Physical biogeochemical interactions and oxygen. drawdown in upwelling systems

Transcription

1 Physical biogeochemical interactions and oxygen drawdown in upwelling systems C. S. Harrison 1, B. Hales 1, S. Siedlecki 2, and R. M. Samelson 1 1 CEOAS, Oregon State University, Corvallis, OR 2 JISAO, University of Washington, Seattle, WA September 15, 214 DRAFT 1

2 Abstract A maximally simple, coupled physical-biogeochemical box model is used to examine carbon cycling and oxygen drawdown in eastern boundary upwelling systems. In this model, oxygen drawdown is controlled by surface particle production and sinking, and is generally greater for weaker upwelling, wider shelf-widths, and larger sinking rates. Retention efficiencies computed from the solutions indicate that up to 5% or more of upwelled nutrient can be retained over the shelf and contribute to oxygen drawdown. A relatively complete characterization of the model system s susceptibility to hypoxia can be obtained by combining two diagnostics: the steady-state oxygen depletion, which is the net drawdown achieved after a (possibly long) transient adjustment to the onset of constant upwelling winds, and an exponential timescale describing the initial transient adjustment. If the sinking rate s is comparable to or larger than the sum of the respiration rate r and the upwelling volume-renewal rate λ, then the system s steady state will generally be hypoxic, provided that the net particle production rate pn r s λ >, where p is the particle production rate when nutrient levels are saturated and N 1 is a constant that depends on the source-water nutrient level. For smaller sinking rates, the system can never reach hypoxia, even for arbitrarily long upwelling seasons. If s r +λ and, in addition, the saturated net particle production rate η λ = p r s λ is greater than roughly (r + λ )/4, hypoxia can be reached within the upwelling season even with relatively small net production rates η λ.3 d 1. Fluctuations in the amplitude of the upwelling forcing can systematically reduce the potential for oxygen depletion below these estimates, especially when sinking rates are large or forcing is strong. 1

3 1 Introduction The coupled dynamics of particulate organic carbon, nutrients and dissolved oxygen (hereafter C, N, and O 2 respectively) in coastal marine systems that result from non-local source waters interacting with the ecosystem biogeochemistry, can lead to environmental stresses like hypoxia (Grantham ea 24; Hales ea 26; ) and enhanced ocean corrosivity (Barton, 212; Harris, 213; Feely et al 28; Feely et al 21.). These conditions can be entirely natural in origin, although human influences including eutrophication (mississippi plume, Baltic Sea references), hydrological cycle alterations (Salisbury, Cai; mississippi plume), rising atmospheric CO 2 (Doney et al.), or climate-change driven shifts in wind forcing and ocean circulation (Bakun; Schwing) are likely contributing factors at some level. Mechanistic understanding of the processes driving variability in coastal systems can appear daunting because of the relatively high-frequency, large amplitude variability there relative to the open ocean, and the added complication of the close-coupling of the terrestrial and benthic ecosystems. Eastern boundary upwelling systems are of broader interest because of their high productivity and vibrant fisheries (Chavez), and of particular interest because their pre-disposition to hypoxia (Hales et al., 26) and the effects of ocean acidification (Feely, 28; Fassbender, 2xx; Harris, 213). Understanding is further complicated by the ephemeral notions of hypoxia and a variety of conventions for expressing the concentrations of O 2 in aqueous media. Hypoxia has been historically defined as any water that holds less than 1% of the saturation content of dissolved O 2 predicted from thermodynamic equilibrium of the aqueous medium with the.2 2

4 atm partial pressure of atmospheric oxygen ( 1s of mmol O 2 m 3 ; Garcia solubility) down to the very low values associated with the transition from dominance of microbial and chemical processes by aerobic processes to dominance by subaerobic processes ( 1 mmol O 2 m 3 ). While these are rigorous definitions, they are of limited application for ocean waters and the organisms that live within them. The variability of ocean-water temperatures means that there is wide variability in the saturation content that is unrelated to biogeochemical processes, and the biological activity within ocean waters means that there is wide deviation from O 2 saturation equilibrium, both positive and negative, even ecosystems with vibrant higher trophic-level populations. The important degree of hypoxia is that at which populations of interest become negatively affected; however, there seems to be little convergence to a single concentration that defines such hypoxia, as different organisms are affected differently and sensitivity is related to environmental conditions such as temperature (Deutsch synthesis; Deutsch et al., pers. com. 214; or, abstract from 213 PICES meeting, or in prep.). Additionally, organisms appear able to optimize for local conditions, such that regardless of the background O 2 levels, any negative deviation in O 2 concentration represents hypoxia (Seibel, 211, J. Exp. Biol.). Attempts to bridge the thermodynamically-rigorous but biologically meaningless extreme ranges with the nebulous but meaningful impact levels have resulted in one-significant-figure estimates of concentration ranges where impacts are likely to a large range of organisms have included 3 mg O 2 L 1 ( 9 mmol m 3 ; NOAA hypoxia page) and 2 mg O 2 L 1 ( 6 mmol m 3 ; NOAA hypoxia page). These have been accorded more significance by virtue of conversion to equally unsavory units (e.g. 2 mg L ml L 1 ; Grantham et al., 24, Science). 3

5 Upwelling systems have two key features that lead to their predilection for these environmental challenges: source waters by definition come from depth, where air-sea ventilation and photosynthesis do not occur and therefore cannot counteract the O 2 -consumptive and CO 2 -productive process of respiration such that source waters are often near hypoxic and corrosive thresholds; and the upwelling of associated nutrients into illuminated surface waters leads to rapid phytoplankton growth fueling a food web that ultimately exports metabolic burden to near-bottom waters of the coastal systems. While coastal ecosystems have complicated and multi-trophic food webs requiring complex simulations, the fundamental question of the sensitivity of the system to hypoxia can be shown to depend on two factors: 1) the flushing of the system with upwelled source waters; and 2) the retention of the metabolic burden resulting from upwelling-fueled primary productivity in local deep waters. The system s operation is bounded by two extreme hypothetical cases: the abiotic, or no-retention case, where little of the potential metabolic burden of upwelling is captured in the local system, and the total-retention state, where it is all focused on the shelf ecosystem. The character of upwelling systems thus suggests that there are two primary factors leading to de-oxygenation: onshore transport of low-o 2 subsurface waters from the ocean interior, and retention of the metabolic potential of upwelled nutrients. The first of these is a trivial matter of flushing near-bottom coastal environments with upwelled source waters and has the obvious asymptotic condition of equivalence with offshore concentrations. The flushing hypothesis has been the most implicated in many previous studies of non-eutrophied 4

6 systems (GGrantham et al 24; Chan et al 28; Pierce et al 212; Adams et al 213; K. Adams thesis.), even though more recent results show the clear coincidence of the most persistent regions of hypoxia with the most intense respiration foci (Siedlecki et al 214). The second is more complicated. In order for upwelled nutrients to lead to near-bottom O 2 depletion, nutrients must reach the euphotic zone, be photosynthetically combined with dissolved inorganic carbon to form particulate organic carbon, sink into near-bottom on-shelf waters, and be respired there, consuming O 2 in the process. If any of these steps are not followed, de-oxygenation will not occur. Nutrients that escape the coastal system and are returned to the open ocean do not contribute; neither do particles that are transported offshore and sink beyond the shelf-break; neither do particles that are retained on the shelf without being respired. The rate of de-oxygenation independent of flushing with offshore water is thus a simple proportionality to the supply of upwelled nutrients: {de-oxygenation rate} = c e W up αn up (1.1) where W up is the upwelling volume transport, N up is the concentration of nutrient in upwelled water, α is a stoichiometric ratio between nutrient and oxygen, and the proportionality factor c e can be thought of as a metabolic-potential retention efficiency. Total retention (c e = 1) implies that all nutrient is converted to organic matter, all of which sinks to near-bottom shelf waters and is respired, taking the concept of nutrient trapping to its logical extreme. No retention (c e = ) simply reduces to the trivial flushing solution discussed above. Total retention implies a deoxygenation rate that increases directly with increasing upwelling and/or increasing nutrient content of upwelled waters. 5

7 As the high rates of deoxygenation implied by c e = 1 are never observed, even in systems prone to hypoxia, the factors that determine the retention efficiency are of interest to the study at hand. Coastal upwelling systems are home to assemblages that include diatoms known to be among the fastest growing (Dugdale) and sinking (...) of any phytoplankton. Fast net growth rates mean nutrients have a greater likelihood of being converted to particulate organic material before they are transported offshore; faster sinking rates mean a greater likelihood of particulate organic material settling on the shelf seafloor rather than in the offshore ocean interior. It is thus the timescales of these processes relative to the timescales of cross-shelf transport that determines the retention efficiency, and therefore the deoxygenation rate. We hypothesize that the de-oxygenation rate can be simulated with a highly simplified biogeochemical model of three tracers oxygen, a representative nutrient, and organic particles that interact via parameterizations of photosynthetic growth, heterotrophic respiration, and particle sinking. In this parameterization, sinking incorporates grazing and pellet excretion, aggregation, ballasting, and self-settling. Specific rates of photosynthesis related to nutrient source or ecosystem structure are incorporated in the adjustable parameters of a basic rate model. Respiration incorporates that by autotrophs, bacteria and higher trophic levels. It is clear that our simple biogeochemical module is not and is not intended to be a full ecosystem simulation and we make no suggestions that the glorious intricacy of these productive systems is lessened in any way by our coarse representations. The net change over time in oxygen concentrations, however, is driven only by the net transports and biogeochemical 6

