Mathematics of the total alkalinity–pH equation – pathway to robust and universal solution algorithms: the SolveSAPHE package v1.0.1

Munhoven, Guy

doi:10.5194/gmd-6-1367-2013

Article (Scientific journals)

Mathematics of the total alkalinity–pH equation – pathway to robust and universal solution algorithms: the SolveSAPHE package v1.0.1

Munhoven, Guy

2013 • In Geoscientific Model Development, 6, p. 1367-1388

Peer Reviewed verified by ORBi

Permalink
https://hdl.handle.net/2268/155792

DOI
10.5194/gmd-6-1367-2013

Files (2)Send to Details Statistics Bibliography Similar publications

Files

Full Text

gmd-6-1367-2013.pdf

Publisher postprint (1.75 MB)

Download

Annexes

gmd-6-1367-2013-supplement.zip

Publisher postprint (2.44 MB)

Supplementary Results and Software

Download

This work is distributed under the Creative Commons Attribution 3.0 License. It is available in open access on the publisher's website under http://www.geosci-model-dev.net/6/1367/2013/gmd-6-1367-2013.html .

All documents in ORBi are protected by a user license.

Send to

RIS BibTex APA Chicago Permalink X Linkedin

Details

Keywords :

pH; Alkalinity; equation solving

Abstract :

[en] The total alkalinity–pH equation, which relates total alkalinity and pH for a given set of total concentrations of the acid–base systems that contribute to total alkalinity in a given water sample, is reviewed and its mathematical properties established. We prove that the equation function is strictly monotone and always has exactly one positive root. Different commonly used approximations are discussed and compared. An original method to derive appropriate initial values for the iterative solution of the cubic polynomial equation based upon carbonate-borate-alkalinity is presented. We then review different methods that have been used to solve the total alkalinity–pH equation, with a main focus on biogeochemical models. The shortcomings and limitations of these methods are made out and discussed. We then present two variants of a new, robust and universally convergent algorithm to solve the total alkalinity–pH equation. This algorithm does not require any a priori knowledge of the solution. SolveSAPHE (Solver Suite for Alkalinity-PH Equations) provides reference implementations of several variants of the new algorithm in Fortran 90, together with new implementations of other, previously published solvers. The new iterative procedure is shown to converge from any starting value to the physical solution. The extra computational cost for the convergence security is only 10–15% compared to the fastest algorithm in our test series.

Research Center/Unit :

Laboratiore de Physique Atmosphérique et Planétaire

Disciplines :

Earth sciences & physical geography
Physical, chemical, mathematical & earth Sciences: Multidisciplinary, general & others

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 :

Mathematics of the total alkalinity–pH equation – pathway to robust and universal solution algorithms: the SolveSAPHE package v1.0.1

Publication date :

2013

Journal title :

Geoscientific Model Development

ISSN :

1991-959X

eISSN :

1991-9603

Publisher :

Copernicus GmbH

Volume :

Pages :

1367-1388

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

http://www.geosci-model-dev.net/6/1367/2013/gmd-6-1367-2013.html
https://doi.org/10.5281/zenodo.3752250

Funders :

F.R.S.-FNRS - Fonds de la Recherche Scientifique

Commentary :

Revised versions of the SolveSAPHE Fortran library with bug fixes are available on Zenodo under https://www.zenodo.org/record/3752250 (DOI: 10.5281/zenodo.3752250).

Available on ORBi :

since 11 September 2013

Statistics

Number of views

340 (18 by ULiège)

