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: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, .
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:
In particular, in order to describe the degree of correlation between and , it is convenient to use the normalized cross-helicity :
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
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 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 :
Since the Hermitian form implies that
It 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
In practice, if co-linear measurements are made along the X direction, the reduced magnetic helicity spectrum is given by:
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
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
and the phase between and is given by
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.
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 as110), 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).
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:
We expect an Alfvèn wave to satisfy the following relations:
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: