Cylindrical Hall – MHD Waves: A Nonlinear Solution

Abstract

The exact nonlinear cylindrical solution for incompressible Hall -magnetohydrodynamic (HMHD) waves, including dissipation, essentially from electron -neutral collisions, is obtained in a uniformly rotating, weakly ionized plasma such as exists in photospheric flux tubes. The ω -k relation of the waves, called here Hall -MHD waves, demonstrates the dispersive nature of the waves, introduced by the Hall effect, at large axial and radial wavenumbers. The Hall -MHD waves are in general elliptically polarized. The partially ionized plasma supports lower frequency modes, lowered by the factor δ ≡ ratio of the ion mass density to the neutral particle mass density, as compared to the fully ionized plasma (δ = 1). The relation between the velocity and the magnetic field fluctuations departs significantly from the equipartition found in Alfvén waves. These short-wavelength and arbitrarily large amplitude waves could contribute toward the heating of the solar atmosphere.

Mean-field dynamo in partially ionized plasmas – I

“…It is instructive to examine the new correlations for the case of say Alfvénic turbulence. Now in the weakly ionized case, the relation between the velocity and the magnetic‐field fluctuations for the Alfvén waves is given by (Krishan & Varghese 2007) with δ=ρ i /ρ n . Substituting these results in the expression for α H , we find where λ H = c /ω pi is the ion‐inertial scale and is the ion‐plasma frequency and is the ratio of the average kinetic helicity and the average enstrophy of the neutral fluid turbulence.…”

“…Although, the ideal magnetohydrodynamics (MHD) is often used as a starting point of an astrophysical investigation, there are many systems with a rather low degree of ionization dominated by the charged particle–neutral collisions and the neutral particle dynamics. A major part of the solar photosphere (Leake & Arber 2006; Krishan & Varghese 2007), the protoplanetary discs (Krishan & Yoshida 2006) and the molecular clouds (Brandenburg & Zweibel 1994) are some of the examples of weakly ionized astrophysical plasmas. The dynamo action in such a plasma would be affected by the multifluid interactions.…”

Kolmogorov dissipation scales in weakly ionized plasmas

“…The dynamics of a weakly ionized plasma can be described with the following equations (Krishan & Yoshida 2006; Krishan & Varghese 2008; Krishan & Gangadhara 2008): the momentum balance of the neutral fluid is given by and the magnetic field B evolves as and where V is the velocity of the neutral fluid and h is the total enthalpy. The Lorentz force in the neutral dynamics appears due to the ion–neutral coupling in that the Lorentz force on the ions is balanced by the ion–neutral collisional force (−ρ i ν in ( V i − V n )) where ν in is the ion–neutral collision frequency.…”

Sweet–Parker current slab in a partially ionized plasma

“…We can now contrast these cases with the physical conditions of the solar atmosphere. First, the condition of a weakly ionized plasma, ν en ≫ν ei , is satisfied in the solar atmosphere up to a height of nearly 800 km above the photosphere (Krishan & Varghese 2008; Krishan & Yoshida 2009). Thus even the Sweet–Parker rate, in the absence of the Hall and the ambipolar effects, with electron–neutral collisions dominating the electron–ion collisions, is enhanced by, for example, two orders of magnitude at a height of 560 km in the solar atmosphere (Singh & Krishan 2009) in comparison to the value obtained by Vishniac & Lazarian (1999).…”

“…There are many astrophysical systems with a fairly low degree of ionization dominated by the charged particle–neutral collisions and the neutral–particle dynamics. A major part of the solar photosphere (Leake & Arber 2006; Krishan & Varghese 2008; Pandey, Vranjes & Krishan 2008), protoplanetary discs (Krishan & Yoshida 2006) and molecular clouds (Brandenburg & Zweibel 1994) are some examples of weakly ionized astrophysical plasmas. The need to study magnetic reconnection in the weakly ionized part of the solar atmosphere has been highlighted by the Hinode observations of the energetic phenomena in the solar chromosphere (Isobe & Shibata 2008).…”