the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Temperature effect on seawater fCO_{2} revisited: theoretical basis, uncertainty analysis and implications for parameterising carbonic acid equilibrium constants
Matthew P. Humphreys
The sensitivity of the fugacity of carbon dioxide in seawater (fCO_{2}) to temperature (denoted υ, reported in % °C^{−1}) is critical for the accurate fCO_{2} measurements needed to build global carbon budgets and for understanding the drivers of air–sea CO_{2} flux variability across the ocean. However, understanding and computing υ have been restricted to either using empirical functions fitted to experimental data or determining it as an emergent property of a fully resolved marine carbonate system, and these two approaches are not consistent with each other. The lack of a theoretical basis and an uncertainty estimate for υ has hindered resolving this discrepancy. Here, we develop a new approach for calculating the temperature sensitivity of fCO_{2} based on the equations governing the marine carbonate system and the van 't Hoff equation. This shows that, to first order, ln (fCO_{2}) should be proportional to $\mathrm{1}/{t}_{\mathrm{K}}$ (where t_{K} is temperature in kelvin), rather than to temperature, as has previously been assumed. This new approach is, to first order, consistent with calculations from a fully resolved marine carbonate system, which we have incorporated into the PyCO2SYS software. Agreement with experimental data is less convincing but remains inconclusive due to the scarcity of direct measurements of υ, particularly above 25 °C. However, the new approach is consistent with field data, performing better than any other approach for adjusting fCO_{2} by up to 10 °C if spatiotemporal variability in its single fitted coefficient is accounted for. The uncertainty in υ arising from only measurement uncertainty in the main experimental dataset where υ has been directly measured is in the order of 0.04 % °C^{−1}, which corresponds to a 0.04 % uncertainty in fCO_{2} adjusted by +1 °C. However, spatiotemporal variability in υ is several times greater than this, so the true uncertainty due to the temperature adjustment in fCO_{2} adjusted by +1 °C using the most widely used constant υ value is around 0.24 %. This can be reduced to around 0.06 % using the new approach proposed here, and this could be further reduced by more measurements. The spatiotemporal variability in υ arises mostly from the equilibrium constants for CO_{2} solubility and carbonic acid dissociation (${K}_{\mathrm{1}}^{\ast}$ and ${K}_{\mathrm{2}}^{\ast}$), and its magnitude varies significantly depending on which parameterisation is used for ${K}_{\mathrm{1}}^{\ast}$ and ${K}_{\mathrm{2}}^{\ast}$. Seawater fCO_{2} can be measured accurately enough that additional experiments should be able to detect spatiotemporal variability in υ and distinguish between different parameterisations for ${K}_{\mathrm{1}}^{\ast}$ and ${K}_{\mathrm{2}}^{\ast}$. Because the most widely used constant υ was coincidentally measured from seawater with roughly global average υ, our results are unlikely to significantly affect global air–sea CO_{2} flux budgets, but they may have more important implications for regional budgets and studies that adjust by larger temperature differences.
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The surface ocean continuously exchanges carbon dioxide (CO_{2}) with the atmosphere. The net direction of exchange varies through space and time, with some ocean areas acting as CO_{2} sources and others as sinks (Takahashi et al., 2009). Whether an area acts as a CO_{2} source or sink depends on the difference between the fugacity of CO_{2} (fCO_{2}) in the sea and in the air (denoted $\mathrm{\Delta}f{\mathrm{CO}}_{\mathrm{2}}=f{\mathrm{CO}}_{\mathrm{2}}\left(\mathrm{sea}\right)f{\mathrm{CO}}_{\mathrm{2}}\left(\mathrm{air}\right)$), with net CO_{2} transfer from high to low fCO_{2}. The fugacity of CO_{2} is similar to its partial pressure (pCO_{2}) but with a small correction for the nonideal behaviour of CO_{2} in air; fCO_{2} is typically about 99.7 % of pCO_{2}. The gas exchange rate is modulated by a set of processes that are commonly parameterised as a function of wind speed (Wanninkhof, 2014). Most of the surface ocean is either a CO_{2} source or sink, rather than being in equilibrium with the atmosphere, because many of the processes that modify seawater fCO_{2} (e.g. seasonal cycles of temperature and primary productivity) operate on faster timescales than air–sea CO_{2} equilibration, which takes from months to a year for the surface mixed layer (Jones et al., 2014).
On top of the natural background variability, the global ocean is a net CO_{2} sink due to rising atmospheric fCO_{2}. About a quarter of the anthropogenic CO_{2} emitted each year, some 2.8 ± 0.4 Gt C in 2022, is taken up by the ocean, mitigating the greenhouse effect of CO_{2} on Earth's climate (Friedlingstein et al., 2023). Together, uptake of CO_{2} and warming drive changes in the marine carbonate system, altering biologically important variables like seawater pH and the saturation states with respect to carbonate minerals (Humphreys, 2017). Quantifying marine CO_{2} uptake requires highly accurate fCO_{2} measurements: it is driven by a global mean ΔfCO_{2} of −4.1 ± 1.6 µatm (Fay et al., 2024), which is around 2 % of mean fCO_{2}. Consequently, small systematic biases in seawater fCO_{2} propagate into large errors in ΔfCO_{2} and, thus, the total ocean CO_{2} sink. The required measurement accuracy has been quantified by the Global Ocean Acidification Observing Network (GOAON), with the network suggesting a minimum accuracy of 0.5 % (i.e. around 2 µatm) for “climate quality” seawater fCO_{2} measurements that can be used to detect decadal trends or 2.5 % (∼10 µatm) for “weather quality” fCO_{2} data to investigate changes of greater magnitude, such as seasonal cycles (Newton et al., 2015).
The observational constraint on marine CO_{2} uptake is based on measurements of surface ocean fCO_{2} from ships, compiled into data products such as the Surface Ocean CO_{2} Atlas (SOCAT; Bakker et al., 2016). Surface seawater is equilibrated with an enclosed headspace, which flows through to a measuring device, and the raw measurements are passed through a series of processing steps to obtain seawater fCO_{2}. One important processing step is the temperature adjustment. While it is being pumped from the surface ocean to the equilibration chamber, seawater may warm up or cool down. As fCO_{2} is highly sensitive to temperature (Weiss, 1974), each measurement must be adjusted back to its in situ temperature in order to obtain environmentally relevant values of fCO_{2} (Pierrot et al., 2009; Pfeil et al., 2013; GoddijnMurphy et al., 2015). For the data in SOCAT with the most stringent quality control flags (A and B), a maximum change in temperature (Δt) of 1 °C between intake and measurement is permitted (although adjustments are usually around 0.2–0.3 °C in practice; Wanninkhof et al., 2022a), in order to minimise uncertainty in fCO_{2} due to the temperature adjustment. Adjustments with greater Δt are often used in other applications. Some groups measure fCO_{2} in discrete samples at a constant temperature, which must then be adjusted by up to 20 °C to return to in situ conditions (Wanninkhof et al., 2022a). Several studies have normalised seawater fCO_{2} observations to a constant temperature in order to investigate the nonthermal (i.e. chemical) drivers of fCO_{2} change on various timescales (e.g. Takahashi et al., 2002, 2003, 2006; Feely et al., 2006). Others have computed the temperature sensitivity in order to isolate the thermal component of observed fCO_{2} change and unravel the drivers of fCO_{2} variability (e.g. Metzl et al., 2010; Fassbender et al., 2022; Rodgers et al., 2023). Data from SOCAT have been adjusted from the intake temperature, typically measured a few metres below the sea surface, to the temperature directly at the sea surface in order to assess how the “cool skin” effect affects computed air–sea CO_{2} fluxes (e.g. Watson et al., 2020; Dong et al., 2022). Finally, climatological fCO_{2} datasets have been adjusted to match different sea surface temperature fields for specific analyses (e.g. Kettle et al., 2009; Land et al., 2013), and this functionality is part of published data processing toolkits (e.g. Shutler et al., 2016; Holding et al., 2019).
The most widely used approach to adjust fCO_{2} data to different temperatures (t), used in all of the examples above, is based on data from Takahashi et al. (1993), who measured the variation in fCO_{2} with temperature in one seawater sample from the North Atlantic Ocean (strictly speaking, they reported pCO_{2}, but the results apply equally to fCO_{2}). Their measurements indicated that the temperature sensitivity of fCO_{2} (denoted υ; Eq. 1) was essentially independent of fCO_{2} and temperature (i.e. temperature and ln (fCO_{2}) have a linear relationship). This apparent nearlinearity was consistent with earlier measurements by Kanwisher (1960) on a single seawater sample collected from the vicinity of Woods Hole (Massachusetts, USA) and with later measurements by Lee and Millero (1995). Takahashi et al. (1993) also claimed that the value of υ was no different for highersalinity waters, citing unpublished preliminary data.
However, it has long been known that υ is not constant across the ocean. For example, Gordon and Jones (1973) computed υ using a finitedifference approach across a range of different temperature, salinity, alkalinity and pH values and found that the variability in υ could be approximately parameterised as a function of pCO_{2}. Using a similar approach but with different parameterisations of the equilibrium constants in their marine carbonate system calculations, Weiss et al. (1982) and CopinMontegut (1988) each fitted υ to functions of fCO_{2} and temperature. It has been noted that υ may vary with the dissolved inorganic carbon (T_{C}) content of seawater (e.g. Takahashi et al., 2002) or, more generally, with the ratio between T_{C} and total alkalinity (A_{T}) (e.g. Goyet et al., 1993; Wanninkhof et al., 2022a).
There are also several lines of evidence that the relationship between temperature and ln (fCO_{2}) is not linear (i.e. υ is not constant for a single, isochemical sample). Takahashi et al. (1993) acknowledged this possibility by fitting ln (fCO_{2}) to a quadratic function of temperature as well as their linear function, although they did not give any reason to choose one over the other. The fits of Weiss et al. (1982), CopinMontegut (1988) and Goyet et al. (1993) included various degrees of polynomial temperature dependence. Calculating how fCO_{2} changes with temperature using software tools like CO2SYS (Lewis and Wallace, 1998) also shows a nonlinear relationship, but its exact form and how much it deviates from linearity both vary widely depending on which parameterisation for the equilibrium constants of carbonic acid is used (McGillis and Wanninkhof, 2006). Data compiled from multiple cruises during which fCO_{2} was measured at two different temperatures in an equivalent pair of samples, often with accompanying A_{T} and T_{C}, showed that a linear (constant υ) adjustment was insufficiently accurate for Δt greater in magnitude than about 1 °C (Wanninkhof et al., 2022a).
Wanninkhof et al. (2022a) recommended calculating υ from A_{T} and T_{C} in order to make accurate adjustments with greater Δt, using the carbonic acid dissociation constants of Lueker et al. (2000) and the boratetochlorinity ratio of Uppström (1974). However, the requirement for a second marine carbonate system parameter (other than fCO_{2}) is a major disadvantage because, due to ships of opportunity and the variety of (semi)autonomous sensors available, seawater fCO_{2} is by far the most widely measured marine carbonate system parameter in the ocean and is often the only one available at a particular observation point. A possible workaround would be to use a parameterisation to generate values for a second parameter. For example, across much of the open ocean, A_{T} can be calculated from temperature and salinity with an uncertainty of less than 10 µmol kg^{−1} (Lee et al., 2006). Under typical conditions, this component alone would propagate through to a less than 0.1 µatm uncertainty in fCO_{2} adjusted by 10 °C (calculated with PyCO2SYS; Humphreys et al., 2022). However, this ignores the much larger contribution of uncertainties in the equilibrium constants used in the calculations. These are not well constrained, but using the set suggested by Orr et al. (2018) makes the uncertainty in the same adjusted fCO_{2} value several hundred times greater, at around 15 µatm. An adjustment method based on direct measurements of fCO_{2} and independent from a second marine carbonate system parameter, avoiding the uncertainties inherent in calculations involving equilibrium constants, is therefore highly desirable.
