typical scatter plot

3. Calibration and Validation

Initially available data from various resources (Diwan et al., 1991; Giardino et al., 2014; Naidu et al., 2013; Sinha et al., 2010; Swamy et al., 1982; Unnikrishnan, 2010; Unnikrishnan et al., 1999; Vora et al., 1980) are collected all along the gulf, and the locations are shown in Fig. 1(a). Part of the above mentioned data used for calibration, which covers the east and west coast of the gulf and rest of the collected data used for validation of the developed numerical model. Model domain is divided into five zones which are based on the bathymetry of the gulf numbered from offshore to nearshore. Based on the sensitivity analysis, Chezy’s bottom friction coefficient for each zone is calibrated using the available data. Fig. 3(a) shows the different zones and corresponding friction coefficient as a function of water depth. It is observed that the friction values are increasing with decrease in water depths from offshore region, except near head of gulf, where a relatively low friction coefficient yield a fair agreement between numerical and available data. The relationship between maximum water depth of each zone and friction coefficients appears to be following a trend, as shown in the Fig. 3(b).

Fig. 3 (a) Adopted friction coefficients as a function of water depth and (b) trend for the same S. Sathish Kumar and R. Balaji / Aquatic Procedia 4 ( 2015 ) 41 – 48 45

Comparisons of estimated tidal level and currents with available data, shown in Fig. 4(a), exhibit a good agreement with the available data. The numerical model results are also extracted at locations, where the data are available for validation. The comparison of results obtained from the present numerical model with that of validation data, shown in Fig. 4(b), clearly demonstrate the agreement. Typical scatter plot shown in Fig. 5, for tidal levels and currents confirms the good agreement. Root mean square errors (RMSE) for the tidal levels and currents are estimated to ensure the quality of the results.

Fig. 4 Typical comparison of tidal amplitude and currents with the available data- (a) calibration (b) validation

Fig. 5 Typical scatter plot comparison of tidal amplitude and current

4. Results and Discussion

Tidal ranges and currents of six major diurnal and semi-diurnal tidal constituents, K1, M2, N2, O1, P1 and S2, are simulated with heterogeneous friction coefficient, as discussed earlier. It is observed from the results that the maximum tidal amplitudes are observed in the region between and . Among the six selected major tidal constituents, M2 is observed to be predominantly influencing the tidal amplitudes along the gulf, followed by K1, S2, O1, N2 and P1. It is also clear from results that the contributions of semi-diurnal constituents, in tidal amplification, are more than that of diurnal constituents. A maximum of about 7.2m of amplitude and about 4ms-1 of velocity are observed for M2 constituent in the water depths ranging from 10 to 20m. A maximum of about 5m, 3.5m, 3m, 2.4m and 2m of tidal amplitudes are observed for K1, S2, O1, N2 and P1 constituents, respectively. A maximum of about 3.4ms-1, 2.6 ms-1, 2.4 ms-1, 1.8 ms-1 and 1.6 ms-1 of tidal velocities are observed for K1, S2, O1, N2 and P1 constituents, respectively. Maximum tidal amplitudes and velocities are observed around the Bhavangar location. Further, a composite tide, inclusive of all six major constituents is applied at offshore boundary of numerical model and tidal levels and currents are obtained at all nodes, as shown in Fig. 6. 46 S. Sathish Kumar and R. Balaji / Aquatic Procedia 4 ( 2015 ) 41 – 48

Fig. 6 Tidal constituents’ amplitude and current variations along the gulf

It is observed that the combined effect of all major constituents further increase the tidal ranges and a maximum of about 10m is observed at Bhavnagar, in a water depth of less than 20m. The currents are observed to reach a maximum of about 4.5ms-1, in similar water depth, as shown in Fig. 7. This behavior of tidal amplifications is attributed to resonance which is function of geometry and bottom friction (Nayak and Shetye, 2003; Shetye, 1999). Maximum tidal amplitude and current are observed in the nearshore region, detailed study has to be done along the shoreline, including all coastal features and ports, harbors, jetties etc.

S. Sathish Kumar and R. Balaji / Aquatic Procedia 4 ( 2015 ) 41 – 48 47

Fig. 7 Composite tidal amplitude and current variation along the gulf

5. Summary and conclusion

Gulf of Khambhat is a shallow and complex natural basin that experiences high tidal variations due to the combined effect of geometry and bottom friction. The present study attempts to establish the spatial variations of tidal amplitudes and currents based on the different bottom friction zones. A two-dimensional finite element based numerical model is developed and calibrated using the available data that covers the entire gulf. The whole domain is split into five different zones with different bottom friction coefficients, and the friction values are increasing with decreasing water depth, except at head of the gulf. The predicted tidal amplitudes and currents are validated with the remaining available data. The tidal amplification is observed along the gulf, in water depths ranging from 10 to 20m, where the seabed contours are relatively steeper. The propagation of major tidal constituents is analyzed to understand their influence on the tidal variations along the gulf. Nomenclature h water depth U, V flow velocities g acceleration due to gravity νt momentum diffusion coefficient Z free surface elevation t time Sh source or sink of fluid in dynamic equations Sx, Sy source or sink terms in dynamic equations Cfr dimensionless friction coefficient Ch Chezy’s friction factor ω angular velocity of the earth λ latitude

