Pope Model Shear Spectrum

Short definition of Pope Model Shear Spectrum
The three-dimensional velocity spectrum proposed by Pope (2009)

This is the common definition for Pope Model Shear Spectrum, but other definitions maybe discussed within the wiki.

The three-dimensional velocity spectrum proposed by Pope (2009) can be used to derive the one-dimensional shear spectrum under the assumption that the turbulence is isotropic. The parameters of the three-dimensional Pope spectrum of velocity, and its form, is based on observations of along-profile velocity fluctuations and their rate of strain. An analytic approximation of this shear spectrum is

[math]\displaystyle{ \Psi_P(\hat{k}_1) = \frac{4}{3} \left(2\pi\right)^{4/3}\, C \frac{18}{55} \hat{k}_1^{1/3} \exp \left(-\beta \left[\left(\hat{k}_1^4 + c_2^4 \right)^{1/4} - c_2 \right]\right) }[/math]

where [math]\displaystyle{ C=1.558 }[/math] is the three-dimensional Kolmogorov constant, [math]\displaystyle{ \beta=38.3 }[/math] comes from observations of one-dimensional along-profile velocity spectra, and [math]\displaystyle{ c_2=0.0273 }[/math] makes the integral of [math]\displaystyle{ \Psi_P }[/math] equal to [math]\displaystyle{ 2/15 }[/math]. The one-dimensional Kolmogorov constant is [math]\displaystyle{ C_1=0.51 }[/math] for this model.

The integral of the Pope shear spectrum can be approximated by

[math]\displaystyle{ I_P(\hat{k}_1) = \frac{15}{2} \int_0^{\hat{k}_1} \Psi_P (\xi)\, \mathrm{d}\xi = \tanh \left(50.5\,\hat{k}_1^{4/3} \right) - 9.02\,\hat{k}_1^{4/3} \exp \left(-113\, \hat{k}_1^{16/9} \right) }[/math]

The wavenumber at which a particular fraction of the variance of shear is resolved by these model spectra are listed in Table 2. The differences between the models are fairly small, and any one of them could be used to estimate the fraction of the variance that is resolved at a particular wavenumber, at the [math]\displaystyle{ 10\% }[/math] level of accuracy. Put another way, for wavenumber larger than
[math]\displaystyle{ \hat{k}_1\gt 0.02 }[/math], which is the range that resolves more than [math]\displaystyle{ 25\% }[/math] of the shear variance, the integral of all models are similar and, in particular, the models [math]\displaystyle{ I_{N_2} }[/math] and [math]\displaystyle{ N_L }[/math] agree to within better than [math]\displaystyle{ 6\% }[/math] over this range. Thus, the choice of model for estimating the fraction of the shear variance that is resolved because of using an upper limit for spectral integration, is not critical. However, we do recommend using either [math]\displaystyle{ I_{N_2} }[/math] or [math]\displaystyle{ I_L }[/math].

Table 2. The non-dimensional wavenumbers, in units of [math]\displaystyle{ \mathrm{cpm} }[/math], at which the model spectra resolve a specified fraction of the variance of shear. Column 1 – the fraction resolved. Columns 2 to 5 – the first and second model of the Nasmyth spectrum, the L spectrum, and the Pope spectrum, respectively.

[math]\displaystyle{ \% }[/math]	[math]\displaystyle{ I_{N_1} }[/math]	[math]\displaystyle{ I_{N_2} }[/math]	[math]\displaystyle{ I_L }[/math]	[math]\displaystyle{ I_P }[/math]
[math]\displaystyle{ 35 }[/math]	[math]\displaystyle{ 0.027 }[/math]	[math]\displaystyle{ 0.026 }[/math]	[math]\displaystyle{ 0.025 }[/math]	[math]\displaystyle{ 0.028 }[/math]
[math]\displaystyle{ 50 }[/math]	[math]\displaystyle{ 0.037 }[/math]	[math]\displaystyle{ 0.036 }[/math]	[math]\displaystyle{ 0.034 }[/math]	[math]\displaystyle{ 0.038 }[/math]
[math]\displaystyle{ 75 }[/math]	[math]\displaystyle{ 0.058 }[/math]	[math]\displaystyle{ 0.057 }[/math]	[math]\displaystyle{ 0.055 }[/math]	[math]\displaystyle{ 0.058 }[/math]
[math]\displaystyle{ 80 }[/math]	[math]\displaystyle{ 0.064 }[/math]	[math]\displaystyle{ 0.065 }[/math]	[math]\displaystyle{ 0.062 }[/math]	[math]\displaystyle{ 0.065 }[/math]
[math]\displaystyle{ 85 }[/math]	[math]\displaystyle{ 0.072 }[/math]	[math]\displaystyle{ 0.075 }[/math]	[math]\displaystyle{ 0.072 }[/math]	[math]\displaystyle{ 0.073 }[/math]
[math]\displaystyle{ 90 }[/math]	[math]\displaystyle{ 0.085 }[/math]	[math]\displaystyle{ 0.091 }[/math]	[math]\displaystyle{ 0.088 }[/math]	[math]\displaystyle{ 0.084 }[/math]
[math]\displaystyle{ 95 }[/math]	[math]\displaystyle{ 0.125 }[/math]	[math]\displaystyle{ 0.120 }[/math]	[math]\displaystyle{ 0.116 }[/math]	[math]\displaystyle{ 0.106 }[/math]