Kinetic bottlenecks to respiratory exchange rates in the deep-sea – Part 1 : Oxygen

Ocean warming is now reducing dissolved oxygen concentrations, which can pose challenges to marine life. Oxygen limits are traditionally reported simply as a static concentration threshold with no temperature, pressure or flow rate dependency. Here we treat the oceanic oxygen supply potential for heterotrophic consumption as a dynamic 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 provide the oxygen supply to the external surface of a respiratory membrane. This general oceanic oxygen supply potential is modulated by further properties such as the diffusive boundary layer thickness to define an upper limit to oxygen supply rates. While the true maximal oxygen uptake rate of any organism is limited by gas transport either across the respiratory interface of the organism itself or across the diffusive boundary layer around an organism, controlled by physico-chemical oceanic properties, it can never be larger than the latter. Here, we define and calculate quantities that describe this upper limit to oxygen uptake posed by physico-chemical properties around an organism and show examples of their oceanic profiles.


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 and declining O 2 , and the phys-ical combination of these properties that best describes the external boundary condition for the rate at which O 2 can be transported across membrane surfaces.Here, we focus on the theoretical background for characterization, description, and mapping of the impacts of ocean deoxygenation from a gas exchange rate perspective.
The rate controls 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 can be calculated.Likewise, the diffusion coefficients of the substances in question and their variation with temperature, as well as generally accepted formulations of the linkages between boundary layer thickness and fluid flow, are known.From these fundamental principles and observations, one can derive combined quantities which embody the essential chemical and physical principles in useful ways to illustrate and predict future changes in the ocean.
The quantities we define here are meant to be used in physico-chemical models of the ocean, for mapping the changing O 2 supply capacity of areas of interest such as for 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., 2011).The O 2 demands of marine organisms have classically been described as a rate problem (e.g.Hughes, 1966) in which diffusive transport across a boundary layer driven by partial pressure differences takes place.As with gas exchange rates at the sea surface (e.g.Wanninkhof, 1992) where wind speed is a critical variable, bulk fluid flow velocity over the animal surface is important.An oxygen concentration value alone does not adequately describe this process.For example, giving only an oxygen concentration as a limit for oxygen supply specifies 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 would be useful.

Published by Copernicus
The challenge in defining generic properties that describe the ocean in a way that is relevant to generic membrane transport of O 2 is 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 (resulting from salinity S, T , and P ), and the basic fluid dynamical form of the dependence of boundary thickness on flow.
There is no debate over the T , [O 2 ], P , and diffusivity terms since these are fundamental, organism-independent ocean chemical properties.The difficulty is in providing a way of including the flow term as a generic, organismindependent principle.There is no doubt of the existence of a diffusive boundary layer that is present over all ocean surfaces, thus also over all respiratory gas exchange interfaces.Generic and simplified descriptions of boundary layers are essential and widely used also for air-sea gas exchange rates, mineral dissolution rates, and phytoplankton nutrient uptake rates.The most widely used formulation is based on the dimensionless Schmidt number (e.g.Santschi et al., 1991;Wanninkhof, 1992).This relationship is standard within the ocean sciences and physiology (e.g.Pinder and Feder, 1990) and is also used here based on a planar surface model for the molecular exchange surface, which works well for all but microscopic scales (Zeebe and Wolf-Gladrow, 2001).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 desired is to provide improved functions and profiles that better allow comparison between different ocean regions undergoing change in their oxygen supply potential as the ocean warms and O 2 levels decline.

Materials and methods
A list of symbols and abbreviations used throughout this paper can be found in Table 1.