8 transformations, and the broad simplifications we explore are helpful in providing contextual constraints in the parameter space defining production, transport and respiration that shapes oxygen budgets. We intend to apply this simplified model of the upwelling biogeochemistry to a realistic physical and bathymetric model to yield insight as to the development of hypoxic conditions on the Oregon coast. In the process of this development, we created the biogeochemical modules and tested them in a simplified box model representation of the surface and nearbottom layers of a coastal upwelling system. These box-model results, presented here, show some interesting features of this simplified system and provide insight into the factors that make some systems tend toward, and others resist, hypoxia. In the box model implementation, we focus on the development of hypoxic conditions in the near-bottom environment. Observations (Adams, Hales, Shearman/Barth, etc.) consistently show that hypoxia develops from the bottom, and not the mid-water column. Labile particles are trapped by the presence of the effectively-impenetrable bottom layer, as observations show that only a vanishingly small proportion of material exported from the surface is ultimately incorporated and preserved in sediments (Devol, Hartnett, McManus, Reimers, Fuchsman). Many observations show near conduit-like connectivity from the upwelled source waters at the shelfbreak and the outcropping upwelled waters in the inner shelf (Van Geen, A. Perlin, Hales), suggesting that both physical flushing and biogeochemical metabolism are focused on this layer. In order to simulate the delivery of labile C to the bottom box as it relates to upwelling, we must link the production in and export from the surface layers of labile 7

9 particulate C with this delivery. This results in our focus on the surface box, where we allow particles to grow as if by photosynthesis with a simple rate formulation dependent on particle and nutrient abundance. The distance between these two surface boxes that simulates a midwater column results in delay and attenuation of the primary particle export from the surface to the bottom, and while we do explicitly model this layer, we discuss only the parameterizations that contribute to the magnitude of the delay and attenuation. 2 Methods The model presented and analyzed here is formulated to represent integral balances over a continental shelf under upwelling conditions, partitioned by depth into three volumes or boxes. The top box, with thickness H, represents a surface boundary layer, to which both wind-driven transport and photosynthetically driven production are confined. The bottom box, taken to have the same thickness H, represents the bottom boundary layer, in which onshore upwelling transport is confined. The interior water column between the surface and bottom boundary layers forms a middle box, with thickness H mid. For nominal surface and bottom boundary layer depths H = 2 m and mid-shelf total depth of 1 m, H mid = 6 m, giving a boxthickness ratio δ H = H/H mid = 1/3. A single cross-shelf width L is specified for all three boxes, and an implicit assumption of alongshore uniformity is made; no explicit along-shore width is specified and all balances and quantities are computed per unit alongshore distance. The main object is to explore the dependence of near-bottom oxygen depletion on general characteristics of the upwelling system. Near-bottom oxygen is represented by the 8

10 oxygen concentration in the bottom box, O bot (mmol m 3 ), equal to total dissolved oxygen in the bottom box divided by the bottom box volume. The rate of change of O bot is proportional to the sum of respiration (remineralization) of bottom-box particles and the balance (onshore minus upward) of upwelling-driven advective oxygen fluxes, do bot dt = α γ rc bot + λ(o up O bot ). (2.1) In (2.1), α = 1 and γ = 7 are stoichiometric ratios representative of the typical proportions of carbon and oxygen, respectively, to nitrogen in marine organic matter (REFs?), r (d 1 ) is the carbon-specific particle respiration (remineralization) rate, C bot (mmol m 3 ) is the particle (particulate organic carbon, or POC) concentration in the bottom box, λ is the upwelling volume-renewal rate,and O up (mmol m 3 ) is the constant oxygen concentration of the deep offshore source waters advected into the bottom box by the upwelling circulation. The equation (2.1) can be rewritten in terms of the oxygen drawdown relative to the offshore value, O bot = O up O bot : d O bot dt = α γ rc bot λ O bot. (2.2) This relative drawdown is independent of O up, which may differ between coastal systems, so the formulation (2.2) is used here. Note that a positive value O bot > corresponds to bottom oxygen depletion. It is not necessary to consider or represent oxygen concentrations in the middle or top boxes, because λ (no downwelling) in all the cases considered here, and mixing between the boxes is neglected, so that oxygen enters the bottom box only from offshore. The neglect of mixing between boxes for all biogeochemical constituents is consistent 9

11 with the focus on advectively driven upwelling systems. We do not attempt to simulate the functioning of the system after oxygen has been entirely depleted (anoxia). The bottom-box particle concentration C bot needed for (2.2) is derived here from a maximally simple particle-nutrient model, Particles : Nutrients : dc top dt dc bot dt dn top dt dn bot dt = ( ) pntop r C top sc top λc top (2.3) K + N top = rc bot + F s (t) (2.4) = ( ) γ 1 pntop r C top + λ(n bot N top ) K + N top (2.5) = γ 1 rc bot + λ(n up N bot ). (2.6) The first term on the right-hand side of (2.3) represents net top-box particle concentration growth deriving from the excess of photosynthetically-driven productivity over particle respiration. Production proceeds via a particle-specific rate constant, p (d 1 ), scaled by a Monodform dependence on a top-box generic nutrient concentration (N top ; mmol m 3 ). The production rate is essentially independent of N top when N top K for a given value of the halfsaturation constant K (mmol m 3 ); for N top K, N top becomes limiting. Top-box particles are lost in proportion to their concentration by respiration at the rate r and by vertical sinking, or settling, at the specific settling rate s = S/H (d 1 ), which is the ratio of the physical sinking velocity S (m d 1 ) to the box thickness H (m). Top-box particles are also lost by offshore advection at a rate given by their concentration multiplied by the volume-renewal rate λ (d 1 ). Bottom-box particles are supplied by the incoming sinking flux F s and lost by respiration. Because most studies show that essentially all particles incorporated in surficial sediments are respired very near the sediment-water interface, with respiration products returned to overly- 1

12 ing waters, sediment respiration is assumed to be incorporated in the bottom-box respiration, so no explicit term is included that would separately represent loss of C bot through long-term sediment burial. The nutrient balances include the terms corresponding to particle growth and respiration seen in (2.3 and (2.4), differing only in sign (production consumes nutrient, respiration releases it) and a scaling by the stoichiometric ratio γ. Onshore and offshore transport terms are as for bottom-box oxygen and top-box particles, respectively, with N up the constant nutrient concentration of the upwelling source waters. The incoming particle flux F s in the bottom-box particle balance (2.4) would be equal to the outgoing flux sc top in (2.3) if the middle box were absent (i.e., for δ 1 H ). More generally (see Appendix), the particle flux is delayed and attenuated by respiration and sinking through the middle box, so that t F s (t) = δs 2 e (r+s )(t t ) C top (t )dt, (2.7) where δ = s /s, and s = S /H mid is the specific settling rate for the middle box. Here S > S would allow for the amplification of particle sinking velocities by aggregation or biological scavenging and repackaging, but we generally take S = S, so that δ = δ H. The incoming flux F bot will differ most (see Appendix) from sc top for high respiration rates (large r) and long settling transit times (small s ). The physical circulation is represented by wind-driven Ekman transport U E = τ y /(ρ f ) per unit alongshore distance, where τ y / f < and U E < corresponds to upwelling conditions, with offshore transport out of the top box; here τ y is poleward alongshore wind stress along an eastern boundary, ρ is a reference density, and f is the Coriolis parameter. The 11

13 offshore Ekman transport is compensated by vertical upwelling transport W up = U E from the bottom box to the top box; in all cases U E and W up, so that downwelling never occurs. After division by the box volume (per unit alongshore distance) HL, this volume flux translates to an upwelling volume-renewal (or cross-shelf transit) rate λ (with units of inverse days, d 1 ), τy λ = U E LH = ρ f LH. (2.8) The wind stress, and thus λ, is taken to have a constant and an oscillatory component, λ(t) = λ λ 1 cos(ωt), (2.9) where λ and λ 1 are constants (d 1 ). In (2.9), the phase is chosen so that the minimum value of λ(t) occurs at t =, and only values λ 1 λ are considered, so that reversals to downwelling (λ < ) do not occur. When λ 1, the forcing period T ω = 2π/ω, and the mean forcing over one forcing cycle is λ (i.e., t+t ω t λ(t )dt /T ω = λ ). An important diagnostic of model behavior is the degree to which upwelled nutrient it retained on the shelf, i.e., in the model boxes. The total model nutrient equivalent per unit alongshore distance, N, is the sum of the particle and nutrient content of each box, with each term in the sum consisting of a concentration multiplied by the corresponding box volume and the particle concentrations converted to nutrient equivalent through scaling by 1/γ: N = LH[γ 1 (C top + δ 1 H C mid +C bot ) + N top + N bot ). (2.1) The sinking fluxes and the top and bottom box respiration terms cancel from the rate of change of N, which [as can be verified by taking the time derivative of (2.1) and substituting (2.3) 12

