Hypergeometric function representation of transport coefficients for drifting bi-Maxwellian plasmas

Abstract

We derive the momentum, parallel energy, and perpendicular energy collisional transport coefficients for drifting bi-Maxwellian plasmas by using the Boltzmann collision integral approach and present them in the form of triple hypergeometric functions. In the derivation, we write the drift velocity u of the bi-Maxwellian plasma in terms of parallel and perpendicular components (i.e., u = u‖ + u⊥), parallel and perpendicular with respect to the ambient magnetic field, and we consider the Coulomb collision interactions. We consider two special cases: first, when the drift velocity is parallel to the ambient magnetic field (i.e., u = u‖), and second, when the drift velocity is perpendicular to the ambient magnetic field (i.e., u = u⊥). For the first case, the transport equations and consequently the transport coefficients are derived and presented in the form of double hypergeometric functions; these results are consistent with the findings of Hellinger and Trávníček [Phys. Plasmas 16(5), 054501 (2009)]. For the second case, the transport coefficients are obtained and found to be in the form of double hypergeometric functions. When we combine these two special cases, i.e., for general u, the transport coefficients are shown to be in the form of triple hypergeometric functions. Also, we investigate the above problem by using another approach, i.e., Fokker Planck approximation. We obtain similar results for both approaches.

Boundary of the Distribution of Solar Wind Proton Beta versus Temperature Anisotropy

“…In Hellinger & Trávníček (2009) and also in a similar analysis by Jubeh & Barghouthi (2017), the foundational equation is that of the Landau collision integral (Landau 1936), which suffers from divergences in the limits of both infinitely short and long wavelengths. The earlier work by Kogan (1961) is also based upon the same Landau collision integral.…”

“…Taking the collisional frequency for proton-proton interactions as derived by Hellinger & Trávníček (2009), which is based upon the Landau collision integral under the assumption of a drifting bi-Maxwellian plasma distribution function, Vafin et al (2019) proceeded to write down the equation for proton temperature relaxation. Note that Jubeh & Barghouthi (2017) have also carried out a similar calculation to what Hellinger & Trávníček (2009) did. In the absence of relative drift in the bi-Maxwellian plasma, the collision frequency derived by Hellinger & Trávníček (2009) and Jubeh & Barghouthi (2017) is actually identical to the well-known formula available in the Naval Research Laboratory (NRL) Plasma Formulary (Richardson 2019), which in turn is taken from Kogan (1961).…”

“…Note that Jubeh & Barghouthi (2017) have also carried out a similar calculation to what Hellinger & Trávníček (2009) did. In the absence of relative drift in the bi-Maxwellian plasma, the collision frequency derived by Hellinger & Trávníček (2009) and Jubeh & Barghouthi (2017) is actually identical to the well-known formula available in the Naval Research Laboratory (NRL) Plasma Formulary (Richardson 2019), which in turn is taken from Kogan (1961). Note that ignoring the secondary protons and relative drift is an idealization.…”

“…Substituting this result into the temperature Equations (A3), we have At this point, we note that if we replace ò(k, ω) by unitywhich amounts to the approximation that leads to the Landau collisional integral-then the ω integration in Equation (A8) can be performed exactly. In fact, Hellinger & Trávníček (2009) and Jubeh & Barghouthi (2017) have done just that, and obtained the result in terms of generalized hypergeometric functions. However, in this case, the k integral diverges at both ends, k = 0 and k = ∞.…”

Expanding-box Quasilinear Model of the Solar Wind

“…This realization has prompted us to reconsider the issue of collisional effects for the solar wind. For this reason, let us simply adopt the more conventional unmagnetized formalism (Barakat & Schunk 1981;Li et al 2004;Hellinger & Trávníček 2009;Jubeh & Barghouthi 2017), is the Debye length. In dimensionless form, the collisional relaxation term for the temperatures can be implemented to the right-hand side of the basic equation by adding the following terms:…”