Kurtosis

Kurtosis is a measure of `peakedness' and `atness' of the PDFs. It is dened by:

F(l) = S4(l)

S₂²(l) (5.8)

hereS4andS2are the fourth order and second order moments respectively. The atness of the Gaussian PDFs is exactly F = 3. High values of F characterize PDFs with sharp peaks and at tails, where as low values identify PDFs with rounded peaks and broad shoulders. Thus the atness F is a useful measure of intermittent features. Two sets of kurtosis curves are drawn for all the quantities. These two sets correspond to the initial state and nal states of the systems that are used in the structure function analysis (see caption of g. 5.2) above. From the analysis of the PDFs , some insight into the intermittent behavior of the MHD ows was obtained. Now from the Kurtosis plots at both initial state and the nal state of the system, the changes in the intermittent behavior of several quantities in the ow can also be understood, while supporting the PDF analysis.

Figures 5.9 a and b represent the initial and nal states of Kurtosis curves of z⁺ for all the three cases. It is seen that the value of Kurtosis is very high for the forced case in comparison with the other two cases, initially. The decaying case appears to show a near Gaussian Kurtosis in the initial stages and in the nal state it has two regions. A region where the value of Kurtosis is at a peak and the curve showing a fall o and a region where it remains close to Gaussian. This is an manifestation of the same trend that was seen in the PDFs: peak values close to the Gaussian curve and broader tails deviating from the Gaussian. The same argument holds for special case too. For the forced case their is an slight increase in Kurtosis, but it attains a plateau in the smalll region and closes towards the Gaussian in the highl regions. From the initial state plots it can be inferred that the initial state of the decaying case is a Gaussian distribution

(as described in the initial conditions in section 4.1). The chosen initial state for the forced case has dominantly small scales which have strong correlations and hence are highly intermittent (i.e. this is a transitionary state) and in the special case, where the forcing was withdrawn after a certain amount of time, it appears that the correlations among the small scales have just begun (since actual initial state of the forced case is also a Gaussian from which this special case has evolved) as does intermittency in its initial state. The nal state plots in general, conrm the PDF analysis where strong intermittency in the small scales and very less or no intermittency in the large scales (i.e. strong correlations in the small scales and little or few correlations at the large scales) are observed.

a)

b)

Figure 5.9: Kurtosis curves in the a) near initial and b) nal state of the systems forz⁺. red: forced case, green: special case, blue: decaying case and black: reference line for Kurtosis of a Gaussian.

Figures 5.10 a and b represent the initial and nal states of Kurtosis curves ofH^M for all the three cases. All these three plots look alike. In the nal state a at region for almost all ofl followed by a small parabola like bend at large scales is seen. This is also consistent with the PDFs seen above for magnetic helicity. They show very few types of structures and the intermediate scales are absent. This transition between the scales is not smooth but abrupt. Thus the two distinct features seen in the PDfs, the Mexican

5.4 Other Statistical Tools 113 hat and an abrupt tail , are corroborated. Also true is the strong intermittency seen in the PDFs as the values of the Kurtosis curves range from 200 in decaying case to about 20 in the forced case. Relatively small value of Kurtosis for the forced case may be because of the relatively atter peak which do not t with the model Gaussian in the PDFs. The initial state Kurtosis curves also look alike (like three parallel lines almost) but they do represent the dual scale nature of the magnetic helicity structures (seen in the PDFs) with values of the Kurtosis as high as10⁵ to as low as 3500. Since this state is not yet completely turbulent, these values do not make any impact.

a)

b)

Figure 5.10: Kurtosis curves in the a) near initial and b) nal state of the systems forH^M. red: forced case, green: special case, blue: decaying case and black: reference line for Kurtosis of a Gaussian.

Figures 5.11 a and b represent the initial and nal states of Kurtosis curves ofj² for all the three cases. The forced case Kurtosis in the initial stage appears like an exponential (only a shape comparison not a mathematical t) but changes to a strange shape with two regions where the value shoots suddenly from 8 to 10 and then falls smoothly, close to 6. The special case starts at a value of about 5 but soon reaches to a constant value 4, which is spread over a large l. In the nal state, it shows a signicant increase in magnitude and a change in curve shape to an exponential like curve (here too only a shape comparison not a mathematical t). The decay case in the initial state appears

to be close to a Gaussian all the while, but changes to an exponential like curve with a large magnitude as in the special case, in the nal state.

j² structures once formed do not change in length (hence no change in correlation length) and almost remain the same. But they do show intermittent behavior as was reported in the PDF studies. These two facts once again are reiterated by the Kurtosis curves where the change in Kurtosis from initial state to nal state is∼ 3. The shapes of these curves in the nal states, do indicate that over the complete scale range, j² structures exhibit intermittency, which the PDFs could capture in totality. In the initial state however, only the chosen initial state of the forced case exhibits signicant intermittency and the other two cases are close to the Gaussian and hence little or no intermittency.

a)

b)

Figure 5.11: Kurtosis curves in the a) near initial and b) nal state of the systems forj². red: forced case, green: special case, blue: decaying case and black: reference line for Kurtosis of a Gaussian.