Ionized Plasma and Neutral Gas Coupling in the Sun’s Chromosphere and Earth’s Ionosphere/Thermosphere

Abstract

We review physical processes of ionized plasma and neutral gas coupling in the weakly ionized, stratified, electromagnetically-permeated regions of the Sun’s chromosphere and Earth’s ionosphere/thermosphere. Using representative models for each environment we derive fundamental descriptions of the coupling of the constituent parts to each other and to the electric and magnetic fields, and we examine the variation in magnetization of the components. Using these descriptions we compare related phenomena in the two environments, and discuss electric currents, energy transfer and dissipation. We present examples of physical processes that occur in both atmospheres, the descriptions of which have previously been conducted in contrasting paradigms, that serve as examples of how the chromospheric and ionospheric communities can further collaborate. We also suggest future collaborative studies that will help improve our understanding of these two different atmospheres, which while sharing many similarities, also exhibit large disparities in key quantities.

Appendix

The two-dimensional simulation results shown in Figs. 14 and 15 were performed with the HiFi spectral-element multi-fluid model (Lukin 2008). Effective grid sizes of 480×1920 and 180×720 were used in the solar-prominence and ionosphere cases, respectively, along the horizontal (y) and vertical (x) directions. Periodic conditions were applied at the side boundaries (y), while closed, reflecting, free-slip, perfect-conductor conditions were applied at the top and bottom boundaries (x), which were placed sufficiently far from the unstable layer to have negligible effect on the Rayleigh–Taylor evolution. Details of the plasma and neutral profiles and parameters used in the simulations are given below.

A.1 Prominence

Normalization constants for this case are number density n ₀=1×10¹⁵ m⁻³, length scale L ₀=1×10⁶ m, and magnetic field B ₀=1×10⁻³ T. Using these in a hydrogen plasma, normalization values for the time t ₀=1.45 s and temperature T ₀=5.76×10⁷ K can be derived. The ion inertial scale is so much smaller than any scale of interest that, in this case, its value has been set explicitly to zero, d _i =(c/ω _pi0)/L ₀=0. This is equivalent to neglecting the Hall term in the Ohm’s law. Using the ratio of Hall term to Pedersen in the center of mass Ohm’s law (47), and the simplifications used in Sect. 4.3, this is equivalent to $\xi_{n}^{2}k_{in} \gg1$, which is valid for these simulations.

The electron and ion density profiles are given by atmospheric stratification,

$$ n_i(x) = n_e(x) = n_0 \exp \biggl( - \frac{x}{x_0} \biggr). $$

(124)

The scale height x ₀ is set by the solar gravitational acceleration, g _S =2.74×10² m s⁻², and the assumed background temperature of the corona, T _b =3.5×10⁻³ T ₀=2.02×10⁵ K, scaled to the normalization length L ₀; its value is x ₀=12.2. The density profile of the neutral fluid that constitutes the prominence is given by a prescribed function of x plus a very low uniform background value,

$$ n_n(x) = n_{n0} \operatorname{sech}^2 ( 2x-1 ) + n_{nb}. $$

(125)

The peak neutral number density enhancement is taken to be n _n0=1×10¹⁶ m⁻³=10n ₀, while n _nb =3.5×10⁻⁷ n _n0, corresponding to the neutral fraction obtained in the HiFi ionization/recombination equilibrium at the background temperature T _b . We chose an artificially low value of T _b (compared to a typical coronal temperature of about 2×10⁶ K) in order to prevent the background neutral density n _nb from being far smaller still.

The electron, ion, and neutral temperatures are all assumed to be equal to each other initially. The temperature profile is given by a prescribed function f(x),

$$ T(x) = T_b f(x) = T_b \frac{\cosh^2(x-0.5)}{\cosh^2(x-0.5) + \lambda}, $$

(126)

which approaches T _b away from the prominence and attains a minimum value T _p =T _b /(1+λ) within the prominence. To obtain a temperature approximately corresponding to that observed on the Sun, with an associated low ionization fraction (n _e0/(n _e0+n _n0))=0.091, we set the parameter λ=20. The resulting prominence temperature T _p =9.60×10³ K.

