- Email: [email protected]
A finite element simulation of tidal circulation in the Gulf of Kutch, India
Estuarine, Coastal and Shelf Science 56 (2003) 131–138
A finite element simulation of tidal circulation in the Gulf of Kutch, India A.S. Unnikrishnana,*, J.L. Luickb a National Institute of Oceanography, Dona Paula, Goa 403004, India National Tidal Facility Australia, Flinders University, GPO Box 2100, Adelaide, SA 5001, Australia
b
Received 13 July 2001; received in revised form 7 December 2001; accepted 7 December 2001
Abstract A finite element (fe) model using the software package ÔADCIRCÕ was developed to simulate the tides and currents in the Gulf of Kutch, located on the northwest coast of India. The surface elevations from the model were analysed for the amplitudes and phases of four major tidal constituents (M2, S2, K1 and O1) and the principal overtide (M4) and compared with observed values. Model currents were compared with observations at three locations along with the corresponding results from a finite difference (fd) model for the same region. The results were found to be consistent, with the fe model providing improved resolution in key areas. Residual circulation evaluated by forcing the fe model with M2 tide and winds (those typical of the southwest monsoon) show currents, which are directed upstream in the shallow region of the Gulf. A better simulation of lateral variations in currents in the fe model helps in identifying relative differences in tidal circulation and residual circulation between the deep and shallow regions of the gulf. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Finite element; Gulf of Kutch; Tidal circulation; Comparison with finite difference model; Residual currents
1. Introduction The Gulf of Kutch (Fig. 1), situated on the west coast of India, experiences very large tidal variations and associated strong tidal currents. The maximum tidal ranges vary from about 3 m near the mouth (Okha) to over 6 m at the upstream stations (Kandla and Navalakhi). Maximum surface currents are in the range of 0.75–1.25 m s1 near the mouth and 1.5–2.5 m s1 in the central region of the Gulf (Anonymous, 1985). One of the characteristic features of tidal propagation within the Gulf is the amplification of the semi-diurnal constituents relative to the diurnal constituents. During the last decade, industrial activities had been increasing in the region surrounding the Gulf and associated discharges into the Gulf can cause concerns for the environment. The Gulf has a fairly deep central region of about 30 m, reaching up to 40 m in the upper reaches. Away
* Corresponding author. E-mail address: [email protected] (A.S. Unnikrishnan).
from the deeper central portion, the depths are quite shallow on both sides. In particular, on the southern part of the lower Gulf, numerous mudflats are present. Upstream of Sikka, the lateral variations in depths are more regular. The Gulf is situated in an arid zone and experiences high rates of evaporation (Baumgartner & Reichel, 1975). There is very little runoff into the Gulf; whatever enters into the Gulf is from seasonal streams, peaking during the southwest monsoon (normally July). As a consequence, salinities are lower near the mouth than at the head of the Gulf, making it an inverse estuary. The available data on salinity show little variations in the vertical, implying that water column is vertically homogeneous within the Gulf. Two early studies (Anonymous, 1985; Shaligram, Ghosh, & Murty, 1988) of the tidal flow in the Gulf of Kutch (or Kachchh) were undertaken following a proposal for a tidal power station near Hansthal Point (Fig. 1). Shaligram et al. (1988) employed a numerical model with successively smaller regions and finer-scale (5, 2.5 and 1 km) grids. The 5 km model was apparently driven by observed sea levels at the open boundary over a 2-day period and the results used to drive the smaller-scale
0272-7714/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0272-7714(02)00135-X
132
A.S. Unnikrishnan, J.L. Luick / Estuarine, Coastal and Shelf Science 56 (2003) 131–138
Fig. 1. Gulf of Kutch (soundings in metres). Location of current meter observations C1 (Navinar Point), C2 (Salaya) and C3 (Sikka).
models. No comparisons with observed currents within the Gulf were reported. Unnikrishnan, Gouveia, and, Vethamony (1999) also used a 2D model to study the tidal propagation in the Gulf. For this study, observations from three different current meter moorings were available to verify the model. An alternate-direction implicit finite difference (fd) scheme with a grid resolution of about 1.5 km was used. By driving the model at the boundary over a range of pure frequencies, a natural period of oscillation close to 10 h was identified, thus explaining the observed amplification of semi-diurnal constituents. Shetye (1999), using analytical methods, further clarified the resonance phenomena, showing the importance of geometric and frictional effects. Finally, Sinha, Dube, Mitra, and Murty (2000) modelled the Gulf using an fd grid of mesh size about 1.7 km. As with the Shaligram et al. (1988) case, the focus was on the effect on circulation of closing off the sub-embayment where a tidal barrage would be located. Based on previous studies, a general understanding of the tidal circulation in the Gulf of Kutch has emerged. However, the circulation is not known at a resolution adequate for many purposes, for example to reliably predict sediment transport and deposition in the shallower areas. To study the currents at higher resolution at specific sites, it was decided to implement a finite element (fe)-based approach. Utilizing a completely different model to that of Unnikrishnan et al. (1999), but with identical forcing, also enables us to compare the results, to help identify and eliminate numerical problems. The flexibility of the fe model allows lower resolution in the centre, but much higher resolution towards the coast
than would model.
