Research on atmospheric turbulence by Wilfried Brutsaert and collaborators


  • Nelson L. Dias

    Corresponding author
    1. Department of Environmental Engineering, Federal University of Paraná, Curitiba, Brazil
    • Corresponding author: N. Dias, Department of Environmental Engineering, Federal University of Paraná, Centro Politécnico da UFPR, Caixa Postal 19100, CEP 81531-990, Curitiba, Paraná 81531-990, Brazil. (

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[1] In this paper, several lines of investigation of atmospheric turbulence, by Wilfried Brutsaert, his students, and collaborators, are reviewed. Overall, we classify these lines as K-theory, surface roughness parameterization dealing with momentum and scalar fluxes, radiative effects on temperature fluctuations, stable conditions, scalar similarity, and atmospheric boundary-layer parameterization. Emphasis is placed on turbulence parameterization. Although these topics are presented more or less in chronological order, this order is broken whenever connections need to be established. Hopefully, these connections are the most interesting part of this review: it is there that Brutsaert's insights and long-range scientific questions may be found. His approach invariably included a careful formulation of the physical and mathematical basis of the problem at hand, and proceeded to focus on some essential issues that allowed analytical, numerical or statistical treatment. There is much to be learned from this approach; it is hoped that some of it can be glimpsed here.

1. Introduction

[2] The atmospheric boundary layer is the region of the atmosphere closest to the surface of the Earth and directly impacted by the surface fluxes of momentum, sensible heat, water vapor, and several gases of interest in air pollution and biogeochemical studies [Brutsaert, 1982; Stull, 1988; Garratt, 1994; Seinfeld and Pandis, 1998]. Therefore, the connection between the ABL features and the regional, or basin-wide, surface fluxes is evident: understanding the ABL is essential for identifying the relevant scales of most hydrological problems; to decide which processes to represent explicitly, and which to parameterize, and more generally for improving our hydrological models, as made clear by Brutsaert [1986].

[3] Most of the time, the ABL is turbulent, particularly in the case of the convective, daytime boundary layer, when the surface fluxes are largest. Then, prompted by the practical needs of hydrology, one is naturally drawn into studying ABL turbulence, a situation not unlike many other cases where engineering demands motivated the advancement of fluid mechanics. The archetypal flow is the turbulent boundary layer (originally studied for pipes and channels, see, e.g., Darrigol [2005]), which not only gives the name to the ABL but is the starting point for many concepts that prove fruitful also for atmospheric turbulence.

[4] As it happens with many researchers, the approaches used by Wilfried Brutsaert and his collaborators to a better understanding of atmospheric turbulence that we will be reviewing here follow many of the early paths taken by fluid dynamicists to study turbulent flows. At the same time, however, they added substantial improvements as these approaches needed adaptation to the particularities of the ABL: its inherent nonstationarity, the surface geometry imposed by nature, the surface spatial inhomogeneities, the effects of buoyancy, and many other facets, all required a concerted effort to adapt, and sometimes create, the concepts needed to model ABL turbulence.

[5] The focus of the present work is atmospheric turbulence itself. Thus, in section 2, we review the results derived from semi-emipircal, or K-theory, for the behavior of surfaces ranging from evaporating pans to natural lakes, as well as insights on the form of the turbulent diffusivity tensor itself. In section 3, we review contributions to the parameterization of scalar mass and heat-transfer coefficients with the identification of scalar roughnesses different from those for momentum. Section 4 gives a brief overview of results concerning the relative importance and direct dissipative effects of longwave radiation in ABL turbulence. Section 5 deals with results for the stable surface layer and section 6 with the important issue of how different the turbulent transport characteristics of two scalars are. In a sense, this is a development of the ideas started with the so-called Reynolds' analogy [Reynolds, 1900] to the study of turbulent transfer of different scalars. Section 7 then deals with results for the convective boundary layer and its interface with the top of the surface layer where local free convective conditions prevail. They are also related to drag, mass, and heat transfer equations for turbulent flow in pipes and channels, as well as by the matching techniques developed for the turbulent boundary layer, starting with Milikan [1938]. Some brief concluding remarks are given in section 8.

2. The Seminal Years: Results From the Semi-Empirical, or K, Theory of Turbulence

[6] In W. Brutsaert's “Evaporation into the Atmosphere” [Brutsaert, 1982], the first reference to one of his own works about the atmospheric boundary layer (more specifically, about evaporation from very small surfaces into the atmosphere) is Brutsaert [1967], which is an attempt at understanding evaporation from a very small surface (such as, it was believed, a pan evaporimeter; however, see the comment below on the limit of validity of the very small surface approximation with regard to the later work of Brutsaert and Yeh [1970a]). But his interest in the subject antedates this paper by at least a couple of years. In three papers published in 1965 [Brutsaert, 1965a-1965c], we find what may be the first results of Brutsaert's long and influential career in atmospheric research. In the present work I will try to give a very brief overview of those efforts, from those early times till present.

[7] In Brutsaert [1965a], he starts a careful analysis of the behavior of turbulence quantities in the surface layer. He is already keenly aware that transfer mechanisms for momentum and scalars may be different. It is also noteworthy that the possibility that radiative heat transfer may affect eddy diffusivities is already taken into account, at least hypothetically, in this paper. Another very interesting point surfaces: the meeting of Dalton and Reynolds in the same Society, but no less than 72 years apart: Dalton [1802] made the observation that “…evaporation is proportional to the difference in vapor pressure at the surface of the water and in the air. He also indicated that this proportionality is affected by the velocity of the wind” [Brutsaert, 1965a]. Reynolds' analogy [Reynolds, 1900], which forms the basis for many of the early equations involving scalar turbulent transport studied by Brutsaert, was published in the same society proceedings in 1874. Reynolds' analogy, in its original form, is disconcertingly simple. It reads: “And various considerations lead to the supposition that A and B in (I) are proportional to inline image and inline image in (II)” (A and B are empirical coefficients in a momentum transfer formulation, whereas inline image and inline image are analogous coefficients in a heat transfer formulation). What these various considerations are, Reynolds does not bother to tell us!

[8] Reynolds' decomposition, introduced in Reynolds' 1895 paper [Reynolds, 1895; Jackson and Launder, 2007], duly figures in Brutsaert [1965a]. Most of Brutsaert's published work on turbulence, including “Evaporation into the Atmosphere,” adopts the same notation as Reynolds' (and so shall we), viz.,

display math(1)

where inline image is the mean, and inline image the fluctuation, of a. Of course, the nature of the averaging in (1) can vary. Originally, Reynolds [1895] considered volume averages [Wyngaard, 2010, p. 30]. In this work, in general we will be referring to ensemble means, but on occasion they may be taken as time or other suitable averages, as is often done in atmospheric turbulence studies [Brutsaert, 1982, pp. 50–51; Wyngaard, 2010, chap. 2].

