[en] The successful and efficient approach at the basis of SolveSAPHE (Munhoven, 2013), which determines the carbonate system speciation by calculating pH from total alkalinity (Alk_T) and dissolved inorganic carbon (C_T), and which converges from any physically sensible pair of such data, has been adapted and further developed for work with AlkT & CO2, AlkT & HCO3 and AlkT & CO3. The mathematical properties of the three modified alkalinity-pH equations are explored. It is shown that the Alk_T & CO2 and Alk_T & HCO3 problems have one and only one positive root for any physically sensible pair of data (i.e, such that, resp., [CO2]>0 and [HCO3]>0). The space of Alk_T & CO3 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 Alk_T-C_T pair as they exhibit strong gradients and are not always monotonous.The two main algorithms used from SOLVESAPHE v. 1 had to be revised in depth to reliably process the three additional data input pairs. The Alk_T & CO2 pair is numerically the most challenging. With the Newton-Raphson based solver, it takes about five times as long to solve as the companion Alk_T & C_T pair, while Alk_T & CO2 requires about four times as much time. All in all, it is nevertheless the secant based solver that offers the best performances. It outperforms the Newton-Raphson based one by up to a factor of four, to reach equation residuals that are up to seven orders of magnitude lower. Just like the pH solvers from routines from the v. 1 series, SolveSAPHE v. 2 includes automatic root bracketing and efficient initialisation schemes for the iterative solvers. For Alk_T & CO3 pairs of data, it also determines the number of roots and calculates non-overlapping bracketing intervals. An open source reference implementation in Fortran 90 of the new algorithms is made publicly available for usage under the GNU Lesser General Public Licence v. 3 or later.
Research Center/Unit :
Sphères - SPHERES
Disciplines :
Earth sciences & physical geography Mathematics
Author, co-author :
Munhoven, Guy ; Université de Liège - ULiège > Département d'astrophys., géophysique et océanographie (AGO) > Labo de physique atmosphérique et planétaire (LPAP)
Language :
English
Title :
SolveSAPHE-r2: revisiting and extending the Solver Suite forAlkalinity-PH Equations for usage with CO2, HCO3 or CO3 input data