Kinetic bottlenecks to chemical exchange rates for deep-sea animals – Part 1 : Oxygen

molecular exchange problem analogous to familiar gas exchange processes at the sea surface. A combination of the purely physico-chemical oceanic properties temperature, hydrostatic pressure, and oxygen concentration defines the ability of the ocean to supply oxygen to any given animal. This general oceanic oxygen supply potential is modulated by animal specific properties such as the diffusive boundary layer thickness 10


Introduction
One of the challenges facing ocean science is the need to make a formal connection between the emerging changes in the physico-chemical properties of the ocean such as rising temperature and CO 2 levels, declining O 2 , and the responses of marine animals.Here, we focus on the theoretical background for characterization, description, and mapping of the impacts of ocean deoxygenation from a kinetic perspective.
The kinetics of diffusive processes are well understood and embodied in Fick's first Law.The solubility of the metabolically important gases is well known as a function of salinity and temperature (Weiss, 1970(Weiss, , 1974;;Garcia and Gordon, 1992).From this and the observed distributions, partial pressures -the essential diffusive quantity across membranes -can be deduced.Likewise, the diffusion coefficients of the substances in question and their variation with temperature, boundary layer thickness and fluid flow are known.From these fundamental principles and observations, one can derive additional parameters which embodied the fundamental chemistry and physics in useful ways to illustrate and predict future changes in the ocean.
The problem of setting limits for aerobic respiration has been traditionally defined as a series of simple oxygen concentration limits.For example coastal ocean "dead zones" are usually defined as having less than 2 mg O 2 l −1 (61 µmol O 2 kg −1 ) and "suboxic" conditions as less than 10 µmol O 2 kg −1 .The term "hypoxia" is used to describe a rather wide range of values; for example very large areas of the ocean are predicted to become hypoxic under climate warming (Shaffer et al., 2009) based upon a concentration threshold of of 80 µmol O 2 kg −1 as defined in Gray et al. (2002).In a major series of coordinated papers reviewing the conditions for hypoxia (Middelburg and Levin, 2009;Gooday et al., 2009;Levin et al., 2009;Kemp et al., 2009;Zhang et al., 2010;Ekau et al., 2010;Rabalais et al., 2010;Pena et al., 2010;Naqvi et al., 2010;Gilbert et al., 2010) simple concentration limits were again cited.
A definition based simply upon concentration has served us well for decades in describing a steady state ocean since it provides a reasonable approximation to the O 2 resource available at a particular site.But it is ill suited to a comparison of ocean regions of different temperatures and pressures, or for an ocean undergoing climate change, since essential temperature, pressure, diffusivity and related factors are not included.This concept of kinetic limitation is not new to marine biologists; for example, Lazier and Mann (1989) and Karp-Boss et al. (1996) have looked at the process for specific organisms.In this paper, we consider only the general case and derive a set of quantities that better describe the kinetic limits on the oxygen supply potential of the ocean for animal respiration than does a simple oxygen concentration value.
The intention is to combine the essential physico-chemical oceanic properties in a mechanistically meaningful, yet generically applicable way.We define quantities that include only the organism-independent physical properties and that can be used to describe, map, and compare various oceanic regions of interest with respect to their oxygen supply potential.Secondarily, we define quantities that do include some organismspecific properties e.g. the respiratory surface diffusive boundary layer thickness and the oxygen partial pressure in direct contact with the animal respiratory tissue surface (essentially venous blood pO 2 ); but we do so only to illustrate the general principles and dependencies, and to investigate the potential for change of purely physical properties under ocean warming and oxygen depletion scenarios.Scientists concerned with particular species of animals can then apply these formulae for specific cases.
The quantities we define here are meant to be used in physico-chemical models of the ocean, for mapping the O 2 supply capacity of areas of interest such as in climate change assessments or for inter-regional comparisons to assist in ecosystem based management decisions as in the designation and knowledge of marine protected areas (e.g.Palumbi et al., 2009;Tallis et al., 2012).

