UNITED STATES

DEPARTMENT OF COMMERCE

Penny Pritzker

Secretary

NATIONAL OCEANIC AND

ATMOSPHERIC ADMINISTRATION

Kathy Sullivan

Acting, Under Secretary for Oceans & Atmosphere

NOAA Administrator

NAT OI NAL OCEANIC AND ATMOSPHERIC ADMINISTRAT OI N

U.S.DEPARTMENT OF COMMERCE

NOAA Technical Memorandum GLERL-167

Reaffirmation of large biases in a long-used method for projecting changes in 

Great Lakes water levels in climate change scenarios

Brent M. Lofgren

Jonathan Rouhana*

NOAA Great Lakes Environmental Research Laboratory, 4840 S. State Rd., Ann Arbor, MI 48108

*current affiliation: University of Notre Dame, South Bend, Indiana 

September 2015, revised May 2016
ii

NOTICE

Mention of a commercial company or product does not constitute an endorsement by the NOAA. 

Use of information from this publication concerning proprietary products or the tests of such 

products for publicity or advertising purposes is not authorized. This is GLERL Contribution No. 

1779.

This publication is available as a PDF file and can be downloaded from GLERL’s web site: 

www.glerl.noaa.gov. Hard copies can be requested from GLERL Information Services, 

4840 S. State Rd., Ann Arbor, MI 48108.

pubs.glerl@noaa.gov.

NOAA’s Mission – To understand and predict changes in Earth’s environment and conserve and 

manage coastal and marine resources to meet our nation’s economic, social, and environmental 

needs 

NOAA’s Mission Goals:

• Protect, restore and manage the use of coastal and ocean resources through an ecosystem 

approach to management

• Understand climate variability and change to enhance society’s ability to plan and respond

• Serve society’s needs for weather and water information

• Support the Nation’s commerce with information for safe, efficient, and environmentally 

sound transportation

• Provide critical support for NOAA’s Mission 
iii

TABLE OF CONTENTS

ABSTRACT........................................................................................................................................................ 1

1. INTRODUCTION ....................................................................................................................................... 1

2. METHODS AND EXPERIMENTAL DESIGN.......................................................................................... 3

2.1 The Large Basin Runoff Model ............................................................................................................ 3

2.2 Experimental Design............................................................................................................................. 4

2.3 Other Model Components..................................................................................................................... 9

2.4 General Circulation Models................................................................................................................ 10

3. RESULTS................................................................................................................................................... 11

3.1 Lake Superior...................................................................................................................................... 11

3.2 Lake Michigan-Huron......................................................................................................................... 13

3.3 Lake Erie............................................................................................................................................. 14

3.4 Lake Ontario ....................................................................................................................................... 16

3.5 Correlation of Discrepancies with Temperature Change .................................................................... 16

3.6 Statistical Significance........................................................................................................................ 16

4. DISCUSSION............................................................................................................................................ 18

5. CONCLUSIONS AND FUTURE PLANS................................................................................................ 20

6. ACKNOWLEDGEMENTS....................................................................................................................... 21

7. REFERENCES .......................................................................................................................................... 21

LIST OF FIGURES

Figure 1. Box and whiskers plots of variables averaged over the Lake Superior basin and over the entire 

year, generated by GCM outputs in combination with LBRM using the temperature adjustment (TA), 

Clausius-Clapeyron (CC), Priestley-Tayor (PT), and energy adjustment (EA) methods. Each data point 

used in making the plots is a single GCM realization. (a) Ratio of PET for future and historical time 

periods, (b) ET for future minus historical time periods (cm/day), (c) runoff for future minus historical 

time periods (cm/day), (d) net basin supply for future minus historical time periods (m3/s), and (e) lake 

level for future minus historical time periods (m). ........................................................................................... 12

Figure 2. As in Fig. 1, but for the Lake Michigan-Huron basin. ..................................................................... 13

Figure 3. Lake Michigan-Huron water levels represented using the temperature adjustment and 

Priestley-Taylor methods for subsets of the GCM realizations. Note that the scale differs among the 

panels. (a) Mid-21st century for RCP 4.5, (b) mid-21st century for RCP 6.0, (c) mid-21st century for 

RCP 8.5, (d) late 21st century for RCP 4.5, (e) late 21st century for RCP 6.0, and (f) late 21st century 

for RCP 8.5. ...................................................................................................................................................... 15

Figure 4. As in Fig. 1, but for the Lake Erie basin. ......................................................................................... 15

Figure 5. As in Figure 1, but for the Lake Ontario basin, and the lake levels are excluded, 

as lake levels were not calculated for Lake Ontario. ....................................................................................... 17

Figure 6. Scatterplot of the difference in lake level (units of m) for Lake Superior, TA 

minus PT (plus signs) and TA minus CC (circles) vs. change in temperature.................................................. 17
iv

LIST OF TABLES

Table 1. LBRM parameter values for each sub-basin of all lakes..................................................................... 5

Table 2. Description of alternative methods of incorporating GCM-based climate projections into 

 the LBRM ............................................................................................................................................ 8

Table 3. Summary of GCM runs used ............................................................................................................. 10

Table 4. Results of the Wilcoxon rank sum test (Hollander and Wolfe 1973) for the lake level of the 

various lakes, comparing the results under the Clausius-Clapeyron method, the Priestley-Taylor 

method, and the energy adjustment method to the temperature adjustment method........................................ 18
1

Reaffirmation of large biases in a long-used method for projecting changes in Great Lakes 

water levels in climate change scenarios

Brent M. Lofgren

Jonathan Rouhana

ABSTRACT. A method for projecting the water levels of the Laurentian Great Lakes under scenarios 

of human-caused climate change, used almost to the exclusion of other methods in the past, relies very 

heavily on the Large Basin Runoff Model (LBRM) as a component for determining the water budget for 

the lake system. This model uses near-surface air temperature as a primary predictor of evapotranspiration 

(ET); in addition to previous published work, we show here again that its very high sensitivity to 

temperature makes it overestimate ET in a way that is highly inconsistent with the fundamental principle 

of conservation of energy at the land surface. Under the traditional formulation, the quantity that has 

been called “energy available for evapotranspiration,” which is proportional to what we call “potential 

evapotranspiration” (PET), is increased by large factors in future scenarios—by a factor of nearly 600 

in the Lake Superior basin under forcing by one GCM case, but more typically by factors between 3 

and 10. Because of the way that LBRM is formulated and calibrated, these factors can be thought of as 

corresponding to the factor of increase in solar radiation incident on the Earth, or, more vividly, as the 

number of Suns present in the sky of the virtual world simulated by LBRM. Therefore, we have created 

alternative formulations, which we regard as more reflective of what is being simulated by the driving 

GCMs, for the way that climate change is ingested into the modeling system that includes LBRM. In 

addition to the energy adjustment method in which PET is increased by an amount proportional to the 

change in net radiative energy at the surface, we add the Priestley-Taylor method, which augments the 

energy adjustment method by inclusion of a temperature-dependent factor with rigorous theoretical 

underpinnings that is a much weaker function of temperature than in the LBRM’s basic formulation, 

as well as the Clausius-Clapeyron method, in which the PET is increased by an amount proportional 

to the increase in water vapor capacity of the atmosphere, again a weaker function of temperature than 

in the LBRM. We establish that all three of these alternative methods show, relative to the traditional 

method, often astoundingly less PET and less ET, more runoff from the land and net basin supply for the 

lake basins, and higher lake water levels in the future. The magnitude of these discrepancies is highly 

correlated with the air temperature change in the driving GCM (larger temperature changes lead to larger 

discrepancies). Using various methods of estimating the statistical significance, we find that, at minimum, 

these discrepancies in results are significant at the 99.998% level.

1. INTRODUCTION

The Laurentian Great Lakes of North America contain approximately 20% of the world’s surface fresh water. They 

support intrinsic and commercial interests for shoreline property owners, for ecosystems in the lakes themselves and 

coastal wetlands, and for commercial and recreational fishing, boating, shipping, and tourism. They are a binational 

resource of the United States and Canada, and over 30 million people live within their drainage basin (Government of 

Canada and U.S. Environmental Protection Agency 1995).

The issue of the potential influence of future climate change on the water levels of the Great Lakes has been a topic of a 

significant body of literature. Many examples have used the method pioneered by Croley (1990). A partial list of this class 

includes Hartmann (1990), Chao (1999), Lofgren et al. (2002), and Angel and Kunkel (2010). Some variants on the more 

common practice of using general circulation model (GCM) output to impose climate change on the regional hydrologic 

system were studies that used transposition of climate from other warmer, and sometimes drier, climates, to the Great 

Lakes region (Croley et al. 1998), and imposition of a somewhat arbitrary range of possible climates with the intention of 

illuminating the possibility of terminal lakes (i.e. having no outflow) approximately 8000 years ago (Croley and Lewis 

2006). 
2

All of these studies feature lake level prominently as an end result of the modeling activity, and generally showed that 

future climate change results in dropping lake levels, with the possibility that such drops will be of large magnitude. One 

of the possible projections, a 1.67-m drop during the 21st century in Lake Michigan-Huron, reported in Lofgren et al. 

(2002), was much quoted in the media long after its publication, regularly without mention that it was published alongside 

an alternative scenario that projected a rise in lake levels. Croley et al. (1998) showed that if the climate of the region 

of Oklahoma were to be transposed over the Great Lakes region, Lake Superior would become a terminal lake. The set 

of results using this method has also been featured in the assessment and synthesis literature (e.g. Great Lakes Regional 

Assessment Group 2000, Kling et al. 2003, International Joint Commission 2006, and Hayhoe et al. 2010).

Other studies have used different approaches to examine the hydrologic budget of the Great Lakes basin. Some have used 

direct analysis of the GCM-based regional water budget (precipitation minus evaporation) in the region (Manabe et al. 

2004, Milly et al. 2005), while others have examined the convergence of water vapor flux in the atmosphere as indicating 

net water budget (Kutzbach et al. 2005). Still others have used regional climate modeling to downscale GCM results and 

project climate both in terms of the atmosphere and the surface (MacKay and Seglenieks 2013, Lofgren 2014, Notaro et 

al. 2015). Most of these studies did not include lake level explicitly as an output, although MacKay and Seglenieks (2013) 

and Notaro et al. (2015) do have such projections based on a limited range of GCM inputs. The group of studies that have 

not been based on the methodology of Croley (1990) have generally not shown large trends toward reduced water budgets 

in the Great Lakes, and even though not always explicitly shown, this necessarily implies small changes in lake levels. 

This group of studies has also been much less influential in the mass media and in assessment reports.