be
practicable
with
the
fd
2. The model A brief description of ADCIRC (Advanced Circulation Model) is given below; details may be found in Luettich, Westerink, and Scheffner (1992). The ADCIRC model has both 2D and 3D versions; however, both the versions are barotropic. Grenier, Luettich, and Westerink (1995) made a comparison between the performance of the 2D and 3D versions and concluded that as far as reproducing the major tidal constituents, the two gave identical results. The 3D version did produce better results for overtides, but for our purposes this was insufficient reason to justify the added complexity, although we were interested in identifying areas in which the overtide, M4, was generated. It is known from the constituents determined for the stations in the Gulf (International Hydrographic Review, Monaco; Admiralty Tide Tables, 1996) that M4 is by far the largest overtide in the Gulf. The model equations consist of the vertically averaged equations of momentum and the equation of continuity. Solving these equations on a fe grid is found to produce spurious solutions. This difficulty has been resolved by re-formulating them into a ÔGeneralized Wave-Continuity Equation (GWCE)Õ (Luettich et al., 1992). The GWCE and the two momentum equations are solved on the triangular fe grid by a Galerkin finite element method on linear triangles in space. The
A.S. Unnikrishnan, J.L. Luick / Estuarine Coastal and Shelf Science 56 (2003) 131–138
procedure, in detail, may be found in Luettich et al. (1992) and briefly in Grenier et al. (1995) and not repeated here. We chose the option of rectangular coordinate system. Though the effects of winds or salinity gradients are non-negligible in the gulf (which is primarily tidal), they are not included in the initial simulation. The solution requires a choice of a Ôweighting factorÕ called Ôs0Õ, on the GWCE; we used s0 ¼ 0:003 as recommended by the authors. The model was tested by using different values of the quadratic friction coefficient from Cf ¼ 0:001 to Cf ¼ 0:005 and a value of 0.003 was found to give a better match in the computed amplitudes and phases shown in Table 2. At Cf\0:002, the solution is found to have some oscillations and therefore, we used Cf ¼ 0:003 during the present simulations. The horizontal eddy viscosity was chosen to be 20 m s2. The fe mesh was constructed using the software package ÔTRIGRIDÕ (Henry & Walters, 1993). The program automatically generates a grid with a low resolution in the deeper central region, and with a high resolution near the coast. The entire Gulf is represented by 1091 points (Fig. 2). The node spacing varies from between about 0.5 and 5 km. An explicit scheme is used in time discretization with a time step of 45 s. The model was forced by prescribing tides along the open boundary. The open boundary conditions were prescribed in terms of variations of water surface elevations using a prediction, according to the Admiralty
133
method (Glenn, 1977). A prediction by this method involves a total of 25 tidal constituents. However, only the four major constituents (M2, S2, K1 and O1) need to be specified while the harmonic constants of the remaining 21 constituents are taken by inference. The tidal constants at Okha (Table 1) were used to construct a composite tide by this method. Predictions based on this simplified method were plotted over a 6-day period along with predictions based on a full set of constants from the Indian Tide Tables (Fig. 3). Agreement was felt to be adequate for the purposes of the model. Mud flats are represented by assigning low values of depths (3.5 m in the lower Gulf and 4.5 m in the upper part of the Gulf). As these values are larger than half the tidal range, the cells are never allowed to dry.
3. Results and discussion 3.1. Amplitudes and phases of four major constituents and the principal overtide The model was run for a period of about 1 month and the computed water surface elevations were analysed harmonically to separate M2, S2, K1, O1, and M4. A comparison between observed and modelled amplitudes and phases of these constituents is shown at selected six stations (Table 2). The mean deviation from the observed amplitudes of M2 and K1 are 5.0 and 2.5 cm, respectively,
Fig. 2. The finite element mesh used in the model.