Given the evidence that the equations of Takahashi et al. (1993) are not sufficiently accurate for the full range of temperature adjustments of fCO_{2} required by ocean scientists, why do they remain so widely used? First, they are straightforward to calculate, relying only on variables that will be available with any fCO_{2} measurement (i.e. fCO_{2} itself and temperature). The more complex existing parameterisations that also fulfil this criterion (CopinMontegut, 1988; e.g. Gordon and Jones, 1973; Weiss et al., 1982) are not based on uptodate parameterisations of the carbonic acid dissociation constants, which can lead to differences in adjusted fCO_{2} values of over 30 µatm for larger Δt (McGillis and Wanninkhof, 2006). However, a more fundamental impediment is not knowing what form the relationship between temperature and ln (fCO_{2}) should be expected to take from a theoretical perspective. Should it be (roughly) linear, with deviations from linearity thus indicating problems with the calculations used to solve the marine carbonate system, or should it be nonlinear, thus indicating some deficiency with the measurements of Takahashi et al. (1993)? With respect to the latter, this then poses the followup question of which of the nonlinear relationships obtained from the different parameterisations of the carbonic acid dissociation constants should be followed.
Here, we aim to provide this missing theoretical basis by developing a new functional form regarding how fCO_{2} and, thus, υ vary with temperature, derived from the equations governing the marine carbonate system and the van 't Hoff equation. We compare this new form with υ calculated from a fully resolved marine carbonate system – to which end, we have added the calculation of υ to PyCO2SYS (Humphreys et al., 2022) – and with laboratory and field data. We compute uncertainties in υ and temperatureadjusted fCO_{2} values due to experimental uncertainties and variability in υ through space and time. We assess the contributions of the different equilibrium constants of the marine carbonate system to υ, especially those for carbonic acid dissociation, and consider the implications for how these constants are parameterised and how these previously separate aspects can be better integrated in future studies for a more accurate and internally consistent understanding of the marine carbonate system.
2.1 Adjusting fCO_{2} to a different temperature
2.1.1 General theory
The temperature sensitivity of seawater fCO_{2} (υ, in units of % °C^{−1}) is the first derivative of the natural logarithm of fCO_{2} with respect to temperature (t, in °C) under constant total alkalinity (A_{T}) and dissolved inorganic carbon (T_{C}) (see Humphreys et al., 2022, and references therein for more detailed definitions of A_{T}, T_{C} and fCO_{2}):
We note that, here and throughout, fCO_{2} must be divided by the unit of fCO_{2} (e.g. 1 µatm if fCO_{2} is in µatm) before taking the logarithm, and “constant A_{T} and T_{C}” implies that all other parameters but temperature are also constant (e.g. salinity and total nutrient contents). We report υ in percent per degree Celsius (% °C^{−1}; i.e. 10^{−2} °C^{−1}).
To adjust an fCO_{2} value from one temperature (t_{0}) to another (t_{1}), υ must be integrated across the relevant temperature range:
where Υ denotes the integral
Equivalent to Eq. (2),
Thus, in order to adjust the temperature of an fCO_{2} value using Eq. (4), a function for υ that can be integrated with respect to temperature is needed.
2.1.2 Experiments to measure υ
Takahashi et al. (1993) attempted to determine υ by measuring how pCO_{2} varies with temperature (eight measurements at seven temperatures from 2.1 to 24.5 °C) in one seawater sample with constant A_{T} (reported as “2270 to 2295” µmol kg^{−1}) and T_{C} (2074 µmol kg^{−1}). The practical salinity was 35.380. Phosphate and silicate concentrations were reported as zero while noting the possibility of silicate contamination from the sample bottles. The sample had been part of the laboratory intercomparison exercise reported by Poisson et al. (1990).
We determined a more precise A_{T} for the sample by finding the value that, when paired with the T_{C} and other parameters above, returned the leastsquares best fit to the reported pCO_{2} values. When using the Lueker et al. (2000) carbonic acid constants and the boratetochlorinity ratio of Uppström (1974), the A_{T} obtained with PyCO2SYS (Humphreys et al., 2022) was 2263 ± 3 µmol kg^{−1}, although each set of carbonic acid constants returned its own unique A_{T}. For consistency with modern measurements and the rest of our analysis, we also converted the Takahashi et al. (1993) pCO_{2} values into fCO_{2}, although this makes no meaningful difference to any of our results.
Takahashi et al. (1993) fitted their pCO_{2} dataset to linear (subscript l) and quadratic (subscript q) equations (here written in terms of fCO_{2}):
where a, b and c are the fitted coefficients. Differentiating with respect to t yields the following for υ:
Finally, integrating across the relevant temperature range yields Υ, following, for example, Takahashi et al. (2009):
Takahashi et al. (1993) reported values of b_{l}=4.23 % °C^{−1}, ${a}_{\mathrm{q}}=\mathrm{4.35}\times {\mathrm{10}}^{\mathrm{5}}$ °C^{−2} and b_{q}=4.33 % °C^{−1} with a rootmeansquare deviation (RMSD) of 1.0 µatm in fCO_{2} for both fits. Reproducing these fits with their reported data gives an RMSD of 1.2 µatm for the linear fit and 0.9 µatm for the quadratic.
Lee and Millero (1995) conducted a similar set of measurements as part of a wider experimental study of the marine carbonate system. Again using only one single sample, they measured the variation in fCO_{2} with temperature at roughly 5 °C intervals from 5.05 to 35 °C (seven data points). The reproducibility of the fCO_{2} measurements was reported as ±3 µatm. The properties of the sample were reported as A_{T}=2320.8 ± 1.6 µmol kg^{−1}, T_{C}=1966.8 ± 1.2 µmol kg^{−1} and salinity=34.95. The experimental setup was a little different; for example, Takahashi et al. (1993) equilibrated the sample with an air headspace that was subsampled for separate measurement, whereas Lee and Millero (1995) continuously circulated the air headspace through the CO_{2} analyser. Lee and Millero (1995) fitted a linear equation of the form of Eq. (5) to their data, finding a bestfitting υ_{l} of 4.26 % °C^{−1}. Refitting their dataset but minimising residuals in fCO_{2} rather than ln (fCO_{2}) returns ${b}_{\mathrm{l}}={\mathit{\upsilon}}_{\mathrm{l}}=\mathrm{4.23}$ % °C^{−1}.
2.1.3 The van 't Hoff equation approach
Takahashi et al. (1993) did not give a theoretical basis for either of the forms (linear and quadratic; Eqs. 5 and 6) that they fitted to their dataset, nor did they give any reason to choose one over the other. However, as will be discussed later, the form chosen has an important effect on adjustments, especially over greater Δt (Sect. 3.1), and on uncertainty propagation (Sect. 3.2). Here, we provide the theoretical basis for an alternative form based on the equations governing the marine carbonate system and the van 't Hoff equation, which is derived from the wellknown thermodynamic equation for the Gibbs free energy of chemical reactions.
Seawater fCO_{2} is related to A_{T} and T_{C} by (e.g. Humphreys et al., 2018)
The asterisks in ${K}_{\mathrm{1}}^{\ast}$ and ${K}_{\mathrm{2}}^{\ast}$ show that they are stoichiometric equilibrium constants, based on the substance contents of the equilibrating species, rather than thermodynamic constants based on their activities; the prime symbol in ${K}_{\mathrm{0}}^{\prime}$ shows that it is a hybrid equilibrium constant, based on the substance content of CO_{2}(aq) and the fugacity of CO_{2}(g) (Zeebe and WolfGladrow, 2001). Following Humphreys et al. (2018), A_{T} and T_{C} can be approximated with A_{x} and T_{x}, which include only the bicarbonate (${{\mathrm{HCO}}_{\mathrm{3}}}^{}$) and carbonate (${{\mathrm{CO}}_{\mathrm{3}}}^{\mathrm{2}}$) terms:
In typical seawater, the approximations in Eqs. (12) and (13) are more than 99 % accurate for T_{C} and more than 97 % accurate for A_{T}. These approximations have previously been used to understand how fCO_{2} responds to changes in A_{T} and T_{C} under constant temperature (Humphreys et al., 2018). Equations (12) and (13) can be rearranged for [${{\mathrm{HCO}}_{\mathrm{3}}}^{}$] and [${{\mathrm{CO}}_{\mathrm{3}}}^{\mathrm{2}}$]:
Under this approximation, both [${{\mathrm{HCO}}_{\mathrm{3}}}^{}$] and [${{\mathrm{CO}}_{\mathrm{3}}}^{\mathrm{2}}$] are constant under constant A_{x} and T_{x}, so temperature affects fCO_{2} only through the ${K}_{\mathrm{2}}^{\ast}/{K}_{\mathrm{0}}^{\prime}{K}_{\mathrm{1}}^{\ast}$ term in Eq. (11). To relate this to υ, we need to quantify this sensitivity to temperature:
The van 't Hoff equation provides a theoretical basis for the firstorder response of equilibrium constants to temperature:
where Δ_{r}H^{⊖} is the standard enthalpy of the reaction, R is the ideal gas constant and t_{K} is the temperature in kelvin. Substituting Eq. (17) into Eq. (16) gives
Equation (18) shows that, if the approximations A_{x} and T_{x} are sufficiently accurate, then (to first order) ln (fCO_{2}) should be proportional to $\mathrm{1}/{t}_{\mathrm{K}}$, rather than to t.
Based on reported standard enthalpies of formation at 298.15 K (Ruscic and Bross, 2023), and assuming that these are constant under the temperature range of interest, the numerator in Eq. (18) is 25 288 ± 98 J mol^{−1} (Appendix A). Given the approximations involved, Eq. (18) is unlikely to provide correct values for υ in seawater directly by using this value, but it does suggest a revised form that could be fitted to experimental data such as those of Takahashi et al. (1993):
Following the same steps taken in Sect. 2.1.2 for the linear and quadratic forms of Takahashi et al. (1993), we find the following:
We fitted Eq. (19) to the Takahashi et al. (1993) dataset, optimising residuals in fCO_{2} rather than ln (fCO_{2}) to avoid implicitly weighting the lowerfCO_{2} values more heavily. The leastsquares bestfit b_{h} was 28 995 ± 235 J mol^{−1}, with an RMSD of 2.9 µatm in fCO_{2}. Fitting Eq. (19) in the same way to the Lee and Millero (1995) dataset gave a leastsquares bestfit b_{h} of 30 794 ± 357 J mol^{−1}, with an RMSD of 2.8 µatm. The different b_{h} values considered here are summarised in Table 1, including the parameterisation developed later in Sect. 2.4, which gives a range of possible values.