[9] The interest in pan devices is clearly present in Brutsaert [1967]: for the case of a very small circular evaporating surface, the advection term of the water vapor transport equation becomes negligible in the presence of the turbulent diffusion terms in three directions x, y, and z. Axisymmetry, and therefore cylindrical (Bessel) functions, are explored. The result is a theoretical investigation of “Dalton-like” formulae for pan evaporation. However, empirical equations of the kind that abounds in hydrology textbooks are missing, except for some useful information about the behavior of the inline image exponent appearing in (2) below, and no attempt is made to compare the results with them: characteristically, Brutsaert seems much more preoccupied with deriving pan evaporation equations on a solid theoretical ground, although experimental data at this time were still inconclusive about what best exponents should be used in the general result obtained in his equation (28), viz.,

display math(2)

where KE is a dimensional coefficient, inline image is the mean specific humidity, the subscripts 0 and a are for the surface value and the overlying air, r0 is the pan radius, and inline image is the friction velocity. More information on inline image and inline image will be obtained from a dedicated experiment at Cornell [Brutsaert and Yu, 1968], with best results found for inline image. A similar approach will be taken again, briefly, in Brutsaert and Yeh [1969] to assess the relative importance of lateral diffusion on the evaporation from small surfaces. Later on, however, it would be found that the upper limit for the very small surface qualification for which the analysis applied is considerably less than 1 square foot [Brutsaert and Yeh, 1970a, p. 1206].

[10] The basis of these early representations of turbulent transport of water vapor is, unavoidably, semi-empirical K-theory. The simplest gradient-turbulent diffusion relationship for the turbulent water vapor flux vector E is

display math(3)

(where inline image is the mean density of the air) which requires, however, that the (scalar) eddy diffusivity K be isotropic. Prompted by the experimental evidence of anisotropy as apparent in the different values of the velocity variances inline image and inline image, Brutsaert [1970] proposed instead to look at

display math(4)

[11] In equation (4), K is a tensor, and the dot denotes a single contraction [Kundu, 1990, chap. 2, sec. 16]. Therefore, unless x, y, and z coincide with its principal axes, the vertical water vapor flux is

display math(5)

[12] In equation (5), x is taken to coincide with the direction of the mean wind. Following Brutsaert [1970], the assumption that y is one of the principal axes, and therefore inline image, was never in doubt, but several authors questioned the neglect of Kzx (one can also argue that by aligning the mean wind with x, the flow can be assumed to be statistically homogeneous in the cross-direction y, and that inline image). The prediction that inline image follows from the insight that measured inline image (θ is potential temperature). Then, in typical style that will appear over and over again, Brutsaert finds order-of-magnitude predictions for the elements of K that turn out to be confirmed by experimental data at the time. Moreover, his predicted Kzx is negative, as follows theoretically from the extension of Taylor's diffusion theory to take it into account and also observationally by inserting inline image (by analogy with temperature) in the x-component of (4), with inline image. The importance of this Kzx in the solution of developing internal boundary layers and the surface fluxes predicted from it would be checked carefully, and shown to be small, by Yeh and Brutsaert [1971a]. In spite of, or maybe because of this, we are not aware that this line of attack has ever been tried again, but inline image is different from zero, and the anisotropic eddy diffusivity may still be a worthy research theme.

[13] Next, in analyzing the power law as a suitable approximation for the wind profile next to the ground, Brutsaert and Yeh [1970b] unveiled an early reference to what amounts to an analytical derivation of Manning's formula by Keulegan [1938]. The derivation is important to explain the apparent dimensional inconsistency in the formula. The subject is explained in Chapter 5 of Brutsaert's Hydrology [2005] but is out of our present scope.

[14] A crowning achievement of these early years is, without doubt, the paper on formulae for evaporation from lakes and pans [Brutsaert and Yeh, 1970a]; the paper is very terse and yet full of insights and detailed analysis of experimental results in the literature. A central device for the good results obtained is a power law for the wind profile of the type

display math(6)

with inline image, inline image, for the mean wind speed at height z as a function of inline image and the roughness length z0 for momentum. Such power laws were at the heart of the early theories by L. Prandtl on turbulent boundary layers [Bodenschatz and Eckert, 2011]; they were later largely replaced by the log law, but their analytical tractability was one of the reasons that Prandtl let the opportunity of claiming the log law pass to von Kármán (see reference above); the same tractability—and not any deeper physical reason—is the main reason for its adoption in a number of papers by Brutsaert and his collaborators. It is of interest, however, to note that the issue is not settled in favor of the log law. By reintroducing the Reynolds number dependency (i.e., by considering large, but not infinite, Reynolds numbers), both modified log laws [Wosnik et al., 2000] and power laws [Barenblatt and Chorin, 1998] can result in turbulent boundary layers.

[15] The joint work of Brutsaert and G. T. Yeh on analytical and numerical solutions of the advection-diffusion equation basically to study “Sutton's problem” [Sutton, 1934] in detail include, still, a detailed perturbation solution for the equation when longitudinal diffusion is included [Yeh and Brutsaert, 1970]; a numerical study along the same lines which also verified the practical validity of power laws in the analytical solutions previously obtained [Yeh and Brutsaert, 1971b]; and a study of the joint behavior of the temperature and humidity fields in the internal boundary layer formed over a lake [Yeh and Brutsaert, 1971c]. In the latter, it is found—after a mathematical analysis of the joint solution of the diffusion equations for heat and water vapor—that adoption of an area average for the water surface temperature is quite reasonable for practical purposes.

[16] Many of the results obtained by Brutsaert using the semi-empirical theory of turbulence and power laws should be revisited with knowledge that has been accumulating on the behavior of the ABL. For example, many of his results could be studied applying more universally accepted similarity functions, at the expense, of course, of analytical simplicity. A technique that seems particularly well suited is the “semi-analytical” approach [see, e.g., Kumar and Sharan, 2010], because it allows great flexibility in the specification of wind profiles and eddy diffusivities throughout the ABL.

[17] Up to this point Brutsaert had pursued the various aspects of lake/pan evaporation under advective conditions. This included neutral and nonneutral conditions, water surface size, the effect of neglecting longitudinal and lateral diffusion, and the coupling between temperature and humidity. Momentum advection still had not been studied, but this was taken up in the sequence. Weisman and Brutsaert [1973] studied numerically the effect of a sudden change in surface humidity and temperature on momentum advection and its impact on lake evaporation. The effect can be significant and can be quantified by proper nondimensionalization of the step changes.

[18] In Weisman and Brutsaert's [1973] work, the roughness length did not change from land to lake. Although wind adjustment to roughness step changes seems to be rather quick [cf. Brutsaert, 1982, pp. 166–167], it is of interest to develop practical methods to predict the wind over large water surfaces. Such methods are painfully lacking, and the still widespread use of over-land meterological data in water quality and evaporation studies should be replaced by sounder approaches.

3. The Scalar Transfer Parameterization

[19] By the simple expedient of parameterizing the rate of dissipation of turbulence kinetic energy (TKE) close to a rough surface by

display math(7)

it is one of the earliest theoretical results by Brutsaert [Brutsaert, 1965c] that the Kolmogorov length microscale close to a rough wall is given by

display math(8)

[20] In (7) and (8), inline image is von Kármán's constant and ν is the kinematic viscosity of air. Equation (7) shows clearly that the rate of dissipation of TKE increases dramatically as the surface is approached, whereas equation (8) provides a convenient estimate of the Kolmogorov microscale from the readily available z0. Brutsaert's 1965 paper, notwithstanding, had limited applicability; in his own words [Brutsaert, 1975a]: “The usefulness of this model for the lower atmosphere was limited, however, since no provision was made to couple the transfer at the interface with that taking place in the fully turbulent dynamic sublayer.”

