Nasmyth spectrum

Short definition of Nasmyth spectrum
Nasmyth shear spectrum

This is the common definition for Nasmyth spectrum, but other definitions maybe discussed within the wiki.

Nasmyth shear spectrum

There is no theoretical derivation for the spectrum of velocity that encompasses the inertial subrange and the range that is affected by viscosity (the dissipation range). Such a spectrum would be a gauge for judging the quality of a measured spectrum. There are only empirical spectra based on the average of many observations. The earliest of these is the Nasmyth spectrum that was later tabulated by Oakey (1982). The data were collected in a tidal channel using a hot-film anemometer that measures the fluctuations of velocity in the direction of profiling.

That is, Nasmyth measured [math]\displaystyle{ E_{11}(k_1) }[/math]. His spectra were non-dimensionalized to form [math]\displaystyle{ F_{11} }[/math] and these spectra were averaged into 15 wavenumber bins that are spread fairly uniformly in logarithmic space (Table 1). Oakey used the isotropic relationship to derive the spectrum of the cross-stream component of velocity, [math]\displaystyle{ F_{22}(\hat{k}_1) }[/math], and hence, the universal shear spectrum [math]\displaystyle{ G_{22}(\hat{k}_1)=(2\pi \hat{k}_1)^2F_{22}(\hat{k}_1) }[/math].

Nasmyth shear spectrum -- [math]\displaystyle{ 1^{\mathrm{st}} }[/math] approximation

Figure 2. The first approximation of the Nasmyth spectrum of shear using the 15 shear spectral values derived by Oakey (red asterisks) and a mathematical model (blue line) of these values. The eighth value encircled in green is assumed to be erroneous. The integral of the spectrum (purple line) and its mathematical model (green line) are normalized to an asymptotic value of unity. The wavenumber at which the integral reaches [math]\displaystyle{ 50 }[/math] to [math]\displaystyle{ 95\% }[/math] of its asymptotic value are indicated by the coloured disks.

Wolk et al (2002) provided a simple mathematical model for the Nasmyth spectrum predicated on the assumption that the eighth spectral point is erroneous (Figure 2). This model is

[math]\displaystyle{ \Psi_{N_1} (\hat{k}_1)= \frac{8.05\,\hat{k}_1^{1/3}}{1+ \left(20.6\,\hat{k}_1\right)^{3.715}} }[/math]

and the model for the integral of this spectrum is

[math]\displaystyle{ I_{N_1}(\hat{k}_1) = \frac{15}{2} \int_0^{\hat{k}_1} \Psi_{N_1} (\xi) \mathrm{d}\xi = \tanh\left(48\,\hat{k}_1^{4/3}\right) - 2.9\,\hat{k}_1^{4/3} \exp\left(-22.3\, \hat{k}_1^{4/3} \right) }[/math]

The one-dimensional Kolmogorov constant of this model is

[math]\displaystyle{ C_1=\frac{3}{4} \frac{8.05}{(2\pi)^{4/3}}=0.52 }[/math]

[math]\displaystyle{ k_1 L_K }[/math]	[math]\displaystyle{ F_{11}(k_1L_K) }[/math]	[math]\displaystyle{ \Psi_N(k_1L_K) }[/math]
[math]\displaystyle{ 2.830\times 10^{-4} }[/math]	[math]\displaystyle{ 1.254\times 10^{5} }[/math]	[math]\displaystyle{ 0.5285 }[/math]
[math]\displaystyle{ 5.0303\times 10^{-4} }[/math]	[math]\displaystyle{ 4.799\times 10^{4} }[/math]	[math]\displaystyle{ 0.6397 }[/math]
[math]\displaystyle{ 8.950\times 10^{-4} }[/math]	[math]\displaystyle{ 1.842\times 10^{4} }[/math]	[math]\displaystyle{ 0.7763 }[/math]
[math]\displaystyle{ 1.592\times 10^{-3} }[/math]	[math]\displaystyle{ 7.050\times 10^{3} }[/math]	[math]\displaystyle{ 0.9404 }[/math]
[math]\displaystyle{ 2.830\times 10^{-3} }[/math]	[math]\displaystyle{ 2.699\times 10^{3} }[/math]	[math]\displaystyle{ 1.138 }[/math]
[math]\displaystyle{ 5.0323\times 10^{-3} }[/math]	[math]\displaystyle{ 1.036\times 10^{3} }[/math]	[math]\displaystyle{ 1.380 }[/math]
[math]\displaystyle{ 8.950\times 10^{-3} }[/math]	[math]\displaystyle{ 3.964\times 10^{2} }[/math]	[math]\displaystyle{ 1.682 }[/math]
[math]\displaystyle{ 1.592\times 10^{-2} }[/math]	[math]\displaystyle{ 1.490\times 10^{2} }[/math]	[math]\displaystyle{ 2.302 }[/math]
[math]\displaystyle{ 2.830\times 10^{-2} }[/math]	[math]\displaystyle{ 3.574\times 10^{1} }[/math]	[math]\displaystyle{ 2.176 }[/math]
[math]\displaystyle{ 5.0330\times 10^{-2} }[/math]	[math]\displaystyle{ 5.600\times 10^{0} }[/math]	[math]\displaystyle{ 1.373 }[/math]
[math]\displaystyle{ 7.977\times 10^{-2} }[/math]	[math]\displaystyle{ 7.214\times 10^{-1} }[/math]	[math]\displaystyle{ 0.5278 }[/math]
[math]\displaystyle{ 1.264\times 10^{-1} }[/math]	[math]\displaystyle{ 6.580\times 10^{-2} }[/math]	[math]\displaystyle{ 0.1342 }[/math]
[math]\displaystyle{ 1.592\times 10^{-1} }[/math]	[math]\displaystyle{ 1.812\times 10^{-2} }[/math]	[math]\displaystyle{ 0.0616 }[/math]
[math]\displaystyle{ 2.004\times 10^{-1} }[/math]	[math]\displaystyle{ 4.552\times 10^{-3} }[/math]	[math]\displaystyle{ 0.0249 }[/math]
[math]\displaystyle{ 2.522\times 10^{-1} }[/math]	[math]\displaystyle{ 1.197\times 10^{-3} }[/math]	[math]\displaystyle{ 0.0101 }[/math]

