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Circulation in the Gulf of Khambhat - A Lagrangian Perspective

The circulation of the Gulf of Khambhat (GoK) is studied from a Lagrangian point of view using a 2D numerical model. The model-predicted tide elevation and current speed are in agreement with the observations. Seasonal variations of advection of particles are simulated by releasing 237 particles homogeneously distributed over the Gulf. After one month of simulation, no particles escaped from the GoK except a few from the southern GoK during southwest monsoon (June-September) and the advection of particles is maximum at the northern part. Residual eddies are present inside the GoK during northeast (October-January) and southwest monsoon seasons. Gulf circulation is studied with the combined forcing of tide and wind for different tidal conditions which had noticeable seasonal difference. The maximum simulated current speeds of 3.4 and 2.8 m/s are noticed during southwest monsoon near to Bhavnagar and Dahej respectively, where the tide elevations are maximum indicating that GoK is a tide-dominated system. A seasonal barrier could be found in the southern Gulf, which not only makes the Gulf circulation distinct from that of the Arabian Sea (AS) but also restricts water-mass exchange between the Gulf and AS during ebb condition. As the Gulf is a dump yard for anthropogenic wastes, the present study puts up an effort to determine the fate of the waste from a hydrodynamic point of view.


Introduction
The western continental shelf of India varied from south to north, and widens off Mumbai, leads into a strongly converging channel, the Gulf of Khambhat (GoK). The GoK, formerly known as the Gulf of Cambay, is a funnel-shaped indentation, situated between Saurashtra peninsula and the mainland of Gujarat (Fig. 1). The GoK has a width of 80 km at the mouth and funnels down to 25 km over the longitudinal reach of 140 km. The gulf is shallow with a maximum water depth of about 30 m and the northern part of the gulf is characterised by water depth less than 10 m and large tidal flats [1].
The tides in GoK are of semi-diurnal type with a large diurnal inequality and varying amplitudes, which increase from the south to north along the GoK. The amplification of tide inside the GoK is due to its shape and varying bottom friction coefficients [2] and the large width of the continental shelf off the central west coast of India [3]. In the north Indian Ocean, the maximum tidal range is found in GoK with an average tidal range of 10 m near to Bhavnagar [4]. The current measurements in GoK indicate a maximum current speed of 3.3 m/s and flows north-northwest during flood tide and south-southeast during ebb tide [5]. The main rivers that drain into the GoK are Narmada, Tapi, Mahi and Sabarmati and these rivers discharge a large amount of suspended sediment (more than 1000 mg/l/d) [6] which makes the Gulf more turbid as this oscillates within this region until a water flushes it towards the open sea [7]. These rivers discharge a substantial freshwater run-off into the gulf, especially during the monsoon seasons. The discharge of the Narmada River, which is the major river, ranges between 10000 and 60000 m 3 /s during monsoonal floods.
The climate of the study region ranges from semi-arid through tropical steppe in the north to tropical rainy in the south with average rainfall exceeding 100 cm per year in the coastal area [8]. The Gulf is exposed to seasonally reversing wind with winds veering from the southwest (SW) during the summer monsoon (June-September) and from the northeast (NE) during the winter monsoon period (October-January). The period between the winter and summer monsoon is the pre-monsoon period (February-May). Higher wind speeds up to 21 m/s are likely to occur during SW monsoon [9]. The Gulf is expected to be influenced by large-scale wind-driven circulation [10]. The jet currents on the proximity of 60 m depth contour are considered as the coastal ocean boundary, which do not allow the gulf-water adverted/flushed out into the open sea directly [11]. The seasonal variability of GoK is due to the seasonality of the main forcing agents -the Arabian Sea at the Gulf's entrance and the monsoon wind regime. The Gulf is homogeneous and displays a one-layer structure and currents at deeper depths inside the Gulf follow the same velocity pattern of the surface due to the shallowness of the depths and high tidal amplitudes [11].
