## B Tools to Analyze MHD Turbulence in Space Plasmas

No matter where we are in the solar wind, short scale data always look rather random as shown in Figure 114.This aspect introduces the problem of determining the time stationarity of the dataset. The concept of stationarity is related to ensembled averaged properties of a random process. The random process is the collection of the samples , it is called ensemble and indicated as .

Properties of a random process can be described by averaging over the collection of all the possible sample functions generated by the process. So, chosen a begin time , we can define the mean value and the autocorrelation function , i.e., the first and the joint moment:

In case and do not vary as time varies, the sample function is said to be weakly stationary, i.e.,

Strong stationarity would require all the moments and joint moments to be time independent. However, if is normally distributed, the concept of weak stationarity naturally extends to strong stationarity.

Generally, it is possible to describe the properties of simply computing time-averages over just one . If the random process is stationary and and do not vary when computed over different sample functions, the process is said ergodic. This is a great advantage for data analysts, especially for those who deals with data from s/c, since it means that properties of stationary random phenomena can be properly measured from a single time history. In other words, we can write:

Thus, the concept of stationarity, which is related to ensembled averaged properties, can now be transferred to single time history records whenever properties computed over a short time interval do not vary from one interval to the next more than the variation expected for normal dispersion.

Fortunately, Matthaeus and Goldstein (1982a) established that interplanetary magnetic field often behaves as a stationary and ergodic function of time, if coherent and organized structures are not included in the dataset. Actually, they proved the weak stationarity of the data, i.e., the stationarity of the average and two-point correlation function. In particular, they found that the average and the autocorrelation function computed within a subinterval would converge to the values estimated from the whole interval after a few correlation times . More recent analysis (Perri and Balogh, 2010) extended the above studies to different parameter ranges by using Ulysses data, showing that the stationarity assumption in the inertial range of turbulence on timescales of 10 min to 1 day is reasonably satisfied in fast and uniform solar wind flows, but that in mixed, interacting fast, and slow solar wind streams the assumption is frequently only marginally valid. If our time series approximates a Markov process (a process whose relation to the past does not extend beyond the immediately preceding observation), its autocorrelation function can be shown (Doob, 1953) to approximate a simple exponential:

from which we obtain the definition given by Batchelor (1970):Just to have an idea of the correlation time of magnetic field fluctuations, we show in Figure 115 magnetic field correlation time computed at 1 AU using Voyager 2’s data.

In this case, using the above definition, .

### B.1 Statistical description of MHD turbulence

When an MHD fluid is turbulent, it is impossible to know the detailed behavior of velocity field and magnetic field , and the only description available is the statistical one. Very useful is the knowledge of the invariants of the ideal equations of motion for which the dissipative terms and are equal to zero because the magnetic resistivity and the viscosity are both equal to zero. Following Frisch et al. (1975) there are three quadratic invariants of the ideal system which can be used to describe MHD turbulence: total energy , cross-helicity , and magnetic helicity . The above quantities are defined as follows:

where and are the fluctuations of velocity and magnetic field, this last one expressed in Alfvén units , and is the vector potential so that . The integrals of these quantities over the entire plasma containing regions are the invariants of the ideal MHD equations:In particular, in order to describe the degree of correlation between and , it is convenient to use the normalized cross-helicity :

since this quantity simply varies between and .

### B.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 ofHowever, 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 might be lost when computing its reduced version since we integrate over the two transverse . 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 :

whereSince the Hermitian form implies that

it follows that andIt has been shown (Batchelor, 1970; Matthaeus and Goldstein, 1982b; Montgomery, 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 componentsIn 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 .

#### B.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.

### B.3 Introducing the Elsässer variables

The Alfvénic character of turbulence suggests to use the Elsässer variables to better describe the inward and outward contributions to turbulence. Following Elsässer (1950); Dobrowolny et al. (1980b); Goldstein et al. (1986); Grappin et al. (1989); Marsch and Tu (1989); Tu and Marsch (1990a); and Tu et al. (1989c), Elsässer variables are defined as

where and are the proton velocity and the magnetic field measured in the s/c reference frame, which can be looked at as an inertial reference frame. The sign in front of , in Equation (110), is decided by . In other words, for an outward directed mean field , a negative correlation would indicate an outward directed wave vector and vice-versa. However, it is more convenient to define the Elsässers variables in such a way that always refers to waves going outward and to waves going inward. In order to do so, the background magnetic field is artificially rotated by every time it points away from the Sun, in other words, magnetic sectors are rectified (Roberts et al., 1987a,b).

#### B.3.1 Definitions and conservation laws

If we express in Alfvén units, that is we normalize it by we can use the following handy formulas relative to definitions of fields and second order moments. Fields:

Second order moments: Normalized quantities:We expect an Alfvèn wave to satisfy the following relations:

#### B.3.2 Spectral analysis using Elsässer variables

A spectral analysis of interplanetary data can be performed using and fields. Following Tu and Marsch (1995a) the energy spectrum associated with these two variables can be defined in the following way:

where are the Fourier coefficients of the -component among , and , is the number of data points, is the sampling time, and , with is the -th frequency. The total energy associated with the two Alfvèn modes will be the sum of the energy of the three components, i.e.,Obviously, using Equations (124 and 125), we can redefine in the frequency domain all the parameters introduced in the previous section.