In this section, we develop a box model that represents, in a simplified manner, the relevant processes in the marine nitrogen cycle: N2 fixation, sediment denitrification, water column denitrification, the tropical/subtropical biological pump, and water mass transport and mixing. We use the model to identify the factors that control the steady state isotopic composition of oceanic nitrate. The nitrate supply to the surface is fully consumed in most of the low-latitude ocean, so the 15N/14N of the sinking flux typically reflects that of the subsurface nitrate [Altabet, 1988; Altabet et al., 1999; Thunell et al., 2004].
 A first approximation to the steady state mean ocean nitrate 15N/14N ratio can be understood by considering a homogeneous nitrate pool influenced by N2 fixation and denitrification. At steady state, N2 fixation (F) must balance water column denitrification (W) plus sedimentary (or benthic) denitrification (B). N2 fixation supplies new fixed N from the large reservoir of atmospheric N2 with little fractionation [Carpenter et al., 1997], so we set the 15N source to Ra × F, where Ra is the 15N/14N ratio of atmospheric N2. This must balance the loss of 15N by water column and sedimentary denitrification, (15N/14N) × (αbB + αwW), where the α's are their respective fractionation factors.
 Water column denitrification expresses a strong preference for 14N (we assume αw = 0.975) [Barford et al., 1999], leaving behind 15N-enriched nitrate [Altabet et al., 1999; Liu and Kaplan, 1989; Sigman et al., 2003; Voss et al., 2001; Brandes et al., 1998]. In contrast, sedimentary denitrification is thought to occur with little apparent isotopic fractionation (αb = 1), due to the effects of porewater diffusion [Brandes and Devol, 1997; Sigman and Casciotti, 2001]. The steady state isotopic balance can be solved (with F = W + B) for the 15N/14N ratio
where the fraction of total denitrification occurring in the water column is denoted fw (fw = W/(W + B)) [Brandes and Devol, 2002]. Thus the steady state 15N/14N ratio (or δ15N, defined as δ15N = ((15N/14N)sample/(15N/14N)reference − 1) × 1000‰, where the reference is atmospheric N2) of a homogeneous ocean with N sinks of different fractionations depends only on the relative magnitude of those sinks (fw) and their fractionation factors. If a given N2 fixation source is balanced predominantly by water column denitrification (fw 1), 14N will be preferentially removed from the ocean, giving a high δ15N for mean ocean nitrate. On the other hand, if most of the denitrification occurs in sediments (fw 0) with little fractionation, the δ15N ratio of the nitrate loss will be closer to that of mean ocean nitrate, so the δ15N of oceanic nitrate will be low (approaching the input ratio, Ra).
 Brandes and Devol  have compiled a N isotope budget for the Holocene ocean. They estimate that additional N sources from atmospheric deposition and rivers together are isotopically similar to N2 fixation, while the additional sink due to organic N burial is isotopically similar to mean ocean nitrate (and in this way similar to sediment denitrification). Therefore these additional smaller budget terms do not significantly alter the conclusion based on N2 fixation and denitrification alone that fw is the dominant control on mean ocean nitrate δ15N.
 However, this homogeneous model of the mean δ15N of ocean nitrate leaves out an important consideration. In the real ocean, water column denitrification is localized, so that nitrate loss occurs in nitrate pools with a δ15N that is different from that of the mean ocean nitrate. As water column denitrification consumes nitrate in the suboxic water mass, it raises the δ15N of the residual nitrate in the suboxic zone. However, because the nitrate δ15N of a mixture of two water parcels is biased toward the parcel with higher nitrate concentration, the depletion of nitrate in the suboxic zone leads to a smaller isotopic influence of its residual nitrate on the ocean's mean δ15N (Figure 3). In the limit of complete nitrate consumption within the suboxic zones of the ocean water column, the suboxic water mass acts only to dilute the nitrate concentration of the rest of the ocean, with no impact on its δ15N. In this limit, there would be no expression of the isotopic fractionation by water column denitrification [Thunell et al., 2004]. Instead, it would be isotopically similar to sediment denitrification, and the sole dependence of mean ocean nitrate δ15N on fw would no longer hold. We refer to this as “the dilution effect.” The dilution effect also results from decreasing total oceanic nitrate. As the mean ocean nitrate concentration decreases, a given amount of denitrification consumes a greater fraction of the nitrate in the suboxic zone, enriching the suboxic nitrate pool but decreasing its influence on the mean ocean δ15N (Figure 3b).
