Numerical Simulation of Vertical Current Sheets in Solar Chromospheric Plasma with the Hall Effect

Abstract

The consequences of the entry of the magnetic field from underlying layers into the Sun’s chromosphere are studied via numerical solution of the 2D MHD problem (with constant physical quantities along the straight horizontal magnetic lines of force). Chromospheric plasma is considered to be collisional; its Joule dissipation and the Hall effect are taken into account. The initial magnetic field is characterized by a value of β = 1.5 and corresponds to an upward current sheet of finite thickness. There are cases in which the coevolution of the magnetic field and plasma leads to the formation of a very thin, vertical current sheet, from which a plasma stream flows downward vertically. The process turned out to be typical of rather large heights, where the influence of the Hall effect on the magnetic field evolution begins to exceed the field-drift effect due to the (partial) freezing of the force lines. The formed thin current sheet then ceases to exist under the action of pinch (sausage and/or kink) instabilities. The reconnection of force lines is not seen under the conditions of this numerical experiment due to the assumed constancy of physical quantities (including velocity) along a force line. Accordingly, the described processes in nature will act as alternative or additional processes with respect to the typically considered scenarios of reconnection.

APPENDIX

The known classical formulas are used for the material parameters of the plasma. The plasma conductivity (Spitzer, 1965; Balescu, 1988)

$$\sigma = \frac{{{{e}^{2}}{{N}_{e}}{{\tau }_{e}}}}{{{{m}_{e}}}}$$

((A1))

(where ${{\tau }_{e}}$ is the time between the electron–ion collisions) enters into the magnetic viscosity

$${\Theta } = \frac{{{{c}^{2}}}}{{4\pi \sigma H{{{v}}_{{*0}}}}},$$

which, with allowance for (3), takes the form ${\Theta }$ in (9), where

$${{\theta }_{*}} = \sqrt {\frac{{{{m}_{i}}{{m}_{e}}}}{{2\pi }}} \frac{{{{c}^{2}}}}{H}\frac{1}{{{{{\left( {k{{T}_{*}}} \right)}}^{2}}}}\frac{{\Lambda {{e}^{2}}}}{{0.75}}.$$

((A2))

Similarly, the thermal conductivity (Aschwanden, 2004) takes the form $K$ in (9),

$${{\kappa }_{{{*}}}} = \frac{{{{k}_{S}}T_{*}^{2}\sqrt {{{m}_{i}}} }}{{32H{{k}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}{{N}_{*}}}},\,\,\,\,{{k}_{S}} = \frac{{3.16 \times 0.75}}{{{\Lambda }\sqrt {2\pi } }}~\frac{{{{k}^{{{7 \mathord{\left/ {\vphantom {7 2}} \right. \kern-0em} 2}}}}}}{{\sqrt {{{m}_{e}}} ~{{e}^{4}}}}$$

((A3))

where ${{k}_{S}}$ is the thermal conductivity by Spitzer and ${\Lambda }$ is the Coulomb logarithm.

We give also several formulas that were not used in this work but may be helpful in a comparison of its content with the works of other authors. The dimensionless electric field (Brushlinskii and Morozov, 1974), which corresponds to the MHD system (4)–(7), is expressed by the values used in this system

$$\mathbf{E} = {\Theta }\mathbf{j}~\, - \left[ {{\mathbf{v}} \times {\mathbf{B}}} \right] + \frac{\xi }{\rho }{{\;}}\left( {\left[ {{\mathbf{j}} \times {\mathbf{B}}} \right] - \frac{{\nabla P}}{2}} \right)~,$$

((A4))

for the field transformation to the dimensional form, the combination ${{B}_{0}}{{{{{v}}_{{*0}}}} \mathord{\left/ {\vphantom {{{{{v}}_{{*0}}}} c}} \right. \kern-0em} c}$ serves as a unit. Expression (A4) sometimes is convenient to use in the identical form (Alekseeva, 1980)

$$\mathbf{j} = {{\sigma }_{P}}\mathbf{E}{{'}} + \frac{{{{\sigma }_{H}}}}{B}\left[ {{\mathbf{B}} \times {\mathbf{E}}{{'}}} \right],$$

((A5))

where

$$\mathbf{E}{{'}} \equiv \mathbf{E} + \left[ {{\mathbf{v}} \times {\mathbf{B}}} \right] + \frac{{\xi ~\nabla P}}{{2\rho }}$$

((A6))

denotes the effective electric field, while ${{\sigma }_{P}}$ and are local (and nondimensionalized) values of the Pedersen and Hall conductivities:

$${{\sigma }_{P}} \equiv \frac{{\Theta }}{{{{{\Theta }}^{2}} + {{A}^{2}}}},\,\,\,\,{{\sigma }_{H}} \equiv \frac{A}{{{{{\Theta }}^{2}} + {{A}^{2}}}},\,\,\,\,A = \frac{{\xi B}}{\rho }~.$$

((A7))

It is also worth noting that parameter $\xi ,$ as defined by Eq. (8), characterizes the difference in the dimensionless macroscopic velocities of the ion and electron gases

$${{{\mathbf{v}}}_{i}} - {{{\mathbf{v}}}_{e}} = \frac{\xi }{\rho }{\mathbf{j}}$$

((A8))

and enters the important relation

$${\xi \mathord{\left/ {\vphantom {\xi {\Theta }}} \right. \kern-0em} {\Theta }} \approx {{\omega }_{e}}{{\tau }_{e}},$$

((A9))

where ${{\omega }_{e}}$ is the frequency of Larmor gyration of an electron (see §§ 1.2 and 2.5 in (Brushlinskii and Morozov, 1974)).