Deep-Sea Research Part II - ePrints@IISc

Deep–Sea Research II 172 (2020) 104684

Available online 6 November 2019

0967-0645/© 2019 Elsevier Ltd. All rights reserved.

Seasonality of surface stirring by geostrophic flows in the Bay of Bengal

Nihar Paul

^a,*

, Jai Sukhatme

^a,b

aCentre for Atmospheric and Oceanic Sciences, Indian Institute of Science, Bangalore, India

bDivecha Centre for Climate Change, Indian Institute of Science, Bangalore, India

A R T I C L E I N F O Keywords:

Stirring in the Bay of Bengal

Physical and spectral characterization of geostrophic flows

Finite time and finite size lyapunov exponents Relative dispersion

Eddy diffusivity

A B S T R A C T

Stirring of passive tracers in the Bay of Bengal driven by altimetry derived daily geostrophic surface currents, is studied on subseasonal timescales. To begin with, Hovm¨oller plots, wavenumber-frequency diagrams and power spectra confirm the multiscale nature of the flow. Advection of latitudinal and longitudinal bands highlights the chaotic nature of stirring in the Bay via repeated straining and filamentation of the tracer field. An immediate finding is that stirring is local, i.e. of the scale of the eddies, and does not span the entire basin. Further, stirring rates are enhanced along the coast of the Bay and are relatively higher in the pre- and post-monsoonal seasons.

Indeed, Finite Time Lyapunov Exponent (FTLE) and Finite Size Lyapunov Exponent (FSLE) maps in all the seasons are patchy with minima scattered through the interior of the Bay. Further, these maps bring out a seasonal cycle wherein rapid stirring progressively moves from the northern to southern Bay during pre- and post-monsoonal periods, respectively. The non-uniform stirring of the Bay is reflected in long tailed probability density functions of FTLEs, that become more stretched for longer time intervals. Quantitatively, advection for a week shows the mean FTLE lies between 0.13±0.07 day⁻¹, while extremes reach almost 0.6 day⁻¹. Averaged over the Bay, Relative dispersion initially grows exponentially, followed by a power-law at scales between approximately 100 and 250 km, which finally transitions to an eddy-diffusive regime. Quantitatively, below 250 km, a scale dependent diffusion coefficient is extracted that behaves as a power-law with cluster size, while above 250 km, eddy-diffusivities range from 6×10³ − 1.6×10⁴ m²s⁻¹ in different regions of the Bay. These estimates provide a useful guide for resolution dependent diffusivities in numerical models that hope to properly represent surface stirring in the Bay.

1. Introduction

Advective transport and mixing are important aspects of geophysical flows (Weiss and Provenzale, 2007). For example, in the oceans, surface stirring plays a key role in determining the fate of chemical and bio- logical fields. Stirring affects dispersal from localized sources (Lekien et al., 2005; Mezi´c et al., 2010; Olascoaga and Haller, 2012), as well the spatial and statistical distribution of large scale inhomogeneities (Abraham, 1998). In the present work, we investigate stirring on intraseasonal timescales via geostrophic surface currents in the Bay of Bengal (BoB), which is a triangular basin spanning 5^∘-22^∘N and 80^∘-100^∘E, centered around 15^∘N. Understanding the stirring and dispersal of fields on the surface of the Bay is important in a variety of contexts; for example, to quantify the influence of fluid advection on biological activity such as plankton blooms (Vinayachandran and Mathew, 2003), possible pathways and timescales in the dispersal of

contaminants, such as the February 2017 oil spill near Chennai,¹ and of fresh water. Specifically, the Bay receives a large inflow of river water in the post-monsoonal period from Ganga-Brahmaputra-Meghna (GBM) and Irrawaddy river basins (Papa et al., 2012; Chaitanya et al., 2014).

This inflow and its transport is clearly seen in measurements of salinity as well as in numerical simulations (Sengupta et al., 2006, 2016; Akhil et al., 2014), and its dispersal and mixing plays an important role in the surface salinity budget of the Bay (Howden and Murtugudde, 2001).

In addition to these practical implications, from a fluid dynamical perspective the surface flow in the Bay is marked by seasonal features that include an intensified western boundary current that flows north- ward (equatorward) before (after) the summer monsoon and relatively steady eddies off the eastern coast of India during the monsoon itself (Vinayachandran et al., 1996; Schott and McCreary, 2001). Further, altimetry data suggest that the Bay has significant intraseasonal vari- ability in surface geostrophic currents (Cheng et al., 2013). These

* Corresponding author.

E-mail address: nihar@iisc.ac.in (N. Paul).

1 https://www.imarest.org/themarineprofessional/item/3104-20-tonnes-of-oil-spilled-in-bay-of-bengal Contents lists available at ScienceDirect

Deep-Sea Research Part II

journal homepage: http://www.elsevier.com/locate/dsr2

https://doi.org/10.1016/j.dsr2.2019.104684

Received 29 December 2018; Received in revised form 30 October 2019; Accepted 31 October 2019

features make for an interesting dynamical setting in which to assess and quantify the stirring of passive tracers. In particular, given the changes in the circulation, it is important to understand the seasonal behavior of stirring in various regions of the BoB.