References

Diwan, S.G., Suryavanshi, A.K., Nayak, B.U., 1991. Wave measurement in severe ocean currents. J. Inst. Eng. 71, 148–152. Fernandes, E.H., Dyer, K.R., Niencheski, L.F.H., 2001. Calibration and Validation of the TELEMAC-2D Model to the Patos Lagoon (Brazil). J. Coast. Res. Spec. Issue 34. Int. Coast. Symp. (ICS 2000) CHALLENGES 21ST CENTURY Coast. Sci. Eng. Environ. (August 2001) 470– 788. Friedrichs, C.T., Aubrey, D.G., 1994. Tidal propagation in strongly convergent channels. J. Geophys. Res. 99, 3321. doi:10.1029/93JC03219 Giardino, A., Elias, E., Arunakumar, A., Karunakar, K., 2014. Tidal modelling in the Gulf of Khambhat based on a numerical and analytical approach, in: Proc. 5th Indian National Conference on Harbour and Ocean Engineering, 5-7 February 2014, CSIR-NIO , India. pp. 106– 111. 48 S. Sathish Kumar and R. Balaji / Aquatic Procedia 4 ( 2015 ) 41 – 48

Hervouet, J.-M., 2007. Hydrodynamics of Free Surface Flows. John Wiley & Sons, Ltd, Chichester, UK. doi:10.1002/9780470319628 Kumar, V.S., Kumar, K.A., 2010. Waves and Currents in Tide-Dominated Location off Dahej, Gulf of Khambhat, India. Mar. Geod. 33, 218–231. doi:10.1080/01490419.2010.492299 Kumar, V.S., Pathak, K.C., Pednekar, P., Raju, N.S.N., Gowthaman, R., 2006. Coastal processes along the Indian coastline. Cureent Sci. 91, 530– 536. Liu, Y., MacCready, P., Hickey, B.M., Dever, E.P., Kosro, P.M., Banas, N.S., 2009. Evaluation of a coastal ocean circulation model for the Columbia River plume in summer 2004. J. Geophys. Res. 114, C00B04. doi:10.1029/2008JC004929 Naidu, V.S., Sukumaran, S., Dubbewar, O., Reddy, G.S., 2013. Operational Forecast of Oil Spill Trajectory and Assessment of Impacts on Intertidal Macrobenthos in the Dahanu Region, West Coast of India. J. Coast. Res. 287, 398–409. doi:10.2112/JCOASTRES-D-12-00071.1 Nayak, R.K., Shetye, S.R., 2003. Tides in the Gulf of Khambhat, west coast of India. Estuar. Coast. Shelf Sci. 57, 249–254. doi:10.1016/S0272- 7714(02)00349-9 Shetye, S.R., 1999. Tides in the , India. Cont. Shelf Res. 19, 1771–1782. doi:10.1016/S0278-4343(99)00038-2 Siddiquie, H.N., Rao, D.G., Wagle, B.G., Vora, K.H., Gujar, A.R., Karisiddaiah, S.M., 1981. The continental shelf in the southern Gulf fo Khambhat - An evaluation of the sea bed for constructions, in: Proc. First Indian National Conference on Harbour and Ocean Engineering, Central Water and Power Research Station, , India. pp. 35–42. Sinha, P.C., Jena, G.K., Jain, I., Rao, A.D., Husain, M.L., 2010. Numerical Modelling of Tidal Circulation and Sediment Transport in the Gulf of Khambhat and Narmada Estuary , West Coast of India. Pertanika J. Sci. Technol. 18, 293–302. Swamy, G.N., Sarma, R.V., Suryanarayana, A., 1982. Physical characteristics of the coastal waters between Navapur and Umbharat, West Coast of India. Part I- Current pattern. Mahasagar-Bulletin Natl. Inst. Oceanogr. 15, 67–83. Unnikrishnan, A.S., 2010. Tidal propagation off the central west coast of India. Indian J. Geo-Marine Sci. 39, 485–488. Unnikrishnan, A.S., Shetye, S.R., Michael, G.S., 1999. Tidal propagation in the Gulf of Khambhat, Bombay high, and surrounding areas. Proc. Indian Acad. Sci. (Earth Planet. Sci.) 108, 155–177. doi:10.1007/BF02842329 Vora, K.H., Gujar, A.R., Karisiddaiah, S.M., 1980. Sandwaves of the Gulf of Khambhat. Indian J. Mar. Sci. 9, 90–93.