Lofgren et al. (2011, hereafter referred to as LHW) attempted to expose some serious problems with the methodology of 

Croley (1990) for projection of Great Lakes water budgets and levels under climate change scenarios. These specifically 

dealt with the Large Basin Runoff Model (LBRM), which is the component of the modeling system that deals with 

the land portion of the Great Lakes basin. The central argument was that the LBRM is overly reliant on near-surface 

air temperature as a predictor of evapotranspiration (ET) to the exclusion of the surface energy budget as a predictor; 

although the LBRM is calibrated such that potential evapotranspiration is constrained by incoming solar radiation during 

the calibration period, this constraint does not apply when considering periods with altered climate regimes. This was 

broken down into three main intersecting lines of argument:

1. The ET projected for future climate scenarios using input from GCMs was much greater than that directly simulated 

by the driving GCMs. The equivalent energy of the difference, in terms of latent heat flux, was shown to be around 30 

W m-2, a significant discrepancy in the surface energy budget.

2. During periods of observation at land-based flux measurement stations based on the eddy covariance principle, the 

near-surface air temperature and ET were decomposed into different temporal classes—the annual cycle, variations 

with a time scale greater than a month but not annual, and variations with a time scale less than a month. For the 

annual time scale, there was a very high correlation between air temperature and ET, while for the other time scales, 

there was almost no correlation. This seems to indicate that the seasonal cycle (i.e. solar energy input) is the major 

driver of both air temperature and ET at the annual scale, while air temperature is a poor predictor of ET variability on 

other time scales. Therefore, thinking of air temperature as either an actual driver of ET, or a proxy that is universally 

applicable, is highly problematic.

3. The potential ET (PET), proportional to what is called by Croley (1983) the “energy available for evapotranspiration,” 

is greater by very large factors in the future scenarios than in the historical base case. The largest factor that was found 

in LHW for a full lake basin in the examples we ran was a 10-fold increase in the PET for all of the land in the Lake 

Superior Basin. For only the basin of the Montreal River, an eastern tributary of Lake Superior, the corresponding 

number was a 60-fold increase. These magnitudes of increase can be taken as changes that are equivalent to increases 

in the incoming sunlight by that factor. These very large increases were due to the formulation of PET as exponential 

functions of air temperature, with calibrated parameters indicating that the Montreal River Basin has PET increasing 

by 90% per degree C, and many of the other sub-basins of Lake Superior increasing in the range of 30-50% per 
3

degree C. This contrasts to the point of comparison of the Clausius-Clapeyron relation, which has an increase in 

saturated water vapor pressure of 7% per degree C, and literature sources such as Held and Soden (2006) and Lorenz 

and DeWeaver (2007), which show smaller increases in large-scale ET under climate change.

These arguments were the central justifications for asserting that the methods that have been repeatedly used over more 

than two decades overestimate future ET. In addition to this, LHW gave a small-scale demonstration of what effect these 

problems might have on projected lake levels. They devised the “energy adjustment” (EA) method, in which the changes 

in PET from a future climate scenario relative to the historical period was given by the ratio of net surface radiation, 

rather than the exponential function of the air temperature change, dubbed the “temperature adjustment” (TA) method. 

This was carried out for two example GCMs taken from the Climate Model Intercomparison Project Phase 3 (CMIP3) 

set of models. One of these was shown in the main body of LHW, and showed a switch from falling lake levels under the 

TA method to rising lake levels under the EA method, a difference of about 90 cm on Lake Michigan-Huron. The other 

example GCM was shown in the supplementary material, and contrasted a large drop in lake levels under the TA method 

with a lesser drop under the EA method, reducing a drop of 2.1 m on Lake Michigan-Huron to only 1 m.

These results received very guarded recognition in the 2014 National Climate Assessment. One of the authors of the 

present paper (Lofgren) acquiesced under protest to the inclusion of the following language: “Recent studies have also 

indicated that earlier approaches to computing evapotranspiration estimates from temperature may have overestimated 

evaporation losses.” [italics added] This statement appears both in the Midwest chapter (Pryor et al. 2014) and in the 

Climate Science Supplement (Walsh et al. 2014). 

Some National Climate Assessment authors suggested that what was missing was an assessment of the statistical 

significance of the difference between the traditional TA method and the more physically-based EA method by using a 

wider selection of GCM data as input (D. Wuebbles and K. Hayhoe, personal communication). Therefore, that is a major 

goal of this study. We also introduce two alternative formulations for PET under climate change in addition to the alreadyused EA method, as additional points of comparison to the TA method. Arguments 1 and 2 from LHW, in the list above, 

stand as already stated there, but our further use of many more GCMs as input leads to further underline argument 3.

Another major application of LBRM is in seasonal-scale projection of Great Lakes water levels within NOAA Great 

Lakes Environmental Research Laboratory’s (GLERL) Advanced Hydrologic Prediction System (AHPS), which is distinct 

from a product with a similar name developed by the National Weather Service’s Office of Hydrology. An assessment 

of the performance of that entire modeling system was given by Gronewold et al. (2011) and shows that the main 

disagreement between AHPS and observed water levels is that AHPS underestimates the amount of water level variability. 

LBRM was also compared to other models in terms of its stream runoff as part of the Great Lakes Runoff Intercomparison 

Project—Lake Michigan (GRIP-M, Fry et al. 2014). Its results were well in line with those of other models. However, 

these assessments did not include the systematic and sustained changes in air temperature associated with changes in the 

climate regime that were investigated in LHW and the present paper.

The rest of this paper is organized as follows. Section 2 will summarize the methodologies that are used here. Section 3 

will present the results, including formal analysis of the statistical significance of the discrepancy among these alternative 

methods. Section 4 will discuss the results and the way that strikingly extreme intermediate results within a model can be 

disguised in the final results. Conclusions are presented in section 5.

2. METHODS AND EXPERIMENTAL DESIGN

2.1. The Large Basin Runoff Model (LBRM)

Key features of the formulation of the LBRM model for purposes of this study include the determination of “energy 

available for evapotranspiration” (for convenience referred to here as PET; see further discussion below) and the 

proportion of PET that is manifested as ET. As stated in Croley (1983), due to considerations of data availability, an a 

priori constraint in developing LBRM was that only daily air temperatures and precipitation be used as input data. 
4

PET is assumed to take the form AeT/Tb, where Tb is a parameter with units of temperature fitted through calibration. 

During the calibration process, the annual total of PET is taken to be such that its latent heat equivalent is equal to the 

annual incoming solar radiative energy:

 LAΣyeareT/Tb = ΣyearS, (1)

where L is the latent heat of evaporation per unit mass, A is a parameter with units of cm/day or other units of 

evapotranspiration, T is daily mean air temperature, and S is the top-of-atmosphere incident solar radiation. The 

summation in eq. (1) uses a daily timestep for the snow-free portion of the year. In the calibration process, for each test 

value of Tb, the value of A is uniquely determined using eq. (1). Thus, eq. (1) constrains the total PET over the year, 

while Tb controls its distribution throughout the year, and fluctuation with other modes of temperature variability. It 

should be noted, however, that outside of the calibration process, including in future projections using adjustments to the 

temperature based on GCM results, the constraint of eq. (1) is not enforced. That is, the PET increases by a factor of 

eΔT/Tb corresponding to the change in temperature between current and future conditions, equivalent to the solar radiation 

also increasing by the same factor. 

To clarify the difference in nomenclature between this paper and Croley (1983), we here use the term potential 

evapotranspiration to refer to the quantity AeT/Tb, because it is the variable in LBRM that best corresponds to the intuitive 

and widely-used definition of PET as the amount of ET that would occur when unlimited soil moisture is available. Within 

the formulation of LBRM, this amount of ET can be approached, but is only reached when soil moisture is infinite. When 

multiplied by L, this quantity is referred to in Croley (1983) as “energy available for evapotranspiration.” Croley uses the 

term PET to refer to the water equivalent of the portion of the energy available for ET that does not actually result in ET.

The full set of calibrated parameters for each sub-basin in LBRM is listed in Table 1. Although the actual calibration that 

led to this set of values is not well documented, the values are generally similar to those of Croley (1983).

2.2 Experimental design

The community’s general understanding of how evapotranspiration occurs concentrates on the surface as its locus of 

evapotranspiration, as well as the surface’s energetics and gradients of temperature and specific humidity across the 

planetary boundary layer as drivers—liquid water in water bodies or soil or water within plant leaves acquires enough 

kinetic energy to change to the gaseous state, a process promoted by absorption of energy (heat) at the surface and a 

gradient of water vapor concentration into the planetary boundary layer. These principles form the paradigm that stands 

behind diagnostic equations of potential evapotranspiration, such as those of Penman and Monteith (Monteith 1973) 

and Priestley and Taylor (1972). They are also the basis for the surface transfer schemes embedded within climate 

models, from the early generation of bucket models (e.g. Manabe and Holloway 1975) to more complex soil-vegetationatmosphere transfer schemes (Dickinson et al. 1986, Sellers et al. 1986) and their newer descendants.

Even most sources of information directed at the general public refrain from directly saying that the increased capacity 

for water vapor in warmer air causes increased evapotranspiration, but often do mention these concepts in close 

proximity to each other, e.g. Brean (2015). Therefore, there can tend to be a simplifying assumption that potential ET 

will increase with temperature in proportion to the Clausius-Clapeyron relationship. An analysis related to this concept 

is found in d’Orgeville et al. (2014), in which they compare regional precipitation changes associated with climate 

change simulations to the “expected” increases, assuming proportionality to the Clausius-Clapeyron relation. They did, 

in fact, find that in the Great Lakes Basin, their regional model showed an increase in precipitation that agreed well 

with this assumption, approximating 7% per degree C. The explanation in http://water.epa.gov/scitech/climatechange/

Water-Impacts-of-Climate-Change.cfm (accessed May 12, 2015) actually seems to reverse the more popularly assumed 

cause-effect relationship: “Warmer temperatures increase the rate of evaporation of water into the atmosphere, in effect 

increasing the atmosphere’s capacity to ‘hold’ water.” So some confusion occurs surrounding the relationship between ET 

and water capacity of the atmosphere.
5

Table 1. LBRM parameter values for each sub-basin of all lakes. For purposes of these parameters, Georgian Bay is 

treated as separate from the rest of Lake Huron. Tb is the parameter used to calculate PET from temperature as in eq. 

(1). Parameter 2 is a coefficient determining snowmelt rate. Parameter 3 gives the rate at which water from the upper 

soil zone percolates into the lower soil zone. Parameter 4 controls the proportion of PET that becomes actual ET from 

the upper soil zone. Parameter 5 is a coefficient for the rate of interflow from the lower soil zone to the surface storage. 

Parameter 6 is a coefficient for the rate of deep percolation from the lower soil zone to the groundwater reservoir. 

Parameter 7 controls the proportion of PET that becomes actual ET from the lower soil zone. Parameter 8 is a coefficient 

for the rate of transfer from the groundwater to the surface storage. Parameter is a coefficient for the rate of outflow from 

the surface storage to the lake. For details, see Croley (1983).

