Meteorological and turbulent heat flux data were collected from four offshore, lighthouse-based monitoring platforms (Fig. 1): Stannard Rock (Lake Superior), White Shoal (Lake Michigan), Spectacle Reef (Lake Huron), and Long Point (Lake Erie). These observations were collected as part of a broader collection of fixed and mobile-based platforms collectively referred to as the Great Lakes Evaporation Network (GLEN; Lenters et al., 2013). The National Data Buoy Center (NDBC) refers to these installations as stations STDM4, WSLM4, and SRLM4 at Stannard Rock, White Shoal, and Spectacle Reef, respectively.

With the exception of Long Point, a footprint analysis indicates that each station is located a sufficient distance from shore, so there is no influence of the land surface on the turbulent flux measurements (Blanken et al., 2011). Long Point, however, is located at the tip of a narrow, 40 km peninsula extending into Lake Erie. As a result, measured fluxes can be influenced by the upwind land surface when the wind direction is from directions between 180∘ S (south) and 315∘ NW (northwest); therefore, the corresponding data were removed when measured wind directions were within this range.

We evaluated five different flux algorithms from three models (hydrodynamic, atmospheric, and hydrologic) that are frequently used for Great Lakes operational and research applications (Fig. 2).

In an early stage of its development, FVCOM required prescribed heat fluxes as forcing variables, rather than being calculated (Chen et al., 2006). In a subsequent version of FVCOM (Version 2.7), turbulent heat fluxes began being calculated using the Coupled Ocean–Atmosphere Response Experiment (COARE) Met Flux Algorithm, version 2.6 (Fairall et al., 1996a, b), which was adopted in the official FVCOM by Chen et al. (2006). The COARE Met Flux Algorithm is one of the most frequently used algorithms in the air–sea interaction community. It was subsequently modified and validated at higher winds in the version known as COARE 3.0 (Fairall et al., 2003) and the latest version, COARE 3.5, (Edson et al., 2013), which includes wave influences on the Charnock parameter (Charnock, 1955). FVCOM has mostly incorporated these updates in their upgraded versions, including the provision for freshwater implementation, except that the latest version of FVCOM (version 4.0) has not yet included wave influences on the Charnock parameter. Hereafter we refer to the COARE implementation in FVCOM as COARE, which is equivalent to COARE 3.0. In version 3 of FVCOM and later, two additional flux calculation algorithms were added (Chen et al., 2013); one was adapted from a flux coupler in the Community Earth System Model (CESM, Jordan et al., 1999; Kauffman and Large, 2002) and was also built into the code of the Los Alamos sea ice model (CICE, Hunke et al., 2015). This algorithm is hereafter referred to as J99 (i.e., Jordan et al., 1999). The other algorithm, hereafter referred to as LS87 (Liu and Schwab, 1987), was originally developed at NOAA's Great Lakes Environmental Research Laboratory (GLERL) and was subsequently used in a variety of Great Lakes research and operational applications (Anderson and Schwab, 2013; Beletsky et al., 2003; Rowe et al., 2015; Wang et al., 2010; and many others). The inclusion of LS87 in FVCOM was tied to the fact that the algorithm was historically part of the real-time nowcasts and forecasts of NOAA's GLOFS, which is based on the Princeton Ocean Model, and that GLOFS is transitioning its physical model to FVCOM.

The WRF model (Skamarock et al., 2008) is increasingly used for regional weather and climate model applications over the Great Lakes (Benjamin et al., 2016; Xiao et al., 2016; Xue et al., 2015). The WRF model includes a one-dimensional lake model that thermodynamically interacts with the overlying atmosphere (WRF-lake, Bonan, 1995; Gu et al., 2015; Henderson-Sellers, 1986; Hostetler and Bartlein, 1990; Hostetler et al., 1993; Subin et al., 2012) and is adapted from the lake component within the Community Land Model, version 4.5 (CLM 4.5, Oleson et al., 2013; Zeng et al., 1998). The algorithm for the turbulent heat flux calculation in the WRF-lake model is mainly based on Zeng et al. (1998), except that roughness length scales for temperature and humidity are constant for its WRF-lake application, while they are updated dynamically in CLM 4.5. Hereafter, this algorithm in its WRF-lake application is referred to as Z98L.