[21] The theory for roughness lengths and transfer coefficients, or scalar transfer parameterization for short, developed by Wilfried Brutsaert rests on a standard matching of the interfacial and the surface sublayers. To save time and space, I will cover only the case of rough-wall turbulent flow (by far the most common in nature). We start with the surface sublayer. Here, Monin-Obukhov similarity theory gives the standard results for the mean wind speed and mean scalar concentration profiles:

display math(9)
display math(10)

where von Kármán's constant κ is assumed to be the same for momentum and all scalars [see Brutsaert, 1982, equation (4.12)]. In (9) and (10), z1 and z2 are any two measurement heights within the surface sublayer, inline image and inline image are the mean wind velocity and concentration at these levels, inline image is Obukhov's [1946, 1971] stability variable at inline image and inline image, d0 is the zero-plane displacement height, and inline image and inline image are the similarity functions for the mean profiles [cf. Brutsaert, 1982, p. 66, equations (4.36)–(4.38)]. Note that Brutsaert [1975a, 1975b] used inline image, which is valid for sparsely placed roughness elements.

[22] The scalar transfer equations for the interfacial layer and for the surface layer are given, respectively, by

display math(11)
display math(12)

where F is the scalar surface flux, inline image is the interfacial transfer coefficient (or Dalton, or Stanton, number, depending on the scalar); inline image is the scalar transfer coefficient between the surface and a reference, or measurement, level za; and h is the height of the border between the interfacial sublayer and the surface sublayer. The objective is to arrive at an expression for inline image in (12) from (9) and (11), matching (and thereby eliminating) the value of inline image at the interface of the two sublayers. In order to do that we rewrite (11), using inline image, as

display math(13)

[23] Now use (10) with inline image, where h is assumed to be the top of the interfacial layer, and inline image (whence inline image):

display math(14)

[24] The height of the interface h can be eliminated from (14) by means of

display math(15)

which results from applying (9) between inline image and inline image. But with

display math(16)

we obtain

display math(17)

which finally gives, with reference to (12),

display math(18)

[25] Equation (18) is one of the most important results from Brutsaert [1975a], where it is equation (23). The significance of (18) stems from the fact that it is possible to obtain a formulation for every term on its right-hand side. The most straightforward is inline image: from (9) and (16),

display math(19)

[26] Next it is remarked that the log law for the velocity profile appears to be valid down to inline image (the height of the roughness elements); using the estimate

display math(20)

[27] [Brutsaert, 1975a, cf. first paragraph after equation (10), but with inline image], (9) with (16) yields

display math(21)

[28] The approach for calculating inline image is the most sophisticated part of the theory. For any kind of surface, the surface flux is modeled in Brutsaert [1965c] and Brutsaert [1975a] as a succession of unsteady-state diffusion problems into eddies, which remain “in contact” with the surface during a time t before being swept away. Each eddy has a lifetime t distributed according to an exponential distribution with parameter s. The flux is given as the average over all eddy lifetimes:

display math(22)

where inline image is the scalar's molecular diffusivity in air. The molecular transfer into an eddy of age t is modeled as a diffusion into a stagnant fluid of infinite depth, an idea from Danckwerts [1951]. Here, the mean rate of renewal is made proportional to the reciprocal of Kolmogorov's time microscale,

display math(23)

where inline image is the thickness of the interfacial layer. The concentration gradient at the surface, on the other hand, is obtained from the solution of the well-known problem

display math(24)
display math(25)
display math(26)
display math(27)

[29] The last condition, an initial condition, makes all the sense because the eddy just “arrived” at the surface bringing its concentration from z = h. The solution of this problem gives

display math(28)

so that the integral in (22) becomes

display math(29)

[30] With (20) and (23), the theory is complete except for a proportionality constant and takes the form

display math(30)

or, equivalently,

display math(31)

where inline image is the roughness Reynolds number, and inline image is the Schmidt number. Given an estimate of inline image and z0, therefore, and if CR can be established, (31) is the last ingredient that was still missing in (18), and a full estimate of the mass (or heat) transfer coefficient inline image is available. From Brutsaert [1975a], inline image.

[31] We should remark that the assumption inline image made above to arrive at (18) may not be valid for tall vegetation [see Brutsaert, 1982, p. 89]. This is a topic worthy of investigation in its own right, given the very large attention that vegetated canopies, in particular forests, are being given with regard to deforestation and greenhouse gas fluxes.

[32] The whole theory as presented in Brutsaert [1975a] is now closed and able to predict mass and heat transfer coefficients between the surface duly characterized by its roughness length z0 and a measurement height za, but it does not give an explicit role for the scalar roughness length inline image. Indeed, from (9) and (10) together one can easily obtain the general transfer equation

display math(32)

where it must be realized that inline image, the scalar roughness length, may be different from z0, the momentum roughness length. This can now be compared with (17) and (18), to find, by means of (21) and (31),

display math(33)

[33] Equation (33) is the main result in Brutsaert [1975b]. The implications of Brutsaert's theory for scalar transfer parameterization are far reaching. From Brutsaert [1975a], we quote: “Thus local evaporation (or heat transfer) in the absence of advection is commonly determined by equations such as (24), but with the assumption that inline image may be replaced by inline image …Although this procedure is probably often adequate in the solution of some problems, this is not the general case.” and then from Brutsaert [1975b]: “the assumption that inline image may introduce marked errors in profile data analyses.” The importance of the difference between momentum roughness z0 and scalar roughness (e.g., inline image) is again called to the attention of the reader in Brutsaert and Mawdsley [1976]. A good illustration of the difference between inline image and inline image is given in Brutsaert [1975a, Figure 6] and is reproduced here in Figure 1. The transfer coefficients are calculated by means of (16) for inline image, inline image and neutral conditions ( inline image), for a rough surface.

Figure 1.

Effect of the roughness length z0 at inline image (under neutral conditions) on the mass transfer coefficient inline image (solid line), and the ratio inline image (dashed line), at inline image. Adapted from Figure 6 in Brutsaert [1975a].

[34] Brutsaert went on to study the roughness over tall vegetation. In a third paper in 1975 [Brutsaert, 1975c], he reviews values of inline image for crops in the literature and summarizes them as

display math(34)

a result which will be highlighted in his book [Brutsaert, 1982, Figure 5.2]. In Brutsaert [1975c], the height of the roughness elements h0 is taken to be the same as the canopy height. In the same paper, the constant used in (20) is 2.7, whereas Garratt [1994, p. 87] gives values in the range of 7.1–14.3, all of which would change the constants in (33). Also, the height of the roughness sublayer can extend up to 1.5 to three times the canopy height h0 [Williams et al., 2007], which also can change the final values of the constants through (20).