14 (2.7) and (A.1)] is given by dn dt = LH{λ[N up (γ 1 C top + N top )] rγ 1 δ 1 H C mid}. (2.11) Thus, nutrients enter the model through onshore transport of nutrients into the bottom box, and leave the model through offshore transport of particles and nutrients out of the top box and through respiration of particles in the middle box. Related retention efficiencies based on (1.1) are defined below (Sec. 3.4). As basic points of reference, two contrasting sets of parameter values are considered (Tables 1-3), denoted by "Fast Particles" (FP) and "Slow Particles" (SP). These cases were chosen to highlight the different bottom-oxygen responses that can occur in response to different sinking rates, even for similar net production rates. This comparison is then supplemented by more general analyses of the dependence of the system response on the physical and biogeochemical control parameters. The FP case, with relatively large values of p, K, and S, represents a system characterized, for example, by a dominance of rapidly growing and sinking plankton, such as large coastal diatoms (Dugdale References). The SP case, with relatively small values of p, K, and S, represents a system with a slower biological response to upwelled nutrients and much slower particle sinking velocities. The respiration rate is taken to be constant and uniform in all boxes, with the nominal value r =.1 d 1. For definiteness, we set bottom-oxygen drawdown ( O bot ) values O hyp = {4,1} mmol m 3 as nominal criteria for hypoxia and anoxia, respectively, corresponding respectively to O bot values of 6 mmol m 3 and mmol m 3 for a nominal deep offshore oxygen concentration O up = 1 mmol m 3. 13

15 3 Results: two example cases 3.1 Overview In this section, the FP and SP cases are considered for two basic physical upwelling scenarios. In the first, "constant-forcing" case, a simple representation of the long-term evolution of the system over an upwelling season is constructed by initializing the model in a state chosen to be representative of conditions just prior to the onset of the upwelling season and then forcing the system for 15 days (comparable to the maximal extent of mid-latitude eastern boundary upwelling seasons) with constant upwelling-favorable winds, i.e., setting λ = λ >, where λ is a constant. In the second, "variable-forcing" case, a similar evolution is considered, but now including an oscillatory component to the upwelling wind stress, so that λ 1 >. In both cases, the initial states have bottom-box oxygen and nutrients equal to their offshore values [ O bot = O up O bot =, N bot = N up = 35], zero bottom-box (and middle-box) particle concentration [C bot = C mid = ], and top-box nutrient and particle concentrations given by N top = 5, C top = 1 (Table 2; here the subscript "" indicates a value at t =, and all values are in units of mmol m 3 ). For the constant-forcing case, the time-evolution of the model system (Figs. 2,3) is characterized by two distinct stages: (i) an initial phase, during which the time-evolution is approximately exponential, followed by (ii) approach toward a long-term, steady-state balance. The initial, exponential phase lasts roughly 7 d for FP (Fig. 2) and over 1 d for SP (Fig. 3). The two cases FP and SP have widely differing bottom oxygen responses. For 14

16 FP, oxygen depletion is rapid and sustained, reaching O bot = 4 mmol m 3 after 64 d, and O bot = 1 mmol m 3 after 84 d (Fig. 2a). For the SP case, oxygen decline is slow and limited, so that O bot is still negligible after 1 d and reaches only 6 mmol m 3 after 15 d (Fig. 3a). In addition, the variable-forcing cases examined below show a large effect on mean bottom oxygen depletion for FP, but little effect for SP. These differences in oxygen responses motivate exploration and comparison of the evolution and detailed balances for the FP and SP cases for both constant and variable forcing. 3.2 Constant forcing Exponential phase Bottom oxygen drawdown is driven by the supply and respiration of sinking particles, and so depends heavily on the temporal evolution of surface particle production. For constant forcing, the initial phase of the response is characterized by top-box nutrient concentrations N top K. Thus, the Monod nutrient uptake term pn top /(K + N top ) p, surface particle production is not nutrient limited and proceeds at the saturated rate η = p r s, and C top grows exponentially with approximate rate η λ. For the parameters chosen here, η λ is.1 d 1 for FP and.5 d 1 for SP (Table 2). The factor of two difference in η λ causes more rapid surface particle growth in FP relative to SP: by 5 d, C top has reached 29 mmol m 3 for FP, but less than 15 mmol m 3 for SP (Figs. 2c,3c). Importantly, while the surface particle load is increasing during the exponential phase, the export of particles to depth is also increasing. Sinking reduces particle accumulation in the 15

17 top box, but it is the sole factor driving accumulation in the bottom box. Thus, as C top increases exponentially, particle sinking drives corresponding exponential growth in C bot, and the latter drives an exponential increase in O bot. In other words, as surface particle concentration grows in response to upwelled nutrients, these particles are exported to depth where they accumulate, drawing down oxygen as they are respired, so that the evolution of the bottom oxygen drawdown follows the top-box particle concentration. The approximate exponential rate of bottom oxygen drawdown is then also η λ, the same as the exponential rate of growth of C top. However, the amplitude of the bottom oxygen response has an additional dependence on parameters, including the sinking, respiration and upwelling rates, as these affect the supply of surface particles to the bottom box and set the flushing rate of bottom oxygen with off-shelf waters. Although the exponential rate of growth η λ for FP is only twice that for SP, the delivery of particles to depth is far more pronounced for FP, for two reasons. The first is that the product of the higher particle production rate and higher settling rate drives a greater sinking flux for FP than for SP, and the second is that the more rapid transit through the middle box for FP results in less attenuation by respiration in the middle box and a higher rate of export to the bottom box, so that respiration is focused in the bottom box for FP. Retention of upwelled nutrients over the shelf is small early in the exponential growth phase, when surface particle concentrations are low. For the FP case, most of the growth in total nutrients N occurs in the late stages of the exponential phase, when N top decreases toward K and the sinking flux is large, so that the offshore loss of N top and C top is reduced 16

18 (Fig. 2bd). For SP, offshore transport of C top compensates reduced offshore transport of N top when N top approaches K, and consequently N grows only through onshore transport of N up during the short initial period when N top N up (Fig. 3bd). The exponential phase ends when nutrient limitation (N top K) slows the growth of C top. Long-term steady-state response While both the FP and the SP oxygen drawdown both initially follow an exponential evolution, the two cases differ fundamentally in the amplitude of oxygen drawdown that is ultimately reached. A drawdown of 1 mmol m 3, which would correspond to anoxic conditions for a nominal offshore oxygen value O up = 1 mmol m 3, is reached for FP in 84 d. In contrast, the SP drawdown never reaches 1 mmol m 3. In both FP and SP, and in general for the constant-forcing scenario, surface nutrients N top are eventually depleted to the point where N top K and the particle production rate drops below η. In the long term, a steady-state balance results, in which the C top growth rate is precisely zero, and the corresponding constant value of C top is such that the difference of respiration and production precisely balances the upwelling flux of nutrients, so that N top is also constant (Figs. 2d,3d). In the case of FP, this steady-state balance has a large oxygen drawdown, which is not reached until well after the hypoxic thresholds are passed. For SP, however, the steady state has only a small oxygen drawdown, which prevents the SP system from approaching hypoxia. The immediate reason for the smaller steady-state value of oxygen drawdown for SP, relative to FP, is the smaller flux of sinking particles into the bottom box in the steady-state balance for SP. The constant-forcing steady states, the associated particle- 17

19 nutrient-oxygen balances, and their general dependence on the physical and biogeochemical controls are explored more fully below, in Sec Variable forcing When an oscillatory component λ 1 > is added to the upwelling forcing, there is a systematic reduction of mean oxygen depletion, relative to the constant-forcing response with the same time-mean forcing. The reduction of bottom oxygen depletion is found to occur when there is a correlation of strong forcing with high surface nutrient levels, causing enhanced mean off-shelf nutrient transport and reducing the fraction of nutrients converted to sinking particles that subsequently deplete bottom oxygen. The mean effect found here depends only on fluctuations in the amplitude of the upwelling (λ ) forcing, and does not involved reversals to downwelling (λ < ) conditions. For the FP case with λ 1 = λ =.3 d 1 and T ω = 2 d (for which upwelling ceases completely at t = {,2,4,...} d 1 ), the time-mean O bot over a forcing cycle at 15 d is reduced by more than 5 mmol m 3, to roughly 1 mmol m 3, relative to the constant forcing value (Fig. 4). For this case, the maximum value of λ(t) is.6 d 1, and the instantaneous surface particle accumulation rate η λ(t) becomes negative during the strong-forcing phase of the cycle. During this phase, C top decreases rapidly and upwelled nutrients are transported off-shelf before they can be consumed by particle production (Fig. 4b). An analogous nutrient loss occurs during the early stages of the response to constant forcing (Figs. 2b,3b), before surface particle levels increase substantially. Consequently, averaged over the forcing cycle, 18

