Returns the Bolin sensitivity \(B_e = \partial DIC/\partial [CO_2^*]\) and the classical Revelle factor \(Rf = DIC/([CO_2^*] B_e)\) at constant total alkalinity, temperature, and salinity, using the condensed closed-form template first proposed by Sundquist, Plummer and Wigley (1979). The routine is named for that paper.
That template predates the other formulations of the Revelle factor / Bolin sensitivity in common use, and Orr et al. (in prep.) show that three later, independently derived formulations -- Zeebe and Wolf-Gladrow (2001, corrected); Egleston, Sabine and Morel (2010, corrected); and Frankignoulle (1994, as implemented in seacarb's buffer) -- each reduce, exactly, to the equation of Sundquist et al. (1979). They differ only in which acid-base systems are collected into the single non-carbonate term \(X\): Sundquist et al. (1979) included borate alone; the other three added water. This routine implements that common equation with the most complete \(X\) available in seacarb, adding phosphate, silicate, fluoride, and, when nonzero, ammonia and sulfide.
Note that this equivalence covers only the Revelle factor / Bolin sensitivity result of each source; Frankignoulle (1994) and Egleston et al. (2010) each define several additional buffer factors (see seacarb's buffer and buffesm) that are not part of this unification.
Unlike buffzwg and buffesm, this routine needs no explicit K1 or K2: the equation is written directly in terms of the three dissolved inorganic carbon species, \(s=[CO_2^*]\), \(b=[HCO_3^-]\), and \(c=[CO_3^{2-}]\), taken from carb()/carbfull()'s own speciation output, with all non-carbonate chemistry isolated in \(X\). buffsun was cross-validated against buffzwg, buffesm, and buffer: for any Pt, Sit, NH4t, and HSt, its Be and Rf agree with buffzwg's Be and RF, buffesm's R, and buffer's BetaD to floating-point precision. It was further checked against buffderiv, which computes \(B_e\) independently and refines [H+] to machine precision; the two agree to about 2e-8 percent, the residual being the convergence tolerance of carb() itself. The buffsun input arguments are identical to those of buffzwg, buffesm, and carb.
buffsun(flag, var1, var2, S = 35, T = 25, Patm = 1, P = 0, Pt = 0, Sit = 0,
NH4t = 0, HSt = 0, k1k2 = "x", kf = "x", ks = "d", pHscale = "T", b = "u74", warn = "y",
eos = "eos80", long = 1e+20, lat = 1e+20)The function returns a data frame containing the following columns:
\(B_e\), the Bolin sensitivity, \(\partial DIC/\partial [CO_2^*]\) at constant total alkalinity, temperature, and salinity (unitless, since both DIC and \([CO_2^*]\) are in mol/kg)
the classic Revelle factor, \(DIC/([CO_2^*] B_e)\), equivalently \((\partial ln[CO_2]/\partial ln DIC)^{-1}\) (unitless); agrees with buffzwg's Be/RF, buffesm's R, and buffer's BetaD to floating-point precision
select the pair of variables available. The flags which can be used are:
flag = 1 pH and CO2 given
flag = 2 CO2 and HCO3 given
flag = 3 CO2 and CO3 given
flag = 4 CO2 and ALK given
flag = 5 CO2 and DIC given
flag = 6 pH and HCO3 given
flag = 7 pH and CO3 given
flag = 8 pH and ALK given
flag = 9 pH and DIC given
flag = 10 HCO3 and CO3 given
flag = 11 HCO3 and ALK given
flag = 12 HCO3 and DIC given
flag = 13 CO3 and ALK given
flag = 14 CO3 and DIC given
flag = 15 ALK and DIC given
flag = 21 pCO2 and pH given
flag = 22 pCO2 and HCO3 given
flag = 23 pCO2 and CO3 given
flag = 24 pCO2 and ALK given
flag = 25 pCO2 and DIC given
enter value of the first variable in mol/kg, except for pH and for pCO2 in \(\mu\)atm
enter value of the second variable in mol/kg, except for pH
Salinity
Temperature in degrees Celsius
Surface atmospheric pressure in atm, default 1 atm
Hydrostatic pressure in bar (surface = 0)
Concentration of total phosphate in mol/kg; set to 0 if NA; when nonzero, accounts for the phosphoric acid system as part of the non-carbonate term X.
