Application of Bulk Richardson Parameterizations of Surface Fluxes to Heterogeneous Land Surfaces

Temple R. Lee aCooperative Institute for Mesoscale Meteorological Studies, Norman, Oklahoma
bNOAA/Air Resources Laboratory Atmospheric Turbulence and Diffusion Division, Oak Ridge, Tennessee

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Michael Buban aCooperative Institute for Mesoscale Meteorological Studies, Norman, Oklahoma
bNOAA/Air Resources Laboratory Atmospheric Turbulence and Diffusion Division, Oak Ridge, Tennessee

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Tilden P. Meyers bNOAA/Air Resources Laboratory Atmospheric Turbulence and Diffusion Division, Oak Ridge, Tennessee

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Abstract

Monin–Obukhov similarity theory (MOST) has long been used to represent surface–atmosphere exchange in numerical weather prediction (NWP) models. However, recent work has shown that bulk Richardson (Rib) parameterizations, rather than traditional MOST formulations, better represent near-surface wind, temperature, and moisture gradients. So far, this work has only been applied to unstable atmospheric regimes. In this study, we extended Rib parameterizations to stable regimes and developed parameterizations for the friction velocity (u*), sensible heat flux (H), and latent heat flux (E) using datasets from the Land-Atmosphere Feedback Experiment (LAFE). We tested our new Rib parameterizations using datasets from the Verification of the Origins of Rotation in Tornadoes Experiment-Southeast (VORTEX-SE) and compared the new Rib parameterizations with traditional MOST parameterizations and MOST parameterizations obtained using the LAFE datasets. We found that fitting coefficients in the MOST parameterizations developed from LAFE datasets differed from the fitting coefficients in classical MOST parameterizations which we attributed to the land surface heterogeneity present in the LAFE domain. Regardless, the new Rib parameterizations performed just as well as, and in some instances better than, the classical MOST parameterizations and the MOST parameterizations developed from the LAFE datasets. The improvement was most evident for H, particularly for H under unstable conditions, which was based on a better 1:1 relationship between the parameterized and observed values. These findings provide motivation to transition away from MOST and to implement bulk Richardson parameterizations into NWP models to represent surface–atmosphere exchange.

Significance Statement

Models used to forecast the weather represent physical processes occurring in Earth’s atmosphere using complex mathematical relationships. Many weather models use an approach developed in the 1950s referred to as Monin–Obukhov similarity theory, or MOST, to determine how heat, moisture, and wind change vertically above the land surface. However, MOST limitations are well known within the scientific community. We implemented a method different from MOST called the bulk Richardson approach. We found the bulk Richardson approach generally yielded predictions of how heat, moisture, and wind change with height near the ground that were just as good as, and in some instances better than, predictions of these quantities obtained using MOST. This result motivates us to consider implementing the bulk Richardson approach into weather models, as doing so is expected to lead to their improvement.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Dr. Temple R. Lee, temple.lee@noaa.gov

Abstract

Monin–Obukhov similarity theory (MOST) has long been used to represent surface–atmosphere exchange in numerical weather prediction (NWP) models. However, recent work has shown that bulk Richardson (Rib) parameterizations, rather than traditional MOST formulations, better represent near-surface wind, temperature, and moisture gradients. So far, this work has only been applied to unstable atmospheric regimes. In this study, we extended Rib parameterizations to stable regimes and developed parameterizations for the friction velocity (u*), sensible heat flux (H), and latent heat flux (E) using datasets from the Land-Atmosphere Feedback Experiment (LAFE). We tested our new Rib parameterizations using datasets from the Verification of the Origins of Rotation in Tornadoes Experiment-Southeast (VORTEX-SE) and compared the new Rib parameterizations with traditional MOST parameterizations and MOST parameterizations obtained using the LAFE datasets. We found that fitting coefficients in the MOST parameterizations developed from LAFE datasets differed from the fitting coefficients in classical MOST parameterizations which we attributed to the land surface heterogeneity present in the LAFE domain. Regardless, the new Rib parameterizations performed just as well as, and in some instances better than, the classical MOST parameterizations and the MOST parameterizations developed from the LAFE datasets. The improvement was most evident for H, particularly for H under unstable conditions, which was based on a better 1:1 relationship between the parameterized and observed values. These findings provide motivation to transition away from MOST and to implement bulk Richardson parameterizations into NWP models to represent surface–atmosphere exchange.

Significance Statement

Models used to forecast the weather represent physical processes occurring in Earth’s atmosphere using complex mathematical relationships. Many weather models use an approach developed in the 1950s referred to as Monin–Obukhov similarity theory, or MOST, to determine how heat, moisture, and wind change vertically above the land surface. However, MOST limitations are well known within the scientific community. We implemented a method different from MOST called the bulk Richardson approach. We found the bulk Richardson approach generally yielded predictions of how heat, moisture, and wind change with height near the ground that were just as good as, and in some instances better than, predictions of these quantities obtained using MOST. This result motivates us to consider implementing the bulk Richardson approach into weather models, as doing so is expected to lead to their improvement.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Dr. Temple R. Lee, temple.lee@noaa.gov

1. Introduction

Monin–Obukhov similarity theory (MOST) (e.g., Monin and Obukhov 1954; Obukhov 1971) has long been used to represent surface-layer exchange in numerical weather prediction (NWP) models (e.g., Grachev and Fairall 1997; Johansson et al. 2001; Foken 2006; Jiménez et al. 2012). In brief, MOST bases the scaling on the fluxes, whereby gradients in wind (U), potential temperature (θ), and specific humidity (q) are functions of a dimensionless stability length ζ:
U¯zκzu*=ϕm(ζ),
θ¯zκzθ*=ϕh(ζ),
q¯zκzq*=ϕq(ζ).
In Eqs. (1)(3), κ is the von Kármán constant; z is the height; and u*, θ*, and q* represent the friction velocity, temperature scale, and moisture scale, respectively. We used 0.40 for κ throughout this study, noting, however, that previous work reported values of κ between 0.35 and 0.42 (e.g., Stull 1988). The overbars above the variables represent the temporal mean values of U, θ, and q. u*, θ*, and q* are computed as
u*=[(uw¯)2+(υw¯)2]0.25,
θ*=(wθυ¯)su*,
q*=(wq)¯su*.
The most common functional forms of ϕm,h,q under unstable atmospheric regimes are
ϕm,h,q=αm,h,q(1βm,h,qζ)γ.
Here, αm,h,q and βm,h,q are coefficients determined empirically; γ is −0.25 for ϕm and is −0.50 in the equations for ϕh and ϕq; and ζ is the dimensionless Monin–Obukhov stability length. Previous work has found αm,h,q ≈ 1, and that βm,h,q vary between ~10 and ~30 (e.g., Dyer and Hicks 1970; Dyer and Bradley 1982; Maronga and Reuder 2017). In Eqs. (1)(3), ζ is defined as
ζ=zdL.
In Eq. (8), d is the displacement height of the vegetation, z is the sampling height, and L is the Monin–Obukhov length scale defined as
L=θ¯υu*3κgwθυ¯.
In Eq. (9), u* is the friction velocity, g is the gravitational acceleration, θυ is the virtual potential temperature, wθυ¯ is the vertical heat flux, and the primes indicate the perturbations from the mean.
Under stable conditions, i.e., where ζ > 0, MOST relationships have been found to have a linear relationship that has the following form:
ϕm,h,q=ηm,h,q+εm,h,qζ.
As in Eqs. (1)(3), in Eq. (10), ηm,h,q and εm,h,q are coefficients determined empirically. Earlier studies had suggested that ϕm = ϕh = ϕq, whereby the coefficients ηm,h,q and εm,h,q are 1 and 5, respectively (e.g., Dyer 1974). Although ηm,h,q = 1 has been reported in other studies (e.g., Businger et al. 1971; Dyer 1974; Högström 1988, 1996; Foken 2006), εm has been determined to range from 4.7 (Businger et al. 1971) to 6 for ϕm (Högström 1988), and εh has been determined to range from 4.7 (Businger et al. 1971) to 8 for ϕh (Högström 1996). It is oftentimes assumed that ϕh = ϕq (e.g., Maronga and Reuder 2017).