As we are dealing with surface currents, in this two-dimensional (2D) setting, it is well known that even relatively simple time dependent flows can lead to complicated tracer patterns (Aref, 1984). An up-to-date review of chaotic mixing, and more broadly mixing implied by multi- scale flows, can be found in Aref et al. (2017) and an overview of ap- plications and methods in an oceanographic context are detailed in Prants et al. (2017). It should be kept in mind that stirring refers to the advection of passive tracers and mixing is sometimes used inter- changeably with stirring (though in a dynamical systems, mixing has a precise definition, Ottino, 1989). Neither of these terms implies ho- mogenization. Along with conventional measures such as relative dispersion, regular and anomalous diffusion, Lagrangian tools from dynamical systems including Finite Time Lyapunov Exponents (FTLEs) and Finite Size Lyapunov Exponents (FSLEs) have proved useful in quantifying stirring via simple advective protocols as well as multiscale turbulent flows. Examples in an oceanic context include, uncovering the mechanisms underlying inter-gyre mixing (Poje and Haller, 1999;

Wiggins, 2005), transport across jets (Samelson, 1992), localized stirring in ocean basins such as the Adriatic (Lacorata et al., 2001), Tasman (Waugh et al., 2005) and Mediterranean seas (García-Olivares et al., 2007), to elucidate the non-uniform nature of surface mixing (Waugh and Abraham, 2008), identification of mesoscale eddies (Beron-Vera et al., 2008) and relative dispersion (Corrado et al., 2017), in the global oceans. In addition to quantifying rates of stirring, these tools also allow for the identification of kinematic transport barriers, i.e. transient structures that inhibit global mixing in geophysical flows (Boffetta et al., 2001; Lehahn et al., 2007a).

In general, the coupling of advective stirring with sinks and sources (other than diffusion, and possibly leading to homogenization) has also proved useful in a geophysical context. For example, advection-linear damping (Chertkov, 1998), to understand the patchiness of biogeo- chemical tracers in the ocean with differing lifetimes (Mahadevan and Campbell, 2002), advection-reaction-diffusion (Neufeld et al., 2002), to elucidate the formation and sustenance of plankton blooms (Hernan- dez-Garcia and Lopez, 2004) and advection-condensation (Pierre- humbert et al., 2005), to probe the large-scale distribution of water vapor in the atmosphere (Sukhatme and Young, 2011). In fact, advection-reaction models have also found use in extraplanetary sce- narios, such as understanding seasonal variations in atmospheric composition (Lebonnois et al., 2001). In these situations, apart from aforementioned measures of stirring, the role of advection and its coupling with various sinks is usually understood in a statistical sense.

Here, a central piece of information that allows one to make robust es- timates is the probability density function of FTLEs. Estimates of these density functions in the Bay have not been previously attempted and would be important in applications such as those mentioned above.

Similarly, does the dispersal of passive tracers suggest an eddy-diffusive parameterization at large scales? If so, estimates of this possibly scale dependent diffusivity would be of use in numerical models that hope to properly represent surface stirring in the BoB.

The outline of this manuscript is as follows. In Section 2, we describe the data used in this study. Section 3 provides an overview of the intraseasonal geostrophic flow from physical and spectral points of view. This provides a detailed view of the flow that is responsible for the intraseasonal advection of the passive tracers. Beginning with the advection of latitudinal and longitudinal stripes, in Section 4, we describe and compare FTLEs and FSLEs, and elucidate the behavior of stirring in various parts of the Bay in different seasons. In Section 5, we describe the statistical measures such as probability density functions of FTLEs, regional scale dependent eddy diffusivities or Finite Size Diffu- sion Coefficients (FSDCs) and Relative Dispersion (RD) in the BoB. A discussion and summary of results concludes the paper.

2. Data

The Ssalto Duacs/gridded multimission altimeter products, which are a part of AVISO project, distributed by the Copernicus Marine and Environment Monitoring Service (CMEMS) (http://marine.copernicus.

eu/), have been used in this study. Specifically, we use MADT-H-UV (Maps of Absolute Dynamic Topography & Absolute Geostrophic Ve- locities) dataset with a spatial resolution of 0.25^∘×0.25^∘, covering 5^∘N to 24^∘N and 80^∘E to 100^∘E on a Cartesian grid. To verify the robustness of our results we have used data of multiple years, specifically, 2008-2017.

Based on its general circulation (Potemra et al., 1991), the natural partition provided by the Andaman Islands and the inflow of fresh water that distinguishes its northern and southern portions, mixing is expected to be spatially heterogeneous and therefore the Bay has been divided into four different regions in this study — Northern Bay (NB → 86.625^∘E-92.625^∘E, 15.125^∘N-20.125^∘N), Central Bay (CB → 81.125^∘E-92.125^∘E, 10.125^∘N-14.875^∘N), Southern Bay (SB → 82.125^∘E-93.125^∘E, 5.125^∘N-9.875^∘N) and Andaman Sea (AS → 93.375^∘E-97.375^∘E, 6.125^∘N-15.125^∘N). The geographic locations of these boxes are shown in Fig. 1.

3. Subseasonal circulation in Bay of Bengal

Given its importance to the regional climate, the seasonal circulation of the BoB has been studied quite extensively (Potemra et al., 1991;

Vinayachandran et al., 1996; Schott and McCreary, 2001). Here, we provide an overview of the subseasonal surface geostrophic circulation that is responsible for advection in the subsequent passive tracer mixing experiments.

3.1. Physical space characterization

On intraseasonal timescales, in the context of geostrophic surface flows, the co-existence of Rossby waves and (nonlinear) eddies is well established throughout the world’s oceans (Chelton et al., 2007). In the BoB too, a rich interplay of Rossby waves and eddies has been noted in numerous studies (Shankar et al., 1996; Babu et al., 2003; Kurien et al., 2010; Chen et al., 2012; Nuncio and Kumar, 2012). It has been suggested that the west coast of the Bay is a critical region for eddy-mean flow interaction with significant baroclinic instability and the production of eddies (Kurien et al., 2010; Chen et al., 2012; Nuncio and Kumar, 2012).

Further, eddies in the BoB have different propagation characteristics in different regions. For example, eddies in the northern and southern Bay

Fig. 1.The Bay of Bengal and subregions [Northern Bay (NB), Central Bay (CB), Southern Bay (SB) and Andaman Sea (AS)].

