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Keywords:

  • gradient-based scaling;
  • SHEBA data;
  • similarity theory;
  • stable boundary layer

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background
  5. 3. Scaling systems
  6. 4. Empirical evaluation
  7. 5. Discussion
  8. 6. Conclusions
  9. Acknowledgements
  10. 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, − rwθ, 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background
  5. 3. Scaling systems
  6. 4. Empirical evaluation
  7. 5. Discussion
  8. 6. Conclusions
  9. Acknowledgements
  10. 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 Ric ≈ 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

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

Let us consider a classic approach of the K-theory, which allows the expression of the turbulent fluxes of momentum equation image and heat H = w′Θ′ in terms of the virtual potential temperature gradient Γ = dΘ/dz > 0 and the wind shear equation image in a stationary, horizontally homogenous, stably-stratified shear flow:

  • equation image(1a)
  • equation image(1b)

where

  • equation image(2a)
  • equation image(2b)

Above, the overbar indicates the ensemble averaging operator, U and V are the components of the wind velocity, fm and fh are empirical functions of the Richardson number, which is defined as Ri = N2/S2, equation image is the Brunt–Väisälä frequency, β = g/T0 is the buoyancy parameter, g is the gravity acceleration, T0 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 Km = l2S of Prandtl (1932), and the mixing length l = κz/(1 + κz/l) of Blackadar (1962), with the ratio of height and the parameter l assumed to be dependent on the Richardson number. The eddy diffusivity Kh is defined analogously.

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

  • equation image(3)

where fε 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 equation image in the form:

  • equation image(4)

where fd 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 fm, fh, fε, and fd 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 σw, σθ. 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background
  5. 3. Scaling systems
  6. 4. Empirical evaluation
  7. 5. Discussion
  8. 6. Conclusions
  9. Acknowledgements
  10. 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, σw, σθ, 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:

  • equation image(5a)
  • equation image(5b)
  • equation image(5c)
  • equation image(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 τ0, H0, to construct scales for length L* = − τ03/2/(κβH0), temperature T* = − H0/u*, and velocity equation image. Based on a dimensional analysis, Monin and Obukhov concluded that the non-dimensional products of statistical moments X in the surface layer (such as σw, σθ, S, Γ), and the flux-based scales, are universal functions φx of a single dimensionless parameter z/L*:

  • equation image(6)

where the exponents a, b, c are chosen in such a way that φx 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):

  • equation image(7a)
  • equation image(7b)
  • equation image(7c)

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

Sorbjan (e.g. 1986a; 1986b; 1988) argued that the functional form of universal similarity functions of the argument z/L* and z* is identical in stable conditions, φx(z/L*) = φx(z*). As a result

  • equation image(8a)
  • equation image(8b)
  • equation image(8c)
  • equation image(8d)

Applying a definition of the Richardson number yields:

  • equation image(9)

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

  • equation image(10a)
  • equation image(10b)
  • equation image(10c)
  • equation image(10d)

where ψm, ψh, ψw, ψθ are the universal similarity functions of the Richardson number. The same result can be formally obtained based on (1)–(4), with equation image, equation image, equation image, equation image, and an additional assumption that Ri/Pr < 1, where PrKm/Kh = fm/fh 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* and Ri are nearly zero, which implies that the values of similarity functions are constant. Specifically, φm(0) = 1 and φh(0) = Pr0, where Pr0 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, φm∼φhz*. With increasing thermal stratification, the parameter equation image. As a result, the similarity functions become singular (large), and strongly impacted by self-correlation.

3.2. The gradient-based σw - scaling

Now, let us consider the similarity scaling based on (5b), which involves the vertical velocity variance equation image, the temperature gradient Γ, and the parameter β (Sorbjan, 2006a):

  • equation image(11a)
  • equation image(11b)
  • equation image(11c)

The above scales are considered only for the case of the Brunt–Väisälä frequency N being sufficiently large. Applying (11) to (1)–(4) yields:

  • equation image(12a)
  • equation image(12b)
  • equation image(12c)

where Ft = fm/Ri, Fh = fh/Ri1/2, equation image, and Ri/Pr < 1.

Equation (3) implies that

  • equation image(13)

where Φw(Ri)∼{fmfε(1 − Ri/Pr)}1/3/Ri1/2 can be seen to be a function of the Richardson number. Using (13), equations (12a)–(12c) can be rewritten in an equivalent form:

  • equation image(14a)
  • equation image(14b)
  • equation image(14c)

where Φm, Φh, and Φθ 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 Φx of a single dimensionless parameter Ri:

  • equation image(15)

3.3. The gradient-based σθ scaling

Yet another similarity scaling can be introduced based on (5c). It involves the temperature variance equation image and the Brunt–Väisälä frequency N:

  • equation image(16a)
  • equation image(16b)
  • equation image(16c)

where the Brunt–Väisälä frequency N is assumed to be sufficiently large.