Number of downloads

996 (3 by ULiège)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

OpenAlex citations

Bibliography

Anderson, N. and Bj̈ork, A.: A new high order method of regula falsi type for computing a root of an equation, BIT, 13, 253-264, 1973.
Antoine, D. and Morel, A.: Modelling the seasonal course of the upper ocean pCO2 (I). Development of a one-dimensional model, Tellus B, 47, 103-121, doi:10.1034/j.1600-0889.47.issue1.11.x, 1995.
Arndt, S., Regnier, P., Godd́eris, Y., and Donnadieu, Y.: GEOCLIM reloaded (v 1.0): a new coupled earth system model for past climate change, Geosci. Model Dev., 4, 451-481, doi:10.5194/gmd-4-451-2011, 2011.
Aumont, O. and Bopp, L.: Globalizing results from ocean in situ iron fertilization studies, Global Biogeochem. Cy., 20, GB2017, doi:10.1029/ 2005GB002591, 2006.
Bacastow, R.: Numerical evaluation of the evasion factor, in: Carbon Cycle Modelling, vol. 16 of SCOPE, chap. 3.4, edited by: Bolin, B., John Wiley & Sons, Chichester, NY, 95-98, 1981.
Bacastow, R. and Keeling, C. D.: Atmospheric carbon dioxide and radiocarbon in the natural carbon cycle. II. Changes from A.D. 1700 to 2070 as deduced from a geochemical model, in: Carbon and the Biosphere, Proceedings of the 24th Brookhaven Symposium in Biology, 16-18 May 1972, edited by: Woodwell, G. M. and Pecan, E. V., US Atomic Energy Commission, 86-135, Upton, NY, 1973.
Bates, R. G.: pH measurements in the marine environment, Pure Appl. Chem., 54, 229-232, doi:10.1351/pac198254010229, 1982.
Bates, R. G. and Culberson, C. H.: Hydrogen ions and the thermodynamic state of marine systems, in: The Fate of Fossil Fuel CO2 in the Oceans, edited by: Andersen, N. R. and Malahoff, A., Plenum Press, New York, NY, 45-61, 1977.
Bolin, B., Bj̈orkstr̈om, A., Holḿen, K., and Moore, B.: The simultaneous use of tracers for ocean circulation studies, Tellus B, 35, 206-236, 1983.
Broecker, W. S. and Peng, T.-H.: Tracers in the Sea, Lamont-Doherty Geological Observatory of Columbia University, Palisades, NY 10964, 1982.
Buck, R. P., Rondinini, S., Covington, A. K., Baucke, F. G. K., Brett, C. M. A., Cam̃oes, M. F., Milton, M. J. T., Mussini, T., Naumann, R., Pratt, K. W., Spitzer, P., and Wilson, G. S.: Measurement of pH. Definition, standards, and procedures (IUPAC Recommendations 2002), Pure Appl. Chem., 74, 2169-2200, doi:10.1351/pac200274112169, 2002.
Bus, J. C. P. and Dekker, T. J.: Two efficient algorithms with guaranteed convergence for finding a zero of a function, ACM T. Math. Software, 1, 330-345, 1975.
Caldeira, K. andWickett, M. E.: Oceanography: anthropogenic carbon and ocean pH, Nature, 425, p. 365, doi:10.1038/425365a, 2003.
Dickson, A. G.: An exact definition of total alkalinity and a procedure for the estimation of alkalinity and total inorganic carbon from titration data, Deep-Sea Res., 28A, 609-623, 1981.
Dickson, A. G.: pH scales and proton-transfer reactions in saline media such as sea water, Geochim. Cosmochim. Ac., 48, 2299-2308, 1984.
Dickson, A. G.: Standard potential of the reaction: AgCl(s) + 12 H2(g) = Ag(s) + HCl(aq), and the standard acidity constant of the ion HSO 4 in synthetic sea water from 273.15 to 318.15 K, J. Chem. Thermodyn., 22, 113-127, doi:10.1016/0021-9614(90)90074-Z, 1990.
Dickson, A. G.: The measurement of sea water pH, Mar. Chem., 44, 131-142, doi:10.1016/0304-4203(93)90198-W, 1993.
