### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Background
- 3. Scaling systems
- 4. Empirical evaluation
- 5. Discussion
- 6. Conclusions
- Acknowledgements
- References

Three gradient-based scaling systems for the stably stratified boundary layer are introduced and examined by using data collected during the SHEBA field programme in the Arctic. The resulting similarity functions for fluxes and variances are expressed in an analytical form, which is expected to be essentially unaffected by self-correlation in a very stable regime. The flux Richardson number *Rf* is found to be proportional to the Richardson number *Ri*, with the proportionality coefficient varying slightly with stability, from 1.11 to 1.47. The Prandtl number decreases from 0.9 in nearly neutral conditions to 0.7 for larger values of *Ri*. The negative correlation coefficient between the vertical velocity and temperature, − *r*_{wθ}, has a local maximum at *Ri* of about 0.08, and monotonically decreases with larger values of the Richardson number. The turbulent kinetic energy budget indicates that for *Ri* > 0.7, turbulence must be non-stationary, i.e. decaying or sporadic. Turbulence within the stably stratified boundary layer can be classified by four regimes: ‘nearly neutral’ (0 < *Ri* < 0.02), ‘weakly stable’ (0.02 < *Ri* < 0.12), ‘very stable’ (0.12 < Ri < 0.7), and ‘extremely stable’ (Ri > 0.7). Copyright © 2010 Royal Meteorological Society

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Background
- 3. Scaling systems
- 4. Empirical evaluation
- 5. Discussion
- 6. Conclusions
- Acknowledgements
- References

The Monin–Obukhov similarity is regarded as the major tool for understanding near-surface turbulence. Since the early 1950 s, its universal framework has been systematically examined and applied in the analysis of numerous field observations. Data accumulated during recent years indicate, however, that the approach has limited utility in very stable conditions. Specifically, the similarity predictions for the gradients can be formally derived only for sub-critical conditions (e.g. Sorbjan, 2006a; 2006b), despite the observational evidence that stable turbulence survives at Richardson numbers exceeding the critical value *Ri*_{c} ≈ 0.2 (e.g. Galperin *et al*, 2007). Similarity scales decrease with thermal stability, causing the similarity functions to become practically singular with strongly scattered values. The definitions of the universal functions and the similarity argument *z*/*L*_{*} contain common divisors (*u*_{*}, *T*_{*}), a property referred to as ‘self-correlation’, where *u*_{*}, *T*_{*}, *L*_{*} are the Monin–Obukhov scales (see section 3.1). As a result, a relationship between the similarity functions and their argument is difficult to be established in very stable conditions with satisfactory confidence (e.g. Klipp and Mahrt, 2004; Baas *et al*, 2006). Moreover, the definition of the stability parameter, *z*/*L*_{*}, includes a ratio of fluxes, which can cause ambiguities, especially when both temperature and wind fluxes are small (Grachev *et al*, 2008). In such cases there are difficulties in differentiating between a nearly neutral state with weak velocity and very stable conditions. By replacing the argument *z*/*L*_{*} by the Richardson number *Ri,* the severity of the self-correlation problem can be reduced (Sorbjan, 2006b).

Some of the aforementioned problems could be resolved by defining similarity scales in terms of the moments of turbulence, which possess ‘better behaviour’ in very stable conditions than do the turbulent fluxes (e.g. Sorbjan, 2006a). Thus, the purpose of this paper is to further explore alternative forms of similarity scales and to examine the resulting similarity laws in the stably stratified boundary layer.

The paper has the following structure. Section 2 discusses the theoretical background in terms of the K-theory. The scaling systems and similarity functions are introduced in section 3. The empirical evaluation is presented in section 4 through the use of the experimental data collected during the SHEBA experiment. A general discussion of our work is presented in section 5. Finally, conclusions are provided in section 6.