Oceanic oxygen supply: physico-chemical vs. organism specific quantities
The O 2 demands of marine organisms have classically been described by physiologists as a rate problem (e.g.Hughes, 1966) in which diffusive transport across a boundary layer driven by partial pressure differences takes place; and, as in knowledge of gas exchange rates at the sea surface (e.g.Wanninkhof, 1992) where wind speed is a critical variable, bulk fluid flow velocity is important.A simple concentration value alone cannot adequately describe this process.For example, giving a concentration as a limit provides no temperature dependent information, so that the same value is assumed to be limiting over a temperature span of possibly 30 • C.This can be overcome to some extent by providing pO 2 as a limiting value (Hofmann et al., 2011), but the rate problem for gas transfer across a diffusive boundary layer contains also terms for diffusivity, and the relationship between boundary layer thickness and velocity over the surface, and thus a much more complete description is required.
Oxygen transport across the respiratory surface diffusive boundary layer is such a critical property that animals typically work to control flow and thus manage the supply of their respiratory rate needs.However there clearly are cases where the physically driven external flow field is controlling (Patterson and Sebens, 1989;Shashar et al., 1993) and is reflected in the physical distribution of organisms.And if O 2 transport is insufficient then either physical work to increase flow and thin the boundary layer is required, or organisms must resort to a temporary draw down of their finite store of alternative internal chemical energy supply.
The challenge here is in providing a useful function that more accurately describes the oceanic supply potential so as to separate as best as possible the purely physical terms which are universally applicable such as temperature T , oxygen concentration [O 2 ], hydrostatic pressure P (together resulting in an oxygen partial pressure pO 2 ), diffusivity, and the basic fluid dynamical form of the dependence of boundary thickness on flow, from those properties that are unique to any one animal such as the size of the respiratory surface area and any unique characteristics of basic shape or mode of swimming or pumping.
There is no debate over combining the T , [O 2 ], P , and diffusivity terms since these are fundamental properties.The difficulty is in providing a better way of including the flow term as a general principle.There is no doubt of the existence of a diffusive boundary layer that is present at all ocean surfaces, and formal descriptions of this are essential for air-sea gas exchange rates, mineral dissolution rates, and phytoplankton nutrient uptake rates.And in cases where the dimensions of the organism exceed about 1 mm (e.g.Zeebe and Wolf-Gladrow, 2001) the relationship between flow rate and boundary layer thickness is found to be strongly non-linear.The most widely used formulation is based on the dimensionless Schmidt number (e.g.Santschi et al., 1991;Wanninkhof, 1992) and this relationship is standard within the ocean sciences.There are other concepts of larger scale attached water volumes to swimming marine animals (Katija and Dabiri, 2009); these affect the bulk fluid flow properties and thus indirectly boundary layer thickness.
As the base case we use a planar surface model for the molecular exchange surface; although considerable fine structure exists the same basic diffusion rates hold.And while the animal will maintain a boundary layer typical for its respiratory needs, the physical forcing required to change the thickness of this layer will follow the same physical laws.We derive all gas flux properties on a per square centimeter scale so as to provide a way to normalize for different respiratory surface areas.
We stress that the end result of our work here cannot be profiles and maps that better describe O 2 rate limits for all organisms since individual requirements vary enormously.Rather it is the provision of relative profiles and maps that in very much the same way that profiles and maps of simple O 2 concentration values are used today can illuminate which regions of the ocean are better able to support aerobic marine life.And to do so in a more quantitative manner that permits incorporation of changes associated with ocean warming and differentiation with depth and temperature.

The oceanic oxygen supply potential SP O 2
Oxygen uptake or respiratory exchange rates by marine animals can be treated as a two step process.Step 1 is the diffusion of dissolved oxygen across the diffusive boundary layer (DBL) which surrounds the gas exchange tissues, whether they be cell walls of single cell organisms or the gill capillaries of large marine fishes or any animal in-between.Step 2 is the diffusion of oxygen through the tissues themselves.The rate for step 1 can be expressed using the standard representation (e.g.Santschi et al., 1991;Boudreau, 1996;Zeebe and Wolf-Gladrow, 2001) of Fick's first law, using gas partial pressures (e.g.Piiper, 1982;Feder and Burggren, 1985;Pinder and Burggren, 1986;Pinder and Feder, 1990;Maxime et al., 1990;Pelster and Burggren, 1996) where E DBL is the diffusive flux of oxygen across the DBL in µmol s −1 cm −2 , D is the molecular diffusion coefficient for O 2 in cm 2 s −1 , L is the thickness of the diffusive boundary layer for O 2 in cm, ρ SW is the density of seawater (the in situ density of seawater ρ SW is calculated according to Millero andPoisson, 1981, as implemented in Hofmann et al., 2010, in kg cm −3 ), ∆pO 2 | DBL is the oxygen partial pressure gradient across the DBL, and K 0 is the "apparent" Henry's Law constant in µmol kg −1 matm −1 .
Here we define the "apparent" Henry's Law constant K 0 as the fraction of a given oxygen concentration over its associated oxygen partial pressure at given conditions, including the hydrostatic pressure dependency according to Enns et al. (1965), in contrast to the classical Henry's Law constant definition which does not include this dependency.[O 2 ] in µmol kg −1 is the oxygen concentration in equilibrium with pO 2 , the associated oxygen partial pressure in matm calculated conventionally with the oxygen saturation concentration (Garcia and Gordon, 1992), using potential temperature (Bryden, 1973;Fofonoff, 1977), and then corrected for hydrostatic pressure P according to (Enns et al., 1965), using P values as calculated from given depth values (Fofonoff and Millard, 1983).Saturation water vapor partial pressure is accounted for as given in Zeebe and Wolf-Gladrow (2001).
The rate for step 2 of the respiratory oxygen uptake can be expressed in a similar fashion where D tissue is the oxygen diffusion coefficient in the tissue, L tissue is the tissue thickness, β is the capacitance for O 2 transfer inside the gas exchange tissue, ∆pO 2 | tissue is the oxygen partial pressure gradient across the gas exchange tissue, and ρ SW is the density of seawater as defined above.
The time for the transfer of a unit mass of oxygen in each step is the inverse of the diffusion rate: ; thus the total time t for the transfer of a unit mass of oxygen into an animal is t = t DBL + t tissue .By inverting this expression we can write the rate equation for the complete process as Our aim is to calculate the external physical constraints on respiratory gas exchange imposed by the oceanic conditions around the organism, the "oceanic oxygen supply potential".Therefore, we assume the oxygen transfer across the DBL to be the limiting step, i.e.E tissue E DBL .This implies that the total oxygen flux E approaches E DBL .And because the oxygen partial pressure difference across the DBL and through the gas exchange tissue is always smaller (equality in Eq. 5 would only hold in the theoretical limit case of the pO 2 in the venous blood being zero) than the free ocean oxygen partial pressure pO 2 | f , we can re-write Eq. ( 4) as Equation ( 5) expresses the fact that the true respiratory gas exchange per unit area of gas exchange surface is always smaller than This expression, however, contains the DBL thickness L, which is an organism specific quantity as it depends on the unique size of the respiratory surface area and any unique characteristics of the organism shape and mode of swimming or pumping.So, in order to arrive at a purely physico-chemical quantity that describes the organism-independent oceanic propensity for supplying oxygen, we multiply Eq. ( 5) by L and arrive at This allows us to define the oceanic oxygen supply potential SP O 2 in µmol s −1 cm −1 as the upper limit (i.e.maximal value) for the product of the respiratory oxygen uptake rate E and the DBL thickness L.
Thus, SP O 2 is a purely oceanic property, not dependent on any organism-specific properties or characteristics.It can be used to generate profiles and maps that illuminate which regions of the ocean are better able to support aerobic marine life than can simple concentration profiles or maps.