2.2 Uncertainty in the experiment
2.2.1 Propagation equations
To quantify uncertainties in fCO_{2} resulting from temperature adjustments, one needs to know the uncertainty in υ and Υ for whichever form is being used. Here, we derive uncertainty propagation equations following JCGM (2008) for the linear (Eq. 5), quadratic (Eq. 6) and van 't Hoff (Eq. 19) forms.
For the linear and van 't Hoff forms, uncertainty propagation is straightforward:
where σ^{2} is uncertainty expressed as a variance.
For the quadratic form, covariances between the fitted coefficients must be included. In general,
where p is the parameter of interest and the uncertainty matrix for the coefficients is
The Jacobian matrices for υ_{q} and Υ_{q} are
Thus, for a single measurement, Eq. (26) gives
Finally, uncertainty in Υ can be converted into uncertainty in exp (Υ) for use in Eq. (4) with
2.2.2 Uncertainties in fitted parameters
To use the equations in Sect. 2.2.1, the uncertainties in the fitted parameters from Eqs. (5), (6) and (19) must be known. We computed these by running Monte Carlo simulations of the Takahashi et al. (1993) experiment, as this dataset is the most widely used in practice.
We used the Takahashi et al. (1993) temperature and fCO_{2} measurements as the starting point for the simulations. For each iteration of the simulation (total iterations = 10^{6}), we generated a random number for each of temperature and fCO_{2}, which were added uniformly across the entire dataset to represent systematic errors (trueness), and then a different random number for each temperature and fCO_{2} data point, to represent random errors in each measurement (precision). The distributions used for the random numbers are shown in Table 2. The simulated temperature and fCO_{2} values (starting point plus systematic and random errors) were then fitted to Eqs. (5), (6) and (19) separately in each iteration. All of the forms of υ and Υ were also computed based on the fit at each iteration.
^{∗} Spread indicates the total range for uniform distributions or the standard deviation for normal distributions, with both distributions centred at zero.
The variances σ^{2}(b_{l}) and σ^{2}(b_{h}) and the variance–covariance matrix U_{q} (Eq. 27) were computed from the fitted coefficients across the full set of iterations.
2.2.3 Validation of uncertainty propagation
The uncertainty propagation equations in Sect. 2.2.1 have all been validated by two comparisons: (1) with the variances of the υ and Υ values computed across all of the simulations in Sect. 2.2.2 and (2) by computing the derivatives required for propagation using automatic differentiation (Maclaurin, 2016) rather than manually. The values from the equations agreed to within 0.01 % with those based on the simulations and to within 10^{−12} % with the calculations based on automatic differentiation. The former difference is greater (albeit still negligible) because of nonlinearities in the forms of υ that are not accounted for in the firstorder derivatives used for uncertainty propagation here. The latter difference is attributable to rounding errors at the limit of computer precision.
2.3 PyCO2SYS extension and calculations
PyCO2SYS is a free, opensource Python package that can be used to solve the marine carbonate system, i.e. to calculate the equilibrium balance of the main acid–base systems in seawater and related properties (Humphreys et al., 2022). PyCO2SYS was originally based on the MATLAB/GNU Octave program CO2SYS.m (v2.0.5; Orr et al., 2018) – itself part of a family of similar software tools beginning with the original CO2SYS for MSDOS (Lewis and Wallace, 1998).
2.3.1 Calculating υ in PyCO2SYS
In PyCO2SYS, υ can be calculated by finite differences or by automatic differentiation (Maclaurin, 2016). The accuracy of the former depends on the size of the difference used, whereas the latter gives the exact result directly (Humphreys et al., 2022). For consistency with the chemical buffer factors in PyCO2SYS, we used automatic differentiation to calculate υ. We constructed a function that evaluates ln (fCO_{2}) from A_{T}, T_{C}, t and all other relevant parameters (e.g. salinity, pressure and nutrients) and applied automatic differentiation to determine its derivative with respect to temperature. The function runs after A_{T} and T_{C} have been determined from whichever pair of known parameters the user provides. This was added to the set of results returned by PyCO2SYS as of v1.8.3 (Humphreys et al., 2024).
Previous versions of PyCO2SYS could already calculate $\partial f{\mathrm{CO}}_{\mathrm{2}}/\partial t$ using forward finitedifference derivatives as part of the uncertainty propagation tool. υ can be calculated from $\partial f{\mathrm{CO}}_{\mathrm{2}}/\partial t$ as follows:
We used the finitedifference $\partial f{\mathrm{CO}}_{\mathrm{2}}/\partial t$ and Eq. (33) to validate the values from automatic differentiation. Across 10^{4} random combinations of A_{T}, T_{C}, salinity, t, pressure, nutrient contents and carbonic acid dissociation constant parameterisation, we found that the two approaches were consistent, with a mean absolute difference of less than 10^{−4} % °C^{−1} (or 0.01 %) and a maximum absolute difference of less than 0.1 % °C^{−1}. This check has been incorporated into the PyCO2SYS test suite to ensure continued consistency in future versions.
2.3.2 Adjusting fCO_{2} to different temperatures
Prior to v1.8.3, PyCO2SYS could already convert fCO_{2} (also pCO_{2}, [CO_{2}(aq)] or xCO_{2}) values from one temperature to another when no second carbonate system variable was provided, although using only Eq. (10), the quadratic fit of Takahashi et al. (1993). In v1.8.3, we have added the option to use the other methods discussed here instead, i.e. the linear fit of Takahashi et al. (1993) and the van 't Hoff form with several different options for b_{h}, including its parameterisation in Sect. 2.4.
As in previous versions of PyCO2SYS, the same adjustment is applied if pCO_{2}, [CO_{2}(aq)] or xCO_{2} is given as the single known marine carbonate system parameter, as these can all be interconverted with each other and with fCO_{2} without knowledge of a second marine carbonate system parameter (Humphreys et al., 2022).
2.3.3 Components of υ
The effect of temperature on fCO_{2} propagates through from all of the different equilibrium constants of the marine carbonate system. The total υ is the sum of the partial derivatives with respect to each constant:
where K is the set of equilibrium constants and other properties that are sensitive to t and involved in calculating fCO_{2} from A_{T} and T_{C}, specifically, the CO_{2} solubility constant ${K}_{\mathrm{0}}^{\prime}$, the carbonic acid dissociation constants ${K}_{\mathrm{1}}^{\ast}$ and ${K}_{\mathrm{2}}^{\ast}$, the borate equilibrium constant ${K}_{\mathrm{B}}^{\ast}$, and the water dissociation constant ${K}_{\mathrm{w}}^{\ast}$. The dissociation constants for phosphoric acid, orthosilicic acid, hydrogen sulfide, ammonia, sulfate and hydrogen fluoride also play a role, but their contributions to υ are negligible, being much less than 1 % in the modern surface ocean, so we did not include them further in our analysis. (Their effects are still included in the total υ calculated with automatic differentiation, described in Sect. 2.3.1.)
We used the forward finitedifference derivatives calculated by PyCO2SYS (Humphreys et al., 2022) to evaluate $\partial f{\mathrm{CO}}_{\mathrm{2}}/\partial K$ and $\partial K/\partial t$ and combined these with Eq. (33) to quantify the influence of each component on υ.
2.3.4 Additional carbonic acid parameterisation
We incorporated the parameterisation of Papadimitriou et al. (2018) for the carbonic acid dissociation constants into PyCO2SYS. This parameterisation covers a range of temperature (−6 to 25 °C) and salinity (33 to 100) conditions appropriate for seaice brines; previously, the lowest valid temperature for any parameterisation in PyCO2SYS was −1.7 °C and the highest salinity was 50. The calculations were tested against the check values provided by Papadimitriou et al. (2018) in their Table 3, and these tests were incorporated into the PyCO2SYS automatic test suite (Humphreys et al., 2022).
2.4 Variability in υ and b_{h}
We used the OceanSODAETZH data product (Gregor and Gruber, 2021; dataset version 5, downloaded 4 December 2023) to investigate spatiotemporal variability in surface ocean υ. OceanSODAETZH provides fields of several marine carbonate system and other hydrographic parameters gridded across the surface ocean (1°×1°) and through time (monthly from 1985 to 2018). The parameter fields were generated using an ensemble cluster–regression approach based on observations in SOCAT (Bakker et al., 2016) and GLODAP (Lauvset et al., 2016). Other similar data products exist, but the main patterns and variability in surface ocean carbonate chemistry are well enough constrained that the choice of a particular data product will not significantly affect the globalscale and timeaveraged analyses conducted here.
First, at each grid point and for each month, we computed the mean across all years of temperature, salinity, A_{T} and T_{C}. We then computed υ from these monthly mean gridded fields using PyCO2SYS v1.8.3 (Sect. 2.3), with the carbonic acid dissociation constants of Lueker et al. (2000), the bisulfate dissociation constant of Dickson (1990a), the hydrogen fluoride dissociation constant of Dickson and Riley (1979), and the total boratetochlorinity ratio of Uppström (1974). OceanSODAETZH does not include nutrient fields, so their concentrations were set to zero, but we do not expect this to affect the results because, as noted in Sect. 2.3.2, their effects on υ are negligible.
We also used the OceanSODAETZH data product to parameterise b_{h}. We calculated fCO_{2} at 50 evenly spaced temperatures from −1.8 to 35.83 °C (i.e. the full range in the data product) at each monthly mean grid point. This simulates running an experiment like that of Takahashi et al. (1993) at each point. Next, we fit Eq. (19) to the generated t and fCO_{2} data to find the bestfitting b_{h} value at each point. Finally, we collated all of the b_{h} values and fitted them using a function of the following form:
where s is practical salinity, f is fCO_{2} and u_{0}–u_{9} are the fitted coefficients. The coefficients and their variance–covariance matrix are provided in the Supplement (Tables S1 and S2). Values of υ computed using Eq. (35) for b_{h} are denoted υ_{p}.
Fitting the parameterisation to a data product has an advantage over using regular grids of the predictors in that it guarantees including all combinations of predictors that do occur in the real ocean and avoids combinations that do not. It also causes the fit to be weighted more strongly by more prevalent combinations of conditions, leading to improved performance in global applications. However, some regions are less well represented by this global fit (Fig. S1 in the Supplement), so regionally specialised fits might work better for some studies.
A difficulty with applying our parameterisation of b_{h} is that in situ fCO_{2} is one of the predictors. This may not always be known (e.g. if discrete measurements were conducted at a fixed temperature in the laboratory), and using a different fCO_{2} will not yield an accurate b_{h}. To mitigate this issue, if in situ fCO_{2} is not known before the conversion, we use the constant b_{h} fitted to the Takahashi et al. (1993) dataset (Sect. 2.1.3) to preliminarily adjust fCO_{2} to the in situ temperature; we then use this preliminary fCO_{2} to compute b_{h} with Eq. (35) before recalculating in situ fCO_{2} with the computed b_{h}. This approach is incorporated into the PyCO2SYS implementation, with the user required to identify whether the input or outputcondition fCO_{2} values represent in situ conditions for this purpose.
2.5 Opensource software
All of the analysis presented here was conducted using the free and opensource Python programming language (Python Software Foundation; https://www.python.org/, last access: 23 October 2024). The main packages used, which are also free and open source, include NumPy (Harris et al., 2020), Autograd (Maclaurin, 2016) and Xarray (Hoyer and Hamman, 2017) for data manipulation and general calculations; SciPy (Virtanen et al., 2020) for curve fitting; Matplotlib (Hunter, 2007) and Cartopy (Met Office, 2010) for data visualisation; and PyCO2SYS (Humphreys et al., 2022) for marine carbonate system calculations. Geographical features on maps were drawn using data that are freely available from Natural Earth (https://www.naturalearthdata.com/, last access: 23 October 2024).
