Introduction

Western continental shelf of India varied from south to north and widest off Bombay, leads into strongly converging channel, the Gulf of Khambhat (GK). This region is of immense interest among researchers considering various ongoing industrial activities (navigational, oil and gas industry etc.) and proposed developmental projects like impounding fresh water, harnessing tidal and thermal energy etc. The GK and its surrounding (Fig. 1) can be broadly divided into inner-outer gulf, continental shelf and continental slope. The gulf extends from upstream to a depth of 30 m with 120 km long and 50 km wide mouth. The continental shelf spans 200–250 km from gulf mouth to 200 m depth contour. The continental slope spans beyond 200 m depth contour extends to the abyssal depth. The inner gulf is gentle curving with irregular surface topography, whereas the outer gulf is straight sloping gentle topography (with funnelling shape). The gulf head is reworked by Mahi sagar river channel and meandering with formation of sand bank (MAL Bank) on left bank of the gulf. GK experiences largest tidal range along the Indian coast. Tides in this region are of mixed semi-diurnal undergoing strong amplification in the gulf. Study of tidal circulation in this region is of great interest to many researchers. Unnikrishnan et al. (1999) developed a barotropic numerical hydrodynamic model for simulation of observed feature of tidal amplification in the gulf. Nayak and Shetye (2003) used the one-dimensional version of the barotropic model to examine the causes of tidal amplification in GK and they concluded that resonance together with geometrical effects and friction setup the amplification of semi-diurnal tides in the Gulf.

Fig. 1
figure 1

Study region with depth contours and location of tidal stations (solid black dots) and two current meter measurement station (C1 and C2)

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Tidal strength is an important agent in driving and modulating net circulation in a semi-enclosed water body. Occurrence of barotropic slope currents following isobaths are common features of shelf seas in non-equatorial regions (Li and McClimans 2000). Tidal flow over the topographical features such as sill and continental shelf slope in a stratified ocean can produce internal waves of tidal frequency (Rattray et al. 1969; Hibiya 2004). Asymmetries in the flood and ebb tide due to bottom friction and non-uniform bathymetry can induce residual flow (Robinson 1981; Geyer and Signell 1990). Several observational studies reported the evidence of intensified residual currents flowing parallel to the continental slope as well as enhanced baroclinic tidal motion and fluctuations corresponding to internal gravity waves and longer period slope waves (Huthnance 1986; Thorpe 1992). The theory of existence of barotropic slope currents under the influence of strong tidal forcing, the process of tidal rectification, was examined by Garreau and Mazé (1992) and Mazé et al. (1998).

We believe that the GK and its adjacent oceans with its complex topography and geometry could manifest complex 3D circulation driven by tides and other external forcing. This circulation pattern has strong influence on the control of various tracer transport and biogeo chemistry occurring in this coastal environment. Thus, there is need of establishment of a 3D circulation model in this region. In view of this, the present study aims to set up a 3D numerical ocean model to understand tidal and residual circulations in the GK and its adjacent oceans. This study further extends the scope for addressing other aspect of coastal circulation in this region.

Methodology

In the present study, we used the Princeton Ocean Model (POM) developed by Blumberg and Mellor (1987). The POM is a terrain following three dimensional ocean circulation model most suited for coastal ocean processes studies. It has been used in several cases for modeling internal tides and residual currents in coastal waters (Holloway 1996; Robertson 2001). The model solves finite difference analogs of the primitive equations, with a non-linear equation of state relating density to the temperature and salinity. Vertical mixing coefficients for momentum and tracers are parameterized based on 2.5 turbulent kinetic energy closer schemes of Mellor and Yamada (1982), whereas horizontal mixing coefficients are modeled with Smogorinsky formulation for diffusion. Bottom frictional stresses are modeled based on logarithmic law of the wall modified for the coastal region for which bottom boundary layer is not well resolved according to Denniss and Middleton (1994). The model employs a C grid in the horizontal and a terrain following sigma coordinate for the vertical. The horizontal time differencing is explicit whereas the vertical differencing is implicit. The model has a free surface and a split time step: the external mode portion of the model is two-dimensional and uses a short time step based on the Courant-Friedrichs-Levvy (CFL) condition and the external wave speed, where as the internal mode is three-dimensional and uses a long time step based on the CFL condition and the internal wave speed. In the present case, model grid of the study region (69.5–72.3°E and 17–22.2 0 N) consists of uniform Cartesian coordinate system at 5 km (~2 min) resolution (121 levels along longitude, 163 level along the latitude) for the horizontal and 24 sigma-layers for the vertical. The bathymetric data used for this purpose are from corrected ETOPO-2 for the Indian coast (www.nio.org). LANDSAT map was used to correct geometry of the gulf, especially around the upstream end of the gulf. Time steps for the external barotropic and internal baroclinic modes are taken as 6 s and 1 min respectively. Temperature and Salinity data to describe initial stratification for the model ocean are taken from the Levitus (1982) climatological database. Temperature and salinity data are extrapolated (by simply using the nearest neighbour valid pixel) to fill the data gaps on the near shore region including the upstream end of the Gulf. The model run is carried out for 62 days by forcing with time varying tidal level and feeble currents at western and southern boundary. Excluding the first 3 days run, remaining 59 days model output were used for the analysis in the paper. It is worth mentioning here that these three days spin up period is enough for the model to reach a stable state as the region is very small and dominated by M2 tides (also inferred from the kinetic energy analysis).