Study of the tidal hydrodynamics of a coastal area i.e. gulfs, estuaries, creeks, bays, lagoons etc. are of paramount importance as these data would be the baseline of any coastal development of that particular area. Tidal hydrodynamics of the Gulf of California was studied by Marinone [12] who inferred that the tide of that area is in oscillation with the Pacific Ocean and the semi-diurnal constituents are near resonance, with head amplitudes four times larger than those at the mouth.
Further study concluded that a depth-averaged circulation could often be used to have an idea about the advection of the passive pollutants [12]. Sheng and Wang [13] used a single-domain coastal circulation model to study the non-linear dynamics of the barotropic tidal circulation in Lunenberg Bay. A study by Allahdadi et al. [14] surmised that currents in the Atchafalaya shelf follow the general circulation pattern of Louisiana coast and is influenced by seasonal wind, tides, river discharge and outer-shelf variations induced by the loop current eddies. A coupled hydrodynamic/ suspended sediment transport model was developed and successfully applied to Cleveland Bay, Australia [15]. Tattersall et al. [16] applied 2D depth-averaged models to simulate the tidal currents and suspended sediment concentrations in the Tamar estuary, England.
Understanding the circulation characteristics of the GoK is extremely important as hydrodynamics have a direct impact on the design of any engineering infrastructure, built in this area.
The tidal strength is an important agent in driving and modulating net circulation in a semi-enclosed water body. Several economic activities (e.g. shipping, oil and gas extraction), as well as flora and fauna, can be directly or indirectly influenced by the tidal variations and associated circulation and the possible changes to those processes. Anthropogenic activities or long-term natural processes can affect those processes with often drastic and unexpected consequences. Studies on hydrodynamics are also essential to understand the interaction of the coastal and inland water in terms of the physical processes as well as the exchange of nutrients or pollutants between the two water bodies. Longuet-Higgins [17] showed the limitations of an Eulerian points of view to explain the long-term transport of water particles: he defined the mass transport velocity as the sum of the Eulerian residual current and the Stokes drift by carrying out a first-order approximation of the Lagrangian residual velocity. Numerical methods based on particle tracking enable calculation of the Lagrangian residual circulation for strongly non-linear system [18]. The ocean exhibits a huge range of dynamical motions, extending scales from millimeters to thousands of kilometers. The Lagrangian analysis is a powerful way to analyse the output of ocean circulation models and other ocean velocity data. This method employs an ensemble of passive Lagrangian particles of zero spatial extent whose trajectories are determined by the velocity field [19]. A large number of industries are present on either bank of the Gulf which discharge an enormous amount of treated effluent into the sea. At present, the fate of the particles discharged in the Gulf is not known. The Lagrangian analysis along with the hydrodynamics could provide an insight to the fate of the sewage/industrial effluent released in the Gulf.
Thus, the main objective of this work is to study the seasonal variation of depth-averaged Lagrangian circulation of the GoK using a calibrated 2D numerical model. Section 2 presents the materials and methods.

Hydrodynamic Model
The currents measured off Dahej at 12 m water depth shows that the currents are similar near the surface and the bottom (Fig. 2) indicating that the water is vertically well mixed. An unsteady and two-dimensional vertically integrated shallow-water equation for continuity and momentum balance given by Stoker [20] has been used to obtain the tides and tide-induced circulation in the GoK. In this study, the 2D model developed for simulating Thane creek tidal circulation [21] and to determine the flushing characteristics of Amba estuary [22] has been applied to the GoK. The model is barotropic, based on shallow water equations and neglects horizontal diffusion terms in the momentum equations.
Due to the seasonal wind reversal in the study region, wind stress terms are introduced in the model.
Wetting and drying in shallow areas were modelled.
where, =H -h, t=time coordinate, x, y = set of horizontal, mutually orthogonal Cartesian coordinates, H=total instantaneous water depth, U, V=horizontal velocity components in x and y direction respectively, mv=vertically integrated average rate of mass infection or withdrawal of fluid per unit volume divided by the fluid density, h=the depth of water measured positively downwards from the reference datum plane.