(i.e. north of 15^∘N and south of 10^∘N) propagate in southwestward and northwestward directions, respectively. While those in the central Bay tend to move in along the same latitude in a westward direction (Chen et al., 2012).

We begin with Hovmoller plots with respect to longitude of the zonal ¨ geostrophic velocity for all the 10 years (meridional velocities show essentially similar features). The climatological mean has been removed from the data to construct anomalies which are subjected to a 10-120 day band pass Lanczos filter. The longitude spans 80.375^∘E to 97.625^∘E, with the latitude taken along the cross section between Irra- waddy river basin and Andaman Islands in the BoB, specifically, aver- aged over 14.125^∘N-15.125^∘N. As seen in Fig. 2, there are westward tilted coherent structures present throughout the year. The tilt of these structures is fairly steady across the different years and yield a westward phase speed of approximately 8-12 cm s⁻¹, consistent with prior esti- mates from the central BoB (Killworth et al., 1997; Chelton et al., 2007;

Chen et al., 2012). Interestingly, these coherent westward tilting struc- tures, or wave packets, also appear to exhibit episodic eastward migration with systematic positive (orange) and negative (blue) anom- alies. Specific examples are: October to December between 80.375^∘E to 90^∘E in 2008, March to June and July to November between 80.375^∘E to 94^∘E in 2009, February to April and June to August between 80.375^∘E to 90^∘E in 2010, January to April between 80.375^∘E to 86^∘E in 2011, February to June and September to December between 80.375^∘E to 90^∘E in 2012, October to December between 80.375^∘E to 90^∘E in 2013, March to July 84^∘E to 90^∘E in 2014, January to May between 80.375^∘E-88^∘E in

2016, January to May and September to November between 80.375^∘E to 90^∘E and June to November between 88^∘E to 94^∘E in 2017. These suggest a small eastward group velocity associated with these wave packets.

Another common feature which we observe is the presence of shorter time scale (<30 days) eddies or waves around 94^∘E-97.625^∘E, i.e. near the Irrawaddy river basin, through out the year (Mukherjee et al., 2019).

We have also constructed Hovm¨oller plots of the zonal velocity with respect to latitude (not shown), these showed a fairly mixed behavior with sporadic instances of northward and southward tilts. By and large, the movement of disturbances in the East-West direction in the Bay is more pronounced and systematic as compared to North-South migration.

3.2. Spectral space characterization

To estimate the length scale and time period of these disturbances we move to spectral space and compute wavenumber-frequency diagrams of the filtered 10-120 day zonal geostrophic velocity anomaly. The re- sults presented are averaged over a latitude band from 14.125^∘N to 15.125^∘N (for consistency with the Hovm¨oller plots). As seen in Fig. 3 (a), we mostly observe westward movement with strong spectral peaks spread over |kx| ≈1− 6 for kx<0, with time periods ranging from about 50 to 120 days. The three solid curves shown in Fig. 3(a) corre- spond to the dispersion relations of baroclinic Rossby modes given by,

Fig. 2. Panels (a)-(j) show Hovm¨oller diagrams of the filtered 10-120 day zonal geostrophic velocity from 2008 to 2017, respectively. The plots are averaged over 14.125^∘N-15.125^∘N from a longitude of 80.375^∘E to 97.625^∘E in the BoB.

σ= − βkx

k²_x+k²_y+¹

L²_R

,for​kx,ky=1,2,3.... (1) Here, kx and ky are the zonal and meridional wavenumber and LR is the Rossby radius of deformation. The wavenumbers have been normalized by the length of the BoB which is equal to 1870 km at 14.125^∘-15.125^∘N. The typical value of LR, lies between 60-140 km over the latitudes spanned by the Bay (Chelton et al., 1998). In fact, following Stammer (1997), we estimated an “eddy length scale” from zero crossing of autocorrelation function of the SSH. This estimate (not shown) is larger than LR (Chen et al., 2012), and varies approximately from 55-200 km over 22^∘N and 6^∘N. The peaks in the left half of Fig. 3(a) correspond to westward phase speeds, and they appear to be guided by Equation (1). This suggests that these disturbances with westward phase speed are related to baroclinic Rossby waves. Further, the position of the maxima in these diagrams show that there is significant power at scales just smaller than the local deformation scale (maxima of the dispersion curves), thus supporting episodic eastward group velocities noted in the longitudinal Hovmoller plots in Fig. 2. ¨

We also computed a dispersion diagram of frequency vs meridional wavenumber, and this is shown in Fig. 3(b). The spectra are averaged over a longitudinal band of 89.625^∘E-90.625^∘E (where we have the largest latitudinal extent of the Bay). The meridional wavenumber has been normalized by the longitudinal cross-section of BoB which is approximately 1842 km at 90^∘E. We note that the most of the power is distributed between ⃒

⃒ky⃒

⃒≈1− 7, with a maximum between 1<⃒

⃒ky⃒

⃒<

6. Here too, largest power is seen in the temporal band of 50-120 days.

Finally, power spectra of the surface currents are examined to see the energy associated with the different length and time scales in the geostrophic velocity field. Specifically, we compute kinetic energy spectra and average over 11.125^∘N to 12.125^∘N which resolves largest zonal scales in the Bay. In a similar manner, spectra for meridional wavenumber have been averaged over 89.675^∘E to 90.675^∘E. The slopes of the spectra, as seen Fig. 4(a), are close to a − 3 power-law between approximately 100 km and 250 km in both the zonal and meridional directions. This is in agreement with a forward enstrophy cascading regime of surface geostrophic currents from about 200 km to 100 km in the global open oceans (Khatri et al., 2018). The temporal spectrum is estimated by calculating the spectrum at each grid point and then averaging. This is done for each year and the results are presented for all the years in log-log and variance preserving form in Fig. 4(b), (c), respectively. The variance preserving form, in agreement with the

wavenumber-frequency plots, shows relatively isolated peaks at scales longer than 30 days. Further, the log-log plot shows signs of an approximate power-law, with a − 3 exponent, for time scales that range from about 10-30 days. The match in temporal and spatial spectral ex- ponents suggests that Taylor’s hypothesis appears to hold at these scales;

specifically, using an annual average speed of 0.2 m s⁻¹, a spatial scale of 200 km maps to approximately 10 days (Ferrari and Wunsch, 2010;