Data of amplitude and phases of five major tidal constituents (M2, S2, N2, O1, and K1) to specify tidal level at the western and southern boundaries are based on global tidal model FES99 (Lefèvre et al. 2002). The feeble currents to specify momentum fluxes at western and southern boundaries were representative of spring season (March and April) and were based on Simple Ocean Data Assimilation climatology (Carton and Giese 2008). This period is selected for the model run for the following reasons. 1) This is the period when wind over this region is weak and tides suppose to have significant control on the net circulation patterns inside the embayment. 2) The study domain is subjected to very small lateral momentum fluxes at the western boundary (+1 Sv = 106 m3 s−1) and out flux (−1 Sv) at the southern boundary. 3) In this period, the sky is mostly cloud free and satellite picture of chlorophyll c2oncentration is available to infer/validate residual currents in the study domain.

Residual circulation is the mean circulation that can be estimated from simulated currents in GK using following procedure. As shown in Eq. 1, the instantaneous current (U) is composed of steady residual flow (U0), tidal-harmonics component (the second term) and perturbation term (Uε). Tidal harmonics were estimated from the simulated currents using least square procedure (Foreman 1978). Then the differences between simulated currents and tidal harmonics were averaged over several days (here entire duration of simulation period excluding first two days) to compute the steady residual circulation.

$$ U={U}_0+{\displaystyle \sum {U}_{ai}} Cos\left({\omega}_i t+{\psi}_i\right)+{U}_{\varepsilon} $$
(1)

Results

Comparison with in Situ Observations and Global Tidal Solutions

Historical tidal observations in the form of amplitude and phases of major tidal constituents at 37 coastal stations across the study region (see Fig. 1) are used for examining the performance of the model results. The observations are taken from special publication by International Hydrographic Bureau (1930), and subsequent publications; and volume 2 of Admiralty tide tables (1996). Spectral analysis of the simulated sea levels at respective coastal stations based on least square procedure (Foreman 1978) is carried out to estimate amplitudes and phases of the major tidal constituents. Comparison between simulated results and the observations is presented in the form of scatter plots (Fig. 2a–b). Root mean square errors (rmse) between predicted and observed tidal levels (due to each tidal component) at all tide gauge stations are calculated based on following expression and are presented in Table 1.

Fig. 2
figure 2

a The scatter plots show the comparison between amplitudes of observed and predicted tides at 37 coastal tide gauge stations around GK. Different symbols: O is for present modelling results (POM), X is for FES99 and + is for FES99. b Same as the (a), but for the phases of major tides instead of amplitudes

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Table 1 Comparison of observed tidal harmonics at coastal tide gauge stations and our model results. “A” represents amplitude (in cm) and “G” represents phase (in degree) of tidal harmonics; associated indices: “o” is for observed and “m” is for model; R is the root mean square error of predicted tidal levels
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$$ rmse=\sqrt{\frac{1}{L}{\displaystyle \sum_i{\left({A}_m Cos\left({\omega}_i t-{\psi}_m\right)-{A}_o Cos\left({\omega}_i t-{\psi}_o\right)\right)}^2}} $$
(2)

Where L is the total temporal segments; Am (Ψm) and Ao (Ψo) are amplitudes (phases) of modelled and observed tides respectively. The index i range between 1 and L.