Figure 3. Illustration of the dilution effect for (a) increased denitrification and (b) decreased N inventory (or mean [NO3−]). As denitrification increases, the δ15N of the denitrification zone (open circle) increases due to Rayleigh fractionation (dark curve). The mixture of denitrified water (open circle) and the rest of the ocean in its initial state (solid circle) falls along a mixing line (light curve) in which the δ15N of the mixed water mass (star) is biased toward the end-member with high nitrate concentration. The decrease in mean δ15N is a measure of the dilution effect (arrow). This effect is produced by either an increase in denitrification (Figure 3a) or a decrease in total nitrate (Figure 3b). Note that higher δ15N values in the denitrification zone are extreme in these illustrative cases, and fall outside the plotted range.
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 From these considerations it can be expected that increasing total denitrification (B + W) or decreasing the total nitrate reservoir in a steady state ocean with constant fw will increase the δ15N of the suboxic zone while decreasing the δ15N of the mean ocean nitrate pool. That is, for a given fw, increased fractional nitrate consumption in the suboxic zone leads to greater nitrate 15N-enrichment in that zone while reducing the 15N-enrichment of mean ocean nitrate (Figure 3). In order to make reliable use of sediment records from both oxic and suboxic regions of the ocean, we therefore require a model that accounts for this spatial structure.
2.2. Model Structure
 In order to account for these potential spatial differences in nitrate δ15N, we construct a box model that divides the ocean horizontally into a suboxic water column and the rest of the ocean (Figure 4). We take the suboxic water column to be the areal extent of waters with annual mean O2 concentrations below 10 μM in the NOAA NESDIS Atlas [Conkright et al., 1994]. This definition includes the eastern tropical North and South Pacific and the Arabian Sea, and comprises ∼2.5% of the global ocean area. The resolution of high-latitude processes has been shown to be a central concern for many paleoceanographic questions [e.g., Sarmiento and Toggweiler, 1984]. However, the focus of this study is the interaction between the suboxic water column and the adjacent waters. This is essentially a low-latitude phenomenon; therefore, we choose not to include the structure of high-latitude water masses in this box model.
Figure 4. Map of O2 concentration (μmol/kg) at the depth of the climatological O2 minimum [Conkright et al., 1994]. Insets: area-averaged nutrient profiles (NO3−, PO43−, and N* = [NO3−] − 16*[PO43−] + 2.9) from the suboxic water column (dark curves) and the rest of the ocean (light curves). The threshold for suboxia is taken to be O2 < 10 μmol/kg so as to include the eastern tropical North and South Pacific and the Arabian Sea.
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 Both horizontal regions are divided into four vertical levels (Figure 5): a surface box (0–100 m) in which nitrate uptake and organic matter export occurs, and three lower boxes in which the organic N flux is remineralized with depth according to a power law (Martin et al. , exponent = 0.858). In the suboxic water column, remineralization is deeper (exponent = 0.40), consistent with the findings of Van Mooy et al. . The depth of the lower thermocline box (275–800 m) is chosen to coincide with the depth of the climatological suboxic water volume (2 × 1014 m3). The deep ocean is taken to be from 800 to 4000 m.
Figure 5. Schematic of model geometry and fluxes. The suboxic water column (∼3% of ocean area) communicates with the rest of the ocean through horizontal mixing and an advective overturning flux (up in the suboxic water column, down in the oxic water column, and from oxic to suboxic below 800 m). Both water columns are divided into four levels (depths indicated at left) which are connected by vertical mixing and the upwelling/downwelling component of overturning. Water mass fluxes, given in Sverdrups (106 m3 s−1), are determined by matching the PO4 distribution (see text).
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 The horizontal regions and depth levels are connected through lateral and vertical mixing and advection designed to represent the key features of ocean circulation that cause the major suboxic zones. Upwelling, which brings high-nutrient waters to the surface in the suboxic water column, stimulates high biological productivity and creates a high oxidant demand below the mixed layer. In addition, thermocline waters in these regions are generally poorly ventilated. These two features are incorporated in the model by adding to the mixing terms an upwelling flux in the suboxic water column (compensated by downwelling in the rest of the ocean), and by removing lateral mixing between the suboxic box and the lower thermocline of the rest of the ocean. The resulting pattern of mixing and transport is shown in Figure 5.
