1. Introduction

Nonlinear steepening and nonhydrostatic dispersion of large disturbances generated during a strong tidal flow over steep ocean topography in a stratified ocean produces internal solitary waves (ISWs). Several generation mechanisms have been reported for the generation of these disturbances [few of them were discussed in Baines (1984), Lai et al. (2010), Buijsman et al. (2010), Da Silva et al. (2015), and Bourgault et al. (2016)].

The Andaman Sea located in the northeastern part of the Indian Ocean is connected to the Bay of Bengal through multiple channels in the west and to the western Pacific through the Malacca Strait in the south. With steep slopes, strong stratification, and barotropic tidal flows, the Andaman Sea is an interesting region to study the characteristics of ISWs. Satellite remote sensing images show multiple ISW hotspots in the Andaman Sea (Alpers et al. 1997; Jackson et al. 2012). Observations from sunglint images of Moderate Resolution Imaging Spectroradiometer (MODIS) (Jackson et al. 2012) show that mode-1 ISWs radiate both toward the west (the southern Bay of Bengal) and east (Andaman Sea) from two different hotspots located between Car Nicobar–Chowra and Katchal–Little Nicobar Islands. Wijeratne et al. (2010) hypothesized that the westward propagating ISWs from the Nicobar Islands propagate across the southern Bay of Bengal and excite seiches in the Trincomalee Bay, Sri Lanka. Jithin et al. (2019) showed that about 80% of internal tidal energy available on the continental slopes of western Bay of Bengal is coming from the Andaman–Nicobar Islands. Solitons from Nicobar Islands contribute significantly to the high-frequency current oscillations in the southern Bay of Bengal (Wijesekera et al. 2019). A numerical study by Jensen et al. (2020) has shown the possible role of internal tides in enhancing the vertical mixing process in the thermocline. Eastward-propagating ISWs from the Nicobar Islands are reported to protect the corals on the continental shelf of eastern Andaman Sea (Wall et al. 2015). Hence, the study of internal waves from Nicobar Islands finds its application in diverse fields such as ocean mixing, biological productivity, offshore structural design, ocean acoustics, and naval operations in the Bay of Bengal and Andaman Sea. Further, Mohanty et al. (2018) showed that about 80% of the total barotropic energy along the Nicobar Islands is converted into baroclinic energy, given the O(8.67) GW of barotropic to baroclinic energy conversion, so region surrounding the Nicobar Islands is of some interest for the global tidal energy budget.

In particular, this study explores the generation mechanism(s) of ISWs from a shallow ridge connecting Car Nicobar–Chowra Islands, their bidirectional propagation and energetics. Batti Malv Island separates this shallow ridge into two parts: (i) north of Batti Malv (referred to hereafter as NBM) and (ii) south of Batti Malv (referred to hereafter as SBM), as shown in Fig. 1. Remote sensing observations show the bidirectional nature of long-living ISWs originated from the ridge between Car Nicobar and Chowra Islands (Jackson et al. 2012; Raju et al. 2019). However, their generation mechanisms are still poorly understood. Hence, for the first time we present a comprehensive analysis of mode-1 and mode-2 ISW generation from NBM and SBM using synthetic aperture radar (SAR) images and high-resolution nonhydrostatic numerical simulations. First, the SAR images over this region are critically investigated and the ISWs are delineated. Three vertical transects across NBM and SBM and also intersecting the prominent surface signatures observed in the SAR images are chosen. Further, the spatiotemporal evolution of ISWs are investigated using nonhydrostatic numerical simulations along these vertical sections.

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Bathymetry for the model domain. The solid (dashed) box indicates the boundaries of the hydrographic charts published by the National Hydrographic Office, India, with chart numbers 407 (409). The filled circle (star) shows the locations of tide gauge stations Nancowry (Campbell Bay). The region between the boundary and black dotted box is the sponge layer used in the model simulations. Shallow ridges north and south of Batti Malv are denoted as NBM and SBM, respectively.

Citation: Journal of Physical Oceanography 51, 1; 10.1175/JPO-D-19-0182.1

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The manuscript is organized as follows. Section 2 presents a detailed discussion on the three selected SAR images. Model setup and validation is presented in section 3. In section 4, discussions are made on the model results and the underlying generation mechanism of ISWs. Energetics of internal tides generating from NBM and SBM are discussed in section 5. Finally, a summary and conclusions are presented in section 6.

2. SAR observations of ISWs

SAR imagery has been widely used to delineate ISWs, identify their hotspots, and provide valuable insights on their propagation characteristics (Da Silva et al. 2015; Magalhaes and da Silva 2018). ISWs in a SAR image appear as a packet of alternating bright and dark bands, with leading bright bands in the propagation direction signifying mode-1 wave, whereas the leading dark bands denotes convex mode-2 wave [refer to Fig. 2 in Magalhaes and da Silva (2018)]. A majority of SAR images covering the study domain show the surface signatures of mode-1 ISWs captured during a strong tidal flow (near to spring tide). Few studies also reported the presence of higher modes (Magalhaes and da Silva 2018). All the available Envisat Advanced SAR (ASAR) images (in Wide Swath Mode and Image Mode obtained from http://hedavi.esa.int/) from 2005 to 2007 and TerraSAR-X images (in ScanSAR and Wide ScanSAR mode obtained from https://terrasar-x-archive.terrasar.com/) from 2010 to 2014 covering the region of interest are carefully analyzed to identify the bidirectional propagation of ISW packets. Three SAR images (listed in Table 1) acquired at different tidal phases are discussed here pertaining to the interpretation of mode-1 and mode-2 ISW generation.

