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  • View in gallery

    (a) Correlation coefficients between the observed and calculated (by the VM and FG) sensible heat fluxes (W m−2), and (b) calculated MAE of the two methods.

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    As in Fig. 1, but for latent heat flux.

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    Observed sensible heat flux vs that calculated by the (a) VM and (b) FG approach for year 2010.

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    As in Fig. 3, but for latent heat flux.

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    Observed sensible heat flux vs that calculated by the (a) VM and (b) FG approach for January 2011.

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    As in Fig. 5, but for latent heat flux.

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    Sensible heat fluxes with an increase trend on Julian day 54 of 2010 and Julian day 103 of 2013 computed by the (a) VM and (b) FG approach.

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    As in Fig. 7, but for latent heat flux.

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    As in Fig. 7, but with a decrease trend on Julian day 277 of 2008 and Julian day 110 of 2013.

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    As in Fig. 9, but for latent heat flux.

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    As in Fig. 7, but with a mixing of increase and decrease trends on Julian days 353 and 343 of 2008.

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    As in Fig. 11, but for latent heat flux.

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    Monin–Obukhov length L (m) on Julian day 343 of year 2008 for the (a) VM and (b) FG approach.

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    Observed heat fluxes vs (a) sensible heat flux and (b) latent heat flux predicted by the operational coupled atmosphere–ocean GEM-NEMO model for the time period of November and December 2014.

  • View in gallery

    Observed heat fluxes vs (a) sensible heat flux and (b) latent heat flux computed by the VM for the time period of November and December 2014.

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    Comparisons between the coupled-model-predicted fluxes and the fluxes computed based on three buoy observations at (a) buoy 45001, (b) buoy 45004, and (c) buoy 45006 over Lake Superior during the time period of 16–18 Nov 2014, when lake-effect snowfall occurred.

  • View in gallery

    Spatial distribution of the sensible heat flux over Lake Superior at 0800 UTC 18 Nov 2014 based on (a) the VM computations using observations at three buoys (indicated by crosses located in the northern portion of the lake; from left to right they are buoys 45006, 45001, and 45004, respectively) and one eddy-covariance system at the Stannard Rock Light (denoted by a cross located in the southern portion of the lake), as well as (b) the coupled model predictions.

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Variational Computation of Sensible and Latent Heat Flux over Lake Superior

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  • 1 Meteorological Research Division, Environment and Climate Change Canada, Toronto, Ontario, Canada
  • | 2 National Hydrology Research Centre, Environment and Climate Change Canada, Saskatoon, Saskatchewan, Canada
  • | 3 Meteorological Research Division, Environment and Climate Change Canada, Dorval, Quebec, Canada
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Abstract

Sensible and latent heat fluxes over Lake Superior are computed using a variational approach with a Bowen ratio constraint and inputs of 7 years of half-hourly temporal resolution observations of hydrometeorological variables over the lake. In an advancement from previous work focusing on the sensible heat flux, in this work computations of the latent heat flux are required so that a new physical constraint of the Bowen ratio is introduced. Verifications are made possible for fluxes predicted by a Canadian operational coupled atmosphere–ocean model due to recent availabilities of observed and model-predicted fluxes over Lake Superior. The observed flux data with longer time periods and higher temporal resolution than those used in previous studies allows for the examination of detailed performances in computing these fluxes. Evaluations utilizing eddy-covariance measurements over Lake Superior show that the variational method yields higher correlations between computed and measured sensible and latent heat fluxes than a flux-gradient method. The variational method is more accurate than the flux-gradient method in computing these fluxes at annual, monthly, daily, and hourly time scales. Under both unstable and stable conditions, the variational method considerably reduces mean absolute errors produced by the flux-gradient approach in computing the fluxes. It is demonstrated with 2 months of data that the variational method obtains higher correlation coefficients between the observed and the computed sensible and latent heat fluxes than the coupled model predicted, and yields lower mean absolute errors than the coupled model. Furthermore, comparisons are made between the coupled-model-predicted fluxes and the fluxes computed based on three buoy observations over Lake Superior.

For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Dr. Zuohao Cao, zuohao.cao@canada.ca

Abstract

Sensible and latent heat fluxes over Lake Superior are computed using a variational approach with a Bowen ratio constraint and inputs of 7 years of half-hourly temporal resolution observations of hydrometeorological variables over the lake. In an advancement from previous work focusing on the sensible heat flux, in this work computations of the latent heat flux are required so that a new physical constraint of the Bowen ratio is introduced. Verifications are made possible for fluxes predicted by a Canadian operational coupled atmosphere–ocean model due to recent availabilities of observed and model-predicted fluxes over Lake Superior. The observed flux data with longer time periods and higher temporal resolution than those used in previous studies allows for the examination of detailed performances in computing these fluxes. Evaluations utilizing eddy-covariance measurements over Lake Superior show that the variational method yields higher correlations between computed and measured sensible and latent heat fluxes than a flux-gradient method. The variational method is more accurate than the flux-gradient method in computing these fluxes at annual, monthly, daily, and hourly time scales. Under both unstable and stable conditions, the variational method considerably reduces mean absolute errors produced by the flux-gradient approach in computing the fluxes. It is demonstrated with 2 months of data that the variational method obtains higher correlation coefficients between the observed and the computed sensible and latent heat fluxes than the coupled model predicted, and yields lower mean absolute errors than the coupled model. Furthermore, comparisons are made between the coupled-model-predicted fluxes and the fluxes computed based on three buoy observations over Lake Superior.