Lake Basin Tb 2 3 4 5 6 7 8 9

Superior 1 3.00E+00 2.27E-01 3.00E-01 5.90E+01 2.10E-10 6.00E-03 1.00E-10 3.50E-02 8.60E-02

Superior 2 7.20E+00 1.90E-01 1.20E+02 1.00E-06 1.00E-01 7.60E-02 7.90E-09 3.30E-02 2.30E+01

Superior 3 4.70E+00 3.21E-01 2.30E+00 1.90E-08 3.90E-02 3.00E-02 3.20E+03 3.00E-02 2.20E-01

Superior 4 6.00E+00 2.64E-01 7.10E-01 8.50E-08 1.10E-10 1.80E-02 3.00E-09 6.70E-01 2.20E-01

Superior 5 6.30E+00 2.57E-01 2.30E+01 1.20E-10 4.00E+00 6.30E+00 2.60E+00 1.40E-02 2.60E-01

Superior 6 4.50E+00 2.98E-01 5.40E-01 4.80E-08 2.10E-10 2.70E-01 9.50E-08 5.80E-02 4.30E-01

Superior 7 5.20E+00 2.13E-01 1.00E+00 7.00E-08 1.10E-02 2.40E-03 3.50E-09 8.50E-03 1.70E-01

Superior 8 9.20E+08 4.10E+01 1.40E+01 4.50E-08 1.00E-10 9.90E-03 7.80E-10 1.40E-02 1.30E-02

Superior 9 2.30E+00 2.25E-01 4.60E-01 3.50E+03 1.40E-03 8.20E-03 4.00E-06 8.90E-04 7.50E-02

Superior 10 2.30E+00 2.25E-01 4.60E-01 3.50E+03 1.40E-03 8.20E-03 4.00E-06 8.90E-04 7.50E-02

Superior 11 2.30E+00 2.20E-01 4.60E-01 3.50E+03 1.40E-03 8.20E-03 4.00E-06 8.90E-04 7.50E-02

Superior 12 5.10E+00 2.57E-01 6.80E-01 2.20E+03 1.10E+03 7.90E+03 9.00E-01 1.40E-02 1.90E-01

Superior 13 1.10E+00 1.44E-01 9.30E+00 9.10E+03 1.10E-03 9.20E-03 6.90E-05 1.00E-03 9.50E-02

Superior 14 3.90E+00 2.44E-01 4.70E+00 9.10E-09 2.70E-03 4.30E-03 2.00E-10 1.70E-03 8.40E-02

Superior 15 9.70E+00 2.53E-01 4.70E+01 5.30E-07 2.00E-02 7.90E-03 3.90E-10 5.30E-01 1.10E-01

Superior 16 5.80E+00 2.33E-01 5.40E-01 8.70E-09 1.00E-10 2.70E-02 3.30E-09 6.00E-02 1.00E-01

Superior 17 1.80E+00 2.10E-01 5.50E-01 4.40E-02 3.20E-03 2.00E-03 9.80E-03 5.90E-03 7.10E-02

Superior 18 1.40E+00 2.01E-01 3.40E-01 6.40E-04 9.90E-04 1.00E-03 9.60E-02 9.30E+00 8.10E-02

Superior 19 2.00E+00 7.13E-02 9.10E+08 2.00E-10 7.90E-03 8.10E-02 2.70E-09 5.30E-03 5.20E-03

Superior 20 2.10E+00 1.88E-01 6.10E-01 1.30E-07 2.20E-03 8.30E-04 5.70E-08 9.90E+08 4.20E-02

Superior 21 2.00E+00 3.06E-01 5.70E+00 7.60E+00 4.20E-03 6.80E-03 4.60E-02 6.70E-03 1.10E-01

Superior 22 3.10E+00 1.58E-01 5.40E-01 6.90E-02 1.10E-10 1.80E-01 6.30E-01 2.60E-02 1.80E-01

Michigan 1 4.50E+00 2.44E-01 2.50E-01 1.60E-07 1.10E-01 2.50E-01 1.00E-10 5.00E-03 5.80E-02

Michigan 2 4.50E+00 2.45E-01 2.50E-01 1.60E-07 1.10E-01 2.50E-01 1.00E-10 5.00E-03 5.80E-02

Michigan 3 7.10E+00 2.09E-01 1.00E+01 9.90E-07 4.70E-02 2.50E-02 8.60E-09 1.20E-02 6.90E-01

Michigan 4 7.10E+00 2.08E-01 1.00E+01 9.90E-07 4.70E-02 2.50E-02 8.60E-09 1.20E-02 6.90E-01

Michigan 5 5.20E+00 2.07E-01 7.20E-01 1.20E-07 9.20E-03 9.00E-08 2.60E-09 3.30E-02 1.10E-01

Michigan 6 5.10E+00 1.90E-01 2.90E-01 1.10E-07 1.00E-05 6.80E-02 8.30E-02 5.30E-02 9.60E-02

Michigan 7 3.90E+00 2.30E-01 3.20E+00 1.10E-08 7.40E-03 4.60E-03 1.50E-09 9.80E-04 9.60E-02

Michigan 8 4.70E+00 2.54E-01 1.70E+00 1.20E-07 8.00E-03 6.20E-03 6.30E-09 5.20E-03 1.20E-01

Michigan 9 5.70E+00 2.66E-01 3.50E+00 2.90E-07 8.90E-03 1.00E-02 5.10E-09 3.40E-03 1.20E-01

Michigan 10 6.00E+00 2.67E-01 1.10E-01 1.80E+01 3.80E-06 4.00E-06 1.00E+01 6.40E-02 2.20E-01

Michigan 11 9.40E+00 7.32E+00 2.70E+02 4.30E-07 1.00E-02 9.10E-08 2.50E-10 4.00E-01 2.10E-01

Michigan 12 5.60E+00 3.82E-01 1.30E+00 2.00E+01 1.80E-02 9.50E-07 7.50E-01 2.50E-06 4.50E-01

Michigan 13 5.90E+00 3.98E-01 1.60E+00 1.90E-01 1.90E-02 8.20E-06 9.90E-03 9.80E-06 1.50E-01

Michigan 14 5.70E+00 3.15E-01 4.00E+00 3.20E-06 2.80E-02 3.40E-02 2.10E-07 1.00E-02 3.80E-01

Michigan 15 8.10E+00 3.22E-01 4.80E-01 3.90E-07 8.00E-06 6.80E+05 3.80E-01 2.20E-02 3.00E-01
6

Table 1. (Continued)

Lake Basin Tb 2 3 4 5 6 7 8 9

Michigan 16 8.30E+00 4.38E-01 1.70E+01 4.70E-08 8.60E-02 2.40E-01 3.90E-08 1.40E-02 1.30E-01

Michigan 17 5.60E+00 1.08E+01 8.40E+05 1.70E-07 4.20E-02 5.00E-02 1.30E-07 1.70E-02 2.60E+01

Michigan 18 8.50E+00 4.96E-01 8.90E+00 3.40E-07 1.10E-02 1.10E-02 1.30E-09 4.30E-03 1.80E-01

Michigan 19 5.90E+00 3.54E-01 1.10E-05 5.70E-03 9.30E+00 6.60E-01 1.10E-04 1.20E-05 5.80E-01

Michigan 20 5.90E+00 3.02E-01 2.00E-01 2.50E-07 1.00E-03 1.90E-02 9.00E-09 2.10E-02 5.70E-02

Michigan 21 4.80E+00 2.92E-01 5.00E+00 4.90E-07 4.40E-03 6.40E-03 5.10E-10 1.70E-04 1.50E-01

Michigan 22 6.00E+00 3.33E-01 3.80E+00 1.60E-07 5.70E-03 4.10E-03 3.90E-10 1.20E-03 1.30E-01

Michigan 23 4.80E+00 2.94E-01 5.00E+00 4.90E-07 4.40E-03 6.40E-03 5.10E-10 1.70E-04 1.50E-01

Michigan 24 7.50E+00 3.94E+01 1.80E+03 6.20E-05 6.00E-03 7.50E-03 2.40E-10 2.60E-04 7.60E+00

Michigan 25 4.80E+00 2.93E-01 5.00E+00 4.90E-07 4.40E-03 6.40E-03 5.10E-10 1.70E-04 1.50E-01

Michigan 26 6.40E+00 1.49E-01 6.00E+01 2.60E-06 2.20E-02 1.20E-01 3.20E-09 2.30E-04 5.10E+01

Michigan 27 6.40E+00 1.47E-01 6.00E+01 2.60E-06 2.20E-02 1.20E-01 3.20E-09 2.30E-04 5.10E+01

Huron 1 4.90E+00 2.44E-01 1.60E-01 2.90E-07 3.10E-02 8.80E-06 5.50E-10 7.50E-02 2.30E-01

Huron 2 6.60E+00 3.29E-01 1.40E+01 2.30E-07 6.50E-03 8.20E-03 1.00E-09 7.40E-04 1.40E-01

Huron 3 6.60E+00 3.30E-01 1.40E+01 2.30E-07 6.50E-03 8.20E-03 1.00E-09 7.40E-04 1.40E-01

Huron 4 6.60E+00 3.28E-01 1.40E+01 2.30E-07 6.50E-03 8.20E-03 1.00E-09 7.40E-04 1.40E-01

Huron 5 5.30E+00 2.97E-01 4.70E+00 3.20E-07 8.90E-03 8.20E-03 9.10E-09 4.20E-03 2.30E-01

Huron 6 5.30E+00 3.03E-01 4.70E+00 3.20E-07 8.90E-03 8.20E-03 9.10E-09 4.20E-03 2.30E-01

Huron 7 6.20E+00 2.13E-01 1.00E+01 2.00E-07 4.60E-03 1.00E-02 4.90E-10 9.60E-04 2.70E-01

Huron 8 5.60E+00 2.97E-01 3.00E+00 4.10E-07 1.00E-02 5.50E-07 8.30E-10 5.20E-04 3.60E-01

Huron 9 5.70E+00 2.67E-01 9.10E-07 6.60E-04 7.10E-01 6.20E-03 7.00E-10 3.00E-02 9.70E-02

Huron 10 5.40E+00 2.86E-01 2.00E-01 3.20E-07 9.70E-04 2.00E-02 5.50E-08 2.60E-02 1.10E-01

Huron 11 5.70E+00 2.94E-01 8.00E-07 2.70E-05 3.00E-01 6.00E-03 7.00E-10 4.10E-02 2.10E-01

Huron 12 6.30E+00 5.63E-01 1.00E-01 4.70E-03 1.70E-02 9.90E-07 2.60E-09 2.50E-03 3.00E-01

Huron 13 6.30E+00 5.63E-01 1.00E-01 4.70E-03 1.70E-02 9.90E-07 2.60E-09 2.50E-03 3.00E-01

Huron 14 4.80E+00 7.40E-01 4.20E-01 9.30E-06 1.50E-02 8.60E-07 1.50E-06 3.50E-04 2.30E-01