There exists good agreement between observations and the simulation for both amplitudes and phases of major diurnal (K1, O1) and semi-diurnal (S2, M2) tidal constituents. Alike the observations, the predicted sea levels in the gulf is dominated by semi-diurnal tides and exhibit large change in phases. Amplitudes of diurnal tides are relatively small and associated with small change in phase. However, there are three stations, namely Bhavanagar, Dahej and Suvali, where model estimated M2 and S2 amplitudes are much smaller than the observations. These stations are located in the tidal flat region in the inner gulf. As the tidal flat regions are poorly represented in the model, large errors are estimated in these stations.

The FES99 and FES04 are global tidal solutions (Lefèvre et al. 2002; Lyard et al. 2006), respectively with 0.25° and 5 min spatial resolution and are available at the public domain along with TOPEX/Poséïdon Sea Surface height (SSH) observations. These tidal solutions are used for different oceanographic applications: tidal correction for satellite measured SSH data for the analysis of sea level change controlled by processes other than the tide; used as forcing in the coastal circulation model to simulate circulation pattern in a tide dominated embayment such as the GK and its surrounding, etc. The validity of FES tidal solutions in GK were also examined in Fig. 2a and b. Prior to this, we applied following correction for the phase angles of FES tide in order to convert them from the reference of Greenwich Mean Time (GMT) to standard local time zone.

$$ {g}_i={G}_i-\frac{360^0}{P_i}{Z}_n $$
(3)

Where gi and Gi are phase angles of the tidal constituent, i, with reference to the local time and GMT respectively. Pi is the period of the tide (in hours) and Zn is the time zone of the location. The present study region belongs to zone number 5. Thus, 75° and 150° corrections required for diurnal and semi-diurnal tides, respectively.

The comparison of FES99 tides with observations suggests that FES tidal amplitudes are comparable with the observations however, small values are over estimated and large values are under estimated. FES99 tidal phases are also comparable with observations. The rmse associated with FES99 tidal levels (Table 1) are significantly larger at the stations located across the gulf regimes and are small for the stations located on the outer domain of the embayment. Errors associated with FES04 are larger than the errors with FES99.

The simulated ocean currents are compared with available observations: Aanderaa Doppler current meter measurement at 72.5°E, 21.66°N (Kumar and Kumar 2010) and moored buoy observation at 71.49°E, 20.8°N deployed by National Institute of Ocean Technology, Chennai (Data sources INCOIS webpage). The former observation is located in the inner gulf regime (*C1 in Fig. 1) and later is located in the northern part of the gulf mouth (*C2 in Fig. 1). Table 2 shows comparison between model and observed currents in the form of amplitude and phases of major tidal harmonics. It revealed that simulated currents compare well with the observations. At the inner gulf, both simulated and model currents are mostly alongshore, dominated by M2 tide with observed amplitude of 165 cm s−1 and 262° phase, while it was 153 cm s−1 and 252° in simulated results. At the station *C2 (gulf-mouth), both model and observation are comparable and have equal magnitude of alongshore and cross-shore components. In this station currents are primarily dominated by M2 tide followed by S2 and N2 tides (see Table 3). The current at this location is weaker than that of the inner gulf station.

Table 2 Root mean square errors (R) in the present model (GK-POM) solution and its comparison with global FES tidal solutions at 37 tide gauze stations
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Table 3 Comparison of simulated currents with observations at two stations: Aanderra Doppler Current Meter measurement station C1 (72.5°E, 21.66°N) (Kumar and Kumar 2010) and at the moored buoy observation station C2 (71.49°E, 20.8°N) (see Fig. 1. U is zonal currents and V is meridional currents. Indices “a” and “b” represent amplitude (in cm s−1) and phase (in degree) of harmonics
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Spatial Pattern of Amplitude and Phases of Major Tides

The spatial pattern of amplitude and phase of major tidal constituents (M2, S2, K1 and O1) are presented in Fig. 3. M2 co-tidal lines (contour of equal phase) are aligned across the axis of the channel suggesting the tidal propagation is along the channel, northeast and southwest direction. It has undergone continuous change by 225° from far-slope regime towards the inner gulf: 325° at the shelf break; 360° at the mouth of the gulf and 105° at the inner gulf. M2 co-range lines (contour of equal amplitudes) parallel the eastern coastal boundary indicating dominance of coastal Kelvin waves on M2 tidal propagation in the region. Amplitude of M2 tide was 40 cm at the far-slope region, and undergoes slow amplification over the continental slope untills the shelf break. On the continental shelf, it has undergone strong amplification, more than 3 times (40 to 150 cm) towards the inner Gulf. It remains uniform inside the inner gulf and decrease further towards gulf head (up to 70 cm). Strong amplification of semidiurnal tides in the continental shelf and in main gulf (except gulf head regions) is an important characteristic of this embayment resulted from the quarter wavelength resonance of the tides owing to its inherent geometrical settings (Nayak and Shetye 2003).