For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Dr. Zuohao Cao, zuohao.cao@canada.ca

1. Introduction

Turbulent latent and sensible heat fluxes are fundamental components for the water and energy balance in a coupled system of the atmosphere and its underlying surfaces such as over the Great Lakes. Precise representation of these fluxes is therefore important to better quantify the air–water interaction in the coupled system (e.g., Dupont et al. 2012; Deacu et al. 2012; Cao et al. 2002; Cao et al. 2004) and to improve model predictability ranging from cloud-resolving models to large-scale weather and climate models (e.g., Jiménez et al. 2012; ECMWF 2016a; Roberts et al. 2012). Such coupled modeling systems over the Great Lakes have been developed for years to better understand physical processes and better predict hydrometeorological variables in the coupled systems (e.g., Dupont et al. 2012; Deacu et al. 2012; Brassington et al. 2015). However, the flux computation in the coupled modeling systems still suffers from substantial inaccuracies due to influences of the surface roughness lengths, the stability functions, and wind speed, which are used in Monin–Obukhov similarity theory (MOST)-based algorithms (e.g., Dupont et al. 2012; Deacu et al. 2012; Gaspar et al. 1990; Blanc 1987, 1985). The MOST-based algorithms are also employed in current numerical weather prediction (NWP) models and general circulation models (GCMs) for the flux computations. The surface flux parameterization schemes in these models are mainly dependent on the surface roughness lengths (e.g., ECMWF 2016a; Noilhan and Mahfouf 1996), the stability functions (e.g., Jiménez et al. 2012), and wind speed (e.g., Richter et al. 2016).

Over a water surface, roughness length is highly dependent on the presence of the wave field, which is in turn dependent on the wind speed, fetch, current, and water depth (e.g., Charnock 1955; Smith 1988). Hence, there appears great disparity in the response of the NWP model-simulated high wind events to the roughness length schemes for heat fluxes (Bao et al. 2002). In high wind situations, it is also difficult to predict bulk flux coefficients, which successively determine the surface sensible and latent fluxes (Richter et al. 2016).

Jiménez et al. (2012) recently employed a revised scheme based on the work of Cheng and Brutsaert (2005) and Fairall et al. (1996) to provide a self-consistent formulation for the full range of atmospheric stability in the surface layer and to reduce or suppress the limits imposed on certain variables in an old scheme. Their applications of the revised scheme to the Weather Research and Forecasting (WRF) Model simulations lead to some improvements in near-surface meteorological variables particularly temperature at 2 m. However, substantial biases are still found in other simulated variables such as surface winds. Furthermore, there is almost no difference in simulated sensible and latent heat fluxes using the old and the new schemes (see Figs. 12c and 12e of Jiménez et al. 2012).

To reduce the inaccuracies of MOST-based flux computations, in this study we propose a variational method (VM) for computations of sensible and latent heat fluxes. In principle, the VM can adjust the computed flux toward the true value by minimizing the differences between the computed and the observed meteorological variables and by utilizing more observed meteorological information over the underlying surface than pure MOST (e.g., Xu and Qiu 1997; Cao and Ma 2005; Cao et al. 2006; Cao and Ma 2009). As an example, the roughness length can be updated utilizing the Charnock (1955) relationship once a new friction velocity is available from the variational iteration.

The objectives of this work are therefore to develop and apply the VM to compute sensible and latent heat fluxes over Lake Superior and to compare them with the MOST-based flux computations that are currently used in the coupled models as well as NWP and GCM models. With recent availabilities of the fluxes predicted by the coupled atmosphere–ocean model operated by the Canadian Meteorological Centre (CMC), we also compare the VM-computed sensible and latent heat fluxes with those directly output by the operational coupled model. Compared with our previous studies that focus on computations of the sensible heat flux only (Cao and Ma 2005; Cao et al. 2006; Cao and Ma 2009), in this study we have included computations of the latent heat flux as well as the sensible heat flux. By doing so, we need more physical constraints for the flux computations. This goal can be achieved through implementation of the variational computations with an additional physical constraint of the Bowen ratio (e.g., Rohli et al. 2004; Noilhan and Mahfouf 1996; ECMWF 2016a,b). Compared with the direct eddy-covariance flux data used in our previous study (Cao and Ma 2005; Cao et al. 2006; Cao and Ma 2009), the observed data used in this work are of longer time periods (7 years) and higher temporal resolution (30-min interval) than those used in the previous study (1–2 years and 1-h interval), which may describe more detailed flux variations.

Direct eddy-covariance observations of sensible and latent heat fluxes over the Great Lakes have become available only recently through the International Upper Great Lakes Study (2012). The eddy-covariance measurements of sensible and latent heat fluxes over the lake are often challenging particularly during the wintertime (e.g., Bourassa et al. 2013), and few direct measurements of these fluxes are available especially with a half-hourly frequency sampling rate (e.g., Spence et al. 2011, 2013; Blanken et al. 2011). In this study, the observed hydrometeorological variables over Lake Superior are employed to compute sensible and latent heat fluxes using the VM, whereas the direct eddy-covariance flux measurements are used to verify the VM, the conventional flux-gradient method (referred to as FG), and the fluxes computed by the coupled model. Usually, it takes time to process quality-controlled observational flux data while conventional measurements of wind, temperature, and moisture become available almost immediately. Hence, the variational computations of fluxes using the observed wind, temperature gradient, and moisture gradient can be utilized for a (near) real-time monitoring of sensible and latent heat fluxes even when the observed fluxes are not available yet. When direct eddy-covariance flux measurements are not available or missing but conventional observations of wind, the temperature gradient, and the moisture gradient are available, the VM-computed fluxes can be a complementary source of information on sensible and latent heat fluxes. Furthermore, the VM-computed fluxes can be employed for (near) real-time assessment of numerical model forecasts of fluxes.

The methods and datasets used in this work are described in sections 2 and 3. The detailed results and conclusions are, respectively, presented in sections 4 and 5.