 Model water mass fluxes are calculated by fitting the steady state phosphate concentrations in the model to annual mean values [Conkright et al., 1994]. We require vertical mixing rates to be the same in the suboxic water column and the rest of the ocean, on a per area basis. This leaves seven fluxes to solve for: three vertical mixing fluxes, three horizontal mixing fluxes, and the upwelling/downwelling flux. The least squares solution is found using the seven linearly independent box conservation equations (the eighth is not independent since it can be derived from total mass conservation).
 Export production is specified in the inverse solution. On the basis of GCM results, we specify a global organic matter export of 10 GtC/yr, with ∼10% occurring in the suboxic water column (about 4 times the areal mean export flux). The resulting vertical and horizontal mixing are high in near-surface waters, decreasing with depth (Figure 5). As with most box models, the magnitude of subsurface vertical mixing required to sustain a realistic export flux is quite high [Matsumoto et al., 2002]. We could have chosen instead to require more realistic (lower) mixing and transport fluxes, at the expense of a realistic export flux. This choice changes only the absolute magnitudes of mass fluxes, not their relative magnitudes. We chose to have a realistic export flux because this will determine the absolute values of the N source/sink terms, which we wish to compare with other estimates.
 On the basis of these mass fluxes, the forward model can be written as a set of equations for N (nitrate), P (phosphate), and 15N (15N-labelled nitrate). In the ith box,
where “Prod” is the uptake of each component by biological production, “Remin” is the release of each component by organic matter remineralization, and “Circ” is the redistribution of each component due to ocean circulation. The sources and sinks of fixed N are denoted F (N2 fixation), W (water column denitrification), and B (sediment, or “benthic,” denitrification). The corresponding 15N terms are multiplied by the 15N/14N ratio (R) of the source pool, as well as the isotope fractionation factors, α. We assume that N2 fixation adds nitrate with the 15N/14N ratio of air (Ra) without fractionation (αf = 1.0) [Carpenter et al., 1997]. In each box, denitrification removes nitrate with the local isotope ratio (Ri) modified by no isotope fractionation in the sediments (αb = 1) [Brandes and Devol, 1997] and 25‰ in the water column (αw = 0.975) [Barford et al., 1999]. In the forward model, export production is calculated by restoring surface phosphate concentration toward annual mean observations. Production and remineralization fluxes of nitrate are calculated as 16 times phosphate fluxes.
 We neglect smaller terms in the N budget, such as atmospheric and riverine sources and the sink due to burial in the sediments. These terms are not clearly isotopically distinct from the primary terms in the budget and are very poorly understood [Brandes and Devol, 2002]. We seek a minimal model here, avoiding complexities that would cloud the important processes.
 Water column denitrification in the model is confined to the suboxic box. Sediment denitrification is distributed between the suboxic water column and the rest of the ocean with equal rates per unit area. In the absence of detailed information about the vertical distribution of sedimentary denitrification, we apply 30% in the seafloor adjacent the top boxes [Middelburg et al., 1996], 30% in the seafloor of both thermocline boxes, and the remaining 10% in the deep ocean. N2 fixation flux follows export and remineralization terms, with 10% occurring in the suboxic water column. However, half of the newly fixed N is released in the surface ocean, while the remaining sinking flux is distributed with depth according to the Martin curve.
 The magnitude of the denitrification terms is tuned to fit two observed characteristics of the nitrate distribution. First, water column denitrification is chosen to match the N* gradient (N* = [NO3−] − 16*[PO43−] + 2.9) in the suboxic water column. The magnitude of water column denitrification required is 70 TgN/yr, which is in the range of observational estimates [Codispoti and Christensen, 1985; Deutsch et al., 2001]. The agreement between the model and observed values for this process suggests that total N supply to the suboxic box (via transport and remineralization) is roughly correct.
 Water column denitrification of this magnitude, balanced entirely by N2 fixation, would yield a mean ocean isotopic composition of >16‰, well above the observed value of 5‰ [Sigman et al., 2000] (although not 25‰, because of the dilution effect). In order to match the observed isotopic range, sediment denitrification of roughly 190 TgN/yr is required. This value lies between the older estimates based on in situ rate measurements [Codispoti and Christensen, 1985] and the upwardly revised estimates based on sediment modeling [Middelburg et al., 1996] and on global isotopic mass balance considerations [Brandes and Devol, 2002]. Our estimate is methodologically most similar to that of Brandes and Devol . However, it is quantitatively lower, partially because we use a slightly smaller isotopic fractionation for sediment denitrification (α = 1.000 versus 0.9985), but mostly because our model includes the dilution effect described above.
