### 13.2 Spectra of the invariants in homogeneous turbulence

Statistical information about the state of a turbulent fluid is contained in the -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
the velocity auto-correlation matrix
and the cross-correlation matrix
At this point, we can construct the spectral matrix in terms of Fourier transform of

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

Then, we define , , and as the reduced spectra of the invariants, depending only on the wave number . 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, 1970). 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 from . We know that the Fourier transform of a real, homogeneous matrix is an Hermitian form , i.e., , and that any square matrix can be decomposed into a symmetric and an antisymmetric part, and :

where

Since the Hermitian form implies that

it follows that
and

It has been shown (Batchelor, 1970Matthaeus and Goldstein, 1982bMontgomery, 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

where has been integrated over the two transverse components

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

where and are the Fourier transforms of and components, respectively.

can be interpreted as a measure of the correlation between the two transverse components, being one of them shifted by in phase at frequency . This parameter gives also an estimate of how magnetic field lines are knotted with each other. 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 and , is usually used in place of alone. This parameter is the normalized magnetic helicity

where is the magnetic spectral power density and varies between and .

#### 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:

where the real part is the coincident spectral density function, and the imaginary part is the quadrature spectral density function (Bendat and Piersol, 1971). While can be thought of as the average value of the product within a narrow frequency band , is similarly defined but one of the components is shifted in time sufficiently to produce a phase shift of at frequency .

In polar notation

In particular,

and the phase between and is given by

Moreover,

so that the following relation holds

This function , called coherence, estimates the correlation between and for a given frequency . Just to give an example, for an Alfvén wave at frequency whose vector is outwardly oriented as the interplanetary magnetic field, we expect to find and , where the indexes and refer to the magnetic field and velocity field fluctuations.