Applying (16) to (1)–(4) yields:

  • equation image(17a)
  • equation image(17b)
  • equation image(17c)
  • equation image(17d)

where Fw∼[fmfε(1 − Ri/Pr)]1/3/Ri1/2, Ψθ∼(fdfh)1/2/[fmfε(1 − Ri/Pr)]1/6, and Ri/Pr < 1.

Using (17d), we obtain from (17a)–(17c):

  • equation image(18a)
  • equation image(18b)
  • equation image(18c)

It can be noted that Ψw≡1/Φθ.

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 Ψx of a single dimensionless parameter Ri:

  • equation image(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:

  • equation image(20a)
  • equation image(20b)
  • equation image(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.

Employing (20), we obtain from (1)–(4):

  • equation image(21a)
  • equation image(21b)
  • equation image(21c)
  • equation image(21d)

where Gtfm/Ri, Gh = fh/Ri1/2, Gw∼[fmfε(1 − Ri/Pr)]1/3/Ri1/2, Gθ∼(fdfh)1/2/[fmfε/(1 − Ri/Pr)]1/6, and Ri/Pr < 1. It can be noted that GtFt, GhFh, Gw≡Φw, Gθ≡Ψθ.

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:

  • equation image(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.

The considered gradient-based and flux-based scaling systems are related. Based on (11), (16) and (20) one can obtain:

  • equation image(23a)
  • equation image(23b)
  • equation image(23c)
  • equation image(23d)

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

  • equation image(24a)
  • equation image(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.

4. Empirical evaluation

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

4.1. Data

The similarity functions formulated in the previous section will now be examined using the data collected during the Surface Heat Budget of the Arctic Ocean (SHEBA) experiment. Detailed overviews of the SHEBA programme and its data can be found in the papers of Andreas et al(1999; 2003; 2006), Persson et al(2002) and Grachev et al(2005; 2007a; 2007b; 2008). Thus, herein, only brief information about SHEBA instruments and data will be provided.

The experiment took place over Arctic pack ice, drifting in the Beaufort Gyre to the north of Alaska (latitude from 74°N to 81°N), from October 1997 through to September 1998. The sub-polar site offered a number of advantages, especially due to the stationarity of weather conditions, and the lack of contamination by drainage or strong advective flows. Except for rare periods, instruments ran almost continuously for 11 months and produced well over 6000 hours of useful data, covering a wide range of stability conditions.

Turbulent and mean meteorological data during SHEBA were obtained on the 20 m main tower (Grachev et al, 2005). Observations were continuously collected at five levels, located at 2.2 m, 3.2 m, 5.1 m, 8.9 m, and 18.2 or 14 m above the surface. The variances and covariances at each level were based on one-hour averaging, and derived through frequency integration of both spectra and cospectra. To prevent a possible flux loss caused by inadequate frequency responses and sensor separations, a prerequisite that the wind velocity U > 1 m/s has been imposed on the sample. Data for the first level, which reflected a relatively large scatter due to local surface effects, were not considered. In addition, data with a temperature difference between the air at median level and the snow surface of less than 0.5 °C were excluded to avoid the large uncertainty in determining the sensible heat flux. Vertical gradients of the mean wind speed and the potential temperature were obtained by fitting a second-order polynomial through the 1-hour profiles followed by an evaluation of the derivative with respect to z for levels 1–5.

The data points presented in this paper are based on a bin-averaging of the individual one-hour data at levels 2, 3, 4 and 5. For this purpose, data were first sorted into bins by using the Richardson number Ri as the sorting parameter. Then, the mean values of relevant parameters were computed for each bin (e.g. Grachev et al, 2008).

It can be noted that in practice there is often a problem with the differentiation between nearly neutral and very stable regimes in cases when both temperature and wind gradients are small. The number of outliers with high values of Ri in nearly neutral conditions is often comparable with the number of points that truly have high values of Ri in very stable conditions. As a result, the Ri-outliers from the nearly neutral range can significantly distort the picture in the very stable regime. To resolve this problem, a special prerequisite was applied to data in order to limit the influence of outliers on the bin-averaging: 0.5Rie < Ri < 2Rie, where the value of the Richardson number Rie was estimated based on an equation analogous to (9), with the analytical form of the Monin–Obukhov similarity functions φh and φm of z/L* obtained by Grachev et al(2007a), and Grachev et al(2008). More specifically, if the actual value of a Richardson number Ri = N2/S2 was not in the range defined by Rie, the data point was rejected. Ultimately, this procedure removed those cases for which there was an inconsistency between the Richardson number calculated directly and analytically.

