We estimated the velocity spectra and co-spectrum of *u* and *w* from the measured time series as

- (1)

where *i*,*j* = 1,2,3; (*u*_{1},*u*_{2},*u*_{3}) = (*u*,*v*,*w*); *f* is the frequency; ⟨⟩ denotes ensemble average operator; the superscript * denotes complex conjugate and is the complex-valued Fourier transform of the *i*th velocity component at height *z*.

We selected the data according to the classification of atmospheric stability in terms of the Obukhov length *L*_{o} following Gryning *et al*., [22] where *L*_{o} is defined as [1]

- (2)

where *u*_{ * } is the surface friction velocity, *κ* = 0.4 is the von Kármán constant, *g* is the acceleration due to gravity, *T* is the mean surface-layer temperature and is the vertical kinematic heat flux density at the surface. The measured spectra and co-spectra given in equation (1) change with atmospheric stability; [8] i.e., *F*_{ij}(*f*,*z*) is a function of *L*_{o}.

#### Spectral tensor model

The velocity-spectrum tensor Φ_{ij}(*k*) contains the information about the second-order statistics of all the three velocity components through indices *i*,*j*, where *k* is a three-dimensional wavenumber vector. Also, by definition, Φ_{ij}(**k**) represents the Reynolds-stress ‘density’ in wavenumber space. [11] The modeled Φ_{ij}(**k**) of M94 is valid in the neutral surface-layer with the assumption of uniform shear d*U* ∕ d*z*. Depending upon their orientation in the plane of uniform shear, the eddies will stretch or compress via *k*(*t*) = (*k*_{1},*k*_{2},*k*_{30} − *k*_{1}(d*U* ∕ d*z*)*t*), where *t* is time (to visualize the motion, see Pope [11]/Chapter 11/Figure 11.5 c on page 407).

The model calculates the evolution of Fourier modes (i.e., three-dimensional Fourier transform of the velocity components for *i* = 1,2 and 3) under the influence of the mean shear from an initial isotropic state, and Φ_{ij}(*k*) was modeled through,

- (3)

The equation for the evolution of the Fourier modes was deduced from the linearized Navier–Stokes equations via RDT, where the time-dependent, random nature of the turbulent field in physical space implies the time dependence and randomness of the field (in Fourier space).

In isotropic turbulence, the velocity-spectrum tensor is

- (4)

where *k*_{0} = *k*(0) and *k* is the length of the vector *k*. The energy spectrum *E*(*k*) given by von Kármán [27] as

- (5)

where *α* ≈ 1.7 is the spectral Kolmogorov constant, *ε* is the rate of viscous dissipation of specific turbulent kinetic energy (TKE) and *L* is a turbulence length scale.

The stretching of eddies due to shear for an infinitely long time is unrealistic, since the eddies must break at some point because of the stretching. The small-scale more isotropic turbulent eddies are not affected by shear. In order to make the spectral tensor stationary, the time dependency in the model was removed by incorporating the general concept of an eddy life time, *τ*(*k*). In the inertial sub-range, the life time of eddies is proportional to *k*^{ − 2 ∕ 3}, and the assumption in the M94 model was, at scales, larger than the inertial sub-range; eddy life time is proportional to *k*^{ − 1} divided by their characteristic velocity such that *τ*(*k*) is proportional to *k*^{ − 2 ∕ 3} for *k* ∞ and *k*^{ − 1} for *k* 0, where *E* was chosen as equation (5). The parameterization of *τ*(*k*) in M94 was

- (6)

where Γ is a parameter to be determined and _{2}*F*_{1} is the Gaussian or ordinary hypergeometric function, which arises from the integration of *E*(*p*). The alternative formulations for eddy life time, which were provided by Mann [10] and the references within, give different *k*-proportionalities for the scales larger than the inertial sub-range, such as *k*^{ − 2} and *k*^{ − 7 ∕ 2} for *k* 0.

The analytical forms of Φ_{ij}(*k*) in M94 can be expressed as

- (7)

where time *t* was substituted by *τ*(*k*). Equation (7) can also be given as

- (8)

and Φ_{ij}(*k*_{0}) = *αε*^{2 ∕ 3}*L*^{11 ∕ 3}Φ_{ ij}(*k**L*,1,1,0). So the model contains three adjustable parameters that were determined from the single-point measurements. These three parameters were as follows:

*αε*^{2 ∕ 3} from equation (5)

*L*, which represents the size of the energy containing eddies

*Γ*, from equation (6), which is a measure of turbulence anisotropy

Using equation (7), the cross-spectrum between any two velocity components can be given as

- (9)

where . Δ*y* and Δ*z* are transverse and vertical separations, respectively. Using equation (9), the single-point power spectrum of the *i*th velocity component (where Δ*y* = Δ*z* = 0) can be given as *F*_{i}(*k*_{1},*αε*^{2 ∕ 3},*L*,Γ) = *χ*_{ ii}(*k*_{1},*αε*^{2 ∕ 3},*L*,Γ,0,0) (with no index summation).

The three parameters at any height *z* were calculated by fitting model *χ*_{ij}(*k*_{1},*αε*^{2 ∕ 3},*L*,Γ,0,0) with measured power spectra (including co-spectrum of *u* (*i* = 1) and *w* (*j* = 3)) from equation (1) and using Taylor's hypothesis: *k*_{1} = 2*πf* ∕ *U*, where *U* is the mean wind speed at *z*. For vertical separations *Δz*, coherences and cross-spectral phases were defined, respectively, as

- (10)

- (11)

where and are the average of *L* and Γ parameters at two heights *z*_{1} and *z*_{2} (so that Δ*z* = *z*_{2} − *z*_{1}), and . The model coherences and cross-spectral phases are independent of *αε*^{2 ∕ 3}, which can be seen from equation (8) and the definitions described earlier.

The M94 model assumes zero Coriolis force and a uniform shear d*U* ∕ d*z*, which is constant with height. We do not expect that the curvature of the atmospheric boundary layer (ABL) velocity profile (i.e., non-zero d^{2}*U* ∕ d*z*^{2}) would alter the results significantly; however, because the three parameters were determined from the single-point measurements, one should expect these parameters to vary with height.

Let us consider the performance of the three parameters with respect to the variances and co-variances. A change in *αε*^{2 ∕ 3} causes a shift of the spectra in the ordinate direction; an increase in *αε*^{2 ∕ 3} results in shifting of *u*,*v* and *w* spectra up and *uw* co-spectrum down and vice-versa. An increase in *L* results in shifting of the spectra both to the left along the abscissa and upward along the ordinate and vice-versa. The model assumes initial isotropic turbulence where Γ = 0, leading to and ⟨*uw*⟩ = 0. For Γ > 0, the turbulence is anisotropic, i.e., and ⟨*uw*⟩ < 0, so Γ describes the anisotropic nature of turbulence. The various length scales of the velocity components can be calculated as functions of *L* and Γ. Higher values of Γ imply larger scale separation between the three velocity components, and the length scale of *u* is greater than that of *v*, which again is greater than that of *w*.