Huron 15 6.60E+00 9.20E-01 1.60E+00 1.30E-02 1.90E-02 9.50E-07 7.60E-03 8.00E-02 7.30E-01

Huron 16 5.30E+00 7.51E-01 1.60E+00 2.20E-07 1.20E-02 5.30E-03 5.00E-09 1.10E-02 1.70E-01

Georgian 1 4.30E+00 7.66E-01 1.10E-01 1.80E-04 2.10E-02 8.30E-07 1.10E-05 8.70E-04 8.10E-02

Georgian 2 6.10E+00 7.64E-01 3.70E+00 2.00E-07 1.50E-02 8.90E-07 4.30E-09 2.00E-03 2.40E-01

Georgian 3 6.00E+00 4.78E-01 3.30E+00 4.10E-07 1.00E-02 7.70E-03 9.40E-09 7.00E-03 3.50E-01

Georgian 4 4.70E+00 3.78E-01 8.40E+00 4.60E-07 7.10E-03 5.70E-03 1.40E-08 1.30E-04 1.30E-01

Georgian 5 4.80E+00 7.30E-01 4.30E+03 8.00E-06 1.60E-02 7.40E-03 2.70E-09 3.10E-04 3.40E-01

Georgian 6 3.90E+00 4.25E-01 5.20E+01 5.90E-05 1.20E-01 5.30E-02 5.90E-07 3.80E-06 7.90E-02

Georgian 7 2.60E+00 1.02E+00 5.70E+05 8.20E-10 7.50E-03 5.80E-03 5.90E-10 5.00E-06 7.90E-02

Georgian 8 3.70E+00 1.24E+00 2.10E+01 2.20E-08 4.60E-03 1.80E-03 8.10E-10 8.50E-04 1.70E-01

Georgian 9 4.40E+00 2.40E-01 1.20E+00 7.50E-09 6.20E-03 3.50E-04 2.40E-10 1.90E-02 7.50E-02

Georgian 10 4.40E+00 3.68E-01 7.40E-01 3.40E-07 2.60E-02 2.70E-06 2.20E-09 4.60E-02 5.40E-02

Georgian 11 2.20E+00 2.32E-01 3.00E+00 7.00E-08 4.80E-03 2.20E-03 5.00E-10 2.60E-04 9.90E-02

Georgian 12 4.90E+00 2.46E-01 1.60E-01 2.90E-07 3.10E-02 8.80E-06 5.50E-10 7.50E-02 2.30E-01

Georgian 13 4.90E+00 2.46E-01 1.60E-01 2.90E-07 3.10E-02 8.80E-06 5.50E-10 7.50E-02 2.30E-01
7

Table 1. (Continued)

Lake Basin Tb 2 3 4 5 6 7 8 9

St. Clair 1 6.40E+00 3.53E-01 9.70E-02 1.10E+00 9.00E-07 1.00E-02 9.80E-01 4.20E-02 2.50E-01

St. Clair 2 8.20E+00 1.66E-01 1.60E+01 4.20E-06 1.00E-01 1.60E-01 1.10E-07 1.50E-02 3.10E+00

St. Clair 3 8.20E+00 2.76E-01 1.60E+01 4.20E-06 1.00E-01 1.60E-01 1.10E-07 1.50E-02 3.10E+00

St. Clair 4 8.20E+00 1.66E-01 1.60E+01 4.20E-06 1.00E-01 1.60E-01 1.10E-07 1.50E-02 3.10E+00

St. Clair 5 8.50E+00 7.46E-01 1.10E-01 5.30E-06 5.10E-02 6.00E-06 8.60E-08 5.10E-02 6.50E-01

St. Clair 6 7.80E+00 4.17E-01 4.10E+04 7.20E+01 3.10E-01 1.10E-01 3.40E-08 2.10E-02 2.80E-01

St. Clair 7 6.40E+00 5.03E-01 4.50E-02 3.40E-05 9.70E-03 1.10E-06 5.10E-09 5.90E-02 1.80E-01

Erie 1 8.00E+00 2.76E-01 1.30E+00 6.70E-07 2.40E-02 5.30E-02 1.40E-07 1.40E-02 1.10E+00

Erie 2 1.00E+01 3.65E-01 3.00E+01 5.70E-06 4.50E-02 1.20E-01 5.90E-08 1.70E-02 4.00E-01

Erie 3 6.20E+00 3.18E-01 7.30E-01 3.70E-03 1.00E-05 1.70E-02 6.50E-04 2.70E-02 3.60E-01

Erie 4 7.90E+00 3.98E-01 8.80E-06 9.50E-03 9.00E-02 7.90E-02 8.80E-05 1.00E-01 1.00E-01

Erie 5 8.10E+00 4.14E-01 5.80E-05 1.10E-06 6.00E+04 9.90E-02 1.00E-04 2.00E-01 3.30E-01

Erie 6 6.60E+00 4.40E-01 3.90E-02 4.50E-08 9.70E-06 3.20E-06 6.40E-08 4.90E-02 1.90E-01

Erie 7 6.40E+00 3.63E-01 4.30E-02 4.70E-07 8.60E-06 6.10E-06 1.20E-06 6.40E-02 4.10E-01

Erie 8 5.90E+00 8.90E-01 4.90E-02 1.20E-06 6.00E-03 2.50E-06 1.60E-06 6.00E-02 2.80E-01

Erie 9 7.10E+00 6.17E-01 9.20E-07 4.90E-06 1.00E-05 3.00E-06 5.90E-08 5.00E-02 1.20E+00

Erie 10 5.20E+00 3.18E-01 1.10E-01 2.90E-07 1.20E-02 6.30E-06 2.30E-07 5.90E-02 6.60E-01

Erie 11 7.60E+00 2.75E-01 7.00E+00 8.40E-07 1.10E-01 5.70E-02 1.60E-08 2.20E-02 3.20E+00

Erie 12 5.80E+00 3.95E-01 1.10E+00 1.20E-06 6.50E-02 5.80E-06 1.90E-08 2.90E-02 3.40E+00

Erie 13 5.10E+00 3.29E-01 6.40E-07 1.30E-06 1.00E-05 3.00E-06 5.90E-08 5.00E-02 3.30E-01

Erie 14 4.50E+00 7.39E+00 6.40E-02 2.30E-06 8.50E-02 4.40E-06 1.90E-04 6.10E-02 6.80E-01

Erie 15 4.40E+00 3.49E-01 3.20E-02 4.90E-07 3.90E-02 5.40E-06 1.00E-10 5.80E-02 6.60E-01

Erie 16 4.30E+00 2.45E-01 1.70E+00 1.10E-06 1.50E-01 1.40E-01 5.30E-08 2.40E-02 6.80E+00

Erie 17 4.60E+00 4.27E-01 1.60E+00 1.40E-05 1.70E-01 2.00E-01 1.50E-06 2.70E-02 6.60E+00

Erie 18 4.60E+00 4.29E-01 1.60E+00 1.40E-05 1.70E-01 2.00E-01 1.50E-06 2.70E-02 6.60E+00

Erie 19 9.40E+00 4.63E-01 2.70E+01 8.00E-03 3.30E-01 4.00E-01 3.80E-08 1.80E-02 3.50E-01

Erie 20 6.90E+00 5.97E-01 2.60E+00 2.80E-06 5.70E-02 7.20E-02 1.60E-07 1.30E-02 5.10E-01

Erie 21 1.40E+01 6.96E-01 9.10E+01 1.10E-04 3.50E-01 4.40E-06 5.40E-08 7.50E-02 1.00E+00

Ontario 1 6.00E+00 3.68E-01 5.10E-02 4.40E-05 6.10E-03 3.30E-06 1.50E-04 2.40E-01 3.30E-01

Ontario 2 6.00E+00 3.68E-01 5.10E-02 4.40E-05 6.10E-03 3.30E-06 1.50E-04 2.40E-01 3.30E-01

Ontario 3 4.20E+00 2.49E-01 9.00E-05 5.60E-08 2.00E-02 7.30E-06 1.40E-10 6.70E-02 6.30E-02

Ontario 4 5.50E+00 2.47E-01 4.90E-01 3.30E-06 1.70E-06 1.90E-01 3.60E-07 2.20E-02 4.00E-01

Ontario 5 4.30E+00 3.81E-01 1.20E-01 1.90E-08 4.10E-02 7.40E-06 9.60E-08 2.90E-01 4.50E-02

Ontario 6 4.30E+00 2.81E-01 1.80E+00 3.30E-06 1.20E-01 8.70E-02 3.90E-08 4.20E-02 3.60E+00

Ontario 7 4.40E+00 3.81E-01 1.00E-06 1.10E-08 2.00E-05 3.00E-06 8.10E-08 5.00E-02 5.90E-02

Ontario 8 3.10E+00 3.39E-01 1.10E-01 1.00E-07 2.10E-02 1.40E-06 1.00E-10 3.60E-02 9.40E-02

Ontario 9 4.40E+00 5.94E-01 8.30E-01 5.50E-02 8.90E-07 1.50E-02 2.00E-03 7.90E+01 1.00E-01

Ontario 10 4.40E+00 5.75E-01 8.30E-01 5.50E-02 8.90E-07 1.50E-02 2.00E-03 7.90E+01 1.00E-01

Ontario 11 6.90E+00 4.88E-01 4.90E+00 5.90E-02 1.50E-01 1.20E-01 2.70E-07 2.20E-02 1.00E-01

Ontario 12 5.60E+00 6.09E-01 5.70E+05 1.10E-10 1.70E-02 4.90E-03 1.90E-09 9.50E-03 5.30E-01

Ontario 13 5.50E+00 3.47E-01 1.90E+00 2.30E-05 9.50E-03 2.40E-02 2.50E-08 7.10E-03 1.50E+00

Ontario 14 5.50E+00 3.19E-01 1.90E+00 3.10E-06 2.00E-02 2.80E-02 2.30E-07 1.20E-02 4.80E-01

Ontario 15 5.50E+00 3.17E-01 1.90E+00 3.10E-06 2.00E-02 2.80E-02 2.30E-07 1.20E-02 4.80E-01
8

The general framework of previous climate change sensitivity experiments using LBRM and the other associated models 

under the method of Croley (1990) was to have one run, the base case, which simply used station-based meteorological 

observations as drivers, for the period 1948-2005 in our case. Then to represent future time periods, the same historical 

meteorology was used as the foundation of the inputs to LBRM, but was adjusted to values intended to represent the 

climate change, based on results from GCMs. Croley (1990) notes that this “change factor” method was used because of 

instructions from the U.S. Environmental Protection Agency, the funding agency for that study. Along with the traditional 

method of treating climate change in the LBRM (here called the temperature adjustment method or “TA”), based on its 

embedded assumptions about the drivers of ET, we will be using three additional methods of applying GCM output to the 

LBRM for evaluation of response to climate scenarios (summarized in Table 2).

Table 2. Description of alternative methods of incorporating GCM-based climate projections into the LBRM.

Name Abbreviation Description Temperature sensitivity of 

PET

Temperature 

adjustment

TA Air temperatures adjusted by GCM 

climate projections are inserted 

directly into LBRM’s equations for 

PET

7%-90% per °C depending 

on sub-basin, most between 

15% and 50%

Energy 

adjustment

EA PET values from LBRM in the base 

case climate are multiplied by the 

ratio of net radiation at the surface in 

the future case vs. the base case

No explicit dependence on 

temperature, only energy 

budget

Priestley-Taylor PT The energy adjustment method is 

applied first, then PET is further 

multiplied by the temperaturedependent factor from the PriestleyTaylor equation (eq. 5)

Explicit temperature 

dependence of ranging 

between 1.2% and 4.0% 

per °C, plus dependence on 

energy budget

ClausiusClapeyronCC PET from the base case is multiplied 

by a factor representing the change 

in water capacity associated with the 

temperature change from the GCM

7% per °C

1. The energy adjustment method (“EA”), as in LHW, uses the sum of the sensible heat flux and latent heat flux given 

by the GCM as an indication of the net radiative energy flux at the surface. The ratio of the GCM’s value in the future 

time period to the historical time period is used as a multiplier for the LBRM’s PET from the base case.

2. The EA method of LHW got some legitimate criticism from Scheff and Frierson (2014) for ignoring the term that is 

directly driven by air temperature in formulations such as that of Priestley and Taylor (1972). The Priestley-Taylor 

formulation is similar to that of Monteith (1973), but uses an empirical constant multiplier in place of an additional 

term for water vapor deficit. We adopt the concept of Priestley-Taylor here and label these runs “PT”, where in 

addition to multiplying PET by the ratio of surface radiative energy as in the EA method, we also multiply it by a 

factor directly driven by the air temperature change. This latter factor, however, ranges between 1.2% per °C (at 30°C) 

and 4.0% per °C (at 0° C), far less sensitive to temperature than the LBRM under the TA method.

3. The causal relationship between ET and atmospheric water holding capacity is not direct and has been a source of 

confusion, but the Clausius-Clapeyron relation seems to set a maximum bound on sensitivity of ET to climate change. 

So we use it as an additional point of comparison, multiplying the PET from the base case by the change in water 

vapor holding capacity of the atmosphere (expressed as a ratio) to project future PET. Even though this method also 

appears to overestimate the sensitivity of PET to climate change (based on e.g. Held and Soden 2006 and Lorenz and 

DeWeaver 2007), we regard it as “less wrong” than the TA method, and therefore find it useful as another point of 

comparison to demonstrate the problems that exist in the TA method.
9

We approximate the Clausius-Clapeyron relationship in an exponential form, in order to create an equation based on the 

Priestley-Taylor equation in closed form. The approximation that we use for saturation vapor pressure is 

(2)

This formula is fitted so that it gives the exact values from the table in Lide (2005) at 5°C and 25°C. In the range of 

0-30°C, this simplified formula lies within 3.4% of the actual value. The Priestley-Taylor equation is

(3)

where Δ is the gradient of es with temperature (0.0407 kPa °C-1 eT/15.51°C), Rn is net radiative flux at the surface, G is 

conductive heat flux into the ground, L is latent heat of vaporization per unit mass, and γ is the psychrometic constant, 

approximately 0.067 kPa °C-1. Filling in the exponential formulas (and dropping units of kPa °C-1, since they cancel out), 

this transforms to

 (4)

To simplify calculations when we are using two sets of numbers—the actual observations and the temperature offset from 

climate change based on GCM simulations—we will use the first-order Taylor expansion of this; i.e. we multiply the 

change in temperature by the derivative of eq, (4) with respect to T (assuming that L is constant with temperature), and 

then divide by eq. (4), in order to get a fractional change in PET due to temperature change. This factor is the multiplier 

that is applied to the PET from the EA method to get the PET from the PT method. Its value is

(5)

2.3 Other Model Components

In addition to the runoff from the land in the lakes’ basins, the other components of net basin supply are the over-lake 

precipitation and evaporation. As in Angel and Kunkel (2010) and preceding studies, over-lake precipitation was taken 

from Thiessen polygon weighting of data from land-based stations for the base case, and adjusted by ratios of GCM 

precipitation for the future cases. Also, as in previous studies, evaporation from the lakes was calculated using the Large 

Lake Thermodynamics Model (LLTM, Croley 1989), which projects water temperatures of the Great Lakes by treating 

them as lumped 1-dimensional lakes. It estimates solar and longwave radiation exchanges at the surface, and evaporation 

(latent heat flux) and sensible heat flux based on bulk aerodynamic formulas. Lake temperature profiles are then 

calculated using this energy budget and formulas for vertical mixing.

Since the over-lake precipitation and evaporation are not affected by the method of adjusting the LBRM for climate 

change, they have only one value for each GCM realization. They are combined with the LBRM results for each of the 

methods (TA, EA, PT, and CC) to get net basin supply (NBS) values that correspond to each GCM realization and each 

method—runoff from LBRM, plus over-lake precipitation, minus over-lake evaporation.

These net basin supplies are then the input to the Coordinated Great Lakes Regulation and Routing Model (CGLRRM), 

developed by the U.S. Army Corps of Engineers and Environment Canada (T. Hunter, personal communication). Each 

set of net basin supply time series was input to the CGLRRM to derive lake levels for Lake Superior, the combined Lake 

Michigan-Huron, Lake St. Clair (not shown), and Lake Erie. Because of the frequent departure of actual operation of the 

dams regulating Lake Ontario from their official plans, CGLRRM does not calculate levels for Lake Ontario.

The values for all variables generated by the GCMs are spatially interpolated to the locations relevant for simulation 

by LBRM, LLTM, and the over-lake precipitation estimate by simple inverse-distance weighting from the nearest four 
10

GCM grid points, without consideration for such factors as elevation or location relative to observing stations with 

special weather characteristics (e.g. lake breeze or lake-effect precipitation). We understand that this is not a state-of-theart way of doing downscaling of GCM output, but we want to replicate the methodology of Angel and Kunkel (2010) 

and predecessor studies, in order to isolate the effects of the portions of the formulation that we expect to be even more 

problematic than the details of downscaling.

2.4 General Circulation Models (GCMs)

A summary of the GCM runs that were used is in Table 3. This group of model realizations was chosen based on the 

availability of the data needed for simulations using our modeling system. Our criterion for choosing the first five 

models listed was that we used the models from the Climate Model Intercomparison Study Phase 5 (CMIP5) for which 

the following variables were available on a monthly basis from the Earth System Grid Federation server at Lawrence 

Livermore National Laboratory for a historical period including 1986-2005 in a single file for each variable, and for future 

periods including 2056-2100 in a single file for each variable: air temperature at 2 m height, humidity at 2 m height, 

precipitation, cloud cover, sensible heat flux, latent heat flux, and wind at 10 m height (either as a scalar quantity or vector 

components). We also accessed the Earth System Grid Federation server at the Geophysical Fluid Dynamics Laboratory 

(GFDL) to access the models generated by GFDL; these output files were in 5-year increments, requiring additional data 

handling. No other criteria were used to select GCMs, such as their accuracy in historically simulating the climate of the 

region. The monthly data from these GCMs were spatially interpolated to the lakes and sub-basins as in previous studies 

using the method of Croley (1990).

For each of the GCM realizations, we calculated the climatological values for each month of the year from the period 

2056-2075 (referred to as the mid-21st century) and 2081-2100 (referred to as the late 21st century) for comparison to 

the historical period 1986-2005. For the historical period, we used only the lowest-numbered model realization that was 

available on the data server. So with the 32 model realizations shown in Table 3 and the two time periods, we had 64 

scenarios to use as input to the hydrologic model suite.

Table 3. Summary of GCM runs used. In the columns under the Representative Concentration Pathways 

(RCP), the number of model realizations using that RCP is indicated. All of these cases were used with 

the lowest-numbered model run from the historical time period.

Institution and country Model name and 

reference RCP 4.5 RCP 6.0 RCP 8.5

National Center for Atmospheric 

Research, USA

CESM1-CAM5, Hurrell et 

al. 2013 3 3 3

Centre National de Recherches 

Météorologiques, France

CM5, Voldoire et al. 2013 1 0 1

Commonwealth Scientific and Industrial 

Research Organization, Australia

Mk3-6-0, Rotstayn et al. 

2012 2 2 2

Hadley Centre for Climate Science and 

Services, United Kingdom

HadGEM2-AO, Collins et 

al. 2011, Martin et al. 2011 1 1 0

Institut Pierre Simon Laplace, France CM5A-LR, Dufresne et al. 

2012 2 1 1

Geophysical Fluid Dynamics Laboratory, 

USA

CM3.0, Delworth et al. 

2006, Donner et al. 2011 1 1 1

Geophysical Fluid Dynamics Laboratory, 

USA

ESM2G, Dunne et al. 

2012, Dunne et al. 2013 1 1 1

Geophysical Fluid Dynamics Laboratory, 

USA

ESM2M, Dunne et al. 

2012, Dunne et al. 2013 1 1 1

 3. RESULTS

All box plots shown here use the default algorithm for creating box plots using the R software package. The heavy 

horizontal line inside the box indicates the median value of all of the data, the upper boundary of the box indicates the 
11

median of the subset of values at or above the overall median (approximating the 75% quantile), and the bottom of the box 

indicates the median of the values at or below the overall median (approximating the 25% quantile). The distance between 

the 25% and 75% quantile is known as the inter-quartile distance (IQD), and the maximum length of the whiskers is 1.5 

times the IQD from the boundary of the box. The whiskers actually extend to the data value that is farthest from the box 

while still being within this maximum distance. Additional data points outside of this maximum length of the whiskers 

(“outliers”) are indicated by individual circles. Each individual data point used in creating the box plots represents one 

GCM realization. While this algorithm for creating box and whiskers plots may not be appropriate for non-Gaussian 

datasets, one point of this study is to highlight the highly irregular and non-Gaussian nature of the data we are dealing

with.

3.1. Lake Superior basin

Lake Superior is the lake whose outflow eventually proceeds to all of the other Great Lakes. Hence its basin’s water 

budget can significantly influence the lake levels of all of the lakes. As an outcome of the vagaries of the calibration 

process of LBRM, the calibrated values of Tb for the sub-basins of Lake Superior are generally smaller than those for the 

other lakes, with many falling in the range of 2-5° C (contrasting with the Clausius-Clapeyron equivalent value of 15.51° 

C), and the most extreme low value being 1.1° C in the basin of the Montreal River at the eastern end of Lake Superior.

