Abstract
Abstract. 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.
Highlights
The CryosphereBiogeochemical models have become indispensable tools to improve our understanding of the cycling of the elements in the Earth system
In ocean carbon cycle models, the air–sea exchange of CO2 is directly linked to the surface ocean [CO2]; the preservation of biogenic carbonates in the surface sediments at the sea floor is closely linked to the deep sea [CO23−] (Broecker and Peng, 1982)
We have explored the mathematical properties of the total alkalinity–pH equation, i.e. the equation that relates [H+] to total alkalinity and the total concentrations of all the acid systems contributing to total alkalinity
Summary
Biogeochemical models have become indispensable tools to improve our understanding of the cycling of the elements in the Earth system. In ocean carbon cycle models, the air–sea exchange of CO2 is directly linked to the surface ocean [CO2]; the preservation of biogenic carbonates in the surface sediments at the sea floor is closely linked to the deep sea [CO23−] (Broecker and Peng, 1982). PH changes in seawater may directly influence air–sea exchange of CO2 or the preservation of carbonates in the deep sea. The dissociation of acids, such as carbonic acid, controls pH: when the ocean takes up or releases CO2 (e.g. as a result of a rise or a decline of the abundance of CO2 in the atmosphere), its pH changes. The currently ongoing ocean acidification due to the massive release of CO2 into the atmosphere by human activity is but one example of such an induced pH change
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