Suhas and Sukhatme, 2015). Note that, given the approximate ten day frequency of repeat satellite passes in AVISO, periods below two weeks suffer from aliasing, and the spectra at these smaller timescales are likely to be unreliable (Arbic et al., 2012). Taken together, the spatial and temporal spectra suggest an uninterrupted distribution of power across mesoscales and from intraseasonal to weekly time scales, respectively.

4. Mixing of passive tracers

As demonstrated, geostrophic flow in the BoB has a multiscale character in both space and time. Specifically, there is strong seasonal dependence of the surface flow (for example, current disruption and reversal, Vinayachandran et al., 1996; Schott and McCreary, 2001), along with significant subseasonal variability consisting of geostrophi- cally balanced disturbances that exhibit predominantly westward phase speeds and align quite well with the dispersion curves for baroclinic Rossby waves. All together, the flow provides a rich environment for the mixing of passive fields. As it happens, aperiodic Rossby waves by themselves have been examined in detail as idealized models of chaotic mixing (Pierrehumbert, 1991). The principal tool used in these mixing calculations is the Lagrangian advection of parcels. This is done using a Runge Kutta fourth order (RK4) scheme. Further, given that the data are on a fixed grid, the flow has been interpolated by a bilinear interpolation scheme.

As our measures of mixing are Lagrangian in nature, it is worth identifying the limits imposed on the calculations due to the resolution of the altimeter data. In general, as discussed by Bartello (2000), coarse velocity field data perform satisfactorily with regard to passive advec- tion when their kinetic energy spectra follow a − 3 power-law. Of course, finer scale data can improve quantification of Lagrangian measures (Beron-Vera, 2010). As seen in Fig. 4(a), currents in the Bay follow this scaling over a range of approximately 100 to 250 km. But, the situation is complicated at smaller scales. Specifically, at the ocean’s surface, scales below approximately 100 km (depending on the region in consideration) have a significant contribution from the divergent Fig. 3. Wavenumber-frequency plots of 10-120 day filtered zonal geostrophic velocity. Panels (a) and (b) show σ− Kx, averaged over 14.125^∘N-15.125^∘N and σ− Ky, averaged over 89.625^∘E-90.625^∘E, respectively. Solid black lines are theoretical linear Rossby wave dispersion curves.

component of the flow (Capet et al., 2008; Bühler et al., 2014, 2017; Qiu et al., 2017), and spectra at these smaller scales also show signs of flattening to shallower power-laws (Callies and Ferrari, 2013). In fact, recent ship-track derived near surface kinetic energy spectra follow an approximate − 5/3 scaling from 10 to 80 km in the Bay (Sukhatme et al., 2019). Not only is the altimeter derived geostrophic data attenuated by filtering below scales of approximately 100 km (Arbic et al., 2013; Dufau et al., 2016), it is unlikely to be a dominant contributor to the actual surface currents. Thus, caution must be exercised in interpreting Lagrangian measures computed from purely geostrophic data below a scale of approximately 100 km.

4.1. A sign of chaos

We begin with a simple numerical experiment where the BoB is divided into zonal and longitudinal sections. Each band is approximately 1^∘ wide and is identified by a separate color as seen in the first panels of Fig. 5(a) and Fig. 5(b). Parcels in each band retain their color and are advected for about ten weeks. Such experiments give a basic feel for the mixing processes at work in a flow (Pierrehumbert and Yang, 1993).

Snapshots of the scalar field are shown every two weeks in Fig. 5. We notice that within the first two weeks, the bands are distorted and form extended filaments and whorls. In fact, the boundaries between the different colors evolve towards highly complicated contours. This pro- cess continues with the repeated wrapping around of progressively thinner scalar filaments, i.e. a cascade of tracer variance to small scales.

Indeed, the geometric picture that emerges is that of chaotic advection of a tracer field (Ottino, 1989), and by week ten it is quite difficult to distinguish between the latitudinal and longitudinal bands. Irrespective of the time in the year whenever such a ten week advection is performed, the chaotic nature of mixing is found to be robust. With this qualitative picture in mind, we now proceed to more quantitative measures of mixing in the BoB.

4.2. Finite time and Finite Size Lyapunov Exponents

A fundamental quantitative characteristic of a chaotic flow is its Lyapunov exponent, this is defined as the exponential rate of separation, averaged over infinite time, of fluid parcels with an initial infinitesimal separation (Benettin et al., 1980). For practical problems, the limit of Fig. 4. Kinetic energy spectra. Panel (a) shows zonal and meridional wavenumber spectra averaged over 11.125^∘N-12.125^∘N and 89.625^∘E-90.625^∘E, respectively (through the widest longitudinal and latitudinal extent of the Central Bay). Panels (b) and (c) contain temporal spectra vs f (frequency) in log-log scale and variance preserving form, respectively. Spectra are estimated at each grid point and then averaged for a year.

time tending to ∞ is not feasible, and the notion is generalized to the so-called Finite Time Lyapunov Exponents (FTLE, λτ),

λτ(x0) = 1

|t− t0|log‖δx(t) ‖

‖δx(t0) ‖. (2)

where, τ is defined as t− t0. For non-autonomous flows, the FTLE is essentially a measure of integrated strain along a parcel’s trajectory. We calculate the right Cauchy-Green Lagrange tensor C^t_t0(x0) associated

with the flow map F_t^t₀(x0), which is defined as, C^t_t

0(x0) = (

∇F_t^t

0(x0) )_T

∇F^t_t

0(x0) (3)

F^t_t0(x0)denotes the position of a parcel at time t, advected by the flow from an initial time and position (t0,x0). C^t_t0(x0)is symmetric and posi- tive definite, its eigenvalues (λ’s) and eigenvectors (ξ’s) can be written as,

Fig. 5. The stirring of latitudinal (first six panels) and longitudinal (last six panels) bands by geostrophic currents from October to December 2012. Snapshots are shown every two weeks for ten week long advection.