20 there is a net reduction of nutrient conversion to sinking particles, and thus of bottom-oxygen depletion. This effect of variable forcing is much less pronounced for SP than for FP. For SP, the forcing and sinking rates are small enough that C top remains high, and N top low, throughout the forcing cycle. The physical and biogeochemical controls on oxygen drawdown under time-dependent forcing are explored more fully below in Sec Retention efficiencies The model-box-integrated budget (2.11) for total model nutrient-equivalent N (2.1) provides a basic descriptor of the amount of total nutrients retained in the model boxes (i.e., "over the shelf"). A more refined measure of retention is the retention efficiency, c e in (1.1), which gives the proportionality between the respiration-based de-oxygenation rate and the supply of upwelled nutrients. This measure can be defined with respect to the supply of nutrients either to the bottom box from offshore (c e ) or to the top box from the bottom box (c e): c e = r t C bot dt γn t up λ(t )dt, c e = r t C bot dt γ t λ(t )N bot dt. (3.1) In (3.1), the respiration-based de-oxygenation rate (αr/γ)c bot has been taken from (2.4), the upwelling transport W up from (1.1) has been converted to the volume-renewal rate λ using (2.8) and W up = U E, and the efficiencies have been computed for time-integrals of (1.1). The efficiencies (3.1) represent the fraction of upwelled nutrients that causes bottom oxygen depletion. It may be shown (see Appendix) that the efficiency c e, which is normalized by N bot, can never exceed one, provided that the initial values of N top and C top are sufficiently 19

21 small. However, the efficiency c e, which is normalized by N up, can exceed one: once brought onshore, the same nutrients can upwell, convert to particles, sink and respire multiple times, each time depleting oxygen without requiring additional onshore nutrient transport. The evolution of the oxygen-depletion efficiencies (3.1) for the constant-forcing FP and SP cases (Fig. 5) resembles that of the bottom oxygen drawdowns much more closely than that of the total nutrient N (Figs. 2,3). The efficiencies are less than.2 for the first 25 days, and then increase rapidly after 5 d to a maximum of order.2 for FP, while remaining small for SP. The relatively large differences in these efficiencies primarily reflects the large difference in sinking rates between FP and SP. 4 Results: controls on response 4.1 Exponential phase and steady states The differences between the FP and SP oxygen responses motivate exploration and comparison of the more general dependence of the oxygen drawdown on the physical and biogeochemical controls. Because of the number of control parameters even in this highly simplified model and the general similarity of the responses to steady (λ 1 =,λ = λ > ) and to timedependent ( < λ 1 λ ) forcing, this dependence is considered here primarily in the context of steady forcing. For the initial, exponential phase with steady forcing, one primary derived parameter describing the timescale of the response, η λ has been introduced above; this is supplemented below by a related estimate t exp [eq. (4.2)] of the time of approach to hypoxia 2

22 based on the exponential-phase solution. It is useful also to derive explicit expressions for the steady-state solutions, for which the dependence on control parameters can then be diagnosed directly. The reduction in oxygen drawdown induced by variable forcing is of special interest and its dependence on the controls is also explored in this section In the exponential phase, during which N top K and pn top /(K +N top ) p, the system is linear and can be solved analytically. This leads to a simple exponential approximation for O bot (t), O bot (t) αc top γ G (e(η λ )t 1), (4.1) where G is a function of the parameters p,r,s,λ,δ [see Appendix; eq. (A.2)]. For the FP and SP cases, initial top box nutrient concentrations (N top = 5 mmol m 3 ) are already substantially larger than K, so that exponential growth begins immediately and (4.1) gives a useful estimate of the initial system evolution (Figs. 2a,3a). An estimate t exp of the time t hyp at which the bottom oxygen drawdown O bot reaches a given threshold value O hyp may be obtained from (4.1): t exp = 1 ( ln 1 + γ O ) hyp G 1 (G 1 + G 2 ), (4.2) η λ αc top η λ where G 1 = ln γ O hyp αc top, G 2 = lng(p,r,s,δ,λ ). (4.3) Thus, the difference η λ determines the basic exponential timescale of the oxygen drawdown response, while the logarithmic factors G 1 and G 2 encode the relative dependence of the amplitude of drawdown on the corresponding parameters. For FP, (4.2) gives useful estimates t exp = {46,55} d of the times t hyp = {64,84} d to O hyp = {4,1} mmol m 3 (Fig. 2a). 21

23 For SP, the exponential approximation (4.1) is accurate for over 1 days (Fig. 3a). However, for SP, O bot never exceeds 1 mmol m 3 and consequently any estimate from (4.2) of t hyp for O hyp 1 mmol m 3 would be fundamentally inaccurate. The reason for this failure of the t exp estimate (4.2) for SP can be found in the steady-state balance that is eventually approached. 4.2 General steady states When the forcing is constant (λ = λ ) and nutrient limitation (N top K) slows the growth of C top, a steady-state balance can result in which the time rates of change of all model variables vanish. The resulting steady states can be written as O bot = O bot,c top = C top,... The adjustment toward these steady states may be slow: in both the FP and SP cases, the steady state is approached only after 1 days. Further, the steady-state oxygen drawdown values O bot may be extremely large: for FP, O bot 17 mmol m 3, much greater than the nominal 1 mmol m 3 offshore deep oxygen concentrations, and for other parameter choices O bot may reach 3-5 mmol m 3, sufficient to drive even fully saturated source waters anoxic. Nonetheless, the steady-state solutions provide basic insight and a perspective that is complementary to the analysis of the initial exponential phase and the t exp estimate (4.2) of the time to hypoxia. For example, as noted above, consideration of the steady states explains why the oxygen drawdown for SP never exceeds 1 mmol m 3 and, therefore, why the t exp estimate fails for SP. Explicit expressions for these steady-state solutions can be obtained by setting the time 22

24 derivatives in (2.2) (2.6) to zero; these are given in the Appendix [equations (A.6) (A.11)]. The steady states prove always to be stable: small perturbations from them decay, so that the perturbed solution evolves back toward the steady state. The rate of approach to the steady state which determines the length of the intermediate transient phase (ii) identified above in Sec. 3 is controlled by the parameter σ 5 = λ r(r + δλ )/[δs(r + λ )]. For FP, σ 5 =.9 d 1, while for SP, σ 5 =.9 d 1, so the SP system adjusts much more rapidly toward steady state than the FP system after the initial exponential phase terminates. For the steady states, the terms in the total-nutrient balance (2.11) must sum to zero. For FP, the onshore nutrient transport into the bottom box is primarily balanced by offshore transport of C top and N top, in roughly equal proportion. For SP, the middle-box respiration loss is comparable to the offshore transport. For the steady states, the retention efficiency c e in (3.1) normalized by N bot is always no greater than one. The steady state values for FP and SP are (c e,c e) = (.48,.32) and c e = c e =.2, respectively. These are approached slowly in the time-dependent solutions (Fig. 5), because they are based on integrals from t =. The differences in these steady-state efficiencies for FP and for SP again reflect the differences in the FP and SP sinking rates. 4.3 Physical controls on constant-forcing response To characterize the response of the system to the physical controls, it is sufficient to consider only variations in the volume-renewal rate λ. (Possible variations in δ H are incorporated into the sinking-rate ratio δ = s /s.) For the constant-forcing scenario, the dependence of the 23

25 initial exponential phase and of the long-term steady-state response on λ, for fixed values of all other (i.e., all biogeochemical) parameters, are examined in this section. For the exponential phase, the characteristic growth rate for surface particle concentration and bottom oxygen drawdown is η λ. The dependence of t exp on λ through G 2 is opposite to, but weaker than, that through η λ, and the constant G 1 in t exp depends only on the biogeochemical parameters. Thus for a given a fixed set of biogeochemical parameters (p,r,s), the forcing level λ exerts a primary control on the exponential response. Because η λ increases linearly as λ decreases, the particle concentration and thus also the bottom oxygen drawdown growth rate over the shelf during the initial exponential phase is greater when upwelling forcing is weaker. This counter-intuitive result holds because the upwelling circulation has two basic, competing effects: while it supplies nutrients to the top box through upwelling of N bot, it also removes nutrients and particles from the top box through advection of C top and N top offshore. Because N bot N up K, relatively weak upwelling (small λ ) is sufficient to maintain N top K and sustain the exponential phase in the top box. Stronger upwelling merely removes particles and nutrients from the top box more rapidly; thus, in a sense, upwelling limits rather than drives the decline of O bot during this initial phase. Consequently, the estimated timescale t exp (4.2) for approach to hypoxia decreases monotonically as upwelling weakens, i.e., as λ decreases (Fig. 6). Numerical solutions show that this trend does not extend to the limit of zero upwelling, λ = (Fig. 6). Instead, for very small λ, the upwelling is too weak to maintain N top K, the exponential phase cannot be supported, and the time to hypoxia 24

26 is much greater than that estimated by t exp (which assumes N top K and is inaccurate when N top K). For sufficiently strong forcing (λ > η), surface particle growth cannot occur over the shelf, and there is no sinking flux and or oxygen depletion. An approximate total-retention response therefore requires proportionately larger values of the intrinsic particle growth rate p (and, as will be seen below, s) for larger values of λ, in order to capture upwelled nutrient and deposit it (as C bot ) in the bottom box. Despite the different values of p, the values of η (Table 3) are similar for the FP (.4 d 1 ) and SP (.35 d 1 ) cases, so POC can accumulate over the shelf for nearly the same range of forcing values λ for FP and SP. For both the FP and SP cases, the long-term steady-state response has a similar dependence on the upwelling timescale λ when all other FP and SP parameters are held fixed: steady-state oxygen depletion is always greater when upwelling is weaker, i.e., when λ is smaller (Fig. 7). For the steady states, the increase of oxygen drawdown with decreasing upwelling extends all the way to the limit of zero upwelling. However, it takes relatively much longer to reach the steady state values as upwelling declines. In the low-upwelling limit, production is nutrient limited, particle concentrations are low throughout the system, surface nutrient levels are low and offshore transport of nutrients minimal. For weak upwelling, upwelling renewal of O bot is also reduced, and this effect is sufficiently strong that the long-term, steady-state oxygen drawdown is largest for the weakest upwelling. As upwelling intensity grows, onshore nutrient flux grows, and particle concentrations through the boxes become higher, but oxygen renewal also increases, and oxygen drawdown decreases. Consequently, 25