2. Methodology

a. Flux-gradient relation

The heat flux can be computed based on the following coupled equations (e.g., Yaglom 1977):
e1
e2
e3
where u is the wind speed at height z, and and are the differences of temperature and specific humidity at two heights z and . Note that in Eqs. (2) and (3), roughness lengths for the temperature and the specific humidity should be and , respectively. However, under low and moderate wind (<20 m s−1) conditions over Lake Superior, replacing and in Eqs. (2) and (3) with momentum roughness has little effect on the computations of sensible and latent heat fluxes, as demonstrated in section 4. Therefore, rather than and is used in Eqs. (2) and (3) throughout this paper. The momentum roughness length over a water surface is a function of the friction velocity (Charnock 1955):
eq1
where the Charnock constant a varies from 0.011 (Smith 1980, 1988) to 0.035 (Table 4.1 of Garratt 1994) while the recommended value of the constant is between 0.014 and 0.0185 (e.g., Garratt 1994). In this work, a value of 0.018 is used. Parameter (= 0.4) is the von Kármán constant, is the friction velocity, and and are the flux temperature scale and the flux specific humidity scale, respectively. They are related to the sensible heat flux H and latent heat flux :
e4a
e4b
where ρ is the air density, cp is the specific heat of air at constant pressure, is the latent heat of evaporation, and E is the evaporation rate. In Eqs. (2)(4), Fh = − and Fq = −. Parameter L in Eqs. (1)(3) is the Monin–Obukhov length defined as
e5
where is the average temperature at two heights, and g is the gravitational acceleration. Moisture correction to L is small and negligible over the lake, as found in this work and our previous work (e.g., Cao et al. 2006). Parameters ψm, ψh, and ψq in Eqs. (1)(3) are integral forms of the departure of wind speed, temperature, and moisture from their neutral values, empirically expressed as
eq2
eq3
for unstable conditions (z/L < 0), where
eq4
and
eq5
for stable conditions (z/L > 0). The field-experiment-based constants γm, γh, βm, and βh are equal to 15, 9, 4.7, and 6.35, respectively (Businger et al. 1971).

If sensible and latent heat fluxes are computed using the FG relation, Eqs. (1)(3) (with inputs of wind speed, the temperature gradient, and the moisture gradient) and three unknown variables , Fh, and Fq consist of a well-determined system where , Fh, and Fq can be solved by an iterative procedure.

b. Flux computation using the variational method

In our previous work (e.g., Cao et al. 2006), two variables of the friction velocity and sensible heat flux are computed using a variational approach with three physical constraints [i.e., Eqs. (1)(3)], in which an overdetermined system is formed. The overdetermined system refers to the system in which the numbers of physical constraints are more than the numbers of unknown variables. Usually, the variational performance is better in overdetermined systems (e.g., Xu and Qiu 1997; Cao et al. 2006). In contrast to our previous work (e.g., Cao et al. 2006), in this study computations of the latent heat flux are included in addition to the friction velocity and sensible heat flux so that other physical constraints are required besides the three physical constraints.

One of the physical constraints could be associated with the surface energy balance
e6
where Rn, H, , and G are, respectively, net radiation, sensible heat flux, latent heat flux, and ground heat flux at a water surface. Parameter G is not available in this study since it is not measured. Because of this, the constraint of the surface energy balance cannot be applied. An alternative physical constraint may be a Bowen ratio B (e.g., Rohli et al. 2004), defined as the ratio of sensible heat flux to latent heat flux:
e7
Using Eq. (4) and assuming that the turbulent diffusivities for temperature and specific humidity are equal (e.g., Philip 1987; Högström 1996), the Bowen ratio B can be written as
e8

In addition to the individual constraint of T and q, the ratio of T and q (proportionally to Bowen ratio B) needs to be constrained, because an inaccurate Bowen ratio could get a wrong sign for the turbulent fluxes and could lead to an inaccurate magnitude in the flux computation (see appendix B for details). Hence, the Bowen ratio B constraint is beneficial for better and consistent flux computations in sign and magnitude.

In the variational framework, Eqs. (1)(3) and (8), together with the three unknown variables of , Fh, and Fq, form an overdetermined system. Because the VM makes use of the additional information of the Bowen ratio in the overdetermined system, it is anticipated to improve the FG in terms of sensible and latent heat flux computation. It is noted that the overdetermined system can be employed in the VM but not in the MOST-based flux-gradient method (Cao et al. 2006; Cao and Ma 2009).

As mentioned earlier, the goal of the VM is to minimize a weighted aggregate of the differences between the calculated and observed wind speed, temperature gradient, moisture gradient, and Bowen ratio, defined as the following cost function J:
e9
where Wu, WT, Wq, and WB are weights for the wind speed, the temperature gradient, the moisture gradient, and the Bowen ratio, respectively, which are generally chosen to be inversely proportional to their respective maximally tolerated observation error variances (e.g., Cao and Ma 2005; Cao et al. 2006; Cao and Ma 2009). Because of uncertainties in measurement errors, the weights are usually difficult to determine precisely, and they can be specified empirically and even arbitrarily (Daley 1996). The estimated weights in this work are set as follows: Wu = 5 m−2 s2, WT = 65 K−2, Wq = 5 kg−2 kg2, and WB = 0.1. As demonstrated in section 4b, the variational computations are not very sensitive to the choice of the weights.

In Eq. (9), uobs, , and are the observed wind speed, temperature gradient, and moisture gradient at height z, whereas u, , and are calculated based on Eqs. (1)(3). The observed Bowen ratio is whereas the calculated Bowen ratio is .

In the variational framework, the cost function J, which is considered as the function of , Fh, and Fq, is minimized to gain optimal estimates of , Fh, and Fq. This requires that the gradients of J with respect to the unknown variables , Fh, and Fq approach zero, which can be approximately expressed as
e10
The detailed analytic formulations of gradient components are presented in appendix A. Similar to Cao et al. (2006), a quasi-Newton method is employed to search the minimum of the cost function J. Listed below is the iterative procedure to calculate the cost function and its gradient and to derive , Fh, and Fq:
  1. Set up initial guesses of unknowns of , Fh, and Fq, say, = 0.3 m s−1, Fh = 0.03 K m s−1, and Fq = 0.1 kg kg−1 m s−1.
  2. Calculate L from Eq. (5).
  3. Calculate u, ∆T, ∆q, and B from Eqs. (1)(3) and (8).
  4. Calculate the cost function in Eq. (9) and its gradients with respect to from Eqs. (A1) and (A4)(A6), with respect to Fh from Eqs. (A2) and (A7)(A10), and with respect to Fq from Eqs. (A3), (A11), and (A12).
  5. Perform the quasi-Newton method to search for zeros of the gradients of J so as to minimize the cost function J; the optimal values of , Fh, and Fq are obtained once the cost function is minimized.
  6. After new values of , Fh, and Fq are obtained, the roughness length is updated using Charnock’s relationship. Repeat steps 2–6 until the procedure converges.
The convergence criterion and requirement are the same as those in Cao et al. (2006). The computations are performed every 30 min with the inputs of observed uobs, , , and and the outputs of calculated u, ∆T, ∆q, and B and estimated , Fh, and Fq.

c. Flux computation using a coupled atmosphere–ocean model

Sensible and latent heat flux forecasts are available at CMC since November 2014 from the coupled atmosphere–ocean model (GEM-NEMO) component of the Water Cycle Prediction System for the Great Lakes and St. Lawrence (WCPS-GLS; Durnford et al. 2018). Here the coupled GEM-NEMO refers to the Global Environmental Multiscale (GEM) regional model (Côte et al. 1998) and Nucleus for European Modeling of the Ocean (NEMO; Madec et al. 1998). Uncoupled GEM turbulent flux forecasts were evaluated by Deacu et al. (2012) at Stannard Rock, Lake Superior. In this study, we verify the forecasts from the coupled GEM-NEMO system at the same location with recently available observations.