Concentration of total silicate in mol/kg; set to 0 if NA; when nonzero, accounts for the silicic acid system (full diprotic treatment) as part of the non-carbonate term X.
Concentration of total ammonium in mol/kg; set to 0 if NA; when nonzero (or when HSt is nonzero), accounts for ammonia alkalinity and carbfull() is used internally instead of carb()
Concentration of total hydrogen sulfide in mol/kg; set to 0 if NA; when nonzero (or when NH4t is nonzero), accounts for sulfide alkalinity and carbfull() is used internally instead of carb()
"cw" for using K1 and K2 from Cai & Wang (1998), "l" from Lueker et al. (2000), "m02" from Millero et al. (2002), "m06" from Millero et al. (2006), "m10" from Millero (2010), "mp2" from Mojica Prieto et al. (2002), "p18" from Papadimitriou et al. (2018), "r" from Roy et al. (1993), "sb21" from Shockman & Byrne (2021), "s20" from Sulpis et al. (2020), and "w14" from Waters et al. (2014). "x" is the default flag; the default value is then "l", except if T is outside the range 2 to 35oC and/or S is outside the range 19 to 43. In these cases, the default value is "w14".
"pf" for using Kf from Perez and Fraga (1987) and "dg" for using Kf from Dickson and Riley (1979 in Dickson and Goyet, 1994). "x" is the default flag; the default value is then "pf", except if T is outside the range 9 to 33oC and/or S is outside the range 10 to 40. In these cases, the default is "dg".
"d" for using Ks from Dickon (1990), "k" for using Ks from Khoo et al. (1977), default is "d"
choice of pH scale: "T" for the total scale, "F" for the free scale and "SWS" for using the seawater scale, default is "T" (total scale)
Concentration of total boron. "l10" for the Lee et al. (2010) formulation or "u74" for the Uppstrom (1974) formulation, default is "u74"
"y" to show warnings when T or S go beyond the valid range for constants; "n" to supress warnings. The default is "y".
"teos10" to specify T and S according to Thermodynamic Equation Of Seawater - 2010 (TEOS-10); "eos80" to specify T and S according to EOS-80.
longitude of data point, used when eos parameter is "teos10" as a conversion parameter from absolute to practical salinity.
latitude of data point, used when eos parameter is "teos10".
James Orr James.Orr@lsce.ipsl.fr
The Lueker et al. (2000) constants for K1 and K2, the Perez and Fraga (1987) constant for Kf and the Dickson (1990) constant for Ks are recommended by Dickson et al. (2007). It is, however, critical to consider that each formulation is only valid for specific ranges of temperature and salinity:
For K1 and K2:
Cai and Wang (1998): S ranging between 0 and 40 and T ranging between 0.2 and 30oC.
Lueker et al. (2000): S ranging between 19 and 43 and T ranging between 2 and 35oC.
Millero et al. (2002): S ranging from 34 to 37 and T ranging between -1.6 and 35oC.
Millero et al. (2006): S ranging between 0.1 and 50 and T ranging between 1 and 50oC.
Millero (2010): S ranging between 1 and 50 and T ranging between 0 and 50oC. Millero (2010) provides a K1 and K2 formulation for the seawater, total and free pH scales. Therefore, when this method is used and if P=0, K1 and K2 are computed with the formulation corresponding to the pH scale given in the flag "pHscale".
Mojica Prieto et al. (2002): S ranging from 5 to 42 and T ranging between 0 and 45oC.
Papadimitriou et al. (2018): S ranging from 33 to 100 and T ranging between -6 to 25oC.
Roy et al. (1993): S ranging between 5 and 45 and T ranging between 0 and 45oC.