Table 1.

Details of the SAR images discussed in section 2.

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Figure 2 presents a part of Envisat ASAR (in Image Mode) image acquired at 1553 UTC 12 October 2007. The most noticeable features in this image are as follows: (i) The surface manifestations of mode-1 ISWs generated from both the hotspot regions, NBM and SBM, are denoted as P1 and P2, respectively. (ii) The surface signatures of mode-2 wave (denoted as P3) is also clearly visible over SBM. (iii) The image contains more than one wave packet (P2, P4, and P5), which seems to be generated from the same location (i.e., SBM) and travels in the westward direction. It is found that the wave packets P1 and P2 (refer Fig. 2) travel a distance of 7 and 13 km from NBM and SBM, respectively, in the west direction. Comparing the distance traveled by P1 and P2 from the crest of sill illustrates a brief idea about their possible generation. At the time of image acquisition, the east–west component of local barotropic tide is close to zero, as computed using TPXO8 tide model. Considering the difference in the distance traveled by these two wave systems, i.e., 6 km, at the time of image acquisition and an average phase speed of 1.9 m s⁻¹ [linear mode-1 wave at a depth of 500 m, computed using Eq. (2)], it can be first guessed that the P2 might have radiated from the upstream side of the sill (at maximum eastward tidal flow) while the P1 is generated on the lee side of sill (toward the end of tidal cycle) as P2 travels a larger distance than P1. P3 is located over the SBM, hence it can be inferred that it might have generated on the lee side of the sill and started to advance in the westward direction after slackening of the barotropic tide. A detailed inspection of this SAR image also reveals the presence of P4 and P5 that are mode-2 in nature.

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Subset of Envisat ASAR image acquired at 1553 UTC 12 Oct 2007. White and black dashed arcs are surface manifestations of ISWs in TerraSAR-X images acquired at 2342 UTC 17 Mar 2014 and 1156 UTC 14 Jul 2010, respectively. P1, P2, P6, and P7 indicates manifestations of mode-1 ISW packets, while P3, P4, P5, P8, and P9 are mode-2 ISW packet signatures. White contours denote bathymetry. White solid lines are transects S1, S2, and S3 along which generation mechanisms are discussed.

Citation: Journal of Physical Oceanography 51, 1; 10.1175/JPO-D-19-0182.1

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Further, enhanced radiometric resolution makes the TerraSAR-X images ideal for identifying and delineating the higher-mode waves. Hence, in addition to an Envisat ASAR image, two TerraSAR-X images (details mentioned in Table 1 and are shown in Fig. S1 in the online supplemental material) are analyzed to study the surface signatures of ISWs at different tidal phases. The surface signatures of leading wave fronts (P6, P7, P8, and P9) in TerraSAR-X image captured at two different dates, 2342 UTC 17 March 2014 and 1156 UTC 14 July 2010, are traced over the Envisat ASAR image shown in Fig. 2. TerraSAR-X image taken on 17 March 2014 show the presence of mode-1 ISWs, P6 and P7 traveling east after being generated from NBM and SBM, respectively, and hence revealing the bidirectional nature of mode-1 ISWs [also mentioned previously in Jackson et al. (2012), Magalhaes and da Silva (2018), and Shimizu and Nakayama (2017)]. The surface manifestations of these waves are shown in Fig. 2. Another TerraSAR-X image acquired on 14 July 2010 show the signatures of westward propagating convex form of mode-2 waves, P8 and P9, appearing to be generated from NBM and SBM, respectively. Thus, this region can be recognized as one of the very few regions in the world oceans for which mode-2 internal waves can be observed propagating some distance away from their generation region.

To study the generation of wave packets P1–P8, three different transects S1, S2, and S3 are taken across NBM, SBM, and observed wave packets in SAR images are considered (Fig. 2). Transect S1 intersects the east–west propagating ISWs from NBM. A shallow sill is present to the east of SBM (Fig. 2). Therefore, two separate transects S2 and S3 are taken to study the west and eastward propagating ISWs generated from SBM. In section 4, the generation mechanism of ISW packets P1–P8 are investigated using the outputs of a nonhydrostatic numerical model along the transects S1, S2, and S3.

3. Numerical model SUNTANS

To simulate the generation of ISWs, we employ nonhydrostatic configuration of the hydrodynamic model, Stanford Unstructured Nonhydrostatic Terrain-Following Adaptive Navier–Stokes Simulator (SUNTANS) (Fringer et al. 2006). SUNTANS is a nonhydrostatic, parallel, and unstructured grid Navier–Stokes simulator. It solves the three-dimensional, Reynolds averaged, Navier–Stokes equation employing the Boussinesq approximation on a grid system that is unstructured in the horizontal and structured in the vertical. Previously, SUNTANS was used successfully in the studies of ISW generation and propagation elsewhere (Rayson et al. 2018; Zhang et al. 2011). It is to be noted that few authors have used hydrostatic simulations to study the generation, propagation, and energetics of ISWs in the Andaman Sea (Jensen et al. 2020; Mohanty et al. 2018). However, a detailed study on the structure of solitary wave packets and their generation mechanism requires nonhydrostatic simulations (Jensen et al. 2020).

a. Model setup

A sketch of model domain with corresponding bathymetry are shown in Fig. 1. The model is used to simulate the solitary wave packets observed in the Envisat ASAR image captured at 1553 UTC 12 October 2007 and also to understand the ISW generation mechanism(s) during a spring–neap tidal cycle. The model was run for 19 days from 1553 UTC 4 October 2007 to 1553 UTC 23 October 2007, which includes a 4-day spinup and a spring–neap tidal cycle of 15 days.