3.1 Functional form of υ
Our first goal is to determine whether the newly proposed form υ_{h} (Eq. 20) can accurately represent the temperature sensitivity of seawater fCO_{2}. To this end, we assess the accuracy of the approximations A_{x} and T_{x} made in developing Eq. (20) by comparing υ_{h} with calculations from a fully resolved marine carbonate system (Sect. 3.1.1). Next, we examine how well the different approaches to calculating υ agree with direct measurements; we do this for (1) laboratory experiments with measurements of fCO_{2} at multiple temperatures for a single seawater sample (Sect. 3.1.2) and (2) across a global dataset with pairs of measurements at two different temperatures (Sect. 3.1.3).
3.1.1 Accuracy of the approximations
We made several approximations to develop Eq. (20) (Sect. 2.1.3). First, all noncarbonate components of A_{T} (Eq. 12) and the contribution of [CO_{2}(aq)] to T_{C} (Eq. 13) were neglected. Second, the simple integrated van 't Hoff equation (Eq. 17) was assumed to accurately represent how the various equilibrium constants vary with temperature. Third, when calculating b_{h} from standard enthalpies of formation (Appendix A), these were assumed to be constant across the relevant temperature range. We must therefore check whether Eq. (20) can still capture the sensitivity of fCO_{2} to temperature accurately enough for our purposes before applying it to oceanographic data.
The first approximation above was also used to investigate the complementary question of how fCO_{2} varies with A_{T} and T_{C} under constant temperature (Humphreys et al., 2018). There, it was observed that the gradient of isocaps (i.e. lines of constant fCO_{2} through A_{T}–T_{C} phase space) was correlated with fCO_{2}. The approximations A_{x} and T_{x} were used as the basis for a simplified representation of the marine carbonate system from which equations were derived to explain the observed correlation. This successful application to a closely related problem provides a first indication that the same approximations may also be suitable here.
We can do a more direct test using software like PyCO2SYS, which does not make any of the approximations above when calculating how fCO_{2} varies with temperature under constant A_{T} and T_{C} (Humphreys et al., 2022). Instead, the complete equations for A_{T} and T_{C} are used, and the equilibrium constants are parameterised with multiple temperature, salinity and pressuresensitive terms, which implicitly include variations in the enthalpies of formation. Such calculations can therefore be used to quantify the effect of the approximations on the accuracy of Eq. (20). Equation (20) cannot be fitted directly to fCO_{2} data because υ represents the derivative with respect to temperature; instead, we need to fit Eq. (19) to obtain the unknown b_{h} which can then be used in Eq. (20) for υ_{h}. If we use PyCO2SYS to calculate fCO_{2} across a range of temperatures with constant A_{T} and T_{C}, denoted fCO_{2}(A_{T},T_{C}), and then fit Eq. (19) to the results (Sect. 2.4), denoted fCO_{2}(υ_{p}), the residuals from the fit should reveal any reduction in accuracy due to the approximations. Large residuals would suggest that at least one approximation was unsuitable for modelling υ, whereas small residuals would mean that the approximated system does still capture the main features of the t–fCO_{2} relationship that are needed to model υ.
For over 97 % of the monthly mean grid points in OceanSODAETZH, the RMSD of the residuals between fCO_{2} calculated from Eq. (19) and from A_{T} and T_{C} is less than 1 µatm, with maximum deviations of ±2 µatm (Fig. 1). At lower ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ ($<\phantom{\rule{0.125em}{0ex}}\sim \mathrm{0.95}$), fCO_{2}(υ_{p}) is generally greater than fCO_{2}(A_{T},T_{C}) at low (<10 °C) and high (>30 °C) temperatures but less than fCO_{2}(A_{T},T_{C}) at intermediate temperatures (10–30 °C), but this switches to the opposite pattern at higher ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ (Fig. 1a). We could not find a relationship between the switchover point, which has a very low RMSD, and some property of chemical buffering in seawater, although the ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ value at which it occurs is correlated with salinity (Fig. 1b).
As context for interpreting this <1 µatm RMSD, the uncertainty in fCO_{2} calculated from A_{T} and T_{C} that propagates from 0.1 % uncertainties in each of A_{T} and T_{C} (i.e. the GOAON climate quality goal; Newton et al., 2015) varies from 2 µatm at −1 °C to 9 µatm at 35 °C. The uncertainties in the equilibrium constants involved in this calculation are poorly known, but using the set proposed by Orr et al. (2018) roughly doubles the total uncertainty in calculated fCO_{2} given above. The <1 µatm RMSD between fCO_{2}(υ_{p}) and fCO_{2}(A_{T},T_{C}) found for 97 % of the surface ocean is thus much smaller than the uncertainties in the marine carbonate system calculations themselves. It is also very similar to the RMSD of the linear and quadratic fits to the Takahashi et al. (1993) dataset (Sect. 2.1.2) and better than the GOAON climate quality goal for fCO_{2} measurements.
In the remaining <3 % of the global surface ocean, the RMSD can increase to over 5 µatm, which is less satisfactory but still within the GOAON weather quality goal (Newton et al., 2015). These higher RMSD values are constrained to ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ values above 0.95 and to certain regions and seasons, primarily the Baltic Sea during winter and the nearcoastal Arctic Ocean (Fig. S1). In these regions, the RMSD is correlated with several marine carbonate system properties, but one of the strongest correlations is with ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ (Fig. 1b). The cause of the increased RMSD is likely inaccuracy in the approximation of T_{x} (Eq. 13). The fraction of T_{C} comprised of [CO_{2}(aq)], which is ignored in T_{x}, increases with ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ (Fig. 1c). Consequently, the approximation that $[{{\mathrm{HCO}}_{\mathrm{3}}}^{}{]}^{\mathrm{2}}/[{{\mathrm{CO}}_{\mathrm{3}}}^{\mathrm{2}}]$ is constant across different temperatures (Eq. 16), which emerges from the definitions of A_{x} and T_{x} (Eqs. 12–15), becomes less accurate with increasing ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$. The fraction of noncarbonate alkalinity, which is ignored in A_{x}, decreases with increasing ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$, which makes A_{x} more accurate at higher ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ (Fig. 1c); thus, A_{x} should not be the cause of the increased RMSD. The RMSD of the b_{h} fit is positively correlated with the difference T_{C}−T_{x} but negatively correlated with A_{T}−A_{x} (Fig. S3), supporting the hypothesis that inaccuracy in T_{x} rather than A_{x} drives the increased RMSD observed at higher ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$.
The ∼59 % of data points with ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}<\mathrm{0.9}$ (in dark purple in Fig. 1a) all have the same pattern of residuals with an RMSD that is virtually independent of ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ and less than 1 µatm. Their independence from ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ suggests that these residuals result from the additional temperaturedependent terms beyond $\mathrm{1}/{t}_{\mathrm{K}}$ from the van 't Hoff equation that are included in parameterisations of the equilibrium constants (e.g. Weiss, 1974; Dickson, 1990b; Lueker et al., 2000). Including such additional terms in Eq. (19) (and thus also in Eqs. 20–21) might allow it to better capture the pattern of ln (fCO_{2}) variation with t, reducing the RMSD. Were υ_{h} to be adapted to include additional fitted parameters, the functional forms included should have a theoretical basis (e.g. Clarke and Glew, 1966) rather than being purely empirical, to ensure accurate uncertainty propagation (Sect. 3.2).
However, to include additional fitted parameters now would be premature. There are major differences between the different carbonic acid parameterisations (e.g. Fig. S2) that are not yet fully understood, and Fig. 1a and b would show different patterns if reproduced using a different parameterisation. Even though it is the parameterisation given by the bestpractice guide (Dickson et al., 2007), it is still wellknown that Lueker et al. (2000) does not perfectly represent the marine carbonate system to the accuracy with which we can measure it (e.g. Álvarez et al., 2020; Carter et al., 2024; Wang et al., 2023; Woosley, 2021; Woosley and Moon, 2023), and there are outstanding issues to resolve with, for example, the contribution of organic matter to total alkalinity (e.g. Ulfsbo et al., 2015; Hu, 2020; Kerr et al., 2021; Lee et al., 2024) and the boratetochlorinity ratio (e.g. Lee et al., 2010; Fong and Dickson, 2019). Furthermore, there are no robust uncertainty estimates for the equilibrium constants that can be propagated through to determine a meaningful uncertainty in υ_{Lu00} (Orr et al., 2018). The carbonic acid and other equilibrium constant parameterisations are not optimised for this derivative, emergent property (υ); direct measurements (like those of Takahashi et al., 1993), with which more subtle variations about the firstorder form in Eq. (20) can be verified, are scarce (Sect. 3.1.2).
Even with these issues remaining unresolved, we conclude that, for over 97 % of the global surface ocean, Eq. (20) with all its approximations (Sect. 2.1.3) can capture the pattern of fCO_{2} sensitivity to temperature according to calculations from A_{T} and T_{C} with the carbonic acid dissociation constants of Lueker et al. (2000) to within an uncertainty of 1 µatm.
3.1.2 Direct measurements of υ
Having established how well υ_{h} from Eq. (20) agrees with calculations from a fully solved marine carbonate system, we next consider how these and other approaches compare with the scarce experimental evidence available.
The linear (υ_{l}) and quadratic (υ_{q}) forms of Takahashi et al. (1993) both agree well with their own measurements, so it is not clear which (if either) is more appropriate. The RMSD for υ_{q} (0.9 µatm) is lower than that for υ_{l} (1.2 µatm), but this reduction is inevitable because υ_{q} has an extra adjustable coefficient (Eqs. 5–8). There is a meaningful difference between υ_{l} and υ_{q} only above ∼25 °C (Fig. 2a), where υ_{q} is lower, meaning that fCO_{2} would increase less with warming than suggested by υ_{l}. For example, adjusting from 25 to 35 °C would give an fCO_{2} that is about 17 µatm lower with υ_{q} than with υ_{l}.
Next, we consider the form proposed here in Eq. (19). When b_{h} is fixed to the value based on standard enthalpies of reaction (Sect. 2.1.3), so only the intercept (c_{h}) can be optimised, the resulting curve υ_{x} has a rather different pattern that is inconsistent with the measurements (Fig. 2a). However, also allowing b_{h} to be fitted to the measurements of Takahashi et al. (1993) (${\mathit{\upsilon}}_{h}^{\text{Ta93}}$) gives a pattern similar to υ_{q}, although with stronger curvature. Even though it agrees with the Takahashi et al. (1993) measurements less well than υ_{l} and υ_{q} do, with an RMSD of 2.9 µatm as opposed to ∼1 µatm, the υ_{h} line still falls within their uncertainty window. As for υ_{q}, the most significant deviation from υ_{l} occurs above 25 °C, where Takahashi et al. (1993) did not report any measurements. Adjusting from 25 to 35 °C would give an fCO_{2} that is about 45 µatm lower with ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$ than with υ_{l}.