Fig. 3
figure 3

Contours of amplitudes (continuous lines) and phases (dash lines) of major tides in the GK and surrounding from the model results

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Spatial pattern of S2 tidal chart is similar to M2 tide; however, S2 amplitude is almost half of M2. On the other hand, diurnal tides (K1, and O1) have small amplitudes, 35 cm and 18 cm at the shelf break respectively for K1 and O1 tides, and they exhibit slow amplification towards the inner gulf. At the gulf head, amplitudes are 65 cm and 30 cm respectively. Unlike the semidiurnal tides, phases of diurnal tides exhibit small change (25°–50° for K1 and 35°–80° for O1) from the shelf break towards the inner gulf. Orientation of the co-tidal lines suggests that semidiurnal tides propagate along south- north direction.

Depth Average Currents and Surface Currents

The typical flood and ebb phases of tidal levels and currents in the study domain as shown in Fig. 4 suggest that the flood phase sea level is high (~75 cm) at the far-slope region, decreases gradually towards the shelf break and became neutral at the gulf mouth (Fig. 4a). It decreases further down to −200 cm in the inner gulf. Head of the gulf manifest a positive sea level isolates from inner gulf and open oceans. Simulated vertical average currents (barotropic currents) mostly flow across the tidal level on the continental shelf and gulf regimes. They are weak on the continental slope regime (less than 10 cm s−1) and are intensified over the shelf regime (50 cm s−1) and proceeds into the gulf where they further intensified up to 150 cm s−1 at the gulf head. During the ebb phase, the sea level increases from negative values (−75 cm) over the slope regime towards the shelf with zero tidal level at the gulf mouth (Fig. 4b). Tidal level further increased up to maximum 200 cm towards the inner gulf and then decreased down to 50 cm at the gulf head. During this time, barotropic currents exhibit westward flow (from GK to open ocean) on the continental shelf. The currents are strong (100 cm s−1) inside the Gulf and their magnitude decreases towards the slope break (10 cm s−1). Surface current features almost resemble the features of the barotropic currents (Fig. 4c–d) over the continental shelf and gulf regions. However, they are stronger than the barotropic currents over the open ocean regime, beyond the shelf break. These are associated with anti-cyclonic and cyclonic eddy features during flood and ebb phases respectively. These circulation features and associated eddies usually occur in deep water beyond the shelf break owing to the consequence of instability of baroclinic currents associated with anomalous velocity shear with respect to surrounding water masses against the steep bathymetry on the continental slope (Barltrop 1998).

Fig. 4
figure 4

Simulated currents and associated tidal levels corresponding to flood and ebb phases: upper panels are for vertical averaged currents and lower panels are for surface currents

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Vertical Structure of Currents

Simulated currents exhibit very pronounce vertical structure. Figure 5 shows the currents along middle of the channel from southwestern corner (Open Ocean regime) to the northeastern corner (inner gulf) at four depths: 10, 30, 80, and 160 m during flood and ebb phases of tides. At the flood phase, surface layer currents over the continental slope are moderate (<40 cm s−1) and exhibit anti clock wise rotation with southeast ward flow on the far slope regime and north and north-east ward flow on the near slope regime. At the proximity of the shelf break, currents are weaker southward (<15 cm s−1). On the shelf, currents are northeastwards and intensified up to 100 cm s−1 at middle of the outer gulf. Inside the inner gulf and head regions, currents are southward and very strong (120 cm s−1). Currents at deeper depths follow the similar flow pattern. At the ebb phase, currents (at all depths) mostly prevailed in opposite direction of the flood phase circulation. This result suggests that currents within upper 160 m depths follow barotropic flow pattern. To investigate further the variation of currents on the continental slope regime, vertical slices of the currents and vertical velocity across the slope from far-slope to shelf break are presented in Fig. 6. It shows that currents have more pronounced vertical two-layered structure. During the flood phase, surface layer currents (0–1200 m depth) exhibit anti clockwise rotation with southeastward orientation in the far slope regime and northwestward orientation in the near slope regime. On the other hand, bottom layer currents (below 1200 m depth) exhibit clockwise rotation with northwestward orientation in the far-slope regime and northeast orientation in the near-slope regime. The flow pattern during the ebb phase is opposite to that of the flood phase circulation: clockwise rotation for the surface layer and anti-clockwise rotation for the bottom layer. This slope circulation is associated with strong upwelling and down welling events. Further investigation of slope circulation in the region is beyond the scope of present study.