C_t^t₀(x0) =λiξ_i, 0<λ1≤λ2,i=1,2; (4) The gradient of the flow map ∇F^tt₀(x0)is computed using an auxiliary grid about the reference point (Onu et al., 2015), and can be written as,

∇F^t_t0(x0) ≈

(α11 α12

α21 α22

)

, (5)

where, αi,j≡xi

(t;t0,x0+δxj

)− xi

(t;t0,x0− δxj

)

2⃒

⃒δxj

⃒⃒ . (6)

Finally, the largest FTLE (Haller, 2002; Haller and Sapsis, 2011) associated with the trajectory x(t,t0,x0)over the time interval [t0, t] is then defined as,

λτ(x0) = 1

|t− t0|log

( ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅

λmax

[ C^t_t0(x0)

√ ])

. (7)

For a flow with multiple scales, such as that in the BoB, the Finite Size Lyapunov Exponent (FSLE) is also a convenient measure to quantify chaos and detect coherent structures (Aurell et al., 1996; Artale et al., 1997). FSLEs are defined via,

Λ(x0,δ0,r) = logr

τ(x0;δ0,r). (8)

This is similar to Equation (2), but here time τ is calculated for a trajectory at a distance δ0 from a reference trajectory at x0 to reach a separation of distance rδ₀, r being defined as the growth factor (d’Ovidio et al., 2004; Lehahn et al., 2007b). Following García-Olivares et al.

(2007), we have also computed the mean FSLE (〈Λ〉), where a set of N tracers with random initial distribution having standard deviation σ are

Fig. 6.Panels (a), (b), (c) and (d) show absolute dynamic topography (with geostrophic quivers), FTLE- 28 days (λ_28d), FSLE (Λ) and mean FSLE (〈Λ〉) computed on

321∘

×₃₂^1∘ grid resolution starting from 18/10/2009. The growth factor r for FSLE (Λ, 〈Λ〉) computation was taken as 32 i.e. δ0 (σ0) and δ_f (σ_f) are ₃₂^1∘ and 1^∘, respectively.

followed in time as they are transported by the velocity field. Defining σ(t)as,

σ (

t )

=〈|xi(t) − 〈xi(t)〉|²〉^1/2, (9) where,

〈xi

( t

)

〉≡〈 {

xi

( t

)

:i=1,2, ...N }

〉=1 N

∑^N

i=1

xi

( t

)

(10) We set the initial size of the cluster σ0 according to Equations (9) and (10), and measure the time τ(x₀;σ0,r)as it takes the growth from σ0 to σ_f =rσ₀ where r is the growth factor and σ_f being the largest scale under consideration (the sub-basin scale). The mean FSLE parameter as a function of the scale is then obtained from,

〈Λ (

x0,σ0,r )

〉= logr

τ(x0,σ0,r), (11)

which is not sensitive when variation in r is close to 1+.

The two measures of mixing, FTLEs and FSLEs, are in broad sense complementary, and their features are compared in detail by Karrasch and Haller (2013). Here, specifically for the BoB we consider a particular example that highlights the features captured by these two metrics. For two-dimensional geostrophic flows, forward FTLEs (FSLEs) indicate the presence of stable manifolds. These manifolds are lines that represent stretching rate where tracers diverge in a flow. Fig. 6 shows a compar- ison of FTLE for τ=28 days (λ_28d), FSLE (Λ) and mean FSLE (〈Λ〉) computed on a ₃₂^1∘×₃₂^1∘ grid for 8^∘×8^∘ box in the central BoB. FSLEs were computed by considering the initial scale as being the size of the computational grid, and the final size to be comparable to the typical size of the eddies in this region. For computing the mean FSLE the number of parcels were taken as 50, which proves to be sufficient to get a steady value at ₃₂^1∘×₃₂^1∘ grid resolution. From the dynamic topography and currents shown in Fig. 6(a), we notice two prominent anicyclonic eddies in the domain. All the three metrics (λ28d, Λ, 〈Λ〉) capture these

Fig. 7. Panels (a) and (d) show the seasonal mean FTLE-14 days (λ14d) and EKE maps for FMA, (b) and (e) for JJAS, (c) and (f) for the OND seasons, respectively.

Panels (g), (h) and (i) show the corresponding seasonal mean FSLE (〈Λ〉) maps for FMA, JJAS and OND seasons, respectively.

features and their filament-like fine scale structure highlights the intri- cate spatial distribution of stretching rates in and around the eddies shown in Fig. 6(b), (c), (d) respectively. But in comparison with the FTLE (λ_28d), the FSLEs provides a more fine scale structure of these stretching rates in physical space.