1) Generation of bathymetry over the model domain

The present study used the modified ETOPO5 bathymetry datasets (Sindhu et al. 2007) and the bathymetry near to the ISW generation regions was further improved using two high-resolution navigational charts [407(7445) and 409(7448)], available from National Hydrographic Office, India. Boundaries of these charts surrounding the generation regions are shown in Fig. 1. The depth contours and sounding depths over the chart area are extracted and digitized. The bathymetry in navigational charts were referenced relative to the local lowest astronomical tide (LAT). These underestimated depth values are corrected using the local LAT extracted from TPXO8 tide model referenced from the mean sea level. Bathymetry data of modified ETOPO5 inside the area covered by the nautical charts is removed and updated with the digitized depth values. This merged bathymetry dataset is interpolated onto the SUNTANS grid using kriging interpolation and further smoothed using a Gaussian filter. An RMSE of 46 m is found for the predictions made by using kriging interpolation. The resultant bathymetry is found to have smooth gradients along NBM and SBM with realistic depth values.

2) Grid generation

Triangular mesh induces numerical error in horizontal divergence in the form of checker board oscillations (Wolfram and Fringer 2013). To reduce this numerical error, the unstructured grid employed here consists of n-sided polygons, dominated by hexagons (about 96.7%). The nodes of hexagonal unstructured mesh are taken as the circumcenters of orthogonal triangulated mesh generated using TOM triangular paving algorithm (Holleman et al. 2013). A brief methodology of grid generation is described in Rayson et al. (2018). The unstructured grid over the study domain is shown in Fig. 3. The horizontal distance between leading wave crest and trough of the P2 ISW packet (discussed in section 2) is about 900 m. The leading wave crest and trough distance of ISW at a horizontal distance of 50 km west of SBM [found in several other SAR images (not shown here)] is around 1.7 km. To better resolve the ISWs packets covering the early stages of generation and development, the horizontal cell center distances is maintained at 110 m nearer to the sill and is stretched to about 500 m east–west of NBM and SBM. The relative high resolution near the generation region is useful to minimize the effects of numerical dispersion (Vitousek and Fringer 2011). Away from the generation area the grid resolution is increased to a maximum length of 12 km near to the domain boundary. ISWs are also generated from the shallow sills in the Great Passage and Ten Degree Channel (Osborne and Burch 1980; Magalhaes and da Silva 2018). To reduce the influence of these ISWs, horizontal grid resolution of about 10 km is maintained around these sites. There are 100 horizontally uniform “z” layers along the vertical, with a surface resolution of 10.37 m and is stretched with a stretching factor of 1.025 to maintain a bottom resolution of 119.57 m. The total number of computational cells used in the present study along the three-dimensional domain are 5 984 017, with only about 35% of active grid cells.

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SUNTANS horizontal unstructured grid used to simulate internal solitary waves around Nicobar Islands. Zoom views show the close-up of the grid around Car Nicobar Island.

Citation: Journal of Physical Oceanography 51, 1; 10.1175/JPO-D-19-0182.1

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3) Forcing parameters

The model is initialized with the temperature and salinity fields for the month of October 2007 for the top 1200 m adapted from Udaya Bhaskar et al. (2007). For greater water depths (depth > 1200 m) the annual mean climatology available from the World Ocean Atlas 2013, version 2, is used. It is to be noted that Udaya Bhaskar et al. (2007) had used the available Argo profiles to generate a monthly gridded temperature and salinity profile using the objective analysis method. Horizontally domain averaged uniform profile obtained from the merged products along with the buoyancy frequency are shown in Fig. 4. The initial velocity field and free surface are set to zero over the whole domain. Model open boundaries are forced with the first eight major tidal constituents, namely, M₂, S₂, N₂, K₂, K₁, O₁, P₁, and Q₁ that are derived from OTIS (Oregon State University Tidal Inversion Software, version 8) global atlas (TPXO8). To reduce the initial transient oscillations in the model, the barotropic velocities are exponentially increased from zero to its original value in 1.5 days using the following equation:

$u_{b_{\text{modified}}}=u_b\left[1-\exp \left(-t/{\tau }_r\right)\right],$(1)

where τ_r = 1.5 days and u_b and $u_{b_{\text{modified}}}$ are the original and modified boundary velocities, respectively.

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Temperature and salinity profiles used to initialize model. Brunt–Väisälä frequency is also shown.

Citation: Journal of Physical Oceanography 51, 1; 10.1175/JPO-D-19-0182.1

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A 50-km width of sponge layer is applied at the open boundaries (shown in Fig. 1) to minimize the reflection of baroclinic energy back into the study domain. A sponge layer is applied to the model by modifying the right-hand side of the momentum equation as explained in Zhang et al. (2011). This sponge layer exponentially dampens the baroclinic horizontal velocities over a distance of 50 km from the nearest open boundary. The eddy viscosity was computed by the Mellor and Yamada (1982) (MY2.5) turbulence closure scheme and the diffusion of scalar quantities is taken to be zero. Bottom drag coefficient is taken to be constant and given by C_D = 0.0025. A constant Coriolis frequency of 2.03 × 10⁻⁵ rad s⁻¹ (at latitude of 8°) is used. For a stable run a time step of 6 s is used.