There are 18 different parameterisations of the carbonic acid dissociation constants in PyCO2SYS v1.8.3, and each one gives a different pattern for υ. However, they are mostly negatively correlated with temperature, like υ_{q} and ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$, so they all give lower fCO_{2} values at higher temperatures than υ_{l} does (Fig. S2). Two of the parameterisations are highlighted here.
The parameterisation of Millero (1979) (υ_{Mi79}) agrees remarkably well with υ_{x} (Fig. 2). This parameterisation was designed for zerosalinity freshwater and excludes many of the noncarbonate equilibria that are active in seawater, such as that of borate.
The parameterisations that are (at least partly) based on the measurements of Mehrbach et al. (1973) fit the Takahashi et al. (1993) dataset better than those that do not, with the Lueker et al. (2000) parameterisation (υ_{Lu00}) having one of the best fits (RMSD=2.6 µatm) and falling within the uncertainty of the measurements (Fig. 2a). The pattern of υ_{Lu00} is very similar to ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$ (Fig. 2b). The similarity was expected, given the results discussed in Sect. 3.1.1 and the ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ of ∼0.92 for the sample used by Takahashi et al. (1993). However, the similarity above 25 °C is more remarkable in this case, as the υ_{h} curves in Sect. 3.1.1 were fitted to data spanning the full temperature range from −1.8 to 35.83 °C, whereas there were no data above 24.5 °C (nor below 2.1 °C) to guide the shape of the fitted ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$ curve here. Thus even without having a wide enough temperature range for the data to show convincing curvature in υ, the extrapolation of Eq. (20) still matches the calculations with PyCO2SYS very well, unlike υ_{l} and υ_{q}.
There are two main reasons for the differences between υ_{x} and ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$ or, equivalently, between υ_{Mi79} and υ_{Lu00}. First, ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$ and υ_{Lu00} include additional equilibria that are not present in υ_{x} and υ_{Mi79}. Some of these extra equilibria are explicitly part of the alkalinity equation in PyCO2SYS (e.g. borate); others are implicit, such as the formation of ${{\mathrm{CaCO}}_{\mathrm{3}}}^{\mathrm{0}}$ and ${{\mathrm{MgCO}}_{\mathrm{3}}}^{\mathrm{0}}$ ion pairs (Millero and Pierrot, 1998), which are incorporated within [${{\mathrm{CO}}_{\mathrm{3}}}^{\mathrm{2}}$] and ${K}_{\mathrm{2}}^{\ast}$ in PyCO2SYS (Humphreys et al., 2022). The approximation that all noncarbonate species are negligible within A_{T} (Eq. 12) is thus more accurate for υ_{x} and υ_{Mi79} (hence their mutual agreement) than it is for ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$ and υ_{Lu00}. Second, the van 't Hoff equation with standard enthalpies of reaction predicts the temperature sensitivity of thermodynamic equilibrium constants. However, chemical equilibria in seawater are modelled using stoichiometric equilibrium constants, which have an additional temperature dependency related to the major ion composition and ionic strength of the solution (Pitzer, 1991). Both reasons would affect ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$ and υ_{Lu00} (and hence their deviation from υ_{x}) but not υ_{Mi79}, which is consistent with υ_{x}.
The strong agreement between υ_{x} and υ_{Mi79} strengthens our conclusion from Sect. 3.1.1 that the approximations made in deriving Eq. (20) are valid for representing the firstorder form of υ. It also means that under certain lowsalinity conditions, Eq. (18) can be used directly with Δ_{r}H^{⊖} values from the literature and no further fitting to quantify υ accurately with respect to calculations with an equilibrium model (PyCO2SYS). The strong agreement between ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$ and υ_{Lu00} suggests that the extra equilibria and ionic strength effects can be adequately accounted for by adjusting the value of b_{h} and without changing the form of the t–ln (fCO_{2}) relationship.
It is unfortunate that the dataset of Takahashi et al. (1993) has a maximum temperature of 24.5 °C, because extra data points above 25 °C could unambiguously distinguish between the different forms of υ. At lower temperatures, the differences between the forms are similar in size to the uncertainty of the measurements, so many more highly accurate measurements would need to be conducted to robustly distinguish between the forms (Fig. 2a). The Kanwisher (1960) measurements covered an even narrower temperature range from 11 to 21.5 °C, which is also insufficient to convincingly show nonlinearity.
One published dataset does cover a wider temperature range, with seven measurements of a single sample from 5.05 to 35 °C (Lee and Millero, 1995). These measurements can be fitted well with both the linear and quadratic forms (Eqs. 5 and 6), with RMSD values of 1.8 µatm (linear) and 1.5 µatm (quadratic). Lee and Millero (1995) reported a constant υ_{l} of 4.26 % °C^{−1}, stating that the difference between this and the value of Takahashi et al. (1993) was due to a different ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ for the sample measured (0.85 versus ∼0.92). As with the Takahashi et al. (1993) dataset, ${\mathit{\upsilon}}_{h}^{\text{LM95}}$ fits less well than either υ_{l} or υ_{q}, with an RMSD of 2.8 µatm, although the fitted curve again falls within the uncertainty window of the data points (±3 µatm). However, in this case, ${\mathit{\upsilon}}_{h}^{\text{LM95}}$ is completely different from the calculation with the Lueker et al. (2000) carbonic acid dissociation constants (Fig. 3). It also predicts significantly less deviation from υ_{l} or υ_{q} than in the Takahashi et al. (1993) case.
The fitted b_{h} value of 30 794 J mol^{−1} for ${\mathit{\upsilon}}_{h}^{\text{LM95}}$ is unusual, falling outside the range fitted across the entire monthly mean OceanSODAETZH data product (25 139 to 30 731 J mol^{−1}; Sect. 2.4; Fig. S2a). Based on the A_{T}, T_{C} and salinity reported by Lee and Millero (1995), and with the carbonic acid dissociation constants of Lueker et al. (2000), we would have expected a b_{h} of 29 825 J mol^{−1}. Unlike Takahashi et al. (1993), which is not inconsistent with Eq. (20) but lacks data at high enough temperatures to convincingly support or reject it, the Lee and Millero (1995) dataset is clearly inconsistent with Eq. (20).
However, the Lee and Millero (1995) measurements are also inconsistent in a number of other ways not related to Eq. (20). We refitted the Lee and Millero (1995) dataset with the linear form (Eq. 5) and found a lower RMSD (1.6 µatm) using the υ_{l} value of 4.23 % °C^{−1} from Takahashi et al. (1993). It appears that Lee and Millero (1995) minimised residuals in ln (fCO_{2}) in their fit rather than minimising residuals in fCO_{2}; the latter approach returns almost exactly the Takahashi et al. (1993) value of 4.23 % °C^{−1}. Minimising residuals in ln (fCO_{2}) implicitly assumes that uncertainty in fCO_{2} is a constant fraction of the fCO_{2} value rather than an absolute fixed amount, but Lee and Millero (1995) reported the “reproducibility” as ±3 µatm, not as a percentage. While, at first glance, it may seem like an improvement that fitting the data consistently with the uncertainty brings the υ_{l} value in line with Takahashi et al. (1993), this in fact contradicts the reasoning of Lee and Millero (1995) themselves in that a different value is expected because of the rather different ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$. For ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ to be the same as that of Takahashi et al. (1993), either T_{C} or A_{T} would need to be adjusted by over 160 µmol kg^{−1}, which is far more than permitted by the measurement accuracy.
The Lee and Millero (1995) dataset is also not consistent with PyCO2SYS calculations. As mentioned above, the carbonic acid dissociation constants of Lueker et al. (2000) do not fit well at all, with an RMSD of 20 µatm (Fig. 3). While Lee and Millero (1995) claim that their data are “thermodynamically consistent” with the constants of Roy et al. (1993) and Goyet and Poisson (1989), neither of these fits very well, with both having an RMSD of ∼6 µatm (Fig. 3). However, they did acknowledge that more studies of the temperature sensitivity of the marine carbonate system are needed, pointing to possible problems with the Weiss (1974) formulation for CO_{2} solubility, which is still used in PyCO2SYS (Humphreys et al., 2022).
In summary, while comparisons with the experimental datasets do not rule out Eq. (20), they are less convincing than the comparisons with PyCO2SYS calculations in Sect. 3.1.1. For the Takahashi et al. (1993) dataset, it is plausible that extra measurements at higher temperatures could have revealed significant deviations from linearity, as predicted independently by both PyCO2SYS and Eq. (20), although the linear υ_{l} still fits the data more closely than υ_{h} does. The Lee and Millero (1995) dataset, which does include highertemperature measurements, does not support this. This latter dataset does feature a number of other inconsistencies that could cast doubt on its accuracy, but, alternatively, there may be some aspect of the equilibrium chemistry of CO_{2} in seawater that is missing from or incorrectly parameterised within existing models such as PyCO2SYS. It is not possible to draw a reliable conclusion either way from only these two small datasets. We need a much bigger body of measurements to be able to robustly distinguish between the different possible forms of υ.
3.1.3 Global data compilation
The Wanninkhof et al. (2022a) compilation provides much more data (∼2000 data points) with which to test the different temperature adjustment approaches, including across variations in salinity and carbonate chemistry (A_{T} and T_{C}). The compilation consists of fCO_{2} measured in discrete seawater samples at 20 °C (irrespective of the in situ temperature), along with fCO_{2} measured at (or close to) the in situ temperature with a continuously flowing underway system at closely corresponding times and locations. Independent measurements of A_{T} and T_{C} are also available for many of the data points.
This dataset goes some way towards fixing the data scarcity problem mentioned in Sect. 3.1.2, but it is still not a complete substitute for additional experiments like that of Takahashi et al. (1993). In particular, each sample has measurements of fCO_{2} at only two different temperatures, so the form of the t–fCO_{2} relationship between those two points cannot be determined for any individual sample. Instead, we must infer the suitability of different forms of υ from how well they work across the entire dataset.
We adjusted the fCO_{2} at 20 °C from the discrete samples to the temperature of the in situ measurement using five different approaches: the linear (υ_{l}) and quadratic (υ_{q}) equations of Takahashi et al. (1993) (Sect. 2.1.2); the van 't Hoff form with b_{h} constant as fitted to the Takahashi et al. (1993) dataset (${\mathit{\upsilon}}_{h}^{\text{Ta93}}$; Sect. 2.1.3) and parameterised as a function of temperature, salinity and fCO_{2} (υ_{p}; Sect. 2.4); and calculation with PyCO2SYS using T_{C} as the second known parameter and the carbonic acid dissociation constants of Lueker et al. (2000) (υ_{Lu00}). To compare the accuracy of these adjustments, we computed the rolling mean of the differences between fCO_{2} adjusted from 20 °C to the in situ temperature and fCO_{2} measured directly at the in situ temperature for each approach, using a ±4 °C window (Fig. 4). Like Wanninkhof et al. (2022a), we excluded outliers falling more than ±2 standard deviations from zero, which was 1.4 % of the total data points.
For negative Δt there are relatively few data (about 14 % of the dataset), such that the different rolling mean lines overlap and mostly fall within the large uncertainty window (Fig. 4). It is not possible to meaningfully distinguish between the different approaches. All appear to overestimate fCO_{2}(t_{1}) for Δt from about −14 to −5 °C, by a similar amount to each other, with some falling just outside the uncertainty window of the rolling mean. The consistency between the different approaches here is expected, as they are most similar to each other in the corresponding t_{1} range from about 10 to 20 °C (Fig. 2a). The minor deviations for negative Δt are therefore probably due to the sparsity of data in this temperature range, which means that errors due to measurement uncertainty and imperfect colocation of samples are not averaged out. We would expect these deviations from zero to diminish were more data added to the compilation.