The range of fitting coefficients derived from the different studies referenced above indicates the lack of agreement of the forms of the equations obtained from MOST and underscores one of several well-known limitations of MOST. Other nontrivial limitations of MOST are that MOST assumes a horizontally homogeneous layer (e.g., Businger et al. 1971), which is a condition seldom met, and the performance of MOST deteriorates above the lowest tens of meters of the surface layer (Sun et al. 2020). As noted by Lee and Buban (2020) and briefly summarized here, MOST suffers from statistical self-correlation because there are not enough scaling variables that are fully independent (e.g., Hicks 1978a,b, 1981; Andreas and Hicks 2002; Hicks 1995). For example, self-correlation in MOST arises because u* is present in the calculation of L, and u* also appears in the bulk-flux equations. Furthermore, MOST can be affected by measurement errors in u*, which are exacerbated because u* is cubed in the calculation L (e.g., Markowski et al. 2019). Similarly, random errors in ζ can be ~40% under unstable conditions (e.g., Salesky and Chamecki 2012; Salesky et al. 2012).

Because of the limitations of MOST, alternatives to MOST have been proposed. One approach is to use a technique based on the Richardson number (Ri), instead of ζ (e.g., Sorbjan 2006, 2010). An advantage of using the Ri instead of ζ is that the Ri has been shown to reduce self-correlation present in MOST (e.g., Sorbjan 2006) and may be better suited for very stable conditions (e.g., Sorbjan 2010). For example, Mauritsen et al. (2007) developed such an approach for neutral and stably stratified ABLs. Their scheme is a total turbulent energy scheme that sums the turbulent kinetic and potential energies and is motivated by Richardson (1920) who found that turbulence decays above a critical limit. The scheme developed by Mauritsen et al. (2007) has since been implemented into the total energy mass flux (TEMF) scheme in the Weather Research and Forecasting (WRF) Model (e.g., Angevine et al. 2010) and more recently into the ECHAM6 climate model (Pithan et al. 2015).

Another approach has been to develop parameterizations relating vertical near-surface wind, temperature, and moisture gradients to the fluxes of these quantities by developing empirical relationships between Ri and the friction coefficient, heat-transfer coefficient, and moisture-transfer coefficient, respectively (Lee and Buban 2020). In their approach, local gradients present in the Ri formulation are approximated by bulk gradients to compute the bulk Richardson number (Rib) following e.g., Stull (1988):
Ri=gθυ¯zθυ¯[(u¯z)2+(υ¯z)2]gΔθυ¯Δzθυ¯[(Δu¯)2+(Δυ¯)2]=Rib.
In Eq. (11), u and υ represent the zonal and meridional wind components, respectively, and the other variables have been previously defined. The bulk gradients are calculated between measurements made at 2 and 10 m above ground level (AGL) which arguably are quantities that are easier and more straightforward to measure than the variables used to compute L. The approach developed was found to yield better predictions than MOST parameterizations of 10-m winds and temperature gradients when evaluated using a fully independent dataset.

However, the similarity relationships from Lee and Buban (2020) were only developed and applied for unstable conditions, i.e., when Rib < 0. If new parameterizations that use Rib as a stability term are to be used to represent surface-layer exchange in NWP models, formulations must be developed across all stability regimes (e.g., Olson et al. 2021). Furthermore, Lee and Buban (2020) used fluxes to compute near-surface gradients in the wind, temperature, and moisture fields. Critical to moving forward with implementing newly suggested bulk Richardson parameterizations into NWP models is to 1) use gradients in the meteorological fields to compute the fluxes, and 2) perform this evaluation for both unstable conditions and stable conditions. Thus, the objective of this study is to apply the Rib framework across both unstable and stable atmospheric regimes using datasets obtained from the Land Atmosphere Feedback Experiment (LAFE) and to evaluate how the Rib parameterizations compare with 1) similarity relationships derived from MOST using the LAFE datasets, and 2) traditional MOST relationships (e.g., those from Högström 1988). We then evaluate the new Rib parameterizations for u*, sensible heat (H) flux, and latent heat (E) flux using fully independent datasets obtained as a component of the Verification of the Origins of Rotation in Tornadoes Experiment-Southeast (VORTEX-SE) campaigns. Performing these evaluations allows us to determine if any improvements result when using the Rib parameterizations which have 1) a different stability term (i.e., Rib rather than ζ), and 2) a different functional form of the similarity equations.

2. Development of M–O and Rib functions

a. Site description

We used data from LAFE, which was a field campaign conducted in August 2017 near Lamont, Oklahoma, at the Department of Energy Atmosphere Radiation Measurement site (36.607°N, 97.488°W, 314 m MSL). This project used micrometeorological towers and surface-based profiling systems to study feedbacks and exchanges between the land surface and overlying atmosphere, with the aim to improve the representation of these processes in NWP models. These instruments were installed over different land covers (i.e., early growth soybean crops, mature soybean crops, grasslands, etc.) to also study the effects of land surface heterogeneities on boundary layer structure. We refer the reader to Wulfmeyer et al. (2018) for more details on the experimental design.

Important to this work are the datasets from three 10-m micrometeorological towers installed along a 1.7-km line and outfitted with an array of instruments to sample temperature, wind, moisture, and sensible and latent heat fluxes. Towers 1, 2, and 3 were installed in an early growth soybean crop, a grassland, and soybean crop, respectively. During the study period, the sites had a mean surface roughness (z0) of around 0.10 m and a canopy displacement height (d) of 0.91, 0.75 m, and 0.96 m at Towers 1, 2, and 3, respectively (see Lee and Buban 2020 for more details on how z0 and d were determined). At all three towers, temperatures were sampled 2 and 10 m AGL; wind, moisture, and fluxes were sampled 3 and 10 m AGL. Further details on the site characteristics, instruments used, data quality control, and data processing appear in Lee and Buban (2020). In the present study, consistent with their work, we used 30-min means of the meteorological and flux datasets to develop the similarity relationships, and we applied least squares regression to the datasets from all three LAFE towers taken together to determine the M–O and Rib similarity relationships.

b. Data filtering

We acknowledge that the footprints of the measurements from 2 and 3 m AGL and the measurements 10 m AGL can, at times, differ and thus have the potential to introduce uncertainties in the results. For this reason, we removed 30-min periods when there was significant flux divergence occurring. To this end, when evaluating the MOST and Rib parameterizations for wind, we did not use any 30-min periods in which the percent difference in u* between the two sampling heights exceeded 15%, which resulted in the removal of about 40% of the 30-min periods. Similarly, when evaluating the MOST and Rib parameterizations for temperature, we omitted time periods in which the percent difference between H measured 3 m AGL and H measured 10 m AGL exceeded 15% following previous work (e.g., Lee et al. 2019b). When evaluating the MOST and Rib parameterizations for moisture, we filtered periods when the percent difference in E measured between 3 and 10 m AGL exceeded 15%. Because we removed time periods with flux divergence occurring, we acknowledge that the parameterizations we develop in the present study are not valid for very shallow surface layers (i.e., <10 m) when there can be significant differences in the fluxes between the two sampling heights. However, we expect for the parameterizations to be valid for surface layer depths exceeding 10 m.

In addition to filtering the LAFE datasets to remove periods of strong flux divergence, we filtered the LAFE datasets by wind direction. The longest fetch at each of the three towers occurred with southerly winds (Lee and Buban 2020); for this reason we omitted observations with a northerly wind, i.e., wind directions >270° or <90°. Winds from these directions occurred 60.5%, 58.6%, and 58.1% of the time during the study period at Towers 1, 2, and 3, respectively. Omitting observations in which northerly winds were occurring was critical at Tower 1, since the land surface immediately to the north consisted of native grassland whereas the land surface to its south was early growth soybean crop, as noted in section 2a.

c. M–O and Rib parameterizations

1) M–O parameterizations

After applying the filtering procedures outlined in the previous section to the LAFE datasets, we performed nonlinear least squares regressions for both unstable conditions and for stable conditions to develop the relationships between ζ and ϕm,h,q (Fig. 1). We used weighted fits when performing the coefficient-fitting process. Following from Markowski et al. (2019), the uncertainty in ϕm, which we define as δϕm, is approximated as
δϕm=[(ϕmΔu¯δΔu¯)2+(ϕmu*δu*)2]0.5[(κzu*ΔzδΔu¯)2+(ϕmu*δu*)2]0.5.
The uncertainty in ϕh and ϕq, defined as δϕh and δϕq, respectively, has the same form as Eq. (12). The uncertainty is used to compute how much each observation is weighted when applying the nonlinear least squares regression. Here, the weight is simply the inverse of the uncertainty.
Fig. 1.
Fig. 1.

