## Abstract

An analysis in a plane of the stability parameters *ζ* = *z*/*L* (where *z* is a height above the ground surface and *L* is the Obukhov length) and bulk Richardson number Ri_{B} is proposed to examine the applicability of Monin–Obukhov similarity (MOS) theory in stable conditions. In this analysis, the data available from two different experimental datasets [(i) Cooperative Atmosphere–Surface Exchange Study-1999 (CASES-99) and (ii) Cardington] are divided into four quadrants. An unexpected behavior of decreasing *ζ* with increasing Ri_{B} was observed with both datasets in quadrant II characterized by *ζ* < 1 and Ri_{B} > 0.2 and in quadrant IV with *ζ* > 1 and Ri_{B} < 0.2. This is in contrast to a commonly expected monotonically increasing behavior between *ζ* and Ri_{B}. It is shown that the MOS theory is consistent for computing the surface fluxes corresponding to the data points lying in quadrants I (with *ζ* > 1 and Ri_{B} > 0.2) and III (with *ζ* < 1 and Ri_{B} < 0.2), whereas it may not be applicable for the points in quadrants II and IV. Thus, a breakdown of the relationship between observed *ζ* and Ri_{B} with growing stability in these quadrants may limit the applicability of MOS theory in stable conditions. Since quadrant IV has very few data points, the applicability of MOS theory needs to be substantiated further with the availability of sufficient data points in this regime.

## 1. Introduction

Monin–Obukhov similarity (MOS) theory is generally used to estimate the surface turbulent fluxes in atmospheric models for weather forecasting, air quality, climate modeling, and wind energy research. The MOS theory is also used in most large-eddy simulation (LES) studies for the surface boundary conditions (Basu et al. 2008). It is well known that MOS theory has limited applicability in very stable conditions because of weak and intermittent turbulence and other complicated factors such as effects of terrain slope, mesoscale eddies, and gravity waves in the relatively shallow surface layer (Nieuwstadt 1984; Sorbjan 2006; Mahrt 2007, 2008, 2010). Despite the unlikely success of local scaling in the stable boundary layer (SBL) due to nonlocal processes like turbulence–wave interaction (Finnigan 1999; Sorbjan 2006), the validity of the MOS theory is still an open-ended question in the SBL owing to interactions with nonturbulent motions and unique features of nocturnal turbulence (Cheng and Brutsaert 2005; Hong et al. 2010).

The MOS theory estimates the turbulent fluxes of momentum and heat from mean wind speed and temperature profiles in surface layer by a flux–gradient relationship using the Obukhov length scale *L* and empirical similarity functions. The stability parameter *ζ* = *z*/*L* (where *z* is the height above the ground surface) is defined by

where *k* is the von Kármán constant, *g* is acceleration due to gravity, is mean potential temperature, *θ*_{*} is a temperature scale, and *u*_{*} is the surface friction velocity. Another stability parameter bulk Richardson number Ri_{B} is defined as

where *U* is the wind speed, *θ* is the potential temperature at a level *z*, *z*_{0m} and *z*_{0h} are the roughness lengths of momentum and heat, respectively, and *θ*_{0} is the surface potential temperature. Note that Ri_{B} is used as a measure of overall stability that is less vulnerable to errors compared to the calculation of the gradient Richardson number for weak wind conditions, where the vertical wind profiles can be complex (Mahrt 2010). In MOS theory, Ri_{B} is interrelated to *ζ* with the following expression:

where and are integrated similarity functions of momentum and heat, respectively, and *ζ*_{0m} = *z*_{0m}/*L* and *ζ*_{0h} = *z*_{0h}/*L*.

The stability parameter *ζ* [Eq. (1)] can be estimated from the turbulent observations, whereas Ri_{B} [Eq. (2)] is computed from the observed mean wind speed and temperature in a bulk layer. Since *ζ* and Ri_{B} both define the stability of the atmosphere, these parameters are expected to describe a monotonic relationship between them with increasing stability. It is noted that the computed *ζ* from mean flows (eventually from Ri_{B}) by MOS theory [Eq. (3)], irrespective of the similarity functions, has a similar monotonic relationship with Ri_{B} with increasing stability. However, the variability in a relationship between observed *ζ* and Ri_{B} increases at several phases with growing stability. The applicability of the MOS theory to estimate the surface turbulent fluxes in various stability conditions depends on this relationship. Moreover, a breakdown of the relationship between observed *ζ* and Ri_{B} with growing stability limits the application of the MOS theory in stable conditions. Thus, for a given observational dataset of turbulent fluxes and mean flow profiles, it is desirable to analyze the observed relationship between *ζ* and Ri_{B} in various parts of the stable regime.