b. Numerical model validation

1) Comparison with sea level data

To better simulate the generation of ISWs, reproduction of barotropic tidal currents in the domain is very essential. One-minute sampled, pressure sensor sea level data at two tide gauge sites, namely, Nancowry and Campbell Bay (locations are marked in Fig. 1) (https://incois.gov.in/portal/datainfo/tidegauge.jsp) are used to validate the tides simulated by SUNTANS. A typical sea level record from a tide gauge site is a summation of various forces, including tides, meteorological forces, swells, etc. Using “UTide” (Codiga 2011), tidal harmonic fitting of the observed tide gauge record from 2011 to 2013 revealed the amplitudes of major tidal constituents listed in Table S1 in the supplemental material. Tidal oscillations at these two locations are reconstructed using tide prediction code of UTide for the same time steps as model output using the first eight dominant frequencies and are treated as observations. Figure 5 shows a comparison between the sea surface height predicted by SUNTANS and the observed tidal amplitudes at Nancowry and Campbell Bay. The model predicted sea surface height shows transient oscillations at the start of model run, which decays within two M₂ tidal cycles. The model well simulates the tidal cycles at both Campbell Bay and Nancowry. However, it is clearly seen that the sea surface height is a little under predicted with a root-mean-square error of 6.63 and 9.29 cm at Campbell Bay and Nancowry, respectively. The amplitude and phase difference between observations and simulations of each individual tidal constituent at Campbell Bay and Nancowry is listed in Tables S1 and S2 of the online supplemental material, respectively. These changes could be attributed to smoothed bathymetry used in the model. The tidal phase is well captured during a spring tide but there is a slight phase shift during neap tide conditions.

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Comparison of modeled and observed sea surface height at (a) Campbell Bay and (b) Nancowry tide gauge stations. The square indicates the sea surface height at which Envisat ASAR (discussed in Fig. 2) is captured. Generation mechanisms along the transects S1, S2, and S3 are discussed in section 4 at the time intervals denoted by rectangles.

Citation: Journal of Physical Oceanography 51, 1; 10.1175/JPO-D-19-0182.1

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2) Comparison with SAR imagery

Horizontal sea surface signatures of ISWs found in a SAR image can be compared with several model simulated parameters for validation. Alternate bright and dark surface signatures in a SAR image are induced by near surface horizontal orbital velocities of ISWs. In the direction of propagation, the leading patch for mode-1 (mode-2) ISW is bright (dark). Leading orbital vertical velocity of mode-1 ISW is downward (upward) for the mode-1 (upper bulge of mode-2) wave. Magnitude of vertical velocities induced by mode-1 ISW is maximum at the depths of highest Brunt–Väisälä frequencies. The buoyancy frequency profile used to initialize the model has a maximum value at a depth of about 80 m. In our model simulations, the bulge center of mode-2 wave is located at a depth range of 100–200 m. Hence, comparing the vertical velocities at a depth of 80 m with the surface manifestations of ISWs clearly reveals the signatures of mode-1 ISWs. However, the signal of mode-2 waves can be weak. Figure 6 depicts the simulated vertical velocities at 1613 UTC 12 October 2007, at a depth of 80 m as compared with the Envisat ASAR image captured at 1553 UTC 12 October 2007. Mode-1 ISW packets (P1 and P2) traveling toward west of NBM and SBM are well captured by the model. Upward (positive) vertical velocity of leading wave packet in the model simulations signify the presence of mode-2 waves. Mode-2 ISW packets (P3, P4, and P5) traveling west of SBM are also very well captured and are consistent with those found in SAR imagery. Hence, all the prominent signatures of mode-1 and mode-2 nonlinear waves identified in the Envisat ASAR image is well simulated by SUNTANS. However, there is a lag in the simulated waves of nearly 1 h and 20 min. This lag occurs due to relatively smooth bathymetry used in the model. However, this lag in simulated barotropic tide over the sill will not affect the generation mechanisms discussed here.

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Model-simulated vertical velocity comparison with the surface signatures of ISWs in a SAR image. (a) ASAR image captured at 1553 UTC 12 Oct 2007 on board Envisat. (b) Model predicted vertical velocity at 80-m depth at 1613 UTC 12 Oct 2007. White regions in (b) are bathymetry. Dashed white and black ellipses denote mode-1 and mode-2 ISWs, respectively.

Citation: Journal of Physical Oceanography 51, 1; 10.1175/JPO-D-19-0182.1

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4. Results and discussions

It is realized that the ISWs observed over the region, are the effect of interaction of the barotropic tide with the bathymetry of two prominent ridges, NBM and SBM. Figure 7 shows the vertical velocity at 80-m depth for different tidal phases, depicting the generation and propagation of ISW packets P1–P8 for a complete M₂ tidal cycle during a spring tide. Inset in Fig. 7c shows the zonal barotropic current over the top of SBM. At the end of flood tide (west to east flow), mode-1 wave (P2) is clearly seen being generated from top of SBM followed by mode-2 wave (P3) just behind it. A weak signal of mode-2 wave (P4) is seen at about 10 km west of SBM. During the initiation of ebb flow (negative zonal barotropic current), P2 disintegrates into an ISW packet and the signatures of mode-1 wave (P1) generating from NBM is clearly visible (Fig. 7b). Mode-2 ISW packets P3 and P4 are observed to move forward toward the west. At the end of ebb tide, Fig. 7c, P1 is found to be disintegrated into ISW and the westward propagating P2 appears to cross P4. From Figs. 7b and 7d–f, it is clear that the ISW packets P3, P5, and P9 are indeed the same mode-2 ISW propagating toward west of ridge SBM. As the tide slackens, mode-1 ISW packet P7 appears from SBM and propagates toward east (Fig. 7d). With progress in time, it merges with mode-1 ISW packet P6 generated from NBM (Fig. 7f). It should be noted that the mode-1 ISWs P6 and P7 in Fig. 7f resembles with the surface signatures found in the TerraSAR-X image (Fig. S1). As discussed in section 2, mode-2 ISW (P8) appears to be generating from NBM and propagates toward west; however, our simulations could not capture the wave packet P8. It can be due to the lack of oceanic currents, use of only the monthly mean stratification of October 2007 and higher spatial resolutions to capture mode-2 ISW away from the ridge. The generation mechanism, baroclinic flow structure and its strength variations during a spring–neap tidal cycle are investigated along the sections S1, S2, and S3 (as shown in Fig. 2).