There are three options for flux computations in NEMO: 1) the Coordinated Ocean–Ice Reference Experiments (CORE) bulk formula, which requires inputs of x and y components of 10-m air velocity, 10-m air temperature, specific humidity, incoming longwave and shortwave radiation, total precipitation (liquid and solid), and solid precipitation (snow); 2) the Coupled Large-Scale Ice–Ocean (CLIO) bulk formula, which requires inputs of x and y components of an ocean stress, wind speed module, 10-m air temperature, specific humidity, cloud cover, total precipitation (liquid and solid), and solid precipitation (snow); and 3) the Mediterranean Forecasting System (MFS) bulk formula, which requires inputs of zonal and meridional components of 10-m wind, total cloud cover, 2-m air temperature, 2-m dewpoint temperature, total precipitation, and mean sea level pressure. Refer to Wang et al. (2015), Lebeaupin Brossier et al. (2015), and Madec et al. (2016) for further details.

d. Assessment method

Performance of abovementioned approaches in computing sensible and latent heat fluxes is evaluated using a correlation coefficient and a mean absolute error. The correlation coefficient, denoted by rxy, is a measure of the strength and the direction of a linear relationship between two variables x and y. The mean absolute error (MAE) is defined as
eq6
where and are ith (i = 1, …, N, where N is the number of the total time level of measurements) calculated and observed sensible and latent heat fluxes.

3. Data

Observations of hydrometeorological variables were made at the Stannard Rock Light (47.183°N, 87.225°W) of Lake Superior during the time period from June 2008 to December 2014 (Blanken et al. 2011; Spence et al. 2011, 2013). This site is 39 km from the nearest shore and 32.8 m above the mean water surface. All the measurements were processed over a 30-min interval with a sampling rate of 10 Hz. Sensible and latent heat fluxes were directly measured using an eddy-covariance system composed of a Campbell Scientific CSAT-3 sonic anemometer and a Licor LI-7500 open-path CO2/H2O gas analyzer deployed at the top of the Stannard Rock lighthouse at a height of 32.8 m above the mean water surface. At the same height, wind speed u and direction were measured using an R. M. Young wind sensor and air temperature T and vapor pressure e were measured with a Vaisala thermohygrometer. The surface water temperature To was measured using an Apogee infrared thermometer (IRT), and the saturation specific humidity qo was calculated based on To, assuming the air immediately above the water was saturated. When the IRT measured obviously erroneous values due to frost in the orifice, To was derived from the Great Lakes Surface Environmental Analysis (http://coastwatch.glerl.noaa.gov) obtained from the National Oceanic and Atmospheric Administration (NOAA) polar-orbiting satellite mounted Advanced Very High Resolution Radiometers. With measurements at two heights, ∆T (= T − To) and ∆q (= qqo) can be readily obtained, where q is computed based on measured temperature and vapor pressure. A detailed description of these measurements and their quality control are given in Blanken et al. (2011) and Spence et al. (2011, 2013).

Besides the data quality control performed in Blanken et al. (2011) and Spence et al. (2011, 2013), additional caution and quality control need to be taken when the Bowen ratio method is applied (e.g., Ohmura 1982; Bertela 1989; see appendix B).

Furthermore, three buoy observations over Lake Superior, operated by the National Data Buoy Center (NDBC), are utilized to evaluate the spatial distribution of heat fluxes over the lake. The observations at three buoy stations and one eddy-covariance system located at the Stannard Rock Light are used by the VM to compute fluxes. These fluxes are then interpolated over Lake Superior using a kriging method.

4. Computations of sensible and latent heat flux

a. Comparison of the variational method with the flux-gradient method

1) Interannual variability

The correlation coefficients between observed and VM-calculated sensible heat fluxes, as shown in Fig. 1a, are greater than those of the FG in all 7 years. During the time period of 2008–14, the correlation coefficients for the VM vary from 0.77 to 0.95, whereas in the FG, the correlation coefficients change from 0.29 to 0.80, with the best performance in 2010. The VM improvements in the correlation coefficients range from 0.15 to 0.48.

Fig. 1.
Fig. 1.

(a) Correlation coefficients between the observed and calculated (by the VM and FG) sensible heat fluxes (W m−2), and (b) calculated MAE of the two methods.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

As demonstrated in Fig. 1b, MAE in the VM-computed sensible heat fluxes is smaller than those in the FG for all 7 years. The FG has the MAE varying from 38 to 66 W m−2, with an average about 52 W m−2, whereas in the variational computation, the MAE changes from 25 to 41 W m−2, with an average around 30 W m−2. The average MAE is reduced more than 20 W m−2 when the VM is applied.

Different from our previous studies in which the eddy-covariance flux data are usually available for 1–2 years with a temporal resolution of about 1 h (Cao and Ma 2005; Cao et al. 2006; Cao and Ma 2009), the observed data used in this work are available for 7 years with a temporal resolution of 30 min. This allows us to comprehensively examine the performance of the two methods in computing the sensible heat flux under unstable and stable conditions, respectively, for all 7 years. As shown in Table 1, the major improvements of the VM over the FG are to reduce MAE. The FG-computed sensible heat fluxes have MAEs of 63 and 31 W m−2 under unstable and stable conditions, respectively, which are significantly reduced to 36 and 15 W m−2 when the VM is employed. Under unstable conditions, the VM-computed sensible heat flux agrees well with the observed flux with a correlation coefficient of 0.78, whereas the FG reaches a correlation coefficient of 0.53. For stable conditions, both methods have a similar and low correlation coefficient with the observed flux, although the VM makes improvements in reducing MAE by about 50%.