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Table 1 Amplitudes and phases (G) at Okha (International Hydrographic Bureau, Monaco)
M2 S2 K1 O1
Amplitude (m)
Phase (deg)
1.11 0.34 0.43 0.2
211.6 235 342.3 346.3
while the computed phases have an average deviation of 4.00 and 3.00 from the observed for M2 and K1 respectively. This was considered quite satisfactory. However, a couple of mismatches were found in the comparison of phases of O1 at the upstream stations. Table 2 illustrates that the amplitudes of the semidiurnal constituents M2 and S2 increase nearly twice from the mouth to the head, whereas the increase in amplification of the diurnal constituents is only marginal. Prandle and Rahman (1980) examined tidal amplification in various estuaries and discussed the importance of variation of depth and breadth in determining the amplification. In our previous study (Unnikrishnan et al., 1999), numerical experiments showed that the natural period of the Gulf is close to 10 h, which explains the observed amplification of constituents having close to semi-diurnal periods. Shetye (1999) showed that in addition to quarter wave length resonance, geometric effects also contribute to the observed amplification of constituents, having close to semi-diurnal periods, in the Gulf of Kutch. The computed M4 amplitudes compare reasonably well with those observed, though they are underestimated by about 10% in the upstream stations Kandla and Navalakhi. One of the possible reasons is the non-inclusion of wetting/drying of mud flats in the model, which could affect the non-linear processes. The observed phase of M4 shows an abrupt change from
Table 2 A comparison of amplitudes and phases of major constituents between observations (International Hydrographic Bureau, Monaco) and those derived from model results Amplitude (m)
Phase (deg)
Observed
Model
Observed
Model
1.25 0.36 0.44 0.23 0.04
1.32 0.36 0.44 0.20 0.05
243 275 352 2 273
236 265 351 356 273
Navinar Point 1.84 M2 S2 0.58 K1 0.47 0.21 O1 M4 0.03
1.85 0.50 0.47 0.21 0.04
244 275 353 356 357
253 288 359 4 311
Tekra M2 S2 K1 O1 M4
2.30 0.67 0.56 0.23 0.17
2.30 0.63 0.50 0.22 0.12
262 294 4 0 50
263 303 5 13 27
Khori M2 S2 K1 O1 M4
2.29 0.63 0.58 0.23 0.15
2.35 0.65 0.50 0.22 0.14
261 296 6 352 34
264 305 6 14 31
Kandla M2 S2 K1 O1 M4
2.32 0.75 0.49 0.24 0.16
2.45 0.65 0.50 0.22 0.14
266 305 4 355 46
264 305 6 20 31
Navalakhi M2 S2 K1 O1 M4
2.48 0.68 0.51 0.24 0.22
2.46 0.68 0.51 0.22 0.19
265 304 15 7 65
266 308 8 17 32
Mandvi M2 S2 K1 O1 M4
Mandvi to Navinar Point, which seems to be inconsistent. Accordingly, the model phase lag of M4 upstream of Navinar Point shows a consistent lag with those observed. The fact that the principal overtide is reproduced fairly well is indicative of the fact that the reproduction of non-linear processes by the model is satisfactory. 3.2. Comparison with observed currents
Fig. 3. Boundary conditions defined at the open boundary along Okha. The continuous line indicates the composite tide constructed from the constituents at Okha used for driving the model and the circles indicate the predicted values during the corresponding period (Indian Tide Tables).
We chose three stations, namely, Navinar Point (C1, Fig. 1), Salaya (C2) and Sikka (C3), where moored current meter data were available. The model was run for the periods during which observations on currents were available. The results are shown in Fig. 4. Also included are the corresponding results of the fd model for the sake of comparison between models.
A.S. Unnikrishnan, J.L. Luick / Estuarine Coastal and Shelf Science 56 (2003) 131–138
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During the period of observations (about 1 month), it was found that the surface currents are affected by winds, while those at mid-depth and below are mostly tidal. The maximum flood currents at the surface, middepth and near bottom are 180, 130 and 115 cm s1 respectively, while the maximum currents during ebb are 68, 105 and 100 cm s1. This shows that southwest monsoon winds are favourable for intensifying the flood currents at the surface but oppose the ebb currents. Examination of available records on current measurements in the other nearby stations showed that during the northeast monsoon period (and at other times), the currents show little vertical variation and are primarily tidal. Though winds are not included in the present simulations (those of Figs. 4 and 5), they are included in the results that follow after. In this section, model currents were compared with observed currents at mid-depth. Out of the three locations, the fe solution is most accurate at Sikka (C3), both in amplitude and phase. At Navinar Point (C1), both the models accurately simulate the phase, while the fe model performs slightly better in simulating the amplitudes. At Salaya (C2), a drawback is found in the fe simulation: the computed v-velocities are shifted in phase with respect to the observations. The CPU time for the fe model was about four times longer than the fd model.
3.3. Tidal circulation in the Gulf The simulated tidal circulation is shown during a typical spring tide. Fig. 5a and b represents the circulation during 2 h before high tide at Okha, using the fe model and the fd model, respectively. During this time, flood currents have maximum intensities with magnitudes up to about 175 cm s1 in the central region of the lower Gulf. The fe simulations showed that there is considerable lateral variation in velocity, which were less conspicuous in our previous fd simulations. It could be argued that this is due to the high viscosity used in the fd model compared to the fe model. However, we ran the fd model with less than half of the viscosity that was earlier used and the results obtained (not shown) did not show a significant difference and do not seem to affect our point that lateral variations are more marked in the fe model. Moreover, near the coast, the circulation pattern in the fe simulation follows the coast closely and the currents follow the curved geometry near Navinar Point closely.
3.4. Residual circulation and role of winds As discussed previously, the circulation in the Gulf is dominated by tides and the effect of winds on circulation
Fig. 4. Comparison of computed and observed currents. (a), (b): off Navinar Point (C1) (22.735°N, 69.711°E). (a): u-velocity, (b): vvelocity. (c), (d): off Salaya (C2) (22.422°N, 69.675°E). (c): u-velocity, (d): v-velocity. (e), (f): off Sikka (C3) (22.561°N, 69.827°E). (e): uvelocity, (): v-velocity. (u ¼ +eastward, v ¼ +northward). circle;, Observed; – – –, fe model; ———, fd model.
is usually minimal. However, the current measurements during southwest monsoon (June–September) (Anonymous, 1996) showed that the surface currents at Sikka was about 30% stronger than those at mid-depth and
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A.S. Unnikrishnan, J.L. Luick / Estuarine, Coastal and Shelf Science 56 (2003) 131–138
Fig. 5. Circulation diagrams during 2 h before high tide at Okha using (a) fe model, (b) fd model.