The generic maximal theoretical oxygen supply rate E max
Although the DBL thickness L is organism-specific, in macroscopic organisms (in microscopic organisms, L is usually taken do be equal to the radius of the sphere of the 13825 microorganism (cf.e.g.Stolper et al., 2010) -in this case, our formulation for E max can still be used with the appropriate L), there is a dependency of L on the fluid flow across gas exchange tissues.This flow will be some combination of the local current velocity and active pumping or swimming by the animal in question.Here, we do not aim to exactly describe the DBL thickness L for any particular animal, but we want to introduce and investigate the general dependency of L on fluid flow to compare various oceanic regions to one another as well as to have a rough measure for the energy, in terms of required flow changes, that might be required to mitigate the effects of global change such as deoxygenation, acidification and warming.To this end, we use a simple planar surface description of the flow dependency of L, as was used to describe the mineral dissolution experiments in Santschi et al. (1991).This yields the generic dependency of L on the fluid flow rate across the interface.(If one desires the organism specific dependency of L, then equations accounting for the fluid flow around the organism must be included in the derivation.This is beyond the scope of this paper.)The DBL thickness L can be expressed as the fraction of the molecular diffusion coefficient D and the mass transfer coefficient K , which is a function of the fluid flow u 100 .
which effectively makes L a function of flow velocity u 100 .Table 1 details this generic description of L as a function of u 100 .Substituting Eq. ( 8) into Eq.( 5) allows us to define an upper limit or "outer envelope" for the oxygen uptake rate E as effectively replacing the dependency on D with the dependency on the transfer coefficient K (see e.g.Santschi et al., 1991), which includes a description of the boundary layer thickness.However, it is important to note here, that we define E max in Eq. ( 9) as thickness L, even a fixed, constant value, while maintaining the important dependency of E max on diffusivity.As mentioned before, the definition of E max as upper limit for the oxygen uptake rate as defined in Eq. ( 9), utilizes a simplified and generic model to describe L. E max values are thus not specific to any particular animal, but are intended solely to compare various oceanic regions in their potential to support aerobic life and in their potential for resilience against global change.At the same time, values of SP O 2 can be converted to an organism specific upper limit for oxygen uptake by dividing by an organism specific boundary layer thickness L and considering the area of the gas exchange surface of the organism in question.