[35] Moreover, the approach described above is strictly valid for bluff roughness elements; for a surface with permeable roughness; however, the situation is not so simple. For this case, Brutsaert again obtained results based on analytical functions [Brutsaert, 1979]. Much later on, this will produce the most appropriate approach to parameterize the sensible heat roughness length inline image over natural prairie [Qualls and Brutsaert, 1996], although the coefficients will need to be adjusted to take into account the remotely sensed (radiometric) surface temperature. Those results also formed the basis to understand heat transfer through an anisothermal canopy [Brutsaert and Sugita, 1996], which allowed to explain how inline image depends on solar elevation; experimental strategies to cope with this fact were then proposed by Sugita and Brutsaert [1996].

[36] Brutsaert's theory on scalar transfer parameterization will have a profound impact, and the list of papers that it influenced is just too long to write down. It has found ample support in later studies, for example, Hignett [1994], Cahill et al. [1997], and De Ridder [2006].

4. Radiation

[37] The work of Brutsaert on the interplay between long-wave radiation and the maintenance of turbulence in the ABL [Brutsaert, 1972] stands by itself; it is not even cited in “Evaporation into the Atmosphere.” The work was followed by the study of the importance of radiation on the temperature spectrum (a related topic) by Dias and Brutsaert [1998]. The conclusion of the latter was that, close to the surface, radiation was relatively unimportant with regard to the fluctuating temperature field. Most of the time, more recent works on stable conditions have not given much attention to radiation effects on the dissipation of temperature fluctuations [e.g., Vickers and Mahrt, 2004; Siqueira and Katul, 2010], and in fact Dias and Brutsaert's conclusion on the relative unimportance of radiation in this respect has been accepted to simplify some analyses of radiative effects in the ABL [Edwards, 2009].

[38] There are some reasons why the subject should be further investigated. A considerable body of data on stable conditions is now available [e.g., Cheng et al., 2005; Duarte et al., 2012] that might allow the study of radiative effects at a few tens of meters above the surface in a stable, nocturnal boundary layer. There, some of the effects predicted by Brutsaert [1972], and Dias and Brutsaert [1998] might be more easily identified under conditions of weak or intermittent turbulence. We note that this section does not deal with the effects of longwave radiation on the cooling of the stable boundary layer, i.e., with radiative effects on the mean temperature field inline image. Rather, in the following, we review some results on the much less frequently studied effects of longwave radiation on the dissipation of temperature fluctuations.

[39] Brutsaert [1972] used Townsend's [1958] pioneering theoretical framework to analyze, for the first time, the interaction of radiation and turbulence in the atmospheric boundary layer. The tool for his analysis was the Richardson number, both in flux and gradient form. Introduced by Richardson [1920], these forms are

display math(35)

[40]  inline image is the ratio of production or destruction of turbulence kinetic energy by buoyancy to its production by wind shear (c.f., (38) below). The second, inline image, is often used when the turbulent fluxes are not directly available; one has inline image where Kh is the eddy diffusivity for heat, and Km the eddy diffusivity for momentum. Using inline image, Richardson [1920] argued that for turbulence to be maintained one should have inline image; on account of molecular dissipation effects, however, the “critical” inline image for the maintenance of turbulence is often smaller [Lyons et al., 1964]; under certain assumptions, Taylor [1931] found inline image, but one should be aware that there are no definite values neither from theory nor from experiment: Zilitinkevich et al. [2007] found a transitional interval, inline image, between strong and weak turbulence, and Galperin et al. [2007] report data supporting the fact that turbulence can survive for inline image.

[41] At any rate, Brutsaert [1972] studied the critical values of the flux Richardson number inline image for the maintenance of turbulence and its dependence on radiation but did the whole analysis with turbulence and radiative models appropriate to the atmospheric boundary layer, instead of the much higher regions studied by Townsend. One important conclusion of Brutsaert's work is that there is no definite (flux) critical Richardson number at which turbulence is suppressed. In the presence of radiation and evaporation, it is found that the critical number can fall in a range between 0.25 and 0.50. In other words, radiation can increase the critical Richardson number, something that had already been qualitatively observed by Townsend.

[42] Coantic and Simonin [1984] applied earlier results from Spiegel [1957], Goody [1964], Simonin et al. [1981], and Schertzer and Simonin [1981] to the Earth's planetary boundary layer; both isotropic and anisotropic spectral temperature budgets were used, and a very thorough analysis of radiative effects was performed, with the inclusion of the continuum absorption, inline image effects, and scattering by water vapor droplets. Some of their conclusions include the fact that the water vapor density is relatively unimportant; the turbulence kinetic energy ( inline image in standard indicial notation), on the other hand, plays an important role, with radiative effects becoming important for sufficiently small values ( inline image). In their analysis of nonisotropic turbulence, they used Kaimal's [1973] curve for near-neutral conditions as an initial value in the calculation of the temperature spectrum, and production was calculated with a near-neutral dimensionless temperature gradient inline image (see next section for expressions for inline image). Different from Brutsaert [1972], the effects of stability were not taken explicitly into account.

[43] All the relatively small amount of work on the dissipation of temperature fluctuations in the wave number domain by radiative effects so far had been based on the following spectral budgets:

display math(36)
display math(37)

where the Ee and inline image are 3-D spherical shell averages of twice the turbulence kinetic energy and the temperature spectra; Ewu and inline image are the same, but for the (vertical velocity)–(horizontal velocity) and (vertical velocity)–(temperature) cospectra; g is the acceleration of gravity, and inline image is the mean temperature (humidity effects and therefore virtual temperatures are not included for the sake of simplicity); inline image is the molecular diffusivity of heat; and N(k) is the spectral dissipation function. Te and inline image are the inertial transfer terms (whose integral is zero over wave number k). Integration of (36) and (37) over k produces the well-known budgets of TKE and temperature variance,

display math(38)
display math(39)

where w is vertical velocity, p is pressure, inline image is the rate of dissipation of temperature semivariance by molecular effects, and inline image the rate of dissipation of temperature semivariance by radiative effects. (Note, however, that the transport terms above cannot be derived from (36) and (37), because those spectral budgets apply for homogeneous turbulence.) Equations (38) and (39) were the starting point of the Townsend's and Brutsaert's analyses of radiation.

[44] Dias and Brutsaert [1998] cast their analysis in dimensionless terms. Their relevant parameters are

display math(40)

(such that inline image as inline image), where inline image is Stefan-Boltzmann's constant, cp is the specific heat of air at constant pressure, inline image is the mean water vapor density, and inline image is Planck's coefficient, which is relatively easy to calculate and a mild function of temperature;

display math(41)

which is a dimensionless Planck wave number, and

display math(42)

which is a new dimensionless parameter that plays a key role in the temperature spectrum: radiative effects on the latter only become important for inline image [see Dias and Brutsaert, 1998].

[45] Dias and Brutsaert [1998] solved (36) and (37) in dimensionless form using the Pao closure adopted by Claussen [1985]; this allowed the analytical calculation of the temperature spectrum inline image; from then, a one-sided, one-dimensional temperature spectrum was calculated by numerical integration using the isotropic relations between the one-dimensional, one-sided temperature spectrum inline image and the three-dimensional temperature spectrum inline image [Monin and Yaglom, 1975, equation (12).(13)].

