Abstract
Abstract. The successful and efficient approach at the basis of the Solver Suite for Alkalinity-PH Equations (SolveSAPHE) (Munhoven, 2013), which determines the carbonate system speciation by calculating pH from total alkalinity (AlkT) and dissolved inorganic carbon (CT), and which converges for any physically sensible pair of such data, has been adapted and further developed to work with AlkT–CO2, AlkT–HCO3-, and AlkT–CO32-. The mathematical properties of the three modified alkalinity–pH equations are explored. It is shown that the AlkT–CO2, and AlkT–HCO3- problems have one and only one positive root for any physically sensible pair of data (i.e. such that [CO2]>0 and [HCO3-]>0). The space of AlkT–CO32- pairs is partitioned into regions where there is either no solution, one solution or where there are two. The numerical solution of the modified alkalinity–pH equations is far more demanding than that for the original AlkT–CT pair as they exhibit strong gradients and are not always monotonous. The two main algorithms used in SolveSAPHE v1 have been revised in depth to reliably process the three additional data input pairs. The AlkT–CO2 pair is numerically the most challenging. With the Newton–Raphson-based solver, it takes about 5 times as long to solve as the companion AlkT–CT pair; the AlkT–CO32- pair requires on average about 4 times as much time as the AlkT–CT pair. All in all, the secant-based solver offers the best performance. It outperforms the Newton–Raphson-based one by up to a factor of 4 in terms of average numbers of iterations and execution time and yet reaches equation residuals that are up to 7 orders of magnitude lower. Just like the pH solvers from the v1 series, SolveSAPHE-r2 includes automatic root bracketing and efficient initialisation schemes for the iterative solvers. For AlkT–CO32- data pairs, it also determines the number of roots and calculates non-overlapping bracketing intervals. An open-source reference implementation of the new algorithms in Fortran 90 is made publicly available for usage under the GNU Lesser General Public Licence version 3 (LGPLv3) or later.
Highlights
Among all the aspects of the ongoing global environmental changes, the solution chemistry of carbon dioxide (CO2) is one of the best known
The difficulties posed by AlkT–CO2 that were at the origin of most of the amendments to the solver algorithms show up in the histograms for the number of iterations required to reach convergence shown in Fig. 6 for solve_at_general which uses the hybrid Newton– Raphson–regula falsi–bisection scheme and in Fig. 7 for solve_at_general_sec which uses the hybrid secant– regula falsi–bisection scheme
The approach adopted in SolveSAPHE (Munhoven, 2013) to safely determine carbonate speciation in particular, and speciation calculations of mixtures of acids in aqueous solution in general, knowing only the total concentrations of the different acid systems and the total alkalinity of the system was adapted and extended here to use [CO2], [HCO−3 ] and [CO23−] instead of the total inorganic carbon concentration, CT
Summary
Among all the aspects of the ongoing global environmental changes (climate change, ocean acidification, etc.), the solution chemistry of carbon dioxide (CO2) is one of the best known. The speciation, i.e. the determination of the concentrations of the individual species, requires four constraints Two of these are given by the equilibrium relationships that characterise the equilibria between dissolved CO2 and HCO−3 on one hand, and between HCO−3 and CO23− on the other hand, assuming that the respective equilibrium constants are known or can be calculated. Direct [CO23−] measurements, which might be most advisable for tracing carbonate mineral saturation states, are best paired with AlkT or CT (Sharp and Byrne, 2019). It can be expected that, once it becomes more widely used, the measurement uncertainty currently affecting that still young measurable can be reduced and eventually become better than that of [CO23−] calculated from AlkT–CT, which is currently the best option (Sharp and Byrne, 2019). The review of Orr et al (2015) offers a systematic analysis of subsisting uncertainties and inconsistencies between 10 such packages, focusing on the sets of
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