The magnetic field is initialized to lie in the out-of-plane direction $\hat{ \mathbf{e}}_{z}$, so that it is perpendicular to both gravity in the $-\hat{\mathbf{e}}_{x}$ direction and the instability wavenumber k in the $\hat{\mathbf{e}}_{y}$ direction. It is given by

$$ \mathbf{B}= B_0 \hat{\mathbf{e}}_z \biggl[ 1 + \beta \biggl\{ \frac {n_e(x)}{n_0} \bigl[ 1 - f(x) \bigr] - \frac{n_n(x)}{2n_0}f(x) - \frac{1}{x_0} \frac{n_{n0}}{2n_0} \bigl[ \tanh ( 2x-1 ) - 1 \bigr] \biggr\} \biggr]^{1/2}, $$

(127)

where β=1.4×10⁻² is the plasma beta evaluated using the background plasma pressure at x=0 and B ₀. The magnetic field profile so constructed accommodates (1) the plasma pressure change from the isothermal hydrostatic profile (2) the neutral pressure, and (3) the gravitational force exerted on the bulk of the neutral fluid (neglecting the small background contribution n _nb ) throughout the atmosphere.

This initial condition is not an exact solution to the multi-fluid model which includes ionization, recombination, and viscous forces, but would be if only gravity, pressure gradients, and Lorentz forces were considered. It is close enough to the full solution, however, that any flows such as those created by pressure gradients driven by ionization/recombination of the initial condition are small compared to the flows initiated by the instability. The instability is initiated by introducing a small neutral density perturbation localized in x on the bottom side of the prominence,

$$ \Delta n_n(x,y) = \delta n_n(x) \exp \bigl[-4x^2\bigr]\frac{1}{5}\sum_{j=1}^5 \sin[j \pi y], $$

(128)

where we chose δ=10⁻², and y is the normalized distance along the gravitational equipotential surface.

A.2 Ionosphere

The normalization constants are number density n ₀=1×10¹⁶ m⁻³, length scale L ₀=2×10⁴ m, and magnetic field B ₀=3×10⁻⁵ T. Using these in a hydrogen plasma, normalization values for the time t ₀=3.06 s, temperature T ₀=5.19×10³ K, and ion inertial scale length d _i =(c/ω _pi0)/L ₀=1.14×10⁻⁴ can be derived.

The neutral density profile is given by atmospheric stratification,

$$ n_n(x) = n_{0} \exp \biggl( -\frac{x}{x_0} \biggr). $$

(129)

The scale height x ₀ is set by Earth’s gravitational acceleration, g _E =9.81 m s⁻², and the assumed background temperature of the ionosphere, T _b =0.225T ₀=1.17×10³ K, scaled to the normalization length L ₀; its value is x ₀=49.1. The electron/ion density profile of the plasma is given by a prescribed function of x plus a low uniform background value,

$$ n_e(x) = n_i(x) = n_{e0} \operatorname{sech}^2 ( 2x-1 ) + n_{eb}. $$

(130)

The peak electron number density is taken to be n _e0=1×10¹² m⁻³=10⁻⁴ n ₀, while n _eb =0.05n _e0. The electron, ion, and neutral temperatures are all assumed to be equal and uniform initially, at T _b =1.17×10³ K.

As before, the magnetic field is initialized to lie in the out-of-plane direction. It is given by

$$ \mathbf{B}= B_0 \hat{\mathbf{e}}_z \biggl[ 1 - \beta \biggl\{ \frac{2n_e(x)}{n_0} + \frac{1}{2x_0} \bigl[ \tanh ( 2x - 1 ) - 1 \bigr] \biggr\} \biggr]^{1/2}, $$

(131)

where β=0.450 is the neutral beta calculated using the background neutral pressure at x=0 and B ₀. The magnetic field profile so constructed accommodates (1) the plasma pressure and (2) the gravitational force exerted on the bulk of the plasma (neglecting the small background contribution n _eb ) throughout the atmosphere.

The instability is initiated with a small electron density perturbation localized in x on the bottom side of the ionization layer,

$$ \Delta n_e(x,y) = -\delta n_e(x) \exp \bigl[-4x^2\bigr]\frac{1}{5}\sum_{j=1}^5 \sin[j \pi y], $$

(132)

where again we chose δ=10⁻², and y is the normalized distance along the gravitational equipotential surface.