5. ANTECEDENTES
5.3 Local (Estudios realizados al interior de la Universidad)
We now relax the assumption that ~B = 0 in Eq. 2.6. Then the 4th-order correlation tensor
Ci j,mn[Eq. 2.12] can be written in terms of the mean field and of the correlation tensors of the fluctuating field: Ci j,mn(~r) = BiBjBmBn+BiBjhb0mb0ni+BmBnhbibji + BiBmhbjb0ni+BiBnhbjb0mi + BjBmhbib0ni+BjBnhbjb0mi + Bihbjb0mb0ni+Bjhbib0mb0ni + Bmhbibjb0ni+Bnhbibjb0mi + hbibjb0mb0ni, (2.32)
where unprimed quantities are evaluated at~xand the primed ones at~x+~r. Due to homogeneity, correlation tensors depend only on~rand not on~x(and the statistical average can be interpreted as a volume average over ~x). This means that the first three terms in Eq. 2.32 have no spatial
dependence at all, while the rest of the tensor can be written in Fourier space as follows: Ci j,mn(~k) = 1 V Z d3~r e−i~k·~rCi j,mn(~r) = BiBmcj,n(~k)+BiBncj,m(~k) + BjBmci,n(~k)+BjBnci,m(~k) + Bic∗mn,j(~k)+Bjc∗mn,i(~k) + Bmci j,n(~k)+Bnci j,m(~k) + ci j,mn(~k). (2.33)
This is the Fourier-space correlation tensor introduced in Eq. 2.13, which has now been expressed in terms of the mean field and the second-, 3rd- and 4th-order correlation tensors of the fluctuating field:
ci,m(~k) = hbi∗(~k)bm(~k)i= 1 V Z d3~r e−i~k·~rhb ib0mi, (2.34) ci j,m(~k) = hhi j∗(~k)bm(~k)i= 1 V Z d3~r e−i~k·~rhb ibjb0mi, (2.35) ci j,mn(~k) = hh∗i j(~k)hmn(~k)i,= 1 V Z d3~r e−i~k·~rhbibjb0mb0ni, (2.36) wherehi j(~k)=(1/V) R d3~x e−i~k·~xb i(~x)bj(~x).
Thus, the presence of the mean field leads to second- and 3rd-order statistics of the fluctuating field appearing alongside the 4th-order ones in the tensor Ci j,mn. Since the Stokes correlators probe the total field, this means that some information about the second- and 3rd-order statistics could be extracted from them, provided the mean field itself can be independently determined and thus used as a “probe” (in fact, it turns out that only its orientation generally has to be known and even that knowledge is not always necessary, although easily obtainable; see Appendix A.2.3).
As before, we need additional symmetry assumptions about the fluctuating field in order to make a transition from the Stokes correlators to theoretically/physically interesting quantities. The technically rigorous choice would be to assume that the statistics of~bwill depend on one special direction, that of the mean field, and be isotropic in the plane perpendicular to it. This, however, leads to a very large number of independent scalar functions appearing in the general form ofCi j,mnand while it is probably worth working them all out, it is quite unlikely that the 6 available Stokes correlators will be sufficient to reconstruct anything of value. Therefore, we make a simplifying assumption (the physical grounds for which are discussed in Sect. 2.4.2) that the mean field is so weak (B2 hb2i) that the fluctuating field remains statistically isotropic. Under this assumption, the case of a weak mean field becomes a straightforward generalisation of the zero-mean-field case considered above. The main gain is that a weak mean field allows
2.4 Method 49
us to use the Stokes correlators to determine not just the power spectrum of the tension force but also the power spectrum of the magnetic field itself: asM(k)=4πk2c
i,i[Eq. 2.2], it is recovered from the second-order terms in Eq. 2.33.
The mathematical details of reconstructing the magnetic-field power spectrum are relegated to Appendix A.2.3. There are many equivalent expressions that can be derived for it; here we display three of them:
M(k) = 2k2 B2⊥cos 2Θ Z 2π 0 dϕReΣIQ = 2k2 B2⊥sin 2Θ Z 2π 0 dϕReΣIU = k 2 2B2⊥sin 2Θ Z 2π 0 dϕ ΣQQ−ΣUU , (2.37)
where Θ is the angle between the x axis and the projection of the mean field onto the plane perpendicular to the line of sight,B⊥is the magnitude of this perpendicular projection. Although
these are easy to measure (Appendix A.2.3), they are manifestly not necessary to determine the functional shape of the spectrum. Thus, we have three independent expressions from which we can deduce this functional shape. That the results should be consistent with one another is a good test of our assumptions (most importantly, the statistical isotropy of the fluctuating part of the field).
The calculation of the tension-force power spectrum is entirely analogous to the zero- mean-field case (see Appendix A.2.3). The result is that Eq. 2.31 still holds subject to two modifications: real part has to be taken of all Stokes correlators and a term proportional toM(k) has to be subtracted, namely
TB,0(k)=ReTB=0(k)− 1
8k 2B2
⊥M(k), (2.38)
whereTB=0(k) is given by Eq. 2.31.
Finally, a disclaimer is in order with regard to the practical applicability of the results obtained for the case of a weak mean field. Since we assumed the mean field to be small compared to the fluctuating field, B2 hb2i, the terms in Eq. 2.32 that contain B
i are small compared tohbibjb0mb0ni. Thus, in order for the second-order statistical information in Eq. 2.32 to be recoverable, the errors associated with the imperfect isotropy of the fluctuating field must be very small—smaller thanO(B2). It is not guaranteed that this is either justified physically or achievable in practice and the verdict on the usefulness of the results of this section will depend on extensive numerical tests, which will not be undertaken in this paper and are left for future work.
MHD Synthetic Gaussian
I,p=3 I,p=3
Q,p=3 Q,p=3
U,p=3 U,p=3
Figure 2.5: Examples of syntheticI, QandU maps generated from an MHD-simulated magnetic field (left panels) and a synthetic Gaussian field (right panels). The same data cubes were used as in Fig. 2.1.
2.5 Numerical Tests 51
Figure 2.6: Left panel: The bold solid red line shows the tension-force power spectrum reconstructed via Eq. 2.31 from the synthetic Stokes maps (Fig. 2.5) based on an MHD-simulated field (saturated state of the Run S4 of Schekochihin et al., 2004). The bold dotted black line is the same spectrum computed directly from the full three-dimensional data (same as Fig. 2.2, right panel). The errors bars on the estimated spectrum are obtained by comparing results from synthetic Stokes maps obtained by integrating along three orthogonal “lines of sight” (the three axes of the data cube). The thin solid blue line with error bars and the thin dotted black line represent analogous information for a synthetic Gaussian field. Right panel: Similar to the left panel, but the reconstructed tension-force spectra are based not on the estimate Eq. 2.31 but on the full information about the projected (line-of-sight integrated) spectra, i.e., they are given by the sum ofΦ1[Eq. 2.28] andΦ2 [Eq. 2.29] calculated in terms
ofCi j,mn(~k⊥) (including its unobservable line-of-sight components).
2.5 Numerical Tests
Having presented the analytical derivation of our method, we now present a proof-of-concept numerical test by analysing two data cubes containing randomly tangled magnetic fields: a saturated magnetic field generated by fluctuation dynamo in an MHD simulation (Run S4 of Schekochihin et al. 2004) and a divergence-free, random-phased Gaussian field synthetically generated to have the same spectrum as the MHD field (Fig. 2.2; snapshots of the two fields are shown in Fig. 2.1). Both fields have zero mean, so the results of Sect. 2.4.7 are not tested here.