(a) ϕm, (b) ϕh, and (c) ϕq as a function of ζ for both unstable and stable conditions. Red line shows the line of best fit.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0047.1

The functions for ϕm,h,q for ζ < 0 take the form of Eqs. (1)(3), and the fitting coefficients αm,h,q and βm,h,q for ϕm,h,q as a function of ζ are shown in Table 1. We note that there are different numbers of data points for ϕm, ϕh, and ϕq because the filtering criteria were applied separately to each variable (cf. section 2b). In the case of stable conditions and as noted earlier, ϕm,h,q as a function of ζ take the form of Eq. (10) (e.g., Businger et al. 1971; Dyer 1974; Högström 1988; Foken 2008). The coefficients εm,h,q and ηm,h,q from these functions are shown in Table 2.

Table 1.

Best-fit parameters using nonlinear least squares for the following equations: ϕm = αm(1 − βmζ)−0.25, ϕh = αh(1 − βhζ)−0.5, and ϕq = αq(1 − βqζ)−0.5 for ζ < 0. All functions apply over the range −2 < ζ < 0; for ϕm,q these functions apply over the range 0 < ϕm,q < 5 and for ϕh these functions apply over the range 0 < ϕh < 10 to remove outliers. The correlation coefficient (r), number of samples (N), and root-mean-square (rms) is also shown for each variable. We also report one standard deviation (i.e., 1σ) for each of the best-fit parameters for each variable. All correlations are significant at the p < 0.05 confidence level.

Table 1.
Table 2.

Best-fit parameters using nonlinear least squares for ϕm,h,q = εm,h,qζ + ηm,h,q. To remove outliers, the functions apply over the range 0 < ζ < 1 and 0 < ζ < 0.25 for ϕm,h and ϕq, respectively; for ϕm,q these functions apply over the range 0 < ϕm,q < 5; for ϕh these functions apply over the range 0 < ϕh < 10. We also report r, N, and rms for each variable, as well as 1σ for each of the best-fit parameters for each variable. All correlations are significant at the p < 0.05 confidence level.

Table 2.

2) Rib parameterizations

To develop the relationships between Rib and Cu,t,r, we compute the weights of Cu,t,r following the procedure summarized in the previous section prior to applying the least squares regressions to the filtered LAFE datasets. Rib is a function of the friction coefficient, heat-transfer coefficient, and moisture-transfer coefficient, which we define as Cu, Ct, and Cr, respectively. Deardorff (1972) defined these terms, assuming a constant flux layer and single-layer ABL model. Lee and Buban (2020) modified Cu, Ct, and Cr such that Cu is a function of u* and the wind speed 10 m AGL, which we define as U10:
Cu=u*U10.
Also from Lee and Buban (2020), Ct and Cr are calculated as
Ct=θ*θυθυs,
Cr=q*qqs.
In the above equations, the subscript s denotes the surface values which we take to be 2 m AGL following e.g., Seidel et al. (2012) and Lee and Buban (2020).
Lee and Buban (2020) found that, for unstable conditions (i.e., Rib < 0), Cu, Ct, and Cr are related to Rib via a 1/3 power law that has the following form:
Cu,t,r=λu,t,r(1ωu,t,rRib)1/3.
As in MOST, the coefficients λu,t,r and ωu,t,r are determined empirically and appear in Table 3. By writing Cu, Ct, and Cr as a function of Rib, these terms have dependence on z; however, we acknowledge these formulations have no explicit dependence of Cu, Ct, and Cr on the roughness lengths for momentum, heat, and moisture, respectively. We revisit this point later in the study.
Table 3.

Best-fit parameters using nonlinear least squares for Cu,t,r = λu,t,r(1 − ωu,t,rRib)1/3. These functions apply for −2 < Rib < 0, 0 < Cu < 0.2, and 0 < Ct,r < 2 to remove outliers. We also report r, N, and rms for each variable, as well as 1σ for each of the best-fit parameters for each variable. All correlations are significant at the p < 0.05 confidence level.

Table 3.
In the present study, we extended the work of Lee and Buban (2020) to stable conditions. We determined that the relationship among Rib, Cu, Ct, and Cr for Rib > 0 also was nonlinear and could be expressed via the following relationship, whereby χu,t,r and γu,t,r are the fitting coefficients determined through least squares regression and appear in Table 4:
Cu,t,r=χu,t,rexp(γu,t,rRib).

3) Discussion

We acknowledge that αm,h,q and βm,h,q obtained from the M–O fits, as well as λu,t,r and ωu,t,r obtained from the Rib fits, are different from the coefficients suggested by Lee and Buban (2020) because 1) Lee and Buban (2020) computed the least squares fits over a different range of ζ and Rib, 2) Lee and Buban (2020) focused solely on unstable conditions in their analyses, and 3) Lee and Buban (2020) did not filter periods with flux divergence.

We also note that the functional fits for ϕm,h,q in the present study are in contrast to the classical relationships presented in the literature (e.g., Businger et al. 1971; Dyer 1974; Dyer and Bradley 1982; Högström 1988). As noted in the section 1 and briefly summarized here, previous studies found that under neutral conditions when ζ = 0, αm,h,qηm,h,q ≈ 1 (e.g., Dyer and Hicks 1970; Dyer 1974; Dyer and Bradley 1982; Maronga and Reuder 2017). In contrast, we found that αm,h,q and ηm,h,q were between 1.05 and 1.64 when ζ = 0. We attribute the lack of agreement between our functional fits and those from the literature to the heterogeneity of the land surfaces in our study, which we argue is much more representative of “real world conditions” than previous studies (e.g., Dyer and Hicks 1970) that developed relationships based upon measurements made over more homogeneous land surfaces than those in the LAFE domain. As noted in section 2a and briefly summarized here, Towers 1, 2, and 3 were installed in an early growth soybean crop, a grassland, and soybean crop, respectively. Even during the 1-month study period, these land surfaces were evolving. For example, the early growth soybean crop south of Tower 1 transitioned to a mature soybean crop by the end of the study period, in part due to the area receiving about 50 mm of rainfall between 10 and 11 August.

We speculate that differences in the land surface type surrounding each of three towers, as well as the growth of the vegetation during the study period, contributed to the scatter present in relationships between ϕm,h,q and ζ (Fig. 1), as well between Cu,t,r, and Rib (Fig. 2). The rms for ϕm,h,q as a function of ζ is consistently larger under stable conditions than under unstable conditions (cf. Tables 1 and 2). In the case of Cu,t,r as a function of Rib, the rms is larger under unstable conditions (cf. Tables 3 and 4). Sensitivity tests (not shown) indicate that the results in this study are more sensitive to the coefficients α and η than to the coefficients β and ε in the functions for ϕm,h,q and to the coefficients ω and γ than to the coefficients λ and χ in the functions for Cu,t,r. The sensitivity is reflected in the 1σ values for each of the best fit parameters that are reported in Tables 14. We also note that nearly all of the relationships have convergence at 0; the exception is the relationship between ϕq and ζ and is caused by the few number of valid data points for stable conditions (Table 2). However, for the remaining functions, since λχ (cf. Tables 3 and 4), there is convergence at 0 which is expected to eliminate potential numerical instabilities when implementing the Rib parameterizations into NWP models.

Fig. 2.
Fig. 2.

(a) Cu, (b) Ct, and (c) Cr as a function of Rib. Red line shows the line of best fit.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0047.1

Table 4.