The first results that we show are the most extreme—the results for changes in PET in the Lake Superior basin. These are 

shown as a ratio of the value in the projected time period to the base case in Figure 1a. Because the figure’s scale is so 

large to accommodate the changes that result from the TA method, the results for the CC, PM, and EA cases are almost 

indistinguishable; they cluster slightly above the value of unity, with just a few values below unity. The largest value is 

1.72 in the case of the CC method for the RCP 8.5 run of the GFDL CM3.0 for the late 21st century.

In sharp contrast to the CC, PM, and EA methods’ values for PET are the values under the TA method in the Lake 

Superior basin. The median value is 5.05, with several values far more extreme than this. The highest value that is shown 

in the figure is 565.4, again for the RCP8.5 run of the GFDL CM3.0 for the late 21st century. Although the median increase 

by a factor of 5.05 may seem small compared to the extreme value of 565.4, it is in fact shockingly large when one 

considers that it implies the equivalent of an increase in the incoming solar radiation by a factor of 5.05.

The actual ET is constrained not only by the PET, but also by the amount of moisture that is available—precipitation and 

its storage as soil moisture. Therefore the results for ET among the different methods scale reasonably to be displayed on 

the same graph, but still differ markedly. The differences in ET between the base case and climate change cases under the 

various methods are shown in Figure 1b. The median change in ET in the Lake Superior basin under the TA method is 

0.024 cm/day, a 20.9% increase over the base case value of 0.115 cm/day. Under the PT method, ET is increased by only 

0.004 cm/day. The fact that this increase is much less than the factor of increase in PET is reflective of the fact that both 

potential and actual ET are small during the winter, when ET is energy-limited, while the summer season’s ET tends to be 

more moisture-limited, especially when PET is increased. Thus the proportional increase in actual ET is much less than 

in PET. Nevertheless, the median value of ET increase under the TA method is greater than the top extent of the whiskers 

for all three of the other methods. Also, the CC, PM, and EA methods’ results are much more similar to each other than 

to the TA method. The difference of 0.020 cm/day between TA and PT is the equivalent of 5.8 W m-2—not a really large 

difference in the surface energy budget, but we need to remember that this is averaged over the entire year and is limited 

by the availability of moisture, and that the total present direct forcing by human-made greenhouse gases is around 1 W 

m-2. 

In terms of runoff (Figure 1c), the results are influenced by those for actual ET, but also reflect changes in precipitation. 

Because there are some GCM runs that project large increases in precipitation, but few project significant decreases, the 

whiskers extending downward are relatively short, especially for the CC, PT, and EA cases, with no outliers on the low 

side. The entire box for the TA method is below the whiskers for the other three cases, although the whisker at the high 

end for the TA method extends into the range of the other boxes. This means that the GCM cases with larger increases in 
12

precipitation and smaller increases in temperature were able to have increased runoff, even under the TA method, although 

most cases have decreased runoff. Under the other methods, though, nearly all cases have increased runoff. 

The changes in runoff propagate directly into changes in NBS (Figure 1d), because the same results for the other 

components of net basin supply (over-lake precipitation and evaporation) were used for all of the methods. Therefore, the 

discrepancies among methods are the same as for runoff.

The lake level of Lake Superior (Figure 1e) is less sensitive to net basin supply than those of Lakes Michigan-Huron and 

Erie. This is in part because there is no other Great Lake upstream to enhance its net basin supply with an altered inflow, 

but more importantly because of the regulation of the outflow into the St. Marys River. One major goal in managing this 

regulated outflow is to maintain a near-constant lake level, thus prohibiting large changes in level. Nevertheless, projected 

lake levels under the TA method are markedly lower than under the other three methods. Although some data points lie 

above zero (rising lake level), the 75% quantile is below zero, so the great majority of GCM cases have falling lake levels 

under the TA method; the median is a drop of 24.5 cm. The CC, PT, and EA methods also have median changes in lake 

level that are negative, but less decisively so (a drop of 3 cm under the PT method), and their 75% quantiles are all greater 

than zero, so they have significant chances of rises in lake level. Also, the results for the CC, PT, and EA methods all 

cluster much closer to one another than to the results of the TA method.

Figure 1. Box and whiskers plots of variables averaged over the Lake Superior basin and over the entire year, 

generated by GCM outputs in combination with LBRM using the temperature adjustment (TA), Clausius-Clapeyron 

(CC), Priestley-Tayor (PT), and energy adjustment (EA) methods. Each data point used in making the plots is a 

single GCM realization. (a) Ratio of PET for future and historical time periods, (b) ET for future minus historical time 

periods (cm/day), (c) runoff for future minus historical time periods (cm/day), (d) net basin supply for future minus 

historical time periods (m3/s), and (e) lake level for future minus historical time periods (m).

a b c

d e

Superior PET ratio

CC EA PT TA

0 100 200 300 400 500

Superior ET

CC EA PT TA

−0.02 0.00 0.02 0.04 0.06

cm/day

Superior runoff

CC EA PT TA

−0.02 0.00 0.01 0.02 0.03

cm/day

Superior NBS

CC EA PT TA

−1000 −500 0 500

cubic m/sec

Superior level

CC EA PT TA

−0.8 −0.4 0.0 0.2 0.4

meters
13

3.2. Lake Michigan-Huron basin

Lakes Michigan and Huron are considered a single unit because their attachment at the Straits of Mackinac is wide and 

deep enough for them to always maintain the same hydraulic level. The results for Lakes Michigan-Huron, Erie, and 

Ontario largely reflect those of Lake Superior, although the results in terms of PET are less extremely dramatic, while the 

results in terms of lake levels are of greater magnitude (lake level results are not available for Lake Ontario).

The ratios of PET in the basin of Lake Michigan-Huron (Fig. 2a) are a muted version of the results for Lake Superior 

(Figure 1a), but again the changes in PET for the CC, PT, and EA methods cluster at values only moderately above unity, 

while under the TA method, the median value is just above 2, while several model runs project increases by factors greater 

than 4. These are not as extreme as the values for Lake Superior, mainly because the calibrated values of Tb are generally 

larger in the Michigan-Huron sub-basins (falling mostly in the range of 5 to 9° C) than those for Superior; the reason for 

this is not clear, and is part of the black-box nature of the calibration of LBRM.

The actual ET from the Michigan-Huron basin (Figure 2b) is again modulated by availability of moisture. The increases 

in ET under the TA method strongly outstrip those under the other three methods, but those under the other three methods 

lean more strongly toward increased ET than in the Lake Superior basin (compare Figure 1b).

In terms of runoff in the Lake Michigan-Huron basin (Figure 2c), as well as net basin supply (Figure 2d), much the 

same story is told again. The runoff and NBS under the TA method are considerably lower than under the other three 

methods. While the CC, PT, and EA methods generally show increased runoff, the balance of over-lake precipitation and 

evaporation place their median changes in NBS at negative values.

a b c

d e

CC EA PT TA

2 4 6 8 10

Michigan−Huron PET ratio

CC EA PT TA

0.00 0.02 0.04 0.06

Michigan−Huron ET

cm/day

−0.03

CC EA PT TA

−0.01 0.00 0.01 0.02 0.03

Michigan−Huron runoff

cm/day

CC EA PT TA

−1500 −500 0 500 1000

Michigan−Huron NBS

cubic m/sec

CC EA PT TA

−2 −1 0 1

Michigan−Huron level

meters

Figure 2. As in Fig. 1, but for the Lake Michigan-Huron basin.
14

Lake levels on Lake Michigan-Huron are more sensitive than Lake Superior levels because of a combination of factors. 

In the model results from all methods, the changes in NBS are generally of the same sign for all lakes. This leads to the 

effect that Lake Superior’s outflow due to its NBS has a reinforcing effect on the change in NBS in Lake MichiganHuron. Furthermore, Lake Michigan-Huron has no regulation at its outlet, in contrast to Lake Superior, so lake level 

variations cannot be artificially damped by direct human intervention; although this can be changed through construction 

of regulation works in the future, we assume the status quo in this respect in order to isolate the effect due to climate. 

Finally, with the accumulation of changes in NBS from the Superior, Michigan-Huron, and Erie basins, the level of Lake 

Erie also changes in the same sense as the change in NBS for those combined basins. Because the difference in level 

between Lake Michigan-Huron and Lake Erie is small (just over 2 m), a significant rise in Lake Erie’s level can inhibit 

outflow from Lake Michigan-Huron, requiring a rise in Michigan-Huron’s level to restore dynamic equilibrium, and viceversa for a drop in lake levels. Thus the change in lake levels for Lake Michigan-Huron (Fig. 2e) is qualitatively similar to 

those for Lake Superior, but the magnitudes are distinctly larger. Under the TA method, the 75% quantile is less than zero, 

indicating a large majority of GCM cases leading to decreased lake levels. The median value is a drop of 67.5 cm, and the 

largest drop is 219 cm in the IPSL RCP8.5 run for the mid-21st century (greater by 6 cm than for the same model run in 

the late 21st century, because of a larger drop in precipitation). The TA results also have a large spread, with a few results 

showing rises in lake level greater than 1 m.

The CC, PT, and EA results again cluster closer to one another than to the TA results, with median values less than zero 

and 75% quantile values greater than zero. The median lake level change under the PT method is a drop of 14.5 cm, 53 

cm higher than the TA method, and the extreme low, again from the IPSL RCP8.5 run for the mid-21st century, is a drop of 

145 cm, 74 cm higher than the extreme under the TA method. The PT and EA methods show a narrower range of changes 

between the 25% and 75% quantiles than the TA method, but also have several outliers above the whiskers. 

Because Lake Michigan-Huron has the largest signal in terms of lake level changes, we include a comparison of lake 

levels for subsets of the GCM realizations in Figure 3, broken down by time period and RCP scenario. The sample sizes 

for these subsets are smaller—there are 12 GCM realizations contributing to the RCP 4.5 sets for each time period, and 10 

each for RCP 6.0 and RCP 8.5. Figure 3 includes only the TA and PT methods, and for all of the combinations of RCP and 

time period, PT results in higher lake levels in all of the following measures: upper and lower extremes, 25% quantile, 

median, and 75% quantile. In some of the panels, the 25% quantile for the PT method is greater than the 75% quantile 

for the TA method, i.e. the boxes do not overlap. The median projected lake levels under the RCP 4.5 scenarios are lower 

than those under the RCP 6.0 scenarios, in both the mid-and late century and using both the TA and PT methods, due to a 

different subset of GCMs included. All combinations of RCP scenario, time period, and method yielded median reduction 

in lake level. Only the late 21st century under RCP 6.0 and using the PT method has a median change in lake level greater 

than zero.

