Geometry for propagating Alfvén wave. The wave vector is in the direction, and the ambient magnetic field is in the plane. The perturbed velocities , , and magnetic field are in the direction.
The solutions of Alfvén waves can be classified into four regions in the plane, where is the ratio of ion and neutral density and is the normalized collisional frequency for neutral particles by ions. In Region 1 (red) and Region 3 (blue), there are two Alfvén modes ( and ) propagating in opposite direction and are purely damping mode with zero real frequency ( ). In the “forbidden zone (FZ)” (Region 2, yellow), the real frequency of all three modes is zero. The green area is the transition zone between different regions.
The real part ( ) and imaginary part ( ) of the wave frequency of three eigen-modes are plotted as a function of the density ratio for . The real frequency is shown by the blue (red) line in the top figure. The damping rates and have the same value and are shown by the same blue line in the bottom figure. The yellow area is FZ with . As the value of approaches the yellow zone from the left, the real frequency of and decreases sharply to zero. On the right hand side of the yellow region, the real frequency of modes 1 and 2 can be expressed as .
Same as Fig. 3 except that FZ (yellow) is very narrow.
The real ( ) and imaginary ( ) part of wave frequency for three eigen-modes are plotted as a function of for .
Same as Fig. 3 .
The eigen-amplitudes, and , and phases of two propagating Alfvén modes with and as a function of for (left) and (right). For , and on the left side of forbidden zone, while . For , and the ion and neutral velocity of mode 1 (mode 2) are out of phase (in phase) with .
The real and imaginary parts of the wavenumbers, and , are plotted as a function of for The curve of for is nearly the same as the curve for in (a). The yellow sections of lines in (a) and (b) correspond to those with values in FZ of Fig. 2 .
The contours of (a) and (b) are plotted in the plane. The contour with is shown by the black line. The region with is termed “the heavy damping zone (HDZ).”
(a)–(c) The spatial profiles of of Alfvén wave injected from the left boundary with and the spatial profile of in (d). The yellow region corresponds to FZ for . (e)–(h) The spatial profiles of for four simulation cares with .
The vertical profiles of (a) neutral and ion density for daytime and nighttime ionosphere, indicated by solid and dashed lines, (b) the collision frequency for a neutral particle to collide with ions in the daytime and nighttime ionosphere, (c), (e) the ratio of neutral density to ions density , and (d), (f) the parameter for the daytime and nighttime ionosphere in blue line and black line for and . The FZ (HDZ) for shear Alfvén waves is enclosed by red solid (blue dashed) line.
(a) The neutral and ion density profiles in the solar chromosphere, indicated by black and blue lines; (b) the vertical profile of collisional frequency for a neutral particle to collide with ions in the solar chromosphere; (c) profile of the ratio of neutral density to ion density, ; and (d) profile of for , , and (solid, dashed, and point-dashed lines). The FZ (HDZ) of Alfvén waves is enclosed by red solid (blue dashed) line.
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