27 the upwelling rates λ that give the greatest oxygen drawdown and the largest total nutrient accumulation N generally do not coincide (Fig. 7aedh). As upwelling increases so that λ η, particle concentration growth over the shelf ceases; consumption of nutrients by particle production cannot keep up with the nutrient supply and surface nutrient levels increase, which in turn causes increased offshore transport of total nutrients at the surface. For the steady states, the "no-growth, no-retention" limit C top = C bot = is reached when λ = η η, where η = pn r s and N = N up /(K + N up ) 1 (see Appendix). The retention efficiencies (3.1) follow the oxygen depletion, with typical values of order.3-.5 for FP and.1 for SP (Fig. 8a). For λ <.1 d 1, the FP system is "superretentive": the N up -normalized efficiency c e > 1, as upwelled nutrients are recycled, causing extreme oxygen drawdown. In the limit of small λ (i.e., as λ ), the steady-state solution (A.6) for bottom-box oxygen drawdown is O bot = s r α(n up N top ) s r αn up, (4.4) where, as before, s = δ s. This is also the maximum steady-state value of O bot with respect to variations in λ with all other parameters held fixed. The timescale 1/σ 5 for approach to this steady state diverges as 1/λ as λ, so that this theoretical limiting steady-state solution will never be reached from general initial conditions. However, the result (4.4) emphasizes that rapid sinking (large s ) increases retention efficiencies and oxygen drawdown. It also suggests that drawdown is large for slow respiration (small r), but this interpretation is less robust, as the large drawdown for small r and λ is realized only after an extremely long, slow approach toward the steady state, with timescale diverging as 1/(λ r) as λ and r. 26

28 4.4 Biogeochemical controls on constant-forcing response Control parameters The biogeochemical controls on the model system consist of the rate parameters p,k,r,s, the sinking-rate ratio δ = s /s, and the deep offshore value of the nutrient concentration, N up. For the initial exponential phase, the initial values of surface particle and nutrient concentrations, C top and N top, constitute additional controls. Exponential phase During the exponential phase, the dependence of the basic rate of response η λ on r and s is the same as that on the physical control λ, while the dependence of η λ on p is precisely opposite to λ : η λ decreases with r and s, and increases with p. Consequently, a rapid initial response rate is favored by large intrinsic particle production rates p and small rates of respiration r and sinking s. Similarly to the forcing rate λ, the sinking or export rate s limits surface particle accumulation, so to support particle growth (η > λ ) for fixed λ and for given p and r, the surface sinking rate s must be less than s max = p r λ. The amplitudes of the C bot and O bot responses during the exponential phase depend not only on η λ but in a more complex way on other combinations of biogeochemical parameters. Thus, for example, during this phase C bot C top for the FP case (Fig. 2), but C top C bot for the SP case (Fig. 3), despite the identical initial conditions and similar η λ values for the two cases. For the exponential approximation t exp to t hyp given above in (4.2), this dependence is encoded in the constants G 1 and G 2. The form of (4.1) and of G 1 in (4.2) shows that, at this level of approximation, the 27

29 timescale of the bottom-oxygen depletion response depends on the hypoxic threshold O hyp and the initial surface particle concentration C top only through their ratio (and the fixed stoichiometric constants α and γ). Thus, with all other parameters fixed, doubling C top will double the bottom oxygen drawdown that is reached at any given, fixed time during the exponential phase. Equivalently, fixing the hypoxia threshold O hyp and doubling C top will reduce the estimated time to hypoxia t exp by t exp = (ln2)/(η λ ). For sufficiently small C top, the system will remain close to the "no-growth, no-retention" limit (C top = C mid = C bot = ) for a sustained period, and there can be large offshore transport of N top before particle levels are high enough for nutrient consumption to balance supply, limiting initial retention of upwelled nutrients and delaying bottom-oxygen drawdown. For both FP and SP, G 1 = {3.3,4.2} for O hyp = {4,1} mmol m 3. The dependence of G 2 on parameters (A.2) is more complex, but its influence on t exp tends to be opposite to that of η λ : for G 2, larger p, smaller r and λ, and especially smaller s tend to increase t exp, slowing the approach to hypoxia rather than accelerating it. Although the dependence is logarithmic, G 2 may have a stronger influence on the oxygen response than η λ itself. For FP, G = 3.52 and G 2 = 1.26, while for SP, G = 15 and G 2 = These differences of a factor of thirty in G and a factor of four in G 2, which arise primarily from the strong, inverse-square dependence of G on the sinking rate s, explain the much greater and more rapid bottom oxygen drawdown for FP relative to SP, despite the difference of only a factor of two in η λ. 28

30 Steady states: sinking rate dependence For variable sinking rates and with other parameters fixed as for FP and SP (with δ = 1/3 as before, so that s = s/3), the steady-state solutions show a strong dependence on s, which controls both the export rate of particles from the surface and the sinking rate through the water column (Fig. 9). For both the smaller and largest values of s, particle accumulation in the bottom box is limited and oxygen drawdown is low. At the surface, low values of s both limit the rate at which particles are exported and increase η λ, resulting in a high C top steady state. However, the small exported fraction of this high particle load is substantially reduced by mid-depth respiration, because of the slow sinking rate, so that particle levels are low in both the middle and bottom boxes. This results in weak bottom oxygen drawdown for small sinking rates, while the offshore flux of total nutrients is dominated by surface particle transport. Very rapid surface export (high s) slows the top box particle concentration growth rate η λ, shutting down production completely when s = s = pn r λ s max (see Appendix). In this "no-growth, no-retention" limit, all particles are flushed out of the system and all variables decay to their offshore values. As this limit is approached, particle levels become low and are not able to respond to the upwelled nutrient flux, surface nutrient levels become high, increasing offshore export of nutrients and limiting retention. Low overall particle levels result in decreased oxygen drawdown at depth, despite higher sinking rates that deliver a higher proportion of sinking surface particles through the middle box to the bottom box. For intermediate values of s, the exported (sinking) fraction of surface particle production increases with s while η λ decreases; particle concentrations decline at the surface but 29

31 increase in the middle and bottom boxes, driving enhanced bottom oxygen depletion. This effect is large for FP, for which s r, but limited for SP, for which the relatively slow sinking rates lead to high levels of respiration in the middle box relative to the upwelled nutrient flux, limiting particle flux to the bottom and resulting in low particle concentrations in both the middle and bottom boxes relative to the surface (Fig. 9). The oxygen-based retention efficiencies (3.1) again follow the oxygen depletion, with maximum values of order.3-.5 for FP and.4 for SP (Fig. 8b). 4.5 Drawdown reduction for variable forcing As illustrated by the results for FP (Sec. 3.3), variable forcing can lead to a significantly reduced average bottom oxygen drawdown relative to constant forcing with the same mean strength. The characteristics of this effect are strongly dependent on the magnitude of the mean forcing λ, the amplitude of the sinusoidal forcing λ 1, and the forcing period T ω. In this section we explore these effects in greater detail. The mean reduction in oxygen drawdown is driven by offshore nutrient transport during strong forcing. High surface nutrient levels can develop when surface particle levels are low, and low surface particle levels can be achieved in two ways. When λ 1 is large enough that the instantaneous surface particle accumulation rate η λ(t) becomes negative for a significant portion of the cycle, particle concentrations decrease rapidly, and upwelled nutrients are swept off-shelf before they can be consumed by particle production. This occurs for the FP case with λ 1 = λ =.3 d 1 (Sec. 3.3). In this case, offshore nutrient transport is largest dur- 3

32 ing the latter stages of strongest upwelling. Alternatively, surface particle growth may instead halt because of the Monod nonlinearity, as C top production draws down surface nutrients when upwelling weakens until N top K. Sinking, rather than offshore transport, then dominates the depletion of C top ; when upwelling resumes, the upwelled nutrients cannot be fully consumed by particle production and are transported offshore. This occurs, for example, for the more weakly forced case with λ 1 = λ =.1 d 1 and other parameters as in FP. In this case, the largest offshore nutrient transport occurs earlier in the upwelling cycle. The dependence of this effect on parameters is somewhat complex. For FP with λ =.3 d 1, bottom oxygen drawdown decreases systematically as the forcing amplitude λ 1 increases from zero toward λ, as the stronger forcing drives larger offshore nutrient transport. For this case, oxygen drawdown also decreases as the forcing period T ω increases (Fig. 1). In contrast, for the FP case with λ =.1 d 1, there is a minimum in the oxygen drawdown at a forcing period T ω 3 d, with less drawdown reduction for T ω > 3 d (Fig. 1). For very long forcing periods, an elevated oxygen drawdown relative to the constantly forced case is possible in some cases; this can be understood as an average over λ of slowly-varying quasi-equilibrium states (e.g., such as those in Fig.7). For the SP case, the sinking and offshore transport rates are sufficiently slow that surface POC is not depleted even for very long forcing periods (T ω > 1 d). Hence, in the SP case, particle levels never become low, upwelled nutrients are more effectively consumed, and surface offshore transport is dominated by particle loss. Consequently, for SP the variable forcing causes only a small reduction of oxygen drawdown. 31