3.3. Lake Erie

The results for Lake Erie are again similar. Most values of Tb are even higher for Lake Erie’s sub-basins than in the 

Michigan-Huron sub-basins, so the PET in the TA method (Figure 4a) does not reach such high extremes, but still is very 

distinct from the distributions for the other methods. Actual ET (Figure 4b) is also distinctly higher under the TA method 

than the others. Runoff (Figure 4c) is mostly decreased under the TA method, but increased under the other methods. For 

NBS (Figure 4d), the TA method has a strong decrease, while the other methods have strong chances of changes in NBS 

both above and below zero.

The lake level for Lake Erie is influenced by the sum of the NBS of Lakes Superior, Michigan-Huron, and Erie, because 

the others flow into Lake Erie. While the outflow from Lake Erie can be variably partitioned between routing to Niagara 

Falls for aesthetic purposes and water turbines for power generation purposes, the overall outflow is not controlled by 

human regulation. Because the head is very large between Lake Erie and Lake Ontario (just under 100 m), there is no 

backwater effect. The sensitivity of Lake Erie’s levels is somewhat diminished relative to Lake Michigan-Huron’s. Figure 

4e shows that under the TA method, lake levels are shown as having well over 75% likelihood of going down, while under 

the other methods, zero change is within the range of the box.
15

a b c

d e

CC EA PT TA

1.0 1.5 2.0 2.5 3.0

Erie PET ratio

CC EA PT TA

0.00 0.02 0.04 0.06

Erie ET

cm/day

CC EA PT TA

−0.02 0.00 0.02 0.04

Erie runoff

cm/day

CC EA PT TA

−400 −200 0 200

Erie NBS

cubic m/sec

CC EA PT TA

−1.5 −1.0 −0.5 0.0 0.5

Erie level

meters

Figure 3. Lake Michigan-Huron water levels represented using the temperature adjustment and Priestley-Taylor 

methods for subsets of the GCM realizations. Note that the scale differs among the panels. (a) Mid-21st century for 

RCP 4.5, (b) mid-21st century for RCP 6.0, (c) mid-21st century for RCP 8.5, (d) late 21st century for RCP 4.5, (e) late 

21st century for RCP 6.0, and (f) late 21st century for RCP 8.5.

Figure 4. As in Fig. 1, but for the 

Lake Erie basin.

a b c

d e

PT TA

−1.0 −0.5 0.0 0.5 1.0

RCP4.5 mid−century

meters

PT TA

−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4

RCP6.0 mid−century

meters

PT TA

−2.0 −1.5 −1.0 −0.5 0.0 0.5

RCP8.5 mid−century

meters

PT TA

−1.5 −1.0 −0.5 0.0 0.5

RCP4.5 late century

meters

PT TA

−1.5 −1.0 −0.5 0.0 0.5

RCP6.0 late century

meters

PT TA

−2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5

RCP8.5 late century

meters

f
16

3.4 Lake Ontario basin

Additional similar results occur on Lake Ontario. The ratios of PET for Lake Ontario (Fig. 5a) occupy a similar range to 

those for Lake Michigan-Huron, due to the values of Tb there. Actual ET (Fig. 5b), runoff (Fig. 5c), and NBS (Fig. 5d) 

all show patterns similar to the other lakes, except that the median values of changes in runoff and NBS under the CC, 

PT, and EA methods are all greater than zero, with the 25% quantile near zero. Largely as a result of lake regulation in 

practice that frequently deviates from the official regulation plans for the outlet of Lake Ontario, there is not a reliable 

segment of the CGLRRM to calculate Lake Ontario outflows and levels, and therefore no lake levels are shown here.

3.5 Correlation of Discrepancies with Temperature Change

The amount by which the TA method overestimates the influence of temperature on the various outputs is linked to the 

magnitude of the temperature change that is a primary input. Figure 6 shows the discrepancy in Lake Superior lake 

level (result from the CC method minus the TA method and PT method minus TA method) as a function of the difference 

between the GCM-simulated historical and future air temperature. Lake Superior was chosen to eliminate the influence 

from other lakes. Figure 6 shows very high correlations—R2=0.904 in the case of CC minus TA and R2=0.869 in the 

case of PT minus TA. This accounts for much of the narrowing of the distributions of all of the variables in Figs. 1-5 

under the CC, PA, and EA methods relative to the TA method; the largest increases in PET and ET under the TA method 

are associated with the largest overestimates and largest temperature increases, while the GCM cases with smaller 

temperature increases have smaller influences under the TA method, and thus lesser disagreement with the other methods. 

The narrowing of the distributions under the other methods relative to TA are actually more distinct in the runoff and, 

especially, ET plots rather than the NBS and lake level plots.

3.6 Statistical significance

The discrepancies between the TA method and all three alternative methods are highly systematic, stemming first from 

the very large differences in the projected changes in PET. This leads to the results being biased in the same direction 

regardless of the GCM that is used as input. Therefore, the TA method’s projected results are greater than those for the 

CC, PT, and EA methods in terms of PET and ET, and less in terms of runoff, NBS, and lake level. 

One simple way to think of the statistical significance of these comparisons is to think of each comparison between the 

TA method and each alternative method as having equal chances of either result being greater or less than the other, under 

the null hypothesis that both distributions are the same. However, this leaves us with the question of how many of these 

comparisons are independent of each other, for which there is not a clear answer. However, we will use a conservative 

guess at the number of independent samples that are present. Since PET controls ET, and in turn, runoff, NBS, and 

lake level, we will consider all of these variables to be dependent on each other, and therefore together constitute 

only one independent variable. Likewise, we consider the separate results from all of the lakes to constitute only one 

independent variable, as well as the various results from different RCP scenarios and realizations of the same GCM, and 

the two different time periods that were compared. Thus, the only dimensions of independence that we have left are the 

comparison between the CC and TA method on one hand, and between the PT and TA method on the other hand (we 

regard PT and EA to not be independent), as well as the eight different GCMs that were used. Thus, we regard the full 

set of experiments as resulting in sixteen independent comparisons between the TA method and alternative methods, all 

of which show discrepancies of the same sign. The probability of this happening under the null hypothesis is (0.5)16 = 

1.53x10-5 (single-tailed test), meaning that in terms of percentage significance, it is around 99.998%.

We also use the Wilcoxon rank sum method (Hollander and Wolfe 1973) on the lake level data. This is a non-parametric 

test for comparison of sets of data samples to find whether their distributions are distinct. It requires no assumptions about 

the shape of the distribution of each, although by combining multiple realizations from each GCM, we are admittedly 

violating the assumption of complete independence of the “observations” (in this case meaning the method to yield 

model results). Using the lake level results of the 64 GCM realizations under the TA method and the 64 results from each 

alternative method, we rank the results from 1 to 128, with lower rank numbers indicating higher lake level, and show 

the sum of the ranks for all of the realizations under the TA method in parentheses in Table 4. Under the theory of the 
17

Figure 5. As in Figure 1, but for the Lake Ontario basin, and the lake levels 

are excluded, as lake levels were not calculated for Lake Ontario.

Figure 6. Scatterplot of the difference in lake level (units of m) for Lake Superior, TA minus 

PT (plus signs) and TA minus CC (circles) vs. change in temperature.

a b

c d

CC EA PT TA

1 2 3 4 5 6 7

Ontario PET ratio

−0.01

CC EA PT TA

0.01 0.02 0.03 0.04 0.05 0.06

Ontario ET

cm/day

CC EA PT TA

−0.02 0.00 0.02 0.04

Ontario runoff

cm/day

CC EA PT TA

−400 −200 0 100 200 300

Ontario NBS

cubic m/sec
18

Wilcoxon rank sum test, the rank sums for two groups of 64 members each, under the null hypothesis of samples being 

drawn from the same distribution, have a Gaussian distribution with a mean of 4128 and a standard deviation of 209.8. 

Thus the complementary error function, indicating the probability that the distributions are the same, can be derived from 

standard Gaussian probability functions, and are shown in Table 4. The smallest rank sum that we found is under the CC 

method for Lake Erie. The rank sum of 5289 is 5.53 standard deviations above the mean value, indicating a probability of 

about five in one quadrillion that the distributions are the same.

These estimates of statistical significance are not perfectly rigorous due to it being unclear how many of the realizations 

should be regarded as independent, but they indicate levels of statistical significance far beyond the often-used standard 

threshold of 95% or even 99%.

4. DISCUSSION

The LBRM was carefully crafted to have features such as a mathematical solution that is entirely analytic within each 

time step (see Croley 1983) and therefore numerically stable under all conditions, as well as a calibration process that 

thoroughly searches the parameter space. It also conditions the energy available for evapotranspiration (which, divided by 

the factor of latent heat of evaporation per unit mass, we refer to as potential ET) on total insolation at the surface during 

the calibration runs. However, it seems that the question was not asked whether that energy constraint would carry over 

to other climatic regimes. One signal that gives a hint at the answer to this is how sensitive the purported energy available 

for evapotranspiration is to air temperature, with the Clausius-Clapeyron relationship of approximately a 7% increase per 

degree C being a significant benchmark as a value that should not be greatly exceeded.

In LHW and here, caution has been used to say that in the TA method, air temperature is taken to be a proxy for PET 

that is equally valid across all time scales and climate regimes (the “universal proxy” assumption), and not to say that 

the TA method assumes that air temperature causes PET (the causation assumption). Now we will examine how these 

two assertions relate to one another. The causation assumption is a specific case of the universal proxy assumption. 

Functionally, in terms of setting up model simulations, these two types of assumption are entirely equivalent; the 

calculations that are used in the modeling are the same whether one regards air temperature as a universal proxy for 

PET or as an actual cause. The assumption that air temperature provides a universally valid proxy across time scales and 

regimes could conceivably also be valid if the same set of mechanisms applied to the relationship between air temperature 

and PET, regardless of time of year, GHG concentration, and other factors. Could this be the case?

To examine some of the mechanisms by which air temperature varies and how this relates to ET, let us first consider 

the annual cycle. It is driven by different amounts of solar radiation being incident to the surface at a given location in 

different seasons. As incident solar radiation is a key constituent of the surface net radiation, which is an explicit factor in 

the process-based Priestley-Taylor PET, the annual cycle yields a very direct forcing on ET. Air temperature variability on 

scales of several days is instead driven most strongly by variability in the direction of atmospheric advection, associated in 

Table 4. Results of the Wilcoxon rank sum test (Hollander and Wolfe 1973) for the lake 

level of the various lakes, comparing the results under the Clausius-Clapeyron method, the 

Priestley-Taylor method, and the energy adjustment method to the temperature adjustment 

method. The statistical significance is stated in terms of the complementary error function 

(probability that the results under the temperature adjustment method and the alternative 

method come from the same distribution), while the rank sum for the temperature adjustment 

method is in parentheses.

