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13.2 Spectra of the invariants in homogeneous turbulence

Statistical information about the state of a turbulent fluid is contained in the n-point correlation function of the fluctuating fields. In homogeneous turbulence these correlations are invariant under arbitrary translation or rotation of the experimental apparatus. We can define the magnetic field auto-correlation matrix
Rbij(r) = < bi(x)bj(x + r) >, (62)
the velocity auto-correlation matrix
Rvij(r) = < vi(x)vj(x + r) >, (63)
and the cross-correlation matrix
vb 1- Rij(r) = 2 < vi(x)bj(x + r) + bi(x)vj(x + r) > . (64)
At this point, we can construct the spectral matrix in terms of Fourier transform of Rij
b 1 integral b -ik.r 3 S ij(k) = --- Rij(r)e d r, (65) 2p
1 integral Svij(k) = --- Rvij(r)e-ik.rd3r, (66) 2p
integral Svb(k) = 1-- Rvb(r)e-ik.rd3r. (67) ij 2p ij

However, in space experiments, especially in the solar wind, data from only a single spacecraft are available. This provides values of Rbij, Rvij, and Rvibj, for separations along a single direction r. In this situation, only reduced (i.e., one-dimensional) spectra can be measured. If r1 is the direction of co-linear separations, we may only determine Rij(r1,0,0) and, as a consequence, the Fourier transform on Rij yields the reduced spectral matrix

r 1 integral -ik.r integral S ij(k1) = --- Rij(r1,0,0)e 1 1dr1 = Sij(k1,k2,k3)dk2dk3. (68) 2p

Then, we define Hrm, Hrc, and Er = Erb + Erv as the reduced spectra of the invariants, depending only on the wave number k1. Complete information about Sij might be lost when computing its reduced version since we integrate over the two transverse k. However, for isotropic symmetry no information is lost performing the transverse wave number integrals (Batchelor, 1970Jump To The Next Citation Point). That is, the same spectral information is obtained along any given direction.

Coming back to the ideal invariants, now we have to deal with the problem of how to extract information about Hm from Rij(r). We know that the Fourier transform of a real, homogeneous matrix R (r) ij is an Hermitian form S ij, i.e., S = S~* - --> s = s* ij ji, and that any square matrix A can be decomposed into a symmetric and an antisymmetric part, s A and a A:

A = As + Aa, (69)
1 As = -(A + A~), (70) 2
1 Aa = -(A - ~A). (71) 2

Since the Hermitian form implies that

* * S = S~ - --> sij = sji, (72)
it follows that
Ss = 1(S + ~S) = 1-(S + S ) = real, (73) 2 2 ij ji
a 1 1 S = -(S - ~S) = -(Sij - Sji) = imaginary. (74) 2 2

It has been shown (Batchelor, 1970Matthaeus and Goldstein, 1982bJump To The Next Citation PointMontgomery, 1983) that, while the trace of the symmetric part of the spectral matrix accounts for the magnetic energy, the imaginary part of the spectral matrix accounts for the magnetic helicity. In particular, Matthaeus and Goldstein (1982b) showed that

Hrm(k1) = 2ImSr23(k1)/k1, (75)
where Hm has been integrated over the two transverse components
integral integral k1- ImS23(k)dk2dk3 = 2 Hm(k)dk2dk3. (76)

In practice, if co-linear measurements are made along the X direction, the reduced magnetic helicity spectrum is given by:

Hr (k ) = 2ImSr (k )/k = 2Im(Y Z*)/k , (77) m 1 23 1 1 1
where Y and Z are the Fourier transforms of By and Bz components, respectively.

Hm can be interpreted as a measure of the correlation between the two transverse components, being one of them shifted by 90o in phase at frequency f. This parameter gives also an estimate of how magnetic field lines are knotted with each other. Hm can assume positive and negative values depending on the sense of rotation of the correlation between the two transverse components.

However, another parameter, which is a combination of Hm and Eb, is usually used in place of Hm alone. This parameter is the normalized magnetic helicity

s (k) = kH (k)/E (k), (78) m m b
where Eb is the magnetic spectral power density and sm varies between +1 and -1.

13.2.1 Coherence and phase

Since the cross-correlation function is not necessarily an even function, the cross-spectral density function is generally a complex number:

Wxy(f ) = Cxy(f ) + jQxy(f ),

where the real part Cxy(f) is the coincident spectral density function, and the imaginary part Qxy(f) is the quadrature spectral density function (Bendat and Piersol, 1971). While C (f ) xy can be thought of as the average value of the product x(t)y(t) within a narrow frequency band (f,f + df ), Qxy(f ) is similarly defined but one of the components is shifted in time sufficiently to produce a phase shift of 90o at frequency f.

In polar notation

W (f) = |W (f)|e-jhxy(f). xy xy

In particular,

V~ ----------------- |Wxy(f )| = C2xy(f ) + Q2xy(f),

and the phase between C and Q is given by

hxy(f) = arctan Qxy(f-). Cxy(f )


|Wxy(f )| 2 < Wx(f )Wy(f ),

so that the following relation holds

|Wxy(f )|2 g2xy(f ) = -------------< 1. Wx(f )Wy(f )

This function g2xy(f), called coherence, estimates the correlation between x(t) and y(t) for a given frequency f. Just to give an example, for an Alfvén wave at frequency f whose k vector is outwardly oriented as the interplanetary magnetic field, we expect to find h (f) = 180o vb and 2 gvb(f ) = 1, where the indexes v and b refer to the magnetic field and velocity field fluctuations.

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