 Modeled nitrate concentration, phosphate concentration, and N* agree with climatological values to within 5% on average (Figure 6). The suboxic water column has higher nutrient levels because the influx of nutrient-rich subsurface water is balanced by an outflow of relatively depleted surface water. Water column denitrification produces low N* in the suboxic zone, with strong gradients toward the surface. In contrast, the oxic water column N* is relatively uniform with depth in both the data and model.
Figure 6. Profiles of model (circles) versus observed (stars) nitrate concentration, phosphate concentration, N*, and nitrate δ15N in the suboxic water column (dark symbols) and the rest of the ocean (light symbols). Deep ocean properties are plotted shallower then their true average depth.
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 We compare model δ15N for the suboxic and oxic water columns with data averaged over several profiles from the Eastern Tropical North Pacific [Sigman et al., 2003] and the Sargasso Sea [Karl et al., 2002; A. N. Knapp et al., The N Isotopic composition of dissolved organic nitrogen and nitrate at the Bermuda Atlantic Time-Series Study site, submitted to Global Biogeochemical Cycles, 2004], respectively. The model captures much of the observed spatial variability in nitrate δ15N in the suboxic water column, similar to N* but with opposite sign. However, model nitrate δ15N is consistently lower than the data by ∼1.0‰ in this region. The upper thermocline of the suboxic zone in the model is too high in N* and too low in nitrate δ15N, probably because denitrification zones reach this depth in the real ocean. Because lateral mixing between the suboxic surface and the global surface is low, most of the surface-thermocline gradient of both N* and δ15N in the suboxic water column is driven by N2 fixation, as has been proposed on the basis of regional isotopic mass balance [Brandes et al., 1998].
 The near-surface waters of the Sargasso Sea exhibit strong vertical δ15N gradients, and these are not well captured by the model. The elevated δ15N observed at the surface occurs because of isotopic fractionation by nitrate assimilation [Sigman et al., 1999], a process not included in the model. This is relatively insignificant since nitrate is completely consumed in this region. More importantly, the low δ15N of nitrate in the upper thermocline is lower in the Sargasso Sea data than in the model. This discrepancy may be due to the fact that the Sargasso Sea near Bermuda, where the nitrate δ15N profiles were collected, deviates from the mean thermocline chemistry of the low- and mid-latitude ocean. The thermocline of the Sargasso Sea, which may be an area of intense N2 fixation, is characterized by both high N* and low nitrate δ15N [Karl et al., 2002], and the global model does not resolve either of these features.
2.3. Deglacial Experiments
 We perform a series of simulations to investigate which climate-N cycle interactions most strongly determine the character of the nitrate isotopic evolution of the mean and suboxic water columns. We aim to identify possible glacial/interglacial scenarios that could explain the basic aspects of the glacial/interglacial records of sediment δ15N, in particular, the remarkable deglacial δ15N maximum observed in many sites underlying or relatively near denitrifying regions. We include two direct forcings of climate change on the N cycle: a deglacial increase in water column denitrification caused by expanded interglacial suboxia [Keigwin and Jones, 1990; Kennett and Ingram, 1995] and a deglacial increase in sediment denitrification linked quantitatively to the observed timing of sea level rise [Christensen, 1994; Bard et al., 1990]. We also include feedback processes that damp perturbations to the N budget.
 Each flux can be written as the product of its modern value and a time-dependent scaling factor that parameterizes the forcings and feedbacks,
The m-subscripted terms are the modern values of water column denitrification, sediment denitrification, and N2 fixation, determined in the previous section. The first scaling terms, denoted by S, are constants that convert the modern fluxes to glacial initial values (see below). The second scaling terms represent N cycle feedbacks. The fraction (N(t) − No)/No is the change in the N inventory, N(t), relative to its initial glacial value, No. The fractional change in N inventory causes changes in fluxes through sensitivity factors α, β and γ. The functions L(t) and H(t) represent the forcing of sedimentary and water column denitrification by sea level rise and suboxia changes, respectively (L for “level”, H for “hydrographic”). These functions take on values between 0 and 1, so that φ and λ are the maximum fractional changes in sedimentary and water column denitrification driven by these processes.