### 2. Background

- Top of page
- Abstract
- 1. Introduction
- 2. Background
- 3. Scaling systems
- 4. Empirical evaluation
- 5. Discussion
- 6. Conclusions
- Acknowledgements
- References

Above, the overbar indicates the ensemble averaging operator, *U* and *V* are the components of the wind velocity, *f*_{m} and *f*_{h} are empirical functions of the Richardson number, which is defined as *Ri* = *N*^{2}/*S*^{2}, is the Brunt–Väisälä frequency, β = *g*/*T*_{0} is the buoyancy parameter, *g* is the gravity acceleration, *T*_{0} is the reference temperature, κ = 0.4 is the von Kármán constant, and *z* is the height. Equation (2a) is based on the expression for eddy viscosity *K*_{m} = *l*^{2}*S* of Prandtl (1932), and the mixing length *l* = κ*z*/(1 + κ*z*/*l*_{∞}) of Blackadar (1962), with the ratio of height and the parameter *l*_{∞} assumed to be dependent on the Richardson number. The eddy diffusivity *K*_{h} is defined analogously.

We will augment the above equations by considering a simplified, steady-state, horizontally homogeneous balance of the vertical velocity variance in the form (e.g. Sorbjan, 1989):

- (3)

where *f*_{ε} is an empirical function of the Richardson number. The left-hand side of (3) describes the shear production and the work against the buoyant force, while the right-hand side represents the dissipation rate. The diffusion and pressure terms are neglected as small in stable conditions. Following Kolmogorov (1941), the dissipation rate in (3) is parametrized as being proportional to the mixing length (the term in the denominator) and the turbulent kinetic energy. In turn, this expression can be expressed as a product of the vertical velocity variance and the anisotropy function dependent on the Richardson number. The von Kármán constant κ is added for convenience.

In addition, we will consider the simplified, steady-state, horizontally homogeneous balance of the temperature variance in the form:

- (4)

where *f*_{d} is an empirical function of the Richardson number. The left-hand side of (4) expresses the production of temperature fluctuations, and the right-hand side is the dissipation rate of the temperature variance. We neglect the diffusion term as small. The dissipation rate for the temperature fluctuations is parametrized in terms of a time-scale, which is assumed to be proportional to the mixing length and the vertical velocity variance.

When the empirical functions *f*_{m}, *f*_{h}, *f*_{ε}, and *f*_{d} are specified, the system (1)–(4) is closed. It describes (for given *S*, Γ, β and *z*) the relationship among the fluxes τ, *H*, and variances σ_{w}, σ_{θ}. We will not attempt to find the solution for the system (1)–(4). Instead, in the following section, some general conclusions about the above system will be derived by employing the approach of dimensional analysis. Such an analysis has a great tradition in meteorology and oceanography, and has been successfully applied to describe the properties of turbulence by Kolmogorov, Obukhov, Monin, Ozmidov and others (e.g. Barenblatt, 1996).

### 3. Scaling systems

- Top of page
- Abstract
- 1. Introduction
- 2. Background
- 3. Scaling systems
- 4. Empirical evaluation
- 5. Discussion
- 6. Conclusions
- Acknowledgements
- References

A simple analysis of the system (1)–(4) indicates that the choice of the similarity scales for the set of eight variables: {τ, *H*, σ_{w}, σ_{θ}, *S*, Γ, *z*, β}, with three independent units, [m], [s], [K], is not unique. Thus, it can be performed in a number of ways. Generally, any three dimensionally independent parameters in the above list can be selected to build a system of three scales for length, temperature and velocity. Below, we will consider scaling systems, based on the following choice of the parameter combinations:

- (5a)

- (5b)

- (5c)

- (5d)

We will refer to the scales derived from the first set of parameters as ‘flux-based scaling’, while the remaining sets will be called the ‘gradient-based scaling’. It should be mentioned that other ‘gradient-based scaling’ systems could also be proposed. For example, one could consider {ε, Γ, β} as governing parameters, where ε is the dissipation rate (Sorbjan and Balsley, 2008).

#### 3.1. The flux-based scaling

Historically, the first scaling system for the atmospheric boundary layer was proposed by Monin and Obukhov (1954), who employed (5a), with the surface values of fluxes τ_{0}, *H*_{0}, to construct scales for length *L*_{*} = − τ_{0}^{3/2}/(κβ*H*_{0}), temperature *T*_{*} = − *H*_{0}/*u*_{*}, and velocity . Based on a dimensional analysis, Monin and Obukhov concluded that the non-dimensional products of statistical moments *X* in the surface layer (such as σ_{w}, σ_{θ}, *S*, Γ), and the flux-based scales, are universal functions φ_{x} of a single dimensionless parameter *z*/*L*_{*}:

- (6)

where the exponents *a*, *b*, *c* are chosen in such a way that φ_{x} is dimensionless. The above result conveys the so-called ‘self-similarity’ property, which manifests itself in the reduction of the number of independent dimensionless variables in comparison to the number of dimensional ones (e.g. Barenblatt, 1996). As a result, self-similarity substantially simplifies the description of phenomena and their experimental, analytical and computational analysis.

