Significance of Bernoulli Integral Terms for the Solar Wind Protons at 1 au

Abstract

The Bernoulli integral describes the energy conservation of a fluid along specific streamlines. The integral is the sum of individual terms that contain the plasma density, speed, temperature, and magnetic field. Typical solar wind analyses use the fluctuations of the Bernoulli integral as a criterion to identify different plasma streamlines from single spacecraft observations. However, the accurate calculation of the Bernoulli integral requires accurately determining the plasma polytropic index from the analysis of density and temperature observations. To avoid this complexity, we can simplify the calculations by keeping only the dominant terms of the integral. Here, we analyze proton plasma and magnetic field observations obtained by the Wind spacecraft at 1 au, during 1995. We calculate the Bernoulli integral terms and quantify their significance by comparing them with each other. We discuss potential simplifications of the calculations in the context of determining solar wind proton thermodynamics using single spacecraft observations.

The Polytropic Behavior of Solar Wind Protons as Observed by the Ulysses Spacecraft during Solar Minimum

“…The thermal and the magnetic terms are considerably (i.e., at least ten times) smaller than the dynamic term of the integral for most of the time we examine here. Similar to the Bernoulli integral of protons at ∼1 au examined by Nicolaou et al (2021a), the thermal term does not become dominant for the range of γ we estimate in our study. Therefore, the estimations of the Bernoulli integral do not suffer significant errors by assuming an adiabatic plasma in our case.…”

“…We have also calculated the Bernoulli integral of the plasma protons, assuming a nearly adiabatic plasma. According to Nicolaou et al (2021a), this is a valid assumption of the solar wind at ∼1 au. The same assumption holds in our case, because the average γ is 1.4, and the solar wind flow speed is almost always ten times (or more) larger than the thermal speed (see Appendix A).…”

“…Following a methodology similar to those of other studies deriving the polytropic index from single-spacecraft observations, we perform a linear regression to log 10 (T) versus log 10 (n) within each subinterval (e.g., Newbury et al 1997;Arridge et al 2009;Livadiotis 2018aLivadiotis , 2018bNicolaou et al 2021aNicolaou et al , 2021bNicolaou et al , 2021c. The slope of the fitted lines determines γ because, according to Equation (1):…”

“…which is a reasonable simplification in our case, as we discuss in the Appendix. The ratio of the standard deviation of the Bernoulli integral over its mean value std BI mean BI ( ) ( ) within each analyzed subinterval is proportional to the homogeneity of the analyzed plasma and indicates whether the data points we analyze correspond to the same streamline or not (e.g., Kartalev et al 2006;Nicolaou et al 2014b;Pang et al 2015;Livadiotis 2018aLivadiotis , 2018bNicolaou et al 2021aNicolaou et al , 2021b. In Figure 3, we show three examples of such subintervals; one with γ ∼ 5/3, another with γ ∼2, and another with γ ∼ 1.2.…”

“…The authors discuss the variations of γ as a function of the solar activity, as well as possible biases due to the instrument providing the observations. Several analyses of Wind spacecraft observations employ a very similar method to characterize the polytropic behavior of solar wind protons at ∼1 au (e.g., Livadiotis & Desai 2016;Livadiotis 2018a;Nicolaou et al 2021aNicolaou et al , 2021bLivadiotis & Nicolaou 2021). These analyses calculate a nearly adiabatic plasma with γ ∼ 5/3 on average.…”