Table 1.

MAE in computing the sensible and latent heat fluxes (W m−2) under unstable and stable conditions for all 7 years.

Table 1.

Computations of latent heat flux are not carried out in our previous work (Cao and Ma 2005; Cao et al. 2006; Cao and Ma 2009) but performed in this study. As demonstrated in Fig. 2, the latent heat fluxes have also shown interannual variations in both computations and observations over the 7 years (2008–14). The correlation coefficients between the observed and the VM-calculated latent heat fluxes are greater than these for the FG in all 7 years (Fig. 2a). In the variational computation, the correlation coefficients vary from 0.62 to 0.85, with the maximum value of 0.85 occurring in 2010, whereas in the FG, the correlation coefficients change from 0.4 to 0.75, with the maximum value of 0.75 occurring in 2010 as well. The VM improvements in the correlation coefficients range from 0.10 to 0.36.

Fig. 2.
Fig. 2.

As in Fig. 1, but for latent heat flux.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

As shown in Fig. 2b, during all 7 years the MAE in the VM-computed latent heat fluxes is smaller than the one of the FG. For the variational computation, the MAE varies from 28 to 45 W m−2, with an average around 37 W m−2, whereas in the flux-gradient computation, the MAE changes from 47 to 85 W m−2, with an average about 64 W m−2. The average MAE is reduced about 27 W m−2 when the VM is applied. These analyses show that the VM considerably improves the FG in computation of the latent heat flux.

The performance of the two methods in computing the latent heat flux is also evaluated with respect to unstable and stable conditions over 7 years. Similarly, the main improvements of the VM over the FG are to reduce MAE. The FG-calculated latent heat fluxes yield MAEs of 69 and 57 W m−2 for unstable and stable conditions, respectively, which are reduced to 39 and 30 W m−2 once the VM is applied (Table 1). Under unstable conditions, the VM-computed latent heat flux agrees well with the observed flux with a correlation coefficient of 0.78, whereas the FG gets a correlation coefficient of 0.57. On the other hand, under stable conditions, both methods obtain a similar and low correlation coefficient with the observed flux.

In the best performance year of 2010, the VM gains the highest correlation coefficient of 0.95 between the observed and the calculated sensible heat flux and the MAE of 28 W m−2, whereas the FG obtains the correlation coefficient of 0.80 and the MAE of 50 W m−2 (Fig. 3). The agreements between the measured and the VM-computed sensible heat flux are better than the FG. Because the correlation points are mostly distributed above the diagonal line, the FG underestimates the sensible heat flux under unstable conditions (Fig. 3b). Furthermore, in the FG, many points deviate from the diagonal line when the measured flux is small (Fig. 3b).

Fig. 3.
Fig. 3.

Observed sensible heat flux vs that calculated by the (a) VM and (b) FG approach for year 2010.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

Similarly, the FG underestimates the observed latent heat flux in the year 2010 since the correlation points are mostly distributed above the diagonal line under unstable conditions (Fig. 4b). Also, numerous points depart from the diagonal line in the FG computation for 2010, especially when the measured latent heat flux is small (Fig. 4b). When the VM is employed, the MAE in computing the latent heat flux is reduced from 47 W m−2 (in the FG) to 30 W m−2 (in the VM), whereas the correlation coefficient increases from 0.75 (in the FG) to 0.85 (in the VM) (cf. Figs. 4a,b).

Fig. 4.
Fig. 4.

As in Fig. 3, but for latent heat flux.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

2) Monthly variation

Usually, the sensible heat flux exhibits relative large values during winter months due to large temperature differences in the vertical direction. Figure 5 shows that the observed sensible heat flux reaches about 400 W m−2 in January 2011. The MAE in VM-computed sensible heat flux is about 50 W m−2 (Fig. 5a), whereas the one in the flux-gradient calculation is about 100 W m−2 (Fig. 5b).

Fig. 5.
Fig. 5.

Observed sensible heat flux vs that calculated by the (a) VM and (b) FG approach for January 2011.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

A similar situation occurs for latent heat flux computations in January 2011. As shown in Fig. 6, the MAE in the VM-computed latent heat flux is about 29 W m−2 (Fig. 6a), while the MAE in the flux-gradient-computed latent heat flux is 72 W m−2 (Fig. 6b).

Fig. 6.
Fig. 6.

As in Fig. 5, but for latent heat flux.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

3) Daily and hourly variation

Flux variations at time scales of hours and a day are usually associated with rapid changes of weather and environment conditions and sometimes are associated with severe weather such as lake-effect snowstorms. Examination of flux computations at those time scales can provide information for seamless weather and environment predictions.

Based on the observations measured over Lake Superior, more than 75% of observed meteorological environments were under unstable conditions. As demonstrated in Table 1, the large error caused by the FG occurs under the unstable atmospheric conditions where the VM substantially improves the FG in the computation of the sensible heat flux. The errors also occur under stable conditions for the flux-gradient-computed sensible heat flux, when the fluxes themselves under stable conditions are usually small in magnitude (e.g., Poulos and Burns 2003). Because of these, in the rest of this subsection we will focus on the unstable conditions for daily evolutions of computed and observed fluxes. Since some of the data were eliminated through the quality control procedure, the results shown here are the samples of typical daily evolutions that passed quality control. There are typically three types of daily evolutions characterized with 1) a tendency of increase, 2) a tendency of decrease, and 3) a mixing of cases 1 and 2.

Evolution of the sensible heat flux on Julian day 54 of year 2010 and Julian day 103 of year 2013 was featured with a tendency of increase. As seen in Fig. 7, the observed increase trends of sensible heat fluxes on these two days are captured by both methods. The VM-computed sensible heat fluxes have MAEs of 14 and 8 W m−2 (Fig. 7a), whereas the FG shows MAEs of 20 and 13 W m−2, respectively (Fig. 7b), indicating that the former is more accurate than the latter in computing the sensible heat flux. Consistent with the finding presented in Table 1, the underestimation error caused by the FG mostly occurs under unstable conditions.

Fig. 7.
Fig. 7.