Best-fit parameters using nonlinear least squares for Cu,t,r=χu,t,reγu,t,rRib. These functions apply over the range 0 < Rib < 0.5, 0 < Cu < 0.1, and 0 < Ct,r < 1 to remove outliers. We also report r, N, and rms for each variable, as well as 1σ for each of the best-fit parameters for each variable. All correlations are significant at the p < 0.05 confidence level.

Table 4.

Following e.g., Mauritsen et al. (2007) we further compare our MOST parameterizations against previous work by computing the turbulent Prandtl number, Pr(0). Under neutral conditions, Pr(0) is defined as the ratio between turbulent viscosity, Km, and the turbulent conductivity, Kh. Its reciprocal has been found to range from 1.00 (e.g., Wieringa 1980) to 1.39 (Kader and Yaglom 1972) (see e.g., Foken 2006 for more details). In our study, when ζ < 0, ϕm = 1.57(1 − 6.71ζ)−0.25 and ϕh = 1.06(1 − 1.10ζ)−0.25 (cf. Table 1); when ζ > 0, ϕm = 4.04ζ + 1.50 and ϕh = 10.9ζ + 1.05 (cf. Table 2). Thus, Pr(0) = 1.48 for ζ < 0 and Pr(0) = 0.37 for ζ > 0 in the present study. These values for Pr(0) are outside range of values reported in from the literature (e.g., Foken 2006) which we attribute to the application of the MOST relationships to heterogeneous land surfaces like the ones in the present study.

3. Evaluation of M–O and Rib parameterizations

a. Datasets

We evaluated the M–O and Rib parameterizations developed in the previous section using datasets obtained from two 10-m micrometeorological towers installed in northern Alabama. One tower was installed near Belle Mina, Alabama (34.693°N, 86.871°W, 189 m MSL), and a second tower was installed near Cullman, Alabama (34.194°N, 86.800°W, 241 m MSL). Both towers were operational from February 2016 through April 2017, and the sensor suite was identical to the sensor suite used on all towers during LAFE. Measurements from the towers at Belle Mina and Cullman were used to support the spring 2016 and spring 2017 campaigns of VORTEX-SE, which was a multiyear study focused on how interactions between the land surface and ABL contribute to severe weather development and tornadogenesis over this region of the United States. See Dumas et al. (2016, 2017), Wagner et al. (2019), Lee et al. (2019a,b), and Markowski et al. (2019) for more details on the meteorological measurements from Belle Mina and Cullman and on the VORTEX-SE experimental design. Most important to this study are the measurements from a CSAT3 sonic anemometer and EC155 closed-path CO2/H2O gas analyzer that were installed 10 and 3 m AGL at both Belle Mina and Cullman and allowed for the computation of u*, H, and E. We used the CSAT3 sonic anemometer and EC155 to sample the three-dimensional wind components and moisture concentration, respectively, at 10 Hz. Following Lee et al. (2019b) and briefly summarized here, we applied postprocessing techniques (see e.g., Meyers 2001 for more details) to the CSAT3 and EC155 datasets, whereby we computed the covariances among the wind, temperature, and moisture, performed the necessary coordinate transformations (e.g., Meyers and Baldocchi 2005), and accounted for angle-of-attack errors (e.g., Kochendorfer et al. 2012). We then used Eqs. (2), (3), and (4) from Lee et al. (2019b) to compute H, E, and u*, respectively, over 30-min periods.

b. Evaluation of M–O parameterizations

1) u* parameterization

To evaluate the M–O parameterization of u*, we integrate Eq. (1) with respect to height z, following from e.g., Jiménez et al. (2012) which enables us to calculate the wind speed at z, which we term ua. In the present study we computed the wind at 10 m AGL:
ua=u*κ[ln(zdz0)ψm(zdL)+ψm(z0L)].

Because the canopy displacement height d at both sites was nonnegligible, i.e., 0.41 and 1.07 m at Belle Mina and Cullman, respectively, we include d in the above calculation for ua, and we used 0.15 and 0.25 m for z0 at Belle Mina and Cullman, respectively. The values for d and z0 are consistent with the expectation that these values are about 60% and 10%, respectively, of the canopy height (e.g., Baldocchi 1997; Lee and Buban 2020).

In Eq. (18) ψm is obtained by integrating the momentum similarity function following e.g., Panofsky (1963) and Jiménez et al. (2012):
ψm=0z/L[1ϕm(ζ)]dζζ.
Note that we evaluate two different forms of ϕm(ζ): 1) the traditional form obtained from legacy MOST relationships (i.e., those from Högström 1988), and 2) the similarity relationship derived from MOST using the LAFE datasets. Following from, e.g., Paulson (1970) and Jiménez et al. (2012), under unstable conditions, we compute ψm as
ψm=2ln{1+[ϕm(ζ)]12}+ln{1+[ϕm(ζ)]22}2tan1ϕm1+π2.
For stable conditions, ψm is computed following Eq. (21) where εm is obtained from the least squares regression:
ψm=εmϕm(ζ).
Rewriting Eq. (18) allows us to compute u* as a function of ua and ψm:
u*=κua[ln(zdz0)ψm(zdL)+ψm(z0L)].

2) H parameterization

To evaluate the M–O parameterization of H, we integrate Eq. (2) so that Δθ is computed as
Δθ=θ*κ[ln(z2dz1d)ψh(z2dL)+ψh(z1dL)].
In Eq. (23), z1 and z2 are the two heights over which we compute Δθ. We note that Jiménez et al. (2012) used z1 as the surface value. However, in the absence of surface measurement we take z1 to be the measurements made 2 m AGL following e.g., Seidel et al. (2012). We obtain ψh by integrating the heat similarity function, again following, e.g., Panofsky (1963) and Jiménez et al. (2012):
ψh=0z/L[1ϕh(ζ)]dζζ.
Like we do for ϕm(ζ), we use the ϕh(ζ) function suggested by Högström (1988) and also the ϕh(ζ) function obtained from the LAFE datasets. When ζ < 0, we compute ψh as
ψh=2ln{1+[ϕh(ζ)]22}.
For stable conditions, ψh is computed following Eq. (26), again where εh is obtained from the least squares regression:
ψh=εhϕh(ζ).
We can combine Eq. (23) for Δθ and Eq. (5) for θ* to compute the kinematic form of the heat flux (wθ)¯s. In the equation below, we compute u* following Eq. (22):
(wθ)¯s=Δθκu*[ln(z2dz1d)ψh(z2dL)+ψh(z1dL)].
The kinematic form of the sensible heat flux can be converted to its dynamic form by multiplying by cp and ρ, which are the specific heat of air and the air density, respectively:
H=Δθκu*cpρ[ln(z2dz1d)ψh(z2dL)+ψh(z1dL)].

3) E parameterization

We used the same approach for the moisture flux as we did for the heat flux and computed (wq)¯s as
(wq)¯s=Δqκu*[ln(z2dz1d)ψq(z2dL)+ψq(z1dL)].
In the above equation, ψq has the same functional form as ψh when ζ < 0 and ζ > 0, respectively [cf. Eqs. (25) and (26), respectively]. The dynamic form of the moisture flux is computed by multiplying the above equation by ρ and the latent heat of vaporization Lυ:
E=Δqκu*ρLυ[ln(z2dz1d)ψq(z2dL)+ψq(z1dL)].
Following e.g., Stull (1988), Lυ is temperature dependent:
Lυ=(2.5010.00237T)×106.

Since ζ is unknown, the equations for ζ [Eq. (8)], u* [Eq. (22)], H [Eq. (28)] and E [Eq. (30)] must be solved iteratively until convergence is achieved. If convergence is not achieved after 100 iterations, that particular data point is removed from the analyses. Sensitivity tests (not shown) indicated that our results are unaffected by our choice for the number of iterations.

c. Evaluation of Rib parameterizations

1) u* parameterization

We compute the wind difference between two levels using the Rib parameterizations such that, for two heights, z1 and z1:
Uz2Uz1=ΔU=u*Cu(Rib).
To be consistent with how we evaluate the M–O parameterization of u*, we make the assumption that the surface winds are 0 m s−1 so that Uz1=0, and we rename this term ua for consistency with wind speed in equations from MOST. Doing so yields Eq. (31) from Lee and Buban (2020), which we have reproduced below albeit with some modifications from their notation for consistency with the notation used in the present study:
ua=u*Cu(Rib).
Eq. (33) needs to be rearranged to evaluate the new Rib parameterizations for u*. Equation (33) can be rewritten so that u* is calculated using the relationship between Cu and Rib [i.e., Cu(Rib)] found in the present study:
u*=uaCu(Rib).