Finally, we include the flux algorithm from the LLTM (Croley, 1989a, b; Croley et al., 2002; Hunter et al., 2015), which is a lumped conceptual lake model developed for hydrological research and forecasting for the Great Lakes. LLTM simulates evaporation and heat fluxes as a lake-wide average, rather than spatial distribution. This algorithm is based primarily on the work of Croley et al. (1989a, b) and is hereafter referred to as C89.

All of the above algorithms are based on applications of Monin–Obukhov similarity theory (Kantha and Clayson, 2000; Obukhov, 1971), where the turbulent fluxes of sensible heat, latent heat, and momentum are expressed with state variable magnitudes associated with surface friction – T*, q*, and u* for air temperature, specific humidity, and horizontal wind velocity, respectively. In each algorithm, the bulk expressions are used to calculate the sensible heat (H) and the latent heat (λE); H=ρacpCHSθw-θa,λE=ρaλCESqw-qa, where ρa is the density of air; cp and λ are the specific heat of air and the latent heat of vaporization, respectively; CH and CE are the bulk transfer coefficients for the sensible and latent heat, respectively; S is the average value of wind speed that includes the effect of the gustiness velocity in addition to horizontal wind speed U (defined later); and θw and θa (qw and qa) are potential temperatures (specific humidity) of the surface water and of air at the measurement height, respectively.

The bulk transfer coefficients have a dependence on atmospheric stability that can be expressed as CD=κln⁡zz0-ΨM(ζ)-2, CH,E=κPrt-1ln⁡zz0-ΨM(ζ)-1ln⁡zz0θ,0q-Ψθ,q(ζ)-1, u*=SCD,T*=ΔθCH/CD,q*=ΔqCECD, where z0, z0θ, and z0q are roughness length scales for momentum, temperature, and humidity, respectively; CD is the drag coefficient; κ is the von Kármán constant (0.40 for COARE, Z98L, and J99, 0.41 for C89, and 0.35 for LS87); Prt is the turbulent Prandtl number (1.0 is used in all the algorithms); and ΨMθ,q(ζ) is the integrated form of stability functions for momentum, temperature, and humidity. All algorithms assume that temperature and humidity have a common value of Ψ; i.e., Ψθ=Ψq=ΨM. ζ=z/L is the stability factor, where L is the Obukhov length and z is the measurement height.

Differences among the algorithms are primarily how they estimate ΨM,θ,q(ζ), z0. and consequently the bulk transfer coefficients. The profile functions ΨM,θ,q(ζ) are typically divided into three regimes, namely unstable, mildly stable, and strongly stable. All the algorithms use Businger-type parameterizations (Businger et al., 1971; Kraus and Businger, 1995) for the unstable regime (Table 1), except COARE, which includes convective behavior in highly unstable conditions by introducing a stability function for a convective limit (Fairall et al., 1996a; Tables S1 and S2 in the Supplement). For stable conditions, Holtslag et al. (1990) is used in LS87, C89, and Z98L, while Beljaars and Holtslag (1991) is used in J99 and COARE (Table 1). Note that there are minor differences in coefficients of ΨM,θ,q(ζ) within the algorithms, which can be found in Tables S1 and S2.

The roughness length scale for momentum, z0, is often parameterized as a function of friction velocity u*. The LS87, C89, and COARE algorithms apply Charnock's formula (Charnock, 1955; Smith, 1988) as follows; z0=αu*2g+0.11νu*, where z0 is the roughness length scale of momentum, α is the Charnock parameter, g is the acceleration due to gravity, and ν is kinematic viscosity. Because the value of z0 feeds back into the value of u* via Eqs. (3) and (5), Eq. (8) must be solved iteratively to arrive at the final values of these variables. Here, COARE calculates the Charnock parameter α as a function of wind speed, while LS87 and C89 use a constant α (Table 1). In contrast to the Charnock formula (Eq. 8), J99 directly calculates z0 as a function of wind speed based on Large and Pond (1981), while Z98L assumes z0 to be a constant 0.001 m. In the original paper of Zeng et al. (1998), inconstant parameterizations for roughness length scales were used, namely in Smith (1988) for momentum and in Brutsaert (1982) for temperature and humidity. The constant value in Z98L is likely related to the fact that the implementation in WRF handles the lake surface as part of various land surface types whose roughness lengths for momentum are often assumed to be constant (Mitchell et al., 2005; Oleson et al., 2013), while the original work of Zeng et al. (1998) assumed ocean surface applications.

Evidence from previous studies suggests that z0 can be significantly larger than z0θ,q, because momentum is transported across the air–sea interface by pressure forces acting on roughness elements, while heat and water vapor must ultimately be transferred by molecular diffusion across the interfacial sub-layer (Brutsaert, 1975; Garratt, 1992; Kantha and Clayson, 2000). However, many land and lake models, including four of the five algorithms used in this study, assume the same roughness length for momentum and heat transfer; for example, Croley (1989b; C89), Liu and Schwab (1987; LS87), Oleson et al. (2013), Zeng et al. (1998; Z98L), the CICE application (J99), the previous NCEP Eta model described in Chen et al. (1997), and the Canadian operational weather and hydrologic models described in Deacu et al. (2012). Deacu et al. (2012) showed that the same value for z0 and z0θ,q resulted in overestimation of turbulent heat fluxes over Lake Superior, and that the overestimation was reduced by using the smooth surface parameterization for z0θ,q, with an empirical coefficient based on Beljaars (1994).

As part of the current study, we intend to conduct a similar experiment to Deacu et al. (2012), namely, updating the original z0θ,q parameterization in the LS87, C89, Z98L, and J99 algorithms to a more realistic parameterization. We conduct this experiment to identify errors in λE and H simulations with these algorithms' original z0θ,q formulation and to evaluate how much the errors could be reduced in this way. We use an alternative z0θ,q formulation that is based on Fairall et al. (2003), which is used in COARE. The formulation utilizes the Liu–Katsaros–Businger model (LKB; Liu et al., 1980), with updates described in Fairall et al. (2003) where a simpler empirical relationship is formulated to represent the LKB model based on a fit to observational data; z0θ,q=min1.6×10-4,5.8×10-5Rr-0.72, where Rr = u*z0/ν is the roughness Reynolds number, which is also updated throughout the iterations. We test both the original and updated parameterizations for z0θ,q in the heat flux simulations.

The velocity of gustiness, wg, is included in Z89L and COARE to account for the additional flux induced by the convective boundary layer in low wind speed regimes. The average value of wind speed S is defined as S=U2+wg2, where U is the mean horizontal wind speed. wg is defined as wg=βgρaHcpT+0.61E, where β is an empirical constant set to β=1.2 in COARE and β=1.0 in Z89L. Further details of the gustiness velocity formulations are described by Fairall et al. (1996a). In LS87, C89, and J99, S is assumed to be identical to U.

All algorithms require meteorological inputs of horizontal wind speed U, potential air temperature θa, potential temperature at the water's surface θw, a humidity-related variable (dew point for LS87, relative humidity for C89, Z98L, and COARE, and specific humidity for J99), air pressure, and sensor height. These meteorological inputs should represent a temporal mean field over the corresponding eddy covariance measurement. U, θa, and θw can be directly used in Eqs. (1) and (2), while qw and qa need to be derived from relative humidity, water surface (or air) temperature, and air pressure.

The following steps were taken to compare and verify simulated sensible and latent heat fluxes against observed fluxes:

The five algorithms were forced by half-hourly meteorological data (U, θa, θw, relative humidity, and air pressure). Missing values were assigned for simulated heat fluxes when any observed values of U, θa, θw, and relative humidity were not available or when lake ice was present at a site.

Figure 3 shows the time series of air temperature, water surface temperature from GLSEA, relative humidity, and wind speed at the four stations. The time series for Stannard Rock were relatively gap-free throughout the 3 years, while there were some data gaps in the time series for the other stations. At all the stations, the air temperature time series were characterized by a typical seasonal cycle (Fig. 3a), with relatively warm and cold winters in 2012–2013 and 2013–2014, respectively. The winter of 2011–2012 was also very warm, but flux data from December 2011 were not analyzed as part of this study. During the two full winters (December–February) of 2012–2013 and 2013–2014, the mean surface air temperatures over the Great Lakes were -1.0 and -5.2 ∘C, respectively, while the long-term (1948–2014) mean was -2.4 ∘C for the same 3-month period. This was also reflected in the water surface temperature time series (Fig. 3b), where only White Shoal and Long Point were affected by ice cover in the winter (January–March) of 2012–2013, shown as gaps in the time series, whereas all four stations were affected by ice cover in the winter of 2013–2014. In addition to the preceding winter, the spring and summer months of 2012 were anomalously warm. The surface air temperature over the Great Lakes for April–September in 2012 was 16.4 ∘C, while the long-term (1948–2014) mean was 14.5 ∘C for the same months. This was also reflected in the 2012 summer water surface temperatures at the stations (Fig. 3b), which showed anomalously warm temperatures compared with the same periods during 2013 and 2014 (particularly at Stannard Rock). Relative humidity generally fluctuated between 50 % and 90 % (Fig. 3c), while wind speed (Fig. 3d) was characterized by a weak seasonal cycle of relatively high wind speeds during fall and winter (October–March) and lower wind speeds during spring and summer (April–September).