[46] The comparison with the Kaimal curve for neutral conditions in the absence of radiation was very good [cf., Dias and Brutsaert, 1998, Figure 2]. The most important result, however, was how the spectral shape (or, more simply, its integral in dimensionless form, inline image) changes with the radiative dimensionless numbers inline image and inline image. Of the two, the one that has the largest influence is inline image; thus, we see in Figure 2 the changes on inline image as inline image increases [from Dias, 1994].

Figure 2.

The dimensionless function inline image computed for several values of the parameter inline image, for inline image, with inline image. From Dias [1994].

[47] The upshot is that values of inline image are uncommon in the stable boundary layer. Hence, radiation is unlikely to play an important role unless very small values of inline image are encountered. This, of course, is the same conclusion arrived at by Coantic and Simonin [1984] recast in dimensionless terms.

5. The Stable Surface Layer

[48] The stable surface layer presents experimental and theoretical questions to micrometeorologists that still haven't found full explanation. From the experimental point of view, scalar fluxes are small, and therefore, their relative errors are probably large, although no systematic approach has been taken to estimate them (as a matter of fact, very few authors put error bars on their flux measurements––see Salesky et al. [2012]). Then there is intermittency and the suppression of turbulence often associated with a critical inline image around 0.2 or 0.25 [Kondo et al., 1978; Cheng et al., 2005] (in the absence of radiative effects; see section 4, and the caveats on the existence of that critical value). This has important practical consequences. For example, in the last 20 years, intermittency and very low or absent turbulence have been found to be a major obstacle in the measurement of nighttime inline image fluxes, and some empirical corrections have been proposed based on a inline image threshold [Goulden et al., 2006].

[49] Apart from general comments in his papers during the 1960s and 1970s, there does not seem to be a systematic study by Brutsaert of stable conditions prior to his 1972 radiation paper, already mentioned above. By the time “Evaporation into the Atmosphere” came out, in 1982, his main views about the stable surface layer, as can be inferred from Brutsaert [1982, sec. 4.2b] were:

[50] 1. Theoretically, under very stable conditions, the Monin-Obukhov flux-profile similarity functions inline image, inline image, and inline image (cf., Brutsaert [1982], equations (4.26)–(4.28)) should be proportional to ζ;

[51] 2. In spite of that, there was a practical limit for that proportionality, and a reasonable compromise for the dimensionless gradients of wind, temperature, and humidity was inline image if inline image, and inline image for inline image (these results were largely confirmed later on with somewhat different stability functions — see Handorf et al. [1999] and Cheng and Brutsaert [2005]; notice that if inline image a log profile again results, which however is different from the one in neutral conditions);

[52] 3. Finally, and again in spite of the provisional summary in section 2, there were doubts about the equality inline image for stable conditions, on theoretical [Warhaft, 1976] as well as experimental [Verma et al., 1978] grounds.

[53] The last issue above translates into the more general problem of similarity (or not) of the behavior of different scalars in turbulent flows. This, of course, is Reynolds' analogy in a new guise, and it is now clear, unlike the early evidence cited above, that it is not a problem of stable conditions only—quite the contrary. Scalar similarity is discussed in the next section.

[54] The theoretical basis for the linearity of the inline image-functions with ζ under stable conditions follows from the following. In near-neutral conditions, the semi-empirical or K-theory of turbulence postulates that a natural length scale in a turbulent boundary layer is the distance z from the surface. As stable stratification sets in, however, and becomes ever stronger, its well-known effect is to limit vertical movements until a point when the distance z to the surface no longer plays any role. Then, the turbulence becomes decoupled with the surface fluxes and follows local similarity: this is called “ z-less stratification” [Wyngaard, 1973; Nieuwstadt, 1984]. Briefly, this can be inferred (among others) from the scalar budget (39) (for any scalar, not only for temperature): if the turbulence structure is independent of z, this means that vertical homogeneity prevails, and that therefore the transport term is null; moreover, as both the flux inline image and the dissipation inline image are independent of z as well, so is inline image, and therefore it must be constant.

[55] The question whether z-less stratification applies was studied by Dias et al. [1995] using turbulence data from FIFE-89 of which a general description can be found in Hall and Sellers [1995]. When data were analyzed in terms of 1 h runs, in an effort to reduce random error effects, all third-order dimensionless statistics (of the type inline image, inline image, and inline image, normalized by the proper combination of inline image and inline image) were found to be independent of ζ. This approach, which permits the analysis of one-level data, had been pioneered by Wyngaard et al. [1978] and allowed Dias et al. [1995] to conclude that z-less stratification holds in the stable surface layer. The subject is still controversial, however: in a later study, the z-less stratification assumption was found not valid [Pahlow et al., 2001], except for the dimensionless variance inline image and the dimensionless rate of dissipation of TKE inline image.

[56] A little later, Cheng and Brutsaert [2005] did return to the dimensionless gradients inline image and inline image using the CASES-99 data set. The results were not very much different from the “classic” functions that already appeared in Brutsaert [1982] and read inline image and inline image valid (before the inline image's leveled off) for inline image. Interestingly, this was obtained with 1 h averages, the same strategy adopted by Dias et al. [1995] to reduce the random error of the third-order statistics. Note that the log-linear inline image functions confirmed by Cheng and Brutsaert [2005] do give indirect support to the z-less hypothesis, so that a definite understanding of stable conditions (in particular, what Cheng and Brutsaert call “strongly stable”) still eludes us, and the subject certainly needs more research.

[57] An equally serious question is whether Monin-Obukhov similarity theory (MOST) itself always holds under stable conditions. The nocturnal boundary layer presents a host of transient phenomena that disturb the classical assumptions of horizontal homogeneity and stationarity of MOST. This has been the object of investigation by Cheng et al. [2005] with the extremely detailed experimental data form CASES-99. Four types of transient phenomena were studied: internal gravity waves, Kelvin-Helmholtz (K-H) shear instability, a density current, and a low-level jet. MOST was found to hold for the nonbreaking internal gravity waves and during the well-developed stages of the density current and the low-level jet. But it did not hold for the K-H instability and during developing stages of the density current.

6. Scalar Similarity

[58] As we have seen in the previous section, there was a warning in Brutsaert [1982] about the validity of the assumption inline image. It is central in the Bowen ratio equation [Bowen, 1926]; and it is very common to assume that all scalars behave as temperature in order to apply the dimensionless inline image functions of MOST, much more easily measured when the scalar is temperature, to admixtures whose measurement is either much more involved or downright impossible with the current technologies.

[59] It is probably fair to assign the beginning of this thread to the theoretical work of Warhaft [1976]: based on the budgets for the heat flux and the moisture flux, he suggested that, for situations when the correlation coefficient inline image between temperature and humidity fluctuations is between −1 and 0, the ratio of their eddy diffusivities is inline image. This was obtained with simple, but reasonable, closures for the transport terms involving third-order moments and pressure terms which are extremely hard to measure in the surface layer [Donaldson, 1972; Launder, 1975].

