[29] The inventories of bomb-^{14}C are calculated following the procedures of *Broecker et al.* [1985]. The area (*A*) between the measured Δ^{14}C depth profile and the reconstructed pre-nuclear Δ^{14}C depth profile (Δ^{14}C^{0}) is calculated as

Here, z^{0} is the depth at which the difference between Δ^{14}C and Δ^{14}C^{0} becomes negligible (<1‰). The mean penetration depth (*Z*) of bomb-^{14}C is obtained as

Here, ΔΔ^{14}C = (Δ^{14}C − Δ^{14}C^{0})_{surface} is the measured excess of surface water Δ^{14}C from its pre-nuclear value. The total column-inventory of bomb-^{14}C is given by

Here ΣCO_{2}is the mean DIC of the water column up to which bomb-^{14}C has penetrated. The proportionality constant ‘*k*’ takes into account density of seawater (average ∼1025 kg m^{−3}), Avogadro's number (6.023 × 10^{23} atoms mol^{−1}), and the ^{14}C/^{12}C atom ratio of the NBS Oxalic Acid-I standard (1.176 × 10^{12} [*Karlen et al.*, 1968]). The inventory of bomb-^{14}C is expressed in units of atoms of ^{14}C per square unit area of the seawater column. Following this procedure, the estimate uncertainty of the calculated inventory is about ±10% [*Peng et al.*, 1998]. To compute the bomb-^{14}C inventories, the measured depth profiles of Δ^{14}C, silicate and ΣCO_{2}values were interpolated to every 1 m. Pre-nuclear Δ^{14}C profiles (Δ^{14}C^{0}) were then reconstructed from the measured values of silicate, using the empirical relationship of *Broecker et al.* [1995], Δ^{14}C^{0} (‰) = −70 − silicate (*μ*mol kg^{−1}). The reconstructed pre-nuclear Δ^{14}C of surface was forced to match with the values obtained from pre-bomb shells. The excess (measured Δ^{14}C − Δ^{14}C^{0}) are then integrated until the difference becomes negligible and the inventory of bomb-^{14}C is obtained using (B1) and (B3).

[30] The air-sea CO_{2} exchange rates were calculated following the procedure outlined by *Stuiver* [1980]The total amount of bomb-^{14}C penetrated in ocean is proportional to the air-sea exchange rate of CO_{2} and the time integrated gradient of Δ^{14}C between the atmosphere (Δ^{14}C_{atm}) and the ocean surface mixed layer (Δ^{14}C_{mix}). Assuming the mean steady state difference in Δ^{14}C values between atmosphere and oceanic mixed layer is −60‰ for the northern Indian Ocean, the total amount of bomb-^{14}C (*Q*_{14}), transferred per unit of ocean surface over time *t*(in years) is related to the air-sea exchange rate of CO_{2} (*F*_{12}) as

The constant term on the right hand side of (B4), takes into account the isotopic fractionation factors for ocean-atmosphere transfer and normalization of^{14}C activity to *δ*^{13}C value of −25‰ [*Stuiver*, 1980]. Here *Q*_{14} is expressed in mol m^{−2} and *F*_{12} in mol m^{−2} yr^{−1}. For the onset of input of bomb-^{14}C to the oceans, 1954 is taken as the initial year (*t* = 0). The values of the integrals for Δ^{14}C_{atm} and Δ^{14}C_{mix} are obtained from Δ^{14}C measurements in the atmosphere and in corals. The tropospheric Δ^{14}C from 1954 and 1968 has been adopted from the reported measurements from the tropical regions [*Nydal and Lövseth*, 1996; *Hua and Barbetti*, 2004; *Chakraborty et al.*, 2008], with peak Δ^{14}C of 700‰ in the year 1965. From 1968 onward, an exponentially decreasing atmospheric Δ^{14}C trend with e-folding time of 17 years has been adopted, fixing the Δ^{14}C for the years 1980 and 1999 at 265‰ and 88‰ respectively. There are too few coral Δ^{14}C measurements in the open northern Indian Ocean, to reconstruct the mixed layer Δ^{14}C history for different regions. Between the years 1954 and 1973, the value of the integral of Δ^{14}C_{mix} is taken as 1400 [*Stuiver*, 1980]. From 1973 onward, a linear decrease of surface ocean Δ^{14}C has been assumed, from a value of 140‰ in 1973 to 50‰ in 2000, to match with the observed values from the GEOSECS measurements during 1977–1978 [*Stuiver and Östlund*, 1983] and during late 1990s [*Dutta et al.*, 2006]. *Q*_{14} are calculated from the bomb ^{14}C inventories for different stations, to determine the air-sea CO_{2} exchange rates (*F*_{12}) using (B4). Based on the uncertainty in bomb-^{14}C inventory (15%) and the integral of the atmosphere-ocean Δ^{14}C gradient (5%), the uncertainty of ^{14}C-based exchange rates estimated to ∼20%.*Druffel and Griffin* [2008] showed that for surface samples, the total uncertainty of a DIC Δ^{14}C value at a given site over a several week period is approximately two times the reported uncertainty (∼7‰). Thus, depending on the application, post-bomb Δ^{14}C data should consider this short-term variability of surface ocean Δ^{14}C values and factor this into their analysis as it might affect the estimates of air-sea CO_{2} exchange rates.

[31] Once the rate of air-sea CO_{2} exchange (*F*_{12}) is known, the net transfer rate (*F*) of atmospheric CO_{2} across the ocean surface can be determined from the difference of the partial pressure of CO_{2} (Δ*p*CO_{2}) in the surface ocean and that in the atmosphere [*Wanninkhof*, 1992]. The net transfer rate of CO_{2} (*F*) is given by

In this case, the sign of *F* (or direction of flux) is *positive upwards.* Thus the net flux of CO_{2} will be from sea to air, in regions where Δ*p*CO_{2} is positive. This condition is common in most tropical oceanic areas due to equatorial upwelling, which brings deeper CO_{2}rich waters to the surface. The exchange rates calculated from bomb-^{14}C inventories are the long-term averaged values. Thus, if average ocean-atmosphere*p*CO_{2} gradient (Δ*p*CO_{2}) over a given oceanic region is known, it is possible to compute the net transfer rate of CO_{2} from the exchange rates. Based on the uncertainties in the ^{14}C-based exchange rates (20%) and in the estimates of Δ*p*CO_{2} (10%) the maximum uncertainty on *F* is expected to be within 30%.