Shockman & Byrne (2021): for K2, S ranging from 19.6 to 41 and T ranging between 15 to 35oC. For K1, formulation is that of Waters et al.
Sulpis et al. (2020): S ranging from 30.7 to 37.6 and T ranging between -1.7 to 31.8oC.
Waters et al.(2014): S ranging between 1 and 50 and T ranging between 0 and 50oC. Waters (2014) provides a K1 and K2 formulation for the seawater, total and free pH scales. Therefore, when this method is used and if P=0, K1 and K2 are computed with the formulation corresponding to the pH scale given in the flag "pHscale".
For Kf:
Perez and Fraga (1987): S ranging between 10 and 40 and T ranging between 9 and 33oC.
Dickson and Riley (1979 in Dickson and Goyet, 1994): S ranging between 0 and 45 and T ranging between 0 and 45oC.
For Ks:
Dickson (1990): S ranging between 5 and 45 and T ranging between 0 and 45oC.
Khoo et al. (1977): S ranging between 20 and 45 and T ranging between 5 and 40oC.
The arguments can be given as a unique number or as vectors. If the lengths of the vectors are different, the longer vector is retained and only the first value of the other vectors is used. It is recommended to use either vectors with the same dimension or one vector for one argument and numbers for the other arguments.
On the underlying formula:
Writing \(s := [CO_2^*]\), \(b := [HCO_3^-]\), \(c := [CO_3^{2-}]\), and \(h := [H^+]\), the Bolin sensitivity is computed as
\(B_e = 1 + c/s + (X b - 4 c^2) \big/ \big( s (b + 4c + X) \big)\)
with the Revelle factor \(Rf = DIC/(s B_e)\), and \(X = h w\), where \(w\) is the same non-carbonate buffering term used by buffzwg and buffesm, built additively, one acid-base system at a time:
\(w = w_{bw} + w_{Si} + w_P + w_F + w_N + w_S\)
with \(w_{bw} = 1 + K_w/h^2 + B_T K_B/(K_B+h)^2\) (borate and water), \(w_{Si} = Si_T K_{Si}(h^2+4K_{2Si}h+K_{Si}K_{2Si})/(h^2+K_{Si}h+K_{Si}K_{2Si})^2\) (silicic acid; see the note below on monoprotic versus diprotic treatment), \(w_P\) the analogous contribution from the three-step phosphoric acid system, \(w_N = NH_{4t} K_N/(K_N+h)^2\) (ammonia), and \(w_S = HS_t K_{HS}/(K_{HS}+h)^2\) (hydrogen sulfide), and \(w_F = F_T K_F'/(K_F'+h)^2\) (hydrogen fluoride, with \(K_F' = K_F (1 + S_T/K_S)\), i.e. Kf placed on the same pH scale as \(h\)). \(-[HF]\) is a species in Dickson's total alkalinity and is carried explicitly by SolveSAPHE, which carb() inverts, so it must appear in \(w = -\partial A_{nc}/\partial h\) as well; it contributes at the ~2e-6 relative level. \(w_N\) and \(w_S\) are automatically and exactly zero unless NH4t or HSt, respectively, are nonzero.
Four independent derivations of the Revelle factor / Bolin sensitivity -- Sundquist, Plummer and Wigley (1979); Zeebe and Wolf-Gladrow (2001, corrected); Egleston, Sabine and Morel (2010, corrected); and Frankignoulle (1994, as implemented in buffer) -- were each rearranged algebraically and shown to reduce, exactly, to this same equation, differing only in which terms are included in \(X\): Sundquist et al. (1979) include only borate (\(X = \gamma\)); the other three include borate and water (\(X = \gamma+\delta\), using the notation of Orr et al., in prep.). This routine's \(X\) is the most complete version of the three, additionally including phosphate, silicate, ammonia, and sulfide.
