Abstract

The turbulence temperature spectrum and structure parameter are related through a widely adopted proportionality coefficient. We formally derive this expression, and present further evidence, to demonstrate that this coefficient is too large by a factor of 2.

1. Introduction

It is customary to quantify the intensity of turbulent fluctuations of meteorological fields through a single parameter called the structure parameter. The structure parameter, which is assumed constant within the inertial subrange of turbulence scales, is relevant to many applications associated with atmospheric boundary layer processes. One of the most studied structure parameters is the (potential) temperature parameter, which is frequently used to describe properties of electromagnetic and acoustic wave propagation in the atmosphere. A relationship between the scalar structure parameter and the scalar spectral density in the inertial subrange was first obtained by Tatarskii (1961) and evaluated for temperature by Wyngaard et al. (1971). The latter form has been widely used in the engineering and atmospheric sciences for nearly fifty years. We formally rederive this relationship and present evidence to demonstrate that the associated proportionality coefficient is too large by a factor of 2.

2. Deriving the integral spectral approximation of the structure parameter

The second-order temperature structure function (Tatarskii 1961; Wyngaard 2010) is defined as

 
DT(r)=[T(x)T(x+r)]2¯,
(1)

where T is temperature or potential temperature when Eq. (1) is used in the atmospheric context, x is the coordinate direction in space, and r = |r| is the separation distance (equal to the magnitude of the separation vector). If we assume that turbulence is locally isotropic and that the separation distance lies within the inertial subrange of spatial scales of turbulent temperature fluctuations, the temperature structure function may be expressed as (Kolmogorov 1941)

 
DT(r)=CT2r2/3,
(2)

where CT2 is the temperature structure-function parameter, often just called the temperature structure parameter. The following relationship between DT and the one-dimensional spectral density of temperature fluctuations ΦT is valid under the assumption of turbulence isotropy (Wyngaard 2010):

 
DT=2[1cos(kr)]ΦT(k)dk=40[1cos(kr)]ΦT(k)dk,
(3)

where k is the wavenumber associated with the x direction. Both Eqs. (2) and (3) are based on the fundamental definition of the structure function and require no additional assumptions beyond their theoretical underpinnings. As an aside, Essenwanger and Reiter [1969, their Eq. (6)]—an oft-cited study focused on structure functions and power spectra of atmospheric velocity increments—incorrectly omits the leading factor of 2 in Eq. (3).

One can evaluate the integral in Eq. (3) analytically by assuming that the entire spectrum has the inertial subrange form

 
ΦT=Ak5/3.
(4)

To do this, we substitute Eq. (4) into Eq. (3) to obtain

 
DT=4A0[1cos(kr)]k5/3dk.
(5)

We next take the integral employing integration by parts:

 
0[1cos(kr)]k5/3dk=32[1cos(kr)]k2/3|0=0+32r0sin(kr)k2/3dk.
(6)

We use the following table integral from Gradshteyn and Ryzhik [2014, their Eq. (3.761), p. 440] for the remaining part on the right-hand side of Eq. (6):

 
0xμ1sin(ax)dx=π2aμΓ(1μ)cos(μπ2),

where x = k, μ = 1/3 according to Eq. (6), and a = r. This yields

 
32r0sin(kr)k2/3dk=32r12r1/3Γ(23)cos(π6)=3πr2/34Γ(23)cos(π6).

Noting that cos(π/6) = sin(π/3) and Γ(2/3) = (3/2)Γ(5/3), we have

 
32r0sin(kr)k2/3dk=3πr2/3432Γ(53)sin(π3)=πr2/32Γ(53)sin(π3).

Thus,

 
0[1cos(kr)]k5/3dk=πr2/32Γ(53)sin(π3)
(7)

and

 
DT=4A0[1cos(kr)]k5/3dk=2πAr2/3Γ(53)sin(π3).
(8)

Making use of Eq. (2) gives us

 
2πAr2/3Γ(53)sin(π3)=CT2r2/3.

Solving for A provides

 
A=CT2Γ(53)sin(π3)2π0.125CT2,
(9)

which results in

 
ΦT=Ak5/30.125CT2k5/3.
(10)

However, Wyngaard et al. (1971) reported the relationship between ΦT and CT2 as

 
ΦT=0.25CT2k5/3.
(11)

The proportionality coefficient value of 0.25 was justified by the following comment in the references of op. cit.:

“The constant 0.25 stands for 2(2π)−1Γ(5/3)sin(π/3). Note that we use a range of 0 to +∞ for k1, whereas a −∞ to +∞ range is used on p. 25 of Ref 1.”

where “Ref. 1” is Tatarskii (1961) and k1 is the analog of our k.

It appears that this value of 0.25 was an error. Since there is no explicit derivation of Eq. (10) presented by Tatarskii (1961), it seems likely that Wyngaard et al. (1971) misinterpreted the limits of integration used to arrive at Tatarskii’s formula [our Eq. (10)] and included the extraneous factor of 2. However, as follows from our derivation demonstrated above, the adjustment is already made in Eq. (3) to change the range from 0 to ∞. This means that the expression (11) from Wyngaard et al. (1971) incorporates a proportionality coefficient that is larger than its actual value by a factor of 2.

3. Discussion

The relationship between ΦT and CT2. in the Wyngaard et al. (1971) form, Eq. (11), has been widely used in structure-parameter calculations (e.g., Kaimal 1973; Asimakopoulos et al. 1976; Wyngaard and LeMone 1980; Moulsley et al. 1981; Kohsiek 1982; Cuijpers and Kohsiek 1989; Beland 1993; Green et al. 1994; Muschinski et al. 2001, 2004; Cheinet and Siebesma 2009; Wilson and Fedorovich 2012; Maronga et al. 2013; Maronga 2014; Maronga et al. 2014; Gibbs et al. 2016). The error reported here may affect conclusions regarding the validity of this relationship as compared with observational and numerical data. Specifically, the majority of these cited studies found reasonable agreement between this relationship and various direct methods. Thus, the theoretical shortcomings of the method were implicitly corrected through the use of an improper proportionality coefficient.

Beyond the derivation presented above, we have further reason to believe that Eq. (10) is the correct form of the integral spectral approximation of the temperature structure parameter. In Wyngaard [2010, their Eqs. (15.43) and (16.64), p. 374], a specific form of our Eq. (4) is considered that is subsequently integrated to arrive at an equivalent expression for Eq. (8). Although the approximation (10) is not presented in op. cit., one can easily show that the corresponding coefficient of proportionality is also 0.125 instead of the widely adopted value of 0.25 from Wyngaard et al. (1971). We recommend that Eq. (10) be used going forward.

Acknowledgments

We thank Alan Shapiro and Brian Greene (University of Oklahoma) for helpful conversations related to the presented derivation and Louis Wicker (NOAA National Severe Storms Laboratory) for reviewing the manuscript and offering helpful feedback. Björn Maronga (Leibniz Universität Hannover) provided especially thoughtful comments related to the various methods described in this paper. Funding was provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA11OAR4320072, U.S. Department of Commerce.

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