Geochemistry, Geophysics, Geosystems
Supporting Information for
Ventilation and dissolved oxygen cycle in Lake Superior: Insights from a numerical model
Katsumi Matsumoto1, Kathy S. Tokos1, and Chad Gregory2
1Dept.
of Earth Sciences, University of Minnesota, 310 Pillsbury Dr. SE, Minneapolis, MN 55455
2University
of Michigan
Contents of this file
Text S1 to S3
Figures S1 to S5
Additional Supporting Information (Files uploaded separately)
Captions for Movies S1 to S3
Introduction
This Supporting Information provides information on four topics: (a) modifications to our
previous model of Lake Superior; (b) model-data comparison; (c) idealized model tracer Dye;
and (d) movies of Dye simulations.
Text S1: Modifications to the Lake Superior Model
The model previously used at the University of Minnesota [White et al., 2012] has been
updated to ROMS version 3.6. The dynamic ice model, as described by White et al. [2012] and
based on Hunke and Dukowicz [1997] and Hunke [2001], was then incorporated into the new
version of ROMS. A modification of the ice code was included that results in a more robust
agreement with observations of both surface temperature and wintertime ice coverage of Lake
Superior.
The net heat flux from the atmosphere to lake ice, which in turn determines the surface
temperature of the ice and its subsequent melting or formation, is calculated by combining the
net fluxes of sensible heat, latent heat, short- and longwave radiation. In the original
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formulation, under boundary conditions described by White et al. [2012], the net longwave
radiative flux (LWnet) from the atmosphere to the ice was calculated as:
LWnet = LWin + 4 Ta3 (Ts - 2Ta) ,
(Eq 1)
where LWin is incoming longwave radiation from the model forcing, and σ, the StefanBoltzmann constant, is set equal to 5.67 x 10-8 W m-2 K-4. The emissivity of ice is ϵ = 0.97, and Ta
and Ts are air and surface temperature. The second term in Eq. 1 appears to be an
approximation of downwelling longwave radiation, and it is unclear why it is added to the
incoming radiation that forces the model. In the new version, we use instead the simple StefanBoltzmann relationship to calculate the outgoing longwave radiation:
LWout =  ϵTs4 ,
(Eq 2)
and calculate LWnet as:
LWnet = LWin -  ϵTs4 .
(Eq 3)
Figure S1 illustrates the resulting difference in terms of ice concentration, which we
define as the percent of the lake that is covered by ice, for the year 1997. Although the
maximum amount of ice formed (blue line) is slightly less than with the previous formulation
(green line), the rapid melting of lake ice is captured much better as seen by comparing with the
ice atlas observations (red line).
Also shown in the figure (black line) is ice concentration from 1997 of White et al. [2012],
with lake ice remaining until well into the summer. Another source of the discrepancy between
White et al. [2012] and the latest version of the model is the forcing. We have chosen to force
the model with the NARR reanalysis, which is more objective and readily available from 1979 to
the present. In addition, rather than an ad hoc application of cloud cover effects to incoming
shortwave radiation, we account for cloud albedo by using the net incoming shortwave
radiation that is included in the NARR output.
Text S2: Model-Data Comparison
With the described modifications to the model, we find good agreement of daily mean
lake surface temperature (LST) and percent ice cover between model output and the
observation-based data. NOAA’s Great Lakes Ice Atlas, obtained from the Environmental
Research Laboratory (GLERL, glerl.noaa.gov) provides daily measurements of the total fraction
of Lake Superior covered by ice from December 1 to May 30 or 31. Daily, lake-wide mean LST is
calculated from Great Lakes Surface Environmental Analysis (GLSEA) maps, also from GLERL.
In Figure S2a is shown daily, lake-wide average LST for the years 1995-2003 from the model
(blue line), observations (red line), and the model of White et al. [2012] (green line). With a root
mean square error (RMSE) value of 1.9°C over the 9 year period, modeled LST maxima and
minima compare well with the seasonal thermal cycle of the lake. The model reproduces
observed LST best in cold winters, at the cost of overestimating LST during following stratified
period. In warmer winters, the model overestimates LST, but with less deviation from
2
observations the following summer. We note that Bennington et al., [2010] also reported a
warm bias in their MITgcm-based model of Lake Superior and attributed the bias to the NARR
air temperatures. Our warm bias may also be related to the NARR forcing, but we believe there
is another more important cause, because LST of White et al. [2012] is also warm. Daily RMSE
between White et al. [2012] and the observations is 2.0°C, similar to that in our current model.