Setting \(X = 0\) collapses this equation exactly to
\(B_e = 1 + bc/(s(b+4c))\)
the result obtained if carbonate alkalinity \(A_c\), rather than total alkalinity \(A_T\), is assumed to remain fixed as CO2 is added to the ocean -- physically incorrect for air-sea CO2 exchange, since borate and water alkalinity (and, if included, phosphate, silicate, ammonia, and sulfide alkalinity) are not actually static: \(A_c\) rises while the non-carbonate alkalinity terms fall in compensation, keeping \(A_T\) fixed. See Orr et al. (in prep.) for a full discussion, including how this collapse relates to the traditional approximation \(B_e \approx [CO_3^{2-}]/[CO_2^*]\) used since Broecker and Peng (1974).
Pressure corrections and pH scale:
For K0, the pressure correction term of Weiss (1974) is used.
For K1, K2, pK1, pK2, pK3, Kw, Kb, Khs and Ksi, the pressure correction was applied on the seawater scale. Hence, if needed, values were first transformed from the total scale to the seawater scale, the pressure correction applied as described by Millero (1995), and the value was transformed back to the required scale (T, F or SWS).
For Kf, the pressure correction was applied on the free scale. The formulation of Dickson and Riley (1979 in Dickson and Goyet, 1994) provides Kf on the free scale but that of Perez and Fraga (1987) provides it on the total scale. Hence, in that case, Kf was first transformed from the total scale to the free scale. With both formulations, the pressure correction was applied as described by Millero (1995), and the value was transformed back to the required scale (T, F or SWS).
For Ks, the pressure correction was applied on the free scale. The pressure correction was applied as described by Millero (1995), and the value was transformed back to the required scale (T, F or SWS).
long and lat are used as conversion parameters from absolute to practical salinity: when seawater is not of standard composition, practical salinity alone is not sufficient to compute absolute salinity and vice-versa. One needs to know the density. When long and lat are given, density is inferred from WOA silicate concentration at given location. When they are not, an arbitrary geographic point is chosen: mid equatorial Atlantic. Note that this implies an error on computed salinity up to 0.02 g/kg.
Cai W. J., and Wang Y., 1998. The chemistry, fluxes, and sources of carbon dioxide in the estuarine waters of the Satilla and Altamaha Rivers, Georgia. Limnology and Oceanography 43, 657-668.
Dickson A. G., 1990 Standard potential of the reaction: AgCI(s) + 1/2H2(g) = Ag(s) + HCI(aq), and the standard acidity constant of the ion HSO4 in synthetic sea water from 273.15 to 318.15 K. Journal of Chemical Thermodynamics 22, 113-127.
Dickson A. G., Sabine C. L. and Christian J. R., 2007 Guide to best practices for ocean CO2 measurements. PICES Special Publication 3, 1-191.
Egleston, E. S., Sabine, C. L. and Morel, F. M. M., 2010 Revelle revisited: Buffer factors that quantify the response of ocean chemistry to changes in DIC and alkalinity, Global Biogeochem. Cycles 24, GB1002, tools:::Rd_expr_doi("10.1029/2008GB003407").
Frankignoulle M., 1994 A complete set of buffer factors for acid/base CO2 system in seawater. Journal of Marine Systems 5, 111-118.
Khoo H. K., Ramette R. W., Culberson C. H. and Bates R. G., 1977 Determination of hydrogen ion concentration in seawater from 5 to 40oC: standard potentials at salinities from 20 to 45. Analytical Chemistry 49, 29-34.
Lee K., Tae-Wook K., Byrne R.H., Millero F.J., Feely R.A. and Liu Y-M, 2010 The universal ratio of the boron to chlorinity for the North Pacific and North Atlantoc oceans. Geochimica et Cosmochimica Acta 74 1801-1811.
Lueker T. J., Dickson A. G. and Keeling C. D., 2000 Ocean pCO2 calculated from dissolved inorganic carbon, alkalinity, and equations for K1 and K2: validation based on laboratory measurements of CO2 in gas and seawater at equilibrium. Marine Chemistry 70 105-119.
Millero F. J., 2010 Carbonate constant for estuarine waters. Marine and Freshwater Research 61: 139-142.