below. The residual circulation in the Gulf was estimated by comparing two cases: first, with tide only, and second, with both tidal and wind forcing. In order to simplify interpretation, for these experiments the model was run with the M2 tide only. For the second experiment, a constant wind speed of 12 knots blowing from southwest throughout the domain was used. This value was obtained from the Indian daily weather report (IDWR) by averaging the winds measured at four stations in the Gulf region for the
month of August, 1996. The winds were mostly southwesterly and averaged about 10–12 knots. Examining the IDWR records of 1996 for the northeast monsoon period (November–February) and the transition months such as April–May and October, the average winds speeds are found to be less than 6 knots. Therefore, we chose the southwest monsoon winds for the simulation of depth-averaged residual circulation. Fig. 6a and b shows the depth-averaged Eulerian residual circulation for (a) tide only (b) tide and winds.
A.S. Unnikrishnan, J.L. Luick / Estuarine Coastal and Shelf Science 56 (2003) 131–138
137
Fig. 6. Depth-averaged Eulerian residual currents by forcing the model with (a) M2 tide only, (b) M2 tide and winds. The winds are southwesterly and have a uniform speed of 6.17 m s1.
In the lower Gulf, the residual currents are unaffected by the inclusion of wind. Midway up the Gulf, wind-driven residual currents appear along the northern and southern coasts (east of Navinar Point and Sikka respectively) (Fig. 6b). These currents are directed towards the head of the Gulf, in the direction of the wind, and they are only significant in the shallow areas near the coasts. The clockwise residual recirculation found between Navinar Point and Sikka is
more intense during the presence of southwesterly winds. The residual circulation has implications for the distribution of nutrients. The seasonally wind-driven flux into shallow regions circulates water over mudflats and through mangroves, where it is likely to become enriched with nutrients, sediments, and organic matter. This enriched water is carried out into deeper water (but still within the photic zone) where primary production
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A.S. Unnikrishnan, J.L. Luick / Estuarine, Coastal and Shelf Science 56 (2003) 131–138
will be enhanced. Near Sikka in the mid-gulf, the offshelf flow takes the form of a narrow jet, which carries the water from the stagnant coastal strip into the strong tidal currents of the centre channel. This jet is likely to be particularly enriched near the start of the monsoon. Similar phenomena have been described elsewhere, for example Hecate Strait, British Columbia (Visser, Bowman, & Crawford, 1990), and the Gulf of Carpentaria in northern Australia (Wolanski, 1993).
4. Conclusions The results of a 1-month model run were analysed to separate M2, S2, K1, O1 and M4, and compared with observations. In this regard, the fe model successfully improved upon earlier studies of the tides and tidal currents in the Gulf of Kutch by better resolving the shallower reaches and sub-embayments. The simulated currents from the fe and fd models were also compared with moored current meter observations at three locations. It is encouraging to see improvements in the accuracies achieved with the fe model in reproducing the observed currents. This occurred in spite of the fact that a relatively lower grid resolution was used to represent the central Gulf in the fe model. Though the performance of the two models is comparable in general, improvement in the circulation found near the coastline makes the fe results more useful in these regions. Flexibility of fe mesh allows better resolution where required, and enables the model circulation pattern to follow the coastline, for example the curvature in the geometry of the Gulf near Navinar Point. The simulated tidal circulation diagrams show marked lateral variations in currents, implying that the tidal propagation in the Gulf has two different regimes: (i) a deep central region, where the currents are strong; and (ii) shallow regions on either side of the central Gulf, with the weaker currents strongly influenced by friction.
Acknowledgments This work was initiated at the National Tidal Facility Australia during the first author’s visit, which was funded by the Department of Science & Technology, Government of India, and the Department of Industry,
Science, and Technology, Australia. The work was supported by grants from the Department of Ocean Development, Government of India (NIO Contribution No. 3712).