Sensible heat fluxes with an increase trend on Julian day 54 of 2010 and Julian day 103 of 2013 computed by the (a) VM and (b) FG approach.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

Daily variations of latent heat fluxes on Julian day 54 of year 2010 and Julian day 103 of year 2013 were representative of the first type of daily evolution with an increase tendency in the observed latent heat flux. Both the VM and the FG well describe the increasing trends in observed latent heat fluxes (Fig. 8). The VM is more accurate than the FG. The former has MAEs of 20 and 10 W m−2 for Julian day 54 of year 2010 and Julian day 103 of year 2013, respectively, whereas the latter has MAEs of 27 and 13 W m−2, respectively.

Fig. 8.
Fig. 8.

As in Fig. 7, but for latent heat flux.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

Figure 9 illustrates the second type of observed sensible heat flux with a decrease tendency, such as on Julian day 277 of year 2008 and Julian day 110 of year 2013. These decreasing tendencies are well described by both methods. The VM-computed sensible heat flux agrees with the observed flux magnitude better than the FG. The former has MAEs of 26 and 41 W m−2, whereas the latter has MAEs of 47 and 71 W m−2 on those two days, respectively.

Fig. 9.
Fig. 9.

As in Fig. 7, but with a decrease trend on Julian day 277 of 2008 and Julian day 110 of 2013.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

When a decrease tendency in the latent heat flux is present, the flux-gradient calculations show larger MAEs in the latent heat flux such as on Julian day 277 of year 2008 and Julian day 110 of year 2013 (Fig. 10). The FG-computed MAEs are 138 and 40 W m−2 on these two days, which are reduced to 61 and 33 W m−2, respectively, when the VM is applied (Fig. 10).

Fig. 10.
Fig. 10.

As in Fig. 9, but for latent heat flux.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

The third type of observed sensible heat flux is characterized with a combination of increase and decrease tendencies as shown in Fig. 11 for Julian days 353 and 343 of year 2008. On Julian day 353 of 2008, the observed sensible heat flux decreased with time in the morning and then slightly oscillated around a value between about 80 and 150 W m−2. On Julian day 343 of 2008, the daily evolutions of the observed sensible heat fluxes were more complicated: the flux decreased from about 150 to 50 W m−2 in the early morning, then increased to about 140 W m−2 in the early afternoon, and finally decreased to about 20 W m−2 in the rest of the afternoon. The VM captures these complicated daily variations very well (Fig. 11a) while the FG has difficulty depicting these tendencies, especially for Julian day 343 of year 2008 (Fig. 11b). Also, the VM-computed sensible heat flux well agrees with the observed flux magnitude except for the first few data points (Fig. 11a), whereas the FG considerably underestimates the magnitude of the observed fluxes for both days (Fig. 11b). The large MAEs are observed in the FG-calculated sensible heat fluxes (103 and 90 W m−2 for Julian days 353 and 343, see Fig. 11b), which are reduced about 2–3 times by the variational computation to 33 and 34 W m−2 for these two days, respectively (Fig. 11a).

Fig. 11.
Fig. 11.

As in Fig. 7, but with a mixing of increase and decrease trends on Julian days 353 and 343 of 2008.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

The FG faces a challenge in computing the latent heat flux when the daily evolution of the observed latent heat flux is characterized with a combination of increase and decrease tendencies such as on Julian day 353 of year 2008 and Julian day 343 of year 2008. As shown in Fig. 12b, the FG is unable to simulate the diurnal variation of the latent heat flux, especially when it experiences multiple ups and downs in daily variations of the latent heat flux. However, the VM captures the observed latent heat flux very well in terms of tendencies and magnitudes (Fig. 12a). The MAEs in the variational computations for Julian day 353 of year 2008 and Julian day 343 of year 2008 are 25 and 19 W m−2, while the MAEs in the flux-gradient calculations are 101 and 80 W m−2. As a result, the VM reduces the large error present in the FG by about a factor of 4. The substantial underestimation of fluxes in the FG is mainly caused by miscalculating an unstable Monin–Obukhov length L (negative) as a stable L (positive) (e.g., see Fig. 13), resulting in small magnitudes in fluxes under stable conditions (e.g., Strub and Powell 1987; Halliwell and Rouse 1989). This sign mismatch of L in the flux-gradient calculation is also observed in Cao et al. (2006).

Fig. 12.
Fig. 12.

As in Fig. 11, but for latent heat flux.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

Fig. 13.
Fig. 13.

Monin–Obukhov length L (m) on Julian day 343 of year 2008 for the (a) VM and (b) FG approach.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

b. Parameter experiments for the variational method

1) Weights

We have performed sensitivity experiments to investigate the effects of the weights in the cost function on the VM-computed sensible and latent heat flux. As shown in Table 2, the original values of four weights (column 1) have been perturbed so that the effects of these weights on the VM-computed sensible and latent heat fluxes can be assessed. The changes in the values of four weights also represent observational errors that are brought in the cost function (e.g., Cao and Ma 2005; Cao et al. 2006). As shown in Table 3, the correlation coefficients of the VM-computed sensible heat flux (using all 7 years of data) for five experiments range from 0.79 to 0.80, which is higher than that of the FG (0.48). The MAE of the VM-computed sensible heat flux (based on all 7 years of data) varies from 31 to 38 W m−2, which is smaller than that of the FG (46 W m−2). Similarly, based on all 7 years of data, the correlation coefficients for the VM-computed latent heat flux (0.72–0.75, see Table 4) are higher than that of the flux-gradient calculated (0.47), whereas the MAE for the VM-computed latent heat flux (38–52 W m−2, see Table 4) is smaller than that of the FG (64 W m−2). Consistent with Cao and Ma (2005) and Cao et al. (2006), our results suggest that the variational computations are not very sensitive to the choice of the weights.

Table 2.

List of experiments and parameters of Wu (m−2 s2), WT (K−2), Wq (kg−2 kg2), and Wb.

Table 2.
Table 3.

Correlation coefficients rxy between the observed sensible heat fluxes (W m−2) and those calculated in the experiments (see Table 2) and MAE in computing the sensible heat fluxes for all 7 years.

Table 3.
Table 4.

As in Table 3, but for the latent heat flux

Table 4.