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(a)–(f) Time evolution of model-simulated vertical velocity (m s⁻¹) at 80-m depth during spring tide. The inset plot in (c) shows the zonal barotropic tidal currents (m s⁻¹) over the sill, where the black dots refer to the phase of tidal cycle for each subplot. White regions are bathymetry. Time mentioned in the x axis of inset is in the format of day:hour. The evolution of different ISW packets P1–P7 (refer to Fig. 2) is also shown.

Citation: Journal of Physical Oceanography 51, 1; 10.1175/JPO-D-19-0182.1

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a. ISW generation along transect S1

Figure 8 shows the temporal sequence of simulated horizontal velocities and isopycnal fluctuations along the transect S1 for a complete tidal cycle (including a flood and an ebb tide, denoted with dash–dot rectangle in Fig. 5). The densimetric or internal Froude number (Fr) is used to examine the criticality of the flow over the sill (Farmer and Smith 1980). The internal Froude number is the ratio of barotropic flow speed with the phase speed of long internal wave given as Fr_i = u/c_i, where c_i is the speed of long internal wave of different modes i = 1, 2, … over the sill and can be obtained by solving the following equation for the vertical normal modes

${\displaystyle \frac{d^2\varphi \left(z\right)}{d{z}^2}}+{\displaystyle \frac{N^2\left(z\right)}{c^2}}\hspace{0.17em}\varphi \left(z\right)=0.$(2)

Here, the boundary conditions are ϕ(0) = ϕ(−H) = 0 and N is the Brunt–Väisälä frequency. It should be noted that the shear induced by tidal current is neglected here in computing the phase speed of different modes.

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(a)–(l) Time evolution of model-simulated vertical section of zonal horizontal velocities (m s⁻¹) along the transect S1 (cross section of S1 is shown in Fig. 2), during strong tidal flow (exact tidal cycle is shown with dash–dotted rectangle in Fig. 5). Contours are isopycnals. White and gray arrows follow the west and east propagating mode-1 ISWs, respectively. The inset plot in (i) shows the zonal barotropic tidal currents (m s⁻¹) over the sill, where the black dots refer to the phase of tidal cycle for each subplot. Time mentioned in the x axis of inset as well as in each subplot is in the format of day:hour.

Citation: Journal of Physical Oceanography 51, 1; 10.1175/JPO-D-19-0182.1

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Densimetric Froude number with respect to mode-1 (Fr₁) and mode-2 (Fr₂) linear internal wave computed over the crest of ridge (distance is zero) is shown in Fig. 8 for corresponding tidal phase. During most of the tidal flow, internal Froude number is greater than one for both mode-1 and mode-2 waves which signifies that the flow at the crest of ridge is supercritical. This means that all the possible internal modes generated over the sill will be advected in the downstream direction. A supercritical ebb flow (east to west, negative zonal velocity about 1.4 m s⁻¹) over the crest of sill leads to the formation of a density front on the lee side (also known as the lee wave), marked with the deepening of the pycnoclines (>24.0 sigma levels) just downstream of sill (Fig. 8d). Approximately 2 h later, this density front is found to be stationed behind the sill and grows in amplitude (pycnoclines deepened and westward velocity increased to 1.8 m s⁻¹) by accumulating the energy from ebb tidal flow (see gray arrow in Figs. 8d and e). As the westward current start decelerating, the density front starts to move eastward (Fig. 8f). Subsequently, the density front moves eastward and the flood tide (west to east flow) starts over the sill (Figs. 8g,h). It should be noted that the isopycnals of lee waves are displaced vertically downward, hence, these waves are clearly of mode-1 in nature. Further, the density front moves eastward and evolves into mode-1 ISW (see gray arrow in Figs. 8k,l). The accelerating flood flow (eastward flow) creates a supercritical condition over the sill and the pycnocline deepens on the lee side of the sill, developing a density front (Figs. 8i,j). This density front intensifies gradually with the tidal flow and as the tide slackens this density front starts to move westward (see white arrow in Figs. 8a,b). Finally, this density front disintegrates into ISW and propagates eastward (see white arrow in Figs. 8c–f). The formation of density front (lee wave) during supercritical flow and evolving into an ISW in the upstream direction is commonly referred to as “lee wave mechanism” (Hibiya 1986; Maxworthy 1979). In fact, the wave packets P1 and P6 shown in Fig. 2 are the mode-1 ISWs along transect S1 propagating westward and eastward, respectively, that are generated by the lee wave mechanism.