4.2. The master scaling

The dependence of the dimensionless fluxes, equation image and Gh = − H/(UsTs) on the Richardson number Ri is shown in Figure 1. The dimensionless moments Gw = σw/Us and Gθ = σθ/Ts are depicted in Figure 2. A clustering of data points in the figures is caused by the fact that the Richardson number Ri is a sorting parameter. The vertical lines with horizontal bars represent the confidence intervals, obtained by perturbing the mean values evaluated at level 5 by one standard deviation. Because the ordinate is logarithmic, the confidence intervals are asymmetric.

thumbnail image

Figure 1. Dependence of the bin-averaged values of the dimensionless (a) momentum flux equation image, (b) heat flux Gh = − H/(UsTs), on the Richardson number Ri. The solid lines are plotted based on Eqs. (27a) and (27b). The vertical lines represent the confidence intervals evaluated at level 5. Data points within the extremely-stable domain are marked by the shaded box.

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thumbnail image

Figure 2. Dependence of the bin-averaged values of the dimensionless standard deviations for (a) vertical velocity Gw = σw/Us[≡Φw = Lw/(κz)], (b) temperature Gθ = σθ/Ts [≡Ψθ = Lθ/(κz)], on the Richardson number Ri. The solid lines are plotted based on Eqs. (27c) and (27d). Data points within the extremely stable domain are marked by the shaded box.

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The scatter in Figure 1(b) is larger than that in Figure 1(a). This can be associated with thermal inhomogeneity around the observational site (e.g. Kukharets and Tsvang, 1998; Tsvang et al, 1998). The ice floe around the main tower was multi-year pack ice, with varying degrees of thickness and a heterogeneous surface of a different type and salinity, snow with a different depth and age, melt-ponds, and even leads (e.g. Sorbjan and Grachev, 2010). The various surface ‘patches’ were characterized by different albedo, thermal capacity and conductivity and, therefore, by different temperatures. Andreas et al(1998) reported analogous behaviour for the case of humidity statistics over a surface with vegetation that was ‘patchy’ at metre scales.

In order to further evaluate the presented results, we note that in nearly neutral conditions, the functions fm, fh, fε, fd are constant, and that following from (1)–(4) τ∼(κzS)2, σw∼κzS, and also H∼(κz)2SN2/β, σθ∼κzN2/β. Combining these relations with (20)–(21) and (23)–(24), we can conclude that for nearly neutral conditions

  • equation image(25a)
  • equation image(25b)
  • equation image(25c)
  • equation image(25d)

Figures 1(a), 1(b) and 2(a) confirm the above predictions for Ri < 0.01. However, the values of the dimensionless temperature variance Gθ in Figure 2(b) are larger than expected in the nearly neutral range. This fact implies that the values of the temperature variance are overestimated for these conditions. Analogous effects can be observed in all of the figures that involve the temperature variance.

In the supercritical range, Ri > Ricr, the values of the similarity functions in Figures 1 and 2 fall off in a coherent fashion with the increasing values of Ri. This indicates the presence of a self-similar regime in very stable conditions. Unfortunately, the dimensional analyses do not allow for the formulation of any constructive similarity prediction. Therefore, we will assume, based on the presented empirical evidence, that the similarity functions obey the following power laws:

  • equation image(26a)
  • equation image(26b)
  • equation image(26c)
  • equation image(26d)

with a validity range of approximately 0.1 < Ri < 0.7. Above this range, the values of the similarity functions are incoherent and scattered. Such behaviour suggests a lack of any general similarity laws for these values of Ri. Consequently, we will limit our analysis to the range Ri < 0.7, and disregard the domain marked by the shaded boxes in Figures 1–2 (and also in the remaining figures).

Taking (25) and (26) into consideration, we adopt the following approximations of the similarity functions:

  • equation image(27a)
  • equation image(27b)
  • equation image(27c)
  • equation image(27d)

The above equations are represented in Figures 1 and 2 by solid curves. The agreement of the curves with data points is generally good.

Using (24a, b) and (27a, b), we also obtain:

  • equation image(28a)
  • equation image(28b)

Further examination of the flux-based similarity functions (28) may be found in the work of Sorbjan and Grachev (2010), who used data collected during both the SHEBA and the CASES-99 field programmes.

4.3. The σw-scaling

Employing (14), (23) and (27) yields:

  • equation image(29a)
  • equation image(29b)
  • equation image(29c)

The dependence of the dimensionless fluxes, Φt and Φh, on the Richardson number Ri is shown in Figure 3(a) and (b).

thumbnail image

Figure 3. Dependence of the bin-averaged values of the dimensionless fluxes for (a) momentum equation image, and (b) heat Φh = − H/(UwTw), on the Richardson number Ri. The solid lines are plotted based on Eqs. (29a) and (29b). The vertical lines represent the confidence intervals evaluated at level 5. Data points within the extremely stable domain are marked by the shaded box.