2) Roughness lengths

To understand the effect of roughness lengths for the temperature and specific humidity on the computations of the sensible and latent heat flux, we employ and in Eqs. (2) and (3), respectively, to replace . The formulation for and is the same as the one used in Deacu et al. (2012):
eq7

where (= 1.5 × 10−5 m2 s−1) is the kinematic viscosity of air. The formulation for the condition of > 0.2 m s−1 is taken from ECMWF (2016a, 2016b), except for the coefficient 1.333. Our results indicate that the computations of sensible and latent heat fluxes are not sensitive to and . The main reason that the computed heat fluxes are not sensitive to choices of these two roughness lengths and is due to low and moderate wind (<20 m s−1) conditions over Lake Superior, rather than high wind (>25 m s−1) conditions over the ocean (e.g., Bao et al. 2002). These results are also consistent with our previous work (Cao et al. 2006), in which and are specified as a function of and the roughness Reynolds number (= ) (e.g., Garratt 1994; Brutsaert 1982).

c. Comparison of the variational method with the coupled model

Comparisons between the observed fluxes and the fluxes predicted by the operational coupled atmosphere–ocean model GEM-NEMO were made possible until recently due to availabilities of the direct eddy-covariance measured fluxes and the outputs of the coupled-model-predicted fluxes over Lake Superior. GEM-NEMO forecasts of sensible and latent heat fluxes are available from November 2014 and onward, and the observed fluxes at Stannard Rock Light of Lake Superior were available up to 2014 at the time this study was performed. As a result, we were able to make comparisons for the time period of November and December 2014.

Figure 13 shows the comparisons between the observed and the coupled-model-predicted sensible and latent heat fluxes. As demonstrated in Fig. 14, the correlation coefficients between the observed and the coupled model are 0.78 and 0.59 for the sensible and latent heat fluxes, respectively. Correspondingly, the MAEs in the coupled-model-predicted sensible and latent heat fluxes are 32 and 43 W m−2 (Fig. 14). Since the correlation points are mainly located below the diagonal line, the coupled model overestimates both sensible and latent heat fluxes (Fig. 14).

Fig. 14.
Fig. 14.

Observed heat fluxes vs (a) sensible heat flux and (b) latent heat flux predicted by the operational coupled atmosphere–ocean GEM-NEMO model for the time period of November and December 2014.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

The performance of the VM has been also evaluated for the same time period. As shown in Fig. 15, the correlation coefficients for the VM-computed sensible and latent heat fluxes are 0.91 and 0.83, respectively. Compared with the coupled model, the VM improvements in the correlation coefficients are 0.13 and 0.24, respectively, for sensible and latent heat fluxes. The MAEs in the VM-computed sensible and latent heat fluxes are 23 and 39 W m−2 (Fig. 15). Hence, the variational computations have smaller MAEs in the sensible and latent heat flux than the coupled model.

Fig. 15.
Fig. 15.

Observed heat fluxes vs (a) sensible heat flux and (b) latent heat flux computed by the VM for the time period of November and December 2014.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

In addition, the FG-computed sensible and latent heat fluxes have been assessed for the same time period. The correlation coefficients in the FG-calculated sensible and latent heat fluxes are 0.79 and 0.77, which are higher than the coupled predicted ones, whereas the MAEs in the FG-calculated sensible and latent heat fluxes are 60 and 65 W m−2, which are larger than the coupled predicted ones. The FG underestimates both sensible and latent heat fluxes.

d. Flux comparison at three buoy stations over Lake Superior

To estimate the spatial distribution of the heat flux over the lake, we have looked for all routine observations in and around Lake Superior, including three buoy stations operated by NDBC and stations along shorelines and at islands. For the stations along shorelines and at islands, only a one-layer measurement is available, so they cannot be used for flux computations due to requirements of two-layer measurements. For the buoy observations, there are two-layer measurements of air and water temperatures whereas there is no moisture (or dewpoint temperature) observation, suggesting that we can employ the VM to compute the sensible heat flux but not the latent heat flux. Over the time period of November and December 2014, the three buoy stations cover observations only for part of November but not for December 2014. Note that the Michigan Technological University station covers observations only until 5 October 2014, when the coupled-model-predicted fluxes were not yet available. Figure 16 presents comparisons between the coupled-model-predicted sensible heat fluxes and the fluxes computed by the VM based on the three buoy observations over Lake Superior during the time period of 16–18 November 2014, when lake-effect snowfall occurred. The fluxes reached a peak value around 0800 UTC 18 November 2014. Of the three buoy stations, buoy 45004 (47.585°N, 86.585°W) is located at the closest site to the Stannard Rock Light (47.183°N, 87.225°W), where the direct eddy-covariance measurements of sensible heat fluxes are available. Buoy 45001 (48.061°N, 87.793°W) and buoy 45006 (47.335°N, 89.793°W) are located roughly north and northwest of the Stannard Rock Light (47.183°N, 87.225°W). As shown in Fig. 16b, the coupled-model-predicted sensible heat flux has an MAE of 46 W m−2 (with a bias of 45 W m−2) in comparison with the sensible heat flux from buoy 45004. This indicates that the coupled model overestimates sensible heat fluxes at the location of buoy 45004, which is consistent with the flux computations at the Stannard Rock Light. On the other hand, the coupled-model-predicted sensible heat fluxes at the locations of buoy 45001 and buoy 45006 have MAEs of 41 W m−2 (with a bias of −3 W m−2) and 61 W m−2 (with a bias of −14 W m−2), respectively, compared with the sensible heat flux based on buoys 45001 and 45006. This suggests that the coupled model underestimates sensible heat fluxes at these two locations.

Fig. 16.
Fig. 16.