However, the forcing for White et al. [2012] was observationally based and not NARR, pointing
a possibly different cause; for example, the coefficients of vertical mixing may need adjusting.
Figure S2b compares daily ice coverage of model output with observations over the
same time period. The model clearly shows the interannual fluctuations of ice coverage seen in
the data. The daily RMSE between the model and observations is 10.2%, while that of White et
al. [2012] and observations is 13.9%. However, this method of comparison could have a very
large RMSE if the ice coverage were identical, but the dates of onset and melting were simply
off by days or weeks. Another way to assess the ice production in the model is to compare the
average amount of ice cover over all days with ice greater than 1% over a calendar year (Figure
S2c). For example, the annual mean ice coverage for 1996 was 40%, although the maximum
coverage for that year is 100%. This allows a better comparison of the thermodynamic
properties of ice formation. The RMSE between the annual averages of the model and
observations is just 5.5%, while that of White et al. [2012] was 10.0%.
Text S3: Tracer Dye
As described by England [1995], tracer Dye is a passive tracer like tracer Age, and the two
tracers are entirely consistent. In the main text, only the description and results of Age are
presented. Here we describe and present tracer Dye.
The time evolution of Dye is governed by:
1, z  0
Dye(x, y, z, t)  
 Dye(x, y, z, t - 1)  {Dye(x, y, z, t)}, z  0
Tracer Dye is conserved in the lake interior and therefore does not age. The Dye
concentration is continuously set to 1 at the surface (Figure S3a). As surface waters penetrate
the lake interior, the interior Dye concentration will increase with time (Figure S3b). As
illustrated, an interior grid box, which is ventilated readily, will quickly reach the maximum
concentration of 1, while those boxes that are poorly ventilated will only slowly increase in Dye
concentration. Eventually the entire lake will reach a concentration of 1 (Figure S3c). From that
time, the Dye concentration will no longer reflect any changes in ventilation and thus becomes
useless. We therefore release Dye on the first of the years of interest.
Before Dye concentration reaches one, its transient penetration into the lake interior
provides information not contained in tracer Age. Specifically, the first derivative of the time
series of Dye concentration at any point in space gives the age spectrum, or the frequency
distribution, of ventilation age (Figure S3d, e). The frequency-weighted mean of the age
spectrum obtained from the Dye tracer would correspond to the value of the Age tracer. In
other words, the spectrum gives information about the make up of an interior water parcel in
terms of multiple surface water source histories.
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As with the Age tracer, the Dye tracer maps on the 100 m level also show that surface
waters penetrate into the interior more readily over undulations in bottom topography in the
eastern basin in both 1997 and 1998 (Figure S4a, b). For example, the cross sections at 47.1c
(Figure S4c, d) show that the Dye concentration is higher in the topographically rougher eastern
arm (e.g., 84-88°W) than in the smoother and shallower western arm (e.g., west of 88°W).
The Dye concentration and its first derivative reveal more detail of the mixing history of
a water parcel. Figure S5 shows two time series for 1998. In the first Dye concentration is shown
at the same water depth of 175 m at three different locations, all where the lake floor is very
deep (220-300m). One is near the north shore in the western arm where there are frequent
wind-driven upwelling events (green line, Figure S5a; see filled circle, main text Figure 1 for
location), one is in the deepest basin (blue line; filled triangle in Figure 1), and one is in the
eastern arm in a region of very rough bottom topography (red line; filled square, Figure 1).
Figure S5a shows that the water on the north shore (green line) remains somewhat isolated
after the initial mixing in January. Then in June, springtime overturning mixes down surface
waters, until almost 80% of the water at that depth has been ventilated. Mixing is nearly
complete by the end of November as another, possibly wind event brings more surface water to
depth. Dye concentration does not reach 1 in 1997 at this location, indicating that overturning is
not complete until the following year. In the very deep basin, the initial mixing is evident (blue
line), as is the spring overturning in June, when most of the dye reaches 175m. In September,
upwelling or horizontal advection brings in water with lower Dye concentration, and the water
here also refrains from mixing completely with the surface until the end of the year. The
deepest basins generally tend to remain free of the Dye the longest, which is consistent with
Age being the oldest in the same basins. But, despite being in a deep part of the lake, surface
waters quickly penetrate to175 m depth in the region of rough topography (red line), so that the
Dye concentration there reaches 1 almost immediately.