The minimal oxygen concentration C f supporting a given, laboratory-determined E
In order to explicitly describe and reveal the influence of temperature and diffusion while comparing various oceanic regions, warm and shallow, as well as deep and cold to one another, we define another quantity in which we remove the effect of the oxygen content of the water.To explicitly include the important dependence of gas exchange on partial pressure (e.g.Piiper, 1982;Pinder and Feder, 1990;Pelster and Burggren, 1996;Childress and Seibel, 1998;Seibel et al., 1999) and the dependency of partial pressure on hydrostatic pressure (Enns et al., 1965), we assume a given oxygen uptake rate E (in µmol s −1 cm −2 ), experimentally determined at diffusivities D and DBL thicknesses L equal to the respective in-situ values, but at one atmosphere.From Eq. ( 5) we can then determine a relation between the free stream oxygen partial pressure pO 2 | f and this given oxygen uptake rate E as which expresses that pO 2 | f has to be larger or equal to the right hand side of Eq. ( 10) to support the given E .Again describing a limiting condition, we define p f as the minimal As mentioned above, we assume experimental conditions at one atmosphere, so the apparent Henry's Law constant K 0 E in Eqs. ( 10) and ( 11) is calculated with one atmosphere.To explicitly include the hydrostatic pressure dependency of partial pressure, i.e. to obtain the oxygen concentration that would result in p f at in-situ hydrostatic pressures, we have to convert p f to a concentration using an apparent Henry's Law constant K 0 IS calculated at in-situ hydrostatic pressure.Thus, we can define C f , the minimum oxygen concentration [O 2 ] in µmol kg −1 the free flowing stream must have to sustain E as Note that, in Eq. ( 12), we retain D and L as independent quantities and do not combine the two parameters to the transfer coefficient K .We do this to provide the opportunity to plug in organism specific values for L. Additionally, the inclusion of the ratio of the Henry's Law constants at one atmosphere and at in-situ pressures explicitly includes the pressure dependency of partial pressures, which makes C f a quantity that can be used to directly compare warm and shallow to cold and deep oceanic regions.
Exemplary metabolic rates are needed to calculate reasonable examples for external O 2 limit values for a given metabolic rate, i.e.C f values using Eq.(12).To obtain example C f values of the appropriate scale, we choose three example oxygen uptake rates, expressed in µmol O 2 s −1 cm −2 , that are roughly consistent with reported values for marine eggs (e.g.Yasumasu and Nakano, 1963), larvae (e.g.Houde and Schekter, 1983), embryos (e.g.Walsh et al., 1989) and adult animals (e.g.Seibel et al., 1999).If no other data was given, the given values of E were converted to per area of exchange 13828 surface values using a standard fish egg representation of a sphere of 1 mm diameter (Chambers and Leggett, 1996).All values have been standardized to a metabolic activity at 5 • C using a Q10 value of 2.5 (van Maaren, 1999).All oxygen uptake values employed are simply examples of the appropriate scale used to illustrate the calculation and use of the C f values defined here.These examples are chosen solely to illustrate the introduced generally applicable physical concept of C f over a range of physically realistic values.

The theoretically required flow velocity change to offset warming and deoxygenation scenarios ∆u 100
Ocean warming not only often increases metabolic demand in animals (i.e.their required E values increase), but it also influences the physical environment that the ocean provides for aerobic life, here expressed as E max .To determine the hospitability of certain ocean regions under ocean warming scenarios, changed E values have to be compared to changed E max values.How E values change with temperature, i.e. how metabolic demand rises with temperature, is very organism specific and is beyond the scope of this paper.Here we focus on the effect of ocean warming on E max , a measure for the physical environment that the ocean provides for aerobic life.E max is influenced by ocean warming in two major ways.Warming enhances diffusion and increases pO 2 for a given [O 2 ], both of which increases E max for otherwise constant conditions.However, warming also entails a decrease in oxygen concentration due to the temperature dependency of oxygen solubility, resulting in a decrease of [O 2 ] which decreases E max .To investigate which effect is dominant and to characterize the physical requirements for offsetting change, i.e. to maintain the same E max values, we calculated the fluid flow velocity change required to offset ocean warming and resulting deoxygenation scenarios of 1 • C, 2 • C, and 3 • C. To this end, we define a quantity ∆T ,E max =ct 100 is calculated with a numercial root finding procedure using Eq. ( 9).This is done by first calculating E max for unchanged conditions and then numerically finding the fluid flow value that yields that same E max under an assumed change in T and [O 2 ], i.e. finding the root of the function ∆T ,E max =ct 100 being the variable.Although we are aware that it will take a considerable amount of time for temperature changes be manifested in the whole ocean, for the sake of simplicity and to illustrate the concept, we use a constant temperature offset at all depths in the watercolumn for plots and tables of ∆u 100 .We note that pronounced ocean warming (Nakanowatari et al., 2007;Lyman et al., 2010) and O 2 depletion (Stramma et al., 2008) has already been observed in the upper 1 km.Oxygen concentration offsets are calculated from temperature dependent solubility changes based on in situ base temperatures.