Fig. 5
figure 5

Simulated flood and ebb phase currents at four depths (10, 30, 80 and 160 m) along the channels of GK and adjacent oceans from south-western corner (far slope regime) to the north-eastern corner (gulf head). Label along X-axis represents indices of the grid numbers along the diagonal of the embayment (see Fig. 1)

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Fig. 6
figure 6

Vertical structure of simulated horizontal currents and vertical velocity (color) on the continental slope. Label along X-axis represents indices of the grid numbers along the diagonal of the model domain

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Residual Circulation

The residual currents determine distribution of various tracers/pollutants etc. in a tidal dominated embayment. The residual circulation resulted from nonlinear interaction between barotropic tides and topography in the study region has been computed from the simulated currents (Methodology), and presented in Fig. 7. The figure also shows the spatial pattern of satellite measured chlorophyll concentration for the month of March based on Sea-Viewing Wide Field-of-View Sensor (SeaWiFS) overlaid with bathymetry. As shown, strong (>5 cms−1) along channel southward jet circulation prevailed inside the inner gulf. On entering the outer gulf, they intensified on the southeastern flank and then move westward. On the vicinity of the gulf mouth, they parallel the 60 m bathymetry contours and move as isobaths jet currents along northwestward direction. On proceeding, they move as alongshore isobaths coastal jet currents on the northwestern flank of the shelf. These jet currents on the proximity of 60 m depth contour considered as the coastal ocean boundary, which does not allow the gulf-water adverted/flushed out into the open ocean directly. Its consequence can be seen on advection of tracers such as Chlorophyll in this region. These coastal boundary jet currents are associated with an anti-cyclonic (clockwise) eddy on its right side (eastern coastal flank) and cyclonic eddy on its left (western) side. On the proximity of the shelf break, residual currents are mostly southward and strong (5 cms−1) over the shelf (shelf-slope) region and relatively weaker on the slope and open ocean (slope-ocean) regime. This asymmetry in slope circulation is associated with cyclonic eddy on the slope-ocean regime. There are several notable features. Eastward cross-shore currents on reflection near the Mumbai coast bifurcates into northwestward isobaths currents on its northern flank that joins the boundary coastal currents at the gulf mouth and southward propagating alongshore currents on its southern flank that further intensified and propagate as southward alongshore currents.

Fig. 7
figure 7

Residual currents in the GK and adjacent coastal oceans based on simulated currents overlaid with spatial distribution of Chlorophyll (Chla) in March 2003 as measured by SeaWIFS

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Most of these features associated with residual currents are seen in the spatial pattern of chlorophyll a concentration (Chla). High intensity Chla adverted along the channelling flow inside the inner and outer gulf and elongated westward at the gulf mouth adjacent to the coastal boundary currents. The anti-cyclonic eddy on the northwestern part of the outer gulf is associated with relatively low intensity Chla and the cyclonic eddy on the left (western) side of isobaths coastal boundary currents, adverted the high intensity Chla water from the coastal boundary system into the core of the eddy. There exist isolated and relatively intense Chla (.4 mg m−3) moving along the slope currents. High intensity Chla features (>1 mg m−3) are observed on the left and right sides of the southward propagating along shore currents on the southeastern flank of the shelf.

Conclusions

We implemented Princeton Ocean Model to study tidal and residual circulation in the Gulf of Khambhat and its adjacent oceans. The model is forced with time varying tidal levels and momentum fluxes at the southern and western boundaries. Simulated tidal levels and circulation pattern in the region compare well the observations. Model currents have pronounced 3D structure, which is matter of further investigation and validation; however, the residual circulations estimated on the region are quite interesting and captured well by the satellite imagery of Chlorophylla concentration on the month of March. This is the period when other forcing such as wind, buoyancy, and freshwater are small and tide dominates the net circulation in the region. This may not be true at all the time, in particular during the monsoonal period. Thus, further investigations, experiments, observation plan and analysis are essential to understand general circulation pattern in the region for different seasons.