The oceanic oxygen supply potential SP O 2
Here we define the oceanic oxygen supply potential as the combination of pO 2 and diffusivity under in situ conditions constituting a theoretical maximal upper limit for respiratory O 2 uptake as provided by a given oceanic region.
Respiratory gas molecules expelled or taken up have to traverse both the respiratory interface and the diffusive boundary layer (DBL) in the medium surrounding the respiratory interface on the outside, i.e. the surrounding ocean.In general, either one of these steps can be the rate-limiting step for respiratory gas transport (see also the companion paper: Hofmann et al., 2013).
We denote E DBL as the maximal diffusive oxygen flux if only the diffusive boundary layer would need to be traversed (hypothetically assuming insignificant respiratory interface barrier).And E int as the maximal diffusive oxygen flux if only the respiratory interface would need to be traversed (hypothetically assuming insignificant boundary layer barrier).We can then write for the true total diffusive flux of oxygen E (in µmol O 2 s −1 cm −2 ) into an animal across both barriers which means the diffusive flux across both barriers E is always smaller than or equal to the maximal flux that the DBL transversion permits (i.e.E DBL ), and we can write E DBL , however, can be expressed by 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) as 1 1 where ρ SW is the in situ density of seawater (calculated according to Millero and Poisson (1981) as implemented in Hofmann et al. (2010)) in kg cm −3 ; D is the molecular diffusion coefficient for O 2 in cm 2 s −1 , calculated from temperature and salinity, e.g. as given in Boudreau (1996, Chapter 4); L is the DBL thickness in cm; pO 2 | DBL in µatm is the oxygen partial pressure difference across the DBL; and where K0 is the apparent Henry's constant for O 2 in mol kg −1 atm −1 (= µmol kg −1 µatm −1 ) at in situ conditions: Combining Eqs. ( 2) and (4) results in With this inequality, we can calculate an upper boundary for the diffusive flux of O 2 into an animal, as posed by the barrier of the diffusive boundary layer and the local pO 2 in the oceanic environment surrounding it.The true oxygen flux into an animal might be limited by respiratory interface transand hydrostatic pressure, first calculated conventionally from the oxygen saturation concentration (Garcia and Gordon, 1992) using potential temperature (θ; Bryden, 1973;Fofonoff, 1977).Resulting pO 2 values are then corrected for hydrostatic pressure (calculated from given depth values according to Fofonoff and Millard (1983)) according to Enns et al. (1965).
port2 , but it is always lower than the upper boundary defined by the right-hand side of Eq. ( 5).
The oxygen partial pressure difference across the DBL pO 2 | DBL can be expressed as with pO 2 | f expressing the oxygen partial pressure in the free stream beyond the DBL and pO 2 | s expressing the oxygen partial pressure directly on the gas exchange surface of the organism.Since, obviously, pO 2 | s ≥ 0 atm, the following inequality always holds true:

A. F. Hofmann et al.: Kinetic bottlenecks to chemical exchange rates -Part 1: Oxygen
Since our aim is to calculate an upper boundary for oxygen flux into an organism, we can simplify Eq. ( 5) by combining it with Eq. ( 7) to Equation ( 8) expresses the fact that the true respiratory gas exchange per unit area of gas exchange surface is always smaller than D ρ SW K0 L pO 2 | f .This expression, however, contains the DBL thickness L, which is an organism-specific quantity as it depends on the unique surface microstructure 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. ( 8) 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 physical 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 O 2 concentration profiles or maps, since it incorporates the essential T , P and further diffusion-related terms.

The generic maximal theoretical oxygen supply rate E max
All respiratory external surfaces will have a diffusive boundary layer that provides the contact with the ocean waters, and the thickness of this will in the macroscopic case3 depend upon the external flow over this surface, whether passive or controlled by organism motions.The most generic case is 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): 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 2 details this generic description of L as a function of u 100 .
For a given free stream oxygen partial pressure pO 2 | f , substituting Eq. ( 11) into Eq.( 8) allows us to define an upper limit or for the oxygen uptake rate E per unit area of respiratory interface supported by the given pO 2 | f as This definition of E max as an upper limit for the oxygen uptake rate utilizes a simplified and generic model to describe L. E max values are thus not biologically specific, but are intended solely to compare various oceanic regions in their potential to supply oxygen under given in situ conditions and to offer a rough estimate as to how flow rates over respiratory interfaces influence this potential.

The minimal oxygen concentration C f supporting a given laboratory-determined E
The traditional use of only the oxygen concentration [O 2 ] as a measure does not take into account the large regional variations in T , P , and further diffusion-related quantities, so it is useful to compare functions containing these properties with [O 2 ] alone.
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 -without loss of generality we assume throughout the paper a generic value of E = 20 × 10 −7 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. ( 8) 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 shows that pO 2 | f has to be greater or equal to the righthand side of Eq. ( 13) to support the given E. Again describing a limiting condition, we define p f as the minimal pO 2 | f that can support E as Table 2. Expressing the DBL thickness L as a function of water flow velocity: a generic planar surface description.
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):  (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) As mentioned above, in order to compare with standard conditions at one atmosphere, we assume laboratory experimental conditions at one atmosphere, so the apparent Henry's Law constant K0 E in Eqs. ( 13) and ( 14) 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 K0 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 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.