In this paper, a quadrantal analysis in a plane of the stability parameters *ζ* (based on observed turbulent observations) and Ri_{B} (based on observed mean flow) components is proposed to evaluate the applicability of MOS theory in stable conditions. The analysis is performed with observations obtained from two experimental datasets that have both turbulent and mean flow variables.

## 2. Observational data

Data used in the present study was obtained from two experimental programs: the Cooperative Atmosphere–Surface Exchange Study 1999 (CASES-99) (Poulos et al. 2002) and the Met Office’s Cardington monitoring facility (http://badc.nerc.ac.uk/data/cardington/).

### a. CASES-99 dataset

The CASES-99 experiment was performed near Leon (37.38°N, 96.14°W) in southeastern Kansas in the United States during the month of October 1999 (Poulos et al. 2002). The terrain of the main experimental site was relatively homogeneously flat and lacked obstacles in the surroundings. The turbulence measurements were taken on a 60-m tower at 1.5 (0.5)-, 5-, 10-, 20-, 30-, 40-, 50-, and 55-m levels by eight sonic anemometers. The turbulence at a sampling rate of 20 Hz was used in the analysis to calculate the fluxes. In general, the fluxes are computed from the hourly observed series from sonic anemometer using the eddy correlation method. However, in very stable conditions, the hourly observed turbulent fluxes based on the classical eddy correlation method might have been influenced by the mesoscale part of the flow, which is highly variable (Mahrt 2008). Thus, for the CASES-99 dataset, true observed turbulent fluxes at 10-m level are determined from sonic observations by using the methodology based on the multiresolution cospectral gap proposed by Vickers and Mahrt (2003, 2006). Hourly fluxes are calculated by dividing the 1-h period into the number of subintervals in accordance with the cospectral gap. In each subinterval, the fluxes are calculated and then their average is taken for the flux corresponding to 1 h. Then, *L* is computed using these fluxes obtained from the multiresolution cospectral gap methodology. After the analysis of turbulence data at 10-m level, 304 hourly averaged data corresponding to stable conditions (i.e., *ζ* > 0) are selected.

To obtain the values of Ri_{B} from Eq. (2) and correspondingly to compute surface fluxes with MOS theory, hourly averaged temperatures at the surface and 10-m level and wind speed at this level are used. The surface temperature is estimated using the mean temperature measured by five downward-looking narrow-beam Everest infrared radiometers, which were within 300 m of the 60-m tower (Poulos and Burns 2003). Note that *z*_{0m} is taken as 0.03 m. Since the traditional similarity theory does not apply when the surface radiative temperature is used instead of aerodynamic temperature at roughness height (Sun and Mahrt 1995), the measured skin temperature or surface radiative temperature in the computations is used with the introduction of the different *z*_{0m} and *z*_{0h} (Sun and Mahrt 1995). Luhar et al. (2009) computed a mean value of *z*_{0h} ≈ 0.0002 m for both CASES-99 and Cardington datasets.

### b. Cardington dataset

Another dataset used in this study is taken from Cardington in Bedfordshire in the United Kingdom (52.06°N, 00.25°W; 29 m above mean sea level). The site is a large grassy field and is reasonably flat. The dataset contains the fast response wind and temperature observations generated from the ultrasonic anemometers at 10, 25, and 50 m above the ground surface and slow response temperature measurements at 1.2, 10, 25, and 50 m obtained from the platinum resistance thermometer. The dataset has recorded measurements averaged over 1-, 10-, and 30-min intervals whereas the turbulence quantities and data from slow response sensors are included in the 10- and 30-min datasets only. Here, we have utilized 3 months of data during August, September, and November 2005 in this study. The roughness length *z*_{0m} is taken as 0.05, 0.025, and 0.03 m for August, September, and November, respectively, and *z*_{0h} ≈ 0.0002 m (Luhar et al. 2009).

For this period, 30-min averaged data at the 10-m level are used to determine the hourly averaged mean wind, temperature, and turbulent quantities. The surface temperature *θ*_{0} is taken as the measured mean grass infrared temperature. After analysis of the data at the 10-m level corresponding to stable conditions (i.e., *ζ* > 0), 971 data hours are selected in the present study.