 While the feedbacks included here are intended to represent those discussed in the introduction, the feedback mechanisms themselves are not explicitly modeled. For example, we are not modeling O2 concentrations or the competitive advantage of N2 fixers. Instead, changes in N fluxes are linked directly to the N inventory such that the net effect of all the implicit steps in each feedback is represented by a single sensitivity parameter. The feedback terms are constructed so that a 1% increase in global N inventory causes an α % increase in W, a β % increase in B, and a γ % decrease in F.
 The fraction of glacial/interglacial sea level rise, L(t), ranges from 0 at the Last Glacial Maximum to 1 at the present time. We fit (by eye) a hyperbolic tangent to sea level reconstruction data from Bard et al.  and Hanebuth et al.  (Figure 7). Sea level rise begins slowly at the onset of deglaciation 20,000 years ago, reaching only 10% of its glacial/interglacial change after ∼4000 years. In contrast, we assume that the deglacial onset of suboxia is rapid, so that a step function, H(t), is appropriate (Figure 7). Many proxy records of suboxia from the eastern tropical Pacific date its onset at ∼15 kya, with a brief return to oxic conditions between 12.9 and 11.6 kya in some areas during the Younger-Dryas [Keigwin and Jones, 1990; Zheng et al., 2000]. The Younger-Dryas interval inferred from the sea level reconstruction [Bard et al., 1990] is coincident with that inferred from sediment laminations [Keigwin and Jones, 1990], suggesting that the timing of our two climate forcings are consistent. Because the Younger-Dryas appears to influence the sediment δ15N in some locations (see Figure 1), we tested the impact of including a brief return to oxic conditions during the Younger-Dryas in the forcing, H(t). While the experiments with more detailed forcing were better able to reproduce short-term features associated with the Younger-Dryas in some sediment cores, the basic trends in glacial/Holocene δ15N were not changed. Since our primary purpose is to elucidate basic processes, rather than to simulate detailed sedimentary records, we represent the onset of interglacial suboxia as a single step change in the results that follow.
Figure 7. Water column denitrification and sea level rise forcing functions, H(t) (dark curve) and L(t) (light curve), as a fraction of their glacial-to-interglacial change (see text). Relative sea level rise is computed by fitting a hyperbolic tangent to the reconstruction data of Bard et al.  and Hanebuth et al. . Water column denitrification increase is assumed to occur at the onset of laminations (∼15 kya) in the eastern tropical Pacific [Keigwin and Jones, 1990].
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 The magnitude of forced deglacial denitrification increases in the sediments and water column are unknown. In order to determine a reasonable magnitude of water column denitrification forcing (i.e., λ), we examined the sensitivity of the deglacial increase in δ15N in the model's suboxic surface waters to the initial increase in water column denitrification. We find that δ15N increases are most realistic when water column denitrification increases by ∼60% upon deglaciation. We therefore adopt a λ value of 0.6. For the impact of sea level rise on sedimentary denitrification, we examine two different forcing magnitudes. In a first set of experiments, we assume that forced denitrification increases in the water column and the sediments are of the same magnitude, so that λ = φ = 0.6. In a second set of experiments, we assume that the climate forcing of sedimentary denitrification is reduced relative to water column forcing. If glacial sediment denitrification was reduced only on the continental shelves, whose area was 75% lower during glacial periods [Christensen et al., 1987], and if ∼40% of modern sediment denitrification occurs on continental shelves [Middelburg et al., 1996], then total sedimentary denitrification would have increased by only ∼30% due to sea level rise. We therefore secondarily examine climate forcings in which φ = 0.3 and λ = 0.6.
 Finally, we note that the time evolution of each flux and the N inventory will depend on all of the feedback and forcing parameters. Specifically, although the climate forcing terms are prescribed, the evolution of each flux will depend on all of the feedbacks as well as the climate forcing. For any complete set of parameter values (α, β, γ, λ, and φ), a unique set of initial values for N inventory, sources, and sinks can be found for which the model will reach modern values 20,000 years after the onset of deglaciation. In other words, the glacial conditions (i.e., the scaling constants Sf, Sb, and Sw) for any model scenario are determined by the values of the forcing and feedback parameters. We perform simulations of the glacial/interglacial transition for all combinations of sensitivity parameters with values 0 (no feedback), 1 (weak feedback), and 9 (strong feedback).