By using a second-order closure scheme, Nieuwstadt (1984) demonstrated that the assumption of the constancy of fluxes with height is not necessary, so that the scales in the stable boundary layer can be height-dependent (local):

- (7a)

- (7b)

- (7c)

Note that a new notation is used to mark the local scales.

Applying a definition of the Richardson number yields:

- (9)

Using (9), one can formally rewrite (8) in the equivalent form:

- (10a)

- (10b)

- (10c)

- (10d)

where ψ_{m}, ψ_{h}, ψ_{w}, ψ_{θ} are the universal similarity functions of the Richardson number. The same result can be formally obtained based on (1)–(4), with , , , , and an additional assumption that *Ri*/*Pr* < 1, where *Pr* ≡ *K*_{m}/*K*_{h} = *f*_{m}/*f*_{h} is the Prandtl number. As a consequence, we conclude that the K-theory formulation (1)–(4) is equivalent to the Monin–Obukhov similarity approach.

In neutral conditions, the parameters *z*/Λ_{*} and *Ri* are nearly zero, which implies that the values of similarity functions are constant. Specifically, φ_{m}(0) = 1 and φ_{h}(0) = *Pr*_{0}, where *Pr*_{0} is a constant, referred to as the neutral value of the Prandtl number. According to the Monin–Obukhov theory, when the temperature gradient Γ is positive and sufficiently large, turbulence is expected to be local and independent on the distance from the underlying surface (the ‘*z*-less regime’). In this case, a dimensional analysis leads to the conclusion that the similarity functions are linear, φ_{m}∼φ_{h}∼*z*/Λ_{*}. With increasing thermal stratification, the parameter . As a result, the similarity functions become singular (large), and strongly impacted by self-correlation.

#### 3.2. The gradient-based σ_{w} - scaling

Equation (3) implies that

- (13)

where Φ_{w}(*Ri*)∼{*f*_{m}*f*_{ε}(1 − *Ri*/*Pr*)}^{1/3}/*Ri*^{1/2} can be seen to be a function of the Richardson number. Using (13), equations (12a)–(12c) can be rewritten in an equivalent form:

- (14a)

- (14b)

- (14c)

where Φ_{m}, Φ_{h}, and Φ_{θ} are universal functions of the Richardson number. Since the above expressions were derived based on the K-theory formulation (1)–(4), and we have found to be equivalent to the Monin–Obukhov similarity approach, we can conclude that (13)–(14) is also formally equivalent to the Monin–Obukhov similarity theory. An analogous result can also be derived by using second-order closure equations (Sorbjan, 2006a).

Using dimensional analysis, (14) can be generalized by asserting that the non-dimensional products of statistical moments *X* in the surface layer and the scales (11), are expected to be universal functions Φ_{x} of a single dimensionless parameter *Ri*:

- (15)

#### 3.3. The gradient-based σ_{θ} scaling

It can be noted that Ψ_{w}≡1/Φ_{θ}.

Again employing a dimensional analysis, we can generalize (18) by stating that the non-dimensional products of statistical moments *X* in the surface layer and the above scales are expected to be universal functions Ψ_{x} of a single dimensionless parameter *Ri*:

- (19)

#### 3.4. The gradient-based master scaling

An alternative similarity scaling can be introduced by using (5d), which involves the temperature gradient Γ, the buoyancy parameter β and height *z*:

- (20a)

- (20b)

- (20c)

where κ the von Kármán constant was added for convenience. As before, we will consider only cases when the Brunt–Väisälä frequency *N* is sufficiently large. We will refer to (20a, b, c) as master scaling.

Analogous to the previous scaling cases, we can generalize the above system by stating that the non-dimensional products of statistical moments *X* in the surface layer and the above scales must be universal functions of a single dimensionless parameter *Ri*:

- (22)

Note that the temperature gradient Γ appears on both sides of (22), within the similarity scales and in the definition of the Richardson number. This fact implies self-correlation, due to errors in the evaluation of Γ. However, it is reasonable to expect that such errors are relatively small when the temperature gradient is sufficiently large, and thus the self-correlations effects related to Γ do not represent a serious issue.