33 4.6 Hypoxic-optimal sinking The steady states depend nonlinearly on several different biogeochemical parameters. Some progress toward a more general characterization of susceptibility to hypoxia within this model framework can be made by restricting attention to solutions with optimal values of the sinking rate s. Because the sinking velocity S, and therefore s, is a parameter that is poorly constrained by observations, and because bottom-oxygen depletion is weak for the smallest and largest values of s and of roughly uniform magnitude for intermediate s (Fig. 9ae), it is natural to simplify the parameter dependence by identifying the value of s that gives the largest steady-state bottom-oxygen drawdown, and then examining the dependence of these "hypoxic-optimal sinking-rate" solutions on the other parameters. From (A.6) (A.12), the steady-state bottom oxygen depletion can be written explicitly: δs 2 O bot = rs + (r + δs)λ ( ) s s α(k + N up ), (4.5) s max s where s and s max are the functions of p, r, λ, K and N up defined above. The expression (4.5) is convenient for examining the dependence of O bot on s for fixed p, r, λ, δ, K and N up. The hypoxic-optimal s is defined here as the sinking rate s hyp that gives the maximal value of O bot in (4.5), holding all other parameters in (4.5) fixed. Assuming N up K, s hyp can be approximated as (see Appendix) where K = K/(K + N up ) = 1 N 1. ( ) ] 1/2 pk s hyp s max [1, (4.6) s max From this hypoxic-optimal sinking rate, a necessary condition for the existence of hy- 32

34 poxic states (i.e., states with O bot = O hyp for a given O hyp ) can be obtained, which has the form p 1 B hyp. (4.7) In (4.7), p = p/(r + λ ), and B hyp < 1 is a constant that is independent of p, r, and λ, and depends only on δ, K, and the ratio A = α(k + N up )/ O hyp (see Appendix). Conversely, a sufficient condition for the non-existence of hypoxic steady states is p < 1/B hyp, where also p > 1 + pk (i.e., s > ) is required for growth, so that all C top > steady states become hypoxic at s hyp as B hyp 1. For fixed values of p and of the sum r + λ, a second condition, p 1 B, (4.8) hyp can be derived, which separates regimes in which the hypoxic-optimal sinking rate s hyp gives hypoxic steady states for any admissable (i.e., < r < λ ) value of r from regimes in which a suitable choice of r would avoid hypoxia (see Appendix). These two criteria indicate that, on the basis of the steady-state response, hypoxia is generally favored for large p and for small r and λ. Note that only the sum r + λ = p s max enters into these criteria, not λ and r separately, which effectively couples, or entangles, the physical and biogeochemical controls. Thus, in general hypoxia is also favored progressively more as the constants B hyp and B hyp increase toward unity, tightening the criteria on p. The dependence of B hyp and B hyp on δ,k and A indicates that the system s basic susceptibility to hypoxia increases with larger δ (faster subsurface sinking or smaller middle-box thickness), larger N up, and smaller K. When scaled by r + λ, the hypoxic-optimal sinking rate s hyp from (4.6) is a function 33

35 only of p [or, implicitly, of η λ = (η λ )/(r + λ ) = p 1 s = s max s], s hyp = s hyp r + λ ( p 1) [ 1 ( ) ] 1/2 pk. (4.9) p 1 This allows a compact representation of the B hyp and B hyp criteria in a simple two-dimensional phase-space ( p, s), or ( η λ, s), on which (for given values of δ, K and A), any system with given values of p, r, s, and λ can be uniquely located, according to the values of the dimensionless rates p, s and η λ (Fig. 11). In this representation, which is based only on the properties of the steady state solutions, the transition to conditions susceptible to anoxia occurs for s.5 1., and is relatively independent of η λ for η λ >.5 (Fig. 11). 5 Discussion The analysis in the preceding section describes separately the basic dependencies of the exponential-phase and the steady-state oxygen-depletion responses on the control parameters. A relatively complete characterization of the system s susceptibility to hypoxia under constant forcing can be obtained by combining these two sets of diagnostics. If the steady state is not hypoxic for a given threshold O hyp, then the time-dependent response will in general never reach hypoxia. However, if the steady state is hypoxic, then the timescale of the exponential response (4.1)-(4.3) determines whether the hypoxic regime can be reached during an upwelling season of given length. The combined characterization is conveniently formulated in terms of the dimensionless rates p, r, λ, s and η (where r + λ = 1). The steady-state solution (4.5) can be written 34

36 with r, λ, s, s, and s max replaced by their corresponding dimensionless equivalents; this is itself equivalent to scaling time t by 1/(r +λ ) prior to computing the steady states. While the hypoxic-optimal s hyp (4.9) depends only on p (for fixed K ), the general steady-state solution (4.5) has an additional dependence on r (or, equivalently, on λ ). For fixed values of r, and of δ,k, and A, the steady-state bottom oxygen depletion increases rapidly with s, and is nearly independent of η λ except when η λ is small (Fig. 12). For values of r,δ,k, and A as in the FP and SP cases, and for the nominal offshore oxygen level O up = 1 mmol m 3, the transition to anoxic steady states or high potential for bottom oxygen depletion occurs for s.5 1., provided that η λ >.5.1 (Fig. 12). These values of normalized sinking rate correspond to dimensional sinking rates that are comparable to the sum of the respiration and upwelling volume-renewal rates. For smaller sinking rates, the system can never reach anoxia, even for arbitrarily long upwelling seasons. For larger sinking rates ( s > 1.), the potential for hypoxia in response to constant forcing is controlled by the exponential timescale t exp (4.2). The corresponding normalized timescale t exp = (r + λ )t exp depends primarily on η λ and s, less strongly on r and λ, and only weakly on δ; it can be long for small η λ, but decreases rapidly for larger η λ (Fig. 12). Typically, t exp 1 for η λ.25 (Fig. 12); thus, if the dimensional exponential growth rate η λ = p r s λ is at least one-fourth the sum r + λ of the respiration and upwelling rates, the dimensional time to hypoxia will be of order 1/(r + λ ) or less. The latter will in turn be less than 1 d if the sum of the respiration and upwelling rates is greater than.1 d 1, as it is for the values of r + λ for FP (.4 d 1 ) and SP (.35 d 1 ). This 35

37 implies that hypoxia can be reached within the upwelling season even with relatively small net particle production rates η λ.3 d 1, if sinking rates are large enough. Fluctuations in the amplitude of the upwelling forcing can systematically reduce the potential for oxygen depletion below these steady state estimates, especially when sinking rates are large or forcing is strong. 6 Summary A maximally simple, coupled physical-biogeochemical box model has been used to examine carbon cycling and oxygen drawdown in eastern boundary upwelling systems. In this model, oxygen drawdown is controlled by surface particle production and sinking, and is generally greater for weaker upwelling, wider shelf-widths, and larger sinking rates. It was found possible to achieve a relatively complete characterization of the bottom oxygen depletion response in this model (Fig. 12). Such a characterization is possible only because of the many simplifications and idealizations in the model formulation. Some aspects of the response of a slightly generalized model have been explored, including the effects of elevated subsurface sinking rates (δ > 1/3; i.e., δ S > 1 or δ H > 1/3) and (see Appendix) of upwelling of middle-box nutrients or of bottom-box particles. Fast subsurface sinking increases particle flux to the bottom layer by reducing particle respiration in the middle box, focusing remineralization and oxygen drawdown in the bottom box, and enhancing both the rate and amplitude of bottom oxygen depletion. Middle-box nutrient upwelling (φ M > 1) results in less flushing of the bottom box with 36

38 offshore oxygen, and similarly enhances oxygen depletion; this dependence may be relevant to comparisons of oxygen depletion in different upwelling systems, as higher rates of mid-depth return flow have been observed, for example, in Oregon and Peru relative to California and NW Africa (Smith, 1981). The primary effect of bottom-box particle upwelling (β > ) is in the limit of small respiration rate r, for which an advectively dominated system with limited bottom-oxygen depletion develops for β = 1; in contrast, when β = (no particle upwelling), C bot always accumulates and measurable oxygen drawdown can result for sufficiently weak upwelling even when r is small ( O bot s/λ ). A non-uniform respiration rate (larger r in the surface box) can be approximately simulated by modifying the production rate p accordingly, so that the difference p r maintains the desired value; this interpretation can be improved by also modifying the effective rate constant K (see Appendix). Many other influences on oxygen drawdown, including effects of three-dimensional circulation and downwelling, and all the many complications of the biological and chemical transformations that are implicit in the model conversions between nutrients and particles, cannot be addressed within the framework of the present model, and require more sophisticated approaches with substantially more complex models. Achieving a similar level of general understanding of these more complex models remains an outstanding scientific challenge. The present work is intended as a preliminary step toward that broader understanding. Acknowledgements This research was supported by the National Science Foundation, Grant OCE

