Journgal of Geophysical Research
Supporting Information for
The Gas Transfer Through Polar Sea Ice experiment: Insights into the rates and
pathways that determine geochemical fluxes.
A. Lovely1, B. Loose1, P. Schlosser2,3,4, W. McGillis4, C. Zappa2, D. Perovich5, S.
Brown2, T. Morell2, D. Hsueh2, R. Friedrich6
1
Graduate School of Oceanography, University of Rhode Island, 215 South Ferry Rd., Narragansett, RI,
02882, USA
2
Department of Earth and Environmental Sciences, Columbia University, Room 106 Geosciences Bldg.,
Palisades, NY 10964, USA
3
Department of Earth and Environmental Engineering, Columbia University, 918 Seeley Mudd Bldg., 500
West 120th St, New York, NY 10027, USA
4
Lamont-Doherty Earth Observatory of Columbia University, 61 Rout 9W, Palisades, NY 10964, USA
5
US Army Corps of Engineers Cold Regions Research and Engineering Laboratory, 72 Lyme Rd.,
Hanover, NH, 03755, USA
6
Institute of Environmental Physics, University of Heidelberg, Heidelberg, Germany 69120
1
Contents of this file
Text S1
Introduction
The contents of this document include the methodology that we used to determine the
level of uncertainty in our estimates of k the gas transfer velocity. This information
follows the standard error propagation procedure.
S1: Uncertainty analysis
The uncertainty in an individual estimate of keff was determined by propagating
both the analytical uncertainty and the model-data misfit through equation 11. Because
equation 11 is a non-linear equation, the Taylor approximation of the form,
𝑦 = 𝑓(𝑎, 𝑏. . )
𝜕𝑓 2
𝜕𝑓 2
(A1)
𝑉𝑎𝑟[𝑦] = 𝑉𝑎𝑟[𝑎] (𝜕𝑎) + 𝑉𝑎𝑟[𝑏] (𝜕𝑏)
is used to linearize the equation and estimate the associated uncertainties. The terms a
and b represent individual random variables in the function f [Glover et al., 2011]. Here
we have assumed that a and b are not correlated. Performing this linearization on
equation 11 yields,
2
2
ℎ
1
ℎ
1
𝑉𝑎𝑟[𝑘𝑒𝑓𝑓 ] = 𝑉𝑎𝑟[𝐶𝑤𝑖 ] ( (
)) − 𝑉𝑎𝑟[𝐶𝑤𝑓 ] ( (
)) + ⋯
𝑑𝑡 𝐶𝑤𝑖 − 𝐶𝑎
𝑑𝑡 𝐶𝑤𝑓 − 𝐶𝑎
2
2
ℎ
𝑉𝑎𝑟[𝐶𝑎 ] [(𝑑𝑡 (𝐶
1
𝑤𝑓 −𝐶𝑎
2
ℎ
)) − (𝑑𝑡 (𝐶
1
𝑤𝑖 −𝐶𝑎
)) ]
(A2)
where Cwi, Cwf and Ca are the random variables that lead to uncertainty in keff.
a. Determining error in Cw
The concentration of tracer in the water is determined with a linear calibration
between GC peak area and gas standards of known concentration. In addition to the
calibration uncertainty, the headspace method has other sources of analytical uncertainty.
We use the mean variance between duplicate headspace samples, collected at the same
time, to estimate the net uncertainty in the headspace method. This is termed Var[CGC].
In addition to the analytical uncertainty, the estimate of gas diffusion through the
ice is another source of variability in Cw. The gas lost to the ice was estimated using a
finite difference approximation of Fickian diffusion and tuned to match the measured
bulk concentration in the ice. To estimate the goodness of fit, we use the residual
variance between the model estimate of gas concentration in the ice and the values
measured from melted ice cores,
1
𝑚𝑜𝑑𝑒𝑙
𝑉𝑎𝑟[𝐶𝑚𝑜𝑑𝑒𝑙 ] = 𝑛−1 √∑𝑛−1
)
1 (𝐶𝑖𝑐𝑒 − 𝐶𝑖𝑐𝑒
2
(A3)
Thus, the net uncertainty in Cwi and Cwf is the sum of the analytical and model-data
uncertainty,
𝑉𝑎𝑟[𝐶𝑤 ] = 𝑉𝑎𝑟[𝐶𝐺𝐶 ] + 𝑉𝑎𝑟[𝐶𝑚𝑜𝑑𝑒𝑙 ]
(A4)
where we have again assumed no correlation between CGC and Cmodel.
b. Determining error in Ca
Air concentration of N2O and SF6 was measured in duplicate every 4 hours. The
variance of these duplicates was used to determine (Var[Ca]) during each scenario.
3
c. Determining error in keff
To perform the uncertainty analysis described above, requires an estimate of the
bulk gas concentration in the ice. This was carried out for N2O and SF6 only, so these
two gases were used in the uncertainty analysis (Table 3).
S2: Brine drainage and solute rejection calculations
Both solute rejection at the time of ice crystal formation, and brine drainage over a longer
period, can be estimated from the salt balance in the tank. As stated in the §4.2, we
cannot estimate both processes at once, but if we refer to all loss during ice formation as
“solute rejection” to distinguish it from residual brine drainage that takes place once the
ice is a constant thickness, then the two can be separated in time.
The inferred solute (salt or gas) concentration in the ice ([Cice]) can be estimated
from the difference between the mass of the solute in the “un-frozen water” that became
ice (Mufw) and the mass of solute removed from the ice (Mr),
éëCice ùû =
Mufw -Mr
Vufw
(A5)
Here, Vufw is the volume of the un-frozen water that has become ice. One could also
express [Cice] in terms of the volume of the ice, but we can avoid worrying about the ice
density if we work in terms of the water equivalent of the ice. Mufw is estimated as the
product of the solute concentration in the water before the freezing began a Vufw. Mr can
be observed by measuring the solute concentration in the water ( d Cw ) over a specified
freezing or brine rejection time interval,
4
Mr = d Cw VT
(A6)
and VT is the volume of the tank.
To estimate melt, we evaluate the change in Cw over a time period when the salt
concentration in the tank decreases,
(
MTf =MTi +CiceVufw
)
Cwf VTi +Vufw =CwiVTi + CiceVufw
Vufw = VTi
(A7)
Cwi - Cwf
Cwf - Cice
Here, the solute mass in the tank at the end of the period balances with the initial mass
and the concentration in the ice times the volume of un-frozen water that turned back to
water (MTf, MTi, CiceVufw). We don’t know Vufw, but all other terms can be estimated.
Cice comes from equation (A5). VTi is the initial tank volume, Cwi and Cwf are the initial
and final tank concentration.
5