Dickson, A. G.: The carbon dioxide system in seawater: equilibrium chemistry and measurements, in: Guide to best practices for ocean acidification research and data reporting, edited by: Riebesell, U., Fabry, V. J., Hansson, L., and Gattuso, J.-P., Publications Office of the European Union, Luxembourg, 17-40, doi:10.2777/58454, 2010.
Dickson, A. G. and Riley, J. P.: The estimation of acid dissolution constants in seawater media from potentiometric titrations with strong base. I. The ionic product of water-Kw, Mar. Chem., 7, 89-99, 1979.
Dickson, A. G., Sabine, C. L., and Christian, J. R., editors.: Guide to Best Practices for Ocean CO2 Measurements, vol. 3 of PICES Special Publication, Carbon Dioxide Information and Analysis Center, Oak Ridge (TN), available at: http://cdiac.ornl.gov/oceans/Handbook 2007.html, last access: 11 September 2012, 2007.
Doney, S. C., Lindsay, K., Fung, I., and John, J.: Natural variability in a stable, 1000-yr global coupled climate-carbon cycle simulation, J. Climate, 19, 3033-3054, doi:10.1175/JCLI3783.1, 2006.
Dowell, M. and Jarrett, P.: A modified regula falsi method for computing the root of an equation, BIT, 11, 168-174, 1971.
Dyrssen, D. W.: Framvaren and the Black Sea-similarities and differences, Aquat. Geochem., 5, 59-73, doi:10.1023/A:1009663704604, 1999.
Follows, M. J., Ito, T., and Dutkiewicz, S.: On the solution of the carbonate chemistry system in ocean biogeochemistry models, Ocean Model., 12, 290-301, doi:10.1016/j.ocemod.2005.05.004, 2006.
Gangstø, R., Joos, F., and Gehlen, M.: Sensitivity of pelagic calcification to ocean acidification, Biogeosciences, 8, 433-458, doi:10.5194/bg-8-433-2011, 2011.
Goosse, H., Brovkin, V., Fichefet, T., Haarsma, R., Huybrechts, P., Jongma, J., Mouchet, A., Selten, F., Barriat, P.-Y., Campin, J.-M., Deleersnijder, E., Driesschaert, E., Goelzer, H., Janssens, I., Loutre, M.-F., Morales Maqueda, M. A., Opsteegh, T., Mathieu, P.-P., Munhoven, G., Pettersson, E. J., Renssen, H., Roche, D. M., Schaeffer, M., Tartinville, B., Timmermann, A., and Weber, S. L.: Description of the Earth system model of intermediate complexity LOVECLIM version 1.2, Geosci. Model Dev., 3, 603-633, doi:10.5194/gmd-3-603-2010, 2010.
Hansson, I.: A new set of pH-scales and standard buffers for sea water, Deep-Sea Res., 20, 479-491, 1973.
Heinze, C., Maier-Reimer, E., and Winn, K.: Glacial pCO2 reduction by the World Ocean: experiments with the Hamburg Carbon Cycle Model, Paleoceanography, 6, 395-430, 1991.
Hoffert, M. I., Wey, Y.-C., Callegari, A. J., and Broecker, W. S.: Atmospheric response to deep-sea injections of fossil-fuel carbon dioxide, Climatic Change, 2, 53-68, doi:10.1007/BF00138226, 1979.
Hofmann, A. F., Soetaert, K., Middelburg, J. J., and Meysman, F. J. R.: AquaEnv: An Aquatic Acid-Base Modelling Environment in R, Aquat. Geochem., 16, 507-546, doi:10.1007/s10498-009-9084-1, 2010.
Hofmann, A. F., Soetaert, K., and Meysman, F. J.: AquaEnv: AquaEnv-an integrated development toolbox for aquatic chemical model generation. R package version 1.0-3., available at: http://cran.r-project.org/web/packages/AquaEnv/ index.html, last access: 29 May 2013, 2012.
Keeling, C. D.: The carbon dioxide cycle: reservoir models to depict the exchange of atmospheric carbon dioxide with the oceans and land plants, in: Chemistry of the Lower Atmosphere, chap. 6, edited by: Rasool, S. I., Plenum Press, New York, NY, 251-329, 1973.
