## 1. Introduction

*U*), potential temperature (

*θ*), and specific humidity (

*q*) are functions of a dimensionless stability length

*ζ*:

*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

*ϕ*

_{m,h,q}under unstable atmospheric regimes are

*α*

_{m,h,q}and

*β*

_{m,h,q}are coefficients determined empirically;

*γ*is −0.25 for

*ϕ*

_{m}and is −0.50 in the equations for

*ϕ*

_{h}and

*ϕ*

_{q}; and

*ζ*is the dimensionless Monin–Obukhov stability length. Previous work has found

*α*

_{m,h,q}≈ 1, and that

*β*

_{m,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

*d*is the displacement height of the vegetation,

*z*is the sampling height, and

*L*is the Monin–Obukhov length scale defined as

*u*

_{*}is the friction velocity,

*g*is the gravitational acceleration,

*θ*

_{υ}is the virtual potential temperature,

*ζ*> 0, MOST relationships have been found to have a linear relationship that has the following form:

*η*

_{m,h,q}and

*ε*

_{m,h,q}are coefficients determined empirically. Earlier studies had suggested that

*ϕ*

_{m}=

*ϕ*

_{h}=

*ϕ*

_{q}, whereby the coefficients

*η*

_{m,h,q}and

*ε*

_{m,h,q}are 1 and 5, respectively (e.g., Dyer 1974). Although

*η*

_{m,h,q}= 1 has been reported in other studies (e.g., Businger et al. 1971; Dyer 1974; Högström 1988, 1996; Foken 2006),

*ε*

_{m}has been determined to range from 4.7 (Businger et al. 1971) to 6 for

*ϕ*

_{m}(Högström 1988), and

*ε*

_{h}has been determined to range from 4.7 (Businger et al. 1971) to 8 for

*ϕ*

_{h}(Högström 1996). It is oftentimes assumed that

*ϕ*

_{h}=

*ϕ*

_{q}(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).

_{b}) following e.g., Stull (1988):

*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 Ri_{b} < 0. If new parameterizations that use Ri_{b} 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 Ri_{b} framework across both unstable and stable atmospheric regimes using datasets obtained from the Land Atmosphere Feedback Experiment (LAFE) and to evaluate how the Ri_{b} 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 Ri_{b} 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 Ri_{b} parameterizations which have 1) a different stability term (i.e., Ri_{b} rather than *ζ*), and 2) a different functional form of the similarity equations.

## 2. Development of M–O and Ri_{b} 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 (*z*_{0}) 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 *z*_{0} 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 Ri_{b} 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 Ri_{b} 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 Ri_{b} 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 Ri_{b} 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 Ri_{b} parameterizations

#### 1) M–O parameterizations

*ζ*and

*ϕ*

_{m,h,q}(Fig. 1). We used weighted fits when performing the coefficient-fitting process. Following from Markowski et al. (2019), the uncertainty in

*ϕ*

_{m}, which we define as

*δϕ*

_{m}, is approximated as

*ϕ*

_{h}and

*ϕ*

_{q}, defined as

*δϕ*

_{h}and

*δϕ*

_{q}, 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.

The functions for *ϕ*_{m,h,q} for *ζ* < 0 take the form of Eqs. (1)–(3), and the fitting coefficients *α*_{m,h,q} and *β*_{m,h,q} for *ϕ*_{m,h,q} as a function of *ζ* are shown in Table 1. We note that there are different numbers of data points for *ϕ*_{m}, *ϕ*_{h}, and *ϕ*_{q} because the filtering criteria were applied separately to each variable (cf. section 2b). In the case of stable conditions and as noted earlier, *ϕ*_{m,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 *ε*_{m,h,q} and *η*_{m,h,q} from these functions are shown in Table 2.