2) H parameterization

Following from Lee and Buban (2020), the potential temperature difference can be expressed as
θz2θz1=Δθ=θ*Ct(Rib).
As with the M–O parameterizations, we used parameterized u* from Eq. (34), which we term u*, to compute (wθ)¯s:
(wθ)¯s=Δθu*Ct(Rib).
Multiplying by cp and ρ yields the dynamic form of the sensible heat flux, shown below:
H=Δθu*Ct(Rib)cpρ.

3) E parameterization

The near-surface moisture difference has the same functional form as the near-surface temperature difference:
qz2qz1=Δq=q*Cr(Rib).
We then compute the kinematic form of the moisture flux as
(wq)¯s=Δqu*Cr(Rib).
Multiplying the above equation by ρ and by Eq. (31) for Lυ yields the following equation for the dynamic form of the latent heat flux:
E=Δqu*Cr(Rib)ρLυ.

4) Use of Rib parameterizations of u*, H, and E in NWP models

If the parameterizations for H and E are to be implemented into surface layer and boundary layer parameterization schemes in NWP models, the Rib parameterizations must be able to ingest data from other sampling heights besides 2 and 10 m, which were the sampling heights in this study, in order to compute Δθ and Δq in the above equations. Doing so is critical because the heights of the lowest model levels vary among models. MOST circumvents this issue by using ζ as a stability parameter. The Rib parameterizations are not limited to the 2- and 10-m sampling heights and can ingest data from other sampling heights because the heights are included in the Δz term in the Rib equation [cf. Eq. (11)]. A NWP model like the High-Resolution Rapid Refresh (HRRR) model (e.g., Benjamin et al. 2016) requires surface values for temperature and moisture [note the terms θυs and qs which appear in Eqs. (14) and (15), respectively, in the calculation of Ct and Cr, respectively (e.g., Olson et al. 2021)]. The gradient between the surface values of θ and q and the lowest model level’s values of θ and q over the difference in height between these two model levels, Δz, could then be used to calculate u*, H, and E so that form of the Rib parameterizations under unstable conditions is
u*=Uz2λu{1ωugΔθυ¯Δzθυ¯[(Δu¯)2+(Δυ¯)2]}1/3,
H=Δθu*cpρλt{1ωtgΔθυ¯Δzθυ¯[(Δu¯)2+(Δυ¯)2]}1/3,
E=Δqρu*Lυλr{1ωrgΔθυ¯Δzθυ¯[(Δu¯)2+(Δυ¯)2]}1/3.
Similarly, u*, H, and E under stable conditions are computed following Eqs. (44)(46):
u*=uaχuexp{γugΔθυ¯Δzθυ¯[(Δu¯)2+(Δυ¯)2]},
H=Δθu*cpρχtexp{γtgΔθυ¯Δzθυ¯[(Δu¯)2+(Δυ¯)2]},
E=Δqρu*Lυχrexp{γrgΔθυ¯Δzθυ¯[(Δu¯)2+(Δυ¯)2]}.

d. Comparison of M–O and Rib parameterizations

As we did when developing the functional fits of the similarity equations, we also considered the impacts of uncertainties when comparing the observed and M–O and Rib-derived u*, H, and E. Following from Markowski et al. (2019), the error in u*, i.e., δu*, is estimated via Eq. (47):
δu*=12|uw¯|1/2δuw¯.
In Eq. (47), δuw¯ is the error in the streamwise vertical momentum flux which, following from the manufacturer and from Markowski et al. (2019), is 0.014 m2 s2.
We estimate the errors in the sensible and latent heat flux, i.e., δH and δE, respectively, following e.g., Jiang et al. (2004):
δH=Hδwθυ¯wθυ¯,
δE=Eδwq¯wq¯.
In the above equations, δwθυ¯ and δwq¯ are the errors in the kinematic forms of the heat and moisture fluxes, respectively. From the manufacturer’s specifications, these terms are approximately 0.019 K m s−1 and 0.087 m s−1, respectively.

To quantify the instrumental uncertainties in the parameterized u*, H, and E, we use the 1σ errors reported in Tables 14 to determine the possible range of fitting coefficients and subsequently the range in u*, H, and E. Doing so allows us to estimate the uncertainties in each of these variables computed using the parameterizations that we developed.

Once we determined the uncertainties in both the observed and parameterized u*, H, and E, we follow the approach by Neri et al. (1989), and which is summarized by Cantrell (2008), and perform a bivariate weighting of the errors from the observations and output from the MOST-derived and Rib-derived parameterizations of u*, H, and E. The bivariate weighting involves iteratively solving the following equation until the first term in Eq. (50) equals the second term in Eq. (50):
Wixi(mbxi+byi) Wi2mb(mbxi+byi)wxi=0.
In Eq. (50), Wi and b are calculated using Eqs. (51) and (52), respectively:
Wi=wxiwyiwxi+wyimb2,
b= Wi(yimbxi) Wi.
In Eqs. (51) and (52), wxi and wyi correspond with the weights in the observations and parameterized values, respectively, and are computed as an inverse of the uncertainties in the observations and parameterizations. xi and yi correspond with the observed u*, H, and E and parameterized u*, H, and E, respectively.

Once the difference between the first and second terms in Eq. (50) ≈0, we use the values of mb and b for the relationship between the VORTEX-SE observations and the parameterizations evaluated in the present study. In addition to reporting the line of best fit, we report the rms and correlation coefficient (r) between the parameterized and observed values (without any weighting) when evaluating the parameterizations.

We acknowledge, though, that the instrumental uncertainties that we compute are only one component of the total uncertainty, and there are other sources of uncertainty (e.g., representativeness and sampling errors) which contribute to the total uncertainty in the observations. However, estimates of representativeness and sampling errors are fraught with uncertainties themselves. For this reason, we present only the instrumental uncertainty, rather than the total uncertainty, acknowledging that the total uncertainty itself is likely to be underestimated.

4. Results and discussion

a. u* parameterizations

We used the formulations developed in section 2 to evaluate how well the parameterized values of u*, H, and E computed using the equations from section 3, compared with the observed values from VORTEX-SE. We compared the new Rib similarity relationships with the MOST similarity relationships from Högström (1988) because Högström reported similarity relationships for both stable and unstable regimes, rather than studies that report similarity relationships for e.g., only unstable regimes. (e.g., Dyer and Hicks 1970). Högström (1988) reported ϕm = (1 − 19.3ζ)−0.25 for ζ < 0 and ϕm = 1 + 6ζ for ζ > 0.

When using Högström’s relationships to compute u*, we found that MOST slightly overpredicted u* at both Belle Mina and Cullman (Figs. 3a, 4a ); the slope of the line of best fit between the parameterized and observed values, which we term mb, was 1.07 and 1.04 at Belle Mina and Cullman, respectively, and r = 0.79 (p < 0.01) and r = 0.88 (p < 0.01), respectively. When using the MOST similarity relationships applied to the LAFE datasets; mb was 1.02 and 1.01 at Belle Mina and Cullman; r = 0.78 (p < 0.01), and r = 0.88 (p < 0.01) at Belle Mina and Cullman, respectively (Figs. 3b, 4b). The use of the Rib functions developed using the LAFE datasets yielded slightly lower values of mb and r, as mb and r were 0.91 and 0.75 (p < 0.01), respectively (Fig. 3c) at Belle Mina. At Cullman, mb and r were 0.75 and 0.79, respectively (Fig. 4c).

Fig. 3.
Fig. 3.

Density plot showing the relationship between the observed u* at Belle Mina, AL, and u* computed using (a) M–O relationships from the literature, (b) M–O relationships developed using the LAFE datasets, and (c) Rib-based relationships developed using the LAFE datasets. (d)–(f),(g)–(i) As in (a)–(c), but for unstable conditions and stable conditions, respectively. The dotted line shows the 1:1 line, and the solid line shows the line of best fit. N, r, rms, and the formula for the best-fit line are shown at the bottom right of each panel.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0047.1

Fig. 4.
Fig. 4.