References Admiralty Tide Tables, Vol 2, (1996). Published by the Hydrographer of the Navy. UK. Anonymous (1985). Investigations and studies for tidal power development in the Gulf of Kutch. Interim Report. Central Electricity Authority, Ministry of Irrigation and Power, Govt. of India, New Delhi, 250. Anonymous (1996). Current and tide measurements for the design of pipeline from Jamnagar to Kandla route. Report submitted to Reliance Petroleum Ltd., Bombay, No. NIO/SP-25/96, 20. Baumgartner, A., & Reichel, E. (1974). The World Water Balance: Annual, Global, Continental and Maritime Precipitation, Evaporation and Runoff. 31 charts (pp. 179). New York: Elsevier Science. Glenn, R. N. (1977). The Admiralty method of tidal prediction, np 159. International Hydrographic Review, Monaco 52(2), 431–450. Grenier, R. R. Jr., Luettich, R. A. Jr., & Westerink, J. J. (1995). A comparison of nonlinear frictional characteristics of two-dimensional and three-dimensional models of a shallow embayment. Journal of Geophysical Research 100(C7), 13719–13735. Henry, R. F., & Walters, R. A. (1993). A geometrically-based, automatic generator for irregular networks. Communications in Numerical Methods in Engineering 9, 555–566. International Hydrographic Bureau (1930). Tides, Harmonic Constants. Special Publication no. 26, Monaco. Luettich, R. A., Jr., Westerink, J. J., & Scheffner, N. W. (1992). ADCIRC: An Advanced Three-dimensional Circulation Model for Shelves, Coasts and Estuaries. I. Theory and Methodology of ADCIRC2DDI and ADCIRC-3DL. DRP Technical Report 1. Prepared for U.S. Army Corps of Engineers, Washington, DC, 137. Prandle, D., & Rahman, M. (1980). Tidal response in estauries. Journal of Physical Oceanography 10, 1552–1573. Shaligram, V. M., Ghosh, L. K., & Murty, K. G. K. (1988). Predicting impact of tidal barrage by 2D model. International Symposium on Tidal Power, 16–18 Feb., New Delhi (pp. 131–136). Shetye, S. R. (1999). Tides in the Gulf of Kutch, India. Continental Shelf Research 19, 1771–1782. Sinha, P. C., Dube, S. K., Mitra, A. K., & Murty, T. S. (2000). A tidal flow model for the Gulf of Kachchh, India. Marine Geodesy 23(2), 117–132. Special Issue on Temporal Sea Level Variations. Unnikrishnan, A. S., Gouveia, A. D., & Vethamony, P. (1999). Tidal regime in the Gulf of Kutch, west coast of India by a twodimensional numerical model. Journal of Waterways, Port and Harbour Engineering, ASCE 125(6), 276–284. Visser, A. W., Bowman, M. J., & Crawford, W. R. (1990). Dynamics of tidally forced basin-wide coastal eddies. In R. T. Cheng (Ed.). Residual Currents and Long-term Transport (544 pp). New York: Springer. Wolanski, E. (1993). Water circulation in the Gulf of Carpenteria. Journal of Marine Systems 4, 401–420.
A finite element simulation of tidal circulation in the Gulf of Kutch, India A.S. Unnikrishnana,*, J.L. Luickb a National Institute of Oceanography, Dona Paula, Goa 403004, India National Tidal Facility Australia, Flinders University, GPO Box 2100, Adelaide, SA 5001, Australia
b
Received 13 July 2001; received in revised form 7 December 2001; accepted 7 December 2001
Abstract A finite element (fe) model using the software package ÔADCIRCÕ was developed to simulate the tides and currents in the Gulf of Kutch, located on the northwest coast of India. The surface elevations from the model were analysed for the amplitudes and phases of four major tidal constituents (M2, S2, K1 and O1) and the principal overtide (M4) and compared with observed values. Model currents were compared with observations at three locations along with the corresponding results from a finite difference (fd) model for the same region. The results were found to be consistent, with the fe model providing improved resolution in key areas. Residual circulation evaluated by forcing the fe model with M2 tide and winds (those typical of the southwest monsoon) show currents, which are directed upstream in the shallow region of the Gulf. A better simulation of lateral variations in currents in the fe model helps in identifying relative differences in tidal circulation and residual circulation between the deep and shallow regions of the gulf. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Finite element; Gulf of Kutch; Tidal circulation; Comparison with finite difference model; Residual currents
1. Introduction The Gulf of Kutch (Fig. 1), situated on the west coast of India, experiences very large tidal variations and associated strong tidal currents. The maximum tidal ranges vary from about 3 m near the mouth (Okha) to over 6 m at the upstream stations (Kandla and Navalakhi). Maximum surface currents are in the range of 0.75–1.25 m s1 near the mouth and 1.5–2.5 m s1 in the central region of the Gulf (Anonymous, 1985). One of the characteristic features of tidal propagation within the Gulf is the amplification of the semi-diurnal constituents relative to the diurnal constituents. During the last decade, industrial activities had been increasing in the region surrounding the Gulf and associated discharges into the Gulf can cause concerns for the environment. The Gulf has a fairly deep central region of about 30 m, reaching up to 40 m in the upper reaches. Away
* Corresponding author. E-mail address: [email protected] (A.S. Unnikrishnan).
from the deeper central portion, the depths are quite shallow on both sides. In particular, on the southern part of the lower Gulf, numerous mudflats are present. Upstream of Sikka, the lateral variations in depths are more regular. The Gulf is situated in an arid zone and experiences high rates of evaporation (Baumgartner & Reichel, 1975). There is very little runoff into the Gulf; whatever enters into the Gulf is from seasonal streams, peaking during the southwest monsoon (normally July). As a consequence, salinities are lower near the mouth than at the head of the Gulf, making it an inverse estuary. The available data on salinity show little variations in the vertical, implying that water column is vertically homogeneous within the Gulf. Two early studies (Anonymous, 1985; Shaligram, Ghosh, & Murty, 1988) of the tidal flow in the Gulf of Kutch (or Kachchh) were undertaken following a proposal for a tidal power station near Hansthal Point (Fig. 1). Shaligram et al. (1988) employed a numerical model with successively smaller regions and finer-scale (5, 2.5 and 1 km) grids. The 5 km model was apparently driven by observed sea levels at the open boundary over a 2-day period and the results used to drive the smaller-scale
0272-7714/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0272-7714(02)00135-X
132
A.S. Unnikrishnan, J.L. Luick / Estuarine, Coastal and Shelf Science 56 (2003) 131–138
Fig. 1. Gulf of Kutch (soundings in metres). Location of current meter observations C1 (Navinar Point), C2 (Salaya) and C3 (Sikka).