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Figure 3(a) indicates that the dimensionless momentum flux Φt is constant for Ri < 0.03, slightly decreases in the range of Ri from 0.03 to 0.1, and more quickly falls off for Ri > 0.1. The dimensionless temperature flux Φh in Figure 3(b) increases with an increasing Richardson number in near-neutral conditions, and decreases for larger values of Ri. At the value of Ri about 0.125, the negative dimensionless flux reaches a maximum, equal to about 0.27. Taking into consideration that the dimensionless flux Figure 1(b) is monotonic, this result may appear surprising. However, further analysis indicates that the result is caused by the inclusion of σw in the definition of the velocity and temperature scales.

4.4. The σθ-scaling

Considering (18), (23) and (27), we obtain:

  • equation image(30a)
  • equation image(30b)
  • equation image(30c)

The dependence of the dimensionless fluxes, Ψt and − Ψh, on the Richardson number Ri, is depicted in Figure 4(a) and (b). The ordinate is in the range from 0.005 to 10. This range is smaller than in the previous figures, as data points for the smallest values of the Richardson number were excluded due to their significant scatter.

thumbnail image

Figure 4. Dependence of the bin-averaged values of the dimensionless (a) momentum flux equation image, and (b) heat flux Ψh = − H/(UθTθ), on the Richardson number Ri. The solid lines are plotted based on Eqs. (30a) and (30b). The vertical lines represent the confidence intervals evaluated at level 5. Data points within the extremely stable domain are marked by the shaded box.

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The scatter of data points in Figure 4(a) and (b) allows for only a tentative evaluation of the above expressions in the nearly neutral range. Equation (30a), plotted in Figure 4(a), has an inflection point around Ri = 0.025. Figure 4(b) shows that expression (30b) possesses a local minimum at about Ri = 0.013, and a local maximum at about Ri = 0.05. For larger values of Ri, the dimensionless momentum and heat fluxes monotonically decrease with an increasing Ri.

5. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background
  5. 3. Scaling systems
  6. 4. Empirical evaluation
  7. 5. Discussion
  8. 6. Conclusions
  9. Acknowledgements
  10. 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 = − βH/(τS). Employing (21a, b) and (27a, b), we obtain:

  • equation image(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.

thumbnail image

Figure 5. Dependence of the bin-averaged values of the flux Richardson number Rf on the gradient Richardson number Ri. The solid line is plotted based on Eq. (31). The vertical lines represent the confidence intervals evaluated at level 5. Data points within the extremely stable domain are marked by the shaded box.

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Taking into consideration that RfRi/Pr and by using (31), we find that

  • equation image(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:

  • equation image(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 Ris = 0.7, steady-state turbulence would not be present. In other words, at Ri > Rs, turbulence is non-stationary, i.e. decaying or sporadic. The inequality Ri < Rs = 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:

  • equation image(34)

The values of − rwθ 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 − rwθ 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.

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Figure 6. Dependence of the bin-averaged values of the negative correlation coefficient − rwθ on the Richardson number Ri. The solid line is plotted based on Eq. (34). Data points within the extremely stable domain are marked by the shaded box.

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Figure 7 depicts the dimensionless components of the doubled turbulent kinetic energy: equation image, equation image, equation image, where equation image, equation image, equation image are the velocity variances, and q2 is the doubled turbulent kinetic energy. The figure indicates that for nearly neutral conditions, all the components are approximately constant, and Au ≈ 0.62, Av ≈ 0.26, and Aw ≈ 0.12. In the range 0.01 < Ri < 0.1, Au decreases to 0.5, and is approximately constant up to Ri = 0.7. The component Av increases to about 0.4 at Ri = 0.7. Consequently, one can observe that Au and Av have a tendency to become nearly equal at Ri > 0.7. The vertical component Aw increases and reaches maximum at Ri = 0.1. Subsequently, it decreases for larger Ri. The vanishing values of Aw 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.

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Figure 7. Dependence of the bin-averaged values of the dimensionless components of the doubled turbulent kinetic energy, equation image, equation image, equation image, on the Richardson number Ri. The vertical lines represent the confidence intervals evaluated at level 5.

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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.

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Figure 8. Sub-regimes of the stably stratified boundary layer.

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6. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background
  5. 3. Scaling systems
  6. 4. Empirical evaluation
  7. 5. Discussion
  8. 6. Conclusions
  9. Acknowledgements
  10. 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 σw and σθ 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, − rwθ, 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 Au and Av have a tendency to become nearly equal for Ri > 0.7. The vertical component Aw 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 > Ris = 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.

Acknowledgements

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

The author's appreciation is directed to A. A. Grachev of CIRES/NOAA for providing the processed SHEBA data. The work has been partly supported by the National Science Foundation grant ATM-0938293.

References

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