Example oceanographic data
All T , S, P , and [O 2 ] data for ocean profiles have been extracted from the Ocean Data View (Schlitzer, 2010)

Results and discussion
As noted above, warming of the ocean (Levitus et al., 2005;Lyman et al., 2010) is reducing the oxygen concentration via the solubility effect, decreasing ventilation and increasing oxygen drawdown from microbial mineralization of organic matter.Thus global ocean oxygen concentrations are declining (Chen et al., 1999;Jenkins, 2008;Stramma et al., 2008), and the combined effects of T and O 2 have impact on aerobic performance (Poertner and Knust, 2007).Yet at the same time, increased temperature enhances diffusion, and results in increased gas partial pressures for the same concentrations, both of which enhance diffusive oxygen uptake rates.The properties we define above allow for a comparison of the relative roles and impacts of these changing oceanic properties.
To illustrate our newly defined quantities, we have chosen a set of example stations from different ocean basins for comparative purposes.Table 2 provides a tabulation for the classical oxygen concentration ([O 2 ]) hydrographic depth profiles as well as depth profiles of all quantities defined in this paper, calculated with data from the World-Ocean-Atlas 2009 (Garcia et al., 2010) oxygen climatolygy for six example stations around the world ocean and the Mediterranean (SC: Southern California (120.5 • W, 29.5 • N); WP: Western Pacific (126.5 • E, 11.5 • N); CH: Chile (75.5 • W, 33.5 • S); WA: Western Africa (6.5 • E, 15.5 • S); BB: Bay of Bengal (87.5 • E, 18.5 • N); MD: Mediterranean (18.5 • E, 35.5 • N)).

Describing the oxygen supply potential of the ocean including temperature effects: [O 2 ] vs. SP O 2
The leftmost column per station in Table 2 and Fig. 1 show classical oxygen concentration profiles and the second column in Table 2 and Fig. 2 show depth profiles of our newly defined oxygen supply potential SP O 2 (Eq.7).It can be clearly seen that, while the general shape of the profiles is similar due to the dominating oxygen signal, there are marked differences especially at depth.Those differences can be attributed to the effect of temperature dependent diffusivity that is included in the definition of SP O 2 .For example, while the oxygen concentration increases again to about 60 % of surface values at 4000 m depth for the three Pacific Ocean stations shown (left panel, Fig. 1), SP O 2 values (left panel, Fig. 2) increase only to about 40 % of surface values due to colder temperatures limiting diffusion at depth.
Similarly, the right panels of Figs. 1 and 2 show that, while the oxygen concentration at the West-Africa station rises above Mediterranean values at a depth of about 1500 m (Fig. 1), SP O 2 values remain higher in the Mediterranean all the way down to 4000 m depth (Fig. 2) due to diffusivity-enhancing higher temperatures.In the profiles shown for the station off Chile (CH, left panels of Figs. 1 and 2), the well known horizontal penetration of an oxygen maximum into the oxygen minimum zone (Wyrtki, 1962) is reflected in the calculated SP O 2 values, however, the local maximum is less pronounced for SP O 2 than for [O 2 ] due to the effect of temperature.The sample station in the Mediterranean (MD, right panels of Figs. 1 and 2), where there is a combination of the least diffusive restriction due to higher temperatures and a mean oxygen concentration being nearly twice as high as the site examined off Southern California (212 vs. 114 µmol kg −1 ), results in the highest deep water oxygen supply potential (SP O 2 ) values for locations we have examined.At the Bay of Bengal station (BB, right panel of Fig. 2), the waters at 800 m and below have a higher oxygen supply potential than the Southern California (SC) station, but at shallower depths the situation is reversed.