By employing (10a), (10b), and (23d), one can also obtain the relationship between the Monin–Obukhov and gradient-based functions:

- (24a)

- (24b)

The master scaling does not involve any higher order moments. Because of dependence only on gradients, its application seems to be the most advantageous. Using the master-scaling similarity functions, all other similarity functions can also be derived.

### 5. Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Background
- 3. Scaling systems
- 4. Empirical evaluation
- 5. Discussion
- 6. Conclusions
- Acknowledgements
- References

All of the gradient-based scaling systems that were considered in this paper are formally equivalent to each other, as well as to the Monin–Obukhov similarity approach in the stable case. There exists, however, an essential difference between the flux-based approach and the gradient-based formulations. The flux-based methodology employs fluxes as external (specified) parameters. As a result, the practical application of the flux-based expressions requires inverting the similarity laws, and calculating surface fluxes from the provided (measured) values of gradients in the surface layer. This procedure is ill-posed in very stable conditions due to the small values of the flux quantities. Moreover, the effective use of the local similarity formulation (8) requires that the fluxes be known *a priori* as functions of height; often quite difficult to accomplish.

Within the gradient-based formulation, the gradients themselves play the role of external parameters and avoid the implied singularities associated with the fluxes. The gradient value can be provided from measurements, or via the use of the differential equations for momentum and heat (with appropriate boundary conditions). As a result, the cross-isobar angle and the depth of the boundary layer, which are otherwise absent in the presented similarity formulations, can be taken into consideration.

Let us consider the flux Richardson number *Rf* = − β*H*/(τ*S*). Employing (21a, b) and (27a, b), we obtain:

- (31)

The above expression is illustrated in Figure 5 as a solid curve. In accordance with (31), *Rf* = 1.11 *Ri* in nearly neutral conditions, and *Rf* = 1.46 *Ri* for large values of *Ri.* Consequently, the curve in the figure only slightly differs from a straight line.

Taking into consideration that *Rf* ≡ *Ri*/Pr and by using (31), we find that

- (32)

Equation (32) indicates that Prandtl number is 0.9 in nearly neutral conditions, and 0.7 for larger values of *Ri*. According to Ohya (2001), Grachev *et al*(2007b), Esau and Grachev (2007), Zilitinkevich *et al*(2007) and Anderson (2009), the Prandtl number increases with *Ri* in supercritical conditions. However, the detailed analysis of Grachev *et al*(2007b) implies that such a result is spurious. When one does not employ the outlier rejection discussed in section 4.1, the resulting SHEBA data do indeed show that *Pr* increases with the increasing values of *Ri*. Conversely, when the outliers are not included, the Prandtl number decreases slightly, as discussed by Sorbjan and Grachev (2010).

Note that the steady-state, turbulent energy budget (3) can also be expressed in the following form:

- (33)

Since the dissipation rate ε is positive-definite, (33) allows us to conclude that the steady state, which results from the balance of shear production and buoyant-dissipative destruction, takes place only for *Rf* < 1. As seen in Figure 5, *Rf* = 1 at *Ri* = 0.7. Thus, for Richardson numbers exceeding the value *Ri*_{s} = 0.7, steady-state turbulence would not be present. In other words, at *Ri* > *R*_{s}, turbulence is non-stationary, i.e. decaying or sporadic. The inequality *Ri* < *R*_{s} = 0.7 is a necessary condition for the presence of steady-state turbulence. It must be satisfied for steady-state turbulence to take place.

Taking (27) into consideration, we derive the negative correlation coefficient:

- (34)

The values of − *r*_{wθ} are depicted in Figure 6. The ordinate in the plots spans the range from 0.005 to 10. We exclude data points for the smallest values of the Richardson number due to their significant scatter. The solid line based on (34) agrees with observations. It implies that the coefficient − *r*_{wθ} tends to a constant value of 0.2 in the near-neutral regime, reaches a local maximum of about 0.39 at about *Ri* = 0.08, and monotonically decreases for larger values of *Ri*.