Figures 4–7 show visual comparisons of 10-day running mean time series of λE and H at each of the four stations. Overall, all five algorithms simulated the general seasonal cycles of λE and H, including the observed high fluxes during fall and winter and low fluxes during spring and summer, which is typical for large North American lakes (Blanken et al., 1997, 2000, 2011; Spence et al., 2011). On the other hand, there were notable overestimations of λE and H by the original algorithms, particularly at Stannard Rock (Fig. 4) in the fall (λE) and winter (H).

The observed λE and H at Stannard Rock were largely gap-free (Fig. 4), showing nearly continuous time series of seasonal cycles, aside from periods of high ice coverage during the cold winter of 2013–2014. A few additional data gaps also occurred, including in the late summer of 2012, a longer data gap during January–May 2014, and a very short data gap during December 2013.

At Stannard Rock, in the late fall (October–December), λE and H were relatively low in 2012 (3-month averages 84 W m-2 for λE and 55 W m-2 for H) and high in 2013 (119 W m-2 in λE, 85 W m-2 for H), indicating the preconditioning of the following mild and severe winters, respectively. During spring and summer of both years (April–September), the observed λE and H were much lower due to the cool lake surface relative to the overlying air. The simulated λE and/or H mostly reproduced these lower values and also showed occasional negative values (Fig. 4), such as during May 2012 and July 2014. During these periods, the air was predominantly warmer than the water's surface (i.e., Tw-Ta<0; Fig. 1), and specific humidity gradients were near zero during May 2012 and reversed (i.e., air-to-water) during July 2014, forcing the algorithms to simulate near-zero and negative (i.e., downward) fluxes, respectively. However, the observed λE and H fluxes remained close to zero but were slightly positive.

The forcing dataset for White Shoal (Fig. 3) was relatively gap-free as well, but there was a missing data period before October 2013 for λE and data gaps in H due to ice cover (Fig. 5). White Shoal tended to be influenced by ice cover even in mild winters, since typical southwesterly winds pushed ice in Lake Michigan downwind, causing ice accumulation in northern parts of the lake near White Shoal. As such, the exclusion of turbulent flux data for this analysis during the mild winter of 2012–2013 at White Shoal was due to ice cover. These observations also showed contrasting heat fluxes in the late fall during the 2 years; the 3-month average H was 40 W m-2 during October–December 2012 and 61 W m-2 during October–December 2013. Some model underestimation of the sensible heat flux (H) occurred during July–September 2013 and June–October 2014.

The Spectacle Reef forcing dataset (Fig. 3) and flux dataset (Fig. 6) both contained a long gap from March 2012 to September 2013 due to electrical problems from lightning strikes. A data gap in λE and H during January–March 2014 was due to ice cover, but unlike White Shoal, Spectacle Reef was less affected by ice cover. This was because winds carried ice that formed nearshore toward the east and offshore in Lake Huron, keeping the area around the flux tower largely in open water. Although ice cover did not affect the station in the winter of 2012–2013 (based on the NIC data), this period was included in the above-referenced long data gap due to electrical power issues.

The dataset at Long Point (Fig. 7) included the largest number of data gaps due to the additional filtering due to wind direction of 180∘ S–315∘ NW, which included typical southwesterly winds in this region. The significant data gaps at Spectacle Reef and Long Point, therefore, did not allow us to compare the fluxes in the late fall between the anomalous 2 years. However, for the purpose of the algorithm verification, the data at the two stations were still valuable, and forcing datasets were largely continuous (Fig. 3).