39 A Appendix Middle-box particle concentration The equation for the middle-box particle concentration is: dc mid dt = rc mid s C mid + s C top where C mid = δ SC mid. (A.1) It is convenient to use here the scaled concentration C mid = δ SC mid, because the box thickness ratio δ H = H/H mid then enters the model equations only through the ratio δ. Multiplication of (A.1) by the integrating factor exp[ (r + s )t] and integration, using C mid =, then gives (2.7) directly. The steady solutions for C mid and F bot may be obtained either by setting the time-derivative equal to zero in (A.1) or letting t in (2.7). The equation (A.1) is needed to obtain the total nutrient balance (2.11) and is also convenient to retain in place of (2.7) for numerical solution. For an impulsive flux sc top = F δ D (t t ) at t = t, where here δ D is the Dirac-δ, the incoming flux F s (t > t ) = s F exp[ (r + s )(t t )] and its total time-integral F s (t)dt = F /(1+r/s ), illustrating that delay and attenuation of the sinking flux are largest for large r and small s. Parameter dependence of t exp constant G The constant G(p,r,s,λ,δ) in (4.1)-(4.3) is: G(p,r,s,λ,δ) = (p r s)(p s λ )(p s λ + δs) δrs 2. (A.2) With p replaced by η = p r s, this may be written equivalently as: G(η,r,s,λ,δ) = η(η λ + r)(η λ + r + δs) δrs 2. (A.3) 38

40 Bounds on retention efficiencies From (2.3) (2.6), it can be shown that c e 1 if N top + γ 1 C top λ min λ max + s N up, (A.4) where λ min λ(t ) λ max for t t. For the efficiencies normalized by N up, this implies the equivalent bounds c e t λ(t )N bot dt N up t λ(t )dt, (A.5) under the same condition on N top + γ 1 C top. The efficiencies for steady-state solutions and the time-integrated efficiencies for time-periodic solutions satisfy this same set of bounds, without any condition on N top or C top. Steady states The explicit (non-trivial) steady state solutions of (2.2) (2.6) are: O bot = αr γλ C bot, C top = C mid = (r + δs)λ γ (N up N top ) rs + (r + δs)λ δs r + δs C top (A.6) (A.7) (A.8) C bot = s r C mid (A.9) N top = r + s + λ p r s λ K N bot = N up + r λ γ C bot. (A.1) (A.11) From (A.7), it follows that N top < N up for all states with C top >. Note that the exponential growth rate η λ from the analysis of the initial transient response appears again in (A.1). 39

41 In (A.7), N up N top = ( 1 pk ) (K + N up ), (A.12) η λ so that N top = N up and C top = C mid = C bot = when pk = η λ, or s = s. The quantities N = N up /(K + N up ) and K = K/(K + N up ) = 1 N may be regarded as dimensionless saturated-growth and growth-limitation nutrient levels, respectively, computed from N up and K. Because, in general, K N up, it follows that K 1 and N 1. Hypoxic-optimal sinking The special steady solution at which O, N bot and C bot are maximum (i.e., where d O/ds = dn bot /ds = dc bot /ds =, with all other parameters fixed) can be obtained as a solution s = s hyp of the cubic equation s 3 + as 2 + bs + c =, (A.13) where a = 2(λ R s max ), b = s s max λ R(s + 3s max ), c = 2λ Rs s max, (A.14) and R = r r + δλ. (A.15) To obtain the cubic equation (A.13), it is helpful to use (4.5) or to write the solution for C bot in the form C bot = δλ γ(k + N up )R r 2 s 2 (s s) (s + λ R)(s max s). (A.16) 4

42 The cubic equation (A.13) can be written, s 3 + as 2 + bs + c = (s 2 2s max s + s s max )(s + 2λ R) + λ RpK s, (A.17) where pk = s max s. For λ RpK sufficiently small (for example, if N up K), a solution of the quadratic part of (A.17) gives the approximation (4.6) for s hyp. This solution for s hyp may be substituted into (A.16) and (A.11) to obtain the corresponding estimates of the maximum values C bot,max and N bot,max of the steady C bot and N bot, C bot,max λ γδ(k + N up )R r 2 N bot,max N up + δ(k + N up)r φ B r s 3 hyp (s hyp + λ R)s max, (A.18) s 3 hyp (s hyp + λ R)s max, (A.19) where the maxima have been taken with respect to s while holding other parameters fixed. Here it has been used that the approximate value s hyp satisfies (s s hyp )/(s max s hyp ) = s hyp /s max. The steady states with O bot = O hyp at the hypoxic-optimal sinking rate s = s hyp satisfy O hyp αλ δ(k + N up )R rλ s 3 hyp (s hyp + λ R)s max. (A.2) The sinking rate s hyp is optimal over s for fixed r and λ. For fixed p and s max, the hypoxic optimal r can be found by substituting λ = p s max r in (A.2) and optimizing over r. Then O hyp 1/F(r), where F(r) = δs hyp (p s max ) + [p s max + (1 δ)s hyp ]r r 2. (A.21) For steady-state hypoxic drawdown O hyp >, F(r) > is required, and because d 2 F/dr 2 <, the maximum value of O hyp must obtain at one of the endpoints r = {, p s max }, while the 41

43 minimum value may obtain at the point r = r q = [p s max +(1 δ)s hyp ]/2, where df/dr =, provided that < r q < p s max. The values of F(r) at these three points are F() = δs hyp (p s max ) (A.22) F(r q ) = δs hyp (p s max ) [p s max + (1 δ)s hyp ] 2 (A.23) F(p s max ) = s hyp (p s max ). (A.24) Thus, the minimum of F and the maximum of O hyp 1/F are at r = for δ < 1, and at r = p s max for δ > 1. The resulting necessary condition for the existence of anoxic states is p s max = r + λ B hyp p, (A.25) where, for δ < 1, B hyp = 1 1 ξ 2, ξ = (1 + N A) 1/2 K 1/2 A, A = α(k + N up). (A.26) 1 K A O hyp Conversely, a sufficient condition for the non-existence of anoxic steady states is p s max = r + λ > B hyp p. (A.27) For δ > 1, A is replaced by δa in the definition of B hyp : B hyp = 1 1 ξ 2 δ, ξ δ = (1 + N δa) 1/2 K 1/2 1 K δa Thus, the expression for the hypoxic-optimal criterion constant B hyp in (4.7) is: δa. (A.28) B hyp = 1 1 ξ 2, ξ = (1 + N A δ ) 1/2 K 1/2 A δ, A 1 K A δ = max{1,δ}a. (A.29) δ 42

44 Substituting r = r q into (A.21) and (A.2) gives a quartic in the variable q = (p/s max ) 1/2, one solution of which gives the condition (4.8) p s max = r + λ B hyp p (or p s max = r + λ > B hypp) (A.3) analogous to (A.25) and (A.27) that describes the regions where hypoxic states do (or do not) exist for r = r q, and therefore for any r, since r q is the least hypoxic r for the given p, s max, and hypoxic-optimal sinking s hyp. The quartic equation for q is q 4 + a q q 3 + b q q 2 + c q q + d q =, (A.31) where a q = 2K 1/2 [2K δa (1 + δ)] (A.32) b q = 2δ + K (1 δ) 2 12K δa (A.33) c q = 2K 1/2 δ(3 δ + 6A) (A.34) d q = 1 2(1 + δ) + (1 δ) 2 4δA, (A.35) and A is defined in (A.29). A solution q = ξ of a quadratic equation in the same variable q is the origin of the previous conditions (A.25) and (A.27). The additional special steady state with C top = C bot occurs when s = s bt, where [ s bt = r ( ) ] 1/2. (A.36) δ Upwelling of C bot (β > ) or N mid (φ M > ) The model equations may be modified to allow upwelling of particles from the bottom to the top box, through a term proportional to the bottom-box particle concentration (C bot ; 43

45 mmol m 3 ), the volume-renewal rate λ, and a dimensionless parameter β that quantifies the area and velocity over which upwelling occurs [controls the fraction of bottom particles to be upwelled]: Top : dc top dt = ( ) pntop r C top sc top λc top + βλc bot K + N top (A.37) Middle : C mid = 1 δ S C mid where Bottom : dc bot dt dc mid dt = rc mid s C mid + s C top (A.38) = rc bot + sc mid βλc bot. (A.39) For β = 1, the steady-state equations (A.9) and (A.1) for C bot and N top are modified to C bot = N top = s C mid r + βλ ( ) p s max Σ 1 K (A.4) (A.41) where ( ) Σ = s 1 β δs, β = βλ r + δs r + βλ (A.42) while the other expressions in (A.6) (A.11) are unchanged. Consequently, the qualitative structure of the solutions is similar to that obtained for β =, but the effective sinking rate Σ in (A.41) is reduced relative to s in (A.1). Thus, growth of C top can be sustained for much larger values of s when β = 1, as the "no-growth" limit for s increases from s to s β, where s β = max { ( ) [ ( 1 s r/δ 2 1 β 1 ± 1 + 4(1 β ) ]} )s r/δ 1/2 (s r/δ) 2. (A.43) For FP and SP, s β = {1.54,.11}, respectively, relative to s = {.57,.1}. If a fraction φ M > of the upwelling volume flux is taken to come from the middle box, so that a reduced, complementary fraction φ B = 1 φ M comes from the bottom box, there 44