models. No comparisons with observed currents within the Gulf were reported. Unnikrishnan, Gouveia, and, Vethamony (1999) also used a 2D model to study the tidal propagation in the Gulf. For this study, observations from three different current meter moorings were available to verify the model. An alternate-direction implicit finite difference (fd) scheme with a grid resolution of about 1.5 km was used. By driving the model at the boundary over a range of pure frequencies, a natural period of oscillation close to 10 h was identified, thus explaining the observed amplification of semi-diurnal constituents. Shetye (1999), using analytical methods, further clarified the resonance phenomena, showing the importance of geometric and frictional effects. Finally, Sinha, Dube, Mitra, and Murty (2000) modelled the Gulf using an fd grid of mesh size about 1.7 km. As with the Shaligram et al. (1988) case, the focus was on the effect on circulation of closing off the sub-embayment where a tidal barrage would be located. Based on previous studies, a general understanding of the tidal circulation in the Gulf of Kutch has emerged. However, the circulation is not known at a resolution adequate for many purposes, for example to reliably predict sediment transport and deposition in the shallower areas. To study the currents at higher resolution at specific sites, it was decided to implement a finite element (fe)-based approach. Utilizing a completely different model to that of Unnikrishnan et al. (1999), but with identical forcing, also enables us to compare the results, to help identify and eliminate numerical problems. The flexibility of the fe model allows lower resolution in the centre, but much higher resolution towards the coast
than would model.
be
practicable
with
the
fd
2. The model A brief description of ADCIRC (Advanced Circulation Model) is given below; details may be found in Luettich, Westerink, and Scheffner (1992). The ADCIRC model has both 2D and 3D versions; however, both the versions are barotropic. Grenier, Luettich, and Westerink (1995) made a comparison between the performance of the 2D and 3D versions and concluded that as far as reproducing the major tidal constituents, the two gave identical results. The 3D version did produce better results for overtides, but for our purposes this was insufficient reason to justify the added complexity, although we were interested in identifying areas in which the overtide, M4, was generated. It is known from the constituents determined for the stations in the Gulf (International Hydrographic Review, Monaco; Admiralty Tide Tables, 1996) that M4 is by far the largest overtide in the Gulf. The model equations consist of the vertically averaged equations of momentum and the equation of continuity. Solving these equations on a fe grid is found to produce spurious solutions. This difficulty has been resolved by re-formulating them into a ÔGeneralized Wave-Continuity Equation (GWCE)Õ (Luettich et al., 1992). The GWCE and the two momentum equations are solved on the triangular fe grid by a Galerkin finite element method on linear triangles in space. The
A.S. Unnikrishnan, J.L. Luick / Estuarine Coastal and Shelf Science 56 (2003) 131–138
procedure, in detail, may be found in Luettich et al. (1992) and briefly in Grenier et al. (1995) and not repeated here. We chose the option of rectangular coordinate system. Though the effects of winds or salinity gradients are non-negligible in the gulf (which is primarily tidal), they are not included in the initial simulation. The solution requires a choice of a Ôweighting factorÕ called Ôs0Õ, on the GWCE; we used s0 ¼ 0:003 as recommended by the authors. The model was tested by using different values of the quadratic friction coefficient from Cf ¼ 0:001 to Cf ¼ 0:005 and a value of 0.003 was found to give a better match in the computed amplitudes and phases shown in Table 2. At Cf\0:002, the solution is found to have some oscillations and therefore, we used Cf ¼ 0:003 during the present simulations. The horizontal eddy viscosity was chosen to be 20 m s2. The fe mesh was constructed using the software package ÔTRIGRIDÕ (Henry & Walters, 1993). The program automatically generates a grid with a low resolution in the deeper central region, and with a high resolution near the coast. The entire Gulf is represented by 1091 points (Fig. 2). The node spacing varies from between about 0.5 and 5 km. An explicit scheme is used in time discretization with a time step of 45 s. The model was forced by prescribing tides along the open boundary. The open boundary conditions were prescribed in terms of variations of water surface elevations using a prediction, according to the Admiralty
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method (Glenn, 1977). A prediction by this method involves a total of 25 tidal constituents. However, only the four major constituents (M2, S2, K1 and O1) need to be specified while the harmonic constants of the remaining 21 constituents are taken by inference. The tidal constants at Okha (Table 1) were used to construct a composite tide by this method. Predictions based on this simplified method were plotted over a 6-day period along with predictions based on a full set of constants from the Indian Tide Tables (Fig. 3). Agreement was felt to be adequate for the purposes of the model. Mud flats are represented by assigning low values of depths (3.5 m in the lower Gulf and 4.5 m in the upper part of the Gulf). As these values are larger than half the tidal range, the cells are never allowed to dry.