Best-fit parameters using nonlinear least squares for the following equations: *ϕ*_{m} = *α*_{m}(1 − *β*_{m}*ζ*)^{−0.25}, *ϕ*_{h} = *α*_{h}(1 − *β*_{h}*ζ*)^{−0.5}, and *ϕ*_{q} = *α*_{q}(1 − *β*_{q}*ζ*)^{−0.5} for *ζ* < 0. All functions apply over the range −2 < *ζ* < 0; for *ϕ*_{m,q} these functions apply over the range 0 < *ϕ*_{m,q} < 5 and for *ϕ*_{h} these functions apply over the range 0 < *ϕ*_{h} < 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., 1*σ*) for each of the best-fit parameters for each variable. All correlations are significant at the *p* < 0.05 confidence level.

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

#### 2) Ri_{b} parameterizations

_{b}and

*C*

_{u,t,r}, we compute the weights of

*C*

_{u,t,r}following the procedure summarized in the previous section prior to applying the least squares regressions to the filtered LAFE datasets. Ri

_{b}is a function of the friction coefficient, heat-transfer coefficient, and moisture-transfer coefficient, which we define as

*C*

_{u},

*C*

_{t}, and

*C*

_{r}, respectively. Deardorff (1972) defined these terms, assuming a constant flux layer and single-layer ABL model. Lee and Buban (2020) modified

*C*

_{u},

*C*

_{t}, and

*C*

_{r}such that

*C*

_{u}is a function of

*u*

_{*}and the wind speed 10 m AGL, which we define as

*U*

_{10}:

*C*

_{t}and

*C*

_{r}are calculated as

*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).

_{b}< 0),

*C*

_{u},

*C*

_{t}, and

*C*

_{r}are related to Ri

_{b}via a 1/3 power law that has the following form:

*λ*

_{u,t,r}and

*ω*

_{u,t,r}are determined empirically and appear in Table 3. By writing

*C*

_{u},

*C*

_{t}, and

*C*

_{r}as a function of Ri

_{b}, these terms have dependence on

*z*; however, we acknowledge these formulations have no explicit dependence of

*C*

_{u},

*C*

_{t}, and

*C*

_{r}on the roughness lengths for momentum, heat, and moisture, respectively. We revisit this point later in the study.

Best-fit parameters using nonlinear least squares for *C*_{u,t,r} = *λ*_{u,t,r}(1 − *ω*_{u,t,r}Ri_{b})^{1/3}. These functions apply for −2 < Ri_{b} < 0, 0 < *C*_{u} < 0.2, and 0 < *C*_{t,r} < 2 to remove outliers. We also report *r*, *N*, and rms for each variable, as well as 1*σ* for each of the best-fit parameters for each variable. All correlations are significant at the *p* < 0.05 confidence level.

_{b},

*C*

_{u},

*C*

_{t}, and

*C*

_{r}for Ri

_{b}> 0 also was nonlinear and could be expressed via the following relationship, whereby

*χ*

_{u,t,r}and

*γ*

_{u,t,r}are the fitting coefficients determined through least squares regression and appear in Table 4:

#### 3) Discussion

We acknowledge that *α*_{m,h,q} and *β*_{m,h,q} obtained from the M–O fits, as well as *λ*_{u,t,r} and *ω*_{u,t,r} obtained from the Ri_{b} 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 Ri_{b}, 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 *ϕ*_{m,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, *α*_{m,h,q} ≈ *η*_{m,h,q} ≈ 1 (e.g., Dyer and Hicks 1970; Dyer 1974; Dyer and Bradley 1982; Maronga and Reuder 2017). In contrast, we found that *α*_{m,h,q} and *η*_{m,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 *ϕ*_{m,h,q} and *ζ* (Fig. 1), as well between *C*_{u,t,r}, and Ri_{b} (Fig. 2). The rms for *ϕ*_{m,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 *C*_{u,t,r} as a function of Ri_{b}, 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 *ϕ*_{m,h,q} and to the coefficients *ω* and *γ* than to the coefficients *λ* and *χ* in the functions for *C*_{u,t,r}. The sensitivity is reflected in the 1*σ* values for each of the best fit parameters that are reported in Tables 1–4. We also note that nearly all of the relationships have convergence at 0; the exception is the relationship between *ϕ*_{q} and *ζ* and is caused by the few number of valid data points for stable conditions (Table 2). However, for the remaining functions, since *λ* ≈ *χ* (cf. Tables 3 and 4), there is convergence at 0 which is expected to eliminate potential numerical instabilities when implementing the Ri_{b} parameterizations into NWP models.