As in Fig. 3, but for Cullman, AL.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0047.1

From the above analyses, we conclude that the new Rib parameterizations we developed for u*, despite having a different functional form, perform similarly to 1) legacy relationships derived from MOST and 2) MOST applied to the same dataset. Given the similarities in the performance of the parameterizations, it is important to understand if there are scenarios in which either MOST or Rib parameterizations perform better. Because there are different parameterizations for unstable and stable conditions, we filtered the parameterized u* and observed u* from Belle Mina and Cullman as a function of stability. Additionally, we filtered the parameterized u* and observed u* by wind speed because it has been shown that models struggle in cases with weak winds (e.g., Zhang and Zheng 2004). The Southeast United States provides a unique testbed for these parameterizations because near-surface wind speeds in the region of the United States are among the lowest in the United States (e.g., Klink 1999). At Belle Mina, wind speeds 10 m AGL < 2.0 m s−1 occur about 50% of the time, and at Cullman wind speeds 10 m AGL < 2.5 m s−1 occur about 50% of the time. We classify these cases as weak winds, and we classify the remainder of cases as strong winds.

When we filtered u* by stability regime, we found that all parameterizations underpredicted u* during unstable regimes (i.e., ζ < 0 and Rib < 0) (Figs. 3d–f, 4d–f). MOST parameterizations at both Belle Mina and Cullman, and r was higher for the Rib parameterizations at Belle Mina. In stable regimes (i.e., ζ >0 and Rib > 0), the Rib parameterizations had the best 1:1 agreement between the parameterized and observed values (Figs. 3g–i); at Cullman, though, mb was lowest for the Rib parameterizations as compared with the MOST parameterizations (Figs. 4g–i).

When we filtered u* by wind speed we found that the Rib u* parameterizations had the largest mb under weak winds at Belle Mina (Figs. 5a–c). In the subset of cases with comparatively high winds speeds, at Belle Mina r was ~0.67 (p < 0.01) and 0.63 (p < 0.01) for the MOST and Rib parameterizations, respectively, and rms was lowest for the Rib parameterizations (Figs. 5d–f). The value of r decreased from ~0.77 (p < 0.01) to ~0.66 (p < 0.01) when using the Rib parameterizations (Figs. 6d–f), but the rms was consistently lower when using the Rib parameterizations at both sites.

Fig. 5.
Fig. 5.

Density plot showing the relationship between the observed u* at Belle Mina, AL, and u* computed using (a) M–O relationships from the literature, (b) M–O relationships developed using the LAFE datasets, and (c) Rib-based relationships developed using the LAFE datasets for wind speeds < 2.0 m s−1. (d)–(f) As in (a)–(c), but for 10-m wind speeds >2.0 m s−1. The dotted line shows the 1:1 line, and the solid line shows the line of best fit. N, r, rms, and the formula for the best-fit line are shown at the bottom right of each panel.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0047.1

Fig. 6.
Fig. 6.

Density plot showing the relationship between the observed u* at Cullman, AL, and u* computed using (a) M–O relationships from the literature, (b) M–O relationships developed using the LAFE datasets, and (c) Rib-based relationships developed using the LAFE datasets for wind speeds <2.5 m s−1. (d)–(f) As in (a)–(c), but for 10-m wind speeds > 2.5 m s−1. The dotted line shows the 1:1 line, and the solid line shows the line of best fit. N, r, rms, and the formula for the best-fit line are shown at the bottom right of each panel.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0047.1

b. H parameterizations

In addition to evaluating the new Rib similarity relationships for u*, we compared the parameterized H from the new Rib similarity relationships with the MOST similarity relationships. For the reasons described in the previous section, we used the MOST relationships for ζ < 0 from Högström (1988). In Högström’s formulation, ϕh = 0.95(1 − 11.6ζ)−0.50 for ζ < 0 and ϕh = 0.95 + 7.8ζ for ζ > 0.

We found that the legacy MOST relationships resulted in r that was 85 (p < 0.01) when applying the legacy MOST parameterizations to the Belle Mina dataset (Fig. 7a) and was 0.77 (p < 0.01) when using the LAFE-derived MOST parameterizations (Fig. 7b). In these cases, mb was 0.88 and 0.56, respectively. There was improvement to mb when using the Rib parameterizations as mb was 1.09 (Fig. 7c). We found significant improvements at Cullman when using the Rib parameterizations to compute H. For the MOST parameterizations, r was ~0.4 but 0.57 for the Rib parameterizations (Fig. 8c). Furthermore, mb was 0.90 when using the Rib parameterizations but 0.58 and 0.41 when using the legacy- and LAFE-derived MOST relationships (Figs. 8a–c).

Fig. 7.
Fig. 7.

As in Fig. 3, but for H.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0047.1

Fig. 8.
Fig. 8.

As in Fig. 4, but for H.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0047.1

As we did for u*, we differentiated between unstable regimes and stable regimes, as well as between weak wind regimes and strong wind regimes and reevaluated the performance of the MOST and Rib H parameterizations. We found that the Rib H parameterizations performed significantly better than the MOST H parameterizations at both sites (Fig. 7d–f, 8d–f); mb was 1.13 at Belle Mina (compared with 0.82 and 0.46, which were the values of mb from the MOST relationships), and at Cullman mb was 0.95, whereas mb was 0.53 and 0.34 for the classical MOST parameterizations and for the MOST parameterizations applied to the LAFE datasets. All H parameterizations performed poorly under stable conditions at Belle Mina and Cullman (Figs. 7g–i, 8g–i).

When filtering H by observed wind speed, we found mb was 0.74 for the relationship using the traditional MOST parameterizations (Fig. 9a) and 0.32 for the relationship using the LAFE MOST parameterizations (Fig. 9b), whereas at Cullman mb was 0.42 and 0.23, respectively (Figs. 10a,b). The Rib parameterizations, however, better predicted H under weak winds; mb was 1.15 and 0.95 at Belle Mina and Cullman (Figs. 9c, 10c). Similar levels of improvement were noted for the subset of cases with higher wind speeds; the relationship between the modeled and observed H was nearest the 1:1 line when using the Rib parameterizations (Figs. 9d–f, 10d–f).

Fig. 9.
Fig. 9.

As in Fig. 5, but for H.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0047.1

Fig. 10.
Fig. 10.

As in Fig. 6, but for H.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0047.1

c. E parameterizations

To evaluate the legacy MOST relationships for E, we used the relationship ϕq = 0.95(1 − 11.6ζ)−0.50 from Högström (1988) for ζ < 0 and ϕq = 0.95 + 7.8ζ for ζ > 0. We found that the scatter in the relationship between the parameterized and observed fluxes was largest for E. The magnitude of E was underestimated at Belle Mina, but overestimated at Cullman. At Belle Mina and Cullman, the Rib parameterizations had the lowest rms and highest r (Figs. 11a–c, 12a–c). When filtering by stability, we found that the Rib parameterizations performed similar to the MOST parameterizations at both sites (Figs. 11d–f,12d–f). Under stable conditions, however, the Rib parameterizations performed worse than the MOST parameterizations both sites (Figs. 11g–i, 12g–i).

Fig. 11.
Fig. 11.

As in Fig. 3, but for E.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0047.1

Fig. 12.
Fig. 12.

As in Fig. 4, but for E.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0047.1

Under low wind speeds, the Rib parameterization performed better than for the MOST parameterizations at Belle Mina (Figs. 13a–c), as mb was 0.22 and 0.52 for the classical MOST parameterizations and the MOST parameterizations derived from the LAFE datasets, respectively. For the Rib parameterizations, however, mb was 0.76, and r was higher than r for the MOST parameterizations. At Cullman, r was highest for the Rib parameterization, but mb was closest to 1 for the classical MOST parameterization (Figs. 14a–c). For the subset of cases with the strong winds, r was highest for the Rib parameterizations at both sites (Figs. 13d–f, 14d–f).

Fig. 13.
Fig. 13.