3. Results and discussion 3.1. Amplitudes and phases of four major constituents and the principal overtide The model was run for a period of about 1 month and the computed water surface elevations were analysed harmonically to separate M2, S2, K1, O1, and M4. A comparison between observed and modelled amplitudes and phases of these constituents is shown at selected six stations (Table 2). The mean deviation from the observed amplitudes of M2 and K1 are 5.0 and 2.5 cm, respectively,
Fig. 2. The finite element mesh used in the model.
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Table 1 Amplitudes and phases (G) at Okha (International Hydrographic Bureau, Monaco)
M2 S2 K1 O1
Amplitude (m)
Phase (deg)
1.11 0.34 0.43 0.2
211.6 235 342.3 346.3
while the computed phases have an average deviation of 4.00 and 3.00 from the observed for M2 and K1 respectively. This was considered quite satisfactory. However, a couple of mismatches were found in the comparison of phases of O1 at the upstream stations. Table 2 illustrates that the amplitudes of the semidiurnal constituents M2 and S2 increase nearly twice from the mouth to the head, whereas the increase in amplification of the diurnal constituents is only marginal. Prandle and Rahman (1980) examined tidal amplification in various estuaries and discussed the importance of variation of depth and breadth in determining the amplification. In our previous study (Unnikrishnan et al., 1999), numerical experiments showed that the natural period of the Gulf is close to 10 h, which explains the observed amplification of constituents having close to semi-diurnal periods. Shetye (1999) showed that in addition to quarter wave length resonance, geometric effects also contribute to the observed amplification of constituents, having close to semi-diurnal periods, in the Gulf of Kutch. The computed M4 amplitudes compare reasonably well with those observed, though they are underestimated by about 10% in the upstream stations Kandla and Navalakhi. One of the possible reasons is the non-inclusion of wetting/drying of mud flats in the model, which could affect the non-linear processes. The observed phase of M4 shows an abrupt change from
Table 2 A comparison of amplitudes and phases of major constituents between observations (International Hydrographic Bureau, Monaco) and those derived from model results Amplitude (m)
Phase (deg)
Observed
Model
Observed
Model
1.25 0.36 0.44 0.23 0.04
1.32 0.36 0.44 0.20 0.05
243 275 352 2 273
236 265 351 356 273
Navinar Point 1.84 M2 S2 0.58 K1 0.47 0.21 O1 M4 0.03
1.85 0.50 0.47 0.21 0.04
244 275 353 356 357
253 288 359 4 311
Tekra M2 S2 K1 O1 M4
2.30 0.67 0.56 0.23 0.17
2.30 0.63 0.50 0.22 0.12
262 294 4 0 50
263 303 5 13 27
Khori M2 S2 K1 O1 M4
2.29 0.63 0.58 0.23 0.15
2.35 0.65 0.50 0.22 0.14
261 296 6 352 34
264 305 6 14 31
Kandla M2 S2 K1 O1 M4
2.32 0.75 0.49 0.24 0.16
2.45 0.65 0.50 0.22 0.14
266 305 4 355 46
264 305 6 20 31
Navalakhi M2 S2 K1 O1 M4
2.48 0.68 0.51 0.24 0.22
2.46 0.68 0.51 0.22 0.19
265 304 15 7 65
266 308 8 17 32
Mandvi M2 S2 K1 O1 M4
Mandvi to Navinar Point, which seems to be inconsistent. Accordingly, the model phase lag of M4 upstream of Navinar Point shows a consistent lag with those observed. The fact that the principal overtide is reproduced fairly well is indicative of the fact that the reproduction of non-linear processes by the model is satisfactory. 3.2. Comparison with observed currents
Fig. 3. Boundary conditions defined at the open boundary along Okha. The continuous line indicates the composite tide constructed from the constituents at Okha used for driving the model and the circles indicate the predicted values during the corresponding period (Indian Tide Tables).
We chose three stations, namely, Navinar Point (C1, Fig. 1), Salaya (C2) and Sikka (C3), where moored current meter data were available. The model was run for the periods during which observations on currents were available. The results are shown in Fig. 4. Also included are the corresponding results of the fd model for the sake of comparison between models.
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During the period of observations (about 1 month), it was found that the surface currents are affected by winds, while those at mid-depth and below are mostly tidal. The maximum flood currents at the surface, middepth and near bottom are 180, 130 and 115 cm s1 respectively, while the maximum currents during ebb are 68, 105 and 100 cm s1. This shows that southwest monsoon winds are favourable for intensifying the flood currents at the surface but oppose the ebb currents. Examination of available records on current measurements in the other nearby stations showed that during the northeast monsoon period (and at other times), the currents show little vertical variation and are primarily tidal. Though winds are not included in the present simulations (those of Figs. 4 and 5), they are included in the results that follow after. In this section, model currents were compared with observed currents at mid-depth. Out of the three locations, the fe solution is most accurate at Sikka (C3), both in amplitude and phase. At Navinar Point (C1), both the models accurately simulate the phase, while the fe model performs slightly better in simulating the amplitudes. At Salaya (C2), a drawback is found in the fe simulation: the computed v-velocities are shifted in phase with respect to the observations. The CPU time for the fe model was about four times longer than the fd model.