Best-fit parameters using nonlinear least squares for _{b} < 0.5, 0 < *C*_{u} < 0.1, and 0 < *C*_{t,r} < 1 to remove outliers. We also report *r*, *N*, and rms for each variable, as well as 1*σ* for each of the best-fit parameters for each variable. All correlations are significant at the *p* < 0.05 confidence level.

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, *K*_{m}, and the turbulent conductivity, *K*_{h}. 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, *ϕ*_{m} = 1.57(1 − 6.71*ζ*)^{−0.25} and *ϕ*_{h} = 1.06(1 − 1.10*ζ*)^{−0.25} (cf. Table 1); when *ζ* > 0, *ϕ*_{m} = 4.04*ζ* + 1.50 and *ϕ*_{h} = 10.9*ζ* + 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 Ri_{b} parameterizations

### a. Datasets

We evaluated the M–O and Ri_{b} 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 CO_{2}/H_{2}O 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

*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

*u*

_{a}. In the present study we computed the wind at 10 m AGL:

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 *u*_{a}, and we used 0.15 and 0.25 m for *z*_{0} at Belle Mina and Cullman, respectively. The values for *d* and *z*_{0} 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).

*ψ*

_{m}is obtained by integrating the momentum similarity function following e.g., Panofsky (1963) and Jiménez et al. (2012):

*ϕ*

_{m}(

*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

*ψ*

_{m}as

*ψ*

_{m}is computed following Eq. (21) where

*ε*

_{m}is obtained from the least squares regression:

*u*

_{*}as a function of

*u*

_{a}and

*ψ*

_{m}:

#### 2) *H* parameterization

*H*, we integrate Eq. (2) so that Δ

*θ*is computed as

*z*

_{1}and

*z*

_{2}are the two heights over which we compute Δ

*θ*. We note that Jiménez et al. (2012) used

*z*

_{1}as the surface value. However, in the absence of surface measurement we take

*z*

_{1}to be the measurements made 2 m AGL following e.g., Seidel et al. (2012). We obtain

*ψ*

_{h}by integrating the heat similarity function, again following, e.g., Panofsky (1963) and Jiménez et al. (2012):

*ϕ*

_{m}(

*ζ*), we use the

*ϕ*

_{h}(

*ζ*) function suggested by Högström (1988) and also the

*ϕ*

_{h}(

*ζ*) function obtained from the LAFE datasets. When

*ζ*< 0, we compute

*ψ*

_{h}as

*ψ*

_{h}is computed following Eq. (26), again where

*ε*

_{h}is obtained from the least squares regression:

*θ*and Eq. (5) for

*θ*

_{*}to compute the kinematic form of the heat flux

*u*

_{*}following Eq. (22):

*c*

_{p}and

*ρ*, which are the specific heat of air and the air density, respectively:

#### 3) *E* parameterization

*ψ*

_{q}has the same functional form as

*ψ*

_{h}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

*L*

_{υ}:

*L*

_{υ}is temperature dependent:

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 Ri_{b} parameterizations

#### 1) ${u}_{*}$ parameterization

_{b}parameterizations such that, for two heights,

*z*

_{1}and

*z*

_{1}:

*u*

_{*}, we make the assumption that the surface winds are 0 m s

^{−1}so that

*u*

_{a}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:

_{b}parameterizations for

*u*

_{*}. Equation (33) can be rewritten so that

*u*

_{*}is calculated using the relationship between

*C*

_{u}and Ri

_{b}[i.e.,

*C*

_{u}(Ri

_{b})] found in the present study:

#### 2) *H* parameterization

*u*

_{*}from Eq. (34), which we term

*c*

_{p}and

*ρ*yields the dynamic form of the sensible heat flux, shown below:

#### 3) *E* parameterization

*ρ*and by Eq. (31) for

*L*

_{υ}yields the following equation for the dynamic form of the latent heat flux:

#### 4) Use of ${\mathrm{Ri}}_{b}$ parameterizations of ${u}_{*}$ , *H*, and *E* in NWP models

*H*and

*E*are to be implemented into surface layer and boundary layer parameterization schemes in NWP models, the Ri

_{b}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 Δ

*θ*and Δ

*q*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 Ri

_{b}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 Δ

*z*term in the Ri

_{b}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

*θ*

_{υs}and

*q*

_{s}which appear in Eqs. (14) and (15), respectively, in the calculation of

*C*

_{t}and

*C*

_{r}, 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, Δ

*z*, could then be used to calculate

*u*

_{*},

*H*, and

*E*so that form of the Ri

_{b}parameterizations under unstable conditions is

*u*

_{*},

*H*, and

*E*under stable conditions are computed following Eqs. (44)–(46):

### d. Comparison of M–O and Ri_{b} parameterizations

_{b}-derived

*u*

_{*},

*H*, and

*E*. Following from Markowski et al. (2019), the error in

*u*

_{*}, i.e.,

*δu*

_{*}, is estimated via Eq. (47):

^{2}s

^{2}.

*δH*and

*δE*, respectively, following e.g., Jiang et al. (2004):

^{−1}and 0.087 m s

^{−1}, respectively.

To quantify the instrumental uncertainties in the parameterized *u*_{*}, *H*, and *E*, we use the 1*σ* errors reported in Tables 1–4 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.

*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 Ri

_{b}-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):

*W*

_{i}and

*b*are calculated using Eqs. (51) and (52), respectively:

*w*

_{xi}and

*w*

_{yi}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.

*x*

_{i}and

*y*

_{i}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 *m*_{b} 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 Ri_{b} 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 *ϕ*_{m} = (1 − 19.3*ζ*)^{−0.25} for *ζ* < 0 and *ϕ*_{m} = 1 + 6*ζ* 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 *m*_{b}, 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; *m*_{b} 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 Ri_{b} functions developed using the LAFE datasets yielded slightly lower values of *m _{b}* and

*r*, as

*m*and

_{b}*r*were 0.91 and 0.75 (

*p*< 0.01), respectively (Fig. 3c) at Belle Mina. At Cullman,

*m*and

_{b}*r*were 0.75 and 0.79, respectively (Fig. 4c).

From the above analyses, we conclude that the new Ri_{b} 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 Ri_{b} 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 Ri_{b} < 0) (Figs. 3d–f, 4d–f). MOST parameterizations at both Belle Mina and Cullman, and *r* was higher for the Ri_{b} parameterizations at Belle Mina. In stable regimes (i.e., *ζ* >0 and Ri_{b} > 0), the Ri_{b} parameterizations had the best 1:1 agreement between the parameterized and observed values (Figs. 3g–i); at Cullman, though, *m _{b}* was lowest for the Ri

_{b}parameterizations as compared with the MOST parameterizations (Figs. 4g–i).

When we filtered *u*_{*} by wind speed we found that the Ri_{b} *u*_{*} parameterizations had the largest *m*_{b} 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 Ri_{b} parameterizations, respectively, and rms was lowest for the Ri_{b} parameterizations (Figs. 5d–f). The value of *r* decreased from ~0.77 (*p* < 0.01) to ~0.66 (*p* < 0.01) when using the Ri_{b} parameterizations (Figs. 6d–f), but the rms was consistently lower when using the Ri_{b} parameterizations at both sites.