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 due to increased stratification 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;Helm et al., 2011), and the combined effects of T and O 2 have an impact on aerobic performance (e.g.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 opposing changes in oceanic properties.
To illustrate our newly defined quantities, we have chosen a set of example stations from different ocean basins for comparative purposes.Table 3 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 climatology for six , and C f for example stations around the world.Details on the tabulated quantities can be found in the text.The units used are as follows: [O 2 ]: µmol kg −1 ; SP O 2 : 10 −7 µmol kg −1 s −1 cm −1 ; E max : 10 −7 µmol kg −1 s −1 cm −2 ; C f : µmol kg −1 .For all locations we assume a constant flow velocity of u 100 = 2 cm s −1 ; more realistic flow profiles or organismspecific descriptions of the DBL thickness L can be employed here.For C f calculations we assume a generic constant value of E = 20 × 10 −7 µmol s −1 cm −2 .T E = 5 • C and S E = 34.NA indicates that values are not available for the respective depth.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 The leftmost column per station in Table 3 and Fig. 1 (both panels) show classical oxygen concentration profiles, and the second column in Table 3 and Fig. 2 show depth profiles of our newly defined oxygen supply potential SP O 2 (Eq.10).
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 in the overall range and 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 Western 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 dissolved gas exchange: E max
The quantity E max (Eq.12, Fig. 3, third column per station in Table 3) 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 mean flow over the animal respiratory surface 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 .While being simplified, the description for the DBL thickness implemented here reproduces the important 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).All animals require some form of fluid flow over their respiratory interfaces, and this, together with the shape of the surface, will determine the boundary layer thickness; and the relationship between external flow and boundary layer thickness will be controlled by the same physical processes.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) physical formalism is the same.The limit case of u 100 → 0 would yield L → ∞.This example formulation for L 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.

Revealing the separate 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 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 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 .A generic flow velocity of u 100 = 2 cm s −1 is assumed for all depths to calculate L. If available, detailed flow profiles can be used here, as well as organism-specific descriptions for L.
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 for free stream oxygen concentration C f is such that there is a low dependency 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).
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.15) with its direct predecessor quantity p f (Eq.14), 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 for 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 3 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 pan- A generic flow velocity of u 100 = 2 cm s −1 is assumed for all depths to calculate L. If available, detailed flow profiles can be used here, as well as organismspecific descriptions for L. For all calculations we assume a generic constant value of E = 20 × 10 −7 µmol s −1 cm −2 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).

Summary and conclusions
In this publication we define new 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, including estimating the impacts of ocean warming and declines in dissolved O 2 .The oxygen supply potential SP O 2 and the maximal oxygen supply rate E max that 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 for oxygen concentration C f supporting a given demand rate E is the quantity that explicitly expresses the effects of temperature, flow and hydrostatic pressure, without the visual obstruction by the dominant oxygen concentration signal.C f values can be used to approximate the hospitability of certain regions for particular animals with known oxygen uptake rate requirements.
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 T and [O 2 ].For example the basic solubility equation always results in lower O 2 concentration from ocean warming, which may be interpreted as more limiting to aerobic life.But when combined with the essential temperature and pressure dependencies of pO 2 and diffusivity, a more complex picture emerges.
Sc −b , with parameters a and b: Santschi et al. (1991): a = 0.078, b = 2 3 ; Shaw and Hanratty 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).

Fig. 4 .
Fig.4.The influence of flow velocity u 100 , temperature T and hydrostatic pressure P on the DBL thickness L and C f , the minimal oxygen concentration supporting a given laboratory-determined oxygen uptake rate E. The dependency of L (and all derived quantities) on u 100 is based on the simple exemplary model description employed here.While individual organism-specific dependencies may vary in detail, the general dependency of L on the flow velocity is captured here.For all calculations we assume a generic constant value of E = 20 × 10 −7 µmol s −1 cm −2 .Unless stated otherwise in the legend, latitude = 29.5 • N, S = 34, T = 5 • C, P = 100 bar, E = 20 × 10 −7 µmol s −1 cm −2 , T E = 5 • C, and S E = 34.

Fig. 5 .Fig. 6 .
Fig. 5.The influence of flow velocity u 100 and examples of given laboratory-determined oxygen uptake rates E (X = µmol s −1 cm −2 ), along depth profiles of temperature T , and hydrostatic pressure P at the Pacific station SC off Southern California (120.5 • W, 29.5 • N).T E = 5 • C, and S E = 34.

Table 1 .
List of symbols and abbreviations -listed in the order of their appearance in the manuscript.