Enstrophy spectrum

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... 

Note that in contrast to the rough velocity field of three-dimensional turbulence, the k 3 spectrum implies an almost everywhere smooth velocity field at small scales such that 5v(l) l (with logarithmic corrections). This also means that at scales below the forcing scale, within the enstrophy cascade range, the flow has a single characteristic timescale t 1) l/5v(l) w r, which is independent of the lengthscale l. 

The theoretically predicted dual cascade with two power-law regimes in the kinetic energy spectrum (Fig. 1.4) has been reproduced in numerical simulations and confirmed by laboratory experiments. In some of the experiments spectra steeper than k 3 was observed in the enstrophy cascade range. This deviation can be related to the presence of additional damping at large scales, the so-called Ekman friction. Since the theoretical description of this regime is very similar to the problem of chemical decay in smooth flows we will return to this later in Chapter 6. 

Since b > 0 the power spectrum is steeper than k l as a consequence of the chemical decay, as has been verified numerically by Nam et al. (1999). Just like the Batchelor spectrum, the spectral form (6.21) is expected to hold for any smooth velocity field that generates chaotic advection. Thus, it remains valid even in the absence of an inertial range, as is the case for unsteady laminar flows or in the enstrophy cascade range of two-dimensional turbulence, which is relevant for geophysical flows. 

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