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Volume 526, February 2011
Article Number A20
Number of page(s) 13
Section The Sun
Published online 15 December 2010

Online material

Appendix A: Cross–correlation functions

In Figs. 9 − 13, we present the cross-correlation functions between different parameters, which we used to perform the correlation scatter plots presented in the paper. All cross-correlation functions are derived up to a time lag of  ±10 days, with a step of 6 h (data resolution) in all investigated cases. The time lags are always expressed in days. Negative lag between two quantities, e.g., V and Ap, hereinafter denoted as the V − Ap correlation, means that V is delayed with respect to Ap.

Appendix B: Correlation between solar-wind parameters

We calculated the cross-correlation functions for all combinations of V, T, n, and B. In Figs. 9 and 14, the cross-correlation functions and the scatter plots for the highest-correlation-coefficient time-lag are presented for the most tightly correlated combinations. All considered correlations are listed in Table 4 where the time lags Δt, the corresponding linear least squares fit parameters a and b, and the correlation coefficient R are presented. A given “X − Y” correlation corresponds to the linear form Y(t) = aX(t) + b, where X(t) represents the value of X that occurred Δt days before the actual value of Y(t), i.e., t is the “retarded time”, t = t − Δt.

Table 4 shows that the most tightly correlated parameters are V and T, having the correlation coefficient R = 0.8, when V is delayed after T for Δt = 0.25 days. Furthermore, Fig. 9 shows that in the V − n and V − B case, the correlation coefficients are higher when the parameters are anti-correlated (the negative correlation coefficient in Table 4). This reflects the main physical property of HSSs, which is a depleted density and magnetic field in the stream itself. The anti-correlations are most significant for the time lags Δt = 0.75 d and Δt = 2.25 d, respectively, meaning that the dips of B and n are delayed with respect to the peak of V. The positive correlations of both the V − B and V − n relationships and their negative time lags (peaks of n and B preceding the peak of V by 2.5 d and 1.75 d), are related to the magnetic field and density compression in the “interacting region” at the frontal edge of HSSs. We note that the time lag in the n − B correlation is Δt =  + 0.5 d, meaning that the peak of B is delayed with respect to the peak in n.

The distribution of data points in Vn and VB graphs in Figs. 14b and c shows two “branches”, one almost horizontal and another almost vertical. The vertical one corresponds to the slow solar wind (V ≈ 300−400 km s-1), and the horizontal one to the fast solar wind (V > 400 km s-1 and low values of n and B). If fitted by the power-law, the relationships are given by n = 1.4  ×  106  ×  V−2.0 ± 0.1, and B = 4710  ×  V−1.1 ± 0.1, with the correlation coefficients R = 0.69 and R = 0.58, respectively. Although the applied power-law fit is obviously more appropriate than the linear fit, it still shows a large deviation from the data in the slow-wind velocity range (v < 400 km s-1). Thus, the obtained power-law relationships can be applied only to the fast solar wind.

Table 4

Relationships T(V), n(V), B(V), T(B), B(n), and T(n).

thumbnail Fig. 9

Cross-correlation functions describing relationships between solar wind parameters (V − T, V − n, and V − B).

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thumbnail Fig. 10

Cross-correlation functions describing the relationship between the solar wind parameters and Dst index (V − Dst, B − Dst, BVDst, and BV2 − Dst).

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thumbnail Fig. 11

Cross-correlation functions describing relationships between the solar wind parameters and Ap index (VAp, BAp, BVAp, and BV2 − Ap).

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thumbnail Fig. 12

Cross-correlation functions describing relationships between the coronal hole fractional areas CH and the solar wind parameters.

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thumbnail Fig. 13

Cross-correlation functions for CHDst and CHAp relationships.

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thumbnail Fig. 14

Correlations VT, Vn, and VB. The linear least squares fits are shown by full lines and the power-law fits by dashed lines. The fit parameters, the correlation coefficient R, and the applied time lag Δt are shown in the insets.

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We note that the first relationship, which can be expressed as nV2 = const., resembles that of kinetic energy conservation. Similarly, the latter correlation (approximately BV = const.) resembles to the magnetic flux conservation. However, we emphasize that these relationships cannot be interpreted in this way, since there is time lag between the adopted values of V and n, as well as V and B. These empirical forms reflect a complex/dynamical time-space relationship in the solar wind, that nevertheless produce a quite simple quantitative relationship. Although their meaning is not clear, they are certainly interesting, and deserve further analysis/interpretation from the theoretical point of view.

© ESO, 2010

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