(b) Momentum balance equations
The vertically integrated momentum balance equations for a shallow water body are given by; Where, g=acceleration due to gravity, η=tide elevation, =fluid mass density, P = fluid pressure ( = ) ,  =Coriolis parameter, Cw=wind stress coefficient, W=wind speed, =wind angle measured from the positive direction of x towards the positive direction of y-axis, Cf=bottom friction coefficient, Su and Sv= average rate of momentum generated or dissipated per unit volume derived by the mass density of water in x and y directions respectively.
The equations 1-3 are solved in Alternate Direction Implicit (ADI) scheme which is a finite difference method for solving parabolic, hyperbolic and elliptic partial differential equations with finite difference grid system. Horizontal diffusion terms in the momentum balance have been neglected since contributions from them are small compared with the other terms. The contour -4 m in the bathymetry describes the inter-tidal region which behaves as part of the landmass during high tide and as water-mass during low tide. Thus, the model considers the wetting condition of it during high water and drying during low water. The input data was smoothed using the spline function to obtain the tide at closer intervals. The governing equations were solved with the initial and boundary conditions on a staggered grid system as described by Leendertse [23].  [25]. Stability of the numerical scheme is governed by t <x/√2 , where x is grid size in meters and t is the time step in seconds and g is the acceleration due to gravity and Hmax is the maximum depth in the model domain. When Hmax is 100 m and x is 500 m, the t is 12 s. It was assumed that initially the sea is at rest, i.e. at t=0, u=v=0. Since the model was initially at rest, it took more than 15 hours for the model to get stabilized. Thus, first one day data from the each model Reanalysis data of zonal and meridional components of wind speed at 10 m height at one hour interval with a resolution of 0.25° x 0.25° for the entire study area were obtained from NCEP / NCAR [26] to use them as the model input for determining the influence of wind on sea level and currents. These data are provided by the NOAA-CIRES Climate Diagnostics Center, Boulder, Colarado at http://www.cdc.noaa.gov/. The river discharge data of four major rivers (Narmada, Tapi, Mahi and Sabarmati) used in this study has been provided by the Central Water Commission (CWC), New Delhi. Due to the high dynamicity of the area, GoK was assumed to be a one layer system, and hence standard seawater density have been used for model simulation in the entire model domain.

Lagrangian circulation
Lagrangian circulation is analysed using a particle tracking method by releasing particles at a fixed time, tracking the path of these particles and the time when they reach the boundary and was performed in a 2-D mode, using a 4th order Runge-Kutta (RK) scheme for particle advection. The trajectories of the particles were obtained by time-integrating the horizontal velocity field.
Lagrangian circulation was calculated using the equation [27] which has previously been used by Naidu et al. [22]. 4-point linear interpolation is adopted to calculate the velocity at an arbitrary point.
where, ̅ is the particle's position, ∆ = time step. In this work, diffusion term is neglected.

Residual circulation
The differences between the simulated currents and that derived from tidal harmonics were averaged over one lunar cycle to compute the steady residual circulation or mean currents. The model was run for one lunar tidal cycle (29.5 days) by storing U and V components of velocity at each grid at 20 min interval for three seasons. Using the following equation the residual velocity Rv is estimated.
The net velocity of u and v components were calculated by integrating the velocity components [22].

Model calibration for Hydrodynamics
Tide and current data from the model were compared with measured data for locations at Surat   (Fig. 2) and hence, a 2D numerical model was applied in this work instead of a 3D model.