Comparisons between the coupled-model-predicted fluxes and the fluxes computed based on three buoy observations at (a) buoy 45001, (b) buoy 45004, and (c) buoy 45006 over Lake Superior during the time period of 16–18 Nov 2014, when lake-effect snowfall occurred.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

Figure 17a presents the spatial distribution of the sensible heat flux over Lake Superior interpolated based on the VM-computed fluxes using observations at three buoy stations and one eddy-covariance system (denoted by crosses), and Fig. 17b presents the coupled-model-predicted sensible heat flux. At 0800 UTC 18 November 2014, both the observation-based and coupled-model-predicted sensible heat fluxes mainly show a decrease of the flux from north to south over Lake Superior. Different from the observation-based fluxes (Fig. 17a), the coupled-model-predicted fluxes display an east–west gradient essentially located in the south and southeast part of Lake Superior (Fig. 17b). Over the north portion of the lake, particularly northwest part of the lake, the model-predicted fluxes are smaller than the observed ones (cf. Figs. 17a,b). The model underestimation in the sensible heat flux also occurs on the southeast corner of the lake. On the other hand, at and to the south of the eddy-covariance system located at the Stannard Rock Light, the model-predicted flux is overestimated. Hence, the VM can be useful for calibrating and/or improving computations of heat fluxes of the coupled model.

Fig. 17.
Fig. 17.

Spatial distribution of the sensible heat flux over Lake Superior at 0800 UTC 18 Nov 2014 based on (a) the VM computations using observations at three buoys (indicated by crosses located in the northern portion of the lake; from left to right they are buoys 45006, 45001, and 45004, respectively) and one eddy-covariance system at the Stannard Rock Light (denoted by a cross located in the southern portion of the lake), as well as (b) the coupled model predictions.

Citation: Journal of Hydrometeorology 19, 2; 10.1175/JHM-D-17-0157.1

5. Conclusions

The variational method is employed for the computation of sensible and latent heat fluxes over Lake Superior owing to recent availability of high temporal resolution hydrometeorological observations over the lake, especially the direct eddy-covariance measurements. The VM takes advantage of utilizing these half-hourly measured data as well as the existing MOST and Bowen ratio information to improve the conventional FG method in computing sensible and latent heat fluxes over Lake Superior.

The computations based on 7 years (2008–14) of half-hourly temporal resolution observations show that the VM yields much higher correlations between the computed and the direct eddy-covariance-measured sensible and latent heat fluxes than the FG. It is demonstrated that the variational computations in sensible and latent heat fluxes are more accurate than the FG at scales of a year, month, day, and hour. For both unstable and stable conditions, the VM substantially reduces the MAE caused by the MOST-based FG in computing sensible and latent heat fluxes.

Sensitivity experiments carried out in this work demonstrate that 1) the variational computations are not sensitive to the choice of the weights and 2) the moisture correction to the Monin–Obukhov length and the roughness lengths for the temperature and the specific humidity have no impacts on the variational computation of sensible and latent heat fluxes over Lake Superior.

Verifications for the coupled atmosphere–ocean model predicted sensible and latent heat fluxes are made using currently available direct eddy-covariance measurements over Lake Superior. The results for the 2-month period (in November and December 2014) show that for the correlation coefficients between the observed and the computed sensible and latent heat fluxes, the VM gains the highest, the FG obtains the second, and the coupled model is the lowest. As for MAEs in computations of sensible and latent heat fluxes, the VM has the lowest, and the coupled model is the second lowest. The FG gets the highest MAEs for both sensible and latent heat fluxes.

Comparisons are also made between the coupled-model-predicted sensible heat fluxes and the fluxes computed based on three buoy observations over Lake Superior.

It is anticipated that the VM can be used to monitor the fluxes in a real-time manner since it usually takes time to process quality-controlled observational flux data, to verify and calibrate model-predicted fluxes in (near) real time, and to develop surface flux datasets, especially when direct eddy-covariance measurements are missing but conventional observations of the wind, temperature gradient, and moisture gradient are available.

Acknowledgments

We thank Stephane Bélair, Pierre Pellerin, and Ralf Staebler for their constructive suggestions and discussions. We also appreciate Dr. Christa D. Peters-Lidard, Editor in Chief, and three anonymous reviewers for their comments and suggestions to improve the quality of this paper.

APPENDIX A

Formulations for the Gradient Components

The analytic expression of the gradient components in Eq. (10) is presented in this appendix. From Eq. (9), we yield
ea1
ea2
ea3
The derivative of the wind speed with respect to is obtained from Eq. (1):
ea4
where
eq8
eq9
The derivatives of ψm with respect to at can be deduced in the same manner.
According to Eq. (2), the derivative of with respect to can be formulated as
ea5
where
eq10
eq11
The derivative of at with respect to can be deduced in the same manner.
The derivative of the specific humidity with respect to can be gained from Eq. (3):
ea6
where
eq12
eq13
The derivative of at with respect to can be deduced in the same manner.
The derivatives of u, , and with respect to the heat flux Fh are given by
ea7
where
eq14
eq15
ea8
where
eq16
eq17
and
ea9
where
eq18
eq19
The derivatives of , , and with respect to Fh at can be deduced in the same manner.
Based on Eq. (8), the derivative of Bowen ratio B with respect to Fh is yielded as
ea10
The derivatives of and Bowen ratio B with respect to Fq are given by
ea11
and
ea12

APPENDIX B

Issues of Bowen Ratio in Flux Computations

The first issue with the Bowen ratio method is that one could get a wrong sign for the turbulent fluxes. This problem can be mathematically formulated through substituting Eq. (8) into Eq. (6) (e.g., Ohmura 1982):
eb1
Rearranging Eq. (B1), one yields (e.g., Ohmura 1982)
eb2
The inequality sign is justified because the direction of the flux is always opposite to that of the gradient. Equation (B2) can be further expressed as the following two conditions:
eb3
If the data do not satisfy either of the above two conditions, the calculated flux using the Bowen ratio method provides a direction of the flux to be the same as that of the gradient, which is not consistent with the definition of the flux–gradient relationship. In this case, the data were excluded.
In addition to the wrong sign problem, the Bowen ratio method could lead to an inaccurate magnitude in the flux computation. This problem can be illustrated through introducing the true temperature gradient and specific humidity gradient by considering the resolution limits of instruments for temperature and specific humidity , respectively:
eb4
eb5
Adding Eq. (B4) multiplied by with Eq. (B5) multiplied by gives
eb6
When the Bowen ratio is equal to −1, that is, = 0, Eq. (B6) becomes
eb7
When the inequality in Eq. (B7) is satisfied, the Bowen ratio will be very likely close to −1. In such a situation, the data and the calculated fluxes were rejected.

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