Khoo, K. H., Ramette, R. W., Culberson, C. H., and Bates, R. G.: Determination of hydrogen ion concentrations in seawater from 5 to 40-C: Standard potentials at salinities from 20 to 45 ‰, Anal. Chem., 49, 29-34, doi:10.1021/ac50009a016, 1977.
Kirby, C. S. and Cravotta III, C. A.: Net alkalinity and net acidity. 1: Theoretical considerations, Appl. Geochem., 20, 1920-1940, doi:10.1016/j. apgeochem.2005.07.002, 2005.
Lavigne, H. and Gattuso, J.-P.: Seacarb: Seawater Carbonate Chemistry with R, R Package Version 2.4., available at: http://cran.r-project.org/web/ packages/seacarb/index.html, last access: 13 December 2012, 2012.
Le Hir, G., Donnadieu, Y., Yves Godd́eris, Y., Pierrehumbert, R. T., Halverson, G. T., Macouin, M., Ńed́elec, A., and Ramstein, G.: The snowball Earth aftermath: exploring the limits of continental weathering processes, Earth Planet. Sc. Lett., 277, 453-463, doi:10.1016/j.epsl.2008.11. 010, 2009.
Lewis, E. and Wallace, D.: Program Developed for CO2 System Calculations, Tech. Rep. 105, Carbon Dioxide Analysis Center, Oak Ridge National Laboratory, Oak Ridge (TN), 1998.
Luff, R., Haeckel, M., and Wallmann, K.: Robust and fast FORTRAN and MATLAB libraries to calculate pH distributions in marine systems, Comput. Geosci., 27, 157-169, 2001.
Maier-Reimer, E.: Geochemical cycles in an Ocean General Circulation Model. Preindustrial tracer distributions, Global Biogeochem. Cy., 7, 645-677, 1993.
Maier-Reimer, E. and Hasselmann, K.: Transport and storage of CO2 in the ocean-an inorganic ocean-circulation carbon cycle model, Clim. Dynam., 2, 63-90, 1987.
Maier-Reimer, E., Kriest, I., Segschneider, J., and Wetzel, P.: The HAMburg Ocean Carbon Cycle Model HAMOCC 5.1-Technical Description Release 1.1, Berichte zur Erdsystemforschung, Reports on Earth System Science 14, Max-Planck-Institut f̈ur Meteorologie, Hamburg, Germany, 2005.
Marchal, O., Stocker, T. F., and Joos, F.: A latitude-depth, circulation-biogeochemical ocean model for paleoclimate studies. Development and sensitivities, Tellus B, 50, 290-316, doi:10.1034/j.1600-0889.1998.t01-2-00006. x, 1998.
Marion, G. M., Millero, F. J., Cam̃oes, M. F., Spitzer, P., Feistel, R., and Chen, C.-T. A.: pH of seawater, Mar. Chem., 126, 89-96, doi:10.1016/j.marchem.2011.04.002, 2011.
Millero, F. J. and Sohn, M. L.: Chemical Oceanography, CRC Press, Boca Raton, Florida, 531 pp., 1992.
Millero, F. J., Feistel, R., Wright, D. G., and McDougall, T. J.: The composition of standard seawater and the definition of the reference-composition salinity scale, Deep-Sea Res. Pt. I, 55, 50-72, doi:10.1016/j.dsr.2007.10.001, 2008.
Millero, F. J., Woosley, R., DiTrolio, B., and Waters, J.: Effect of ocean acidification on the speciation of metals in seawater, Oceanography, 22, 72-85, doi:10.5670/oceanog.2009.98, 2009.
M̈uller, S. A., Joos, F., Plattner, G.-K., Edwards, N. R., and Stocker, T. F.: Modeled natural and excess radiocarbon: sensitivities to the gas exchange formulation and ocean transport strength, Global Biogeochem. Cy., 22, GB3011, doi:10.1029/2007GB003065, 2008.
Munhoven, G.: Modelling Glacial-Interglacial Atmospheric CO2 Variations: The Role of Continental Weathering, Ph.D. thesis, Universit́e de Lìege, Lìege, available at: http://www.astro. ulg.ac.be/-munhoven/ en/PhDThesis.pdf (last access: 21 February 2013), 1997.