General circulation
Model was run for different scenarios, viz., the tide-driven, wind-driven and the combined tide Areas like GoK, where current speed is high are potential sites for extraction of current energy.
Power can be generated commercially if the mean peak current exceeds 2 m/s [28]. The current speed exceeds 1 m/s in the western GoK (off Bhavnagar) for 40.5% during southwest monsoon and around 20% during remaining period. Whereas in the northern GoK (mouth of Mahi estuary), currents are low and exceeds 1 m/s only for 0.4% of the time in a year. In the eastern GoK (mouth of Narmada estuary), during 10% of the time in a year current speed is more than 1 m/s ( Table 2).
The influence of Narmada river discharge was found to be noticeable from the magnitude of u-component of current speed which exceeded the magnitude of v-component near to the Narmada mouth. To understand the horizontal extent of the river discharge, three locations were selected from central GoK, Narmada mouth and southern GoK to obtain the u-v component of current speed at these locations during SW-monsoon. In the central and southern GoK (Figs. 11a, 11c), dominance of v-component was observed but near to the Narmada mouth, it was the opposite (Fig. 11b). Thus it could be inferred that the influence of Narmada discharge on the circulation of GoK is quite a localized phenomena and it does not impact the overall circulation of GoK. The area of extent of the river discharge was determined by obtaining the u-v components at different locations starting from Narmada mouth towards offshore (not shown in the figure). U-component was dominating at the Narmada mouth and it was found that at a distance of 13 km from the river mouth, the v-component came into action and from that point onwards the dominance of v-component was observed. This result also manifests the limiting extent of the Narmada discharge into the overall circulation of GoK. Thus, horizontal stratification was not taken into consideration in the present study.

Lagrangian circulation
Lagrangian particle tracks were simulated by model forced by the combined action of tide and wind. Track results were obtained by the simulation of one month by releasing 237 particles in the model domain among which 27 particle trails are presented in the figures. The track results revealed that after one month of simulation, none of the particles came out of GoK during SW monsoon except for a few from the southernmost portion of the Gulf (Fig. 12). It can be noticed from the particle trajectories that maximum distances are travelled by those which are released near to Dahej at the Narmada mouth due to the high current in the area. Particles those are released in the mouths of Sabarmati and Mahi River, have a tendency to circulate at those particular locations. Particles released in the western coast of GoK have alongshore movement and those from mid GoK have circulatory movements. The particles which are released near to Mumbai coast have alongshore movement as well.
The particle trajectories extracted for the NE monsoon also have similar tendencies, but the distances covered in this case are quite smaller. Particles released in the northern GoK especially near to the mouths of Sabarmati and Mahi, could not come out of the northern part of the Gulf. Particles from mid GoK, traversed quite more due to the high current speed. They mostly had circulatory motions but particles released near to the coast have alongshore movement even in the eastern GoK, which is quite different from the SW monsoon. Particle trajectories of the southern GoK were found to be small and none of them came out of the Gulf after one month of simulation (Fig. 12). Near to Mumbai, particle trails were parallel to the coast. The trajectories are not very different in case of pre-monsoon. But during pre-monsoon, the particles from the southern GoK have tendencies to move further downward but none of the particles came out of it after one month of simulation. Particles released in either side of GoK near to the coastline revealed circulatory movement. Particles released inside the Narmada estuary had longshore movement and travelled to the northern GoK according to the tidal condition. A particle released near to the Mahi estuary travelled across the Gulf and moved further southward, but could not had an escape. Particles released off Mumbai have shown alongshore movement even in this case and those released inside the gulf travelled larger distances than those released outside in each of the three seasons.

Residual circulation
The residual circulation during SW monsoon revealed two residual eddies inside the Gulf, in the western GoK and mid GoK (Fig. 13a). Similar features are absent for other two seasons, but residual eddy was identified in the mid GoK during NE monsoon (Fig. 13b). No residual eddy is noticed during the pre-monsoon season (Fig. 13c). Maximum residual current speed is noticed inside the Gulf, during SW monsoon.