[60] As Brutsaert noted in “Evaporation into the Atmosphere,” the subject was controversial and still on purely theoretical grounds. Brost [1979] made the observation that using the budgets for inline image and inline image variance and inline image- inline image covariance, however, would result in inline image regardless of inline image.

[61] The apparent contradiction was solved by Dias and Brutsaert [1996], who showed that under the validity of MOST those same budgets implied inline image. They also showed that existing experimental evidence and standard assumptions from MOST could be used to derive, analytically, the almost perfect similarity between inline image and inline image.

[62] A slightly updated approach to the subject can be made in the following terms: start with the Cauchy-Schwarz inequality, whose application to the issue of scalar similarity was pioneered by Hill [1989]. The inequality can be used to prove the following theorem [Steele, 2004]:

display math(43)

where a and b are any two scalars. From (43), it is straightforward to prove that all Monin-Obukhov similarity functions for a and b are then equal.

[63] Dias and Brutsaert [1996] used scalar budgets such as (39) but for inline image, inline image, and inline image, where inline image and b = q. Here I will rewrite their results in terms of any two scalars a and b and the Lewis number (see below), given the growing interest in other substances such as inline image, inline image, etc. The scalar budgets are very simple, with the transport terms involving third-order moments neglected as Dias et al. [1995] had found to hold for FIFE-89 data in stable conditions. It is noteworthy, however, that the same has been found to be true in unstable conditions [Wyngaard et al., 1978; Fairall and Larsen, 1986; Sempreviva and Højstrup, 1998]. In this respect, their results should be equally applicable to the unstable surface layer. With this simplification, the three budgets of inline image, inline image, and inline image can be used to derive

display math(44)

the approximation holding if the Lewis number inline image.

[64] This results in a second-degree equation in the variable x / z, where inline image and inline image, that only admits real roots if inline image. Above, one has

display math(45)

namely the correlation coefficient between the gradients inline image and inline image. From Dias and Brutsaert [1996] two results follow, namely inline image and inline image. Given the growing number of substances whose dispersion in the surface layer must be predicted for environmental reasons, it is of practical interest to study the range of the inline image number for which it is reasonable to assume scalar similarity. This is straightforward from (44), as the expression involving inline image has a minimum at inline image, as can be seen in Figure 3. Setting an acceptable maximum of 1.1 for this term gives inline image, for which one has inline image.

Figure 3.

A practical range of acceptability for the Lewis number for the scalar similarity hypothesis to be valid: the highlighted range is inline image, for which one has inline image.

[65] It remains to prove that inline image itself is equal to one. From (43) and inline image, it follows that there is a constant λ, such that inline image; then,

display math(46)

by virtue of Reynolds' postulates. One assumption missing in Dias and Brutsaert [1996] to complete the proof is that inline image and inline image must be ergodic processes; for then we can write inline image for all individual realizations of inline image and inline image, from which inline image and inline image.

[66] McNaughton and Laubach [1998] studied the effect of the nonstationarity of the wind on the eddy diffusivities of heat and water vapor in a dry-to-wet transition within a stable boundary layer; they concluded that two causes existed for the nonequality of the eddy diffusivities: lack of complete adjustment of the turbulent processes to the new surface (hence local advection) and temporal variations in the external wind with accompanying variations in the scalar fluxes. Local advection had been argued by Bertela [1989] to spoil the Bowen ratio method, but the most striking example of its effects on scalar similarity probably comes from the experiment of Wesely and Hicks [1978]. Figure 4 shows their calculated values of inline image as a function of distance over a step change from land to a warm cooling pond. As turbulence adjusts to the surface and advection terms become less important, the scalar correlation coefficient reaches the theoretical value of inline image.

Figure 4.

inline image over a warm pond measured by Wesely and Hicks [1978] [from Dias and Brutsaert, 1996, with permission].

[67] Even with these strong theoretical results in favor of perfect scalar similarity under idealized conditions, by the mid-1990s it was becoming widely recognized that, in actual field conditions, scalar data series always exhibit at least some degree of dissimilarity. The subject was tackled, for unstable conditions, by Asanuma and Brutsaert [1999a], who reported a wide range of inline image measured from an aircraft over the Landes forest during the HAPEX experiment. This impacted the humidity dimensionless variance function and was attributed to the spatial variability of the humidity sources and to the passive role of humidity as opposed to temperature (strictly speaking, humidity is not a passive scalar; therefore, the statements on humidity being passive should be taken in context, as meaning that most of the buoyancy flux is due to the sensible, rather than to the latent, heat flux). Clearly, MOST by itself and the results from Hill [1989] and Dias and Brutsaert [1996] are not enough to explain this fact.

[68] Similar findings had been obtained by Katul et al. [1995], and in fact Katul and Hsieh [1999] argued, on the basis of values of inline image often reported, and following the line of reasoning of Warhaft [1976], that under unstable conditions the dimensionless variances for temperature and humidity are unequal even if the corresponding dimensionless mean scalar profiles are equal.

[69] With the adoption of a relatively simple conceptual model to understand the intricacies of the real world, it was then shown by Asanuma and Brutsaert [1999b] that “chessboard-like” surface scalar flux variability, combined with local free convective scaling [Wyngaard et al., 1971; Kader and Yaglom, 1990], did provide further explanation to the often observed dissimilarity under unstable conditions between temperature and humidity.

[70] It should be stressed that there are various possible causes for scalar dissimilarity, and they appear to grow in number as time goes by and different experiments give, or require, their identification. Dissimilarity of the entrainment fluxes at the top of the atmospheric boundary layer is one of them, as proposed in de Bruin et al. [1993] and explained in McNaughton and Laubach [1998] [see also Katul et al., 2008; Li et al., 2011]. Even over uniform surfaces, the dissimilarity of the entrainment fluxes and the large-scale ABL processes still do influence the scalar behavior in the surface layer, as suggested by Asanuma et al. [2007] (in that work, the influence of Kader and Yaglom's [1990] paper is clear, as for that matter it is in Bernardes and Dias [2010]; this is no coincidence, since Brutsaert was much impressed by these results from the “Russian school” of thought and transmitted that impression to his graduate students). More recently, it has been found that another cause of scalar dissimilarity and the breakdown of MOST is the strength of the surface flux forcing [Cancelli et al., 2012].

[71] In Katul et al. [1996] and Asanuma et al. [2007], an alternative similarity descriptor to inline image, the relative transfer efficiency

display math(47)

is used to analyze scalar flux similarity. This is important, because inline image and inline image “tell different stories”: while inline image is influenced by the scalar cospectrum throughout its frequency range, inline image depends on the (vertical velocity)–(scalar) cospectra, which tend to be shifted to higher frequencies and therefore give different results. Thus, inline image is more apt to identify how the corresponding scalar fluxes are similar, and consequently how applicable such concepts as the Bowen ratio are. Results from Cancelli et al. [2012] show that surface fluxes are often more similar than the scalars themselves. Interestingly, similar findings were reached by Asanuma and Brutsaert [1999b] (but with surface flux variability effects) regarding the Bowen ratio concept.