To characterize stirring in the Bay, we compute FTLEs for τ =7, 14, 21 and 28 days on a grid of ¹₈^∘×¹₈^∘ for all days of years of data (2008- 2017). For seasonal estimates, FTLEs are averaged for Feb-Mar-Apr (FMA), Jun-Jul-Aug-Sep (JJAS) and Oct-Nov-Dec (OND). Fig. 7(a), (b), (c) show these seasonal averages of FTLE for τ=14 days (λ14d). The seasonal behavior for other τ values is qualitatively similar and is not shown here. In the pre-monsoon season (FMA), the FTLEs after are high along the western boundary and in the northern Bay with values of approximately 0.1-0.16 day⁻¹ and 0.11-0.15 day⁻¹, respectively. The mouth of Ganga-Brahmaputra-Meghna (GBM), Irrawaddy river basins and southwest portion of the Bay (near the Andaman Islands) also show relatively high stirring rates ranging between 0.9-0.11 day⁻¹. The pic- ture changes in the monsoon period (JJAS) with higher FTLEs on the western boundary moving systematically equatorward and emergence of a localized pocket of high FTLEs off the east coast of Sri Lanka, this region of enhanced stirring during the monsoon is due to the formation of the so-called Sri Lankan dome (Vinayachandran and Yamagata, 1998). Further, the central region of the Bay shows relatively smaller FTLEs, suggesting of kinematic barriers to basin wide mixing between

the eastern and western boundaries of the Bay. In the monsoon season, we also notice enhancement in mixing in the southern BoB just above the northwestern edge of Sumatra coast. Finally, in OND, the southern Bay lights up with relatively high FTLEs. A close inspection of the maps in OND shows signs of a ring of relatively high FTLEs between 82^∘E-86^∘E and 7^∘N-11^∘N, a feature sometimes referred to as the BoB dome, the western rim of the gyre is equatorward connected to EICC (Vinaya- chandran and Yamagata, 1998). The western boundary of the BoB shows very high stirring rates in this season owing to a strong reversed EICC.

The northwest boundary is now relatively quiescent with low FTLE values. Overall, the coastal regions always show stronger mixing than the interior of the Bay. In addition, we see a seasonal cycle with rapid stirring progressively moving from the northern to southern Bay, from pre-monsoonal to post-monsoonal periods, respectively.

The eddy kinetic energy (EKE) in the Bay (where eddies are defined as deviation from a climatological mean of 10 years) in each season is shown Fig. 7(d), (e), (f). Clearly, the EKE maps align quite closely with that of the FTLEs in each season. This has been noted on global (Waugh and Abraham, 2008), as well more local scales (Waugh et al., 2005), in the world’s oceans. In a similar manner, following Equation (11), using number of parcels in the cluster as 100, daily mean FSLEs (〈Λ〉) are computed on a grid of ¹₈^∘×¹₈^∘ for all days of 10 years and averaged for FMA, JJAS and OND seasons. The growth factor has been set as r=8, such that final standard deviation becomes σ_f =1^∘, i.e. the typical scale

Fig. 8.Histogram of FTLEs (with different increments) over the Bay; (a) whole year (b) in different seasons. Panel (c) shows the daily time series of mean FTLE (normalized by mean λ of all the ten years) through the year. Panel (d) shows a fit to the FTLE distributions (normalized by mean λ) for different τ by a Weibull distribution.

of eddies. These seasonal mean FSLE values are shown Fig. 7(g), (h), (i) for FMA, JJAS and OND, respectively and are in good agreement with seasonal mean FTLE and EKE maps. In fact, the lower stirring rates in the central Bay, especially in the monsoon, are more starkly evident in the FSLE maps.

5. Statistical measures of stirring

At a fundamental level, the action of a chaotic flow on the transport or evolution of passive fields requires knowledge of the probability density function of the FTLEs (Balkovsky and Fouxon, 1999; Sukhatme and Pierrehumbert, 2002; Fereday and Haynes, 2004; Haynes and Vanneste, 2004). The histogram of FTLEs is shown in Fig. 8(a). This distribution reflects the spatially non-uniform nature of stirring induced by the geostrophic currents in the Bay. Indeed, the right tail represents regions that experience rapid stirring, while the left tail is indicative of slow rates of stirring. Also, the shape of the distribution changes for different τ. Specifically, the histogram becomes taller (mean decreases) with more of a stretched exponential with lighter tail for larger τ. In essence, the non-uniformity of mixing is highlighted more starkly for longer time intervals. Even though the regions of strong mixing vary from season to season, as seen in Fig. 8(b), there is not much of a sea- sonal dependence in the FTLE distributions, i.e. the Bay is non-uniformly chaotic throughout the year. This can also be seen in Fig. 8(c), which shows a daily time series of the mean FTLE, normalized by mean λ of all ten years over the Bay, through the year. The average stirring rate is quite uniform, though there is a marginal drop below 4 % during the JJAS season. Keeping in mind that coastal regions show higher rates of mixing, the absence of a strong boundary current during JJAS is likely responsible for this drop in the normalized mean FTLE.

For consistency with other parts of the world’s oceans (Waugh and Abraham, 2008; Waugh et al., 2012), we note that a Weibull distribution accurately fits the FTLE histogram normalized by the mean FTLE. The specific expression plotted in Fig. 8(d) is,

PW(λ) =b a

(λ a

)_b−1 exp

(

− λ^b a^b )

, (12)

with a=1.13 and b=1.92.

5.1. Finite size diffusion coefficient

Another advantage of FSLE is that, from a statistical point of view, they allow for an estimation of a scale dependent diffusion coefficient (Lacorata et al., 2001). This Finite Size Diffusion Coefficient (FSDC, D) is defined as,

D(σ) =σ²〈Λ〉(σ). (13)

As the cluster size grows, if it is driven by chaos at very small scales (≪lu) and by (eddy) diffusion at very large scales (≫ lu), then 〈Λ〉(σ)has the following asymptotic behavior,

〈Λ〉(σ) =

⎧

⎨

⎩

〈Λ〉_max, ifσ≪lu, D

σ^{2, if}σ≫lu. (14)

Though, for real world data, there usually exist other intermediate regimes for 〈Λ〉(σ)(Corrado et al., 2017).

