Passive scalar spectra

However, there are important differences between the velocity and the passive scalar spectra (e.g., [68, 13, 14, 4, 8]). The spectra of passive scalars also varies for different systems depending on the value of Sc. 

Figure 2.23 Schematic power spectrum of the passive scalar in turbulent flows in the case of large Schmidt number.

Figure 6.2 Sketch of the spectrum (6.19) for two different values of the decay rate (b is larger (smaller) for the top (bottom) curve). Dashed line indicates the k 5D Obukhov spectrum of a non-reactive passive scalar for comparison.

Here we neglected intermittency corrections which are indeed present due to the distribution of Lyapunov exponents. This will be discussed later. In the special case b = 0 we recover the Batchelor k l spectrum of a passive scalar with a cut-off at the diffusive scale Id = y/DjX. 

Although Eq. (6.52) is very similar to the equation for transport of the decaying passive scalar, an important difference is that in this case the flow v is not independent of the transported scalar field u>. Thus, the vorticity field is a dynamically active scalar related to the flow field through the relationship w = z (V x v), where z is the unit vector perpendicular to the plane of the flow. Nevertheless, Nam et al. (2000) have shown that the friction has a similar effect on the energy spectrum as the decay in the case of the scalar spectrum. The kinetic energy spectrum is related to the enstrophy spectrum Z(k) = (ul) by E(k) = Z(k)/k2, and therefore in the enstrophy cascade range... 

K. Nam, T.M. Antonsen, P.N. Guzdar, and E. Ott. k spectrum of finite lifetime passive scalars in Lagrangian chaotic fluid flows. Phys. Rev. Lett., 83 3426-3429, 1999. 

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