46 are two main effects on the system. First, the equation (2.2) for the bottom oxygen drawdown is modified to d O bot dt = α γ rp bot λ φ B O bot, (A.44) so that the flushing of bottom oxygen for a given value of λ is reduced by the factor φ B < 1. Second, the middle box receives and supplies, respectively, fractions φ M of the onshore and upward upwelling fluxes of nutrients, so that all three nutrient equations are modified, and the middle-box nutrient equation becomes coupled to the other variables: Top : Middle : Bottom : dn top dt dn mid dt dn bot dt ( ) = γ 1 pntop r C top + λ(φ B N bot + φ M N mid N top ) (A.45) K + N top = γ 1 rc mid + λφ M (N up N mid ) (A.46) = γ 1 rc bot + λφ B (N up N bot ). (A.47) This also modifies the total nutrient budget, so that (2.11) is replaced by dn dt = LH{λ[N up (γ 1 C top + N top )]}, (A.48) as particle respiration in the middle box is no longer a loss of total nutrients. p r equivalence (large surface r) Suppose that a solution with given (p,k,r) is also intended to represent a solution with the same subsurface r but a larger respiration rate ˆr = r + δr in the top box. Then for N top K, the same net difference p r of production and respiration is obtained if p is 45

47 replaced by ˆp = p + δr: N top p r p r; (A.49) K + N top N top N top ˆp ˆr = (p + δr) (r + δr) K + N top K + N top (A.5) (p + δr) (r + δr) = p r. (A.51) Thus, if δr =.4, for example, the FP and SP cases (p,k,r) = {(1,1,.1),(.5,.25,.1)} can also represent ( ˆp,K, ˆr) = {(1.4,1,.5),(.95,.25,.5)}. This works well for N top K. However, the two production-respiration differences are not the same in the range N top K, which is important for the steady-state response. Better agreement can be obtained if K is also replaced by ˆK, where ˆK is chosen to make the slopes agree for N top K: d dn top ( p N ) top r K + N top N top K = d ( ˆp dn top N ) top ˆr ˆK = ˆK + N top N top ˆK pˆp K. (A.52) Then the FP and SP cases can be more accurately taken also to represent, again with δr =.4 for example, ( ˆp, ˆK, ˆr) = {(1.4,.71,.5),(.95,.13,.5)}. 46

48 List of Tables 1 Case-specific FP and SP parameters Common FP and SP parameters Derived FP and SP parameters

49 Table 1: Case-specific FP and SP parameters FP SP [units] S = sh 1 1 m d 1 s.5.5 d 1 p 1.5 d 1 K 1.25 mmol m 3 48

50 Table 2: Common FP and SP parameters value [units] r.1 d 1 λ.3 d 1 λ d 1 δ = s /s 1/3 δ H = H/H mid 1/3 δ S = S /S 1 N up 35 mmol m 3 N top 5 mmol m 3 C top 1 mmol m 3 O hyp {4,1} mmol m 3 H 2 m L 2 km U E = λ HL -1.4 m 2 s 1 τ y = ρ fu E -.14 N m 2 ( τ y is computed using ρ = 125 kg m 3 and f = 1 4 s 1 ) 49

51 Table 3: Derived FP and SP parameters FP SP [units] η.4.35 d 1 η λ.1.5 d 1 K.28.7 N s max.6.1 d 1 s d 1 s hyp d 1 p s η r λ B hyp ( O hyp = 1) B hyp ( O hyp = 1) B hyp ( O hyp = 4) B hyp ( O hyp = 4)

52 List of Figures 1 Schematic of box model. Upwelling favorable windstress (τ y ) drives offshore surface transport, moving both nutrients (N) and particles (C) offshore. This is compensated by onshore subsurface transport, bringing nutrients on-shelf. Upwelled nutrients drive particle production, some of which can be transported to depth, while others are exported offshore. Upwelling is parameterized by λ, the inverse shelf-transit or volume-renewal time, which is a function of the wind stress as well as the shelf width L and the depth of the surface layer H Model solutions vs. time t (d 1 ) for case FP. (a) O bot (mmol m 3 ), with the exponential approximation (4.1) (dotted) and the hypoxic/anoxic levels O hyp = {4,1} indicated. (b) (LH) 1 dn /dt (thick solid) with individual terms: λ N up (solid), λ N top (dashed), λ γ 1 C top (dotted), rc mid /(δγ) (dash-dot). (c) C bot (solid), C mid (dashed), C top (dash-dot). (d) N (thick solid), N bot (solid), N top (dash-dot). Steady state values are shown at t = 15 d (large dots) Model solutions as in Fig. 2, but for case SP Solutions for FP as in Fig. 2 but for sinusoidal time-dependent forcing (2.9) with λ 1 = λ =.3 d 1. The constant forcing (λ 1 = ) steady state values (large dots, at t = 15 d 1 ) and time-dependent solutions (dotted) for (a) O bot and (c) C bot are also shown Retention efficiencies (3.1) vs. time t (d 1 )for the FP (thick) and SP (thin) cases: c e (solid blue), c e (dashed blue). The steady state values for FP and SP are (c e,c e) = (.48,.32) and c e = c e =.2, respectively. [Ignore black lines] 58 51

53 6 Bottom oxygen drawdown time t hyp (solid) and exponential estimate t exp (dashed) for O hyp = {1,4} mmol m 3 ({thick,thin}) vs. λ for case FP with constant forcing. The basic exponential timescale (η λ ) 1 is also shown (dotted) Steady state solutions vs. λ with other parameters as in the (a-d) FP and (e-h) SP cases. (a,e) O bot (mmol m 3 ), with the hypoxic/anoxic levels O hyp = {4,1} indicated (dotted lines). (b,f) Terms of (LH) 1 dn /dt: λ N up (solid), λ (γ 1 C top + N top ) (dashed), rc mid /(δγ) (dash-dot). (c,g) C bot (solid), C mid (dashed), C top (dash-dot). (d,h) N (thick solid), N bot (solid), N top (dash-dot). The FP and SP (λ =.3 d 1 ) solutions are indicated (large dots) Retention efficiencies c e (solid) and c e (dashed) vs. (a) λ and (b) s, with other parameters as in the FP (thick) and SP (thin) cases. The efficiencies for the FP and SP values of λ and s are indicated (black dots) Steady state solutions vs. s with other parameters as in the FP case, and panels otherwise as in Fig. 7. The FP (s =.5) and SP (s =.5) solutions are indicated (large dots). The quantities s,s bt, and the hypoxic-optimal values s hyp, O bot (s hyp ),C bot (s hyp ),N bot (s hyp ) are also indicated (dotted lines in a,c,d,e,g,h) Cycle-mean bottom oxygen drawdown (top) and top box budget fractions (bottom; normalized) as a function of upwelling period T ω (d) for λ = λ 1 =.3 (left panels) and.1 (right) d 1 and other parameters as in FP

54 11 (a) Steady-state regime diagrams vs. ( p, s) for δ = 1/3, O hyp = O up = 1 mmol m 3. The critical lines p = 1/B hyp (green solid) and p = 1/B hyp (red solid) are shown for.2 K 2 (.57 K.541). The hypoxicoptimal sinking lines s = s hyp ( p) (dashed) and the no-growth critical lines s = s max (black dotted) and s = s (black dashed) are shown for K = {1,.25} (thick and thin, respectively). Along the optimal-sinking lines, for p < 1/B hyp (blue) there are no anoxic ( O bot O up ) steady states; for 1/B hyp < p < 1/B hyp (green) anoxic steady states exist for certain r and λ ; and for p > 1/B hyp (red) the s = s hyp steady states are anoxic for all r and λ. The FP and SP values of ( p, s) are indicated (black dots). (b) Same as (a), but vs. ( η λ, s) Steady state solutions O bot = O up O bot (mmol m 3 ; contours) or estimated dimensionless time to anoxia t exp = (r +λ )t exp (color) vs. ( η λ, s) for values of (K,δ, r, λ ) as in (a) FP and (b) SP, with O hyp = O up = 1 mmol m 3, where r = r/(r + λ ), λ = λ /(r + λ ) = 1 r. O bot is shown where O bot O up and t exp is shown where O bot > O up. The FP and SP values of ( η λ, s) are indicated (black dots). The dimensionless time t = 5 corresponds to 125 d for the FP and SP values of r + λ

55 windstress :" y < #$!" Top #" H 6&77%$)!" H M #$%" Bottom H L!"#$%%&'()"*+*,$-$+).&'/$+$)1$%2)-+*'&-)3,$45)! =!" y / (# flh ) < Figure 1: Schematic of box model. Upwelling favorable windstress (τ y ) drives offshore surface transport, moving both nutrients (N) and particles (C) offshore. This is compensated by onshore subsurface transport, bringing nutrients on-shelf. Upwelled nutrients drive particle production, some of which can be transported to depth, while others are exported offshore. Upwelling is parameterized by λ, the inverse shelf-transit or volume-renewal time, which is a function of the wind stress as well as the shelf width L and the depth of the surface layer H. 54