[72] As mentioned above, the concept of scalar similarity extends well beyond temperature and humidity. From the early work of Ohtaki [1985], to name but a few, it has been applied to study scalar transport over canopies by Williams et al. [2007] and Cava et al. [2008], and by Ruppert et al. [2006] for a relaxed eddy accumulation method.

7. The Parameterization of the Atmospheric Boundary Layer Under Convective Conditions

[73] A similarity theory for turbulence in the ABL under convective conditions (e.g., a similarity theory for the Convective Boundary Layer) is put forth by Brutsaert and Mawdsley [1976]. Two alternative scales are proposed for the thickness of the ABL: inline image (f is the Coriolis parameter and kr is a coefficient), and inline image, the height of the elevated stable layer or capping inversion [Garratt, 1994, p. 3]. The former is the so-called Rossby-Montgomery formula [Rossby and Montgomery, 1935; cf., Mironov and Fedorovich, 2010]; it is a consequence of a similarity theory for the ageostrophic wind in a neutrally stratified planetary boundary layer [Tennekes, 1982], but it was used in early similarity models for a nonneutral atmosphere as well [cf., Brutsaert and Mawdsley, 1976]. When inline image is chosen as the height of the boundary layer, Brutsaert and Mawdsley propose that ABL similarity functions should depend on

display math(48)
display math(49)
display math(50)
display math(51)
display math(52)

[74] The last two parameters account for baroclinicity and were introduced for the first time; as pointed out by Brutsaert and Mawdsley [1976], inline image and inline image had been introduced by Zilitinkevich and Deardorff [1974] a couple of years before. Matching of the outer (or mixed) and surface layer profiles for inline image, inline image, inline image, and inline image in a way inspired by Milikan [1938] (but actually simpler) gives the main result of this paper, as stated by the authors —the obtention of a set of similarity relationships for the whole ABL:

display math(53)
display math(54)
display math(55)
display math(56)

[75] Above, inline image is a dimensionless height; surface values inline image and inline image (instead of inline image and inline image) can also be accommodated, in which case the inline images are negligible. Later on, a further relation will be proposed, viz.,

display math(57)

[76] Here, inline image is a representative wind speed (as opposed to the velocity vector in (53) and (54)), which can be a suitable average within the ABL [e.g., Sugita and Brutsaert, 1992] or the geostrophic wind speed [e.g., Crago and Brutsaert, 1994].

[77] In general, it is expected that inline image be functions of inline image and inline image. In this first paper, only the dependence on inline image was considered, and preliminary values for those functions were obtained on the basis of comparison with basin-wide monthly evapotranspiration. Similar equations also exist for the case where inline image is the most important height scale, and a subscript r usually goes with them. To keep this section within a reasonable length, they will not be detailed here; the reader is referred to Brutsaert and Mawdsley [1976] and Brutsaert [1982]. Moreover, although ABL similarity formulations for convective conditions involving the coriolis parameter f and the corresponding length scale inline image have become much less common in recent years, some interesting ideas, in fact, containing equations similar to (53)–(56) as a particular case, were advanced by Zilitinkevich [1975], whose formulations were partly inspired by directional dimensional analysis [cf. Zilitinkevich, 1973; Kader and Yaglom, 1990].

[78] Remarks were made by Brutsaert and Mawdsley [1976] on the fact that the theory applies for steady conditions over a uniform surface, so that advective conditions and frontal activity (among others) might hinder its applicability and that the C and D functions, for temperature and water vapor, were probably different.

[79] Further estimates of the water vapor D similarity function were made in the sequel work by Mawdsley and Brutsaert [1977]; in terms of turbulence in the ABL, the main finding was that D is much smaller than C, which is a consequence of the lack of similarity between temperature and humidity in most of the ABL.

[80] A more accurate determination of the D function was by then clearly in order, and with better data from the AMTEX experiment, this was the central objective of Brutsaert and Chan [1978]: once more it was confirmed that inline image, with a general conclusion that inline image; the use of inline image as the preferred scale (over inline image) for the ABL thickness was by then becoming clear.

[81] Equally clear, however, was the fact that the scatter of the A, B, C, D functions was inevitably very large: for example, Figure 5 shows the C function obtained by Brutsaert and Sugita [1991]. Therefore, in order to gain insight into ABL physics —but equally important, in order to obtain results of practical value —it was also necessary to perform alternative analyses where some of the complicating factors of that physics could be somewhat removed or at least attenuated. The next line of attack would be to look in detail at near-neutral conditions; in the words of Kustas and Brutsaert [1986]: “The rationale of investigating nearly adiabatic conditions is that buoyancy effects due to density stratification can be neglected; thus the similarity formulations are in their simplest form, and the surface parameters, which are in fact also needed for nonneutral conditions, can be obtained much more easily.”

Figure 5.

The bulk similarity inline image, adapted (with data points redigitized) from Brutsaert and Sugita [1991]; also from which, the solid line is inline image; the dashed line is equation (58).

[82] Thus, the rationale for looking at simpler conditions is clearly stated. The same argument will be again invoked in Brutsaert and Kustas [1985]. The “surface parameters” are z0 (momentum roughness) and d0 (zero-plane displacement height), and they will be estimated alongside with the verification of the validity of wind and humidity logarithmic profiles in the ABL under near-neutral conditions.

[83] Shortly thereafter, the issue of the validity of the “log law” will be brought up again by Parlange and Brutsaert [1989] with data from the HAPEX-MOBILHY experiment [André et al., 1986], measured over the Landes Forest, and by Sugita and Brutsaert [1990b] with FIFE data. The conclusion from these studies will be that there is a fairly consistent range in the ABL where the log law is applicable (from Kustas and Brutsaert [1986]: inline image and similar ranges from Parlange and Brutsaert [1989] and Sugita and Brutsaert [1990b]) and that under near-neutral conditions the range of validity extends to approximately inline image, which is well beyond the accepted limits of the surface layer. These results will also be largely confirmed using sodar data from HAPEX-MOBILHY [Parlange and Brutsaert, 1990].

[84] Over complex terrain, the logarithmic law holds to considerable heights (up to several times the mean height of the hills) in the lower region of the ABL not only for the wind speed, but for scalars as well. This was well established by Brutsaert and Kustas [1985] and Brutsaert et al. [1990]. Importantly, in both cases the scalar roughness inline image was obtained by selection of a few days when both neutral conditions and soil saturation were present. The value of inline image was intermediate between the predictions for permeable and bluff roughness elements. Radiometric scalar roughness obtained from infrared-derived surface temperatures will fall in this range as well [Sugita and Brutsaert, 1990a] and be found to be a function mainly of solar elevation.