Phase speed of eastward and westward propagating mode-1 ISW can be inferred by comparing their propagating distances captured at different times. It is found that the mode-1 ISW propagating both in the east and west directions is about 2.1 m s⁻¹ at about 1100–1300-m depth, which is close to phase speed of linear wave. It is to be noted that a strong convex mode-2 wave found at about 26 km east of the sill in Fig. 8b is not the wave generated from the sill. It may be generated elsewhere and is propagating in the westward direction, which will be investigated separately.

b. ISW generation along transect S2

Transect S2 intersects only the westward propagating mode-1 and mode-2 ISWs generated from SBM (Fig. 2). The generation mechanism of these waves during the moderate and strong barotropic tidal flow are discussed in this section. Here, the barotropic flood tidal flow (eastward) over SBM during a spring tide (denoted with a rectangle filled with circles in Fig. 5) is treated as strong, while the flow after six M₂ tidal cycles from spring tide (denoted with a hatched rectangle in Fig. 5) is treated as moderate flow. The internal Froude number calculated over the SBM region shows that the barotropic flow throughout the spring–neap tidal cycle is subcritical (Fr₁ < 1) with respect to mode-1 linear internal wave. But, the flow becomes supercritical (Fr₂ > 1) for higher modes as it approaches the spring tide. In general, during the subcritical flow the disturbances created either upstream or over the crest of the sill can freely propagate upstream of the flow while in supercritical flow disturbances are advected farther away from sill in the downstream direction. At neap tide, the barotropic flow over the sill is minimum resulting in a featureless isopycnals.

The evolution of ISWs during a moderate flood tide along the transect S2 is shown in Fig. 9. It displays the modeled zonal velocities overlaid with isopycnals. The subcritical flow over the sill lowers the isopycnals over the crest and this depression is maintained during the flood tide (zonal flow is positive over the sill) (see white arrow in Figs. 9a–c). Simultaneously, a density front is generated in the downstream direction of the flow near to the sill (Figs. 9b,c). At the peak of flood tide, it can be noticed that the isopycnals in the upstream direction are found to be displaced a little upward (Figs. 9a,b). This uplifting of isopycnals signifies the blocking of eastward flow on the upstream side of sill. As the flood tide starts decreasing, the uplifted isopycnals propagate in the upstream direction as long elevated first mode wave. During propagation in the upstream direction this first-mode long wave steepens and evolves into a density front (see white arrows in the Figs. 9c–f) [consistent with the findings of Grimshaw and Helfrich (2018)]. Nonhydrostatic dispersion further breaks this density front into an ISW (see white arrow in Figs. 9e,f). Owing to the balance between nonlinearity and dispersion these ISWs can propagate large distances from the sill.

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(a)–(f) Time evolution of model-simulated vertical section of zonal horizontal velocities (m s⁻¹) along transect S2 (cross section of S2 is shown in Fig. 2), during moderate tidal flow (exact tidal cycle is shown with hatched rectangle in Fig. 5). Contours are isopycnals. White and gray arrows follow the west propagating mode-1 and mode-2 ISWs, respectively. The inset plot in (a) shows the zonal barotropic tidal currents (m s⁻¹) over the sill, where the black dots refer to the phase of tidal cycle for each subplot. Time mentioned in the x axis of inset and also in each subplot is in the format of day:hour.

Citation: Journal of Physical Oceanography 51, 1; 10.1175/JPO-D-19-0182.1

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During a strong tidal flow, the barotropic tidal flow increases. With this enhanced tidal flow, chocking occurs below the crest in the upstream direction and results in the shoaling of isopycnals upstream of sill. Shoaling of isopycnals found during a moderate tidal flow is further enhanced during strong flow. This shoaling raises the bottom isopycnals and creates a strong vertical density gradient near to 23.4 sigma level. This strong near surface vertical stratification in the upstream flow just ahead of the crest (Fig. 10b) can be assumed as an interface in two-layer flow. It can be reasonably assumed as a two-layer hydraulic flow. In a two-layer fluid flow the composite Froude number is expressed as $G^2=F_1^2+F_2^2$, which determines the criticality of the flow (Armi 1986); $F_i=u_i/\sqrt{g^{\prime }{h}_i}$, and F_i, u_i, and h_i for i = 1, 2 are Froude number, flow velocity, and the thickness of fluid layers, respectively, while g′ denotes the interfacial reduced gravity. For G² < 1, G² = 1, and G² > 1, the flow is internally subcritical, critical, and supercritical, respectively. For a subcritical two layered flow ($G^2\approx F_1^2<1$; just upstream of sill and with the interface at 23.4 sigma level, $F_1^2=0.6$ and $F_2^2=0.03$) the fluid interface over the crest of the sill deepens (Fig. 10a). As the flow increases during a strong tide, the upstream interface shoals through the initial subcritical adjustment. With increased tidal flow, this uplifted interface ahead of the sill permits a supercritical condition ($G^2\approx F_1^2>1$; just upstream of sill and with the interface at 23.4 sigma level, $F_1^2=1.3$ and $F_2^2=0.04$) in the upper layer. It should be noted that the supercritical upstream flow due to the upward tilt of isopycnals upstream of crest is broadly consistent with those found in Farmer and Denton (1985), Cummins et al. (2006), and Cummins and Armi (2010). However, the flow is still subcritical over the crest. This transition from supercritical upper layer in the upstream direction to the subcritical flow over the crest creates an internal hydraulic jump (Fig. 10b). This phenomena is referred to as the “upstream internal hydraulic jump” (Cummins et al. 2006). As soon as the tidal flow weakens, the internal jump advances forward in the west direction (see white arrow in Fig. 10c) and rapidly disperses into an ISW packet P2 (see white arrow in Fig. 10d). Hence, the ISW packet P2 seen in Fig. 2 represents the mode-1 ISW formed due to upstream internal hydraulic jump.

