## 2 Data
### Variation of the Bernoulli Integral  
The Bernoulli integral is often used as a criterion to identify streamlines within the solar wind (e.g., [4; 5; 6; 7; 8; 9]). Such analyses usually examine the standard deviation of the Bernoulli integral within selected time intervals. Ideally, if the plasma measurements within the selected data interval represent the same streamline, the standard deviation of the calculated Bernoulli integral should be zero. However, in data analyses we may classify observations with standard deviations smaller than a reasonable threshold value as streamlines.  
Figure 3: (**a**) The term \(\frac{\gamma}{\gamma-1}\) as a function of \(\gamma\). The grey ribbon indicates the range of \(\gamma\) for which the absolute value of \(\frac{\gamma}{\gamma-1}\) exceeds 5 (red dashed). (**b**) the histograms of (grey) the dynamic term and (red) the absolute value of the thermal term of the Bernoulli integral assuming plasma with \(\frac{\gamma}{\gamma-1}=5\) (or \(-5\)).  
Figure 2: (**a**) Time series of the Bernoulli integral for different polytropic index values; (solid magenta) \(\gamma=0\), (dashed green) \(\gamma=1/2\), (dashed dotted blue) \(\gamma=3/2\), (dotted grey) \(\gamma=5/3\), and (dotted black) \(\gamma\rightarrow\infty\). (**b**) The Bernoulli integral for the considered \(\gamma\) values in a short sub-interval from day 305 to day 310 of 1995. (**c**) Histograms of the Bernoulli integral in 1995 for the different polytropic indices we assume here.  
Here, we examine the standard deviation of the Bernoulli integral in short consecutive time intervals in 1995. As in [9; 14; 15], we analyze only time intervals that are shorter than 15 min. The choice of short intervals reduces the possibility to analyze data-points from different streamlines. However, the intervals should be large enough to have enough data-points for accurate calculations (for more see [9; 14; 15]). We calculate the standard deviations over the absolute mean values of the Bernoulli integral \(\frac{\sigma_{Bernoulli}}{|Bernoulli|}\) in each interval. We repeat the calculations, each time assuming a different polytropic index of the plasma protons. We specifically consider \(\gamma=0\), \(\gamma=5/6\), and \(\gamma=5/4\). We also calculate the relative standard deviation of the dynamic term \(\frac{\sigma_{Bernoulli}}{dynamic}\) within each interval. Figure 4a shows the histograms of the calculated relative deviations, while Figure 4b plots the histograms of the differences \(\frac{\sigma_{Bernoulli}}{|Bernoulli|}-\frac{\sigma_{dynamic}}{dynamic}\), for each \(\gamma\) value.  
Our result indicates that the second and the third term of the integral do not significantly modify its relative deviation. For \(\gamma=0\), the second term of the integral vanishes, since \(\frac{\gamma}{\gamma-1}=0\). In this case, the relative standard deviation of the integral does not differ by more than 5% of the relative deviation of the dynamic term. This means that the magnetic term contributes no more than \(\sim\)5% to the relative deviations of the integral. For \(\gamma=5/6\) (\(\frac{\gamma}{\gamma-1}=-5\)), the second term becomes negative and reduces the mean value of the Bernoulli integral. This seems to have some minor effects on the relative standard deviations. We observe that the corresponding histogram of \(\frac{\sigma_{Bernoulli}}{|Bernoulli|}-\frac{\sigma_{dynamic}}{dynamic}\) is slightly wider (by \(\sim\)1%) than that for \(\gamma=0\). Finally, for \(\gamma=5/4\), we do not observe any apparent difference from the case with \(\gamma=0\). According to this, we infer that even when \(\gamma\) is very close to unity, we may completely ignore the second term of the Bernoulli integral with almost no effect on the accuracy of the calculated relative standard deviations.