Incorporating a generic description of the influence of fluid flow velocity on gas exchange: E max
The quantity E max (Eq.9, Fig. 3, third column per station in table 2) determines the combined influences of oxygen concentration, temperature, and fluid flow velocity over gas exchange surfaces, so as to provide an estimate of the maximal diffusive transport rates per unit gas exchange area that the ocean can support.Using a canonical constant current velocity of u 100 = 2 cm s −1 we can visualize example profiles of our quantity E max at our example stations in the world oceans (Fig. 3), which are, due to the constant u 100 , very similar in shape to the SP O 2 profiles in Fig. 2. To calculate E max values and profiles for specific purposes, detailed flow fields and organism-specific descriptions for the DBL thickness L should be used or more specific molecular exchange models employed (see, e.g.Lazier and Mann, 1989;Karp-Boss et al., 1996).
Here, we define E max with a generic description of its dependency on L and thus fluid flow velocity u 100 to be able to define the quantity ∆u 100 to estimate the relative energy requirement needed to mitigate certain ocean warming and deoxygenation scenarios, as will be discussed in Sect.4.4.While being simplified, the description for the DBL thickness implemented here reproduces the general non-linear dependency of L on the fluid flow velocity u 100 , and the dependency of this relation on temperature (leftmost panel in Fig. 4).Although for real animals, considerable fine structure of gas exchange tissues may exist, and the animal will maintain a boundary layer typical for its respiratory needs, the physical forcing required to change the thickness of this layer will follow the same physical laws.While equivalently one could plot the mass transfer coefficient K , we plot the DBL thickness L here (similar to Santschi et al., 1991), as it can be better visualized and intuitively understood.K however, is equivalent to the "gas transfer velocity" or "piston velocity" commonly used to describe air-sea gas exchange, both for the open ocean (Wanninkhof, 1992) and coastal seas (e.g.Gypens et al., 2004) and estuaries (Hofmann et al., 2008), so the underlying simplified physical formalism is the same.The limit case of u 100 → 0 would yield L → ∞.This example formulation for L (or better K ) as a function of u 100 is thus not defined in a physically meaningful way for zero flow velocity and should not be used for the stagnant water case.Here we plot L with a minimum of u 100 = 0.5 cm s −1 , which can be seen as an operational lower limit.additionally incorporate the effect of hydrostatic pressure when comparing various oceanic regions in their ability to support given laboratory-determined respiratory rates E , we have defined the quantity C f .C f profiles can then be compared to [O 2 ] profiles to determine oceanic regions that support the given oxygen demand E .Due to the nonlinear dependency of the thickness of the diffusive boundary layer L on the current velocity u 100 , the minimal oxygen concentration C f supporting a given oxygen uptake rate E is also a nonlinear function of u 100 .The middle panel of Fig. 4 explicitly visualizes this dependency for different temperatures but constant pressure and salinity, and the right panel of Fig. 4 depicts this dependency for different hydrostatic pressures.The left panel of Fig. 5 shows depth profiles of C f for the Southern California (SC) example hydrographical station (i.e. with in-situ temperature, salinity and pressure) for three different values for u 100 .This further illustrates the sensitivity of C f with respect to u 100 .It can be shown for the range of u 100 values in the middle panel of Fig. 4, that a u 100 decrease by half results in approximately a doubling of C f (i.e.C f is roughly proportional to the inverse of u 100 ).The nonlinear character of the relation between the fluid flow velocity u 100 and the limit free stream oxygen concentration C f is such that there is a low dependency when when u 100 > 2 cm s −1 and very high dependency when u 100 < 1 cm s −1 .This is a result of the respective behavior of the thickness of the diffusive boundary layer L on u 100 , as also described by e.g.Garmo et al. (2006).

Revealing
The right panel of Fig. 5 shows example depth profiles of C f at the SC station for three different respiratory rates E , on realistic orders of magnitude, to express the sensitivity of C f with respect to E .
The general shape of C f depth profiles in Fig. 5 can be explained by comparison of C f (Eq.12) with its direct predecessor quantity p f (Eq.11), the minimal oxygen partial pressure required to support E .The quantity C f directly includes effects of diffusivity, boundary layer thickness, and temperature and hydrostatic pressure effects on partial pressure.The predecessor quantity p f expresses only effects on diffusivity and boundary layer thickness.As temperature decreases with depth, so does the "efficiency" of diffusion, with the result that L increases.This translates into a higher p f with depth, i.e. a higher necessary pO 2 to sustain the given oxygen consumption rate E .The limit oxygen concentration C f initially follows suit, but as hydrostatic pressure increases with depth, so does the pO 2 for a given [O 2 ] (Enns et al., 1965).This means that p f , the pO 2 necessary to sustain E , which is more or less constant with depth from below 2000 m, as temperature does not change anymore, is sustained by a smaller and smaller oxygen concentration: C f decreases from about 2000 m on.
Different ocean basins exhibit markedly different temperature and salinity profiles; these differences affect the quantity C f since this subsumes the influences of temperature, salinity and hydrostatic pressure on diffusive gas transport.Figure 6 and the fourth column in Table 2 show C f depth profiles for our example hydrographical stations, assuming a constant current velocity of 2 cm s −1 .In the Pacific, the profiles are rather similar, while warm enclosed seas like the Mediterranean differ markedly.Here, due to warm temperatures throughout the water column, the entire profiles are shifted towards lower C f values.It is remarkable that in the Mediterranean, the oxygen concentration at 4000 m depth required to sustain a given oxygen consumption rate is lower than at the surface.In the Atlantic the C f maximum is sharply defined at around 1000 m depth; in the Pacific the maximum is more broadly defined at around the same depth, in keeping with classical hydrographic profiles.In the Indian Ocean (Bay of Bengal), the C f maximum is deeper at ≈ 2000 m.Given the marked similarity of C f profiles in the Pacific (left panel of Fig. 6), one can conclude that the differences in oxygen supply potential SP O 2 between various stations in the Pacific (left panel of Fig. 2) are mainly due to differences in oxygen concentration profiles.This is not the case for the example stations that are not located in the Pacific (right panels of Figs. 2 and 6), where the effects of temperature and pressure considerably contribute to the differences in SP O 2 profiles.
Even though our exact description of the dependency is generalized, the fact that the limit oxygen concentration C f is dependent on current velocity makes it necessary that water flow rates be included in the set of variables that are controlled or reported when comparing different systems or different animal responses (e.g.Stachowitsch et al., 2007;Riedel et al., 2008;Haselmair et al., 2010).This dependency also indicates that sessile benthic animals in high current environments have an advantage in low oxygen conditions over free swimming fish and over zooplankton that are suspended relatively stationary with respect to the water mass surrounding it.It also explains why under the same low oxygen conditions, continental slopes or the flanks of seamounts experiencing comparatively high mixing rates and currents (van Haren and Gostiaux, 2010) can be more hospitable environments for animals than comparatively stagnant waters.