Millero F. J., Graham T. B., Huang F., Bustos-Serrano H. and Pierrot D., 2006 Dissociation constants of carbonic acid in seawateras a function of salinity and temperature. Marine Chemistry 100, 80-84.
Orr J. C. et al., in prep. Declining timescale for air-sea CO2 equilibration.
Perez F. F. and Fraga F., 1987 Association constant of fluoride and hydrogen ions in seawater. Marine Chemistry 21, 161-168.
Roy R. N., Roy L. N., Vogel K. M., Porter-Moore C., Pearson T., Good C. E., Millero F. J. and Campbell D. M., 1993. The dissociation constants of carbonic acid in seawater at salinities 5 to 45 and temperatures 0 to 45oC. Marine Chemistry 44, 249-267.
Schockman, K.M., Byrne, R.H., 2021. Spectrophotometric determination of the bicarbonate dissociation constant in seawater, Geochimica et Cosmochimica Acta.
Sundquist E. T., Plummer L. N. and Wigley T. M. L., 1979 Carbon dioxide in the ocean surface: the homogeneous buffer factor. Science 204, 1203-1205.
Uppstrom L.R., 1974 The boron/chlorinity ratio of the deep-sea water from the Pacific Ocean. Deep-Sea Research I 21 161-162.
Zeebe R. E. and Wolf-Gladrow D. A., 2001 CO2 in seawater: equilibrium, kinetics, isotopes. Elsevier Oceanography Series 65, Amsterdam, 346 pp.
## Computation with a pair of variables
buffsun(flag=15, var1=0.00230, var2=0.00200, S=35, T=18, P=0, Pt=0,
Sit=0, pHscale="T", kf="pf", k1k2="l", b="u74")
## Using vectors as arguments
flag <- c(15, 15, 15)
var1 <- c(0.00230, 0.00230, 0.00230)
var2 <- c(0.00200, 0.00200, 0.00200)
S <- c(35, 35, 30)
T <- c(18, 25, 18)
P <- c(0, 0, 0)
Pt <- c(0, 5e-6, 5e-6)
Sit <- c(0, 0, 20e-6)
kf <- c("pf", "pf", "pf")
k1k2 <- c("l", "l", "l")
pHscale <- c("T", "T", "T")
b <- c("l10", "l10", "l10")
buffsun(flag=flag, var1=var1, var2=var2, S=S, T=T, P=P, Pt=Pt,
Sit=Sit, kf=kf, k1k2=k1k2, pHscale=pHscale, b=b)
## Compare against buffzwg, buffesm, and buffer (should all agree to
## floating-point precision, for any Pt and Sit)
o <- buffsun(flag=15, var1=0.00230, var2=0.00200, S=35, T=18, Pt=5e-6, Sit=20e-6)
z <- buffzwg(flag=15, var1=0.00230, var2=0.00200, S=35, T=18, Pt=5e-6, Sit=20e-6)
e <- buffesm(flag=15, var1=0.00230, var2=0.00200, S=35, T=18, Pt=5e-6, Sit=20e-6)
u <- buffer(flag=15, var1=0.00230, var2=0.00200, S=35, T=18, Pt=5e-6, Sit=20e-6)
o$Rf - z$RF # should be ~0 (floating-point precision)
o$Rf - e$R # should be ~0 (floating-point precision)
o$Rf - u$BetaD # should be ~0 (floating-point precision)
## The X=0 (Ac-constant) collapse: an internal, non-exported illustration
## of how buffsun's general result reduces to the constant-carbonate-
## alkalinity special case (see Details); not run as part of normal use.
if (FALSE) {
cf <- carbfull(flag=15, var1=0.00230, var2=0.00200, S=35, T=18)
h <- 10^(-cf$pH); s <- cf$CO2; bb <- cf$HCO3; cc <- cf$CO3
Be_X0 <- 1 + bb*cc/(s*(bb+4*cc))
Be_X0
}
Run the code above in your browser using DataLab