Practically, a cluster with 10⁴ parcels was released in the geographical regions shown in Fig. 1, and these were advected for approximately 200 days. Here, we have chosen a cluster whose initial standard deviation is ¹₈^∘ and followed the parcels until the standard de- viation increases by a factor of 1.2. The size of the initial cluster and the number of parcels in the cluster have been chosen in such a way the mean FSLE computed becomes independent of all initial standard de- viation. This experiment was repeated for all days of each year, and an average was taken over all the years.

The behavior of the mean FSLE with cluster size can be seen in Fig. 9 (a). The smoothly interpolated flow results in an exponential separation, i.e. a relatively flat mean FSLE, for scales below approximately 100 km.

For the NB an eddy-diffusive regime is observed for scales greater than 200-250 km (via a − 2 scaling of the mean FSLE in Fig. 9(a)). The SB region shows signs of an eddy-diffusive regime at the largest scales, i.e.

near 300 km). At intermediate scales, i.e. 100 to 300 km, the mean FSLE transitions between these two end regimes. The Andaman Sea region shows a diffusive regime emerging at smaller scales between 140 and 200 km. This overall picture is consistent with the calculated “eddy length scale” and the local chaotic mixing seen in Fig. 5. The CB is distinct in that the mean FSLE never really transitions to an eddy- diffusive regime. It is possible that, especially in the intermediate regime, finer data may yield different power-laws as seen by Corrado et al. (2017) in other parts of the world’s oceans. An estimation of a scale-dependent diffusion coefficient, the FSDC, is presented in Fig. 9(b).

It is interesting to note that the FSDC scales as a power-law with cluster size (exponent of 1.78); this is potentially a useful reference which provides a resolution-diffusivity relation for use in models. In particular, the FDSC scaling can be employed in numerical ocean models that hope to faithfully represent surface mixing in the Bay. At large scales, for the SB, NB and Andaman Sea region, the eddy diffusivity is approximately 1.6×10⁴, 8×10³ and 6×10³ m²s⁻¹, respectively. These numbers are higher (by a factor of 2) than the minimum Osborn-Cox eddy diffusivity estimates as determined by Abernathey and Marshall (2013) (see in particular the Bay of Bengal in their global Fig. 5), but are of the same order as diffusivities estimated via mean FSLE in other parts of the world’s oceans (Corrado et al., 2017).

5.2. Relative dispersion

Another commonly used mixing diagnostic is relative dispersion (RD). This is defined as (LaCasce, 2010),

〈R²(t)〉= 1 N(N− 1)

∑

i≠j

R²_ij(t), (15)

where, 〈R²(t)〉 is the mean relative dispersion of an ensemble of N pairs having the same initial separation with random orientation. RD, along with notions such as normal or anomalous diffusion, helps in quanti- fying the homogenization of tracers (see, for example, Weeks et al., 1996; LaCasce and Speer, 1999; Sukhatme, 2005, for ideal, oceanic and atmospheric applications, respectively). Recently, Waugh et al. (2012) have compared RD and FTLEs, and highlighted how they explore different aspects of the mixing process. Quite starkly, even in the case when the FTLE distribution collapses to a single point, the RD exhibits a wide spread (Waugh et al., 2012).

In practice, we advected 10³ pairs of parcels at a given grid location for 120 days starting on the first day of each month in all the ten years.

Two suites of experiments were conducted with pairs in a circle that are initially 13.75 and 27.5 km apart, respectively. The evolution of RD with these two initial separations is shown in Fig. 10(a) and (b). At small scales, the advecting flow is a smoothly interpolated geostrophic flow, hence the observed exponential separation is not surprising. As dis- cussed earlier, at small scales (below approximately 100 km, depending on the region in consideration), the surface ocean flow appears to have a relatively shallow kinetic energy spectrum (Callies and Ferrari, 2013), and a significant divergent component (Capet et al., 2008; Bühler et al., 2014; Qiu et al., 2017). In fact, pair separation on the order of a few km, estimated using drifter data from the Gulf of Mexico, appears to conform with the classic Richardson 〈R²〉∼t³ prescription (Poje et al., 2017).

Indeed, as the currents scale with an approximate − 5/3 power law over approximately 10-80 km in the Bay (Sukhatme et al., 2019), it would be interesting to ascertain the behavior of RD in this range via high reso- lution surface ocean models.

At scales between 100 and 250 km, we observe that the RD is

reasonably well approximated by a power-law growth in time. While we anticipate exponential separation in an enstrophy cascading regime, it should be remembered that this is under the assumption of an inertial range with a constant enstrophy flux (Lin, 1972). Computation of the enstrophy flux using altimeter data does show a dominant forward enstrophy regime at these scales, but the enstrophy flux is not constant, and is actually scale dependent (Khatri et al., 2018). Therefore, the power-law growth of RD is not inconsistent with a forward enstrophy transfer regime. Finally, at scales larger than 250 km, the growth of RD slows down and tends to 〈R²〉∼t, indicative of eddy diffusive behavior.

This is seen in the emergent flat portion of a compensated RD plot (for the 27.5 km initial separation case) beyond approximately 100 days of evolution as shown in Fig. 10(c). Note that, consistent with the FSDC estimates, the eddy diffusivity (averaged over the Bay) as seen from this RD calculation is also of the order of 10⁴ m²s⁻¹.

As with the FTLEs, the distribution of RD at a given time is of importance in assessing the non-uniformity of mixing. The distribution of RD (actually, the square root of RD, normalized by its rms value), at various times, when the mean square root RD lies between 100 and 250 km is shown in Fig. 10(d) and (e) for 13.75 and 27.5 km initial separations, respectively. Quite clearly, the distributions are not Gaussian, rather they show signs of a log-normal behavior.

