13 Appendix 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.

Figure 97:

$BY$ component of the IMF recorded within a high velocity stream.

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 $N$ samples $x(t)$ , it is called ensemble and indicated as ${x(t)}$ .

Properties of a random process ${x(t)}$ can be described by averaging over the collection of all the $N$ possible sample functions $x(t)$ generated by the process. So, chosen a begin time $t1$ , we can define the mean value $mx$ and the autocorrelation function $Rx$ , i.e., the first and the joint moment:

In case $mx(t1)$ and $Rx(t1,t1 + t)$ do not vary as time $t1$ varies, the sample function $x(t)$ is said to be weakly stationary, i.e.,

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

Generally, it is possible to describe the properties of ${x(t)}$ simply computing time-averages over just one $x(t)$ . If the random process is stationary and $mx(k)$ and $Rx(t, k)$ 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 $tc$ .

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 98 magnetic field correlation time computed at $1 AU$ using Voyager’s 2 data.

Figure 98:

Magnetic field auto-correlation function at $1 AU$ (adopted from Matthaeus and Goldstein, 1982b

In this case, using the above definition, $t - ~ 3.2 .103 s c$ .

13.1 Statistical description of MHD turbulence
13.2 Spectra of the invariants in homogeneous turbulence
  13.2.1 Coherence and phase
13.3 Introducing the Elsässer variables
  13.3.1 Definitions and conservation laws
  13.3.2 Spectral analysis using Elsässer variables