As in Fig. 5, but for E.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0047.1

Fig. 14.
Fig. 14.

As in Fig. 6, but for E.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0047.1

5. Discussion

In the previous section, we described the relationships between the parameterized and observed values of u*, H, and E but have not discussed in detail the significant amount of scatter that is present between the parameterized and observed u*, H, and E, particularly in the case of E. For example, the scatter in E is consistent with Lee and Buban (2020), who reported the low correlations between the parameterized and observed near-surface moisture gradients. The findings from Lee and Buban (2020) and those reported in the present study underscore the difficulties associated with parameterizing moisture fluxes, which arises because moisture is nearly passive and the moisture flux above the surface layer is unconstrained. This dissimilarity has been recently documented in the literature and, as summarized by, e.g., Li and Bou-Zeid (2011), has been attributed to, e.g., advection (e.g., Lee et al. 2004) and land surface heterogeneities (e.g., Detto et al. 2008).

Another possible reason for the discrepancies between the parameterized and observed fluxes is that we developed the parameterizations over the semiarid U.S. Central Plains and applied them to the Southeast United States where E is a typically a larger component of the surface energy balance. In the Southeast United States, we expect for processes such as moisture advection and land surface heterogeneity to be more important drivers of moisture dissimilarity, yet these processes are not represented in the MOST or Rib parameterizations. Yet, advection and land surface heterogeneities can play an important role in energy exchange between the land surface and atmosphere and contribute to the absence of closure in the surface energy balance (e.g., Foken 2008; Xu et al. 2017; Butterworth et al. 2021). To estimate the impact these errors may have, we evaluate the surface energy balance closure. The observed net radiation Rn1 is calculated following Eq. (53):
Rn1=SWinSWout+LWinLWout.
In Eq. (53), SWin, SWout, LWin, and LWout are the incoming shortwave radiation, outgoing shortwave radiation, incoming longwave radiation, and outgoing longwave radiation, respectively. If we assume surface energy balance closure, we can also compute net radiation, Rn2, following Eq. (54):
Rn2=H+E+G.
In Eq. (54), G is the ground heat flux, and the remaining terms have been previously defined. We computed G using the gradient method (e.g., Sauer and Horton 2005), whereby G is a function of the soil temperature difference measured between 2 and 5 cm below ground and the soil thermal conductivity (see Lee et al. 2019b for more details).

In the LAFE tower datasets, Rn1Rn2, indicating that the assumption of surface energy balance closure was correct. However, we do not observe full energy balance closure at Belle Mina and Cullman, as Rn1 is typically about 10% larger than Rn2, which suggests the greater importance of e.g., moisture advection and land surface heterogeneities on surface-layer exchange in this region and which may contribute to the scatter present between the parameterized and observed values of u*, H, and E.

6. Summary and outlook

In the present study we expanded recent work that suggested that Rib parameterizations better represent near-surface wind and temperature gradients under unstable regimes. We did this by developing Rib parameterizations for both unstable and stable regimes and evaluated how well the parameterizations predicted u*, H, and E using near-surface observations from LAFE. We used observations from a fully independent dataset obtained in a different region of the United States and compared parameterizations of u*, H, and E against 1) traditional parameterizations derived from MOST, and 2) parameterizations that we obtained when applying MOST to the LAFE datasets. We found that fitting coefficients in the MOST parameterizations that we developed using the LAFE datasets differed from those in classical MOST parameterizations, which we attributed to the land surface heterogeneity present in the area surrounding each of the LAFE micrometeorological towers. Despite deviations from classical studies, we found that the Rib parameterizations generally performed as well as or, in some cases, better than the traditional MOST parameterizations and also the MOST parameterizations developed using the LAFE datasets. The improvement was more evident for H, and was most notable for H under unstable conditions, based on the increase in the slope of the relationship between the observed and parameterized values. The largest amount of scatter between the parameterized and observed values was for the E parameterizations, which we suspect arises because the parameterizations were developed using observations over a semiarid region and then were evaluated in a region where the latent heat flux is a larger term in the surface energy balance.

Overall, we note that the MOST parameterizations and Rib parameterizations capture the same underlying physics. In fact, there is nothing explicitly new about the physics represented in the Rib parameterizations; instead, these are simply an alternative way to represent near-surface exchanges of heat, moisture, and momentum and that use a different stability term (Rib rather than ζ). The Rib parameterizations themselves are not immune to the self-correlation present in MOST. As noted in e.g., Lee and Buban (2020) and briefly summarized here, in MOST, the self-correlation is present in u*. In the Rib parameterizations, the self-correlation occurs in the wind gradients; however, as previously noted, wind gradients are easier to measure than u* which requires high-frequency wind measurements. As sampling wind gradients is easier than sampling vertical u*, this allows the potential to test and extend the Rib parameterizations above 10 m, which is the maximum sampling height in the present study. Although it is encouraging that other studies (e.g., Grachev et al. 2018) that had measurements from additional sampling heights obtained similar results as the present study, in the future it will be important to extend the newly suggested similarity relationships above 10 m AGL in order to test their applicability within deeper surface layer depths. To this end, in a follow up study we will use small unmanned aircraft systems (sUAS) outfitted with onboard sensors for sampling temperature, moisture, and wind (e.g., Lee et al. 2017, 2019a,b) to evaluate the newly suggested parameterizations within and above the surface layer. This will be done by performing vertical sUAS profiles up to and exceeding 1 km AGL conducted multiple times per day across different meteorological regimes.

Overall, this work in combination with work by Lee and Buban (2020), suggests the need to consider 1) using Rib, rather than ζ, as a stability parameter, and 2) modifying the similarity equations used to represent momentum, heat, and moisture fluxes in NWP models. Thus, in addition to evaluating the newly suggested parameterizations within and above the surface layer, future research efforts should focus on applying Rib parameterizations to areas with different land surface types (i.e., different surface roughnesses) to make theses parameterizations more generalizable to NWP models. For example, long-term datasets from the NOAA Air Resources Laboratory (ARL) Atmospheric Turbulence and Diffusion Division (ATDD) Surface Energy Balance Network (SEBN) and recent field studies, e.g., the Chequamegon Heterogeneous Ecosystem Energy-balance Study Enabled by a High-density Extensive Array of Detectors (CHEESEHEAD) in northern Wisconsin (Butterworth et al. 2021), provide rich datasets over different land surfaces with which to further hone the new Rib parameterizations. Doing so will allow to us to make possible corrections to Cu, Ct, and Cr that are a function of the roughness lengths of momentum, heat, and moisture, respectively.

Furthermore, it will be important to test the new Rib parameterizations in large-eddy simulation models, e.g., the Collaborative Model for Multiscale Atmospheric Simulation (COMMAS) (e.g., Wicker and Wilhelmson 1995; Buban et al. 2012) and experimental versions of the HRRR and its successors, i.e., the Rapid Refresh Forecast System (RRFS) (Ladwig et al. 2019). Doing so will ultimately permit new Rib parameterizations to be implemented into the next generation of weather forecasting models such as the Unified Forecast System (UFS) (Tallapragada 2018) and future versions of operational NWP models (e.g., Olson et al. 2021).

Acknowledgments

We thank Mr. Randy White and Mr. Mark Heuer from NOAA/ARL/ATDD for helping us to install and maintain the meteorological instruments obtained from LAFE and VORTEX-SE that were used in this study. We gratefully acknowledge Dr. David D. Turner of the NOAA/Global Systems Laboratory (GSL) and Dr. Volker Wulfmeyer of the University of Hohenheim for organizing LAFE and for helpful discussions regarding this work. We thank the three anonymous reviewers whose comments and suggestions significantly helped us improve this work. Last, we note that the results and conclusions of this study, as well as any views expressed herein, are those of the authors and do not necessarily reflect those of NOAA or the Department of Commerce.

Data availability statement

Datasets from LAFE are available from the Department of Energy Atmospheric Radiation Measurement (ARM) website at https://www.arm.gov/research/ campaigns/sgp2017lafenoaaarlatdd. The datasets obtained from Belle Mina and Cullman during the 2016 and 2017 VORTEX-SE field campaigns are available at https://data.eol.ucar.edu/dataset/527.008 and https://data.eol.ucar.edu/dataset/541.021, respectively.