Table 1. The Nasmyth non-dimensional velocity and shear spectrum after Oakey (1982). The one-dimensional wavenumber is in units of [math]\displaystyle{ \mathrm{cpm} }[/math] and is non-dimensionalized by the Kolmogorov length, [math]\displaystyle{ L_K }[/math].

Nasmyth shear spectrum -- [math]\displaystyle{ 2^{\mathrm{nd}} }[/math] approximation

The notion that the eighth value of the Nasmyth spectrum is erroneous may be a false one. Observations of all three velocity components in a high Reynolds number wind tunnel by Saddoughi and Veeravalli (1994) indicate that all three components have a spectral rise above the [math]\displaystyle{ k_1^{1/3} }[/math] tendency of the inertial subrange, shortly before the spectrum falls rapidly due to viscosity, and that the rise is larger in the shear than in the strain components (see Figure 13 of Saddoughi and Veeravalli, 1994). However, the rise seen in the Nasmyth shear spectrum is not evident in the Nasmyth strain spectrum [math]\displaystyle{ G_{11} }[/math].

Figure 3. The second approximation of the Nasmyth shear spectrum (blue) and the adjusted values from Oakey (1982) (red asterisks). The grey shading is the range that contains 90% of the shear variance. The coloured disks mark the fraction of the variance that is resolved by integrating the spectrum from zero to the indicated wavenumber. The spectrum is only 25% resolved at its peak.

Figure 4. Same as previous figure but in linear space.

Lueck (2021b) re-examined the spectrum tabulated by Oakey (1982) and noticed that the strain spectrum, [math]\displaystyle{ \left(2\pi \hat{k}_1\right)^2 F_{11} }[/math] integrates to a value that is [math]\displaystyle{ 2\% }[/math] higher than [math]\displaystyle{ 1/15 }[/math]. Therefore, the tabulated values for [math]\displaystyle{ F_{11} }[/math] should be reduced by this factor. A second model that takes into account the spectral rise displayed by the eighth point, and uses the reduced values for [math]\displaystyle{ F_{11} }[/math], is

[math]\displaystyle{ \begin{equation} \begin{split} \Psi_{N_2}(\hat{k}_1) &= \left(\frac{7.89\,\hat{k}_1^{1/3}}{1+ \left(21.2\,\hat{k}_1 \right)^3 }\right) \left( \frac{1}{1+ \left(6\,\hat{k}_1\right)^{5/2} }\right) \left(1+ \frac{0.11 y}{(y-1)^2 + y/2} \right) \\ y &= \left(\frac{\hat{k}_1}{0.019}\right)^2 \end{split} \end{equation} }[/math]

which integrates to a value that is only [math]\displaystyle{ 0.1\% }[/math] larger than [math]\displaystyle{ 15/2 }[/math]. The one-dimensional Kolmogorov constant is [math]\displaystyle{ C_1=0.51 }[/math] for this model. An approximation for the integral of this spectrum is

[math]\displaystyle{ I_{N_2}(\hat{k}_1) = \frac{15}{2} \int_0^{\hat{k}_1} \Psi_{N_2} (\xi)\,\mathrm{d}\xi = \tanh\left(61.5\,\hat{k}_1^{4/3}\right) - 18.1\,\hat{k}_1^{4/3} \exp\left(-52.5\,\hat{k}_1^{4/3}\right) }[/math]

This approximation is shown in log-log space in Figure 3, and in linear space in Figure 4.