## 3. Results and discussions

In very stable conditions, where Ri_{B} exceeds from a critical value of 0.2, vertical turbulent fluxes vanish using the bulk formula with existing similarity functions in MOS theory (Louis 1979; Beljaars and Holtslag 1991; Howell and Sun 1999). However, large values of Ri_{B} are found in the observational studies in very stable conditions (Poulos and Burns 2003; Aditi and Sharan 2007; Luhar et al. 2009), especially in weak-wind stable conditions. It is well known that the turbulence can become extremely weak at very large Richardson numbers; it does not vanish due to a partial result of the shear generation of turbulence by submesoscale motions (Mahrt 2008). It is now well recognized that organized turbulent exchange exists even under very stable conditions, and that it relates relatively well to larger-scale quantities, such as vertical gradients. The main reason why most previous studies failed to recognize that fact is that the temporal and spatial scales of such fluxes become largely reduced under very stable stratification, reaching values as low as 5 or 10 s for the temporal scale (Mahrt and Vickers 2006). Beyond that scale, the exchange is performed by mesoscale fluxes, less organized and not directly associated with vertical gradients. While studying similarity relationships, one might attempt to remove all nonturbulent contributions to the fluxes, while for balancing surface energy budgets one might want to include heat fluxes at larger time scales, regardless of their origins. A good portion of the problem, therefore, becomes finding the correct scale that characterizes the turbulent exchange.

The turbulent quantities are not well predicted by the MOS theory for these large Richardson numbers in very stable conditions. For an observed value of Ri_{B} [Eq. (2)], the MOS theory computes *ζ* [Eq. (3)] by utilizing a similarity function as a functional form of *ζ*. This computed *ζ* is used to obtain surface turbulent fluxes in the surface layer. The reliability of computed turbulent fluxes by the MOS theory depends on a similarity function that is generally obtained from mean and turbulent variables. However, in very stable conditions, the attainment of true observed turbulent fluxes is still a matter of discussion (Vickers and Mahrt 2003). Since the fluxes tend to be smaller at night in very stable conditions, it often becomes difficult to have reliable turbulence measurements in these conditions. As a result, several aspects of stably stratified turbulence have remained poorly understood and even controversial until now, and they continue to be the subject of intense research (see Cheng and Brutsaert 2005, and references therein). In following quadrantal analysis, we find that the observational data in stable conditions can be analyzed by separating into four regimes using *ζ* and Ri_{B}.

### a. Quadrantal analysis

Quadrantal analysis is based on a one-to-one relationship between two stability parameters *ζ* (from observed turbulent fluxes), and Ri_{B} (obtained from the wind and temperature profiles) with increasing stability. Computations of *ζ* from the observed values of Ri_{B} depend on monotonic behavior of Ri_{B} with increasing stability; thus, both Ri_{B} and *ζ* are expected to increase with growing stability in stable conditions. To analyze the characteristics observational features of Ri_{B} with stability parameter *ζ* in stable conditions, the data are classified into a two-dimensional space of *ζ* and Ri_{B} components. In general, both Ri_{B} and *ζ* are stability parameters and they are supposed to be intercorrelated.

In this quadrantal analysis, observed values of Ri_{B} and *ζ* from both experimental datasets were classified into four quadrants (Fig. 1). Classification of the quadrants is carried out by assuming a fact that a critical value of Ri_{B} (≈0.2) in stable conditions corresponds to *ζ* = 1. The data points with Ri_{B} > 0.2 and *ζ* > 1 are considered to be representing the very stable regime where turbulence is weak and intermittent. Similarly, a regime with 0 < Ri_{B} < 0.2 and 0 < *ζ* < 1 is classified for weak to moderately stable conditions. Thus, observed values of Ri_{B} and *ζ* in each quadrant are classified as follows: (i) first quadrant (i.e., I) *ζ* > 1 and Ri_{B} > 0.2, (ii) second quadrant (II) *ζ* < 1 and Ri_{B} > 0.2, (iii) third quadrant (III) *ζ* < 1 and Ri_{B} < 0.2, and (iv) fourth quadrant (IV) *ζ* > 1 and Ri_{B} < 0.2. The characteristics features of the observed *ζ* with Ri_{B} are analyzed in each quadrant. Based on the proposed classification for Cardington dataset, ~22%, ~8%, ~69%, and ~1% points were observed in quadrants I, II, III, and IV, respectively. Similarly, ~12%, ~23%, ~58%, and ~7% data hours were observed respectively in quadrants I, II, III, and IV in the CASES-99 dataset.