Clausius-Clapeyron Priestley-Taylor Energy adjustment

Lake Superior 6.00x10-16 (5328) 2.85x10-17 (5382) 1.76x10-20 (5504)

Lake Michigan-Huron 4.29x10-15 (5292) 1.11x10-15 (5317) 1.46x10-20 (5507)

Lake Erie 5.03x10-15 (5289) 9.87x10-16 (5319) 1.14x10-20 (5511)
19

the mid-latitudes with the frequent passage of fronts and air masses. This does not directly force ET in the way that solar 

radiation does, and in fact warmer air is regularly moister air, thus inhibiting ET. These contrasts between the annual and 

daily scale were demonstrated in LHW’s Fig. 2, which shows high correlation between air temperature and ET for the 

annual cycle, but almost none at other time scales. This invalidates the universal proxy assumption, as in assertion 2 listed 

in the present paper’s Introduction. 

Some other methods are popular for assessment of climate change’s impact on water resources and ET, using a oneway coupling of data feeding from a GCM to a hydrologic model. One of the most popular uses the PET equation 

of Thornthwaite (1948), often within the context of the Palmer Drought Severity Index. Also sometimes used is the 

formulation of Hamon (1961). These formulations use aspects of the seasonal cycle, often in terms of length of daylight, 

as explicit predictors within their formulation. This contrasts with LBRM, in which all of the variability in PET is 

attributed to air temperature as the predictor (regardless of whether this is framed as the universal proxy assumption or the 

causation assumption). By attributing much of the variability within historical times directly to the seasonal cycle, instead 

of to air temperature, the sensitivity to air temperature under these other formulations is much less than in LBRM.

Warming due to greenhouse gases has yet another mechanism (aside from solar input and advection), with a seminal 

explanation demonstrated in Manabe and Wetherald (1967). The warming can be thought of as being “seeded” by a small 

increase in downward longwave radiation due to the presence of enhanced CO2 in the atmosphere. The resultant warming 

of the surface then mixes through the entire depth of the troposphere via convection and circulations driven by baroclinic 

instability. This warmer troposphere further enhances downward longwave radiation, so the surface and the troposphere 

mutually warm each other. Furthermore, a warmer atmosphere has more water vapor in it, which is also a greenhouse 

gas, and this further enhances downward longwave radiation, and further magnifies the warming. However, as the surface 

warms, its upward longwave radiation also increases, so that the change in the surface net radiation is minimal, despite 

substantial changes in temperature. The way that this is manifested in the GCMs that we used as forcing can be seen in the 

relatively small increases in PET that resulted under the EA and PT methods.

Another problematic assumption underlying the LBRM and the TA method is that the constraint of eq. (1), enforced in 

the calibration runs of LBRM as a means of keeping PET in line with solar radiative heating, universally ensures that 

the LBRM conserves energy. Eq. (1) does assure is that the total PET over the year is entirely unaffected by the value of 

Tb, and that actual ET is only minimally affected. But outside of the calibration runs and particularly in altered climate 

regimes, PET is crucially affected by Tb, as demonstrated by the contrast in the results of the TA and CC runs.

The extreme changes in PET demonstrated under the TA method are easily verified by simple calculations. The lowest 

calibrated value of Tb is 1.1° C for Lake Superior sub-basin 13 (the Montreal River near eastern Lake Superior). With a 

high-end increase in temperature of 7°, the factor of increase in PET is e7.0/1.1=580.4. Even with a lesser increase, such 

as 3°, the factor is 15.3. Most of the other Lake Superior sub-basins have Tb values between about 2° and 5° C, yielding 

lesser but still large increases in PET. Because the base case’s PET is constrained to be equivalent to the incident sunlight 

by eq. (1), these factors of increase in PET can be considered as factors of increase in equivalent incident sunlight, or, 

more vividly, as the equivalent number of suns in the sky of the virtual world created in the LBRM.

The assessments by Gronewold et al. (2011) and Fry et al. (2014) of LBRM and its application to seasonal to multiseasonal projection, as well as hindcasting, do not involve large and systematic shifts of air temperature as in this paper. 

Therefore, the results found there and here do not impact each other very strongly. Additional assessment of LBRM in the 

context of the connection between air temperature anomalies and errors in runoff in historical periods is called for.

Other strange features of LBRM become evident by examining the set of calibrated parameters in Table 1. For example, 

parameters 4 and 7 from that table are used in the formula:

ETUSZ = AeT/Tb P4USZM/(1 + P4USZM + P7LSZM), (6)
20

where ETUSZ is ET from the upper soil zone, AeT/Tb is as in eq. (1), P4 and P7 are calibrated parameters 4 and 7, and USZM

and LSZM are soil moistures values of the upper soil zone and lower soil zone. There is a similar formula for ET from 

the lower soil zone. The values of parameters 4 and 7 are shown to vary by more than ten orders of magnitude among 

the various sub-basins. The other parameters also vary widely, as do the resultant values of storage in the upper soil 

zone, lower soil zone, groundwater, and surface storage reservoirs. Physical interpretation of this amount of variance is 

impossible, and is evidence that there is a high amount of compensation among the values of the calibrated parameters 

in optimizing the runoff against observations. However, as already shown, the sensitivity to climate change under the TA 

method is crucially dependent on the calibrated value of Tb.

It has been convenient to use the term “paradigm” to describe the repeated use of the LBRM under the TA method to 

represent response to climate change. However, comparison with Kuhn’s (2012) definition of “paradigm” reveals some 

telling contrasts with his understanding of how a paradigm is created and used. The basic element of Kuhn’s definition 

is that a paradigm is an agreed and shared set of principles that are understood among the practitioners of a branch of 

science, and which enables further research to refine that understanding. Once the elements of the paradigm are deemed 

valid in the minds of nearly all practitioners and as long as they remain that way, they become material for textbooks 

rather than research articles, in order to initiate new practitioners into the paradigm. Because of this need for initiation, 

reports of the research done by those in the field are largely unintelligible to learned outsiders. Godfrey-Smith (2003) 

reinforces the idea of paradigms supporting progress, speaking of the need for scientists to rely on the knowledge amassed 

by the larger scientific community: “If each individual insisted on testing everything himself, science would never 

advance beyond the most rudimentary ideas.”

In contrast to this, use of the LBRM under the TA method does not generate new understanding, but only yields numbers 

for runoff, which contributes to calculations of NBS and lake level. Instead of requiring hard-earned background 

knowledge, it has been designed to have inputs and outputs that are easily understood, and internal workings that are 

invisible to its users, and, to a large extent, to its developers; it is a black box. Its reasonable outcomes for replicating the 

past were taken as sufficient evidence that its underlying assumptions would carry it to success under contrasting future 

conditions. The system has two key pieces whose conceptual development and subjection to peer review were carried 

out separately: LBRM itself and the change factor method that lies behind the TA method. Unlike a paradigm as seen 

by Kuhn, this combined system’s use has not been contingent on critical evaluation by a community, but rather has been 

uncritically accepted by a collection of users who find an opportunity in extracting the final level outputs that the model 

suite including LBRM puts forward, correct or not, verifiable or not. We have chosen instead to lift the hood of this black 

box and evaluate more of the intermediate results that lead to the final results, and whose connection between past and 

future scenarios can be more easily traced.

The paradigms underlying all fields of science combine objects, their characteristics, and the processes that govern those 

objects and the interactions among them. It may be that cross-disciplinary research tends to lack true paradigms, and that 

the substitutes are paradigms that exist within only one field (in this case hydrology), and that recognize only some objects 

and their properties from a neighboring field (in this case meteorology), but not the relevant processes in that neighboring 

field’s paradigm. In the case of LBRM, even this doesn’t account for the problems. The conservation of energy at the 

surface is a part of the paradigm of hydrology, and is given cursory recognition in the calibration stage of LBRM via eq. 

(1), but full conservation is not enforced throughout, and the consequences of this were apparently not investigated until 

LHW.

The climate modeling community does have a strong paradigm for interaction between the atmosphere and surface, whose 

principles are directly derived from hydrology (it has the same roots as the Priestley-Taylor equation). But the application 

of LBRM under the TA method has summarily rejected the results of the climate models’ calculation of surface fluxes, 

rather than engaging them and attempting to justify corrections to them.

5. CONCLUSIONS AND FUTURE PLANS

A method for projecting the water levels of the Laurentian Great Lakes under scenarios of human-caused climate change, 
21

used almost to the exclusion of other methods in the past, relies very heavily on the Large Basin Runoff Model (LBRM) 

as a component for determining the water budget for the lake system. LBRM’s exclusive reliance on air temperature as 

a predictor of PET and its very high sensitivity to air temperature cause it to overestimate future ET and cause the entire 

modeling system to underestimate future lake levels. This excessive sensitivity is at least partially due to LBRM’s failure 

to distinguish between the ET-air temperature correlation associated with the annual cycle (therefore driven largely by 

changes in incoming solar radiation and net surface radiation) and variability in ET and air temperature associated with 

other time scales and other causes. 

We created three alternative methods for transferring climate outputs from GCMs into the LBRM: the energy adjustment 

method (EA), the Priestley-Taylor method (PT), and the Clausius-Clapeyron method (CC). We deem all of these 

alternative methods, especially the PT method, to be more reflective of what is actually happening within the GCM than 

the TA method. All three of these alternative methods show, relative to the temperature adjustment method (TA), less PET 

and ET, more runoff from the land and net basin supply for the lake basins, and higher lake water levels in the future. The 

results from the EA, PT, and CC methods cluster much closer to one another than to the TA method results. The median 

change in lake level of Lake Michigan-Huron among all the GCM runs shifts from a drop of 67.5 cm under the TA method 

to a drop of only 14.5 cm under the PT method. This contrast makes the better guess at the median value within a range 

that is quite small compared to the natural variability that has been observed historically, although the possibility of other 

outcomes for future multi-decadal mean values has also been shown.

The magnitude of the discrepancies between the TA and other methods is highly correlated with the air temperature 

change in the driving GCM (larger temperature changes lead to larger discrepancies). This narrows the distributions 

among the driving GCM runs in changes of all of the variables, although the spread still remains sizable. 

Using various methods of estimating the statistical significance, we find that, at minimum, these discrepancies in results 

are significant at the 99.998% level. Because of the highly systematic sources of error within the formulation of the 

LBRM under the TA method, we regard these extremely high values of statistical significance as entirely inevitable and 

largely beside the point—simply the result of a model formulation that, intentionally or not, yields excessive sensitivity of 

ET to air temperature.

The alternative methods of transferring GCM-predicted variables into the LBRM for future projections are relatively 

quick and easy modifications to the existing modeling system, for purposes of demonstration of the problems with the 

traditional method, and are not final answers. As a next step, therefore, we would like to put forward a larger effort to 

model the Great Lakes basin’s water budget using a hydrologic model with a strong basis in the surface energy budget, 

and driven by a variety of GCM data using a more modern climate downscaling technique.

6. ACKNOWLEDGEMENTS

This paper benefited from discussions with A. D. Gronewold, D. H. Lee, and R. Bolinger, and corrections from 

anonymous reviewers of its companion paper, to appear in the Journal of Hydrometeorology.

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