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(a)–(g) Time evolution of model-simulated vertical section of zonal horizontal velocities (m s⁻¹) along transect S2 (cross section of S2 is shown in Fig. 2), during strong tidal flow (exact tidal cycle is shown with circle filled rectangle in Fig. 5). Contours are isopycnals. White and gray arrows follow the west propagating mode-1 and mode-2 ISWs, respectively. The inset plot in (d) shows the zonal barotropic tidal currents (m s⁻¹) over the sill, where the black dots refer to the phase of tidal cycle for each subplot. Time mentioned in the x axis of inset and also in each subplot is in the format of day:hour.

Citation: Journal of Physical Oceanography 51, 1; 10.1175/JPO-D-19-0182.1

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The convex mode-2 ISW is characterized by a bulge in isopycnals and a symmetrical zonal current at the vertical edges with a reverse current at the center (see gray arrow in Figs. 9 and 10). During moderate or strong flood tide, the flow over the sill (SBM) is supercritical (Fr > 1) with respect to linear mode-2 wave. This strong flow advects the mode-2 wave generated over the crest in the downstream direction (Figs. 9a and 10a,b). However, the Froude number of mode-2 wave on the lee side of sill (at about 3 km) is about one (figure not shown) for both moderate and strong tidal flow. Hence, the wave is arrested on the lee side of sill and grows in amplitude during flood tide (Figs. 9b and 10b). As the flood tide starts decelerating the mode-2 wave starts propagating along the upstream direction of the flow (see gray arrows in Figs. 9c–f and 10c–f). The same has been observed in the model simulated vertical velocity at 80 m depth (P3 in Fig. 6) and Envisat ASAR image (P3 in Fig. 2). Westward-propagating ISW packets P3 and P5 in Fig. 2 are the mode-2 waves generated from SBM.

Phase speeds of mode-1 and mode-2 waves are found to be 2 and 1.2 m s⁻¹, respectively, at the depths of about 1000–1200 m. It is very interesting to note that a mode-2 wave is noticed at about 30–35 km west of the sill (Figs. 9e and 10e) and is found to be propagating westward (Figs. 11f and 12f). This matches with the P4 ISW packet identified in the Envisat ASAR image (Fig. 2) and model simulated vertical velocity (Figs. 7a–c). This could be generated locally by the interaction of the internal tidal beam with the pycnocline, which require a detailed separate study.

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(a)–(f) Time evolution of model-simulated vertical section of zonal horizontal velocities (m s⁻¹) along transect S3 (cross section of S3 is shown in Fig. 2), during strong tidal flow (exact tidal cycle is shown with dashed rectangle in Fig. 5). Contours are isopycnals. White and gray arrows follow the east propagating mode-1 and mode-2 ISWs, respectively. The inset plot in (a) shows the zonal barotropic tidal currents (m s⁻¹) over the sill, where the black dots refer to the phase of tidal cycle for each subplot. Time mentioned in the x axis of inset and also in each subplot is in the format of day:hour.

Citation: Journal of Physical Oceanography 51, 1; 10.1175/JPO-D-19-0182.1

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Depth-integrated, 15-day time-averaged baroclinic energy flux. Two boxes indicate the subdomains A (dotted) and B (dashed) used for energy budget calculations. Gray contours denote bathymetry and solid black lines denote the transects S1, S2, and S3.

Citation: Journal of Physical Oceanography 51, 1; 10.1175/JPO-D-19-0182.1

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5. Energetics

It should be noted that the eddy viscosity and diffusion coefficients in the numerical simulations are purely chosen for the stability requirements of numerical differencing. Hence, these terms are neglected in both the barotropic [Eq. (9)] and baroclinic flux [Eq. (10)] equations as compared with those of complete equations used in Kang and Fringer (2012).

Figure 12 illustrates the horizontal distribution of depth and time integrated baroclinic energy flux $\langle \overline{F^{\prime }}\rangle$. Time integration is carried out for a spring–neap tide period, i.e., 15 days. It can be clearly seen that the baroclinic energy flux radiates away from NBM and SBM toward east and west directions. A significant amount of energy flux of about 30–40 kW m⁻¹ is seen to be radiating from the ridge slopes south of Batti Malv. It is noticeable that a shallow sill located east of SBM influences the baroclinic flux and the energy is concentrated/diverted between these two regions. This supports the selection of transect S3 to understand the generation mechanism of eastward propagating ISWs as discussed in section 4c. It is very interesting to note that less energy is radiated from the NBM region, which brings out the dissimilarity between these two regions.

To analyze and understand the radiation and conversion of energy over the region, the domain is subdivided into two subdomains. Subdomain A covers the NBM, while subdomain B covers the SBM region (see Fig. 12). To quantitatively understand the power contribution of both subdomains for tidal cycles including a spring and neap tide, each term in energy budget Eqs. (3)–(8) are averaged for 15 days. These terms integrated over subdomains A and B along with tidal energy budget are shown in Fig. 13. In subdomain A, nearly 1.6 GW (59.3%) of 2.7 GW barotropic tidal input is converted into baroclinic energy, out of which 0.98 GW (61.3%) is radiated toward east and west of NBM. About 0.62 GW (38.8%) of converted baroclinic energy and about 1.07 GW of barotropic energy is dissipated locally. In subdomain B, approximately 4.1 GW (77.4%) of 5.3 GW barotropic energy is converted into baroclinic energy, and only about 2.1 GW (51.2%) of this baroclinic energy is radiated east and west of SBM. The Baroclinic energy dissipation (2.02 GW) is roughly twice than that of the barotropic dissipation (1.3 GW). Compared to subdomain A, subdomain B is highly efficient in converting barotropic energy to baroclinic energy. However, in the subdomain B about half of the converted baroclinic energy is dissipated locally, which signifies the energy loss due to higher-mode internal waves. While subdomain A is highly efficient in radiating low-mode baroclinic energy and relatively less significant higher modes. This difference in radiating baroclinic energy between subdomain A and B is due to relatively steeper slopes at SBM (that favors higher modes) and a shallow ridge situated to the east of SBM. In total, the region surrounding SBM is a major hotspot for both generating low-mode internal waves and dissipating higher modes.