Figure 7 depicts the dimensionless components of the doubled turbulent kinetic energy: , , , where , , are the velocity variances, and *q*^{2} is the doubled turbulent kinetic energy. The figure indicates that for nearly neutral conditions, all the components are approximately constant, and *A*_{u} ≈ 0.62, *A*_{v} ≈ 0.26, and *A*_{w} ≈ 0.12. In the range 0.01 < *Ri* < 0.1, *A*_{u} decreases to 0.5, and is approximately constant up to *Ri* = 0.7. The component *A*_{v} increases to about 0.4 at *Ri* = 0.7. Consequently, one can observe that *A*_{u} and *A*_{v} have a tendency to become nearly equal at *Ri* > 0.7. The vertical component *A*_{w} increases and reaches maximum at *Ri* = 0.1. Subsequently, it decreases for larger *Ri*. The vanishing values of *A*_{w} at larger values of *Ri* can be interpreted as the rising of an extremely stable–sporadic regime, in which turbulence becomes horizontal and concentrated in vertically uniform layers.

Referring now to the ensemble of Figures 1–6, one can identify four sub-regimes in the stable boundary layer. As shown in Figure 8, they can be listed in the following order: nearly neutral (*Ri* < 0.02), stable (0.02 < *Ri* ≤ 0.12), very stable (0.12 < *Ri* < 0.7) and extremely stable (*Ri* > 0.7). In the nearly neutral regime, the master similarity functions follow the power laws (25). The stable regime is the transition between nearly neutral and very stable conditions. In the very stable regime, the master similarity functions are described by the empirical power laws (26). The potential generality of any scaling laws in the extremely stable conditions is doubtful. Figure 8 shows that when the wind shear *S* increases, turbulence becomes stronger and the turbulent regime less stable. On the other hand, when the Brunt–Väisälä frequency *N* increases, turbulence weakens and the turbulent regime becomes more stable.

### 6. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Background
- 3. Scaling systems
- 4. Empirical evaluation
- 5. Discussion
- 6. Conclusions
- Acknowledgements
- References

Properties of the stably stratified boundary layer have been examined through the use of a gradient-based similarity approach. This formulation is formally equivalent to the Monin–Obukhov similarity approach, but possesses improved properties in very stable conditions. Specifically, it avoids the singularity imposed by small values of scales and, therefore, is less affected by self-correlation. While the Monin–Obukhov similarity scales are based on two second-order moments (momentum and heat flux), each of the variance-gradient-based scaling systems is defined through a single second-order moment (temperature or the vertical velocity variance). The master scaling formulation does not employ second-order moments, and thus its application is most advantageous.

The gradient-based similarity functions for fluxes and variances have been evaluated by using tower data collected during the SHEBA field programme. We found that the derived analytical expressions for the similarity functions are consistent with these data. The analytical expressions for the similarity functions that are conveyed in terms of the σ_{w} and σ_{θ} scales can be derived from the master scale similarity functions.

The flux Richardson number *Rf* is found to be proportional to the Richardson number, with the proportionality coefficient varying slightly with stability, from 1.11 to 1.47. The Prandtl number is 0.9 in nearly neutral conditions, and 0.7 for larger values of *Ri*. The negative correlation coefficient between the vertical velocity and the temperature, − *r*_{wθ}, has a local maximum of about 0.39 at *Ri* ≈ 0.08, and monotonically decreases at larger values of *Ri*. The dimensionless components of the doubled turbulent kinetic energy *A*_{u} and *A*_{v} have a tendency to become nearly equal for *Ri* > 0.7. The vertical component *A*_{w} reaches maximum at *Ri* ≈ 0.1 and subsequently decreases for larger values of the Richardson number. The budget of the turbulent kinetic energy indicates that at the Richardson number *Ri* > *Ri*_{s} = 0.7, turbulence is non-stationary and decaying, or sporadic.

Stable turbulence can be classified into four sub-regimes: ‘near neutral’ regime for 0 < *Ri* < 0.02, the ‘stable’ regime for 0.02 < *Ri* < 0.12, the ‘very stable’ regime for 0.12 < *Ri* < 0.7, and the ‘extremely stable’ regime for *Ri* > 0.7. In the nearly neutral regime, the master similarity functions follow the exact power laws. The stable regime is the transition between nearly neutral and stable conditions. In the very stable-continuous regime, the master similarity functions *G* can be described by empirical power laws. The generality of any scaling laws in extremely stable conditions is problematic. The specified regimes are controlled by a local stability parameter *Ri*, and thus can generally occur at any level within the stably stratified boundary layer.