6. Summary and discussion

Using surface geostrophic currents derived from altimetry data, we studied the stirring of passive fields along the surface of the Bay of Bengal. In particular, our primary focus was on stirring that takes place on intraseasonal time scales. To begin with, we examined the flow by means of Hovm¨oller plots and wavenumber-frequency diagrams. It was seen that the geostrophic currents in the Bay are dominated by westward progressing disturbances that have temporal scales between 50 and 120 days. In fact, the power in these systems aligned well with the theoret- ical dispersion curves for linear baroclinic Rossby waves. Interestingly, some of these have length scales that are smaller than the local defor- mation scale, and show an eastward group velocity which was noted in the Hovm¨oller plots. Temporal and spatial power spectra were seen to follow approximate power-laws (− 3 scaling, from 100-250 km and 10- 30 days, respectively) and suggested an uninterrupted distribution of power across these length and subseasonal time scales.

The advection of latitudinal and longitudinal bands by the multiscale surface geostrophic flow immediately hinted at the presence of chaotic mixing. In particular, the repeated folding and filamentation of stripes brought forth a complicated geometry to the mixing process, and by the end of approximately 10 weeks, it was difficult to distinguish between the two initial conditions. In addition, stirring was not basin wide but was seen to be restricted to the scale of eddies. To quantify stirring rates, different metrics, namely FTLEs and FSLEs, were then employed and compared. These bring out various facets of advective stirring, and at the same time provide consistency in that the results are not artifacts of any one diagnostic tool.

Maps of the FTLEs and FSLEs suggested an equatorward movement of regions of enhanced stirring from pre-monsoonal to post-monsoonal periods. In each season, stirring along the coasts is vigorous while the central Bay had relatively low FTLE values, suggestive of the presence of kinematic barriers that were consistent with the eddy scale of stirring noted above. The low stirring rates in the central Bay are especially stark in the monsoon season. Specific seasonal features, such as enhanced stirring at the mouth of GBM, the appearance of the Sri Lankan dome in the monsoon and the BoB dome in winter, were captured by high FTLE and mean FSLE pockets. Also, overall, the spatial maps of seasonal mean FTLE and mean FSLE were in tune with those of EKE, with high EKE aligned with regions of rapid stirring.

Variations in FTLEs are known to be important in statistical char- acterization of passive fields; here, the non-uniform nature of surface stirring in the Bay was manifested in probability density functions of FTLEs that had long tails and their shape, and as in other parts of the world’s oceans, was captured by a Weibull distribution. In terms of a domain average, the FTLE for a week’s increment was approximately 0.13 day⁻¹, while the spread captured by the histogram ranged up to 0.6 day⁻¹. In addition, with longer time increments, the distribution of FTLEs became taller (and smaller mean), but with progressively more stretched exponential with lighter tails. Thus, the non-uniformity of mixing was further highlighted at longer time intervals. This quantita- tive estimate of the distribution of FTLEs is potentially useful in devel- oping kinematic models, such as for the dispersal of pollutants in the Bay.

The relative dispersion (RD) of parcels provided a complimentary view to the FTLEs. Below 100 km, the smoothly interpolated nature of the data results in pair separation that was exponential in time. From Fig. 9.Panels (a) and (b) show FSLE and eddy diffusivity as a function of initial standard deviation with expansion factor r=1.2 for the Northern Bay, Central Bay, Southern Bay and Andaman Sea.

Fig. 10. Panels (a) and (b) show the RD with time for initial separations of 13.75 and 27.5 km, respectively. Panel (c) shows the compensated RD (by t⁻¹) as a function of time for the initial separation of 27.5 km. Panels (d) and (e) show histograms of the square root of RD (denoted by r, normalized by its rms value) at different days when the mean RD is between 100 and 250 km.

100 to 250 km, i.e. the RD followed a power-law in time, which is consistent a forward enstrophy transfer regime, but with a variable enstrophy flux. At larger scales, the pair separation took on an eddy- diffusive growth, i.e. 〈R²〉∼t, with a diffusivity of the order of 10⁴ m²s⁻¹ when averaged over the BoB.

Apart from seasonal features, mean FSLEs provided a quantitative measure of the scale up to which tracers experience chaotic stirring.

Averaging the growth of clusters for all days across 10 years, the mean FSLE was relatively constant (up to 100 km), transitions (from 100 to 250 km) and then enters an eddy-diffusive regime (from 250-300 km).

The Andaman Sea was seen to enter an eddy-diffusive regime at rela- tively smaller scales (between 140-200 km) while the central Bay showed no signs of this transition. The large scale eddy-diffusivity, with each eddy acting independently and inducing a random walk, estimated from the mean FSLE plots was about 1.6×10⁴ m²s⁻¹ in the Southern Bay, 8×10³ m²s⁻¹ in the Northern Bay and approximately 6× 10³ m²s⁻¹ in the Andaman Sea region. Interestingly, before the emergence of an eddy-diffusive regime, the finite size diffusion coefficient showed a similar power-law behavior in all regions of the Bay (exponent of 1.78 with cluster size). These estimates can be used as a guideline for ocean models, being run at a given resolution, that hope to capture the stirring at the surface of the Bay in an accurate manner.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to thank Copernicus Marine and Environment Monitoring Service (CMEMS) for distributing AVISO’s Ssalto/Duacs altimeter product MADT-H-UV dataset. The authors would like to ex- press their gratitude to Prof. Debasis Sengupta, Prof. Anirban Guha and Prof. Amit Tandon for helpful discussions. The authors would also like to thank the two anonymous reviewers for their comments which helped to improve the manuscript. We also thank the Divecha Centre for Climate Change, IISc for research support. JS would like to acknowledge support from the University Grants Commission (UGC) for funding via 6-3/2018 under the 4th cycle of the Indo-Israel joint research program. The au- thors also acknowledge support from the Ministry of Earth Sciences under the National Monsoon Mission.

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