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Tidal energy budget along with barotropic energy percentage (bold) and baroclinic energy percentage (italic) for subdomains A and B as shown in Fig. 12.

Citation: Journal of Physical Oceanography 51, 1; 10.1175/JPO-D-19-0182.1

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6. Summary and conclusions

Bidirectional long-living mode-1 and mode-2 ISWs generating from shallow ridges connecting various Islands in Andaman and Nicobar archipelago, India are reported in several studies using MODIS and SAR images. This article investigates the east–west propagating mode-1 and mode-2 ISWs from the steep ridges to the north and south of Batti Malv Island (shallow ridges connecting Car Nicobar–Batti Malv and Batti Malv–Chowra Islands) using SAR observations and a three-dimensional, high-resolution, nonhydrostatic model SUNTANS. To study the generation, evolution and energetics of the ISWs, SUNTANS simulations are performed for 15 days that include a spring–neap tidal cycle. The numerical output of SUNTANS are compared and validated with tide gauge data and SAR images.

All the available SAR images of Envisat ASAR and TerraSAR-X covering the study domain are analyzed for the surface signatures of mode-1 and mode-2 ISWs from 2005 to 2014. Three typical SAR images, including one Envisat ASAR (1553 UTC 12 October 2007) and two TerraSAR-X (2342 UTC 17 March 2014 and 1156 UTC 14 July 2010) are selected to study the most prominent surface signatures of mode-1 and mode-2 ISWs in this region. From Envisat ASAR image, it is observed that the west propagating mode-1 ISW from SBM is generated ahead of NBM indicating the different generation mechanism(s). Three vertical transects S1 (across NBM), S2 and S3 (across SBM) are considered for a detailed model study. Model results are analyzed along these transects to understand the generation mechanism(s) of these waves. Transect S1 is used to study east–west propagating ISWs from NBM, while the transects S2 and S3 are used for west and east propagating ISWs, respectively, across SBM.

In terms of a densimetric Froude number the zonal flow at the crest of NBM along the transect S1 is found to be supercritical for most of the spring–neap tidal cycle. In such conditions, all the possible modes generated over the sill advects in the downstream direction. It is found that the mode-1 wave develops on the lee side of the sill. As the tidal flow slackens, this mode-1 lee wave propagate along the upstream direction and disintegrates into an ISW. The phase speed of mode-1 ISW is about 2.1 m s⁻¹ at the depths of about 1100–1400 m. This process repeats for every ebb and flood tidal flow generating waves in both east and west directions. This type of wave generation is commonly referred as lee wave generation. Unlike the flow over NBM, the zonal flow along the transect S2 at the crest of SBM is subcritical with respect to mode-1 and supercritical for higher modes. Subcritical flow depresses the isopycnals over the crest of sill, and shoals upstream. As the tidal flow relaxes this uplifted isopycnals propagate in the upstream direction as long-elevated mode-1 wave, which steepness and further disintegrates into a mode-1 ISW. However, during a spring tide, supercritical conditions are found near the surface, leading to the formation of a hydraulic jump over the crest of sill. During spring tides, an internal hydraulic jump forms upstream of the sill crest that connects the upstream supercritical flow associated with the shallow surface layer to the subcritical flow closer to the crest. Subsequently, this jump disintegrates into an upstream propagating ISW. Along the transect S3, the tidal flow is also subcritical with respect to mode-1 and supercritical for higher modes. Subcritical flow over the sill creates depressed isopycnals over the crest. These depressed isopycnals leads to a formation of density front and during the decelerating ebb tide this density front advances eastward and disintegrates into a mode-1 ISW. There is a weak hydraulic jump found along S3 during a spring tide. Mode-2 waves along the transects S2 and S3 are generated by lee wave formed in the downstream side of sill. The phase speed of mode-1 and mode-2 ISWs is about 2 and 1.2 m s⁻¹ at depths of 1000–1200 m. Hence, the model results show the evidence of a hydraulically controlled flows developing over SBM [similar to the flows discussed in Farmer and Armi (1999)]. In total, the generation of mode-1 ISWs at NBM is by lee wave mechanism, while the generation of mode-1 ISWs at SBM is due to “upstream influence” and mode-2 ISWs are generated by lee wave mechanism. Table 2 presents the summary on the characteristics of all the ISWs discussed above.

Table 2.

Summary of the characteristics of ISWs generated from the north and south of Batti Malv Island.

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Finally, the time and domain averaged tidal energy budget for input is converted into baroclinic energy from which nearly 0.98 GW is radiated as baroclinic energy. In subdomain B, 4.1 GW of 5.3 GW barotropic energy is converted into baroclinic energy from which 2.1 GW radiates as baroclinic energy and the remaining is dissipated locally. Comparing these two regions reveal that subdomain B is highly efficient in converting barotropic to baroclinic energy while subdomain A is efficient in radiating baroclinic energy. In conclusion, ridge connecting the Car Nicobar and Chowra Islands could provide unique prospects in studying the generation and dissipation of both low- and high-mode internal waves. Generation of several modes of internal waves in close proximity can provide an unique environment for studying the interactions between them.

Data availability statement

Model setup and modified bathymetry can be available from the corresponding author on a reasonable request.