### b. Characteristic observational features in each quadrant

Based on the abovementioned quadrantal classification of the observed data in stable conditions, Figs. 1a and 1b show scatter diagrams between Ri_{B} and *ζ* dividing into four quadrants for CASES-99 and the Cardington experiment, respectively. A well-defined relationship between Ri_{B} and *ζ* with increasing stability in stable conditions was considered a basis of the proposed analysis. The values of *ζ* are also computed from the observed Ri_{B} using a similarity function of Beljaars and Holtslag (1991). The relationship between the computed *ζ* with Ri_{B} is plotted along with the observed *ζ* and Ri_{B} (Figs. 1a,b). In fact, the computed values of the *z*/*L* from observed Ri_{B} using a similarity function of Beljaars and Holtslag (1991) show a monotonically increasing behavior with Ri_{B} (Figs. 1a,b) in stable conditions.

In first quadrant where *ζ* is greater than unity and Ri_{B} > 0.2 for both datasets, a large scatter between Ri_{B} and *ζ* is observed with increasing stability. This quadrant defines a regime of very stable atmosphere. Despite a large scatter in this quadrant between Ri_{B} and *ζ*, both stability parameters have an increasing trend with growing stability. Thus, both stability parameters Ri_{B} and *ζ* in this quadrant describe the atmospheric stability in very stable conditions. A reasonable correlation between computed surface fluxes by the MOS theory and observed fluxes is expected for the points lie in this quadrant of very stable conditions. However, computed fluxes from the MOS theory have a large underestimation with the observed one (Aditi and Sharan 2007) for these points in very stable regime. In this quadrant, the MOS theory may have limitations in predicting the surface fluxes as it does not include some external features such as mesoscale motions, gravity waves, intermittency, low-level jets, and meandering in the theory.

Data points in quadrant II with Ri_{B} > 0.2, implying very stable regime of the atmosphere, correspond to the values of *ζ* smaller than unity representing weakly to moderate stable conditions. In this quadrant, large values of Ri_{B} define a weak turbulence with strong stratification, whereas corresponding small values of *ζ* lead from near-neutral to moderately stable conditions. Thus, both stability parameters define two different states of the atmospheric stability at the same time in this quadrant. Figure 1 reveals an observed behavior of decreasing *ζ* with increasing Ri_{B} for both the datasets in this quadrant and this is in contrast to commonly observed increasing nature of *ζ* with increasing Ri_{B}. At the moment, we do not have a physical explanation for this. In this quadrant, *ζ* decreases with increasing Ri_{B} and MOS theory leads to a monotonically increasing *ζ* with Ri_{B} (Figs. 1a,b) and thus, MOS theory is of questionable value in this regime. As MOS theory is not appropriate for computing the surface fluxes corresponding to the observations in quadrant II, it is necessary to evolve alternative techniques to describe the fluxes in these conditions. In this quadrant, weak turbulence, strong stratification, and a large Richardson number sometimes corresponded to small near-neutral values of *ζ*. The behavior of decreasing *ζ* with increasing Ri_{B} cannot be described by a monotonic relationship between *ζ* and Ri_{B} arising from MOS theory. From analysis of CASES-99 dataset, Ha et al. (2007) observed that this behavior was not sensitive to choice of averaging time, although sampling errors associated with the weak fluxes may be important.

These cases of very strong stability with small values of *ζ* in quadrant II have often been neglected because of the variety of observational difficulties for very weak turbulence. On the one hand, intermittent turbulence in very stable conditions produces strongly nonstationary events during which the validity of turbulent transport and storage measurements is uncertain. Fluxes calculated using a conventional averaging time of several minutes or longer are severely contaminated by poorly sampled mesoscale motions in very weak turbulence (Vickers and Mahrt 2006). One needs to calculate the true observed turbulent fluxes by removing the nonturbulent part of the flow from the turbulent measurements. However, with extremely stable conditions, it is difficult to separate the turbulence from nonturbulent motions.

The third quadrant, where *ζ* is less than unity and Ri_{B} < 0.2, can be classified as a regime of weak to moderately stable conditions. In this quadrant, both *ζ* and Ri_{B} are well correlated with increasing stability. Because of a better relationship between *ζ* and Ri_{B} in this quadrant, the MOS theory works well in weak to moderately stable conditions.