First-order estimate of the energy required for mitigation of global change scenarios: ∆u 100
In order to investigate which of the multiple effects of warming and deoxygenation will be dominant for certain locations in the ocean, we define the quantity ∆u 100 ; this describes the flow offset from the canonical value of 2 cm s −1 needed to keep E max constant under global warming and deoxygenation scenarios.It is essentially the work that must be done to thin the boundary layer so as to restore a required O 2 uptake rate.
For a first order estimate we assume that the oxygen concentration decreases for a given constant temperature offset by only the difference in the oxygen saturation concentration at every location in the profile.We are aware that this will underestimate true ocean changes, as increased T will also result in higher microbial oxygen consumption rates fueled by the large mass of organic carbon stored in ocean organic matter in sediments, seawater, and sinking particles.Figure 7 and the fifth to seventh columns in Table 2 show ∆u 100 values for all our example stations and 1 • C, 2 • C, and 3 • C warming scenarios.
It can be seen that within the oxygen minimum zone, especially off the Southern California (SC) and the Bay of Bengal (BB) locations, a relatively high flow increase would be needed to compensate for the warming and deoxygenation scenario.As the oxygen concentration becomes small, very high flow rates are necessary to adequately 13836 decrease L and thus to maintain E max at its original value.The reason for this is the rather flat dependency of diffusive transport on flow for flow values higher than 2 cm s −1 (see Fig. 4).These then will clearly be very highly stressed regions under even modest ocean warming scenarios.
In marked contrast in warm regions at depths shallower than the oxygen minimum zone and particularly in the warm Mediterranean Sea, the diffusion enhancement of increased temperatures overcompensates for the oxygen concentration loss.Thus ∆u 100 values are negative, which means a flow less than 2 cm s −1 would suffice to provide the same E max as in the base case.But recall that we assume here only a solubility reduction in [O 2 ] and in practice ∆[O 2 ] will be significantly higher.
The effect is principally due to the fact that the solubility, and thus the oxygen concentration difference, is smaller for warm waters than it is for cold waters and thus temperature enhanced diffusion overcomes the effect of deoxygenation.The absolute values of negative ∆u 100 flow rates are smaller than the positive ones since the relationship of L with flow is much steeper as flow rates decrease below 2 cm s −1 (see Fig. 4).
The estimates provided here are of the physico-chemical properties for inter-regional comparison and actual impacts will be species specific.Furthermore, we are only investigating the effect of ocean warming on the oceanic supply of oxygen.The effect of increased temperature on the oxygen demand, i.e. the increased metabolic demand of the organism now living at higher temperature, which presumably is very organism specific, is not treated here.