The first derivatives of the Dye time series reveal the composition of the ventilation age
distribution (Figure S5b). Where Dye reaches 1 almost immediately, such as in shallow waters or
over rough topography, the first derivative has a single high peak at the very beginning,
indicating a large fractional contribution of surface water at the beginning. Elsewhere there
may be multiple peaks that indicate multiple contributions of surface waters to give the value of
Age. For example, the first derivatives of Dye concentration on both the North Shore and in the
deepest basin have two large peaks (Figure S5b): one early in January when the model starts in
nearly isothermal conditions, and a second in June when late spring overturning finally reaches
the 175 m. This indicates that the water in each of these regions is composed primarily of two
sources of water: one is surface water from the fall overturning; and a second source comes
from the surface during spring overturning. Since the area under the first peak is larger than the
one in June, the fractional contribution of the fall overturning is larger than of the spring
overturning at this depth for this year. The smaller peaks, especially for the North Shore water
(green line), indicate many smaller mixing events, each contributing a small volume of surface
water to 175m depth. This is in contrast to the region over rough bottom topography. There is
only one peak of surface water injection (red line), when the water column was thoroughly
mixed, and no more information of surface water history is possible. Similarly, Figure S5b
depicts the initial major mixing event to all depths in January, and the lesser event in June. The
deeper waters are able to receive a greater contribution from the surface during June than the
shallower ones, since the shallower ones are already nearly completely mixed.
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Figure S1. Simulations and observations of ice concentration, defined as the percent of the lake
that is covered by ice, for 1997. Red line is from NOAA’s Great Lakes Ice Atlas; blue line is ROMS
v3.6 with the current ice modification, black line is ROMS v3.6 without ice modification, and the
green line is from White et al. [2012].
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Figure S2. Daily, lake-wide average LST for the years 1995-2003 from the model (blue line),
observations from GLSEA (red line), and the model of White et al. [2012], (green line): a) daily,
lake-wide mean surface temperature, b) total fraction of lake covered by ice, and c) annual
mean of fraction of lake >1% covered by ice.
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Figure S3. How tracer Dye works. The effect of vertical mixing in the interior is to increase Dye
concentration (a-c). Given a time series of Dye concentration (d) of some water parcel in the
model domain, the first derivative of the time series (e) is the age spectrum or the frequency
distribution of ventilation age. In this example, this water parcel is largely composed of waters
Figure S2
that left the surface at times centered on t1 and t2. The frequency weighted mean age would
correspond to the value of tracer Age.
d)
Dye =1
Dye = 0
b)
Dye concentration
a)
Dye = 1
c)
Dye = 1
e)
Frequency
0 < Dye < 1
Dye = 1
t1
Time
t2
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Figure S4. Model-simulated Dye distribution in 1997 (a, c) and 1998 (b, d) at 100 m water depth
(maps) and along 47.1° N (cross sections). The lines in the maps indicate location of the
sections.
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Figure S5. Time series of Dye (a) and its first time derivative (b) at three locations at 175 m
depth. See Figure 1 in the main text for the three locations.
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Movie S1. Model output of percent of Lake Superior covered by ice in 1997 (upper panel) and
1998 (lower panel).
Movie M2. Model output of tracer Age, in days, at 100 m water depth in 1997 (upper panel) and
1998 (lower panel).
Movie M3. Model output of tracer Age, in days, at 47.1 N in 1997 (upper panel) and 1998 (lower
panel).
References
Bennington, V., G. McKinley, N. Urban, and C. P. McDonald (2012), Can spatial heterogeneity
explain the perceived imbalance in Lake Superior’s carbon budget? A model study, J.
Geophys. Res., 117(G03020), doi:10.1029/2011JG001895.
England, M. H. (1995), The age of water and ventilation timescales in a global ocean model, J.
Phys. Oceanogr., 25(11), 2756–2777.
Hunke, E. C. (2001), Viscuous-plastic sea ice dynamics with the EVP model: Linearization issues,
J. Comput. Phys., 170(1), 18–38.
Hunke, E. C., and J. K. Dukowicz (1997), An elastic–viscous–plastic model for sea ice dynamics,
J. Phys. Oceanogr., 27(9), 1849–1867, doi:10.1175/1520-0485.
White, B., J. Austin, and K. Matsumoto (2012), A three dimensional model of Lake Superior with
ice and biogeochemistry, J. Gt. Lakes Res., 38, 61–71.
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