Improving Stable Boundary-Layer Height Estimation Using a Stability-Dependent Critical Bulk Richardson Number

Abstract

For many decades, attempts have been made to find the universal value of the critical bulk Richardson number ($Ri_{Bc}$; defined over the entire stable boundary layer). By analyzing an extensive large-eddy simulation database and various published wind-tunnel data, we show that $Ri_{Bc}$ is not a constant, rather it strongly depends on bulk atmospheric stability. A (qualitatively) similar dependency, based on the well-known resistance laws, was reported by Melgarejo and Deardorff (J Atmos Sci 31:1324–1333, 1974) about forty years ago. To the best of our knowledge, this result has largely been ignored. Based on data analysis, we find that the stability-dependent $Ri_{Bc}$ estimates boundary-layer height more accurately than the conventional constant $Ri_{Bc}$ approach. Furthermore, our results indicate that the common practice of setting $Ri_{Bc}$ as a constant in numerical modelling studies implicitly constrains the bulk stability of the simulated boundary layer. The proposed stability-dependent $Ri_{Bc}$ does not suffer from such an inappropriate constraint.

Appendix A

1.1 A.1 Estimation of $h$ from LES

There is a general consensus in the literature that the estimation of $h$ from observational data is a challenging task. Over the years, several variables have been explored for this purpose, including (but not limited to): wind speed, wind-speed profile curvature, streamwise velocity variance, vertical velocity variance, momentum flux, buoyancy flux, and gradient Richardson number (see Vickers and Mahrt 2004; Steeneveld et al. 2007; Pichugina and Banta 2010 and references therein). Even though, in the case of observational data, different profiles could lead to significantly different estimates of $h$, this is not the case for the LES-generated data. In Fig. 6, we plot vertical profiles of velocity components, (resolved) variances, (total) momentum fluxes and (total) sensible heat fluxes. It is clear that $h$ estimated from most of these profiles is more-or-less similar. Sensible heat-flux profiles usually lead to marginally higher estimates of $h$ (see also Basu and Porté-Agel 2006; Stoll and Porté-Agel 2008).

Fig. 6

The panels from top to bottom correspond to the vertical profiles of horizontal velocity components, variances, momentum fluxes, and sensible heat fluxes, respectively. The simulation shown on the left panel used $G = 12\,\text{ m }\,\text{ s }^{-1}$ and $C = 0.75\,\text{ K }\,\text{ h }^{-1}$, whereas the right one used $G = 6\, \text{ m }\,\text{ s }^{-1}$ and $C = 1.25\,\text{ K }\,\text{ h }^{-1}$. All these panels represent the last 6 h of the simulations. The 25th, 50th, and 75th percentiles are shown to depict the temporal variabilities. On each panel, $h$ estimated from the peak of LLJ is overlayed

Similar to Fig. 2 (top-left panel) and Fig. 4, the dependencies of $Ri_{Bc}$ on $h/L$ are shown in Fig. 7. In these panels, the vertical profiles of momentum flux (left panel) and (resolved) horizontal velocity variance (right panel) are utilized for the estimation of $h$. The locations where the momentum flux and (resolved) horizontal velocity variance decrease to less than 10 % of their respective surface value are denoted as $h$. These estimates of $h$ in conjunction with Eq. 2 are used for the calculation of $Ri_{Bc}$. The scatter and the slopes of the linear regression in these figures are slightly larger than those in the top-left panel of Fig. 2. However, the trend is once again clear. Without any doubt, $Ri_{Bc}$ increases with increasing stability.

Fig. 7

Variation of $Ri_{Bc}$ with respect to $h/L$ based on LES-generated samples. The red lines denote the best-fit linear regression line (with the constraint of passing through the origin). $h$ is estimated from the profiles of (total) momentum flux (left panel) and (resolved) horizontal velocity variance (right panel), respectively

1.2 A.2 Sensitivity Experiments

In this appendix, we document the effects of grid resolution and longwave radiation on the LES-generated data. We consider three cases representing distinct stability regimes: (i) weakly stable ($h/L \approx 3$), (ii) moderately stable ($h/L \approx 6$), and (iii) very stable ($h/L \approx 9$). We perform several 6-h long simulations using a range of grid sizes (5–10 m). In line with the LES database, some simulations use a domain size of 800 m $\times$ 800 m $\times$ 790 m, while a few others use a smaller domain size of 400 m $\times$ 400 m $\times$ 400 m. In some simulations, a radiation scheme is utilized. Recently, we incorporated the Column Radiation Model (CRM; version 2.1.7) in our MATLES code. CRM is a stand-alone version of the radiation code utilized by the NCAR Community Climate Model (Kiehl et al. 1998). See Table 2 for the specifications of all the these simulations.

Table 2 List of sensitivity experiments

The results from the sensitivity experiments are reported in Table 3 and Fig. 8. All the profiles and statistics are computed from the last 30 min of the simulations. The following statements can be made based on these results:

• for the weakly stable case, the surface fluxes are almost independent of grid resolution. For all other cases, the surface fluxes minutely decrease with increasing resolutions;

• for all the cases, $h$ decreases with increasing resolution;

• for all the cases, the values of $h/L$ estimated from the fine-resolution runs are within 10 % of their corresponding coarse-resolution values;

• $Ri_{Bc}$ decreases with increasing resolution. For the very stable case, $Ri_{Bc}$ decreases by almost 25 %; whereas, in the case of the weakly stable case, it decreases by only $\approx 8$ %;

• $\alpha$ varies with grid-resolution;

• the overall impact of longwave radiation on the simulated statistics is marginal; and

• most importantly, $Ri_{Bc}$ increases with increasing stability. This trend is not masked by the effects of grid resolution and radiation.

Table 3 Basic characteristics of the simulated SBL

Fig. 8

The panels from top to bottom correspond to the vertical profiles of wind speed, vertical profiles of potential temperature, time series of friction velocity, and time series of sensible heat fluxes, respectively. All the profiles are computed from the last 30 min of the simulations. The simulation shown on the left panel used $G = 12\,\text{ m } \,\text{ s }^{-1}$ and $C = 2.00\,\text{ K }\,\text{ h }^{-1}$, whereas the right one used $G = 6\,\text{ m } \,\text{ s }^{-1}$ and $C = 1.25\, \text{ K } \,\text{ h }^{-1}$. See Table 2 for the specifications of the simulations