Summary and conclusions
In this publication we define quantities that describe the ocean's ability to supply oxygen, based on diffusive boundary transport rate limitations.These quantities subsume well known oceanic physical properties relevant to diffusive boundary transport into functions that may be used for various purposes.The oxygen supply potential SP O 2 and the maximal oxygen supply rate E max an environment can sustain are the more general parameters that could be used to map specific oceanic regions according to their ability to supply oxygen.The limit oxygen concentration C f supporting a given demand rate E is the quantity that explicitly expresses the effects of tempertature, flow and hydrostatic pressure, without the obstruction by the oxygen concentration signal.C f values can be used to roughly estimate the hospitability of certain regions for particular animals with known oxygen uptake rate requirements.We also define the quantity ∆u 100 that describes the significance of ocean warming and deoxygenation scenarios to aerobic life by specifying the flow offset necessary to counteract the presented scenarios.In cases with adverse effects to animals, the required increase in flow as expressed by positive ∆u 100 values is an indicator of the physical energy required to maintain the same maximal oxygen supply rate (E max ).Whether this required energy will be supplied by the energy contained in ambient flow or is a result of animal activity like pumping or swimming is deliberately not distinguished here.
All our newly defined quantities express the requirements and limitations imposed only by the oceanic physical environment.The results from example oceanographic stations around the world strongly suggest a greater diversity of regions and a more complex response of biogeochemical cycles to ocean warming than anticipated from the simple change in O 2 concentration alone.It should not be surprising that the fields produced appear superficially to resemble traditional O 2 concentration profiles and maps; descriptions of the formation of the [O 2 ] minimum and well established gradients along major ocean circulation pathways are dominant features and powerful drivers that have long been described (Wyrtki, 1962).It is for this reason that the relatively crude representation of various limits by simple concentration values has been in use for so long; they are familiar and have served as reasonable approximations.But the basic kinetic rate representation given here allows for much greater insight, in particular for different oceanic depth realms and for an ocean changing simultaneously in The DBL thickness L can be expressed as the fraction of the temperature-dependent molecular diffusion coefficient D for O 2 in cm 2 s −1 , calculated from temperature and salinity as given in Boudreau (1996, Chapter 4) using the implementation in the R package marelac (Soetaert et al., 2010), and the mass transfer coefficient K (Santschi et al., 1991;Boudreau, 1996) K can be calculated for O 2 from the water-flow induced shear velocity u in cm s −1 and the dimensionless Schmidt number Sc for O 2 (as calculated by linearly interpolating two temperature dependent formulations for S = 35 and S = 0 in Wanninkhof, 1992, with respect to given salinity) with parameters a and b: Santschi et al. (1991): a = 0.078, b = 2 3 ; Shaw and Hanratty (1977) (also given in Boudreau, 1996, a = 0.0889, b = 0.704); Pinczewski and Sideman (1974) as given in Boudreau (1996): a = 0.0671, b = 2 3 ; Wood and Petty (1983) as given in Boudreau (1996): a = 0.0967, b = 7 10 .Due to small differences we use averaged results of all formulations.u can be calculated from the ambient current velocity at 100 cm away from the exchange surface u 100 and the dimensionless drag coefficient c 100 (Sternberg, 1968;Santschi et al., 1991;Biron et al., 2004) c 100 is calculated from the water flow velocity u 100 as (Hickey et al., 1986;Santschi et al., 1991)  E max : 10 −7 µmol kg −1 s −1 cm −2 ; C f : µmol kg −1 ; ∆u 100 : cm s −1 .For all locations we assume a constant flow velocity of u 100 = 2 cm s −1 , more realistic flow profiles or organism specific descriptions of the DBL thickness L can be employed here.T E = 5 • C and S E = 34.NA indicates that values are not available for the respective depth, Inf indicates situations where ∆u 100 is not defined since subtracting the saturation concentration difference associated with the temperature increase from the actual oxygen concentration would result in negative oxygen concentrations.SC: Southern California (120.5 • W, 29.5 • N); WP: Western Pacific (126.5 • E, 11.5 • N); CH: Chile (75.5 • W, 33.5 • S); WA: Western Africa (6.5     ] is assumed to decrease according to the offset in oxygen saturation concentration, using in-situ T as base value and applying the respective assumed temperature change.Positive values of ∆u 100 indicate that the flow would have to be increased from the base value of 2 cm s −1 to compensate for the combined warming and deoxygenation effects, negative values indicate that a slower flow would suffice to provide the same E max as in the base case.To not loose details on the small scale and to limit the plots to meaningful values, we truncated the abscissa to ∆u 100 values ≤ 1.65 cm s −1 .

13819
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Discussion
Paper | Discussion Paper | Discussion Paper | Discussion Paper |

13821
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

) 13823 Discussion
Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Discussion
Paper | Discussion Paper | Discussion Paper | Discussion Paper |

SP O 2 L
, which makes our definition of E max generic as it allows any description of the DBL Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

13827
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | pO 2 | f that can support E as Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | that expresses the required flow change under a given warming and deoxygenation 13829 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | scenario as ∆u 100 := u ∆T ,E max =ct 100 − u base 100 (13) with u base 100 being the canonical mean background flow value of 2 cm s −1 used throughout this paper (also used as the typical flow rate across the gill surface in Hughes, 1966) and u ∆T ,E max =ct 100 being the bulk fluid flow value that results under changed conditions (∆T and resulting ∆[O 2 ]) in the same E max value as u 100 = u base 100 under standard T , [O 2 ] conditions.u version of the World-Ocean-Atlas 2009 oxygen climatology (Garcia et al., 2010).Calculations have been performed in the open-source programming language R (R Development Core Team, 2010).13830 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

13831
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Discussion
Paper | Discussion Paper | Discussion Paper | Discussion Paper | the influences of temperature, flow, and hydrostatic pressure on gas exchange: C f To explicitly single out the influence of temperature and fluid flow on gas exchange, which is co-mingled with the oxygen signal in profiles of SP O 2 and E max , and to 13833 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

13835
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Discussion
Paper | Discussion Paper | Discussion Paper | Discussion Paper | 13837 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | T and [O 2 ].For example the basic solubility equation always results in lower O 2 concentration from Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | areas: synthesis and future development, Biogeosciences, 7, 1443-1467, doi:10.5194/bg-7-1443-2010,2010.13819 13845 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |Table 1. Expressing the DBL thickness L as a function of water flow velocity: a generic planar surface description.

Table 2 .
Depth profiles of [O 2 ], SP O 2 , E max , C f , ∆u 100 for example stations around the world.Details on the tabulated quantities can be found in the text.∆u 100 values are calculated for three ocean warming and associ-