Munhoven, G.: Glacial-interglacial rain ratio changes: implications for atmospheric CO2 and ocean-sediment interaction, Deep-Sea Res. Pt. II, 54, 722-746, doi:10.1016/j.dsr2.2007.01.008, 2007.
Munhoven, G.: Future CCD and CSH variations: deep-sea impact of ocean acidification, Geochim. Cosmochim. Ac., 73, p. A917, 2009.
Munhoven, G. and François, L. M.: Glacial-interglacial variability of atmospheric CO2 due to changing continental silicate rock weathering: a model study, J. Geophys. Res., 101, 21423-21437, doi:10.1029/96JD01842, 1996.
Opdyke, B. N. and Walker, J. C. G.: Return of the coral reef hypothesis: basin to shelf partitioning of CaCO3 and its effect on atmospheric CO2, Geology, 20, 733-736, 1992.
Orr, J., Najjar, R., Sabine, C., and Joos, F.: Abiotic-HOWTO, available at: http://ocmip5.ipsl.jussieu.fr/OCMIP/phase2/simulations/Abiotic/HOWTO- Abiotic.html, last access: 20 July 2012, 2000.
Palmer, J. R. and Totterdell, I. J.: Production and export in a global ocean ecosystem model, Deep-Sea Res. Pt. I, 48, 1169-1198, doi:10.1016/S0967- 0637(00)00080-7, 2001.
Park, P. K.: Oceanic CO2 system: an evaluation of ten methods of investigation, Limnol. Oceanogr., 14, 179-186, 1969.
Peng, T.-H., Takahashi, T., Broecker, W. S., and Olafsson, J.: Seasonal variability of carbon dioxide, nutrients and oxygen in the northern North Atlantic surface water: observations and a model, Tellus B, 39, 439-458, 1987.
Press,W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling,W. T.: Numerical Recipes (FORTRAN Version), Cambridge University Press, Cambridge, 1989.
Ridgwell, A. J.: Glacial-Interglacial Pertubations of the Global Carbon Cycle, Ph.D. thesis, University of East Anglia, Norwich, 2001.
Ridgwell, A. and Schmidt, D. N.: Past constraints on the vulnerability of marine calcifiers to massive carbon dioxide release, Nat. Geosci., 3, 196-200, doi:10.1038/ngeo755, 2010.
Ridgwell, A., Hargreaves, J. C., Edwards, N. R., Annan, J. D., Lenton, T. M., Marsh, R., Yool, A., andWatson, A.: Marine geochemical data assimilation in an efficient Earth System Model of global biogeochemical cycling, Biogeosciences, 4, 87-104, doi:10.5194/bg-4-87-2007, 2007.
Shaffer, G., Malskær Olsen, S., and Pepke Pedersen, J. O.: Presentation, calibration and validation of the low-order, DCESS Earth System Model (Version 1), Geosci. Model Dev., 1, 17-51, doi:10.5194/gmd-1-17-2008, 2008.
Walker, J. C. G. and Opdyke, B. N.: Influence of variable rates of neritic carbonate deposition on atmospheric carbon dioxide and pelagic sediments, Paleoceanography, 10, 415-427, 1995.
Waters, J. F. and Millero, F. J.: The free proton concentration scale for seawater pH, Mar. Chem., 149, 8-22, doi:10.1016/j.marchem.2012.11.003, 2013.
Wolf-Gladrow, D. A., Zeebe, R. E., Klaas, C., K̈ortzinger, A., and Dickson, A. G.: Total alkalinity: the explicit conservative expression and its application to biogeochemical processes, Mar. Chem., 106, 287-300, doi:10.1016/j.marchem.2007.01.006, 2007.
Yao,W. and Millero, F. J.: The chemistry of the anoxic waters in the Framvaren Fjord, Norway, Aquat. Geochem., 1, 53-88, 1995.
Zeebe, R. E.: LOSCAR: Long-term Ocean-atmosphere-Sediment CArbon cycle Reservoir Model v2.0.4, Geosci. Model Dev., 5, 149-166, doi:10.5194/gmd-5-149- 2012, 2012.
Zeebe, R. E. and Wolf-Gladrow, D.: CO2 in seawater: equilibrium, kinetics, isotopes, vol. 65 of Elsevier Oceanography Series, Elsevier, Amsterdam (NL), 2001.