Very few data hours were observed in the fourth quadrant characterized by *ζ* > 1 and Ri_{B} < 0.2. In case of availability of data in this quadrant, *ζ* is expected to increase with decreasing Ri_{B} whereas the application of MOS theory in this regime leads to a monotonically increasing behavior between *ζ* and Ri_{B} (Figs. 1a,b). Thus, MOS theory may not be applicable for computing the fluxes for the observations corresponding to the quadrant IV. Since this quadrant has very few data points, the applicability of MOS theory needs to be substantiated further with the availability of sufficient data points in this regime.

It is illuminating that for both the CASES-99 and Cardington observations, there are more cases in quadrant II than quadrant IV. Relative to similarity theory, quadrant II consists of weaker heat flux and larger temperature gradient, and also corresponds to larger momentum flux and smaller wind speed gradient relative to the similarity theory. From another point of view, quadrant II corresponds to smaller thermal diffusivity *K _{H}* and larger momentum diffusivity

*K*(and thus larger turbulent Prandtl number Pr

_{M}_{t}). Also it is noticed that for both datasets, there is a lot of scatter for weak stability in quadrants II and IV. In quadrant II, a majority of data hours correspond to weak wind stable conditions. These weak wind conditions are most common in very stable conditions (Sharan et al. 2003; Mahrt 2011). Weak wind cases have been often eliminated from existing analysis by imposed minimum conditions on turbulence intensity or nonstationarity, or they are simply eliminated as outliers (Mahrt 2011). The scatter in the very stable quadrant (I) is more than the scatter in the weakly stable quadrant (III). Despite the uncertainty in the strongly stable boundary layer, the theoretical behavior of the stable boundary layer using simple dynamical systems has, in fact, shown (McNider et al. 1995; Van de Wiel et al. 2002a,b) that there may be abrupt transitions as the SBL transitions from strongly stable to weakly stable conditions and their analysis shows multiple solutions may exist. Thus, this may be the reason for scatter in weakly stable conditions.

Notice that the scatter to the left of the line is relatively more than that to its right (Figs. 1a,b). The analysis of data based on different quadrants provides a break in different stability regimes as well as the interesting characteristic features in both datasets. For example, in quadrant II, despite very strong stratification, the stability in terms of *ζ* is weak. This analysis, in spite of scattering, shows that if the points lie in quadrant II, MOS theory may not be consistent for computing the surface fluxes. Although the observational points falling in the quadrant IV are relatively small, we expect the computed values of *ζ* from Eq. (3) for observed Ri_{B} in this quadrant to be smaller than one, which is in contrast to their corresponding observed values larger than 1.

## 4. Summary

This study examines relationships between the traditional Monin–Obukhov stability parameter *z*/*L* and the bulk Richardson number Ri_{B} for two datasets. A quadrantal analysis in a plane of the stability parameters *ζ* and Ri_{B} components is proposed to evaluate the applicability of MOS theory in stable conditions. The analysis is performed with observations obtained from two experimental datasets: (i) CASES-99 and (ii) Cardington. An unexpected behavior of decreasing *ζ* with increasing Ri_{B} is observed with both datasets in quadrant II characterized by *ζ* < 1 and Ri_{B} > 0.2 This is in contrast to a commonly expected monotonically increasing behavior between *ζ* and Ri_{B}. A similar trend is found for the observations lying in quadrant IV with *ζ* > 1 and Ri_{B} < 0.2. Classification of the observations in the quadrants assumes a critical value of Ri_{B} (≈0.2) corresponding to *ζ* = 1. However, the computed value of *ζ* is found to be larger than 1 corresponding to the lower values of Ri_{B} in the vicinity of 0.2 (Figs. 1a,b). It is shown that MOS theory is consistent for computing the surface fluxes corresponding to the data points lying in the quadrants I and III, whereas it may not be applicable for the points in II and IV quadrants. Thus, a breakdown of the relationship between observed *ζ* and Ri_{B} with growing stability in these quadrants may limit an evaluation of the applicability of MOS theory in stable conditions. However, very few points lie in the quadrant IV. The conclusion drawn, here, needs to be substantiated further in the quadrant IV with the availability of more data points in this regime.

## Acknowledgments

The authors gratefully acknowledge the U.K. Met Office for use of Cardington measurements and National Center for Atmospheric Research (NCAR) for CASES-99 observations. The authors thank Dr. Larry Mahrt for his valuable comments. The authors also thank the reviewers for their valuable comments and suggestions.

## REFERENCES

## Footnotes

Current affiliation: Center for Air Resource Engineering and Science, Clarkson University, Potsdam, New York.