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

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

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10.5194/gmd-6-1367-2013

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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 :

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 [BE]

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

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