3.3. Tidal circulation in the Gulf The simulated tidal circulation is shown during a typical spring tide. Fig. 5a and b represents the circulation during 2 h before high tide at Okha, using the fe model and the fd model, respectively. During this time, flood currents have maximum intensities with magnitudes up to about 175 cm s1 in the central region of the lower Gulf. The fe simulations showed that there is considerable lateral variation in velocity, which were less conspicuous in our previous fd simulations. It could be argued that this is due to the high viscosity used in the fd model compared to the fe model. However, we ran the fd model with less than half of the viscosity that was earlier used and the results obtained (not shown) did not show a significant difference and do not seem to affect our point that lateral variations are more marked in the fe model. Moreover, near the coast, the circulation pattern in the fe simulation follows the coast closely and the currents follow the curved geometry near Navinar Point closely.
3.4. Residual circulation and role of winds As discussed previously, the circulation in the Gulf is dominated by tides and the effect of winds on circulation
Fig. 4. Comparison of computed and observed currents. (a), (b): off Navinar Point (C1) (22.735°N, 69.711°E). (a): u-velocity, (b): vvelocity. (c), (d): off Salaya (C2) (22.422°N, 69.675°E). (c): u-velocity, (d): v-velocity. (e), (f): off Sikka (C3) (22.561°N, 69.827°E). (e): uvelocity, (): v-velocity. (u ¼ +eastward, v ¼ +northward). circle;, Observed; – – –, fe model; ———, fd model.
is usually minimal. However, the current measurements during southwest monsoon (June–September) (Anonymous, 1996) showed that the surface currents at Sikka was about 30% stronger than those at mid-depth and
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Fig. 5. Circulation diagrams during 2 h before high tide at Okha using (a) fe model, (b) fd model.
below. The residual circulation in the Gulf was estimated by comparing two cases: first, with tide only, and second, with both tidal and wind forcing. In order to simplify interpretation, for these experiments the model was run with the M2 tide only. For the second experiment, a constant wind speed of 12 knots blowing from southwest throughout the domain was used. This value was obtained from the Indian daily weather report (IDWR) by averaging the winds measured at four stations in the Gulf region for the
month of August, 1996. The winds were mostly southwesterly and averaged about 10–12 knots. Examining the IDWR records of 1996 for the northeast monsoon period (November–February) and the transition months such as April–May and October, the average winds speeds are found to be less than 6 knots. Therefore, we chose the southwest monsoon winds for the simulation of depth-averaged residual circulation. Fig. 6a and b shows the depth-averaged Eulerian residual circulation for (a) tide only (b) tide and winds.
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Fig. 6. Depth-averaged Eulerian residual currents by forcing the model with (a) M2 tide only, (b) M2 tide and winds. The winds are southwesterly and have a uniform speed of 6.17 m s1.
In the lower Gulf, the residual currents are unaffected by the inclusion of wind. Midway up the Gulf, wind-driven residual currents appear along the northern and southern coasts (east of Navinar Point and Sikka respectively) (Fig. 6b). These currents are directed towards the head of the Gulf, in the direction of the wind, and they are only significant in the shallow areas near the coasts. The clockwise residual recirculation found between Navinar Point and Sikka is
more intense during the presence of southwesterly winds. The residual circulation has implications for the distribution of nutrients. The seasonally wind-driven flux into shallow regions circulates water over mudflats and through mangroves, where it is likely to become enriched with nutrients, sediments, and organic matter. This enriched water is carried out into deeper water (but still within the photic zone) where primary production
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will be enhanced. Near Sikka in the mid-gulf, the offshelf flow takes the form of a narrow jet, which carries the water from the stagnant coastal strip into the strong tidal currents of the centre channel. This jet is likely to be particularly enriched near the start of the monsoon. Similar phenomena have been described elsewhere, for example Hecate Strait, British Columbia (Visser, Bowman, & Crawford, 1990), and the Gulf of Carpentaria in northern Australia (Wolanski, 1993).
4. Conclusions The results of a 1-month model run were analysed to separate M2, S2, K1, O1 and M4, and compared with observations. In this regard, the fe model successfully improved upon earlier studies of the tides and tidal currents in the Gulf of Kutch by better resolving the shallower reaches and sub-embayments. The simulated currents from the fe and fd models were also compared with moored current meter observations at three locations. It is encouraging to see improvements in the accuracies achieved with the fe model in reproducing the observed currents. This occurred in spite of the fact that a relatively lower grid resolution was used to represent the central Gulf in the fe model. Though the performance of the two models is comparable in general, improvement in the circulation found near the coastline makes the fe results more useful in these regions. Flexibility of fe mesh allows better resolution where required, and enables the model circulation pattern to follow the coastline, for example the curvature in the geometry of the Gulf near Navinar Point. The simulated tidal circulation diagrams show marked lateral variations in currents, implying that the tidal propagation in the Gulf has two different regimes: (i) a deep central region, where the currents are strong; and (ii) shallow regions on either side of the central Gulf, with the weaker currents strongly influenced by friction.
Acknowledgments This work was initiated at the National Tidal Facility Australia during the first author’s visit, which was funded by the Department of Science & Technology, Government of India, and the Department of Industry,
Science, and Technology, Australia. The work was supported by grants from the Department of Ocean Development, Government of India (NIO Contribution No. 3712).
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