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  • Benjamin, S. G., and Coauthors, 2016: A North American hourly assimilation and model forecast cycle: The Rapid Refresh. Mon. Wea. Rev., 144, 16691694, https://doi.org/10.1175/MWR-D-15-0242.1.

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  • Butterworth, B. J., and Coauthors, 2021: Connecting land–atmosphere interactions to surface heterogeneity in CHEESEHEAD19. Bull. Amer. Meteor. Soc., 102, E421E445, https://doi.org/10.1175/BAMS-D-19-0346.1.

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  • Cantrell, C. A., 2008: Review of methods for linear least-squares fitting of data and application to atmospheric chemistry problems. Atmos. Chem. Phys., 8, 54775487, https://doi.org/10.5194/acp-8-5477-2008.

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    • Search Google Scholar
    • Export Citation
  • Buban, M. S., C. L. Ziegler, E. R. Mansell, and Y. P. Richardson, 2012: Simulation of dryline misovortex dynamics and cumulus formation. Mon. Wea. Rev., 140, 35253551, https://doi.org/10.1175/MWR-D-11-00189.1.

    • Search Google Scholar
    • Export Citation
  • Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971: Flux-profile relationships in the atmospheric surface layer. J. Atmos. Sci., 28, 181189, https://doi.org/10.1175/1520-0469(1971)028<0181:FPRITA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Butterworth, B. J., and Coauthors, 2021: Connecting land–atmosphere interactions to surface heterogeneity in CHEESEHEAD19. Bull. Amer. Meteor. Soc., 102, E421E445, https://doi.org/10.1175/BAMS-D-19-0346.1.

    • Search Google Scholar
    • Export Citation
  • Cantrell, C. A., 2008: Review of methods for linear least-squares fitting of data and application to atmospheric chemistry problems. Atmos. Chem. Phys., 8, 54775487, https://doi.org/10.5194/acp-8-5477-2008.

    • Search Google Scholar
    • Export Citation
  • Deardorff, J. W., 1972: Parameterization of the planetary boundary layer for use in general circulation models. Mon. Wea. Rev., 100, 93106, https://doi.org/10.1175/1520-0493(1972)100<0093:POTPBL>2.3.CO;2.

    • Search Google Scholar
    • Export Citation
  • Detto, M., G. Katul, M. Mancini, N. Montaldo, and J. D. Albertson, 2008: Surface heterogeneity and its signature in higher-order scalar similarity relationships. Agric. For. Meteor., 148, 902916, https://doi.org/10.1016/j.agrformet.2007.12.008.

    • Search Google Scholar
    • Export Citation
  • Dumas, E. J., T. R. Lee, M. Buban, and B. Baker, 2016: Small Unmanned Aircraft System (sUAS) measurements during the 2016 Verifications of the Origins of Rotation in Tornadoes Experiment Southeast (VORTEX-SE). NOAA Tech. Memo. OAR ARL-273, 29 pp., https://doi.org/10.7289/V5/TM-OAR-ARL-273.

  • Dumas, E. J., T. R. Lee, M. Buban, and B. Baker, 2017: Small Unmanned Aircraft System (sUAS) measurements during the 2017 Verifications of the Origins of Rotation in Tornadoes Experiment Southeast (VORTEX-SE). NOAA Tech. Memo. OAR ARL-274, 49 pp., https://doi.org/10.7289/V5/TM-OAR-ARL-274.

  • Dyer, A. J., 1974: A review of flux-profile relationships. Bound.-Layer Meteor., 7, 363372, https://doi.org/10.1007/BF00240838.

  • Dyer, A. J., and B. B. Hicks, 1970: Flux-gradient relationships in the constant flux layer. Quart. J. Roy. Meteor. Soc., 96, 715721, https://doi.org/10.1002/qj.49709641012.

    • Search Google Scholar
    • Export Citation
  • Dyer, A. J., and E. F. Bradley, 1982: An alternative analysis of flux-gradient relationships at the 1976 ITCE. Bound.-Layer Meteor., 22, 319, https://doi.org/10.1007/BF00128053.

    • Search Google Scholar
    • Export Citation
  • Foken, T., 2006: 50 years of the Monin–Obukhov similarity theory. Bound.-Layer Meteor., 119, 431447, https://doi.org/10.1007/s10546-006-9048-6.

    • Search Google Scholar
    • Export Citation
  • Foken, T., 2008: The energy balance closure problem: An overview. Ecol. Appl., 18, 13511367, https://doi.org/10.1890/06-0922.1.

  • Grachev, A. A., and C. W. Fairall, 1997: Dependence of the Monin–Obukhov stability parameter on the bulk Richardson number over the ocean. J. Appl. Meteor., 36, 406414, https://doi.org/10.1175/1520-0450(1997)036<0406:DOTMOS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Grachev, A. A., and Coauthors, 2018: Air–sea/land interaction in the coastal zone. Bound.-Layer Meteor., 167, 181210, https://doi.org/10.1007/s10546-017-0326-2.

    • Search Google Scholar
    • Export Citation
  • Hicks, B. B., 1978a: Some limitations of dimensional analysis and power laws. Bound.-Layer Meteor., 14, 567569, https://doi.org/10.1007/BF00121895.

    • Search Google Scholar
    • Export Citation
  • Hicks, B. B., 1978b: Comments on ‘the characteristics of turbulent velocity components in the surface layer under convective conditions’, by H. A. Panofsky, H. Tennekes, D. H. Lenschow, and J. C. Wyngaard. Bound.-Layer Meteor., 15, 255258, https://doi.org/10.1007/BF00121926.

    • Search Google Scholar
    • Export Citation
  • Hicks, B. B., 1981: An examination of turbulence statistics in the surface boundary layer. Bound.-Layer Meteor., 21, 389402, https://doi.org/10.1007/BF00119281.

    • Search Google Scholar
    • Export Citation
  • Hicks, B. B., 1995: Monin–Obukhov similarity—An historical perspective. Preprints, 11th Symp. on Boundary Layers and Turbulence, Charlotte, NC, Amer. Meteor. Soc., 1–4.

  • Högström, U., 1988: Non-dimensional wind and temperature profiles in the atmospheric surface layer: A re-evaluation. Topics in Micrometeorology: A Festschrift for Arch Dyer, B. B. Hicks, Ed., Springer, 152 pp.

  • Högström, U., 1996: Review of some basic characteristics of the atmospheric surface layer. Bound.-Layer Meteor., 78, 215246, https://doi.org/10.1007/BF00120937.

    • Search Google Scholar
    • Export Citation
  • Jiang, L., S. Islam, and T. N. Carlson, 2004: Uncertainties in latent heat flux measurement and estimation: Implications for using a simplified approach with remote sensing data. Can. J. Remote Sens., 30, 769787, https://doi.org/10.5589/m04-038.

    • Search Google Scholar
    • Export Citation
  • Jiménez, P. A., J. Dudhia, J. F. González-Rouco, J. Navarro, J. P. Montávez, and E. García-Bustamante, 2012: A revised scheme for the WRF surface layer formulation. Mon. Wea. Rev., 140, 898918, https://doi.org/10.1175/MWR-D-11-00056.1.

    • Search Google Scholar
    • Export Citation
  • Johansson, C., A. S. Smedman, U. Högström, J. G. Brasseur, and S. Khanna, 2001: Critical test of the validity of Monin–Obukhov similarity during convective conditions. J. Atmos. Sci., 58, 15491566, https://doi.org/10.1175/1520-0469(2001)058<1549:CTOTVO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kader, B. A., and A. M. Yaglom, 1972: Heat and mass transfer laws for fully turbulent wall flows. Int. J. Heat Mass Transf., 15, 23292351, https://doi.org/10.1016/0017-9310(72)90131-7.

    • Search Google Scholar
    • Export Citation
  • Klink, K., 1999: Climatological mean and interannual variance of United States surface wind speed, direction and velocity. Int. J. Climatol., 19, 471488, https://doi.org/10.1002/(SICI)1097-0088(199904)19:5<471::AID-JOC367>3.0.CO;2-X.

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