The Δt range above 0 °C contains the majority of the dataset (86 %) and sufficient data for the approaches to diverge clearly from each other and for some to move well outside the uncertainty window (Fig. 4). Here, the linear form (υ_{l}) significantly overestimates fCO_{2}(t_{1}) as Δt increases. Wanninkhof et al. (2022a) tried fitting υ_{l} directly to this data compilation to improve the accuracy of the adjustment but were not able to fix this issue. Instead, they concluded that calculations with a fully resolved carbonate system were still required for Δt greater than about 1 °C, recommending the Lueker et al. (2000) carbonic acid dissociation constants with the Uppström (1974) boratetochlorinity ratio for the best agreement (υ_{Lu00}). Like υ_{l}, the quadratic form (υ_{q}) overestimates fCO_{2}(t_{1}) for positive Δt. The error is about half of that for υ_{l} but still well outside the uncertainty window. The van 't Hoff form with constant b_{h} (${\mathit{\upsilon}}_{h}^{\text{Ta93}}$) has a similar error to υ_{q} but with the opposite sign; i.e. fCO_{2}(t_{1}) is underestimated. However, the van 't Hoff form with a variable, parameterised b_{h} (υ_{p}) gives almost perfect agreement, even for the largest Δt, performing even better than the calculation with a fully resolved carbonate system (υ_{Lu00}), which also falls just within the uncertainty window.
This analysis shows that the van 't Hoff form is at least as capable as the other approaches of adjusting fCO_{2} to different temperatures across this data compilation. When variability in b_{h} is accounted for by the parameterisation proposed in Sect. 2.4, which is completely independent of this data compilation, this approach (υ_{p}) can be used for accurate Δt adjustments of up to at least 10 °C. More data for negative Δt are still essential to improve our confidence in this conclusion across the full temperature range found in the ocean.
Part of the motivation for the new approach to υ developed here (Sect. 2.1.3) was to avoid needing to know a second marine carbonate system parameter when adjusting fCO_{2} to different temperatures. υ_{p} with fixed b_{h} does not meet this requirement, because, as for υ_{l} and υ_{q}, the adjusted fCO_{2} values are not accurate enough. Initially it appears that υ_{p} does meet the requirement – its only inputs are fCO_{2}, temperature and salinity (Eq. 35), and it is equally or more accurate than calculations with PyCO2SYS (Fig. 4). However, arguably, the need for a second parameter has been sidestepped rather than truly eliminated. The parameterisation for b_{h} works because, to first order, b_{h}, υ_{h} and fCO_{2} are closely correlated with the ratio ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ but less sensitive to the absolute values of either T_{C} or A_{T} (Goyet et al., 1993; Wanninkhof et al., 2022a). In other words, the fCO_{2} term in the parameterisation acts as a proxy for ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$, implicitly providing the information about T_{C} and A_{T} necessary to calculate υ.
It would be possible to parameterise υ_{l} and υ_{q} across the ocean, as we have done for υ_{h} (i.e. υ_{p}), which would improve their agreement with the data compilation too. However, we have already established that there is no theoretical reason to expect the t–ln (fCO_{2}) relationship to be linear, especially at higher t, and Wanninkhof et al. (2022a) tried fitting υ_{l} to this dataset directly without satisfactory results. Further benefits of υ_{h} over υ_{q} are discussed in Sect. 3.2 on uncertainty propagation.
3.2 Experimental uncertainty
We performed Monte Carlo simulations of the Takahashi et al. (1993) experiment (Sect. 2.2.2) to determine the uncertainty in υ computed from that dataset using the linear (υ_{l}; Eq. 5), quadratic (υ_{q}; Eq. 6) and van 't Hoff (${\mathit{\upsilon}}_{h}^{\text{Ta93}}$; Eq. 19) fits.
Were the linear fit a valid representation of the t–ln (fCO_{2}) relationship, then the 1σ uncertainty in experimentally determined υ_{l} would be 0.035 % °C^{−1}. The biggest contributor (0.028 % °C^{−1} on its own) is the precision of the fCO_{2} measurement, with 0.020 % °C^{−1} from systematic bias in fCO_{2} alone. Systematic biases can affect υ because it is the gradient of the natural logarithm of fCO_{2} (Eq. 1), which is a nonlinear transformation. Temperature uncertainties have a much smaller impact, with 0.001 % °C^{−1} uncertainty in υ arising from t precision alone and no effect from t bias. Propagating through to exp (Υ_{l}) for a Δt of +1 °C gives an uncertainty of just under 0.04 % (Figs. 5 and S4). Following Eq. (4), the percentage uncertainty in exp (Υ) propagates through to the same percentage uncertainty in the adjusted fCO_{2} value at t_{1}, so this calculation is most relevant for understanding how uncertainties in υ affect temperatureadjusted fCO_{2} values.
With the quadratic fit (Eq. 6), the 1σ uncertainty in the slope (b_{q}) is 0.127 % °C^{−1}, whereas the corresponding value in the squared term (a_{q}) is $\mathrm{41.2}\times {\mathrm{10}}^{\mathrm{6}}$ °C^{−2}, and there is a covariance between these of $\mathit{\sigma}({a}_{\mathrm{q}},{b}_{\mathrm{q}})=\mathrm{51}\times {\mathrm{10}}^{\mathrm{9}}$ °C^{−3}. Propagating through to υ_{q} and exp (Υ_{q}) gives an uncertainty that is similar to that for the linear fit in the middle of the t range but is up to 4 times greater in colder and warmer waters (Figs. 5 and S4). It is essential to include the covariance term for propagation; failure to do so wrongly inflates the final uncertainty.
The van 't Hoff fit has a 1σ uncertainty in the b_{h} of 216.4 J mol^{−1}. Propagating through to ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$ and $\mathrm{exp}\left({\mathrm{{\rm Y}}}_{h}^{\text{Ta93}}\right)$ gives a similar uncertainty to that for υ_{l}, decreasing slightly with increasing temperature (Figs. 5 and S4).
The uncertainties calculated here, for $\mathrm{\Delta}t=+\mathrm{1}$ °C, are all lower than the climate quality target uncertainty for seawater fCO_{2} measurements of 0.5 % proposed by GOAON (Newton et al., 2015). This means that this uncertainty component alone would not cause an fCO_{2}(t_{1}) value with Δt up to ±1 °C to breach this target, regardless of which form was correct, although it is probably not small enough to ignore in a complete uncertainty budget. Also, its contribution scales roughly in proportion with Δt; therefore, for greater Δt, this component would become a greater part of the overall uncertainty in fCO_{2}(t_{1}).
Why do the different approaches have different uncertainties and how do we choose which approach and corresponding uncertainty to use? The linear and van 't Hoff forms both appear to have similarly low uncertainties, but the temperature adjustments that they predict differ from each other by more than the apparent uncertainty windows, especially above 25 °C (Fig. 2). The quadratic form has a completely different uncertainty profile with apparently much greater uncertainty at the edges of the temperature range (Fig. 5). The assumption implicit within any uncertainty calculation – that the model being fitted to the data is valid and that all of its adjustable parameters are truly unknown and necessary – is key to resolving this.
As we have shown in Sect. 3.1, it is not consistent with existing models of the marine carbonate system for υ to be constant and independent from temperature, so the uncertainty calculated for υ_{l} is valid only under this premise. In other words, the assumptions that “υ is constant” and “υ varies with temperature” are mutually incompatible. This is how υ_{l} and ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$ can deviate from each other by more than their apparent uncertainties.
It makes sense that υ_{q} has greater uncertainty than υ_{l} and ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$, as υ_{q} has an extra coefficient to be constrained by the same number of data points. Given more data at higher t, υ_{q} could plausibly be fitted to a shape almost identical to υ_{h}, yet it would still appear to have greater uncertainty for this reason. When using υ_{q}, we accept that there is some curvature in the t–ln (fCO_{2}) relationship, but we implicitly assume that we do not know what form that curvature takes, other than it can be represented by some secondorder polynomial. The adjustable parameters that represent the curvature are thus constrained empirically only by the available data, leading to much higher apparent uncertainty towards its edges. However, we now know that, to first order, υ should follow a particular curvature that can be represented with only one adjustable parameter (b_{h}); it is not an unknown that must be found empirically. The lower uncertainty in ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$ and $\mathrm{exp}\left({\mathrm{{\rm Y}}}_{h}^{\text{Ta93}}\right)$ relative to υ_{q} and exp (Υ_{q}) is therefore more realistic (Fig. 5), with the caveat that more experimental work is still needed to determine whether the expected relationship from Eq. (20) can be produced in laboratory measurements (Sect. 3.1.2).
While small, this uncertainty component is probably not negligible for the most accurate analyses. More independent reproductions of the Takahashi et al. (1993) experiment, ideally including temperatures up to 35 °C, could reduce this component into a negligible term. However, this component reflects only the experimental uncertainties in fCO_{2} and temperature. To use this value as the total uncertainty in υ and/or exp (Υ) would be to assume that the single sample measured by Takahashi et al. (1993) is representative of the global ocean. It does not account for spatiotemporal variability in υ, which we discuss in Sect. 3.3.
3.3 Spatial and temporal variability
3.3.1 Variability in υ_{Lu00}
The timeaveraged theoretical value of surface ocean υ_{Lu00} in OceanSODAETZH (Gregor and Gruber, 2021) follows a latitudinal distribution, with lower values near the Equator (Fig. 6a). The range in υ_{Lu00} is more than 15 % of its mean, from less than 4.0 % °C^{−1} to over 4.6 % °C^{−1}. The pattern is dominantly driven by temperature. The other variables (A_{T}, T_{C} and salinity) together act in the opposite sense, slightly reducing the latitudinal gradient in υ_{Lu00}. This can be seen from the distribution of ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$, which uses the constant b_{h} found by fitting to the Takahashi et al. (1993) dataset (Sect. 2.1.3) and has a similar but stronger latitudinal gradient (Fig. 6b). A counterintuitive consequence of this is that, were the Takahashi et al. (1993) experiment to be repeated systematically across the globe, fitting with a linear equation at each location, this would return the opposite pattern of υ_{l} to what we see in Fig. 6a: higher values near the Equator and lower near the poles. The variability in υ_{l} as determined following Takahashi et al. (1993) is independent from the original in situ temperature of the sample, so the experiment would reveal variability in υ only due to the nonthermal parameters.
The seasonal range in υ is a similar size to, although slightly less than, the spatial range in its timeaveraged value. It is less than 0.2 % °C^{−1} across most of the open ocean, reaching a maximum of just over 0.4 % °C^{−1}, or around 10 % of the mean, in some nearcoastal areas with strong seasonal variability in temperature and/or ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ (Fig. 6c).