Figures 4–7 also show model results using both the original and updated z0θ,q parameterizations (Eq. 9). The original results of LS87, C89, Z98L, and J99 showed the overestimations of λE and H at Stannard Rock of anywhere from 33 %–50 % for most of the algorithms to ∼80 % overestimation for Z98L (both λE and H) and LS87 (λE) (Fig. 4, Table 2). These overestimations were particularly obvious during high flux events in fall and winter (October–March). The overestimation at Stannard Rock was significantly lessened to roughly 24 %–33 % error by using the updated z0θ,q formula (Eq. 9). This was consistent with the findings of Deacu et al. (2012), who showed improvements in latent and sensible heat flux simulation by updating the roughness length scale parameterization at Stannard Rock for the December 2008 simulation period. Similar improvements were found at White Shoal (Fig. 5), Spectacle Reef (Fig. 6), and Long Point (Fig. 7).

While the 10-day running mean time series of λE and H provided an effective way to illustrate the overall cycle (Figs. 4–7), abrupt changes in λE and H often occur on daily timescales, caused by the passage of frontal systems and cold-air outbreaks (Blanken et al., 2008). Thus, we further evaluated the performance of the various algorithms at daily timescales by means of scatter plots of observed and modeled daily mean heat fluxes (Figs. 8 and 9). Data points of λE (Fig. 8) diverged more from the 1:1 line than H (Fig. 9), showing both overestimated fluxes (at Stannard Rock and Long Point with Z98L) and underestimated fluxes (at Spectacle Reef). Overall, the updated z0θ,q formula reduced simulated λE, generally bringing the fluxes into better agreement with observations. An exception to this occurred for λE at Spectacle Reef, where the agreement became slightly worse with the updated formulation. The error reduction ratio of λE at Spectacle Reef was negative, and the mean bias was more negative with the updated formulation at this station (Table 2). This was also represented in the 10-day running mean time series (Fig. 6a and b). For H (Fig. 9, Table 3), notable overestimation was seen in the original J99, LS87, and Z98L, particularly at relatively large heat loss values (>∼300 W m-2). At Stannard Rock, Spectacle Reef, and Long Point, this overestimation was improved with the updated z0θ,q formula according to error reduction ratios (Table 3). At White Shoal, however, the improvement was not as significant, at 28 % compared to 58 % for Stannard Rock, 69 % for Spectacle Reef, and 50 % for Long Point. At Long Point, despite the notable error reduction, H was still overestimated in the high flux range.

Stannard Rock showed small groups of λE and H around the origin, where the simulated fluxes underestimated the observed fluxes (i.e., below the 1:1 line; Figs. 8 and 9). These data represented two summer periods when the observed fluxes were near zero, but the simulated fluxes were negative (Fig. 4; see the discussion in Sect. 3.1). At White Shoal, there was a population of H values below the 1:1 line (Fig. 9), representing periods when the simulation results underestimated the observations during July 2013–September 2013 and June 2014–October 2014 (Fig. 5c and d; see the discussion in Sect. 3.1).

Figures 10 and 11 show the magnitude of error in simulated daily λE and H (i.e., difference from observations) as functions of θw-θa, qw-qa, CH, CE, U, and ζ=z/L for the five algorithms at Stannard Rock. Similar results were observed in the error and bias analyses at the other sites (Figs. S1–S6 in the Supplement). There were several features common in all the algorithms; the λE (H) errors were positively correlated with qw-qa (θw-θa), especially with the original algorithms, the amplitudes of the errors became large (both positive and negative) as wind speed increased, and the majority of data were in the range -2<ζ<0 (unstable). Most notably, the transfer coefficients CH and CE were significantly reduced with the updated z0θ,q formula, which also reduced the error in the λE and H simulations. This was to be expected, since z0θ and z0q are directly translated into CH and CE, respectively. The study period did not include the occurrence of highly unstable conditions (ζ≪-1; Figs. 10, 11, and S1–S6); therefore, the period was not sufficient in evaluating the convective behavior treatment in COARE. Also, the study period did not include sustained low wind speeds. According to Fairall et al. (1996a), the gustiness parameterization has only a modest effect until the wind speed becomes less than 2–3 m s-1. The wind speeds during our study period were mostly greater than 3 m s-1 (Figs. 10, 11 and S1–S6) and, therefore, did not allow us to evaluate the influence of the gustiness parameterizations in COARE and Z98L.