[85] It is, of course, tempting to go beyond near-neutral conditions; in point of fact, since most of the time the daytime ABL is non-neutral, it is only natural to expect that there is a connection between the regional surface fluxes and the lower region of the ABL that extends the log law in a natural way. In other words, perhaps not the whole ABL, but only its lowest region, might be able to give reasonable surface fluxes. Unfortunately, while this is true if “ground-truth,” reliably measured surface fluxes are available, the rawinsonde profiles turned out to be unable to disclose these regional fluxes by themselves under nonneutral conditions [Brutsaert and Sugita, 1990]. Brutsaert and Sugita obtained better results for the estimation of regional scalar fluxes when they used surface temperatures measured with infra-red thermometry and the radiometric scalar roughness mentioned above, both using surface-layer similarity functions [Sugita and Brutsaert, 1990a] and bulk ABL similarity functions [Brutsaert and Sugita, 1991], and Brutsaert and Parlange were successful to estimate fluxes without surface temperatures over the Landes Forest [Brutsaert and Parlange, 1992].

[86] A different and simpler approach to understand the ABL is to assume simple constant or linear profiles for the scalars from z = 0 to the height of the inversion inline image in so-called slab models. The scalar concentration usually displays a fast variation with height across inline image. Zero-order models approximate this behavior by a “jump” of inline image across an infinitesimally small region inline image. First-order models fit a linear continuous function to a region of finite depth inline image. Integration of the scalar profile across the ABL provides budget equations that can be used to infer the general behavior of the so-called entrainment fluxes, i.e., the turbulent fluxes across inline image. By taking this approach with data from the AMTEX experiment, Brutsaert [1987] studied perhaps for the first time the application of slab models for water vapor. He concluded that the entrainment constant for water vapor has a similar value to that found for temperature and that first-order models performed much better.

[87] The first-order model was then the natural choice by Kustas and Brutsaert [1987] for the study of entrainment fluxes in an ABL over rough terrain (the Swiss Pre-Alpine region), where it was found that inline image could be of the order of 1/3 inline image. The relatively simple model for the entrainment flux proposed by Tennekes [1973] was found to perform well when extended to the virtual potential temperature. The mechanical generation of turbulence was found to be significant for the entrainment processes not only in the morning but also in the afternoon when the surface sensible heat fluxes are weak, on account of the rugged nature of the terrain.

[88] The importance of baroclinicity in the bulk ABL formulation was investigated by Crago and Brutsaert [1994] and Crago and Brutsaert [1995]. The former study found that the geostrophic drag coefficient, as well as the functions B and Bw, displayed little dependence on stability when inline image or inline image were derived from the geostrophic wind. Similarly, a constant geostrophic drag coefficient was found adequate to estimate inline image, although it required site-specific calibration. In Crago and Brutsaert [1995], it was found by means of an error propagation analysis that, although baroclinicity effects can be large, random errors and acceleration effects usually overwhelmed stability and baroclinicity effects (again, when the geostrophic wind is used in the bulk ABL formulations).

[89] In 1992, a paper was published in the Journal of Fluid Mechanics by Kader and Yaglom [Kader and Yaglom, 1990]. It employed an approach pioneered by the Russian school of fluid mechanics, called directional dimensional analysis, whereby different directions in space are assigned different fundamental dimensions (thus, for the atmospheric surface layer, we have two fundamental lengths, Lx and Lz, instead of only one). In terms of the Π-theorem [cf. Barenblatt, 1996, pp. 41–42], this means that there will be, correspondingly, one less dimensionless parameter. Indeed, in the case of the atmospheric surface layer, Kader and Yaglom propose a rational classification of three sublayers (the dynamic, dynamic-convective, and convective sublayers) and for each of these the smaller number of degrees of freedom means that instead of an unknown function of ζ (as in classical Monin-Obukhov similarity theory) only a constant needs to be determined experimentally.

[90] Brutsaert was duly impressed. First, a paper was published where convenient interpolation formulae were proposed for the stability functions inline image [Brutsaert, 1992]; the new inline images “allow a more reliable estimation of the surface fluxes under extreme conditions of instability than was previously possible,” and the consequence was that the matching of these functions with the formulation for the bulk ABL then produced improved results as well for surface flux estimates [Sugita and Brutsaert, 1992].

[91] It is interesting to note that in the previous paper on the subject [Brutsaert and Sugita, 1991], a best-performing C function (in terms of its ability to predict the surface sensible heat flux) had been

display math(58)

whereas in Sugita and Brutsaert [1992] it is given by

display math(59)

[92] Therefore, by 1992 it had been concluded that matching the predictions of Kader and Yaglom [1990] for the convective sublayer of the surface layer with the bulk ABL formulations was clearly superior to fitting an independently proposed equation. In terms of the quality of the surface sensible heat flux predictions, the improvement can be assessed by comparing Brutsaert and Sugita [1991, Figure 8] with Sugita and Brutsaert [1992, Figure 2]; the correlation coefficient between observed and estimated surface heat fluxes went from 0.910 to 0.986. Probably for the first time, equation (59) shows an explicit dependence on the surface momentum roughness z0.

[93] These new stability formulations will be equally helpful for the estimation of inline image when suitable near-surface measurements are available for comparison; the surface layer version will be investigated by Parlange and Brutsaert [1993] and a bulk ABL approach by Brutsaert and Parlange [1996]. In both cases, good correlation with the near-surface eddy covariance measurements above forest was obtained.

[94] These successes prompted further theoretical work. Brutsaert's 1999 paper in Reviews of Geophysics [Brutsaert, 1999] gives the state-of-the art in the parameterization of the ABL and a summary of his efforts started some two decades earlier. A better interpolation equation for inline image, again based on the previous work by Kader and Yaglom [1990], is developed, and an approach for the derivation of C and Bw, which is firmly grounded on the Π-Theorem, gives the basis for the dependence of C on inline image and Bw on inline image. Perhaps the key remark in this work is this: “This shows that because the largest gradients occur near the surface, a sound ASL formulation is the key to a reliable bulk transport description for the entire ABL. It also explains the marked dependency of Bw on surface roughness that has been mostly ignored in past applications.” [Brutsaert, 1999].

[95] In the course of 20 years, Brutsaert went from an insight about linking the ABL profiles to the regional surface fluxes to a full methodology that besides providing practical ways to estimate these fluxes also advanced considerably our knowledge of turbulence in the ABL. This was hard work because of the nature of the data available for most of these studies (from radiosoundings) and their inherent noise, but succeeded admirably.

8. Concluding Remarks

[96] It is next to impossible to summarize the breath and depth of the contributions reviewed briefly here. They encompass a very large range of space and time scales and definitely an impressive range of different approaches to turbulence: analytical, numerical, and statistical methods are applied as needed. But if there were one single aspect to emphasize, it would probably be the rationality that Brutsaert helped to bring to the analysis of environmental flows. Complicated geometry and boundary conditions, nonstationarity, randomness and chaotic behavior cannot, and should not, prevent us from understanding atmospheric turbulence: they certainly did not prevent Brutsaert from doing so.


[97] I wish to thank the conveners of the Brutsaert-Parlange Symposium, in the persons of Marc Parlange and John Selker (and all the people at Cornell that made it happen) for getting us together in a rare occasion to discuss science so freely, and to thank Jean-Yves Parlange and Wilfried Brutsaert for all their teachings. The financial support by Brazil's CNPq through grant 450502/2012-0 to participate in the symposium is gratefully acknowledged. Finally, I would like to acknowledge the contributions of three anonymous reviewers, which helped to fix many deficiencies of, and to improve, this paper.