Together, the spatial and seasonal variability in υ_{Lu00} has a standard deviation of 0.23 % °C^{−1}. This could be considered to represent the 1σ uncertainty in using a constant υ_{l} globally, keeping in mind that it has strong regional systematic biases rather than being randomly distributed. It is significantly greater than the apparent experimental uncertainty in υ_{l} or ${\mathit{\upsilon}}_{h}^{\text{Ta93}}$ of around 0.04 % °C^{−1} (Sect. 3.2). Experiments like that of Takahashi et al. (1993) should be sufficiently accurate to detect the variability in υ if conducted at different locations around the global ocean. However, this also means that spatiotemporal variability that is unaccounted for would lead to uncertainty in υ and exp (Υ) that is several times greater in practice than that computed in Sect. 3.2.
On multidecadal timescales, υ has been decreasing globally at an average rate of 0.001 $\mathrm{\%}\phantom{\rule{0.125em}{0ex}}\mathrm{\xb0}{\mathrm{C}}^{\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{\mathrm{1}}$ (Fig. 6d). The decrease has been primarily driven by increasing ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ with a smaller contribution from warming of the surface ocean. This trend is consistent with that identified by Wanninkhof et al. (2022a), and it should lead to a minor decrease in the temperature dependency of surface ocean fCO_{2} while atmospheric CO_{2} levels continue to rise.
Wanninkhof et al. (2022a) pointed out that υ is greater at lower temperatures and lower at lower ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ and that temperature and ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ are, to first order, negatively correlated in the global surface ocean (largely due to the influence of temperature on CO_{2} solubility), leading to them having opposing effects on υ and thus reducing its variability. We find that this effect does reduce spatial variability in υ, although by 30 % at most relative to a constant${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ scenario. For a temperature change from −1.7 to 30 °C, the main body of the OceanSODAETZH monthly mean data product has a range in υ_{Lu00} from about 3.97 % to 4.60 % °C^{−1}, whereas at a constant ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ of 0.9, υ_{Lu00} varies from 3.85 % to 4.72 % °C^{−1} (Fig. 7). However, this partial cancellation of the effect of temperature on υ by ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ is not relevant for making temperature adjustments; it does not mean that the temperature sensitivity of υ can be ignored or is any less important. This is because adjustments are made on single samples under constant ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$. The sensitivity of υ to temperature for any individual sample is not lessened by the negative correlation between temperature and ${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ observed across the global dataset. In other words, coupled temperature–${T}_{\mathrm{C}}/{A}_{\mathrm{T}}$ variations in the surface ocean reduce the spatiotemporal variability in υ under in situ conditions, but they do not affect by how much υ changes with temperature for a particular sample. Consequently, the apparent reduced variability in υ is relevant only for small (e.g. Δt<1 °C) temperature adjustments.
3.3.2 Components of υ
Variability in υ through space and time arises primarily from the temperature dependence of the equilibrium constants of the marine carbonate system. Across the contemporary global surface ocean, the biggest contributor to υ is ${K}_{\mathrm{0}}^{\prime}$, the solubility constant for CO_{2}, which typically represents 58 %–89 % of total υ (Fig. 8). The carbonic acid equilibrium constants ${K}_{\mathrm{1}}^{\ast}$ and ${K}_{\mathrm{2}}^{\ast}$ contribute a similar size effect on υ to each other and slightly less than ${K}_{\mathrm{0}}^{\prime}$. However, the ${K}_{\mathrm{1}}^{\ast}$ contribution is always negative, whereas the ${K}_{\mathrm{2}}^{\ast}$ contribution is always positive; therefore, they partly cancel each other out. Smaller contributions are made by the borate (9 %–21 %) and water (1 %–8 %) equilibria (${K}_{\mathrm{B}}^{\ast}$ and ${K}_{\mathrm{w}}^{\ast}$, respectively). All other equilibrium constants have a negligible influence on υ in the contemporary ocean, although they might become important at pH values closer to their pK.
Each parameterisation of the carbonic acid dissociation constants gives a different υ distribution, but there are some similarities (Fig. 8). We could not identify a link between the functional forms of the parameterisations (i.e. which combinations of t and salinity terms were used) and the distributions shown in Fig. 8; rather, the type of seawater, specific dataset(s) and methodology used to determine the constants were the main controls.
In Fig. 8, the two options labelled “GEOSECS” (6 and 7; Me73 and Me73P) both use the measurements of Mehrbach et al. (1973), as parameterised by Peng et al. (1987) (because Mehrbach et al., 1973, did not provide parameterised functions for their measurements); curiously, total υ with these options has a roughly opposite spatial distribution to all of the other options, with the lowest υ found at high latitudes. Although the GEOSECS options also use different ${K}_{\mathrm{B}}^{\ast}$ and ${K}_{\mathrm{w}}^{\ast}$ parameterisations from all of the other options (Humphreys et al., 2022), which modulates the pattern, the inversion is caused by ${K}_{\mathrm{1}}^{\ast}$ and ${K}_{\mathrm{2}}^{\ast}$. In any case, these options are present in (Py)CO2SYS to reproduce calculations from the GEOSECS era and are not intended for use with modern measurements. Aside from this pair, all other parameterisations based at least in part on the Mehrbach et al. (1973) dataset (labelled “Mehrbach”) have similar distributions for ${K}_{\mathrm{1}}^{\ast}$, ${K}_{\mathrm{2}}^{\ast}$ and total υ. The SB21 parameterisation (Schockman and Byrne, 2021) consists of new, spectrophotometric measurements of the product ${K}_{\mathrm{1}}^{\ast}{K}_{\mathrm{2}}^{\ast}$ which were paired with the Waters and Millero (2013) ${K}_{\mathrm{1}}^{\ast}$ to find ${K}_{\mathrm{2}}^{\ast}$, resulting in virtually zero variability in total υ overall. This low variability in total υ is echoed by the Su20 parameterisation (Sulpis et al., 2020), which is based on field observations during which A_{T}, T_{C}, pH and fCO_{2} were measured simultaneously, but the low variability is arrived at in a different way, with rather different distributions for the individual ${K}_{\mathrm{1}}^{\ast}$ and ${K}_{\mathrm{2}}^{\ast}$ effects. However, this does not mean that these parameterisations support the use of a constant, temperatureinsensitive υ, rather just that the effects of temperature and seawater chemistry (e.g. ${A}_{\mathrm{T}}/{T}_{\mathrm{C}}$) on in situ υ cancel each other out across the surface ocean more completely than they do for Lueker et al. (2000) (Fig. 7). Indeed, neither SB21 nor Su20 can be fitted to the Takahashi et al. (1993) dataset very well (Fig. S2). Parameterisations Pa18 (Papadimitriou et al., 2018), focused on highsalinity, lowtemperature seaice brines, and Mi02 (Millero et al., 2002), based on field measurements, both have similar variability in υ to the “Mehrbach” group.
There is a noticeable difference in the distributions, especially for ${K}_{\mathrm{1}}^{\ast}$, when moving to the group of parameterisations based on synthetic seawater (Dickson and Millero, 1987; Goyet and Poisson, 1989; Roy et al., 1993; DM87H and DM87HM, GP89, and Ro93, respectively). This suggests that some components of natural seawater that were not replicated in the synthetic seawater used in their experiments may affect the apparent temperature sensitivity of the equilibrium constants being measured and thus υ. For example, the synthetic seawater recipes used by Goyet and Poisson (1989) and Roy et al. (1993) both did not include any borate, unlike real seawater (Lee et al., 2010; Uppström, 1974). Suspected interactions between borate and carbonate species (Mojica Prieto and Millero, 2002) were not explicitly modelled when evaluating the equilibrium constants, so their effects would be implicitly included within the ${K}_{\mathrm{1}}^{\ast}$ and ${K}_{\mathrm{2}}^{\ast}$ values in real seawater but not in the synthetic seawater. Dissolved organic matter could also have an equivalent effect. Ultimately, any dissolved species that reacts in an equilibrium with water (i.e. any weak acid or base) or with any component of T_{C} (i.e. CO_{2}(aq), ${{\mathrm{HCO}}_{\mathrm{3}}}^{}$ or ${{\mathrm{CO}}_{\mathrm{3}}}^{\mathrm{2}}$) could affect υ.
The parameterisation of Mi79 is also shown for completeness, although its distribution is not meaningful because it is intended for zerosalinity freshwater (Millero, 1979).
The parameterisations of ${K}_{\mathrm{1}}^{\ast}$ and ${K}_{\mathrm{2}}^{\ast}$ differ from each other sufficiently that, given the high accuracy with which we can measure fCO_{2} and thus experimentally determine υ (Sect. 3.2), more experiments like that of Takahashi et al. (1993) but including a wider range of temperatures (i.e. at least up to 35 °C) would be a powerful tool to help distinguish between the different parameterisations of the equilibrium constants (Fig. S2) and other options (e.g. boratetochlorinity ratio; Humphreys et al., 2022) used when solving the marine carbonate system. Taking this a step further, experimental data like those of Takahashi et al. (1993), Woosley (2021), and Woosley and Moon (2023) could be incorporated when generating these parameterisations. These data constrain the derivatives of the equilibrium constants with respect to temperature, which the parameterisations are not normally optimised for. Constraining these derivatives in the fit would mean that the parameterisation can more accurately calculate derivative properties like υ. Such an approach is common in other fields, such as the Pitzer model for chemical activities in aqueous solutions (Pitzer, 1991). There, different measurable properties of the solution constrain the derivatives of fitted coefficients with respect to temperature, pressure and composition, so a single set of coefficients can be found that can accurately calculate all of these properties (e.g. Archer, 1992).
At present, including derivative properties in parameterisations would be limited by the scarcity of measurements like those of Takahashi et al. (1993). One possible alternative would be to use data like those compiled by Wanninkhof et al. (2022a) and datasets in which the marine carbonate system has been overdetermined (i.e. more than two parameters measured) in the field to better parameterise the equilibrium constants (e.g. Sulpis et al., 2020). A practical challenge with this would be that fCO_{2} measurements are archived in different compilations (e.g. SOCAT; Bakker et al., 2016) from other carbonate system variables (e.g. GLODAP; Lauvset et al., 2024), and it is not straightforward to match them back together. However, even if this were overcome, this approach would be less useful for parameterising equilibrium constants than using dedicated experiments in which wellcharacterised samples are manipulated under a range of different conditions. In the ocean, temperature is correlated with most other properties, so it would be difficult to isolate which of several correlated drivers was responsible for observed variability. Using these field data to generate parameterisations would also mean that they could no longer be used for independent validation. A better approach would be to reserve field data compilations for testing parameterisations that are based on more carefully controlled laboratory experiments. Combining new experimental measurements with field data in this way and interpreting the results through the lens of the new theoretical basis for υ developed in this study will lead to more accurate temperature adjustments of seawater fCO_{2} measurements and contribute towards a more internally consistent understanding of the marine carbonate system.
3.3.3 Propagation through to temperature adjustments
Expressions for υ, such as υ_{l}, υ_{q} and υ_{h}, are of interest because they can be used to adjust fCO_{2} data to different temperatures without knowing a second carbonate system parameter. The variability in υ discussed so far in Sect. 3.3 is therefore most important in terms of how it affects exp (Υ) and consequent uncertainty and biases in the adjusted fCO_{2} values if ignored.