Discussion
The GoK is basically a not-so-simple system with extreme tidal conditions, river discharges and wind impacts. The tidal height is maximum near to Bhavnagar which is associated with the maximum current speed of more than 3 m/s. This behavior of tidal amplification is attributed to resonance which is the function of geometry and bottom friction [2,29]. Thus it could be inferred that the current speed in this area is modulated by the tidal amplitude as well as the local geometry effects. The very simplistic depth-averaged numerical model predicted the tidal amplitude as well as amplitude of the current speed quite accurately although minute variation in the predicted and observed phase was itself. Although the barrier formation is a seasonal phenomenon, it could be associated with the mixing characteristics of GoK. It was hypothesised that this type of barrier formation is mostly associated with the bathymetry of the region which is purely a regional phenomenon [30][31]. It should also be noted that the minimal exchange of water-mass between the Arabian Sea and GoK could also be a result of this kind of formation [32][33] which would be the future aspect of the study. The barrier formation is associated with the topography of the Gulf as well as the tidal currents and a schematic diagram well-reflects this phenomena (Fig. 14) where it has been shown that due to high flood current, water could overcome the barrier and goes inside GoK but during ebb period the water exchange between GoK and AS is hampered. Mostly during the ebb condition, the barrier was noticed while during flood either it was vague or absent. During pre-monsoon season, the barrier was noticed for both the tidal condition. Thus it could be surmised that due to the high flood current speed, the whole water parcel surrounding GoK goes inside the Gulf but due to the comparatively smaller ebb current speed, the Gulf restricts the water exchange with the northeastern Arabian Sea during ebb; hence forms a barrier. This not only would restrict the water-mass to come out but also the suspended sediment as well as anthropogenic wastes, which could have direct/indirect impacts on the biological community of the Gulf. The treated effluent discharged by the industries present on either bank of the Gulf will affect the Gulf-ecology since the Gulf has a tendency to accumulate particles which cloud include suspended sediment, pollutants, industrial or domestic wastes, spill of oil etc. Alongshore component of current dominated in the northern and western Gulf but the dominance of the cross-shore component of current was identified in the eastern GoK at the Narmada mouth which is due to the influence of Narmada river discharge into the sea. The influence of the river could be seen up to a distance of 13 km from the river mouth on an average and its impact was found to be purely localised. along the west coast of India is located on the shelf break which could exert some influence at the circulation of GoK [38]. Thus, the wind forcing of this region was taken into consideration. The time series data at the three locations of the Gulf suggested a considerable riverine influence on eastern GoK and as a result the u-component of current speed dominated in that region in each of the three seasons, which was found in a limited area of extent and does not influence the Gulf circulation as a whole. During SW-monsoon a considerable difference between the tidal current and combined tide and wind-driven current was noticed which is due to the presence of monsoonal winds. Due to weak wind forcing during the other two seasons, this difference is quite negligible. On a nutshell, the results indicate that even though there is an effect of atmospheric forcing on the circulation of GoK, it is mostly driven by the tidal forcing. There are residual eddies present inside the Gulf, which might act like systems to trap the particles inside GoK and these flow depict large asymmetry in flood and ebb flow of the Gulf. The seasonal barrier, formed by the combined action of tide and bathymetry could be of serious concern as it isolates the Gulf from the surroundings during ebb condition. Thus, suspended sediment, effluents or any anthropogenic waste could be accumulated inside GoK which could be a serious threat to the aquatic organisms.

Conclusion
The Lagrangian circulation of Gulf of Khambhat was simulated using a 2D numerical model for three seasons, viz, pre-monsoon, monsoon and post-monsoon. The model results depicted the presence of residual eddies inside the Gulf during SW monsoon. The Lagrangian track results have shown that the particles released inside the Gulf, does not leave the system except for a few from southern GoK.
This could be associated with the presence of the residual eddies which do not allow particles to leave the Gulf. The combined tide and wind-driven circulation of GoK have exhibited that though the two systems are connected, there is a clear dissimilarity between the Gulf circulation with that of the Arabian Sea. The present work infers that tidal domination is maximum in GoK than any other physical forcing. As the Gulf restricts water parcel exchange with the Arabian Sea, the present study manifests a serious concern on the Gulf health in future. Future work will concentrate on the suspended sediment transport as well as the mixing characteristics of the GoK, a highly turbid water body of the west coast of India.