4.4 Spectral Properties
4.4.1 Spectral Behavior of Magnetic Helicity
As discussed in section 1.2, magnetic helicity is one of the important ideal invari-ant of the system and hence its spectral behavior helps in gaining some insight into the MHD turbulent ow. Normalized magnetic helicity spectra for both the decaying case and the forced case are shown below (g. 4.5). Here it is observed that the initial helicity present in high k region, moves to lowk region, with the progress of time It was realized in this work that the power law exponent does not satisfy thek−2 power law for the inverse cascade of magnetic helicity but varies in the two cases studied here signicantly (see next section and table 4.4). In view of this, the prefactor Hd also is dropped as the whole expression (see footnote below) 2 goes together and cannot be separated out. Hence a normalization fac-tor of kαis used, whereα is the actual power law exponent obtained for each case by trial and error (taking the value at which the spectra gets compensated the best). The gure below i.e. g. 4.5 represents the set of such normalized curves that depict the `inverse cascade of magnetic helicity', rst seen in the numerical simulations of EDQNM equations [7]. Although the same aect was observed in several DNS methods [10, 8, 69, 9], never in any of these works a transfer from such a highk to lowk was reported. However in 2D-hydrodynamics, such trans-fers over a vast region, are seen in the inverse cascade of energy e.g.[37]. The compensated spectra show inertial ranges, the horizontal line here indicating the
2The normalization factor that should have been used here is2/3H
dk−2, withHd the dissipation of magnetic helicity. The suggested power law ofk−2 is the one obtained from numerical simulations of EDQNM equations [7] and the power law toHd is obtained from dimensional analysis [1]. This dimensional analysis is reproduced in Appendix A for academic interest.
compensated spectral power. In the forced case, the power law exponents are -3.3 and -1.7 for lowk (7 - 30) and highk (250 - 400) inertial ranges respectively (see g. 4.6 and table 4.4). The inertial range in the low k region, is close to the Kolmogorov type scaling. While for decaying case the power law exponent
a)
k
b)
k
Figure 4.5: Normalized magnetic helicity spectra in a10243simulation. a) forced case and b) decay case. Forced case t=0 to 6.9. Decay case t= 0.03 to 10. Inverse cascade is clearly seen in both the cases. Note that in the decaying case, the initial condition shown here is at a time slightly away from t=0 (i.e. at t=0.03), where already the initial spectrum (which was limited to a band of wave numbers) is stretched over the entire spectral width available to the system. But the majority of the energy is still contained in the initial band of wavenumbers which is making the initial state shown here look like a spectrum from the forced case. Normalization of the typekα is used where αis the power law exponent, in both the cases.
has a value of -3.6. It is interesting to note that exponents in any case do not satisfy the prediction of EDQNM (power law exponent of -2)(see [7]). The dis-sipation region lies beyond the second inertial range in the forced case and the only inertial range in the case of decaying turbulence. The cascade regions seen in the above gures can be well understood by looking at the ux of magnetic helicity. To obtain the ux of magnetic helicity, the rst term on the r.h.s. of the equation (1.33) and r.h.s. of the equation (1.36) are Fourier transformed after accounting for the hyperviscosity, in their non-dimensional form, to obtain H˙M(k) = 2Re
. In this equation, the rst term on the r.h.s. is the nonlinear transfer term and second is the dissipation term of mag-netic helicity. The plots below (g. 4.7 a and b), show the transmission spectra
4.4 Spectral Properties 69
a)
k
b)
k
c)
k
Figure 4.6: Compensated magnetic helicity spectra in a10243 simulation. a) lowk inertial range for forced case b) highk inertial range for forced case and c) inertial range for decay case.
of magnetic helicity (ΠHM(k) = Rk
0 d3k02Re[b˜∗.v^×b]), over all wavenumbers for both the cases. This quantity represents the conservative ux of magnetic helicity as is seen in the gures. These plots have two regions. A region of posi-tive ux and other a region of negaposi-tive ux. The negaposi-tive ux (absolute of the negative ux is plotted in `magenta') is the ux moving in towards the smaller wavenumber shells (inverse cascade), while the positive ux (plotted in `blue') is the ux moving out towards the large wavenumber shells (normal direct cascade).
In the forced turbulence case, the uxes are constant over a large wavenumber region for both inverse and direct cascades. This constant ux is a indicator of a
sustained cascade process, maintained by the input of magnetic helicity through the forcing of a band of wavenumbers at high k (k = 203 - 209). Here magnetic helicity is moving towards the lowk regions, while dissipation is concentrated in the high k regions. Thus the magnetic helicity experiences very low dissipation and is transmitted towards the lowk regions. In the decaying case, though, there is an established inverse transfer process, the ux is not a constant as there is no active injection of magnetic helicity. In the high k regions, the inverse cascade is not present and here in general, the direct energy cascade process dominates, thus only a direct downward cascade is seen, in both forced and decay cases. It can be seen that the contribution to the ux is mainly due to the transmission term alone. This is because of the negligible values of the dissipation term, in comparison to the transmission term. This fact is true for both the decaying and forced cases and thus a spectrum representative of this fact is only shown (see g. 4.7c).