After a 1 °C warming adjustment from mean surface ocean conditions in OceanSODAETZH, both the υ_{l} and υ_{p} forms have a global mean bias (relative to the change calculated from A_{T} and T_{C}) of less than 0.1 µatm, which is negligible. That the global mean bias is close to zero for υ_{l} is a coincidence, due to Takahashi et al. (1993) using seawater with roughly average υ_{Lu00} (Fig. 6a). However, there are stronger regional biases in υ_{l}, with 95 % of the residuals falling in the range from −1.3 to +1.1 µatm, leading to a global RMSD of 0.8 µatm, or equivalently 0.24 % (Fig. 9a). The van 't Hoff form with parameterised b_{h} (υ_{p}) has much lower regional biases, with a global RMSD of 0.16 µatm (0.04 %) and with 95 % of the residuals falling from −0.26 to +0.24 µatm (Fig. 9b). The Takahashi et al. (1993) quadratic fit (υ_{q}) and the van 't Hoff form with constant b_{h} fitted to the Takahashi et al. (1993) dataset (${\mathit{\upsilon}}_{h}^{\text{Ta93}}$) (both not shown) had negligible global mean biases and intermediate regional biases, with RMSD values of around 0.4 µatm.
The experimental 1σ uncertainty in exp (Υ_{l}) and $\mathrm{exp}\left({\mathrm{{\rm Y}}}_{h}^{\text{Ta93}}\right)$ fitted to the Takahashi et al. (1993) dataset was around 0.04 % for a 1 °C warming adjustment in both cases (Sect. 3.2). For υ_{l}, the additional uncertainty from spatiotemporal variability that was unaccounted for, as well as the fact that the t–ln (fCO_{2}) relationship is not linear, is therefore 4 times greater than the experimental uncertainty. For υ_{h}, the experimental and spatiotemporal uncertainties are about equal in size. Combining these components gives a 1σ uncertainty in adjusted fCO_{2} of around 0.06 % for υ_{p}, or 0.24 % for υ_{l}. A 1 °C adjustment with either approach could therefore still meet the GOAON climate quality goal of 0.5 % (Newton et al., 2015), if other uncertainty components (e.g. in the original fCO_{2} measurement at t_{0}) were small enough. However, the spatiotemporal component of the uncertainty in υ_{l} results from a known bias with strong regional patterns. Such biases should be eliminated by using a more realistic form of υ (e.g. υ_{h} rather than υ_{l} or υ_{q}, and υ_{p} or similar to account for variability in υ_{h}), rather than incorporating them into an overall uncertainty value. However, if applying υ_{l} globally, 0.24 % is a reasonable value for the component of uncertainty due to a +1 °C adjustment. It is important to remember that this component is a systematic bias with strong regional patterns when combining it with other components that may be independent for each measurement, because these two types of uncertainty have very different statistical consequences when propagated through to other calculations.
The global biases and the amplitude of the spatiotemporal variability scale approximately, but not exactly, with Δt (Fig. 10). The υ_{p} approach has the lowest RMSD throughout the Δt range, whereas υ_{l} has the greatest. υ_{q} and υ_{h} have intermediate RMSD values, with that for υ_{h} being significantly lower than that for υ_{q} for negative Δt. The RMSD values are generally greater for positive Δt than for negative Δt, which emphasises the need to better understand the temperature sensitivity of fCO_{2} and the wider marine carbonate system, particularly at higher temperatures (e.g. Sect. 3.1.1; Woosley, 2021). For a Δt of 10 °C, as might be used in studies unravelling the drivers of seasonal fCO_{2} changes, the adjusted fCO_{2} would have an uncertainty due to spatiotemporal variability of around 0.8 % for υ_{p}, which is outside the GOAON climate quality goal (0.5 %) but well within the weather quality goal of 2.5 % (Newton et al., 2015). However, with υ_{l}, this component of the uncertainty in adjusted fCO_{2} would be over 3 %, also exceeding the weather quality goal.
Seawater fCO_{2} data can be adjusted to different temperatures using either Eq. (2) or Eq. (4) (which are mathematically identical) with an appropriate expression for Υ, which is υ integrated over the temperature range of interest. The new approach proposed here uses Υ_{h} as defined in Eq. (21), which has one unknown coefficient (b_{h}). The value of b_{h} can be found by fitting to experimental data where available (e.g. Table 1), but spatiotemporal variability in b_{h} should be accounted for – for example, with the parameterisation in Eq. (35). This approach and the earlier linear and quadratic approaches of Takahashi et al. (1993) have been built into the PyCO2SYS software as of v1.8.3.
There is no theoretical reason to expect a linear relationship between temperature and ln (fCO_{2}) under isochemical conditions (constant A_{T} and T_{C}), as was proposed empirically by Takahashi et al. (1993). Rather, calculations of υ with PyCO2SYS and a theoretical analysis based on the equations governing the marine carbonate system and the van 't Hoff equation each independently suggest that ln (fCO_{2}) should be proportional to $\mathrm{1}/{t}_{\mathrm{K}}$ to first order (Sect. 3.1.1). Although this means that the t–ln (fCO_{2}) relationship does seem to be approximately linear within the temperature range of 2.1 to 24.5 °C measured by Takahashi et al. (1993), it should diverge rapidly from linearity, especially at temperatures above 25 °C, although there are some discrepancies with the very few existing laboratory data which require more measurements to resolve (Sect. 3.1.2). Nevertheless, the new form of the relationship between temperature and fCO_{2} proposed here (υ_{h} with a parameterised b_{h} coefficient, i.e. υ_{p}; Eq. 19 and Sect. 2.4) agrees better with fieldbased datasets than any other approach previously used to adjust seawater fCO_{2} to different temperatures (Sect. 3.1.3).
The parameterisation for b_{h} (and thus υ_{p}) is based on a specific set of carbonate system parameterisations. We know that these are not perfect (e.g. Woosley and Moon, 2023; Carter et al., 2024), so the b_{h} parameterisation is an intermediate solution. It does agree very well with the available field data (Wanninkhof et al., 2022a) with which it can be tested (and from which it is independent), but it may need to be updated when new data and/or more accurate parameterisations of the marine carbonate system become available. For some applications, accuracy may be improved by developing parameterisations of b_{h} that are optimised for any particular region(s) of interest.
The uncertainty in υ depends on the method being used to compute it. The Takahashi et al. (1993) experiment gave a 1σ uncertainty of about 0.04 % °C^{−1} for the linear form (Sect. 3.2), but this does not account for variability in υ with temperature and seawater composition, nor for the fact that the relationship between temperature and ln (fCO_{2}) is not linear. A 1σ in υ of 0.23 % °C^{−1} is therefore more appropriate if using this linear approach (Sect. 3.3.1), which corresponds to 0.24 % for fCO_{2} adjusted by +1 °C (Sect. 3.3.3). This uncertainty can be reduced by using the new form of the t–ln (fCO_{2}) relationship developed here and parameterising its fitted constant (b_{h}) as a function of temperature, salinity and fCO_{2} (Sect. 2.4), which gives an uncertainty of 0.06 % °C^{−1} for fCO_{2} after the same +1 °C adjustment (Sect. 3.3.3). While these uncertainties are probably not large enough to have major consequences for global air–sea CO_{2} flux budgets, they are not negligible and may be important in regional studies. They should be incorporated into uncertainty budgets and their systematic components should be eliminated by using a more meaningful and accurate form to model the t–ln (fCO_{2}) relationship, like that proposed here. However, the uncertainty budget is still incomplete: we cannot propagate uncertainties from the equilibrium constants of the marine carbonate system through to υ, as those uncertainties are not yet adequately defined (Orr et al., 2018). However, if υ can be fitted directly to measurements like those of Takahashi et al. (1993), the uncertainties in the equilibrium constants can be ignored when adjusting fCO_{2} values to different temperatures.
More measurements like those conducted by Takahashi et al. (1993) are still needed to empirically confirm whether the υ_{h} form, and indeed calculations from A_{T} and T_{C}, are sufficiently accurate, especially above 25 °C where the deviation from υ_{l} is expected to be greatest. If further measurements appear to confirm the hightemperature linearity observed by Lee and Millero (1995), there must be either (1) a major error in a parameterisation or a missing component of the equilibrium model for seawater chemistry implemented in software tools like PyCO2SYS at higher temperatures or (2) some technical problem with the experimental approach that means it does not measure quite what we think. More measurements could show whether discrepancies between υ_{p} and υ_{Lu00} are due to needing a more sophisticated equation for υ_{h} with extra fitted parameters or whether they represent inaccuracies in the Lueker et al. (2000) and other carbonate system parameterisations. In principle, we can measure the effect of temperature on fCO_{2} accurately enough to distinguish between these different parameterisations. Coupling such new experiments with other measurements such as pH on the same samples may also help to understand and rectify wider discrepancies in equilibrium models of seawater chemistry. Together with field measurements for independent validation, and using a theoretical basis to interpret the results rather than working only with empirical fits, such a holistic approach should help to better understand υ and its application to adjusting fCO_{2} data to different temperatures and should also lead to a more accurate and internally consistent model of the marine carbonate system.
The reactions corresponding to the equilibrium constants of interest are as follows (Humphreys et al., 2022):
The standard enthalpies of formation (Δ_{f}H^{⊖}) for the relevant species are given in Table A1.
Using Hess's law, the standard enthalpies of reaction for Reactions (AR1)–(AR3), as used in Eq. (18), are as follows:
Summing Eqs. (A1)–(A3) as required by Eq. (18) and propagating the uncertainties listed in Table A1 gives a value of ${\mathrm{\Delta}}_{\mathrm{r}}{H}_{\mathrm{2}}^{\ominus}{\mathrm{\Delta}}_{\mathrm{r}}{H}_{\mathrm{0}}^{\ominus}{\mathrm{\Delta}}_{\mathrm{r}}{H}_{\mathrm{1}}^{\ominus}=\mathrm{25}\phantom{\rule{0.125em}{0ex}}\mathrm{288}$ ± 98 kJ mol^{−1}.
The code for the updated version of PyCO2SYS (v1.8.3) is freely available on GitHub and is archived on Zenodo (Humphreys et al., 2024); it can be installed in Python using either pip or conda (via condaforge). The code used for the rest of the analysis in this paper is available via Zenodo (https://doi.org/10.5281/zenodo.13492773, Humphreys, 2024).
All datasets used in this study are from third parties. Data for the measurements of Takahashi et al. (1993) and Lee and Millero (1995) were obtained from tables printed within the respective publications. The surface fCO_{2} compilation of Wanninkhof et al. (2022a) is available from the US National Centers for Environmental Information (NCEI accession 0247018, https://doi.org/10.25921/8c3kvz52, Wanninkhof et al., 2022b). The OceanSODAETZH data product (Gregor and Gruber, 2021) is also available from the NCEI (NCEI accession 0220059, https://doi.org/10.25921/m5wxja34, Gregor and Gruber, 2020).
The supplement related to this article is available online at: https://doi.org/10.5194/os2013252024supplement.
The author has declared that there are no competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.
This article was inspired by discussions during the “Series of workshops on surface ocean pCO_{2} observations, synthesis and data products” held at the Flanders Marine Institute (VLIZ) in November 2023. We are indebted to the authors of the previous studies that collected and made the data used here available, especially the measurements of Takahashi et al. (1993), the compilation of Wanninkhof et al. (2022) and the OceanSODAETZH data product (Gregor and Gruber, 2021). We thank Rik Wanninkhof and Fiz Perez for providing reviews and Mario Hoppema for editorial comments which helped to improve the paper.
This paper was edited by Mario Hoppema and reviewed by Fiz F. Perez and Rik Wanninkhof.
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