### b. H parameterizations

In addition to evaluating the new Ri_{b} similarity relationships for *u*_{*}, we compared the parameterized *H* from the new Ri_{b} 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, *ϕ*_{h} = 0.95(1 − 11.6*ζ*)^{−0.50} for *ζ* < 0 and *ϕ*_{h} = 0.95 + 7.8*ζ* 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, *m*_{b} was 0.88 and 0.56, respectively. There was improvement to *m*_{b} when using the Ri_{b} parameterizations as *m*_{b} was 1.09 (Fig. 7c). We found significant improvements at Cullman when using the Ri_{b} parameterizations to compute *H*. For the MOST parameterizations, *r* was ~0.4 but 0.57 for the Ri_{b} parameterizations (Fig. 8c). Furthermore, *m*_{b} was 0.90 when using the Ri_{b} parameterizations but 0.58 and 0.41 when using the legacy- and LAFE-derived MOST relationships (Figs. 8a–c).

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 Ri_{b} *H* parameterizations. We found that the Ri_{b} *H* parameterizations performed significantly better than the MOST *H* parameterizations at both sites (Fig. 7d–f, 8d–f); *m*_{b} was 1.13 at Belle Mina (compared with 0.82 and 0.46, which were the values of *m*_{b} from the MOST relationships), and at Cullman *m*_{b} was 0.95, whereas *m _{b}* 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 *m _{b}* 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

*m*

_{b}was 0.42 and 0.23, respectively (Figs. 10a,b). The Ri

_{b}parameterizations, however, better predicted

*H*under weak winds;

*m*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

_{b}*H*was nearest the 1:1 line when using the Ri

_{b}parameterizations (Figs. 9d–f, 10d–f).

### c. E parameterizations

To evaluate the legacy MOST relationships for *E*, we used the relationship *ϕ*_{q} = 0.95(1 − 11.6*ζ*)^{−0.50} from Högström (1988) for *ζ* < 0 and *ϕ*_{q} = 0.95 + 7.8*ζ* 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 Ri_{b} parameterizations had the lowest rms and highest *r* (Figs. 11a–c, 12a–c). When filtering by stability, we found that the Ri_{b} parameterizations performed similar to the MOST parameterizations at both sites (Figs. 11d–f,12d–f). Under stable conditions, however, the Ri_{b} parameterizations performed worse than the MOST parameterizations both sites (Figs. 11g–i, 12g–i).

Under low wind speeds, the Ri_{b} parameterization performed better than for the MOST parameterizations at Belle Mina (Figs. 13a–c), as *m _{b}* was 0.22 and 0.52 for the classical MOST parameterizations and the MOST parameterizations derived from the LAFE datasets, respectively. For the Ri

_{b}parameterizations, however, m

_{b}was 0.76, and

*r*was higher than

*r*for the MOST parameterizations. At Cullman,

*r*was highest for the Ri

_{b}parameterization, but

*m*was closest to 1 for the classical MOST parameterization (Figs. 14a–c). For the subset of cases with the strong winds,

_{b}*r*was highest for the Ri

_{b}parameterizations at both sites (Figs. 13d–f, 14d–f).

## 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).

*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 Ri

_{b}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

*R*

_{n1}is calculated following Eq. (53):

_{in}, SW

_{out}, LW

_{in}, and LW

_{out}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,

*R*

_{n2}, following 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, *R*_{n1} ≈ *R*_{n2}, 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 *R*_{n1} is typically about 10% larger than *R*_{n2}, 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 Ri_{b} parameterizations better represent near-surface wind and temperature gradients under unstable regimes. We did this by developing Ri_{b} 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 Ri_{b} 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 Ri_{b} parameterizations capture the same underlying physics. In fact, there is nothing explicitly new about the physics represented in the Ri_{b} 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 (Ri_{b} rather than *ζ*). The Ri_{b} 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 Ri_{b} 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 Ri_{b} 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 Ri_{b}, 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 Ri_{b} 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 Ri_{b} parameterizations. Doing so will allow to us to make possible corrections to *C*_{u}, *C*_{t}, and *C*_{r} that are a function of the roughness lengths of momentum, heat, and moisture, respectively.

Furthermore, it will be important to test the new Ri_{b} 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 Ri_{b} 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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